Research ArticleVibrational Energy Flow Model for a High Damping Beam withConstant Axial Force

Xiaoyan Teng1 Nan Liu1 and Jiang Xudong 2

1School of Mechanical and Electrical Engineering Harbin Engineering University Harbin 150001 China2School of Mechanical and Power Engineering Harbin University of Science and Technology Harbin 150080 China

Correspondence should be addressed to Jiang Xudong xudongjiangsinacom

Received 4 January 2020 Revised 27 May 2020 Accepted 22 June 2020 Published 17 July 2020

Academic Editor Ivano Benedetti

Copyright copy 2020 Xiaoyan Teng et al)is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

)e energy flow analysis (EFA) method is developed to predict the energy density of a high damping beam with constant axialforce in the high-frequency range )e energy density and intensity of the beam are associated with high structural damping lossfactor and axial force and introduced to derive the energy transmission equation For high damping situation the energy lossequation is derived by considering the relationship between potential energy and total energy)en the energy density governingequation is obtained Finally the feasibility of the EFA approach is validated by comparing the EFA results with the modalsolutions for various frequencies and structural damping loss factors )e effects of structural damping loss factor and axial forceon the energy density distribution are also discussed in detail

1 Introduction

With the development of high-speed aircrafts and trans-portation vehicles high-frequency vibration of componentsis of great concern for both academic and industrial com-munities in recent years As one of the typical structuralcomponents beams with high damping treatment for vi-brational reduction are extensively used in mechanical andaerospace engineering Moreover these beam-type struc-tures sometimes experience axial forces arising from initialstress and temperature variation which can significantlyinfluence the dynamic characteristics of structures Conse-quently high-frequency dynamic response prediction of ahigh damping beam with axial force is of great significancefor a beam-type structure design

Both the statistical energy analysis (SEA) [1] and theenergy flow analysis (EFA) [2] are popular methods topredict structural time and space averaged energy densitiesin high-frequency range SEA only can provide a singleaveraged energy value for each subsystem of a built-upstructure As an alternative method to SEA EFA can providethe locally smoothed energy value at every interested pointin a built-up structure As a consequence EFA is more

advantageous over SEA in solving high-frequency dynamicresponses due to its ability to obtain detailed local responseinformation

Wohlever et al [3] derived the energy density governingequation of hysterically damped rods and beams fromtraditional displacement solutions Chen et al [4] calculatedthe transient response of beams under high-frequency shockload by energy finite element method and virtual modesynthesis method Liu et al [5] developed the vibrationalenergy flow model for functionally graded beams for high-frequency response prediction Kong et al [6] theoreticallytested the validity region and criterion of EFA from theformation of reverberant plane wave field Nokhbatolfog-hahai et al [7] and Navazi et al [8] verified the vibrationalenergy flow model of a uniform beam and plate from ex-perimental data Zhu et al [9] developed a hybrid methodfor midfrequency vibrational problems which combinedFEA for the stiff member and EFA for the reverberant field ofthe flexible member Sadeghmanesh et al [10] presented amethod for defining a criterion to select a proper structuraltheory based on the order of shear deformation and rotaryinertia for low- to high-frequency vibration analyses Fur-thermore the influences of the fluid loading on the dynamic

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 3584048 11 pageshttpsdoiorg10115520203584048

behaviors of structures were considered such that EFA wasapplied to model the interaction of the fluid with structures[11ndash13] )ere usually exist the discontinuities which referto material and structural configuration variation for thecoupled structures in engineering )e energy density is notcontinuous on coupling boundaries where wave reflectionand transmission arise from encountering the discontinu-ities In the conventional EFA coupling approach FEM isused to solve the energy density governing equation and thewave-based method is utilized to obtain the power trans-ferring coefficients [14ndash16] Park and Hong [17] and Zhihuiet al [18] developed a hybrid power flow analysis methodusing coupling loss factor of SEA to solve the couplingproblems in EFA Kwon et al [19] and Zheng et al [20]developed an energy flow model for high-frequency vibro-acoustic analysis of complex structures using EFA method

For structures with initial stress Zhang et al [21] and Diet al [22] investigated the effects of the axial force due tothermal expansion on the high-frequency vibration of beamsand plates Recently Zhang et al [23] have derived the energydensity governing equation by considering the energydensity and intensity associated with axial force which isneglected in [21 22] Unfortunately the developed EFA canonly be used to analyze the high-frequency vibration oflightly damping beams In this paper we extend the EFAmodel developed by Zhang et al [23] to high damping beamswith axial force )is paper is organized as follows Firstlythe energy density governing equation is derived from theenergy transmission equation and the energy loss equationfor a high damping beam with axial force Particularly therelationship between energy density and intensity ofstructures and the dissipated energy equation are derivedfrom the wavenumber obtained without approximation ofthe real and imaginary term )en the accuracy of thepresented formulation is verified by comparing the devel-oped EFA results with modal analysis results of a pinned-pinned beam with high damping loss factor and axial force)e effects of damping loss factor and axial force on theenergy density distribution are also discussed Finally thepresent EFA method is applied to beams with differentboundary conditions and conclusions on the new EFAmodel are given

2 EnergyDensityGoverning Equation of aHighDamping Beam with Constant Axial Force

21EnergyTransmissionEquation For a beamwith constantaxial force driven by a harmonic point force as depicted inFigure 1 the flexible motion equation can derived as [24 25]

Dc

z4w

zx4 minus Nc

z2w

zx2 + ρSz2w

zt2 F0e

jωtδ x minus x0( 1113857 (1)

where w is the transverse displacement of the vibratingbeam Dc D(1 + jη) is the complex bending stiffness withD representing the real part of Dc and η the structuraldamping loss factor Nc N(1 + jη) is the complex axialload with N representing the real constant part ofNc ρ is themass density S is the cross-section area F0e

jωtδ(x minus x0) is

the transverse harmonic load applied at point x0 with F0denoting its amplitude and ω is its circular frequency δ(x minus

x0) is the Delta function L is the length of the beamSubstituting the general solution w(x t) Cej(kx+ωt)

into equation (1) gives the dispersion relation

Dck4

+ Nck2

minus ρSω2 0 (2)

)en the complex wavenumber k can be solved fromequation (2) and be presented as

k plusmnkc1 plusmnjkc2

kc1

minusNc

2Dc

+

Nc

2Dc

1113888 1113889

2

+ρSω2

Dc

1113888 1113889

11139741113972

111397411139731113972

kc2

Nc

2Dc

1113888 1113889 +

Nc

2Dc1113874 11138751113874 1113875

2+

ρSω2

Dc

1113888 1113889

111397111139741113972

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(3)

where kc1 k1 + jk2 is the complex wavenumber k1 and k2are the corresponding real and imaginary parts respectively

In terms of equation (2) the group velocity cg can beexpressed as equation (4) by implicit derivative with respectto the real part k1 of the complex wavenumber kc1

cg zωzk1

D 2k3

1 minus 6k1k22 minus 6ηk2

1k2 + 2ηk32( 1113857 + N k1 minus ηk2( 1113857

ρSω

(4)

In a lightly damped system from equations (3) and (4)the higher-order terms with respect to the structuraldamping can be neglected and the flexural wavenumber andgroup velocity can be approximated as [21 23]

k1 asymp

minusN

2D+

N

2D1113874 1113875

2+ρSω2

D

111397111139741113972

k2 asymp minus ηρSω2

4Dk31 + 2Nk1

(5)

cg asymp2Dk31 + Nk1

ρSω (6)

As a result the general solution of equation (1) isexpressed as

w(x t) Aeminusjkc1x+ Bejkc1x

+ Geminus kc2x+ Hekc2x

1113872 1113873ejωt (7)

x

x0L

N NF0δ (x ndash x0)e jωt

②①

Figure 1 A pined-pinned transversely vibrating beam with con-stant axial force

2 Mathematical Problems in Engineering

where A B G and H are complex coefficients that can becalculated from the boundary conditions

)e right first two terms of equation (7) are far-fieldsolutions representing the propagating wave while the rightlast two ones of equation (7) are near-field solutions rep-resenting the evanescent wave For high-frequency vibra-tion the flexible wavelength is quite short )e near-fieldsolutions are neglected beyond one wavelength away fromthe boundaries and driving points where they are dissipatedrapidly )us only the far-field solutions are taken intoaccount for EFA and the expression of displacement isrewritten as

w(x t) Aeminusjkc1x

+ Bejkc1x

1113872 1113873ejωt

(8)

)e energy density in a vibrating beam is the sum of thekinetic and potential energy densities which are associatedwith bending strain and constant axial force respectively)e time averaged energy density can be represented as[3 26]

langerang 14Re ρS

zw

zt1113888 1113889

zw

zt1113888 1113889

lowast

+ Dc

z2w

zx21113888 1113889z2w

zx21113888 1113889

lowast

1113890

+ Nc

zw

zx1113888 1113889

zw

zx1113888 1113889

lowast

1113891

(9)

where the bracket operator langmiddotrang denotes the time average overone period and the superscript lowast denotes the complexconjugate operator )e total intensity is associated with theshear force the bending moment and the constant axialforce )e time averaged intensity of the vibrating beam ispresented as [3 26]

langIrang 12Re Dc

z3w

zx31113888 1113889zw

zt1113888 1113889

lowast

minus Dc

z2w

zx21113888 1113889z2w

zx zt1113888 1113889

lowast

1113890

minus Nc

zw

zx1113888 1113889

zw

zt1113888 1113889

lowast

1113891

(10)

Substituting equation (8) into equations (9) and (10) andspace averaging over a wavelength the time and space av-eraged energy density and intensity can be obtained as

langerang 14

ρSω2+ N kc1

111386811138681113868111386811138681113868111386811138682

+ D kc11113868111386811138681113868

11138681113868111386811138684

1113872 1113873 α1|A|2e2k2x

+ α2|B|2e

minus 2k2x1113872 1113873 (11)

langIrang 12ωRe Dck

3c1 + Dckc1 kc1

111386811138681113868111386811138681113868111386811138682

+ Nckc11113960 1113961 α1|A|2e2k2x

minus α2|B|2e

minus 2k2x1113872 1113873 (12)

where langmiddotrang denotes the time and space averaged operator toeliminate the spatially harmonic terms α1 and α2 are thespace averaging values of e2k2x and eminus 2k2x

From equations (11) and (12) the relationship betweenthe time and space averaged energy density and intensity so-called energy transmission equation is derived as

langIrang ωRe Dck

3c1 + Dckc1 kc1

111386811138681113868111386811138681113868111386811138682

+ Nckc11113960 1113961

k2 ρSω2 + N kc11113868111386811138681113868

11138681113868111386811138682

+ D kc11113868111386811138681113868

11138681113868111386811138684

1113872 1113873

dlangerang

dx (13)

22 Energy Loss Equation In the case of hystericdamping the dissipated power of per volume smoothedby time and space averaging is proportional to thepotential energy density smoothed by time and space av-eraging for an elastic system harmonically vibrating withfrequency ω

langπdissrang 2ηωlangeprang (14)

where ep is the potential energy density In a lightly dampedsystem the potential energy density is equal to the kineticenergy and hence the dissipated power is expressed aslangπdissrang ηωlangerang

In a vibrating beam with constant axial force the timeaveraged potential energy density can be expressed as [27]

langeprang 14Re Dc

z2w

zx21113888 1113889z2w

zx21113888 1113889

lowast

+ Nc

zw

zx1113888 1113889

zw

zx1113888 1113889

lowast

1113890 1113891

(15)

Substituting equation (8) into equation (15) and smoothedby local space averaging equation (15) can be rewritten as

langeprang 14

D kc11113868111386811138681113868

11138681113868111386811138684

+ N kc11113868111386811138681113868

11138681113868111386811138682

1113872 1113873 α1|A|2e2k2x

+ α2|B|2e

minus 2k2x1113872 1113873

(16)

Combining equations (11) (14) and (16) the relationshipbetween time and locally space averaged energy density andthe dissipated power of per volume can be represented as

langπdissrang 2ηω N kc1

111386811138681113868111386811138681113868111386811138682

+ D kc11113868111386811138681113868

11138681113868111386811138684

1113872 1113873

ρSω2 + N kc11113868111386811138681113868

11138681113868111386811138682

+ D kc11113868111386811138681113868

11138681113868111386811138684 langerang (17)

23 Energy Density Governing Equation For the elasticmedium the power balance equation at the steady state isexpressed as

Mathematical Problems in Engineering 3

dlangIrang

dx+langπdissrang langπinrangδ x minus x0( 1113857 (18)

where πin refers to the input power at the driving pointx x0

Substituting equations (13) and (17) into equation (18)the energy density governing equation for the high dampingbeam with axial force can be derived as

ωRe Dck3c1 + Dckc1 kc1

111386811138681113868111386811138681113868111386811138682

+ Nckc11113960 1113961

k2 ρSω2 + N kc11113868111386811138681113868

11138681113868111386811138682

+ D kc11113868111386811138681113868

11138681113868111386811138684

1113872 1113873

d2langerang

dx2 +2ηω N kc1

111386811138681113868111386811138681113868111386811138682

+ D kc11113868111386811138681113868

11138681113868111386811138684

1113872 1113873

ρSω2 + N kc11113868111386811138681113868

11138681113868111386811138682

+ D kc11113868111386811138681113868

11138681113868111386811138684 langerang langπinrangδ x minus x0( 1113857 (19)

Unlike the lightly damping system in [23] the energytransition equation is derived using the exact wavenumberwithout neglecting higher-order damping terms )e energytransition equation represents the relationship between timeand space averaged energy density and intensity withoutgroup velocity because the damping effect is not neglectedOn account of high structural damping the potential energydensity is not the same as the kinetic energy density of thestructure )us the energy loss equation is derivedaccording to the relationship between potential energy andtotal energy

3 Solution of Energy DensityGoverning Equation

As shown in Figure 1 the beam is divided into two regions atx0 From equation (19) the general solution of the flexuralenergy density governing equation is expressed as

langefrangi Ci1eminusλfx

+ Ci2eλfx

(i 1 2) (20)

λf

minus2ηk2 N kc1

111386811138681113868111386811138681113868111386811138682

+ D kc11113868111386811138681113868

11138681113868111386811138684

1113872 1113873

Re Dck3c1 + Dckc1 kc1

111386811138681113868111386811138681113868111386811138682

+ Nckc11113960 1113961

11139741113972

(21)

where the subscript i denotes the ith region of the beam )econstants Ci1 Ci2 can be determined by the energy boundaryconditions

From equations (13) and (20) the energy intensity of thevibrating beam is represented as

langIfrangi Pf Ci1eminus λfx

minus Ci2eλfx

1113872 1113873 (22)

Pf minusλf

ωRe Dck3c1 + Dckc1 kc1

111386811138681113868111386811138681113868111386811138682

+ Nckc11113960 1113961

k2 ρSω2 + N kc11113868111386811138681113868

11138681113868111386811138682

+ D kc11113868111386811138681113868

11138681113868111386811138684

1113872 1113873 (23)

Because there are no energy outflow and inflow at bothends of the pinned-pinned beam the following energy in-tensity boundary conditions can be determined as

langIfrang1(0) 0 langIfrang2(L) 0 (24)

At the interface between the regions ① and ② theenergy density is continuous and the energy intensity issubject to conservation of energy )e resulting boundarycondition can be written as

langefrang1 x0( 1113857 langefrang2 x0( 1113857 (25)

langIfrang2 x0( 1113857 langIfrang1 x0( 1113857 +langπinrang (26)

)e input power of the vibrating beam is obtained by theimpedance method and expressed as [23 28]

langπinrang 12

F01113868111386811138681113868

11138681113868111386811138682Re

1Z

1113874 1113875 (27)

Z 2Dckc1k

2c2

ω1 + j

kc1

kc21113888 1113889 (28)

where Z is the impedance at the driving point of the vi-brating beam with axial force

Substituting equation (20) into equations (24)sim(26) thematrix equation can be obtained as

1 minus1 0 0

0 0 eminusλfL minuseminusλfL

eminusλfx0 eλfx0 minuseminusλfx0 minuseλfx0

minusPfeminusλfx0 Pfeλfx0 Pfeminusλfx0 minusPfeλfx0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C11

C12

C21

C22

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

0

0

0

langπinrang

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(29)

According to equation (29) the flexural energy densitycan be analytically calculated for the vibrating beam withhigh structural damping loss factor and constant axial force

4 Verification and Discussion

To validate the proposed EFA formulation and investigatethe effects of structural damping loss factor and axial forcevarious energy flow analyses are performed for a pinned-pinned beam at both ends shown in Figure 1 )e resultsfrom analytical solutions of EFA governing equation arecompared with exact modal solutions (Appendix A andAppendix B) which are defined as time averaged energydensity obtained from the displacement solutions For eachmodal analysis the first 5000 modes are extracted to ensurethe accuracy of the solution In addition the present vi-brational energy flow method is applied to beams withdifferent boundary conditions

41 Model Verification )e beam is made of aluminumalloy with density ρ 2700kgm3 and elastic modulusE 70GPa and its dimensions are L 3mS 2 times 10minus4m2Ib 2 times 10minus10m4 )e structural damping loss factor η

4 Mathematical Problems in Engineering

ranges from 005 to 08 and the axial force N from 0 to 20 kNrespectively )e pinned-pinned beam as shown in Figure 1is excited by a transverse harmonic force located at x0 L2

Figure 2 shows the energy density distribution of pin-ned-pinned beams for the case with the axial forceN 20 kN the amplitude of transverse harmonic forceF0 1N and the structural damping loss factor η 05 whenthe analysis frequencies are 2 4 6 and 8 kHz Except for thatthe developed EFA results differ near both ends of the beamthe developed EFA results is analogous to the modal analysisresults because the evanescent wave is neglected in theenergy flow models Moreover it can be observed that thedifference of the EFA solution from the modal solution issmaller at higher frequencies and hence the accuracy of theEFA model is improved at higher frequencies )e energydensity distributions with different amplitudes of transverseharmonic force (F0 01 1 10 kN) as shown in Figure 3 arecalculated when the excitation frequency is f 6 kHz It isalso found that the EFA solutions are smooth representingthe overall trend of modal solutions and in good agreementwith modal solutions except the vicinity of boundaries Inaddition the global variation of energy density increaseswith larger amplitudes of harmonic point force

)e exact energy density distribution obtained by modalanalysis method oscillates spatially without space averagingAt high-frequency ranges the variance of classical solutionsis uniform resulting from short wavelength )erefore thedeveloped EFA results obtained with space averaging areanalogous to the classical solutions as the analysis frequencyincreases

Figures 4 and 5 show the energy density distributions ofpinned-pinned beams withN 20 kN F0 1N and f 6 kHzwhen the structural damping loss factor is η 005 and 05respectively If the structural damping loss factor is small thewavenumber and group velocity given in equations (3) and(4) in the developed energy flow model are well approxi-mated by those in equations (5) and (6) in the traditionalenergy flow model Hence in a low damping beam thedeveloped EFA solution is close to the traditional EFAsolution However in a high damping beam the developedEFA result represents the trend of exact solution frommodalanalysis and the developed EFA result concurs more withthe exact solution than the traditional EFA result becausehigher-order damping terms are no longer neglected

42 Effects of Constant Axial Force Figure 6 shows theenergy density distributions of pinned-pinned beams withdifferent axial forces 0kNleNle 20kN amplitude of trans-verse harmonic force F0 1N and structural damping lossfactor η 05 when the analysis frequency is 6 kHz )eenergy density varies more dramatically along x positionwith increasing axial force)e total energy can be calculatedfrom integrating the energy density over the length of thebeam Figure 7 shows that the total energy is higher at largeraxial force

)e effects of constant axial force can be explained by theinput power and group velocity Figures 8 and 9 show theinput power and group velocity respectively It is observed

that the input power increases and the group velocity de-creases with increasing axial force Because the total energyis proportional to the input power the total energy is higherfor larger axial force arousing larger input power )e lowergroup velocity makes energy propagate more slowly fromthe driving point to the boundary It means that more energydissipates near the driving point and less energy propagatesto the boundary As a result the energy density changesmore dramatically with larger axial force along the xposition

Ener

gy d

ensit

y (d

B)

0 05 1 15 2 25 3Position (m)

ndash60

ndash40

ndash20

0

20

40

60

80

100

120

140

EFA solution F0 = 1kN

Modal solution F0 = 1kNEFA solution F0 = 01kN

EFA solution F0 = 10kNModal solution F0 = 10kNModal solution F0 = 01kN

Figure 3 Energy density distribution of pinned-pinned beamswith N 20 kN η 05 and f 6 kHz when F0 01 1 10 kN

0 05 1 15 2 25 3ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

0

20

40

60

Ener

gy d

ensit

y (d

B)

Position (m)

EFA solution f = 2kHzModal solution f = 2kHzEFA solution f = 4kHzModal solution f = 4kHz

EFA solution f = 6kHzModal solution f = 6kHzEFA solution f = 8kHzModal solution f = 8kHz

Figure 2 Energy density distribution of pinned-pinned beamswith N 20 kN F0 1N and η 05 when f 2 4 6 8 kHz

Mathematical Problems in Engineering 5

43 Effects of Structural Damping Loss Factor Figure 10shows the energy density distributions of pinned-pinnedbeams with different structural damping loss factor0le ηle 08 amplitude of transverse harmonic force F0 1Nand axial force N 20 kN when the analysis frequency is6 kHz )e developed EFA results represent the globalvariation of the modal solutions at each structural dampingloss factor Owing to the exclusion of the evanescent wavethe developed energy flow solutions more closely approachthe modal solutions as the structural damping loss factorincreases)e energy density changes more suddenly along x

position with increasing structural damping loss factorFigure 11 shows that the total energy is lower at largerstructural damping loss factor

Figures 12 and 13 show the input power and groupvelocity at various structural damping loss factors respec-tively It is found that the input power decreases and thegroup velocity increases with an increase of structuraldamping loss factor Since the total energy is proportional tothe input power the total energy is lower for larger structuraldamping loss factor leading to less input power In fact thelarger group velocity contributes to flatter change of theenergy density along the x position But the energy densitydistribution changes more dramatically with larger struc-tural damping loss factor along the x position It attributes tothat the energy density is mainly affected by the input powerwhile the effect of group velocity is minor

Ener

gy d

ensit

y (d

B)

0 05 1 15 2 25 3Position (m)

ndash20

ndash10

0

10

20

30

40

50

60

1556

60

N = 20kNN = 5kN

N = 1kNN = 0kN

Figure 6 Energy density distribution of pinned-pinned beamswith f 6 kHz F0 1N and η 05 when N 0 1 5 20 kN

0 2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

12

14

Axial force (kN)

Tota

l ene

rgy

(μW

)

Figure 7 Total energy of pinned-pinned beams versus axial forcewith F0 1N η 05 and f 6 kHz

0 05 1 15 2 25 335

40

45

50

55

60

Developed EFA solution Modal solution Traditional EFA solution

Ener

gy d

ensit

y (d

B)

Position (m)

Figure 4 Energy density distribution of pinned-pinned beamswith N 20 kN F0 1N and η 005 when f 6 kHz

Ener

gy d

ensit

y (d

B)

0 05 1 15 2 25 3

Developed EFA solution Modal solution Traditional EFA solution

Position (m)

ndash100

ndash80

ndash60

ndash40

ndash20

0

20

40

60

Figure 5 Energy density distribution of pinned-pinned beamswith N 20 kN F0 1N and η 005 when f 6 kHz

6 Mathematical Problems in Engineering

44 Application to Beams with Different BoundaryConditions )e analysis model consists of a beam fixed atboth ends as shown in Figure 14 and a beam fixed at one endand free at the other as shown in Figure 15 respectivelyBoth beams share the common dimensions of L

3mS 2 times 10minus4m2Ib 2 times 10minus10m4 material properties ofρ 2700kgm3 and E 70GPa and the structuraldamping loss factor η 05 Since the energy outflow at thefixed point is zero and the energy flow satisfies theequilibrium condition at the driving point the energyintensity boundary conditions for the fixed-fixed beam isidentical to those for the pinned-pinned beam Howeverthe impedance of fixed-fixed beam at the driving point isobtained by substituting equations (B5) and (B6) inAppendix B into equation (A8) in Appendix A For fixed-free beam the subdomain ② is absent so that the coef-ficients C21 C22 in equation (29) vanish the followingmatrix equation is rewritten as

1 minus1

minusPfeminusλfx0 Pfeλfx0⎡⎣ ⎤⎦

C11

C121113890 1113891

0

langπinrang1113890 1113891 (30)

Similarly the impedance of fixed-free beam at drivingpoint is determined by substituting the equation (B8) and(B9) in Appendix B into equation (A8) in Appendix AFrom equation (30) the energy density for the fixed-fixedbeam can be evaluated

Figure 14 shows a fixed-fixed beam with constant axialforce Figures 16 and 17 show the energy density distributionof fixed-fixed beams and fixed-free beams with F0 1NN 20 kN and η 05 when the analysis frequencies are2 kHz 4 kHz 6 kHz and 8 kHz respectively )e developedEFA results coincide well with the overall trend of the exactsolutions from modal analysis at each frequency Similar toprevious cases the developed energy flow model gives moreaccurate result at high frequencies despite the exclusion of

0 2 4 6 8 10 12 14 16 18 20075

080

085

090

095

010

0105

Axial force (kN)

Inpu

t pow

er (m

W)

Figure 8 Input power of pinned-pinned beams versus axial forcewith F0 1N η 05 and f 6 kHz

0 5 10 15 20 25300

350

400

450

500

550

Axial force (kN)

Gro

up v

eloc

ity (m

s)

Figure 9 Group velocity versus axial force with η 05 andf 6 kHz

0 05 1 15 2 25 3ndash190

ndash140

ndash90

ndash40

10

60

Ener

gy d

ensit

y (d

B)

Position (m)

EFA solution η = 005

Modal solution η = 005

EFA solution η = 020

Modal solution η = 020

EFA solution η = 050

Modal solution η = 050

EFA solution η = 080

Modal solution η = 080

Figure 10 Energy density distribution of pinned-pinned beamswith N 20 kN F0 1N and f 6 kHz when η 005 020 050080

0 01 02 03 04 05 06 07 080

005

010

015

020

025

030

Tota

l ene

rgy

(μW

)

Structural damping loss factor

Figure 11 Total energy of pinned-pinned beams versus structuraldamping loss factor with F0 1N N 20 kN and f 6 kHz

Mathematical Problems in Engineering 7

the evanescent wave As a consequence it is feasible topredict the vibrational energy density of beams with variousboundary conditions in the high-frequency range by theproposed energy flow model

5 Conclusion

An energy flow model for a beam with axial force thatincludes high damping effect is established to evaluate theenergy density in the high-frequency range )e wave-number is expressed as the function of the damping lossfactor and the axial force Particularly the energy trans-mission equation and the energy dissipation equation arederived from the wavenumber without approximation of itsreal and imaginary term)en the energy density governing

0 01 02 03 04 05 06 07 08042

044

046

048

050

052

054

056

058

Inpu

t pow

er (m

W)

Structural damping loss factor

Figure 12 Input power of pinned-pinned beams versus structuraldamping loss factor with F0 1N N 20 kN and f 6 kHz

0 01 02 03 04 05 06 07 08Structural damping loss factor

825

830

835

840

845

850

855

860

865

870

Gro

up v

eloc

ity (m

s)

Figure 13 Group velocity versus structural damping loss factorwith N 20 kN and f 6 kHz

x

x0

L

F0δ (x ndash x0)ejωt

②①N N

Figure 14 A fixed-fixed beam with constant axial force

x

L

NF0δ (x ndash x0)ejωt

①

Figure 15 A fixed-free beam with constant axial force

0 05 1 15 2 25 3ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

0

20

40

60

EFA solution f = 2kHz

EFA solution f = 4kHzEFA solution f = 6kHz

EFA solution f = 8kHz

Modal solution f = 2kHz

Modal solution f = 4kHzModal solution f = 6kHz

Modal solution f = 8kHz

Ener

gy d

ensit

y (d

B)

Position (m)

Figure 16 Energy density distribution of fixed-fixed beams withF0 1N N 20 kN and η 05 when f 2 4 6 8 kHz

0 05 1 15 2 25 3

EFA solution f = 2kHz

EFA solution f = 4kHzEFA solution f = 6kHz

EFA solution f = 8kHz

Modal solution f = 2kHz

Modal solution f = 4kHzModal solution f = 6kHz

Modal solution f = 8kHz

Position (m)

ndash250

ndash200

ndash150

ndash100

ndash50

0

50

100

Ener

gy d

ensit

y (d

B)

Figure 17 Energy density distribution of fixed-free beams withF0 1N N 20 kN and η 05 when f 2 4 6 8 kHz

8 Mathematical Problems in Engineering

equation is obtained for the high damping beam with axialforce

To verify the developed energy flow model numericalanalyses are performed for the simply supported beam withvarious damping loss factor and axial force )e EFA so-lutions are greatly consistent with the modal solutions in allcases Furthermore it is found that both the high dampingfactor and large axial force can significantly alter the level ofenergy density as well as the distribution Due to theresulting deceasing input power the increasing dampingloss factor makes the energy density decay more dramati-cally along axial position However that is the contrary casefor the axial force )e proposed method is expected to beuseful for the prediction of the high-frequency vibration ofhigh damping structures with initial stress

Appendix

A Impedance and Exact Solution from ModalAnalysis for High Damping Beams withAxial Force

When a high damping beam with axial force is excited at x0by a transverse harmonic force the governing equation ofmotion can be expressed as

Dc

z4w

zx4 minus Nc

z2w

zx2 + ρSz2w

zt2 F0δ x minus x0( 1113857e

jωt (A1)

)e steady-state solution of equation (A1) can be foundby modal superposition as

w(x t) 1113944infin

i1ϕi(x)qi(t) (A2)

where ϕi(x) is the ith natural mode of the beam and qi(t) isthe ith mode displacement

By substituting equation (A2) into the motion equationof the beam equation (A1) and using the orthogonality ofnatural modes we obtain

Kiqi(t) + Mi

d2qi(t)

dt2 Fi (A3)

where

Fi 1113946L

0F0δ x minus x0( 1113857ϕi(x)dx (A4)

represents the so-called ldquomodal forcerdquo and

Mi 1113946L

0ρSϕ2

i dx

Ki ω2i (1 + jη) 1113946

L

0ρSϕ2

i dx

(A5)

are the so-called ldquomodal massrdquo and ldquomodal stiffnessrdquo )eeffect of structural damping is modeled by assuming ahysteretic damping model which introduces an additionalcomplex term (1 + jη) to modal stiffness )e natural fre-quency of the ith mode ωi can be determined from thecorresponding wavenumber in equation (2) which can bewritten as

k2i minus

N

2D+

N

2D1113874 1113875

2+ρSω2

i

D

1113971

(A6)

)e forced vibration response of the beam can beexpressed as the linear combination of the normal modes

w(x t) 1113944infin

i1

F0 middot ejωt

Mi ω2i (1 + jη) minus ω21113858 1113859

ϕi(x)ϕi x0( 1113857 (A7)

)e impedance at the driving point of the finite beamwith axial force can be obtained from its definition

Z F0e

jωt

_w x0 t( 1113857

minus11113936infini1 ωϕ

2i x0( 1113857Mi ω2

i (1 + jη) minus ω21113858 1113859

(A8)

where _w(x0 t) is the velocity at driving point which can becalculated as the time derivative of the displacement atx x0 Once the mode function is determined by theboundary condition the impedance at driving point of thefinite beam with axial force is derived from equation (A8)

)e time averaged energy density is given by

langerang 14

ρSzw

zt1113888 1113889

zw

zt1113888 1113889

lowast

+ Dz2w

zx21113888 1113889z2w

zx21113888 1113889

lowast

+ Nzw

zx1113888 1113889

zw

zx1113888 1113889

lowast

1113890 1113891 (A9)

Substituting equation (A7) into equation (A9) theenergy density can be obtained based on the modal solution

B Mode Shapes for High Damping Beams withAxial Force

B1 BeamPinned at Both Ends )e transverse displacementand the bending moment are zero at a simply supported endHence the boundary conditions can be stated as

w(0) 0

z2w(0)

zx2 0

w(L) 0

z2w(L)

zx2 0

(B1)

Mathematical Problems in Engineering 9

)e general solution of equation (A1) is shown inequation (7) whereas the frequency equation with bothsimply supported ends is derived as

sin kiL 0 (B2)

)e vibration mode of the beam is given by

ϕi(x) sin kix (B3)

B2 Beam Fixed at Both Ends At a fixed end the transversedisplacement and the slope of the displacement are zeroHence the boundary conditions are given by

w(0) 0

zw(0)

zx 0

w(L) 0

zw(L)

zx 0

(B4)

By virtue of general solution in equation (7) andboundary conditions with both fixed ends the frequencyequation is obtained as

cos kiL cosh kiL minus 1 0 (B5)

)e vibration mode of the beam is expressed as

ϕi(x) cos kix minus cosh kix( 1113857 +cos kiL minus cosh kiL

sin kiL minus sinh kiL

middot sin kix minus sinh kix( 1113857

(B6)

B3BeamFixedatOneEndandFreeat theOther If the beamis fixed at x 0 and free at x L the transverse deflectionand its slope must be zero at x 0 and the bending momentand shear force must be zero at x L )us the boundaryconditions become

w(0) 0

zw(0)

zx 0

z3w(L)

zx3 0

z2w(L)

zx2 0

(B7)

According to the general solution in equation (7) andboundary conditions with one fixed end and the other freeend the frequency equation is derived as

cos kiL cosh kiL + 1 0 (B8)

)e vibration mode of the beam is written as

ϕi(x) cos kix minus cosh kix( 1113857 +cos kiL + cosh kiL

sin kiL + sinh kiL

middot sin kix minus sinh kix( 1113857

(B9)

Data Availability

)e data used to support the finding of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that there no conflicts of interest

Acknowledgments

)e presented work was supported by National NaturalScience Foundation of China (Grant no 51505096) andNatural Science Foundation of Heilongjiang Province ofChina (Grant no QC2016056)

References

[1] L Bot ldquoEntropy in sound and vibration towards a newparadigmrdquo Proceedings of the Royal Society A MathematicalPhysical and Engineering Science vol 473 no 2197 pp 1ndash92017

[2] A Acri ldquoA literature review on energy flow analysis of vi-brating structuresrdquo ASRO Journal of Applied Mechanicsvol 1 no 1 pp 12ndash15 2017

[3] J C Wohlever and R J Bernhard ldquoMechanical energy flowmodels of rods and beamsrdquo Journal of Sound and Vibrationvol 153 no 1 pp 1ndash19 1992

[4] Z Chen Z Yang Y Wang and X Wang ldquoA high-frequencyshock response analysis method based on energy finite ele-ment method and virtual mode synthesis and simulationrdquoActa Aeronautica et Astronautica Sinica vol 39 no 08pp 182ndash191 2018

[5] Z Liu and J Niu ldquoVibrational energy flow model for func-tionally graded beamsrdquo Composite Structures vol 186pp 17ndash28 2018

[6] X Kong H Chen D Zhu and W Zhang ldquoStudy on thevalidity region of energy finite element analysisrdquo Journal ofSound and Vibration vol 333 no 9 pp 2601ndash2616 2014

[7] A Nokhbatolfoghahai H Navazi Y Ghobaad andH Haddadpour ldquoExperimental measurement of energydensity of a vibrating beamrdquo Journal of Vibration and Controlvol 24 no 24 pp 5735ndash5746 2018

[8] H M Navazi A Nokhbatolfoghahaei Y Ghobad andH Haddadpour ldquoExperimental measurement of energydensity in a vibrating plate and comparison with energy finiteelement analysisrdquo Journal of Sound and Vibration vol 375pp 289ndash307 2016

[9] D Zhu H Chen X Kong and W Zhang ldquoA hybrid finiteelement-energy finite element method for mid-frequencyvibrations of built-up structures under multi-distributedloadingsrdquo Journal of Sound amp Vibration vol 333 no 22pp 5723ndash5745 2014

[10] M Sadeghmanesh H Haddadpour M T Abadi andH M Navazi ldquoA method for selection of structural theories

10 Mathematical Problems in Engineering

for low to high frequency vibration analysesrdquo EuropeanJournal of Mechanics-A Solids vol 75 pp 27ndash40 2019

[11] W Zhang AWang N Vlahopoulos and KWu ldquoA vibrationanalysis of stiffened plates under heavy fluid loading by anenergy finite element analysis formulationrdquo Finite Elements inAnalysis and Design vol 41 no 11-12 pp 1056ndash1078 2005

[12] J-B Han S-Y Hong and J-H Song ldquoEnergy flow model forthin plate considering fluid loading with mean flowrdquo Journalof Sound and Vibration vol 331 no 24 pp 5326ndash5346 2012

[13] H-W Kwon S-Y Hong D-K Oh et al ldquoExperimental studyon the energy flow analysis of underwater vibration for thereinforced cylindrical structurerdquo Journal of Mechanical Sci-ence and Technology vol 28 no 9 pp 3405ndash3410 2014

[14] S Wang and R J Bernhard ldquoPrediction of averaged energyfor moderately damped systems with strong couplingrdquoJournal of Sound and Vibration vol 319 no 1-2 pp 426ndash4442009

[15] V S Pereira and J M C Dos Santos ldquoCoupled plate energymodels at mid- and high-frequency vibrationsrdquo Computers ampStructures vol 134 pp 48ndash61 2014

[16] H-W Kwon S-Y Hong and J-H Song ldquoVibrational energyflow analysis of coupled cylindrical thin shell structuresrdquoJournal of Mechanical Science and Technology vol 30 no 9pp 4049ndash4062 2016

[17] Y-H Park and S-Y Hong ldquoHybrid power flow analysis usingcoupling loss factor of SEA for low-damping system-part Iformulation of 1-D and 2-D casesrdquo Journal of Sound andVibration vol 299 no 3 pp 484ndash503 2007

[18] Z Liu J Niu and X Gao ldquoAn improved approach for analysisof coupled structures in Energy Finite Element Analysis usingthe coupling loss factorrdquo Computers amp Structures vol 210pp 69ndash86 2018

[19] H-W Kwon S-Y Hong and J-H Song ldquoEnergy flowmodels for underwater radiation noise prediction in medium-to-high-frequency rangesrdquo Proceedings of the Institution ofMechanical Engineers Part M Journal of Engineering for theMaritime Environment vol 230 no 2 pp 404ndash416 2016

[20] X Zheng W Dai Y Qiu and Z Hao ldquoPrediction and energycontribution analysis of interior noise in a high-speed trainbased on modified energy finite element analysisrdquoMechanicalSystems and Signal Processing vol 126 pp 439ndash457 2019

[21] W Zhang H Chen D Zhu and X Kong ldquo)e thermal effectson high-frequency vibration of beams using energy flowanalysisrdquo Journal of Sound and Vibration vol 333 no 9pp 2588ndash2600 2014

[22] W Di X Miaoxia and L Yueming ldquoHigh-frequency dy-namic analysis of plates in thermal environments based onenergy finite elementmethodrdquo Shock and Vibration vol 2015pp 1ndash14 2015

[23] Z Chen Z Yang N Guo and G Zhang ldquoAn energy finiteelement method for high frequency vibration analysis ofbeams with axial forcerdquo Applied Mathematical Modellingvol 61 no 5 pp 521ndash539 2018

[24] P M Morse and K U Indard Ceoretical AcousticsPrinceton University Press Princeton NJ USA 1986

[25] W Weaver S P Timoshenko and D H Young VibrationProblems in Engineering John Wiley amp Sons Hoboken NJUSA 1990

[26] M N Ichchou A Le Bot and L Jezequel ldquoEnergy models ofone-dimensional multi-propagative systemsrdquo Journal ofSound and Vibration vol 201 no 5 pp 535ndash554 1997

[27] Y Lase M N Ichchou and L Jezequel ldquoEnergy flow analysisof bars and beams theoretical formulationsrdquo Journal of Soundand Vibration vol 192 no 1 pp 281ndash305 1996

[28] D J Nefske and S H Sung ldquoPower flow finite elementanalysis of dynamic systems basic theory and application tobeamsrdquo Journal of Vibration and Acoustics vol 111 no 1pp 94ndash100 1989

Mathematical Problems in Engineering 11

where A B G and H are complex coefficients that can becalculated from the boundary conditions

)e right first two terms of equation (7) are far-fieldsolutions representing the propagating wave while the rightlast two ones of equation (7) are near-field solutions rep-resenting the evanescent wave For high-frequency vibra-tion the flexible wavelength is quite short )e near-fieldsolutions are neglected beyond one wavelength away fromthe boundaries and driving points where they are dissipatedrapidly )us only the far-field solutions are taken intoaccount for EFA and the expression of displacement isrewritten as

w(x t) Aeminusjkc1x

+ Bejkc1x

1113872 1113873ejωt

(8)

)e energy density in a vibrating beam is the sum of thekinetic and potential energy densities which are associatedwith bending strain and constant axial force respectively)e time averaged energy density can be represented as[3 26]

langerang 14Re ρS

zw

zt1113888 1113889

zw

zt1113888 1113889

lowast

+ Dc

z2w

zx21113888 1113889z2w

zx21113888 1113889

lowast

1113890

+ Nc

zw

zx1113888 1113889

zw

zx1113888 1113889

lowast

1113891

(9)

where the bracket operator langmiddotrang denotes the time average overone period and the superscript lowast denotes the complexconjugate operator )e total intensity is associated with theshear force the bending moment and the constant axialforce )e time averaged intensity of the vibrating beam ispresented as [3 26]

langIrang 12Re Dc

z3w

zx31113888 1113889zw

zt1113888 1113889

lowast

minus Dc

z2w

zx21113888 1113889z2w

zx zt1113888 1113889

lowast

1113890

minus Nc

zw

zx1113888 1113889

zw

zt1113888 1113889

lowast

1113891

(10)

Substituting equation (8) into equations (9) and (10) andspace averaging over a wavelength the time and space av-eraged energy density and intensity can be obtained as

langerang 14

ρSω2+ N kc1

111386811138681113868111386811138681113868111386811138682

+ D kc11113868111386811138681113868

11138681113868111386811138684

1113872 1113873 α1|A|2e2k2x

+ α2|B|2e

minus 2k2x1113872 1113873 (11)

langIrang 12ωRe Dck

3c1 + Dckc1 kc1

111386811138681113868111386811138681113868111386811138682

+ Nckc11113960 1113961 α1|A|2e2k2x

minus α2|B|2e

minus 2k2x1113872 1113873 (12)

where langmiddotrang denotes the time and space averaged operator toeliminate the spatially harmonic terms α1 and α2 are thespace averaging values of e2k2x and eminus 2k2x

From equations (11) and (12) the relationship betweenthe time and space averaged energy density and intensity so-called energy transmission equation is derived as

langIrang ωRe Dck

3c1 + Dckc1 kc1

111386811138681113868111386811138681113868111386811138682

+ Nckc11113960 1113961

k2 ρSω2 + N kc11113868111386811138681113868

11138681113868111386811138682

+ D kc11113868111386811138681113868

11138681113868111386811138684

1113872 1113873

dlangerang

dx (13)

22 Energy Loss Equation In the case of hystericdamping the dissipated power of per volume smoothedby time and space averaging is proportional to thepotential energy density smoothed by time and space av-eraging for an elastic system harmonically vibrating withfrequency ω

langπdissrang 2ηωlangeprang (14)

where ep is the potential energy density In a lightly dampedsystem the potential energy density is equal to the kineticenergy and hence the dissipated power is expressed aslangπdissrang ηωlangerang

In a vibrating beam with constant axial force the timeaveraged potential energy density can be expressed as [27]

langeprang 14Re Dc

z2w

zx21113888 1113889z2w

zx21113888 1113889

lowast

+ Nc

zw

zx1113888 1113889

zw

zx1113888 1113889

lowast

1113890 1113891

(15)

Substituting equation (8) into equation (15) and smoothedby local space averaging equation (15) can be rewritten as

langeprang 14

D kc11113868111386811138681113868

11138681113868111386811138684

+ N kc11113868111386811138681113868

11138681113868111386811138682

1113872 1113873 α1|A|2e2k2x

+ α2|B|2e

minus 2k2x1113872 1113873

(16)

Combining equations (11) (14) and (16) the relationshipbetween time and locally space averaged energy density andthe dissipated power of per volume can be represented as

langπdissrang 2ηω N kc1

111386811138681113868111386811138681113868111386811138682

+ D kc11113868111386811138681113868

11138681113868111386811138684

1113872 1113873

ρSω2 + N kc11113868111386811138681113868

11138681113868111386811138682

+ D kc11113868111386811138681113868

11138681113868111386811138684 langerang (17)

23 Energy Density Governing Equation For the elasticmedium the power balance equation at the steady state isexpressed as

Mathematical Problems in Engineering 3

dlangIrang

dx+langπdissrang langπinrangδ x minus x0( 1113857 (18)

where πin refers to the input power at the driving pointx x0

Substituting equations (13) and (17) into equation (18)the energy density governing equation for the high dampingbeam with axial force can be derived as

ωRe Dck3c1 + Dckc1 kc1

111386811138681113868111386811138681113868111386811138682

+ Nckc11113960 1113961

k2 ρSω2 + N kc11113868111386811138681113868

11138681113868111386811138682

+ D kc11113868111386811138681113868

11138681113868111386811138684

1113872 1113873

d2langerang

dx2 +2ηω N kc1

111386811138681113868111386811138681113868111386811138682

+ D kc11113868111386811138681113868

11138681113868111386811138684

1113872 1113873

ρSω2 + N kc11113868111386811138681113868

11138681113868111386811138682

+ D kc11113868111386811138681113868

11138681113868111386811138684 langerang langπinrangδ x minus x0( 1113857 (19)

Unlike the lightly damping system in [23] the energytransition equation is derived using the exact wavenumberwithout neglecting higher-order damping terms )e energytransition equation represents the relationship between timeand space averaged energy density and intensity withoutgroup velocity because the damping effect is not neglectedOn account of high structural damping the potential energydensity is not the same as the kinetic energy density of thestructure )us the energy loss equation is derivedaccording to the relationship between potential energy andtotal energy

3 Solution of Energy DensityGoverning Equation

As shown in Figure 1 the beam is divided into two regions atx0 From equation (19) the general solution of the flexuralenergy density governing equation is expressed as

langefrangi Ci1eminusλfx

+ Ci2eλfx

(i 1 2) (20)

λf

minus2ηk2 N kc1

111386811138681113868111386811138681113868111386811138682

+ D kc11113868111386811138681113868

11138681113868111386811138684

1113872 1113873

Re Dck3c1 + Dckc1 kc1

111386811138681113868111386811138681113868111386811138682

+ Nckc11113960 1113961

11139741113972

(21)

where the subscript i denotes the ith region of the beam )econstants Ci1 Ci2 can be determined by the energy boundaryconditions

From equations (13) and (20) the energy intensity of thevibrating beam is represented as

langIfrangi Pf Ci1eminus λfx

minus Ci2eλfx

1113872 1113873 (22)

Pf minusλf

ωRe Dck3c1 + Dckc1 kc1

111386811138681113868111386811138681113868111386811138682

+ Nckc11113960 1113961

k2 ρSω2 + N kc11113868111386811138681113868

11138681113868111386811138682

+ D kc11113868111386811138681113868

11138681113868111386811138684

1113872 1113873 (23)

Because there are no energy outflow and inflow at bothends of the pinned-pinned beam the following energy in-tensity boundary conditions can be determined as

langIfrang1(0) 0 langIfrang2(L) 0 (24)

At the interface between the regions ① and ② theenergy density is continuous and the energy intensity issubject to conservation of energy )e resulting boundarycondition can be written as

langefrang1 x0( 1113857 langefrang2 x0( 1113857 (25)

langIfrang2 x0( 1113857 langIfrang1 x0( 1113857 +langπinrang (26)

)e input power of the vibrating beam is obtained by theimpedance method and expressed as [23 28]

langπinrang 12

F01113868111386811138681113868

11138681113868111386811138682Re

1Z

1113874 1113875 (27)

Z 2Dckc1k

2c2

ω1 + j

kc1

kc21113888 1113889 (28)

where Z is the impedance at the driving point of the vi-brating beam with axial force

Substituting equation (20) into equations (24)sim(26) thematrix equation can be obtained as

1 minus1 0 0

0 0 eminusλfL minuseminusλfL

eminusλfx0 eλfx0 minuseminusλfx0 minuseλfx0

minusPfeminusλfx0 Pfeλfx0 Pfeminusλfx0 minusPfeλfx0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C11

C12

C21

C22

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

0

0

0

langπinrang

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(29)

According to equation (29) the flexural energy densitycan be analytically calculated for the vibrating beam withhigh structural damping loss factor and constant axial force

4 Verification and Discussion

To validate the proposed EFA formulation and investigatethe effects of structural damping loss factor and axial forcevarious energy flow analyses are performed for a pinned-pinned beam at both ends shown in Figure 1 )e resultsfrom analytical solutions of EFA governing equation arecompared with exact modal solutions (Appendix A andAppendix B) which are defined as time averaged energydensity obtained from the displacement solutions For eachmodal analysis the first 5000 modes are extracted to ensurethe accuracy of the solution In addition the present vi-brational energy flow method is applied to beams withdifferent boundary conditions

41 Model Verification )e beam is made of aluminumalloy with density ρ 2700kgm3 and elastic modulusE 70GPa and its dimensions are L 3mS 2 times 10minus4m2Ib 2 times 10minus10m4 )e structural damping loss factor η

4 Mathematical Problems in Engineering

ranges from 005 to 08 and the axial force N from 0 to 20 kNrespectively )e pinned-pinned beam as shown in Figure 1is excited by a transverse harmonic force located at x0 L2

Figure 2 shows the energy density distribution of pin-ned-pinned beams for the case with the axial forceN 20 kN the amplitude of transverse harmonic forceF0 1N and the structural damping loss factor η 05 whenthe analysis frequencies are 2 4 6 and 8 kHz Except for thatthe developed EFA results differ near both ends of the beamthe developed EFA results is analogous to the modal analysisresults because the evanescent wave is neglected in theenergy flow models Moreover it can be observed that thedifference of the EFA solution from the modal solution issmaller at higher frequencies and hence the accuracy of theEFA model is improved at higher frequencies )e energydensity distributions with different amplitudes of transverseharmonic force (F0 01 1 10 kN) as shown in Figure 3 arecalculated when the excitation frequency is f 6 kHz It isalso found that the EFA solutions are smooth representingthe overall trend of modal solutions and in good agreementwith modal solutions except the vicinity of boundaries Inaddition the global variation of energy density increaseswith larger amplitudes of harmonic point force

)e exact energy density distribution obtained by modalanalysis method oscillates spatially without space averagingAt high-frequency ranges the variance of classical solutionsis uniform resulting from short wavelength )erefore thedeveloped EFA results obtained with space averaging areanalogous to the classical solutions as the analysis frequencyincreases

Figures 4 and 5 show the energy density distributions ofpinned-pinned beams withN 20 kN F0 1N and f 6 kHzwhen the structural damping loss factor is η 005 and 05respectively If the structural damping loss factor is small thewavenumber and group velocity given in equations (3) and(4) in the developed energy flow model are well approxi-mated by those in equations (5) and (6) in the traditionalenergy flow model Hence in a low damping beam thedeveloped EFA solution is close to the traditional EFAsolution However in a high damping beam the developedEFA result represents the trend of exact solution frommodalanalysis and the developed EFA result concurs more withthe exact solution than the traditional EFA result becausehigher-order damping terms are no longer neglected

42 Effects of Constant Axial Force Figure 6 shows theenergy density distributions of pinned-pinned beams withdifferent axial forces 0kNleNle 20kN amplitude of trans-verse harmonic force F0 1N and structural damping lossfactor η 05 when the analysis frequency is 6 kHz )eenergy density varies more dramatically along x positionwith increasing axial force)e total energy can be calculatedfrom integrating the energy density over the length of thebeam Figure 7 shows that the total energy is higher at largeraxial force

)e effects of constant axial force can be explained by theinput power and group velocity Figures 8 and 9 show theinput power and group velocity respectively It is observed

that the input power increases and the group velocity de-creases with increasing axial force Because the total energyis proportional to the input power the total energy is higherfor larger axial force arousing larger input power )e lowergroup velocity makes energy propagate more slowly fromthe driving point to the boundary It means that more energydissipates near the driving point and less energy propagatesto the boundary As a result the energy density changesmore dramatically with larger axial force along the xposition

Ener

gy d

ensit

y (d

B)

0 05 1 15 2 25 3Position (m)

ndash60

ndash40

ndash20

0

20

40

60

80

100

120

140

EFA solution F0 = 1kN

Modal solution F0 = 1kNEFA solution F0 = 01kN

EFA solution F0 = 10kNModal solution F0 = 10kNModal solution F0 = 01kN

Figure 3 Energy density distribution of pinned-pinned beamswith N 20 kN η 05 and f 6 kHz when F0 01 1 10 kN

0 05 1 15 2 25 3ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

0

20

40

60

Ener

gy d

ensit

y (d

B)

Position (m)

EFA solution f = 2kHzModal solution f = 2kHzEFA solution f = 4kHzModal solution f = 4kHz

EFA solution f = 6kHzModal solution f = 6kHzEFA solution f = 8kHzModal solution f = 8kHz

Figure 2 Energy density distribution of pinned-pinned beamswith N 20 kN F0 1N and η 05 when f 2 4 6 8 kHz

Mathematical Problems in Engineering 5

43 Effects of Structural Damping Loss Factor Figure 10shows the energy density distributions of pinned-pinnedbeams with different structural damping loss factor0le ηle 08 amplitude of transverse harmonic force F0 1Nand axial force N 20 kN when the analysis frequency is6 kHz )e developed EFA results represent the globalvariation of the modal solutions at each structural dampingloss factor Owing to the exclusion of the evanescent wavethe developed energy flow solutions more closely approachthe modal solutions as the structural damping loss factorincreases)e energy density changes more suddenly along x

position with increasing structural damping loss factorFigure 11 shows that the total energy is lower at largerstructural damping loss factor

Figures 12 and 13 show the input power and groupvelocity at various structural damping loss factors respec-tively It is found that the input power decreases and thegroup velocity increases with an increase of structuraldamping loss factor Since the total energy is proportional tothe input power the total energy is lower for larger structuraldamping loss factor leading to less input power In fact thelarger group velocity contributes to flatter change of theenergy density along the x position But the energy densitydistribution changes more dramatically with larger struc-tural damping loss factor along the x position It attributes tothat the energy density is mainly affected by the input powerwhile the effect of group velocity is minor

Ener

gy d

ensit

y (d

B)

0 05 1 15 2 25 3Position (m)

ndash20

ndash10

0

10

20

30

40

50

60

1556

60

N = 20kNN = 5kN

N = 1kNN = 0kN

Figure 6 Energy density distribution of pinned-pinned beamswith f 6 kHz F0 1N and η 05 when N 0 1 5 20 kN

0 2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

12

14

Axial force (kN)

Tota

l ene

rgy

(μW

)

Figure 7 Total energy of pinned-pinned beams versus axial forcewith F0 1N η 05 and f 6 kHz

0 05 1 15 2 25 335

40

45

50

55

60

Developed EFA solution Modal solution Traditional EFA solution

Ener

gy d

ensit

y (d

B)

Position (m)

Figure 4 Energy density distribution of pinned-pinned beamswith N 20 kN F0 1N and η 005 when f 6 kHz

Ener

gy d

ensit

y (d

B)

0 05 1 15 2 25 3

Developed EFA solution Modal solution Traditional EFA solution

Position (m)

ndash100

ndash80

ndash60

ndash40

ndash20

0

20

40

60

Figure 5 Energy density distribution of pinned-pinned beamswith N 20 kN F0 1N and η 005 when f 6 kHz

6 Mathematical Problems in Engineering

44 Application to Beams with Different BoundaryConditions )e analysis model consists of a beam fixed atboth ends as shown in Figure 14 and a beam fixed at one endand free at the other as shown in Figure 15 respectivelyBoth beams share the common dimensions of L

3mS 2 times 10minus4m2Ib 2 times 10minus10m4 material properties ofρ 2700kgm3 and E 70GPa and the structuraldamping loss factor η 05 Since the energy outflow at thefixed point is zero and the energy flow satisfies theequilibrium condition at the driving point the energyintensity boundary conditions for the fixed-fixed beam isidentical to those for the pinned-pinned beam Howeverthe impedance of fixed-fixed beam at the driving point isobtained by substituting equations (B5) and (B6) inAppendix B into equation (A8) in Appendix A For fixed-free beam the subdomain ② is absent so that the coef-ficients C21 C22 in equation (29) vanish the followingmatrix equation is rewritten as

1 minus1

minusPfeminusλfx0 Pfeλfx0⎡⎣ ⎤⎦

C11

C121113890 1113891

0

langπinrang1113890 1113891 (30)

Similarly the impedance of fixed-free beam at drivingpoint is determined by substituting the equation (B8) and(B9) in Appendix B into equation (A8) in Appendix AFrom equation (30) the energy density for the fixed-fixedbeam can be evaluated

Figure 14 shows a fixed-fixed beam with constant axialforce Figures 16 and 17 show the energy density distributionof fixed-fixed beams and fixed-free beams with F0 1NN 20 kN and η 05 when the analysis frequencies are2 kHz 4 kHz 6 kHz and 8 kHz respectively )e developedEFA results coincide well with the overall trend of the exactsolutions from modal analysis at each frequency Similar toprevious cases the developed energy flow model gives moreaccurate result at high frequencies despite the exclusion of

0 2 4 6 8 10 12 14 16 18 20075

080

085

090

095

010

0105

Axial force (kN)

Inpu

t pow

er (m

W)

Figure 8 Input power of pinned-pinned beams versus axial forcewith F0 1N η 05 and f 6 kHz

0 5 10 15 20 25300

350

400

450

500

550

Axial force (kN)

Gro

up v

eloc

ity (m

s)

Figure 9 Group velocity versus axial force with η 05 andf 6 kHz

0 05 1 15 2 25 3ndash190

ndash140

ndash90

ndash40

10

60

Ener

gy d

ensit

y (d

B)

Position (m)

EFA solution η = 005

Modal solution η = 005

EFA solution η = 020

Modal solution η = 020

EFA solution η = 050

Modal solution η = 050

EFA solution η = 080

Modal solution η = 080

Figure 10 Energy density distribution of pinned-pinned beamswith N 20 kN F0 1N and f 6 kHz when η 005 020 050080

0 01 02 03 04 05 06 07 080

005

010

015

020

025

030

Tota

l ene

rgy

(μW

)

Structural damping loss factor

Figure 11 Total energy of pinned-pinned beams versus structuraldamping loss factor with F0 1N N 20 kN and f 6 kHz

Mathematical Problems in Engineering 7

the evanescent wave As a consequence it is feasible topredict the vibrational energy density of beams with variousboundary conditions in the high-frequency range by theproposed energy flow model

5 Conclusion

An energy flow model for a beam with axial force thatincludes high damping effect is established to evaluate theenergy density in the high-frequency range )e wave-number is expressed as the function of the damping lossfactor and the axial force Particularly the energy trans-mission equation and the energy dissipation equation arederived from the wavenumber without approximation of itsreal and imaginary term)en the energy density governing

0 01 02 03 04 05 06 07 08042

044

046

048

050

052

054

056

058

Inpu

t pow

er (m

W)

Structural damping loss factor

Figure 12 Input power of pinned-pinned beams versus structuraldamping loss factor with F0 1N N 20 kN and f 6 kHz

0 01 02 03 04 05 06 07 08Structural damping loss factor

825

830

835

840

845

850

855

860

865

870

Gro

up v

eloc

ity (m

s)

Figure 13 Group velocity versus structural damping loss factorwith N 20 kN and f 6 kHz

x

x0

L

F0δ (x ndash x0)ejωt

②①N N

Figure 14 A fixed-fixed beam with constant axial force

x

L

NF0δ (x ndash x0)ejωt

①

Figure 15 A fixed-free beam with constant axial force

0 05 1 15 2 25 3ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

0

20

40

60

EFA solution f = 2kHz

EFA solution f = 4kHzEFA solution f = 6kHz

EFA solution f = 8kHz

Modal solution f = 2kHz

Modal solution f = 4kHzModal solution f = 6kHz

Modal solution f = 8kHz

Ener

gy d

ensit

y (d

B)

Position (m)

Figure 16 Energy density distribution of fixed-fixed beams withF0 1N N 20 kN and η 05 when f 2 4 6 8 kHz

0 05 1 15 2 25 3

EFA solution f = 2kHz

EFA solution f = 4kHzEFA solution f = 6kHz

EFA solution f = 8kHz

Modal solution f = 2kHz

Modal solution f = 4kHzModal solution f = 6kHz

Modal solution f = 8kHz

Position (m)

ndash250

ndash200

ndash150

ndash100

ndash50

0

50

100

Ener

gy d

ensit

y (d

B)

Figure 17 Energy density distribution of fixed-free beams withF0 1N N 20 kN and η 05 when f 2 4 6 8 kHz

8 Mathematical Problems in Engineering

equation is obtained for the high damping beam with axialforce

To verify the developed energy flow model numericalanalyses are performed for the simply supported beam withvarious damping loss factor and axial force )e EFA so-lutions are greatly consistent with the modal solutions in allcases Furthermore it is found that both the high dampingfactor and large axial force can significantly alter the level ofenergy density as well as the distribution Due to theresulting deceasing input power the increasing dampingloss factor makes the energy density decay more dramati-cally along axial position However that is the contrary casefor the axial force )e proposed method is expected to beuseful for the prediction of the high-frequency vibration ofhigh damping structures with initial stress

Appendix

A Impedance and Exact Solution from ModalAnalysis for High Damping Beams withAxial Force

When a high damping beam with axial force is excited at x0by a transverse harmonic force the governing equation ofmotion can be expressed as

Dc

z4w

zx4 minus Nc

z2w

zx2 + ρSz2w

zt2 F0δ x minus x0( 1113857e

jωt (A1)

)e steady-state solution of equation (A1) can be foundby modal superposition as

w(x t) 1113944infin

i1ϕi(x)qi(t) (A2)

where ϕi(x) is the ith natural mode of the beam and qi(t) isthe ith mode displacement

By substituting equation (A2) into the motion equationof the beam equation (A1) and using the orthogonality ofnatural modes we obtain

Kiqi(t) + Mi

d2qi(t)

dt2 Fi (A3)

where

Fi 1113946L

0F0δ x minus x0( 1113857ϕi(x)dx (A4)

represents the so-called ldquomodal forcerdquo and

Mi 1113946L

0ρSϕ2

i dx

Ki ω2i (1 + jη) 1113946

L

0ρSϕ2

i dx

(A5)

are the so-called ldquomodal massrdquo and ldquomodal stiffnessrdquo )eeffect of structural damping is modeled by assuming ahysteretic damping model which introduces an additionalcomplex term (1 + jη) to modal stiffness )e natural fre-quency of the ith mode ωi can be determined from thecorresponding wavenumber in equation (2) which can bewritten as

k2i minus

N

2D+

N

2D1113874 1113875

2+ρSω2

i

D

1113971

(A6)

)e forced vibration response of the beam can beexpressed as the linear combination of the normal modes

w(x t) 1113944infin

i1

F0 middot ejωt

Mi ω2i (1 + jη) minus ω21113858 1113859

ϕi(x)ϕi x0( 1113857 (A7)

)e impedance at the driving point of the finite beamwith axial force can be obtained from its definition

Z F0e

jωt

_w x0 t( 1113857

minus11113936infini1 ωϕ

2i x0( 1113857Mi ω2

i (1 + jη) minus ω21113858 1113859

(A8)

where _w(x0 t) is the velocity at driving point which can becalculated as the time derivative of the displacement atx x0 Once the mode function is determined by theboundary condition the impedance at driving point of thefinite beam with axial force is derived from equation (A8)

)e time averaged energy density is given by

langerang 14

ρSzw

zt1113888 1113889

zw

zt1113888 1113889

lowast

+ Dz2w

zx21113888 1113889z2w

zx21113888 1113889

lowast

+ Nzw

zx1113888 1113889

zw

zx1113888 1113889

lowast

1113890 1113891 (A9)

Substituting equation (A7) into equation (A9) theenergy density can be obtained based on the modal solution

B Mode Shapes for High Damping Beams withAxial Force

B1 BeamPinned at Both Ends )e transverse displacementand the bending moment are zero at a simply supported endHence the boundary conditions can be stated as

w(0) 0

z2w(0)

zx2 0

w(L) 0

z2w(L)

zx2 0

(B1)

Mathematical Problems in Engineering 9

)e general solution of equation (A1) is shown inequation (7) whereas the frequency equation with bothsimply supported ends is derived as

sin kiL 0 (B2)

)e vibration mode of the beam is given by

ϕi(x) sin kix (B3)

B2 Beam Fixed at Both Ends At a fixed end the transversedisplacement and the slope of the displacement are zeroHence the boundary conditions are given by

w(0) 0

zw(0)

zx 0

w(L) 0

zw(L)

zx 0

(B4)

By virtue of general solution in equation (7) andboundary conditions with both fixed ends the frequencyequation is obtained as

cos kiL cosh kiL minus 1 0 (B5)

)e vibration mode of the beam is expressed as

ϕi(x) cos kix minus cosh kix( 1113857 +cos kiL minus cosh kiL

sin kiL minus sinh kiL

middot sin kix minus sinh kix( 1113857

(B6)

B3BeamFixedatOneEndandFreeat theOther If the beamis fixed at x 0 and free at x L the transverse deflectionand its slope must be zero at x 0 and the bending momentand shear force must be zero at x L )us the boundaryconditions become

w(0) 0

zw(0)

zx 0

z3w(L)

zx3 0

z2w(L)

zx2 0

(B7)

According to the general solution in equation (7) andboundary conditions with one fixed end and the other freeend the frequency equation is derived as

cos kiL cosh kiL + 1 0 (B8)

)e vibration mode of the beam is written as

ϕi(x) cos kix minus cosh kix( 1113857 +cos kiL + cosh kiL

sin kiL + sinh kiL

middot sin kix minus sinh kix( 1113857

(B9)

Data Availability

)e data used to support the finding of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that there no conflicts of interest

Acknowledgments

)e presented work was supported by National NaturalScience Foundation of China (Grant no 51505096) andNatural Science Foundation of Heilongjiang Province ofChina (Grant no QC2016056)

References

[1] L Bot ldquoEntropy in sound and vibration towards a newparadigmrdquo Proceedings of the Royal Society A MathematicalPhysical and Engineering Science vol 473 no 2197 pp 1ndash92017

[2] A Acri ldquoA literature review on energy flow analysis of vi-brating structuresrdquo ASRO Journal of Applied Mechanicsvol 1 no 1 pp 12ndash15 2017

[3] J C Wohlever and R J Bernhard ldquoMechanical energy flowmodels of rods and beamsrdquo Journal of Sound and Vibrationvol 153 no 1 pp 1ndash19 1992

[4] Z Chen Z Yang Y Wang and X Wang ldquoA high-frequencyshock response analysis method based on energy finite ele-ment method and virtual mode synthesis and simulationrdquoActa Aeronautica et Astronautica Sinica vol 39 no 08pp 182ndash191 2018

[5] Z Liu and J Niu ldquoVibrational energy flow model for func-tionally graded beamsrdquo Composite Structures vol 186pp 17ndash28 2018

[6] X Kong H Chen D Zhu and W Zhang ldquoStudy on thevalidity region of energy finite element analysisrdquo Journal ofSound and Vibration vol 333 no 9 pp 2601ndash2616 2014

[7] A Nokhbatolfoghahai H Navazi Y Ghobaad andH Haddadpour ldquoExperimental measurement of energydensity of a vibrating beamrdquo Journal of Vibration and Controlvol 24 no 24 pp 5735ndash5746 2018

[8] H M Navazi A Nokhbatolfoghahaei Y Ghobad andH Haddadpour ldquoExperimental measurement of energydensity in a vibrating plate and comparison with energy finiteelement analysisrdquo Journal of Sound and Vibration vol 375pp 289ndash307 2016

[9] D Zhu H Chen X Kong and W Zhang ldquoA hybrid finiteelement-energy finite element method for mid-frequencyvibrations of built-up structures under multi-distributedloadingsrdquo Journal of Sound amp Vibration vol 333 no 22pp 5723ndash5745 2014

[10] M Sadeghmanesh H Haddadpour M T Abadi andH M Navazi ldquoA method for selection of structural theories

10 Mathematical Problems in Engineering

for low to high frequency vibration analysesrdquo EuropeanJournal of Mechanics-A Solids vol 75 pp 27ndash40 2019

[11] W Zhang AWang N Vlahopoulos and KWu ldquoA vibrationanalysis of stiffened plates under heavy fluid loading by anenergy finite element analysis formulationrdquo Finite Elements inAnalysis and Design vol 41 no 11-12 pp 1056ndash1078 2005

[12] J-B Han S-Y Hong and J-H Song ldquoEnergy flow model forthin plate considering fluid loading with mean flowrdquo Journalof Sound and Vibration vol 331 no 24 pp 5326ndash5346 2012

[13] H-W Kwon S-Y Hong D-K Oh et al ldquoExperimental studyon the energy flow analysis of underwater vibration for thereinforced cylindrical structurerdquo Journal of Mechanical Sci-ence and Technology vol 28 no 9 pp 3405ndash3410 2014

[14] S Wang and R J Bernhard ldquoPrediction of averaged energyfor moderately damped systems with strong couplingrdquoJournal of Sound and Vibration vol 319 no 1-2 pp 426ndash4442009

[15] V S Pereira and J M C Dos Santos ldquoCoupled plate energymodels at mid- and high-frequency vibrationsrdquo Computers ampStructures vol 134 pp 48ndash61 2014

[16] H-W Kwon S-Y Hong and J-H Song ldquoVibrational energyflow analysis of coupled cylindrical thin shell structuresrdquoJournal of Mechanical Science and Technology vol 30 no 9pp 4049ndash4062 2016

[17] Y-H Park and S-Y Hong ldquoHybrid power flow analysis usingcoupling loss factor of SEA for low-damping system-part Iformulation of 1-D and 2-D casesrdquo Journal of Sound andVibration vol 299 no 3 pp 484ndash503 2007

[18] Z Liu J Niu and X Gao ldquoAn improved approach for analysisof coupled structures in Energy Finite Element Analysis usingthe coupling loss factorrdquo Computers amp Structures vol 210pp 69ndash86 2018

[19] H-W Kwon S-Y Hong and J-H Song ldquoEnergy flowmodels for underwater radiation noise prediction in medium-to-high-frequency rangesrdquo Proceedings of the Institution ofMechanical Engineers Part M Journal of Engineering for theMaritime Environment vol 230 no 2 pp 404ndash416 2016

[20] X Zheng W Dai Y Qiu and Z Hao ldquoPrediction and energycontribution analysis of interior noise in a high-speed trainbased on modified energy finite element analysisrdquoMechanicalSystems and Signal Processing vol 126 pp 439ndash457 2019

[21] W Zhang H Chen D Zhu and X Kong ldquo)e thermal effectson high-frequency vibration of beams using energy flowanalysisrdquo Journal of Sound and Vibration vol 333 no 9pp 2588ndash2600 2014

[22] W Di X Miaoxia and L Yueming ldquoHigh-frequency dy-namic analysis of plates in thermal environments based onenergy finite elementmethodrdquo Shock and Vibration vol 2015pp 1ndash14 2015

[23] Z Chen Z Yang N Guo and G Zhang ldquoAn energy finiteelement method for high frequency vibration analysis ofbeams with axial forcerdquo Applied Mathematical Modellingvol 61 no 5 pp 521ndash539 2018

[24] P M Morse and K U Indard Ceoretical AcousticsPrinceton University Press Princeton NJ USA 1986

[25] W Weaver S P Timoshenko and D H Young VibrationProblems in Engineering John Wiley amp Sons Hoboken NJUSA 1990

[26] M N Ichchou A Le Bot and L Jezequel ldquoEnergy models ofone-dimensional multi-propagative systemsrdquo Journal ofSound and Vibration vol 201 no 5 pp 535ndash554 1997

[27] Y Lase M N Ichchou and L Jezequel ldquoEnergy flow analysisof bars and beams theoretical formulationsrdquo Journal of Soundand Vibration vol 192 no 1 pp 281ndash305 1996

[28] D J Nefske and S H Sung ldquoPower flow finite elementanalysis of dynamic systems basic theory and application tobeamsrdquo Journal of Vibration and Acoustics vol 111 no 1pp 94ndash100 1989

Mathematical Problems in Engineering 11

dlangIrang

dx+langπdissrang langπinrangδ x minus x0( 1113857 (18)

where πin refers to the input power at the driving pointx x0

Substituting equations (13) and (17) into equation (18)the energy density governing equation for the high dampingbeam with axial force can be derived as

ωRe Dck3c1 + Dckc1 kc1

111386811138681113868111386811138681113868111386811138682

+ Nckc11113960 1113961

k2 ρSω2 + N kc11113868111386811138681113868

11138681113868111386811138682

+ D kc11113868111386811138681113868

11138681113868111386811138684

1113872 1113873

d2langerang

dx2 +2ηω N kc1

111386811138681113868111386811138681113868111386811138682

+ D kc11113868111386811138681113868

11138681113868111386811138684

1113872 1113873

ρSω2 + N kc11113868111386811138681113868

11138681113868111386811138682

+ D kc11113868111386811138681113868

11138681113868111386811138684 langerang langπinrangδ x minus x0( 1113857 (19)

Unlike the lightly damping system in [23] the energytransition equation is derived using the exact wavenumberwithout neglecting higher-order damping terms )e energytransition equation represents the relationship between timeand space averaged energy density and intensity withoutgroup velocity because the damping effect is not neglectedOn account of high structural damping the potential energydensity is not the same as the kinetic energy density of thestructure )us the energy loss equation is derivedaccording to the relationship between potential energy andtotal energy

3 Solution of Energy DensityGoverning Equation

As shown in Figure 1 the beam is divided into two regions atx0 From equation (19) the general solution of the flexuralenergy density governing equation is expressed as

langefrangi Ci1eminusλfx

+ Ci2eλfx

(i 1 2) (20)

λf

minus2ηk2 N kc1

111386811138681113868111386811138681113868111386811138682

+ D kc11113868111386811138681113868

11138681113868111386811138684

1113872 1113873

Re Dck3c1 + Dckc1 kc1

111386811138681113868111386811138681113868111386811138682

+ Nckc11113960 1113961

11139741113972

(21)

where the subscript i denotes the ith region of the beam )econstants Ci1 Ci2 can be determined by the energy boundaryconditions

From equations (13) and (20) the energy intensity of thevibrating beam is represented as

langIfrangi Pf Ci1eminus λfx

minus Ci2eλfx

1113872 1113873 (22)

Pf minusλf

ωRe Dck3c1 + Dckc1 kc1

111386811138681113868111386811138681113868111386811138682

+ Nckc11113960 1113961

k2 ρSω2 + N kc11113868111386811138681113868

11138681113868111386811138682

+ D kc11113868111386811138681113868

11138681113868111386811138684

1113872 1113873 (23)

Because there are no energy outflow and inflow at bothends of the pinned-pinned beam the following energy in-tensity boundary conditions can be determined as

langIfrang1(0) 0 langIfrang2(L) 0 (24)

At the interface between the regions ① and ② theenergy density is continuous and the energy intensity issubject to conservation of energy )e resulting boundarycondition can be written as

langefrang1 x0( 1113857 langefrang2 x0( 1113857 (25)

langIfrang2 x0( 1113857 langIfrang1 x0( 1113857 +langπinrang (26)

)e input power of the vibrating beam is obtained by theimpedance method and expressed as [23 28]

langπinrang 12

F01113868111386811138681113868

11138681113868111386811138682Re

1Z

1113874 1113875 (27)

Z 2Dckc1k

2c2

ω1 + j

kc1

kc21113888 1113889 (28)

where Z is the impedance at the driving point of the vi-brating beam with axial force

Substituting equation (20) into equations (24)sim(26) thematrix equation can be obtained as

1 minus1 0 0

0 0 eminusλfL minuseminusλfL

eminusλfx0 eλfx0 minuseminusλfx0 minuseλfx0

minusPfeminusλfx0 Pfeλfx0 Pfeminusλfx0 minusPfeλfx0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C11

C12

C21

C22

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

0

0

0

langπinrang

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(29)

According to equation (29) the flexural energy densitycan be analytically calculated for the vibrating beam withhigh structural damping loss factor and constant axial force

4 Verification and Discussion

To validate the proposed EFA formulation and investigatethe effects of structural damping loss factor and axial forcevarious energy flow analyses are performed for a pinned-pinned beam at both ends shown in Figure 1 )e resultsfrom analytical solutions of EFA governing equation arecompared with exact modal solutions (Appendix A andAppendix B) which are defined as time averaged energydensity obtained from the displacement solutions For eachmodal analysis the first 5000 modes are extracted to ensurethe accuracy of the solution In addition the present vi-brational energy flow method is applied to beams withdifferent boundary conditions

41 Model Verification )e beam is made of aluminumalloy with density ρ 2700kgm3 and elastic modulusE 70GPa and its dimensions are L 3mS 2 times 10minus4m2Ib 2 times 10minus10m4 )e structural damping loss factor η

4 Mathematical Problems in Engineering

ranges from 005 to 08 and the axial force N from 0 to 20 kNrespectively )e pinned-pinned beam as shown in Figure 1is excited by a transverse harmonic force located at x0 L2

Figure 2 shows the energy density distribution of pin-ned-pinned beams for the case with the axial forceN 20 kN the amplitude of transverse harmonic forceF0 1N and the structural damping loss factor η 05 whenthe analysis frequencies are 2 4 6 and 8 kHz Except for thatthe developed EFA results differ near both ends of the beamthe developed EFA results is analogous to the modal analysisresults because the evanescent wave is neglected in theenergy flow models Moreover it can be observed that thedifference of the EFA solution from the modal solution issmaller at higher frequencies and hence the accuracy of theEFA model is improved at higher frequencies )e energydensity distributions with different amplitudes of transverseharmonic force (F0 01 1 10 kN) as shown in Figure 3 arecalculated when the excitation frequency is f 6 kHz It isalso found that the EFA solutions are smooth representingthe overall trend of modal solutions and in good agreementwith modal solutions except the vicinity of boundaries Inaddition the global variation of energy density increaseswith larger amplitudes of harmonic point force

)e exact energy density distribution obtained by modalanalysis method oscillates spatially without space averagingAt high-frequency ranges the variance of classical solutionsis uniform resulting from short wavelength )erefore thedeveloped EFA results obtained with space averaging areanalogous to the classical solutions as the analysis frequencyincreases

Figures 4 and 5 show the energy density distributions ofpinned-pinned beams withN 20 kN F0 1N and f 6 kHzwhen the structural damping loss factor is η 005 and 05respectively If the structural damping loss factor is small thewavenumber and group velocity given in equations (3) and(4) in the developed energy flow model are well approxi-mated by those in equations (5) and (6) in the traditionalenergy flow model Hence in a low damping beam thedeveloped EFA solution is close to the traditional EFAsolution However in a high damping beam the developedEFA result represents the trend of exact solution frommodalanalysis and the developed EFA result concurs more withthe exact solution than the traditional EFA result becausehigher-order damping terms are no longer neglected

42 Effects of Constant Axial Force Figure 6 shows theenergy density distributions of pinned-pinned beams withdifferent axial forces 0kNleNle 20kN amplitude of trans-verse harmonic force F0 1N and structural damping lossfactor η 05 when the analysis frequency is 6 kHz )eenergy density varies more dramatically along x positionwith increasing axial force)e total energy can be calculatedfrom integrating the energy density over the length of thebeam Figure 7 shows that the total energy is higher at largeraxial force

)e effects of constant axial force can be explained by theinput power and group velocity Figures 8 and 9 show theinput power and group velocity respectively It is observed

that the input power increases and the group velocity de-creases with increasing axial force Because the total energyis proportional to the input power the total energy is higherfor larger axial force arousing larger input power )e lowergroup velocity makes energy propagate more slowly fromthe driving point to the boundary It means that more energydissipates near the driving point and less energy propagatesto the boundary As a result the energy density changesmore dramatically with larger axial force along the xposition

Ener

gy d

ensit

y (d

B)

0 05 1 15 2 25 3Position (m)

ndash60

ndash40

ndash20

0

20

40

60

80

100

120

140

EFA solution F0 = 1kN

Modal solution F0 = 1kNEFA solution F0 = 01kN

EFA solution F0 = 10kNModal solution F0 = 10kNModal solution F0 = 01kN

Figure 3 Energy density distribution of pinned-pinned beamswith N 20 kN η 05 and f 6 kHz when F0 01 1 10 kN

0 05 1 15 2 25 3ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

0

20

40

60

Ener

gy d

ensit

y (d

B)

Position (m)

EFA solution f = 2kHzModal solution f = 2kHzEFA solution f = 4kHzModal solution f = 4kHz

EFA solution f = 6kHzModal solution f = 6kHzEFA solution f = 8kHzModal solution f = 8kHz

Figure 2 Energy density distribution of pinned-pinned beamswith N 20 kN F0 1N and η 05 when f 2 4 6 8 kHz

Mathematical Problems in Engineering 5

43 Effects of Structural Damping Loss Factor Figure 10shows the energy density distributions of pinned-pinnedbeams with different structural damping loss factor0le ηle 08 amplitude of transverse harmonic force F0 1Nand axial force N 20 kN when the analysis frequency is6 kHz )e developed EFA results represent the globalvariation of the modal solutions at each structural dampingloss factor Owing to the exclusion of the evanescent wavethe developed energy flow solutions more closely approachthe modal solutions as the structural damping loss factorincreases)e energy density changes more suddenly along x

position with increasing structural damping loss factorFigure 11 shows that the total energy is lower at largerstructural damping loss factor

Figures 12 and 13 show the input power and groupvelocity at various structural damping loss factors respec-tively It is found that the input power decreases and thegroup velocity increases with an increase of structuraldamping loss factor Since the total energy is proportional tothe input power the total energy is lower for larger structuraldamping loss factor leading to less input power In fact thelarger group velocity contributes to flatter change of theenergy density along the x position But the energy densitydistribution changes more dramatically with larger struc-tural damping loss factor along the x position It attributes tothat the energy density is mainly affected by the input powerwhile the effect of group velocity is minor

Ener

gy d

ensit

y (d

B)

0 05 1 15 2 25 3Position (m)

ndash20

ndash10

0

10

20

30

40

50

60

1556

60

N = 20kNN = 5kN

N = 1kNN = 0kN

Figure 6 Energy density distribution of pinned-pinned beamswith f 6 kHz F0 1N and η 05 when N 0 1 5 20 kN

0 2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

12

14

Axial force (kN)

Tota

l ene

rgy

(μW

)

Figure 7 Total energy of pinned-pinned beams versus axial forcewith F0 1N η 05 and f 6 kHz

0 05 1 15 2 25 335

40

45

50

55

60

Developed EFA solution Modal solution Traditional EFA solution

Ener

gy d

ensit

y (d

B)

Position (m)

Figure 4 Energy density distribution of pinned-pinned beamswith N 20 kN F0 1N and η 005 when f 6 kHz

Ener

gy d

ensit

y (d

B)

0 05 1 15 2 25 3

Developed EFA solution Modal solution Traditional EFA solution

Position (m)

ndash100

ndash80

ndash60

ndash40

ndash20

0

20

40

60

Figure 5 Energy density distribution of pinned-pinned beamswith N 20 kN F0 1N and η 005 when f 6 kHz

6 Mathematical Problems in Engineering

44 Application to Beams with Different BoundaryConditions )e analysis model consists of a beam fixed atboth ends as shown in Figure 14 and a beam fixed at one endand free at the other as shown in Figure 15 respectivelyBoth beams share the common dimensions of L

3mS 2 times 10minus4m2Ib 2 times 10minus10m4 material properties ofρ 2700kgm3 and E 70GPa and the structuraldamping loss factor η 05 Since the energy outflow at thefixed point is zero and the energy flow satisfies theequilibrium condition at the driving point the energyintensity boundary conditions for the fixed-fixed beam isidentical to those for the pinned-pinned beam Howeverthe impedance of fixed-fixed beam at the driving point isobtained by substituting equations (B5) and (B6) inAppendix B into equation (A8) in Appendix A For fixed-free beam the subdomain ② is absent so that the coef-ficients C21 C22 in equation (29) vanish the followingmatrix equation is rewritten as

1 minus1

minusPfeminusλfx0 Pfeλfx0⎡⎣ ⎤⎦

C11

C121113890 1113891

0

langπinrang1113890 1113891 (30)

Similarly the impedance of fixed-free beam at drivingpoint is determined by substituting the equation (B8) and(B9) in Appendix B into equation (A8) in Appendix AFrom equation (30) the energy density for the fixed-fixedbeam can be evaluated

Figure 14 shows a fixed-fixed beam with constant axialforce Figures 16 and 17 show the energy density distributionof fixed-fixed beams and fixed-free beams with F0 1NN 20 kN and η 05 when the analysis frequencies are2 kHz 4 kHz 6 kHz and 8 kHz respectively )e developedEFA results coincide well with the overall trend of the exactsolutions from modal analysis at each frequency Similar toprevious cases the developed energy flow model gives moreaccurate result at high frequencies despite the exclusion of

0 2 4 6 8 10 12 14 16 18 20075

080

085

090

095

010

0105

Axial force (kN)

Inpu

t pow

er (m

W)

Figure 8 Input power of pinned-pinned beams versus axial forcewith F0 1N η 05 and f 6 kHz

0 5 10 15 20 25300

350

400

450

500

550

Axial force (kN)

Gro

up v

eloc

ity (m

s)

Figure 9 Group velocity versus axial force with η 05 andf 6 kHz

0 05 1 15 2 25 3ndash190

ndash140

ndash90

ndash40

10

60

Ener

gy d

ensit

y (d

B)

Position (m)

EFA solution η = 005

Modal solution η = 005

EFA solution η = 020

Modal solution η = 020

EFA solution η = 050

Modal solution η = 050

EFA solution η = 080

Modal solution η = 080

Figure 10 Energy density distribution of pinned-pinned beamswith N 20 kN F0 1N and f 6 kHz when η 005 020 050080

0 01 02 03 04 05 06 07 080

005

010

015

020

025

030

Tota

l ene

rgy

(μW

)

Structural damping loss factor

Figure 11 Total energy of pinned-pinned beams versus structuraldamping loss factor with F0 1N N 20 kN and f 6 kHz

Mathematical Problems in Engineering 7

the evanescent wave As a consequence it is feasible topredict the vibrational energy density of beams with variousboundary conditions in the high-frequency range by theproposed energy flow model

5 Conclusion

An energy flow model for a beam with axial force thatincludes high damping effect is established to evaluate theenergy density in the high-frequency range )e wave-number is expressed as the function of the damping lossfactor and the axial force Particularly the energy trans-mission equation and the energy dissipation equation arederived from the wavenumber without approximation of itsreal and imaginary term)en the energy density governing

0 01 02 03 04 05 06 07 08042

044

046

048

050

052

054

056

058

Inpu

t pow

er (m

W)

Structural damping loss factor

Figure 12 Input power of pinned-pinned beams versus structuraldamping loss factor with F0 1N N 20 kN and f 6 kHz

0 01 02 03 04 05 06 07 08Structural damping loss factor

825

830

835

840

845

850

855

860

865

870

Gro

up v

eloc

ity (m

s)

Figure 13 Group velocity versus structural damping loss factorwith N 20 kN and f 6 kHz

x

x0

L

F0δ (x ndash x0)ejωt

②①N N

Figure 14 A fixed-fixed beam with constant axial force

x

L

NF0δ (x ndash x0)ejωt

①

Figure 15 A fixed-free beam with constant axial force

0 05 1 15 2 25 3ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

0

20

40

60

EFA solution f = 2kHz

EFA solution f = 4kHzEFA solution f = 6kHz

EFA solution f = 8kHz

Modal solution f = 2kHz

Modal solution f = 4kHzModal solution f = 6kHz

Modal solution f = 8kHz

Ener

gy d

ensit

y (d

B)

Position (m)

Figure 16 Energy density distribution of fixed-fixed beams withF0 1N N 20 kN and η 05 when f 2 4 6 8 kHz

0 05 1 15 2 25 3

EFA solution f = 2kHz

EFA solution f = 4kHzEFA solution f = 6kHz

EFA solution f = 8kHz

Modal solution f = 2kHz

Modal solution f = 4kHzModal solution f = 6kHz

Modal solution f = 8kHz

Position (m)

ndash250

ndash200

ndash150

ndash100

ndash50

0

50

100

Ener

gy d

ensit

y (d

B)

Figure 17 Energy density distribution of fixed-free beams withF0 1N N 20 kN and η 05 when f 2 4 6 8 kHz

8 Mathematical Problems in Engineering

equation is obtained for the high damping beam with axialforce

To verify the developed energy flow model numericalanalyses are performed for the simply supported beam withvarious damping loss factor and axial force )e EFA so-lutions are greatly consistent with the modal solutions in allcases Furthermore it is found that both the high dampingfactor and large axial force can significantly alter the level ofenergy density as well as the distribution Due to theresulting deceasing input power the increasing dampingloss factor makes the energy density decay more dramati-cally along axial position However that is the contrary casefor the axial force )e proposed method is expected to beuseful for the prediction of the high-frequency vibration ofhigh damping structures with initial stress

Appendix

A Impedance and Exact Solution from ModalAnalysis for High Damping Beams withAxial Force

When a high damping beam with axial force is excited at x0by a transverse harmonic force the governing equation ofmotion can be expressed as

Dc

z4w

zx4 minus Nc

z2w

zx2 + ρSz2w

zt2 F0δ x minus x0( 1113857e

jωt (A1)

)e steady-state solution of equation (A1) can be foundby modal superposition as

w(x t) 1113944infin

i1ϕi(x)qi(t) (A2)

where ϕi(x) is the ith natural mode of the beam and qi(t) isthe ith mode displacement

By substituting equation (A2) into the motion equationof the beam equation (A1) and using the orthogonality ofnatural modes we obtain

Kiqi(t) + Mi

d2qi(t)

dt2 Fi (A3)

where

Fi 1113946L

0F0δ x minus x0( 1113857ϕi(x)dx (A4)

represents the so-called ldquomodal forcerdquo and

Mi 1113946L

0ρSϕ2

i dx

Ki ω2i (1 + jη) 1113946

L

0ρSϕ2

i dx

(A5)

are the so-called ldquomodal massrdquo and ldquomodal stiffnessrdquo )eeffect of structural damping is modeled by assuming ahysteretic damping model which introduces an additionalcomplex term (1 + jη) to modal stiffness )e natural fre-quency of the ith mode ωi can be determined from thecorresponding wavenumber in equation (2) which can bewritten as

k2i minus

N

2D+

N

2D1113874 1113875

2+ρSω2

i

D

1113971

(A6)

)e forced vibration response of the beam can beexpressed as the linear combination of the normal modes

w(x t) 1113944infin

i1

F0 middot ejωt

Mi ω2i (1 + jη) minus ω21113858 1113859

ϕi(x)ϕi x0( 1113857 (A7)

)e impedance at the driving point of the finite beamwith axial force can be obtained from its definition

Z F0e

jωt

_w x0 t( 1113857

minus11113936infini1 ωϕ

2i x0( 1113857Mi ω2

i (1 + jη) minus ω21113858 1113859

(A8)

where _w(x0 t) is the velocity at driving point which can becalculated as the time derivative of the displacement atx x0 Once the mode function is determined by theboundary condition the impedance at driving point of thefinite beam with axial force is derived from equation (A8)

)e time averaged energy density is given by

langerang 14

ρSzw

zt1113888 1113889

zw

zt1113888 1113889

lowast

+ Dz2w

zx21113888 1113889z2w

zx21113888 1113889

lowast

+ Nzw

zx1113888 1113889

zw

zx1113888 1113889

lowast

1113890 1113891 (A9)

Substituting equation (A7) into equation (A9) theenergy density can be obtained based on the modal solution

B Mode Shapes for High Damping Beams withAxial Force

B1 BeamPinned at Both Ends )e transverse displacementand the bending moment are zero at a simply supported endHence the boundary conditions can be stated as

w(0) 0

z2w(0)

zx2 0

w(L) 0

z2w(L)

zx2 0

(B1)

Mathematical Problems in Engineering 9

)e general solution of equation (A1) is shown inequation (7) whereas the frequency equation with bothsimply supported ends is derived as

sin kiL 0 (B2)

)e vibration mode of the beam is given by

ϕi(x) sin kix (B3)

B2 Beam Fixed at Both Ends At a fixed end the transversedisplacement and the slope of the displacement are zeroHence the boundary conditions are given by

w(0) 0

zw(0)

zx 0

w(L) 0

zw(L)

zx 0

(B4)

By virtue of general solution in equation (7) andboundary conditions with both fixed ends the frequencyequation is obtained as

cos kiL cosh kiL minus 1 0 (B5)

)e vibration mode of the beam is expressed as

ϕi(x) cos kix minus cosh kix( 1113857 +cos kiL minus cosh kiL

sin kiL minus sinh kiL

middot sin kix minus sinh kix( 1113857

(B6)

B3BeamFixedatOneEndandFreeat theOther If the beamis fixed at x 0 and free at x L the transverse deflectionand its slope must be zero at x 0 and the bending momentand shear force must be zero at x L )us the boundaryconditions become

w(0) 0

zw(0)

zx 0

z3w(L)

zx3 0

z2w(L)

zx2 0

(B7)

According to the general solution in equation (7) andboundary conditions with one fixed end and the other freeend the frequency equation is derived as

cos kiL cosh kiL + 1 0 (B8)

)e vibration mode of the beam is written as

ϕi(x) cos kix minus cosh kix( 1113857 +cos kiL + cosh kiL

sin kiL + sinh kiL

middot sin kix minus sinh kix( 1113857

(B9)

Data Availability

)e data used to support the finding of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that there no conflicts of interest

Acknowledgments

)e presented work was supported by National NaturalScience Foundation of China (Grant no 51505096) andNatural Science Foundation of Heilongjiang Province ofChina (Grant no QC2016056)

References

[1] L Bot ldquoEntropy in sound and vibration towards a newparadigmrdquo Proceedings of the Royal Society A MathematicalPhysical and Engineering Science vol 473 no 2197 pp 1ndash92017

[2] A Acri ldquoA literature review on energy flow analysis of vi-brating structuresrdquo ASRO Journal of Applied Mechanicsvol 1 no 1 pp 12ndash15 2017

[3] J C Wohlever and R J Bernhard ldquoMechanical energy flowmodels of rods and beamsrdquo Journal of Sound and Vibrationvol 153 no 1 pp 1ndash19 1992

[4] Z Chen Z Yang Y Wang and X Wang ldquoA high-frequencyshock response analysis method based on energy finite ele-ment method and virtual mode synthesis and simulationrdquoActa Aeronautica et Astronautica Sinica vol 39 no 08pp 182ndash191 2018

[5] Z Liu and J Niu ldquoVibrational energy flow model for func-tionally graded beamsrdquo Composite Structures vol 186pp 17ndash28 2018

[6] X Kong H Chen D Zhu and W Zhang ldquoStudy on thevalidity region of energy finite element analysisrdquo Journal ofSound and Vibration vol 333 no 9 pp 2601ndash2616 2014

[7] A Nokhbatolfoghahai H Navazi Y Ghobaad andH Haddadpour ldquoExperimental measurement of energydensity of a vibrating beamrdquo Journal of Vibration and Controlvol 24 no 24 pp 5735ndash5746 2018

[8] H M Navazi A Nokhbatolfoghahaei Y Ghobad andH Haddadpour ldquoExperimental measurement of energydensity in a vibrating plate and comparison with energy finiteelement analysisrdquo Journal of Sound and Vibration vol 375pp 289ndash307 2016

[9] D Zhu H Chen X Kong and W Zhang ldquoA hybrid finiteelement-energy finite element method for mid-frequencyvibrations of built-up structures under multi-distributedloadingsrdquo Journal of Sound amp Vibration vol 333 no 22pp 5723ndash5745 2014

[10] M Sadeghmanesh H Haddadpour M T Abadi andH M Navazi ldquoA method for selection of structural theories

10 Mathematical Problems in Engineering

for low to high frequency vibration analysesrdquo EuropeanJournal of Mechanics-A Solids vol 75 pp 27ndash40 2019

[11] W Zhang AWang N Vlahopoulos and KWu ldquoA vibrationanalysis of stiffened plates under heavy fluid loading by anenergy finite element analysis formulationrdquo Finite Elements inAnalysis and Design vol 41 no 11-12 pp 1056ndash1078 2005

[12] J-B Han S-Y Hong and J-H Song ldquoEnergy flow model forthin plate considering fluid loading with mean flowrdquo Journalof Sound and Vibration vol 331 no 24 pp 5326ndash5346 2012

[13] H-W Kwon S-Y Hong D-K Oh et al ldquoExperimental studyon the energy flow analysis of underwater vibration for thereinforced cylindrical structurerdquo Journal of Mechanical Sci-ence and Technology vol 28 no 9 pp 3405ndash3410 2014

[14] S Wang and R J Bernhard ldquoPrediction of averaged energyfor moderately damped systems with strong couplingrdquoJournal of Sound and Vibration vol 319 no 1-2 pp 426ndash4442009

[15] V S Pereira and J M C Dos Santos ldquoCoupled plate energymodels at mid- and high-frequency vibrationsrdquo Computers ampStructures vol 134 pp 48ndash61 2014

[16] H-W Kwon S-Y Hong and J-H Song ldquoVibrational energyflow analysis of coupled cylindrical thin shell structuresrdquoJournal of Mechanical Science and Technology vol 30 no 9pp 4049ndash4062 2016

[17] Y-H Park and S-Y Hong ldquoHybrid power flow analysis usingcoupling loss factor of SEA for low-damping system-part Iformulation of 1-D and 2-D casesrdquo Journal of Sound andVibration vol 299 no 3 pp 484ndash503 2007

[18] Z Liu J Niu and X Gao ldquoAn improved approach for analysisof coupled structures in Energy Finite Element Analysis usingthe coupling loss factorrdquo Computers amp Structures vol 210pp 69ndash86 2018

[19] H-W Kwon S-Y Hong and J-H Song ldquoEnergy flowmodels for underwater radiation noise prediction in medium-to-high-frequency rangesrdquo Proceedings of the Institution ofMechanical Engineers Part M Journal of Engineering for theMaritime Environment vol 230 no 2 pp 404ndash416 2016

[20] X Zheng W Dai Y Qiu and Z Hao ldquoPrediction and energycontribution analysis of interior noise in a high-speed trainbased on modified energy finite element analysisrdquoMechanicalSystems and Signal Processing vol 126 pp 439ndash457 2019

[21] W Zhang H Chen D Zhu and X Kong ldquo)e thermal effectson high-frequency vibration of beams using energy flowanalysisrdquo Journal of Sound and Vibration vol 333 no 9pp 2588ndash2600 2014

[22] W Di X Miaoxia and L Yueming ldquoHigh-frequency dy-namic analysis of plates in thermal environments based onenergy finite elementmethodrdquo Shock and Vibration vol 2015pp 1ndash14 2015

[23] Z Chen Z Yang N Guo and G Zhang ldquoAn energy finiteelement method for high frequency vibration analysis ofbeams with axial forcerdquo Applied Mathematical Modellingvol 61 no 5 pp 521ndash539 2018

[24] P M Morse and K U Indard Ceoretical AcousticsPrinceton University Press Princeton NJ USA 1986

[25] W Weaver S P Timoshenko and D H Young VibrationProblems in Engineering John Wiley amp Sons Hoboken NJUSA 1990

[26] M N Ichchou A Le Bot and L Jezequel ldquoEnergy models ofone-dimensional multi-propagative systemsrdquo Journal ofSound and Vibration vol 201 no 5 pp 535ndash554 1997

[27] Y Lase M N Ichchou and L Jezequel ldquoEnergy flow analysisof bars and beams theoretical formulationsrdquo Journal of Soundand Vibration vol 192 no 1 pp 281ndash305 1996

[28] D J Nefske and S H Sung ldquoPower flow finite elementanalysis of dynamic systems basic theory and application tobeamsrdquo Journal of Vibration and Acoustics vol 111 no 1pp 94ndash100 1989

Mathematical Problems in Engineering 11

ranges from 005 to 08 and the axial force N from 0 to 20 kNrespectively )e pinned-pinned beam as shown in Figure 1is excited by a transverse harmonic force located at x0 L2

Figure 2 shows the energy density distribution of pin-ned-pinned beams for the case with the axial forceN 20 kN the amplitude of transverse harmonic forceF0 1N and the structural damping loss factor η 05 whenthe analysis frequencies are 2 4 6 and 8 kHz Except for thatthe developed EFA results differ near both ends of the beamthe developed EFA results is analogous to the modal analysisresults because the evanescent wave is neglected in theenergy flow models Moreover it can be observed that thedifference of the EFA solution from the modal solution issmaller at higher frequencies and hence the accuracy of theEFA model is improved at higher frequencies )e energydensity distributions with different amplitudes of transverseharmonic force (F0 01 1 10 kN) as shown in Figure 3 arecalculated when the excitation frequency is f 6 kHz It isalso found that the EFA solutions are smooth representingthe overall trend of modal solutions and in good agreementwith modal solutions except the vicinity of boundaries Inaddition the global variation of energy density increaseswith larger amplitudes of harmonic point force

)e exact energy density distribution obtained by modalanalysis method oscillates spatially without space averagingAt high-frequency ranges the variance of classical solutionsis uniform resulting from short wavelength )erefore thedeveloped EFA results obtained with space averaging areanalogous to the classical solutions as the analysis frequencyincreases

Figures 4 and 5 show the energy density distributions ofpinned-pinned beams withN 20 kN F0 1N and f 6 kHzwhen the structural damping loss factor is η 005 and 05respectively If the structural damping loss factor is small thewavenumber and group velocity given in equations (3) and(4) in the developed energy flow model are well approxi-mated by those in equations (5) and (6) in the traditionalenergy flow model Hence in a low damping beam thedeveloped EFA solution is close to the traditional EFAsolution However in a high damping beam the developedEFA result represents the trend of exact solution frommodalanalysis and the developed EFA result concurs more withthe exact solution than the traditional EFA result becausehigher-order damping terms are no longer neglected

42 Effects of Constant Axial Force Figure 6 shows theenergy density distributions of pinned-pinned beams withdifferent axial forces 0kNleNle 20kN amplitude of trans-verse harmonic force F0 1N and structural damping lossfactor η 05 when the analysis frequency is 6 kHz )eenergy density varies more dramatically along x positionwith increasing axial force)e total energy can be calculatedfrom integrating the energy density over the length of thebeam Figure 7 shows that the total energy is higher at largeraxial force

)e effects of constant axial force can be explained by theinput power and group velocity Figures 8 and 9 show theinput power and group velocity respectively It is observed

that the input power increases and the group velocity de-creases with increasing axial force Because the total energyis proportional to the input power the total energy is higherfor larger axial force arousing larger input power )e lowergroup velocity makes energy propagate more slowly fromthe driving point to the boundary It means that more energydissipates near the driving point and less energy propagatesto the boundary As a result the energy density changesmore dramatically with larger axial force along the xposition

Ener

gy d

ensit

y (d

B)

0 05 1 15 2 25 3Position (m)

ndash60

ndash40

ndash20

0

20

40

60

80

100

120

140

EFA solution F0 = 1kN

Modal solution F0 = 1kNEFA solution F0 = 01kN

EFA solution F0 = 10kNModal solution F0 = 10kNModal solution F0 = 01kN

Figure 3 Energy density distribution of pinned-pinned beamswith N 20 kN η 05 and f 6 kHz when F0 01 1 10 kN

0 05 1 15 2 25 3ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

0

20

40

60

Ener

gy d

ensit

y (d

B)

Position (m)

EFA solution f = 2kHzModal solution f = 2kHzEFA solution f = 4kHzModal solution f = 4kHz

EFA solution f = 6kHzModal solution f = 6kHzEFA solution f = 8kHzModal solution f = 8kHz

Figure 2 Energy density distribution of pinned-pinned beamswith N 20 kN F0 1N and η 05 when f 2 4 6 8 kHz

Mathematical Problems in Engineering 5

43 Effects of Structural Damping Loss Factor Figure 10shows the energy density distributions of pinned-pinnedbeams with different structural damping loss factor0le ηle 08 amplitude of transverse harmonic force F0 1Nand axial force N 20 kN when the analysis frequency is6 kHz )e developed EFA results represent the globalvariation of the modal solutions at each structural dampingloss factor Owing to the exclusion of the evanescent wavethe developed energy flow solutions more closely approachthe modal solutions as the structural damping loss factorincreases)e energy density changes more suddenly along x

position with increasing structural damping loss factorFigure 11 shows that the total energy is lower at largerstructural damping loss factor

Figures 12 and 13 show the input power and groupvelocity at various structural damping loss factors respec-tively It is found that the input power decreases and thegroup velocity increases with an increase of structuraldamping loss factor Since the total energy is proportional tothe input power the total energy is lower for larger structuraldamping loss factor leading to less input power In fact thelarger group velocity contributes to flatter change of theenergy density along the x position But the energy densitydistribution changes more dramatically with larger struc-tural damping loss factor along the x position It attributes tothat the energy density is mainly affected by the input powerwhile the effect of group velocity is minor

Ener

gy d

ensit

y (d

B)

0 05 1 15 2 25 3Position (m)

ndash20

ndash10

0

10

20

30

40

50

60

1556

60

N = 20kNN = 5kN

N = 1kNN = 0kN

Figure 6 Energy density distribution of pinned-pinned beamswith f 6 kHz F0 1N and η 05 when N 0 1 5 20 kN

0 2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

12

14

Axial force (kN)

Tota

l ene

rgy

(μW

)

Figure 7 Total energy of pinned-pinned beams versus axial forcewith F0 1N η 05 and f 6 kHz

0 05 1 15 2 25 335

40

45

50

55

60

Developed EFA solution Modal solution Traditional EFA solution

Ener

gy d

ensit

y (d

B)

Position (m)

Figure 4 Energy density distribution of pinned-pinned beamswith N 20 kN F0 1N and η 005 when f 6 kHz

Ener

gy d

ensit

y (d

B)

0 05 1 15 2 25 3

Developed EFA solution Modal solution Traditional EFA solution

Position (m)

ndash100

ndash80

ndash60

ndash40

ndash20

0

20

40

60

Figure 5 Energy density distribution of pinned-pinned beamswith N 20 kN F0 1N and η 005 when f 6 kHz

6 Mathematical Problems in Engineering

44 Application to Beams with Different BoundaryConditions )e analysis model consists of a beam fixed atboth ends as shown in Figure 14 and a beam fixed at one endand free at the other as shown in Figure 15 respectivelyBoth beams share the common dimensions of L

3mS 2 times 10minus4m2Ib 2 times 10minus10m4 material properties ofρ 2700kgm3 and E 70GPa and the structuraldamping loss factor η 05 Since the energy outflow at thefixed point is zero and the energy flow satisfies theequilibrium condition at the driving point the energyintensity boundary conditions for the fixed-fixed beam isidentical to those for the pinned-pinned beam Howeverthe impedance of fixed-fixed beam at the driving point isobtained by substituting equations (B5) and (B6) inAppendix B into equation (A8) in Appendix A For fixed-free beam the subdomain ② is absent so that the coef-ficients C21 C22 in equation (29) vanish the followingmatrix equation is rewritten as

1 minus1

minusPfeminusλfx0 Pfeλfx0⎡⎣ ⎤⎦

C11

C121113890 1113891

0

langπinrang1113890 1113891 (30)

Similarly the impedance of fixed-free beam at drivingpoint is determined by substituting the equation (B8) and(B9) in Appendix B into equation (A8) in Appendix AFrom equation (30) the energy density for the fixed-fixedbeam can be evaluated

Figure 14 shows a fixed-fixed beam with constant axialforce Figures 16 and 17 show the energy density distributionof fixed-fixed beams and fixed-free beams with F0 1NN 20 kN and η 05 when the analysis frequencies are2 kHz 4 kHz 6 kHz and 8 kHz respectively )e developedEFA results coincide well with the overall trend of the exactsolutions from modal analysis at each frequency Similar toprevious cases the developed energy flow model gives moreaccurate result at high frequencies despite the exclusion of

0 2 4 6 8 10 12 14 16 18 20075

080

085

090

095

010

0105

Axial force (kN)

Inpu

t pow

er (m

W)

Figure 8 Input power of pinned-pinned beams versus axial forcewith F0 1N η 05 and f 6 kHz

0 5 10 15 20 25300

350

400

450

500

550

Axial force (kN)

Gro

up v

eloc

ity (m

s)

Figure 9 Group velocity versus axial force with η 05 andf 6 kHz

0 05 1 15 2 25 3ndash190

ndash140

ndash90

ndash40

10

60

Ener

gy d

ensit

y (d

B)

Position (m)

EFA solution η = 005

Modal solution η = 005

EFA solution η = 020

Modal solution η = 020

EFA solution η = 050

Modal solution η = 050

EFA solution η = 080

Modal solution η = 080

Figure 10 Energy density distribution of pinned-pinned beamswith N 20 kN F0 1N and f 6 kHz when η 005 020 050080

0 01 02 03 04 05 06 07 080

005

010

015

020

025

030

Tota

l ene

rgy

(μW

)

Structural damping loss factor

Figure 11 Total energy of pinned-pinned beams versus structuraldamping loss factor with F0 1N N 20 kN and f 6 kHz

Mathematical Problems in Engineering 7

the evanescent wave As a consequence it is feasible topredict the vibrational energy density of beams with variousboundary conditions in the high-frequency range by theproposed energy flow model

5 Conclusion

An energy flow model for a beam with axial force thatincludes high damping effect is established to evaluate theenergy density in the high-frequency range )e wave-number is expressed as the function of the damping lossfactor and the axial force Particularly the energy trans-mission equation and the energy dissipation equation arederived from the wavenumber without approximation of itsreal and imaginary term)en the energy density governing

0 01 02 03 04 05 06 07 08042

044

046

048

050

052

054

056

058

Inpu

t pow

er (m

W)

Structural damping loss factor

Figure 12 Input power of pinned-pinned beams versus structuraldamping loss factor with F0 1N N 20 kN and f 6 kHz

0 01 02 03 04 05 06 07 08Structural damping loss factor

825

830

835

840

845

850

855

860

865

870

Gro

up v

eloc

ity (m

s)

Figure 13 Group velocity versus structural damping loss factorwith N 20 kN and f 6 kHz

x

x0

L

F0δ (x ndash x0)ejωt

②①N N

Figure 14 A fixed-fixed beam with constant axial force

x

L

NF0δ (x ndash x0)ejωt

①

Figure 15 A fixed-free beam with constant axial force

0 05 1 15 2 25 3ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

0

20

40

60

EFA solution f = 2kHz

EFA solution f = 4kHzEFA solution f = 6kHz

EFA solution f = 8kHz

Modal solution f = 2kHz

Modal solution f = 4kHzModal solution f = 6kHz

Modal solution f = 8kHz

Ener

gy d

ensit

y (d

B)

Position (m)

Figure 16 Energy density distribution of fixed-fixed beams withF0 1N N 20 kN and η 05 when f 2 4 6 8 kHz

0 05 1 15 2 25 3

EFA solution f = 2kHz

EFA solution f = 4kHzEFA solution f = 6kHz

EFA solution f = 8kHz

Modal solution f = 2kHz

Modal solution f = 4kHzModal solution f = 6kHz

Modal solution f = 8kHz

Position (m)

ndash250

ndash200

ndash150

ndash100

ndash50

0

50

100

Ener

gy d

ensit

y (d

B)

Figure 17 Energy density distribution of fixed-free beams withF0 1N N 20 kN and η 05 when f 2 4 6 8 kHz

8 Mathematical Problems in Engineering

equation is obtained for the high damping beam with axialforce

To verify the developed energy flow model numericalanalyses are performed for the simply supported beam withvarious damping loss factor and axial force )e EFA so-lutions are greatly consistent with the modal solutions in allcases Furthermore it is found that both the high dampingfactor and large axial force can significantly alter the level ofenergy density as well as the distribution Due to theresulting deceasing input power the increasing dampingloss factor makes the energy density decay more dramati-cally along axial position However that is the contrary casefor the axial force )e proposed method is expected to beuseful for the prediction of the high-frequency vibration ofhigh damping structures with initial stress

Appendix

A Impedance and Exact Solution from ModalAnalysis for High Damping Beams withAxial Force

When a high damping beam with axial force is excited at x0by a transverse harmonic force the governing equation ofmotion can be expressed as

Dc

z4w

zx4 minus Nc

z2w

zx2 + ρSz2w

zt2 F0δ x minus x0( 1113857e

jωt (A1)

)e steady-state solution of equation (A1) can be foundby modal superposition as

w(x t) 1113944infin

i1ϕi(x)qi(t) (A2)

where ϕi(x) is the ith natural mode of the beam and qi(t) isthe ith mode displacement

By substituting equation (A2) into the motion equationof the beam equation (A1) and using the orthogonality ofnatural modes we obtain

Kiqi(t) + Mi

d2qi(t)

dt2 Fi (A3)

where

Fi 1113946L

0F0δ x minus x0( 1113857ϕi(x)dx (A4)

represents the so-called ldquomodal forcerdquo and

Mi 1113946L

0ρSϕ2

i dx

Ki ω2i (1 + jη) 1113946

L

0ρSϕ2

i dx

(A5)

are the so-called ldquomodal massrdquo and ldquomodal stiffnessrdquo )eeffect of structural damping is modeled by assuming ahysteretic damping model which introduces an additionalcomplex term (1 + jη) to modal stiffness )e natural fre-quency of the ith mode ωi can be determined from thecorresponding wavenumber in equation (2) which can bewritten as

k2i minus

N

2D+

N

2D1113874 1113875

2+ρSω2

i

D

1113971

(A6)

)e forced vibration response of the beam can beexpressed as the linear combination of the normal modes

w(x t) 1113944infin

i1

F0 middot ejωt

Mi ω2i (1 + jη) minus ω21113858 1113859

ϕi(x)ϕi x0( 1113857 (A7)

)e impedance at the driving point of the finite beamwith axial force can be obtained from its definition

Z F0e

jωt

_w x0 t( 1113857

minus11113936infini1 ωϕ

2i x0( 1113857Mi ω2

i (1 + jη) minus ω21113858 1113859

(A8)

where _w(x0 t) is the velocity at driving point which can becalculated as the time derivative of the displacement atx x0 Once the mode function is determined by theboundary condition the impedance at driving point of thefinite beam with axial force is derived from equation (A8)

)e time averaged energy density is given by

langerang 14

ρSzw

zt1113888 1113889

zw

zt1113888 1113889

lowast

+ Dz2w

zx21113888 1113889z2w

zx21113888 1113889

lowast

+ Nzw

zx1113888 1113889

zw

zx1113888 1113889

lowast

1113890 1113891 (A9)

Substituting equation (A7) into equation (A9) theenergy density can be obtained based on the modal solution

B Mode Shapes for High Damping Beams withAxial Force

B1 BeamPinned at Both Ends )e transverse displacementand the bending moment are zero at a simply supported endHence the boundary conditions can be stated as

w(0) 0

z2w(0)

zx2 0

w(L) 0

z2w(L)

zx2 0

(B1)

Mathematical Problems in Engineering 9

)e general solution of equation (A1) is shown inequation (7) whereas the frequency equation with bothsimply supported ends is derived as

sin kiL 0 (B2)

)e vibration mode of the beam is given by

ϕi(x) sin kix (B3)

B2 Beam Fixed at Both Ends At a fixed end the transversedisplacement and the slope of the displacement are zeroHence the boundary conditions are given by

w(0) 0

zw(0)

zx 0

w(L) 0

zw(L)

zx 0

(B4)

By virtue of general solution in equation (7) andboundary conditions with both fixed ends the frequencyequation is obtained as

cos kiL cosh kiL minus 1 0 (B5)

)e vibration mode of the beam is expressed as

ϕi(x) cos kix minus cosh kix( 1113857 +cos kiL minus cosh kiL

sin kiL minus sinh kiL

middot sin kix minus sinh kix( 1113857

(B6)

B3BeamFixedatOneEndandFreeat theOther If the beamis fixed at x 0 and free at x L the transverse deflectionand its slope must be zero at x 0 and the bending momentand shear force must be zero at x L )us the boundaryconditions become

w(0) 0

zw(0)

zx 0

z3w(L)

zx3 0

z2w(L)

zx2 0

(B7)

According to the general solution in equation (7) andboundary conditions with one fixed end and the other freeend the frequency equation is derived as

cos kiL cosh kiL + 1 0 (B8)

)e vibration mode of the beam is written as

ϕi(x) cos kix minus cosh kix( 1113857 +cos kiL + cosh kiL

sin kiL + sinh kiL

middot sin kix minus sinh kix( 1113857

(B9)

Data Availability

)e data used to support the finding of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that there no conflicts of interest

Acknowledgments

)e presented work was supported by National NaturalScience Foundation of China (Grant no 51505096) andNatural Science Foundation of Heilongjiang Province ofChina (Grant no QC2016056)

References

[1] L Bot ldquoEntropy in sound and vibration towards a newparadigmrdquo Proceedings of the Royal Society A MathematicalPhysical and Engineering Science vol 473 no 2197 pp 1ndash92017

[2] A Acri ldquoA literature review on energy flow analysis of vi-brating structuresrdquo ASRO Journal of Applied Mechanicsvol 1 no 1 pp 12ndash15 2017

[3] J C Wohlever and R J Bernhard ldquoMechanical energy flowmodels of rods and beamsrdquo Journal of Sound and Vibrationvol 153 no 1 pp 1ndash19 1992

[4] Z Chen Z Yang Y Wang and X Wang ldquoA high-frequencyshock response analysis method based on energy finite ele-ment method and virtual mode synthesis and simulationrdquoActa Aeronautica et Astronautica Sinica vol 39 no 08pp 182ndash191 2018

[5] Z Liu and J Niu ldquoVibrational energy flow model for func-tionally graded beamsrdquo Composite Structures vol 186pp 17ndash28 2018

[6] X Kong H Chen D Zhu and W Zhang ldquoStudy on thevalidity region of energy finite element analysisrdquo Journal ofSound and Vibration vol 333 no 9 pp 2601ndash2616 2014

[7] A Nokhbatolfoghahai H Navazi Y Ghobaad andH Haddadpour ldquoExperimental measurement of energydensity of a vibrating beamrdquo Journal of Vibration and Controlvol 24 no 24 pp 5735ndash5746 2018

[8] H M Navazi A Nokhbatolfoghahaei Y Ghobad andH Haddadpour ldquoExperimental measurement of energydensity in a vibrating plate and comparison with energy finiteelement analysisrdquo Journal of Sound and Vibration vol 375pp 289ndash307 2016

[9] D Zhu H Chen X Kong and W Zhang ldquoA hybrid finiteelement-energy finite element method for mid-frequencyvibrations of built-up structures under multi-distributedloadingsrdquo Journal of Sound amp Vibration vol 333 no 22pp 5723ndash5745 2014

[10] M Sadeghmanesh H Haddadpour M T Abadi andH M Navazi ldquoA method for selection of structural theories

10 Mathematical Problems in Engineering

for low to high frequency vibration analysesrdquo EuropeanJournal of Mechanics-A Solids vol 75 pp 27ndash40 2019

[11] W Zhang AWang N Vlahopoulos and KWu ldquoA vibrationanalysis of stiffened plates under heavy fluid loading by anenergy finite element analysis formulationrdquo Finite Elements inAnalysis and Design vol 41 no 11-12 pp 1056ndash1078 2005

[12] J-B Han S-Y Hong and J-H Song ldquoEnergy flow model forthin plate considering fluid loading with mean flowrdquo Journalof Sound and Vibration vol 331 no 24 pp 5326ndash5346 2012

[13] H-W Kwon S-Y Hong D-K Oh et al ldquoExperimental studyon the energy flow analysis of underwater vibration for thereinforced cylindrical structurerdquo Journal of Mechanical Sci-ence and Technology vol 28 no 9 pp 3405ndash3410 2014

[14] S Wang and R J Bernhard ldquoPrediction of averaged energyfor moderately damped systems with strong couplingrdquoJournal of Sound and Vibration vol 319 no 1-2 pp 426ndash4442009

[15] V S Pereira and J M C Dos Santos ldquoCoupled plate energymodels at mid- and high-frequency vibrationsrdquo Computers ampStructures vol 134 pp 48ndash61 2014

[16] H-W Kwon S-Y Hong and J-H Song ldquoVibrational energyflow analysis of coupled cylindrical thin shell structuresrdquoJournal of Mechanical Science and Technology vol 30 no 9pp 4049ndash4062 2016

[17] Y-H Park and S-Y Hong ldquoHybrid power flow analysis usingcoupling loss factor of SEA for low-damping system-part Iformulation of 1-D and 2-D casesrdquo Journal of Sound andVibration vol 299 no 3 pp 484ndash503 2007

[18] Z Liu J Niu and X Gao ldquoAn improved approach for analysisof coupled structures in Energy Finite Element Analysis usingthe coupling loss factorrdquo Computers amp Structures vol 210pp 69ndash86 2018

[19] H-W Kwon S-Y Hong and J-H Song ldquoEnergy flowmodels for underwater radiation noise prediction in medium-to-high-frequency rangesrdquo Proceedings of the Institution ofMechanical Engineers Part M Journal of Engineering for theMaritime Environment vol 230 no 2 pp 404ndash416 2016

[20] X Zheng W Dai Y Qiu and Z Hao ldquoPrediction and energycontribution analysis of interior noise in a high-speed trainbased on modified energy finite element analysisrdquoMechanicalSystems and Signal Processing vol 126 pp 439ndash457 2019

[21] W Zhang H Chen D Zhu and X Kong ldquo)e thermal effectson high-frequency vibration of beams using energy flowanalysisrdquo Journal of Sound and Vibration vol 333 no 9pp 2588ndash2600 2014

[22] W Di X Miaoxia and L Yueming ldquoHigh-frequency dy-namic analysis of plates in thermal environments based onenergy finite elementmethodrdquo Shock and Vibration vol 2015pp 1ndash14 2015

[23] Z Chen Z Yang N Guo and G Zhang ldquoAn energy finiteelement method for high frequency vibration analysis ofbeams with axial forcerdquo Applied Mathematical Modellingvol 61 no 5 pp 521ndash539 2018

[24] P M Morse and K U Indard Ceoretical AcousticsPrinceton University Press Princeton NJ USA 1986

[25] W Weaver S P Timoshenko and D H Young VibrationProblems in Engineering John Wiley amp Sons Hoboken NJUSA 1990

[26] M N Ichchou A Le Bot and L Jezequel ldquoEnergy models ofone-dimensional multi-propagative systemsrdquo Journal ofSound and Vibration vol 201 no 5 pp 535ndash554 1997

[27] Y Lase M N Ichchou and L Jezequel ldquoEnergy flow analysisof bars and beams theoretical formulationsrdquo Journal of Soundand Vibration vol 192 no 1 pp 281ndash305 1996

[28] D J Nefske and S H Sung ldquoPower flow finite elementanalysis of dynamic systems basic theory and application tobeamsrdquo Journal of Vibration and Acoustics vol 111 no 1pp 94ndash100 1989

Mathematical Problems in Engineering 11

43 Effects of Structural Damping Loss Factor Figure 10shows the energy density distributions of pinned-pinnedbeams with different structural damping loss factor0le ηle 08 amplitude of transverse harmonic force F0 1Nand axial force N 20 kN when the analysis frequency is6 kHz )e developed EFA results represent the globalvariation of the modal solutions at each structural dampingloss factor Owing to the exclusion of the evanescent wavethe developed energy flow solutions more closely approachthe modal solutions as the structural damping loss factorincreases)e energy density changes more suddenly along x

position with increasing structural damping loss factorFigure 11 shows that the total energy is lower at largerstructural damping loss factor

Figures 12 and 13 show the input power and groupvelocity at various structural damping loss factors respec-tively It is found that the input power decreases and thegroup velocity increases with an increase of structuraldamping loss factor Since the total energy is proportional tothe input power the total energy is lower for larger structuraldamping loss factor leading to less input power In fact thelarger group velocity contributes to flatter change of theenergy density along the x position But the energy densitydistribution changes more dramatically with larger struc-tural damping loss factor along the x position It attributes tothat the energy density is mainly affected by the input powerwhile the effect of group velocity is minor

Ener

gy d

ensit

y (d

B)

0 05 1 15 2 25 3Position (m)

ndash20

ndash10

0

10

20

30

40

50

60

1556

60

N = 20kNN = 5kN

N = 1kNN = 0kN

Figure 6 Energy density distribution of pinned-pinned beamswith f 6 kHz F0 1N and η 05 when N 0 1 5 20 kN

0 2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

12

14

Axial force (kN)

Tota

l ene

rgy

(μW

)

Figure 7 Total energy of pinned-pinned beams versus axial forcewith F0 1N η 05 and f 6 kHz

0 05 1 15 2 25 335

40

45

50

55

60

Developed EFA solution Modal solution Traditional EFA solution

Ener

gy d

ensit

y (d

B)

Position (m)

Figure 4 Energy density distribution of pinned-pinned beamswith N 20 kN F0 1N and η 005 when f 6 kHz

Ener

gy d

ensit

y (d

B)

0 05 1 15 2 25 3

Developed EFA solution Modal solution Traditional EFA solution

Position (m)

ndash100

ndash80

ndash60

ndash40

ndash20

0

20

40

60

Figure 5 Energy density distribution of pinned-pinned beamswith N 20 kN F0 1N and η 005 when f 6 kHz

6 Mathematical Problems in Engineering

44 Application to Beams with Different BoundaryConditions )e analysis model consists of a beam fixed atboth ends as shown in Figure 14 and a beam fixed at one endand free at the other as shown in Figure 15 respectivelyBoth beams share the common dimensions of L

3mS 2 times 10minus4m2Ib 2 times 10minus10m4 material properties ofρ 2700kgm3 and E 70GPa and the structuraldamping loss factor η 05 Since the energy outflow at thefixed point is zero and the energy flow satisfies theequilibrium condition at the driving point the energyintensity boundary conditions for the fixed-fixed beam isidentical to those for the pinned-pinned beam Howeverthe impedance of fixed-fixed beam at the driving point isobtained by substituting equations (B5) and (B6) inAppendix B into equation (A8) in Appendix A For fixed-free beam the subdomain ② is absent so that the coef-ficients C21 C22 in equation (29) vanish the followingmatrix equation is rewritten as

1 minus1

minusPfeminusλfx0 Pfeλfx0⎡⎣ ⎤⎦

C11

C121113890 1113891

0

langπinrang1113890 1113891 (30)

Similarly the impedance of fixed-free beam at drivingpoint is determined by substituting the equation (B8) and(B9) in Appendix B into equation (A8) in Appendix AFrom equation (30) the energy density for the fixed-fixedbeam can be evaluated

Figure 14 shows a fixed-fixed beam with constant axialforce Figures 16 and 17 show the energy density distributionof fixed-fixed beams and fixed-free beams with F0 1NN 20 kN and η 05 when the analysis frequencies are2 kHz 4 kHz 6 kHz and 8 kHz respectively )e developedEFA results coincide well with the overall trend of the exactsolutions from modal analysis at each frequency Similar toprevious cases the developed energy flow model gives moreaccurate result at high frequencies despite the exclusion of

0 2 4 6 8 10 12 14 16 18 20075

080

085

090

095

010

0105

Axial force (kN)

Inpu

t pow

er (m

W)

Figure 8 Input power of pinned-pinned beams versus axial forcewith F0 1N η 05 and f 6 kHz

0 5 10 15 20 25300

350

400

450

500

550

Axial force (kN)

Gro

up v

eloc

ity (m

s)

Figure 9 Group velocity versus axial force with η 05 andf 6 kHz

0 05 1 15 2 25 3ndash190

ndash140

ndash90

ndash40

10

60

Ener

gy d

ensit

y (d

B)

Position (m)

EFA solution η = 005

Modal solution η = 005

EFA solution η = 020

Modal solution η = 020

EFA solution η = 050

Modal solution η = 050

EFA solution η = 080

Modal solution η = 080

Figure 10 Energy density distribution of pinned-pinned beamswith N 20 kN F0 1N and f 6 kHz when η 005 020 050080

0 01 02 03 04 05 06 07 080

005

010

015

020

025

030

Tota

l ene

rgy

(μW

)

Structural damping loss factor

Figure 11 Total energy of pinned-pinned beams versus structuraldamping loss factor with F0 1N N 20 kN and f 6 kHz

Mathematical Problems in Engineering 7

the evanescent wave As a consequence it is feasible topredict the vibrational energy density of beams with variousboundary conditions in the high-frequency range by theproposed energy flow model

5 Conclusion

An energy flow model for a beam with axial force thatincludes high damping effect is established to evaluate theenergy density in the high-frequency range )e wave-number is expressed as the function of the damping lossfactor and the axial force Particularly the energy trans-mission equation and the energy dissipation equation arederived from the wavenumber without approximation of itsreal and imaginary term)en the energy density governing

0 01 02 03 04 05 06 07 08042

044

046

048

050

052

054

056

058

Inpu

t pow

er (m

W)

Structural damping loss factor

Figure 12 Input power of pinned-pinned beams versus structuraldamping loss factor with F0 1N N 20 kN and f 6 kHz

0 01 02 03 04 05 06 07 08Structural damping loss factor

825

830

835

840

845

850

855

860

865

870

Gro

up v

eloc

ity (m

s)

Figure 13 Group velocity versus structural damping loss factorwith N 20 kN and f 6 kHz

x

x0

L

F0δ (x ndash x0)ejωt

②①N N

Figure 14 A fixed-fixed beam with constant axial force

x

L

NF0δ (x ndash x0)ejωt

①

Figure 15 A fixed-free beam with constant axial force

0 05 1 15 2 25 3ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

0

20

40

60

EFA solution f = 2kHz

EFA solution f = 4kHzEFA solution f = 6kHz

EFA solution f = 8kHz

Modal solution f = 2kHz

Modal solution f = 4kHzModal solution f = 6kHz

Modal solution f = 8kHz

Ener

gy d

ensit

y (d

B)

Position (m)

Figure 16 Energy density distribution of fixed-fixed beams withF0 1N N 20 kN and η 05 when f 2 4 6 8 kHz

0 05 1 15 2 25 3

EFA solution f = 2kHz

EFA solution f = 4kHzEFA solution f = 6kHz

EFA solution f = 8kHz

Modal solution f = 2kHz

Modal solution f = 4kHzModal solution f = 6kHz

Modal solution f = 8kHz

Position (m)

ndash250

ndash200

ndash150

ndash100

ndash50

0

50

100

Ener

gy d

ensit

y (d

B)

Figure 17 Energy density distribution of fixed-free beams withF0 1N N 20 kN and η 05 when f 2 4 6 8 kHz

8 Mathematical Problems in Engineering

equation is obtained for the high damping beam with axialforce

To verify the developed energy flow model numericalanalyses are performed for the simply supported beam withvarious damping loss factor and axial force )e EFA so-lutions are greatly consistent with the modal solutions in allcases Furthermore it is found that both the high dampingfactor and large axial force can significantly alter the level ofenergy density as well as the distribution Due to theresulting deceasing input power the increasing dampingloss factor makes the energy density decay more dramati-cally along axial position However that is the contrary casefor the axial force )e proposed method is expected to beuseful for the prediction of the high-frequency vibration ofhigh damping structures with initial stress

Appendix

A Impedance and Exact Solution from ModalAnalysis for High Damping Beams withAxial Force

When a high damping beam with axial force is excited at x0by a transverse harmonic force the governing equation ofmotion can be expressed as

Dc

z4w

zx4 minus Nc

z2w

zx2 + ρSz2w

zt2 F0δ x minus x0( 1113857e

jωt (A1)

)e steady-state solution of equation (A1) can be foundby modal superposition as

w(x t) 1113944infin

i1ϕi(x)qi(t) (A2)

where ϕi(x) is the ith natural mode of the beam and qi(t) isthe ith mode displacement

By substituting equation (A2) into the motion equationof the beam equation (A1) and using the orthogonality ofnatural modes we obtain

Kiqi(t) + Mi

d2qi(t)

dt2 Fi (A3)

where

Fi 1113946L

0F0δ x minus x0( 1113857ϕi(x)dx (A4)

represents the so-called ldquomodal forcerdquo and

Mi 1113946L

0ρSϕ2

i dx

Ki ω2i (1 + jη) 1113946

L

0ρSϕ2

i dx

(A5)

are the so-called ldquomodal massrdquo and ldquomodal stiffnessrdquo )eeffect of structural damping is modeled by assuming ahysteretic damping model which introduces an additionalcomplex term (1 + jη) to modal stiffness )e natural fre-quency of the ith mode ωi can be determined from thecorresponding wavenumber in equation (2) which can bewritten as

k2i minus

N

2D+

N

2D1113874 1113875

2+ρSω2

i

D

1113971

(A6)

)e forced vibration response of the beam can beexpressed as the linear combination of the normal modes

w(x t) 1113944infin

i1

F0 middot ejωt

Mi ω2i (1 + jη) minus ω21113858 1113859

ϕi(x)ϕi x0( 1113857 (A7)

)e impedance at the driving point of the finite beamwith axial force can be obtained from its definition

Z F0e

jωt

_w x0 t( 1113857

minus11113936infini1 ωϕ

2i x0( 1113857Mi ω2

i (1 + jη) minus ω21113858 1113859

(A8)

where _w(x0 t) is the velocity at driving point which can becalculated as the time derivative of the displacement atx x0 Once the mode function is determined by theboundary condition the impedance at driving point of thefinite beam with axial force is derived from equation (A8)

)e time averaged energy density is given by

langerang 14

ρSzw

zt1113888 1113889

zw

zt1113888 1113889

lowast

+ Dz2w

zx21113888 1113889z2w

zx21113888 1113889

lowast

+ Nzw

zx1113888 1113889

zw

zx1113888 1113889

lowast

1113890 1113891 (A9)

Substituting equation (A7) into equation (A9) theenergy density can be obtained based on the modal solution

B Mode Shapes for High Damping Beams withAxial Force

B1 BeamPinned at Both Ends )e transverse displacementand the bending moment are zero at a simply supported endHence the boundary conditions can be stated as

w(0) 0

z2w(0)

zx2 0

w(L) 0

z2w(L)

zx2 0

(B1)

Mathematical Problems in Engineering 9

)e general solution of equation (A1) is shown inequation (7) whereas the frequency equation with bothsimply supported ends is derived as

sin kiL 0 (B2)

)e vibration mode of the beam is given by

ϕi(x) sin kix (B3)

B2 Beam Fixed at Both Ends At a fixed end the transversedisplacement and the slope of the displacement are zeroHence the boundary conditions are given by

w(0) 0

zw(0)

zx 0

w(L) 0

zw(L)

zx 0

(B4)

By virtue of general solution in equation (7) andboundary conditions with both fixed ends the frequencyequation is obtained as

cos kiL cosh kiL minus 1 0 (B5)

)e vibration mode of the beam is expressed as

ϕi(x) cos kix minus cosh kix( 1113857 +cos kiL minus cosh kiL

sin kiL minus sinh kiL

middot sin kix minus sinh kix( 1113857

(B6)

B3BeamFixedatOneEndandFreeat theOther If the beamis fixed at x 0 and free at x L the transverse deflectionand its slope must be zero at x 0 and the bending momentand shear force must be zero at x L )us the boundaryconditions become

w(0) 0

zw(0)

zx 0

z3w(L)

zx3 0

z2w(L)

zx2 0

(B7)

According to the general solution in equation (7) andboundary conditions with one fixed end and the other freeend the frequency equation is derived as

cos kiL cosh kiL + 1 0 (B8)

)e vibration mode of the beam is written as

ϕi(x) cos kix minus cosh kix( 1113857 +cos kiL + cosh kiL

sin kiL + sinh kiL

middot sin kix minus sinh kix( 1113857

(B9)

Data Availability

)e data used to support the finding of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that there no conflicts of interest

Acknowledgments

)e presented work was supported by National NaturalScience Foundation of China (Grant no 51505096) andNatural Science Foundation of Heilongjiang Province ofChina (Grant no QC2016056)

References

[1] L Bot ldquoEntropy in sound and vibration towards a newparadigmrdquo Proceedings of the Royal Society A MathematicalPhysical and Engineering Science vol 473 no 2197 pp 1ndash92017

[2] A Acri ldquoA literature review on energy flow analysis of vi-brating structuresrdquo ASRO Journal of Applied Mechanicsvol 1 no 1 pp 12ndash15 2017

[3] J C Wohlever and R J Bernhard ldquoMechanical energy flowmodels of rods and beamsrdquo Journal of Sound and Vibrationvol 153 no 1 pp 1ndash19 1992

[4] Z Chen Z Yang Y Wang and X Wang ldquoA high-frequencyshock response analysis method based on energy finite ele-ment method and virtual mode synthesis and simulationrdquoActa Aeronautica et Astronautica Sinica vol 39 no 08pp 182ndash191 2018

[5] Z Liu and J Niu ldquoVibrational energy flow model for func-tionally graded beamsrdquo Composite Structures vol 186pp 17ndash28 2018

[6] X Kong H Chen D Zhu and W Zhang ldquoStudy on thevalidity region of energy finite element analysisrdquo Journal ofSound and Vibration vol 333 no 9 pp 2601ndash2616 2014

[7] A Nokhbatolfoghahai H Navazi Y Ghobaad andH Haddadpour ldquoExperimental measurement of energydensity of a vibrating beamrdquo Journal of Vibration and Controlvol 24 no 24 pp 5735ndash5746 2018

[8] H M Navazi A Nokhbatolfoghahaei Y Ghobad andH Haddadpour ldquoExperimental measurement of energydensity in a vibrating plate and comparison with energy finiteelement analysisrdquo Journal of Sound and Vibration vol 375pp 289ndash307 2016

[9] D Zhu H Chen X Kong and W Zhang ldquoA hybrid finiteelement-energy finite element method for mid-frequencyvibrations of built-up structures under multi-distributedloadingsrdquo Journal of Sound amp Vibration vol 333 no 22pp 5723ndash5745 2014

[10] M Sadeghmanesh H Haddadpour M T Abadi andH M Navazi ldquoA method for selection of structural theories

10 Mathematical Problems in Engineering

for low to high frequency vibration analysesrdquo EuropeanJournal of Mechanics-A Solids vol 75 pp 27ndash40 2019

[11] W Zhang AWang N Vlahopoulos and KWu ldquoA vibrationanalysis of stiffened plates under heavy fluid loading by anenergy finite element analysis formulationrdquo Finite Elements inAnalysis and Design vol 41 no 11-12 pp 1056ndash1078 2005

[12] J-B Han S-Y Hong and J-H Song ldquoEnergy flow model forthin plate considering fluid loading with mean flowrdquo Journalof Sound and Vibration vol 331 no 24 pp 5326ndash5346 2012

[13] H-W Kwon S-Y Hong D-K Oh et al ldquoExperimental studyon the energy flow analysis of underwater vibration for thereinforced cylindrical structurerdquo Journal of Mechanical Sci-ence and Technology vol 28 no 9 pp 3405ndash3410 2014

[14] S Wang and R J Bernhard ldquoPrediction of averaged energyfor moderately damped systems with strong couplingrdquoJournal of Sound and Vibration vol 319 no 1-2 pp 426ndash4442009

[15] V S Pereira and J M C Dos Santos ldquoCoupled plate energymodels at mid- and high-frequency vibrationsrdquo Computers ampStructures vol 134 pp 48ndash61 2014

[16] H-W Kwon S-Y Hong and J-H Song ldquoVibrational energyflow analysis of coupled cylindrical thin shell structuresrdquoJournal of Mechanical Science and Technology vol 30 no 9pp 4049ndash4062 2016

[17] Y-H Park and S-Y Hong ldquoHybrid power flow analysis usingcoupling loss factor of SEA for low-damping system-part Iformulation of 1-D and 2-D casesrdquo Journal of Sound andVibration vol 299 no 3 pp 484ndash503 2007

[18] Z Liu J Niu and X Gao ldquoAn improved approach for analysisof coupled structures in Energy Finite Element Analysis usingthe coupling loss factorrdquo Computers amp Structures vol 210pp 69ndash86 2018

[19] H-W Kwon S-Y Hong and J-H Song ldquoEnergy flowmodels for underwater radiation noise prediction in medium-to-high-frequency rangesrdquo Proceedings of the Institution ofMechanical Engineers Part M Journal of Engineering for theMaritime Environment vol 230 no 2 pp 404ndash416 2016

[20] X Zheng W Dai Y Qiu and Z Hao ldquoPrediction and energycontribution analysis of interior noise in a high-speed trainbased on modified energy finite element analysisrdquoMechanicalSystems and Signal Processing vol 126 pp 439ndash457 2019

[21] W Zhang H Chen D Zhu and X Kong ldquo)e thermal effectson high-frequency vibration of beams using energy flowanalysisrdquo Journal of Sound and Vibration vol 333 no 9pp 2588ndash2600 2014

[22] W Di X Miaoxia and L Yueming ldquoHigh-frequency dy-namic analysis of plates in thermal environments based onenergy finite elementmethodrdquo Shock and Vibration vol 2015pp 1ndash14 2015

[23] Z Chen Z Yang N Guo and G Zhang ldquoAn energy finiteelement method for high frequency vibration analysis ofbeams with axial forcerdquo Applied Mathematical Modellingvol 61 no 5 pp 521ndash539 2018

[24] P M Morse and K U Indard Ceoretical AcousticsPrinceton University Press Princeton NJ USA 1986

[25] W Weaver S P Timoshenko and D H Young VibrationProblems in Engineering John Wiley amp Sons Hoboken NJUSA 1990

[26] M N Ichchou A Le Bot and L Jezequel ldquoEnergy models ofone-dimensional multi-propagative systemsrdquo Journal ofSound and Vibration vol 201 no 5 pp 535ndash554 1997

[27] Y Lase M N Ichchou and L Jezequel ldquoEnergy flow analysisof bars and beams theoretical formulationsrdquo Journal of Soundand Vibration vol 192 no 1 pp 281ndash305 1996

[28] D J Nefske and S H Sung ldquoPower flow finite elementanalysis of dynamic systems basic theory and application tobeamsrdquo Journal of Vibration and Acoustics vol 111 no 1pp 94ndash100 1989

Mathematical Problems in Engineering 11

44 Application to Beams with Different BoundaryConditions )e analysis model consists of a beam fixed atboth ends as shown in Figure 14 and a beam fixed at one endand free at the other as shown in Figure 15 respectivelyBoth beams share the common dimensions of L

3mS 2 times 10minus4m2Ib 2 times 10minus10m4 material properties ofρ 2700kgm3 and E 70GPa and the structuraldamping loss factor η 05 Since the energy outflow at thefixed point is zero and the energy flow satisfies theequilibrium condition at the driving point the energyintensity boundary conditions for the fixed-fixed beam isidentical to those for the pinned-pinned beam Howeverthe impedance of fixed-fixed beam at the driving point isobtained by substituting equations (B5) and (B6) inAppendix B into equation (A8) in Appendix A For fixed-free beam the subdomain ② is absent so that the coef-ficients C21 C22 in equation (29) vanish the followingmatrix equation is rewritten as

1 minus1

minusPfeminusλfx0 Pfeλfx0⎡⎣ ⎤⎦

C11

C121113890 1113891

0

langπinrang1113890 1113891 (30)

Similarly the impedance of fixed-free beam at drivingpoint is determined by substituting the equation (B8) and(B9) in Appendix B into equation (A8) in Appendix AFrom equation (30) the energy density for the fixed-fixedbeam can be evaluated

Figure 14 shows a fixed-fixed beam with constant axialforce Figures 16 and 17 show the energy density distributionof fixed-fixed beams and fixed-free beams with F0 1NN 20 kN and η 05 when the analysis frequencies are2 kHz 4 kHz 6 kHz and 8 kHz respectively )e developedEFA results coincide well with the overall trend of the exactsolutions from modal analysis at each frequency Similar toprevious cases the developed energy flow model gives moreaccurate result at high frequencies despite the exclusion of

0 2 4 6 8 10 12 14 16 18 20075

080

085

090

095

010

0105

Axial force (kN)

Inpu

t pow

er (m

W)

Figure 8 Input power of pinned-pinned beams versus axial forcewith F0 1N η 05 and f 6 kHz

0 5 10 15 20 25300

350

400

450

500

550

Axial force (kN)

Gro

up v

eloc

ity (m

s)

Figure 9 Group velocity versus axial force with η 05 andf 6 kHz

0 05 1 15 2 25 3ndash190

ndash140

ndash90

ndash40

10

60

Ener

gy d

ensit

y (d

B)

Position (m)

EFA solution η = 005

Modal solution η = 005

EFA solution η = 020

Modal solution η = 020

EFA solution η = 050

Modal solution η = 050

EFA solution η = 080

Modal solution η = 080

Figure 10 Energy density distribution of pinned-pinned beamswith N 20 kN F0 1N and f 6 kHz when η 005 020 050080

0 01 02 03 04 05 06 07 080

005

010

015

020

025

030

Tota

l ene

rgy

(μW

)

Structural damping loss factor

Figure 11 Total energy of pinned-pinned beams versus structuraldamping loss factor with F0 1N N 20 kN and f 6 kHz

Mathematical Problems in Engineering 7

the evanescent wave As a consequence it is feasible topredict the vibrational energy density of beams with variousboundary conditions in the high-frequency range by theproposed energy flow model

5 Conclusion

An energy flow model for a beam with axial force thatincludes high damping effect is established to evaluate theenergy density in the high-frequency range )e wave-number is expressed as the function of the damping lossfactor and the axial force Particularly the energy trans-mission equation and the energy dissipation equation arederived from the wavenumber without approximation of itsreal and imaginary term)en the energy density governing

0 01 02 03 04 05 06 07 08042

044

046

048

050

052

054

056

058

Inpu

t pow

er (m

W)

Structural damping loss factor

Figure 12 Input power of pinned-pinned beams versus structuraldamping loss factor with F0 1N N 20 kN and f 6 kHz

0 01 02 03 04 05 06 07 08Structural damping loss factor

825

830

835

840

845

850

855

860

865

870

Gro

up v

eloc

ity (m

s)

Figure 13 Group velocity versus structural damping loss factorwith N 20 kN and f 6 kHz

x

x0

L

F0δ (x ndash x0)ejωt

②①N N

Figure 14 A fixed-fixed beam with constant axial force

x

L

NF0δ (x ndash x0)ejωt

①

Figure 15 A fixed-free beam with constant axial force

0 05 1 15 2 25 3ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

0

20

40

60

EFA solution f = 2kHz

EFA solution f = 4kHzEFA solution f = 6kHz

EFA solution f = 8kHz

Modal solution f = 2kHz

Modal solution f = 4kHzModal solution f = 6kHz

Modal solution f = 8kHz

Ener

gy d

ensit

y (d

B)

Position (m)

Figure 16 Energy density distribution of fixed-fixed beams withF0 1N N 20 kN and η 05 when f 2 4 6 8 kHz

0 05 1 15 2 25 3

EFA solution f = 2kHz

EFA solution f = 4kHzEFA solution f = 6kHz

EFA solution f = 8kHz

Modal solution f = 2kHz

Modal solution f = 4kHzModal solution f = 6kHz

Modal solution f = 8kHz

Position (m)

ndash250

ndash200

ndash150

ndash100

ndash50

0

50

100

Ener

gy d

ensit

y (d

B)

Figure 17 Energy density distribution of fixed-free beams withF0 1N N 20 kN and η 05 when f 2 4 6 8 kHz

8 Mathematical Problems in Engineering

equation is obtained for the high damping beam with axialforce

To verify the developed energy flow model numericalanalyses are performed for the simply supported beam withvarious damping loss factor and axial force )e EFA so-lutions are greatly consistent with the modal solutions in allcases Furthermore it is found that both the high dampingfactor and large axial force can significantly alter the level ofenergy density as well as the distribution Due to theresulting deceasing input power the increasing dampingloss factor makes the energy density decay more dramati-cally along axial position However that is the contrary casefor the axial force )e proposed method is expected to beuseful for the prediction of the high-frequency vibration ofhigh damping structures with initial stress

Appendix

A Impedance and Exact Solution from ModalAnalysis for High Damping Beams withAxial Force

When a high damping beam with axial force is excited at x0by a transverse harmonic force the governing equation ofmotion can be expressed as

Dc

z4w

zx4 minus Nc

z2w

zx2 + ρSz2w

zt2 F0δ x minus x0( 1113857e

jωt (A1)

)e steady-state solution of equation (A1) can be foundby modal superposition as

w(x t) 1113944infin

i1ϕi(x)qi(t) (A2)

where ϕi(x) is the ith natural mode of the beam and qi(t) isthe ith mode displacement

By substituting equation (A2) into the motion equationof the beam equation (A1) and using the orthogonality ofnatural modes we obtain

Kiqi(t) + Mi

d2qi(t)

dt2 Fi (A3)

where

Fi 1113946L

0F0δ x minus x0( 1113857ϕi(x)dx (A4)

represents the so-called ldquomodal forcerdquo and

Mi 1113946L

0ρSϕ2

i dx

Ki ω2i (1 + jη) 1113946

L

0ρSϕ2

i dx

(A5)

are the so-called ldquomodal massrdquo and ldquomodal stiffnessrdquo )eeffect of structural damping is modeled by assuming ahysteretic damping model which introduces an additionalcomplex term (1 + jη) to modal stiffness )e natural fre-quency of the ith mode ωi can be determined from thecorresponding wavenumber in equation (2) which can bewritten as

k2i minus

N

2D+

N

2D1113874 1113875

2+ρSω2

i

D

1113971

(A6)

)e forced vibration response of the beam can beexpressed as the linear combination of the normal modes

w(x t) 1113944infin

i1

F0 middot ejωt

Mi ω2i (1 + jη) minus ω21113858 1113859

ϕi(x)ϕi x0( 1113857 (A7)

)e impedance at the driving point of the finite beamwith axial force can be obtained from its definition

Z F0e

jωt

_w x0 t( 1113857

minus11113936infini1 ωϕ

2i x0( 1113857Mi ω2

i (1 + jη) minus ω21113858 1113859

(A8)

where _w(x0 t) is the velocity at driving point which can becalculated as the time derivative of the displacement atx x0 Once the mode function is determined by theboundary condition the impedance at driving point of thefinite beam with axial force is derived from equation (A8)

)e time averaged energy density is given by

langerang 14

ρSzw

zt1113888 1113889

zw

zt1113888 1113889

lowast

+ Dz2w

zx21113888 1113889z2w

zx21113888 1113889

lowast

+ Nzw

zx1113888 1113889

zw

zx1113888 1113889

lowast

1113890 1113891 (A9)

Substituting equation (A7) into equation (A9) theenergy density can be obtained based on the modal solution

B Mode Shapes for High Damping Beams withAxial Force

B1 BeamPinned at Both Ends )e transverse displacementand the bending moment are zero at a simply supported endHence the boundary conditions can be stated as

w(0) 0

z2w(0)

zx2 0

w(L) 0

z2w(L)

zx2 0

(B1)

Mathematical Problems in Engineering 9

)e general solution of equation (A1) is shown inequation (7) whereas the frequency equation with bothsimply supported ends is derived as

sin kiL 0 (B2)

)e vibration mode of the beam is given by

ϕi(x) sin kix (B3)

B2 Beam Fixed at Both Ends At a fixed end the transversedisplacement and the slope of the displacement are zeroHence the boundary conditions are given by

w(0) 0

zw(0)

zx 0

w(L) 0

zw(L)

zx 0

(B4)

By virtue of general solution in equation (7) andboundary conditions with both fixed ends the frequencyequation is obtained as

cos kiL cosh kiL minus 1 0 (B5)

)e vibration mode of the beam is expressed as

ϕi(x) cos kix minus cosh kix( 1113857 +cos kiL minus cosh kiL

sin kiL minus sinh kiL

middot sin kix minus sinh kix( 1113857

(B6)

B3BeamFixedatOneEndandFreeat theOther If the beamis fixed at x 0 and free at x L the transverse deflectionand its slope must be zero at x 0 and the bending momentand shear force must be zero at x L )us the boundaryconditions become

w(0) 0

zw(0)

zx 0

z3w(L)

zx3 0

z2w(L)

zx2 0

(B7)

According to the general solution in equation (7) andboundary conditions with one fixed end and the other freeend the frequency equation is derived as

cos kiL cosh kiL + 1 0 (B8)

)e vibration mode of the beam is written as

ϕi(x) cos kix minus cosh kix( 1113857 +cos kiL + cosh kiL

sin kiL + sinh kiL

middot sin kix minus sinh kix( 1113857

(B9)

Data Availability

)e data used to support the finding of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that there no conflicts of interest

Acknowledgments

)e presented work was supported by National NaturalScience Foundation of China (Grant no 51505096) andNatural Science Foundation of Heilongjiang Province ofChina (Grant no QC2016056)

References

[1] L Bot ldquoEntropy in sound and vibration towards a newparadigmrdquo Proceedings of the Royal Society A MathematicalPhysical and Engineering Science vol 473 no 2197 pp 1ndash92017

[2] A Acri ldquoA literature review on energy flow analysis of vi-brating structuresrdquo ASRO Journal of Applied Mechanicsvol 1 no 1 pp 12ndash15 2017

[3] J C Wohlever and R J Bernhard ldquoMechanical energy flowmodels of rods and beamsrdquo Journal of Sound and Vibrationvol 153 no 1 pp 1ndash19 1992

[4] Z Chen Z Yang Y Wang and X Wang ldquoA high-frequencyshock response analysis method based on energy finite ele-ment method and virtual mode synthesis and simulationrdquoActa Aeronautica et Astronautica Sinica vol 39 no 08pp 182ndash191 2018

[5] Z Liu and J Niu ldquoVibrational energy flow model for func-tionally graded beamsrdquo Composite Structures vol 186pp 17ndash28 2018

[6] X Kong H Chen D Zhu and W Zhang ldquoStudy on thevalidity region of energy finite element analysisrdquo Journal ofSound and Vibration vol 333 no 9 pp 2601ndash2616 2014

[7] A Nokhbatolfoghahai H Navazi Y Ghobaad andH Haddadpour ldquoExperimental measurement of energydensity of a vibrating beamrdquo Journal of Vibration and Controlvol 24 no 24 pp 5735ndash5746 2018

[8] H M Navazi A Nokhbatolfoghahaei Y Ghobad andH Haddadpour ldquoExperimental measurement of energydensity in a vibrating plate and comparison with energy finiteelement analysisrdquo Journal of Sound and Vibration vol 375pp 289ndash307 2016

[9] D Zhu H Chen X Kong and W Zhang ldquoA hybrid finiteelement-energy finite element method for mid-frequencyvibrations of built-up structures under multi-distributedloadingsrdquo Journal of Sound amp Vibration vol 333 no 22pp 5723ndash5745 2014

[10] M Sadeghmanesh H Haddadpour M T Abadi andH M Navazi ldquoA method for selection of structural theories

10 Mathematical Problems in Engineering

for low to high frequency vibration analysesrdquo EuropeanJournal of Mechanics-A Solids vol 75 pp 27ndash40 2019

[11] W Zhang AWang N Vlahopoulos and KWu ldquoA vibrationanalysis of stiffened plates under heavy fluid loading by anenergy finite element analysis formulationrdquo Finite Elements inAnalysis and Design vol 41 no 11-12 pp 1056ndash1078 2005

[12] J-B Han S-Y Hong and J-H Song ldquoEnergy flow model forthin plate considering fluid loading with mean flowrdquo Journalof Sound and Vibration vol 331 no 24 pp 5326ndash5346 2012

[13] H-W Kwon S-Y Hong D-K Oh et al ldquoExperimental studyon the energy flow analysis of underwater vibration for thereinforced cylindrical structurerdquo Journal of Mechanical Sci-ence and Technology vol 28 no 9 pp 3405ndash3410 2014

[14] S Wang and R J Bernhard ldquoPrediction of averaged energyfor moderately damped systems with strong couplingrdquoJournal of Sound and Vibration vol 319 no 1-2 pp 426ndash4442009

[15] V S Pereira and J M C Dos Santos ldquoCoupled plate energymodels at mid- and high-frequency vibrationsrdquo Computers ampStructures vol 134 pp 48ndash61 2014

[16] H-W Kwon S-Y Hong and J-H Song ldquoVibrational energyflow analysis of coupled cylindrical thin shell structuresrdquoJournal of Mechanical Science and Technology vol 30 no 9pp 4049ndash4062 2016

[17] Y-H Park and S-Y Hong ldquoHybrid power flow analysis usingcoupling loss factor of SEA for low-damping system-part Iformulation of 1-D and 2-D casesrdquo Journal of Sound andVibration vol 299 no 3 pp 484ndash503 2007

[18] Z Liu J Niu and X Gao ldquoAn improved approach for analysisof coupled structures in Energy Finite Element Analysis usingthe coupling loss factorrdquo Computers amp Structures vol 210pp 69ndash86 2018

[19] H-W Kwon S-Y Hong and J-H Song ldquoEnergy flowmodels for underwater radiation noise prediction in medium-to-high-frequency rangesrdquo Proceedings of the Institution ofMechanical Engineers Part M Journal of Engineering for theMaritime Environment vol 230 no 2 pp 404ndash416 2016

[20] X Zheng W Dai Y Qiu and Z Hao ldquoPrediction and energycontribution analysis of interior noise in a high-speed trainbased on modified energy finite element analysisrdquoMechanicalSystems and Signal Processing vol 126 pp 439ndash457 2019

[21] W Zhang H Chen D Zhu and X Kong ldquo)e thermal effectson high-frequency vibration of beams using energy flowanalysisrdquo Journal of Sound and Vibration vol 333 no 9pp 2588ndash2600 2014

[22] W Di X Miaoxia and L Yueming ldquoHigh-frequency dy-namic analysis of plates in thermal environments based onenergy finite elementmethodrdquo Shock and Vibration vol 2015pp 1ndash14 2015

[23] Z Chen Z Yang N Guo and G Zhang ldquoAn energy finiteelement method for high frequency vibration analysis ofbeams with axial forcerdquo Applied Mathematical Modellingvol 61 no 5 pp 521ndash539 2018

[24] P M Morse and K U Indard Ceoretical AcousticsPrinceton University Press Princeton NJ USA 1986

[25] W Weaver S P Timoshenko and D H Young VibrationProblems in Engineering John Wiley amp Sons Hoboken NJUSA 1990

[26] M N Ichchou A Le Bot and L Jezequel ldquoEnergy models ofone-dimensional multi-propagative systemsrdquo Journal ofSound and Vibration vol 201 no 5 pp 535ndash554 1997

[27] Y Lase M N Ichchou and L Jezequel ldquoEnergy flow analysisof bars and beams theoretical formulationsrdquo Journal of Soundand Vibration vol 192 no 1 pp 281ndash305 1996

[28] D J Nefske and S H Sung ldquoPower flow finite elementanalysis of dynamic systems basic theory and application tobeamsrdquo Journal of Vibration and Acoustics vol 111 no 1pp 94ndash100 1989

Mathematical Problems in Engineering 11

the evanescent wave As a consequence it is feasible topredict the vibrational energy density of beams with variousboundary conditions in the high-frequency range by theproposed energy flow model

5 Conclusion

An energy flow model for a beam with axial force thatincludes high damping effect is established to evaluate theenergy density in the high-frequency range )e wave-number is expressed as the function of the damping lossfactor and the axial force Particularly the energy trans-mission equation and the energy dissipation equation arederived from the wavenumber without approximation of itsreal and imaginary term)en the energy density governing

0 01 02 03 04 05 06 07 08042

044

046

048

050

052

054

056

058

Inpu

t pow

er (m

W)

Structural damping loss factor

Figure 12 Input power of pinned-pinned beams versus structuraldamping loss factor with F0 1N N 20 kN and f 6 kHz

0 01 02 03 04 05 06 07 08Structural damping loss factor

825

830

835

840

845

850

855

860

865

870

Gro

up v

eloc

ity (m

s)

Figure 13 Group velocity versus structural damping loss factorwith N 20 kN and f 6 kHz

x

x0

L

F0δ (x ndash x0)ejωt

②①N N

Figure 14 A fixed-fixed beam with constant axial force

x

L

NF0δ (x ndash x0)ejωt

①

Figure 15 A fixed-free beam with constant axial force

0 05 1 15 2 25 3ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

0

20

40

60

EFA solution f = 2kHz

EFA solution f = 4kHzEFA solution f = 6kHz

EFA solution f = 8kHz

Modal solution f = 2kHz

Modal solution f = 4kHzModal solution f = 6kHz

Modal solution f = 8kHz

Ener

gy d

ensit

y (d

B)

Position (m)

Figure 16 Energy density distribution of fixed-fixed beams withF0 1N N 20 kN and η 05 when f 2 4 6 8 kHz

0 05 1 15 2 25 3

EFA solution f = 2kHz

EFA solution f = 4kHzEFA solution f = 6kHz

EFA solution f = 8kHz

Modal solution f = 2kHz

Modal solution f = 4kHzModal solution f = 6kHz

Modal solution f = 8kHz

Position (m)

ndash250

ndash200

ndash150

ndash100

ndash50

0

50

100

Ener

gy d

ensit

y (d

B)

Figure 17 Energy density distribution of fixed-free beams withF0 1N N 20 kN and η 05 when f 2 4 6 8 kHz

8 Mathematical Problems in Engineering

equation is obtained for the high damping beam with axialforce

To verify the developed energy flow model numericalanalyses are performed for the simply supported beam withvarious damping loss factor and axial force )e EFA so-lutions are greatly consistent with the modal solutions in allcases Furthermore it is found that both the high dampingfactor and large axial force can significantly alter the level ofenergy density as well as the distribution Due to theresulting deceasing input power the increasing dampingloss factor makes the energy density decay more dramati-cally along axial position However that is the contrary casefor the axial force )e proposed method is expected to beuseful for the prediction of the high-frequency vibration ofhigh damping structures with initial stress

Appendix

A Impedance and Exact Solution from ModalAnalysis for High Damping Beams withAxial Force

When a high damping beam with axial force is excited at x0by a transverse harmonic force the governing equation ofmotion can be expressed as

Dc

z4w

zx4 minus Nc

z2w

zx2 + ρSz2w

zt2 F0δ x minus x0( 1113857e

jωt (A1)

)e steady-state solution of equation (A1) can be foundby modal superposition as

w(x t) 1113944infin

i1ϕi(x)qi(t) (A2)

where ϕi(x) is the ith natural mode of the beam and qi(t) isthe ith mode displacement

By substituting equation (A2) into the motion equationof the beam equation (A1) and using the orthogonality ofnatural modes we obtain

Kiqi(t) + Mi

d2qi(t)

dt2 Fi (A3)

where

Fi 1113946L

0F0δ x minus x0( 1113857ϕi(x)dx (A4)

represents the so-called ldquomodal forcerdquo and

Mi 1113946L

0ρSϕ2

i dx

Ki ω2i (1 + jη) 1113946

L

0ρSϕ2

i dx

(A5)

are the so-called ldquomodal massrdquo and ldquomodal stiffnessrdquo )eeffect of structural damping is modeled by assuming ahysteretic damping model which introduces an additionalcomplex term (1 + jη) to modal stiffness )e natural fre-quency of the ith mode ωi can be determined from thecorresponding wavenumber in equation (2) which can bewritten as

k2i minus

N

2D+

N

2D1113874 1113875

2+ρSω2

i

D

1113971

(A6)

)e forced vibration response of the beam can beexpressed as the linear combination of the normal modes

w(x t) 1113944infin

i1

F0 middot ejωt

Mi ω2i (1 + jη) minus ω21113858 1113859

ϕi(x)ϕi x0( 1113857 (A7)

)e impedance at the driving point of the finite beamwith axial force can be obtained from its definition

Z F0e

jωt

_w x0 t( 1113857

minus11113936infini1 ωϕ

2i x0( 1113857Mi ω2

i (1 + jη) minus ω21113858 1113859

(A8)

where _w(x0 t) is the velocity at driving point which can becalculated as the time derivative of the displacement atx x0 Once the mode function is determined by theboundary condition the impedance at driving point of thefinite beam with axial force is derived from equation (A8)

)e time averaged energy density is given by

langerang 14

ρSzw

zt1113888 1113889

zw

zt1113888 1113889

lowast

+ Dz2w

zx21113888 1113889z2w

zx21113888 1113889

lowast

+ Nzw

zx1113888 1113889

zw

zx1113888 1113889

lowast

1113890 1113891 (A9)

Substituting equation (A7) into equation (A9) theenergy density can be obtained based on the modal solution

B Mode Shapes for High Damping Beams withAxial Force

B1 BeamPinned at Both Ends )e transverse displacementand the bending moment are zero at a simply supported endHence the boundary conditions can be stated as

w(0) 0

z2w(0)

zx2 0

w(L) 0

z2w(L)

zx2 0

(B1)

Mathematical Problems in Engineering 9

)e general solution of equation (A1) is shown inequation (7) whereas the frequency equation with bothsimply supported ends is derived as

sin kiL 0 (B2)

)e vibration mode of the beam is given by

ϕi(x) sin kix (B3)

B2 Beam Fixed at Both Ends At a fixed end the transversedisplacement and the slope of the displacement are zeroHence the boundary conditions are given by

w(0) 0

zw(0)

zx 0

w(L) 0

zw(L)

zx 0

(B4)

By virtue of general solution in equation (7) andboundary conditions with both fixed ends the frequencyequation is obtained as

cos kiL cosh kiL minus 1 0 (B5)

)e vibration mode of the beam is expressed as

ϕi(x) cos kix minus cosh kix( 1113857 +cos kiL minus cosh kiL

sin kiL minus sinh kiL

middot sin kix minus sinh kix( 1113857

(B6)

B3BeamFixedatOneEndandFreeat theOther If the beamis fixed at x 0 and free at x L the transverse deflectionand its slope must be zero at x 0 and the bending momentand shear force must be zero at x L )us the boundaryconditions become

w(0) 0

zw(0)

zx 0

z3w(L)

zx3 0

z2w(L)

zx2 0

(B7)

According to the general solution in equation (7) andboundary conditions with one fixed end and the other freeend the frequency equation is derived as

cos kiL cosh kiL + 1 0 (B8)

)e vibration mode of the beam is written as

ϕi(x) cos kix minus cosh kix( 1113857 +cos kiL + cosh kiL

sin kiL + sinh kiL

middot sin kix minus sinh kix( 1113857

(B9)

Data Availability

)e data used to support the finding of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that there no conflicts of interest

Acknowledgments

)e presented work was supported by National NaturalScience Foundation of China (Grant no 51505096) andNatural Science Foundation of Heilongjiang Province ofChina (Grant no QC2016056)

References

[1] L Bot ldquoEntropy in sound and vibration towards a newparadigmrdquo Proceedings of the Royal Society A MathematicalPhysical and Engineering Science vol 473 no 2197 pp 1ndash92017

[2] A Acri ldquoA literature review on energy flow analysis of vi-brating structuresrdquo ASRO Journal of Applied Mechanicsvol 1 no 1 pp 12ndash15 2017

[3] J C Wohlever and R J Bernhard ldquoMechanical energy flowmodels of rods and beamsrdquo Journal of Sound and Vibrationvol 153 no 1 pp 1ndash19 1992

[4] Z Chen Z Yang Y Wang and X Wang ldquoA high-frequencyshock response analysis method based on energy finite ele-ment method and virtual mode synthesis and simulationrdquoActa Aeronautica et Astronautica Sinica vol 39 no 08pp 182ndash191 2018

[5] Z Liu and J Niu ldquoVibrational energy flow model for func-tionally graded beamsrdquo Composite Structures vol 186pp 17ndash28 2018

[6] X Kong H Chen D Zhu and W Zhang ldquoStudy on thevalidity region of energy finite element analysisrdquo Journal ofSound and Vibration vol 333 no 9 pp 2601ndash2616 2014

[7] A Nokhbatolfoghahai H Navazi Y Ghobaad andH Haddadpour ldquoExperimental measurement of energydensity of a vibrating beamrdquo Journal of Vibration and Controlvol 24 no 24 pp 5735ndash5746 2018

[8] H M Navazi A Nokhbatolfoghahaei Y Ghobad andH Haddadpour ldquoExperimental measurement of energydensity in a vibrating plate and comparison with energy finiteelement analysisrdquo Journal of Sound and Vibration vol 375pp 289ndash307 2016

[9] D Zhu H Chen X Kong and W Zhang ldquoA hybrid finiteelement-energy finite element method for mid-frequencyvibrations of built-up structures under multi-distributedloadingsrdquo Journal of Sound amp Vibration vol 333 no 22pp 5723ndash5745 2014

[10] M Sadeghmanesh H Haddadpour M T Abadi andH M Navazi ldquoA method for selection of structural theories

10 Mathematical Problems in Engineering

for low to high frequency vibration analysesrdquo EuropeanJournal of Mechanics-A Solids vol 75 pp 27ndash40 2019

[11] W Zhang AWang N Vlahopoulos and KWu ldquoA vibrationanalysis of stiffened plates under heavy fluid loading by anenergy finite element analysis formulationrdquo Finite Elements inAnalysis and Design vol 41 no 11-12 pp 1056ndash1078 2005

[12] J-B Han S-Y Hong and J-H Song ldquoEnergy flow model forthin plate considering fluid loading with mean flowrdquo Journalof Sound and Vibration vol 331 no 24 pp 5326ndash5346 2012

[13] H-W Kwon S-Y Hong D-K Oh et al ldquoExperimental studyon the energy flow analysis of underwater vibration for thereinforced cylindrical structurerdquo Journal of Mechanical Sci-ence and Technology vol 28 no 9 pp 3405ndash3410 2014

[14] S Wang and R J Bernhard ldquoPrediction of averaged energyfor moderately damped systems with strong couplingrdquoJournal of Sound and Vibration vol 319 no 1-2 pp 426ndash4442009

[15] V S Pereira and J M C Dos Santos ldquoCoupled plate energymodels at mid- and high-frequency vibrationsrdquo Computers ampStructures vol 134 pp 48ndash61 2014

[16] H-W Kwon S-Y Hong and J-H Song ldquoVibrational energyflow analysis of coupled cylindrical thin shell structuresrdquoJournal of Mechanical Science and Technology vol 30 no 9pp 4049ndash4062 2016

[17] Y-H Park and S-Y Hong ldquoHybrid power flow analysis usingcoupling loss factor of SEA for low-damping system-part Iformulation of 1-D and 2-D casesrdquo Journal of Sound andVibration vol 299 no 3 pp 484ndash503 2007

[18] Z Liu J Niu and X Gao ldquoAn improved approach for analysisof coupled structures in Energy Finite Element Analysis usingthe coupling loss factorrdquo Computers amp Structures vol 210pp 69ndash86 2018

[19] H-W Kwon S-Y Hong and J-H Song ldquoEnergy flowmodels for underwater radiation noise prediction in medium-to-high-frequency rangesrdquo Proceedings of the Institution ofMechanical Engineers Part M Journal of Engineering for theMaritime Environment vol 230 no 2 pp 404ndash416 2016

[20] X Zheng W Dai Y Qiu and Z Hao ldquoPrediction and energycontribution analysis of interior noise in a high-speed trainbased on modified energy finite element analysisrdquoMechanicalSystems and Signal Processing vol 126 pp 439ndash457 2019

[21] W Zhang H Chen D Zhu and X Kong ldquo)e thermal effectson high-frequency vibration of beams using energy flowanalysisrdquo Journal of Sound and Vibration vol 333 no 9pp 2588ndash2600 2014

[22] W Di X Miaoxia and L Yueming ldquoHigh-frequency dy-namic analysis of plates in thermal environments based onenergy finite elementmethodrdquo Shock and Vibration vol 2015pp 1ndash14 2015

[23] Z Chen Z Yang N Guo and G Zhang ldquoAn energy finiteelement method for high frequency vibration analysis ofbeams with axial forcerdquo Applied Mathematical Modellingvol 61 no 5 pp 521ndash539 2018

[24] P M Morse and K U Indard Ceoretical AcousticsPrinceton University Press Princeton NJ USA 1986

[25] W Weaver S P Timoshenko and D H Young VibrationProblems in Engineering John Wiley amp Sons Hoboken NJUSA 1990

[26] M N Ichchou A Le Bot and L Jezequel ldquoEnergy models ofone-dimensional multi-propagative systemsrdquo Journal ofSound and Vibration vol 201 no 5 pp 535ndash554 1997

[27] Y Lase M N Ichchou and L Jezequel ldquoEnergy flow analysisof bars and beams theoretical formulationsrdquo Journal of Soundand Vibration vol 192 no 1 pp 281ndash305 1996

[28] D J Nefske and S H Sung ldquoPower flow finite elementanalysis of dynamic systems basic theory and application tobeamsrdquo Journal of Vibration and Acoustics vol 111 no 1pp 94ndash100 1989

Mathematical Problems in Engineering 11

equation is obtained for the high damping beam with axialforce

To verify the developed energy flow model numericalanalyses are performed for the simply supported beam withvarious damping loss factor and axial force )e EFA so-lutions are greatly consistent with the modal solutions in allcases Furthermore it is found that both the high dampingfactor and large axial force can significantly alter the level ofenergy density as well as the distribution Due to theresulting deceasing input power the increasing dampingloss factor makes the energy density decay more dramati-cally along axial position However that is the contrary casefor the axial force )e proposed method is expected to beuseful for the prediction of the high-frequency vibration ofhigh damping structures with initial stress

Appendix

A Impedance and Exact Solution from ModalAnalysis for High Damping Beams withAxial Force

When a high damping beam with axial force is excited at x0by a transverse harmonic force the governing equation ofmotion can be expressed as

Dc

z4w

zx4 minus Nc

z2w

zx2 + ρSz2w

zt2 F0δ x minus x0( 1113857e

jωt (A1)

)e steady-state solution of equation (A1) can be foundby modal superposition as

w(x t) 1113944infin

i1ϕi(x)qi(t) (A2)

where ϕi(x) is the ith natural mode of the beam and qi(t) isthe ith mode displacement

By substituting equation (A2) into the motion equationof the beam equation (A1) and using the orthogonality ofnatural modes we obtain

Kiqi(t) + Mi

d2qi(t)

dt2 Fi (A3)

where

Fi 1113946L

0F0δ x minus x0( 1113857ϕi(x)dx (A4)

represents the so-called ldquomodal forcerdquo and

Mi 1113946L

0ρSϕ2

i dx

Ki ω2i (1 + jη) 1113946

L

0ρSϕ2

i dx

(A5)

are the so-called ldquomodal massrdquo and ldquomodal stiffnessrdquo )eeffect of structural damping is modeled by assuming ahysteretic damping model which introduces an additionalcomplex term (1 + jη) to modal stiffness )e natural fre-quency of the ith mode ωi can be determined from thecorresponding wavenumber in equation (2) which can bewritten as

k2i minus

N

2D+

N

2D1113874 1113875

2+ρSω2

i

D

1113971

(A6)

)e forced vibration response of the beam can beexpressed as the linear combination of the normal modes

w(x t) 1113944infin

i1

F0 middot ejωt

Mi ω2i (1 + jη) minus ω21113858 1113859

ϕi(x)ϕi x0( 1113857 (A7)

)e impedance at the driving point of the finite beamwith axial force can be obtained from its definition

Z F0e

jωt

_w x0 t( 1113857

minus11113936infini1 ωϕ

2i x0( 1113857Mi ω2

i (1 + jη) minus ω21113858 1113859

(A8)

where _w(x0 t) is the velocity at driving point which can becalculated as the time derivative of the displacement atx x0 Once the mode function is determined by theboundary condition the impedance at driving point of thefinite beam with axial force is derived from equation (A8)

)e time averaged energy density is given by

langerang 14

ρSzw

zt1113888 1113889

zw

zt1113888 1113889

lowast

+ Dz2w

zx21113888 1113889z2w

zx21113888 1113889

lowast

+ Nzw

zx1113888 1113889

zw

zx1113888 1113889

lowast

1113890 1113891 (A9)

Substituting equation (A7) into equation (A9) theenergy density can be obtained based on the modal solution

B Mode Shapes for High Damping Beams withAxial Force

B1 BeamPinned at Both Ends )e transverse displacementand the bending moment are zero at a simply supported endHence the boundary conditions can be stated as

w(0) 0

z2w(0)

zx2 0

w(L) 0

z2w(L)

zx2 0

(B1)

Mathematical Problems in Engineering 9

)e general solution of equation (A1) is shown inequation (7) whereas the frequency equation with bothsimply supported ends is derived as

sin kiL 0 (B2)

)e vibration mode of the beam is given by

ϕi(x) sin kix (B3)

B2 Beam Fixed at Both Ends At a fixed end the transversedisplacement and the slope of the displacement are zeroHence the boundary conditions are given by

w(0) 0

zw(0)

zx 0

w(L) 0

zw(L)

zx 0

(B4)

By virtue of general solution in equation (7) andboundary conditions with both fixed ends the frequencyequation is obtained as

cos kiL cosh kiL minus 1 0 (B5)

)e vibration mode of the beam is expressed as

ϕi(x) cos kix minus cosh kix( 1113857 +cos kiL minus cosh kiL

sin kiL minus sinh kiL

middot sin kix minus sinh kix( 1113857

(B6)

B3BeamFixedatOneEndandFreeat theOther If the beamis fixed at x 0 and free at x L the transverse deflectionand its slope must be zero at x 0 and the bending momentand shear force must be zero at x L )us the boundaryconditions become

w(0) 0

zw(0)

zx 0

z3w(L)

zx3 0

z2w(L)

zx2 0

(B7)

According to the general solution in equation (7) andboundary conditions with one fixed end and the other freeend the frequency equation is derived as

cos kiL cosh kiL + 1 0 (B8)

)e vibration mode of the beam is written as

ϕi(x) cos kix minus cosh kix( 1113857 +cos kiL + cosh kiL

sin kiL + sinh kiL

middot sin kix minus sinh kix( 1113857

(B9)

Data Availability

)e data used to support the finding of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that there no conflicts of interest

Acknowledgments

)e presented work was supported by National NaturalScience Foundation of China (Grant no 51505096) andNatural Science Foundation of Heilongjiang Province ofChina (Grant no QC2016056)

References

[1] L Bot ldquoEntropy in sound and vibration towards a newparadigmrdquo Proceedings of the Royal Society A MathematicalPhysical and Engineering Science vol 473 no 2197 pp 1ndash92017

[2] A Acri ldquoA literature review on energy flow analysis of vi-brating structuresrdquo ASRO Journal of Applied Mechanicsvol 1 no 1 pp 12ndash15 2017

[3] J C Wohlever and R J Bernhard ldquoMechanical energy flowmodels of rods and beamsrdquo Journal of Sound and Vibrationvol 153 no 1 pp 1ndash19 1992

[4] Z Chen Z Yang Y Wang and X Wang ldquoA high-frequencyshock response analysis method based on energy finite ele-ment method and virtual mode synthesis and simulationrdquoActa Aeronautica et Astronautica Sinica vol 39 no 08pp 182ndash191 2018

[5] Z Liu and J Niu ldquoVibrational energy flow model for func-tionally graded beamsrdquo Composite Structures vol 186pp 17ndash28 2018

[6] X Kong H Chen D Zhu and W Zhang ldquoStudy on thevalidity region of energy finite element analysisrdquo Journal ofSound and Vibration vol 333 no 9 pp 2601ndash2616 2014

[7] A Nokhbatolfoghahai H Navazi Y Ghobaad andH Haddadpour ldquoExperimental measurement of energydensity of a vibrating beamrdquo Journal of Vibration and Controlvol 24 no 24 pp 5735ndash5746 2018

[8] H M Navazi A Nokhbatolfoghahaei Y Ghobad andH Haddadpour ldquoExperimental measurement of energydensity in a vibrating plate and comparison with energy finiteelement analysisrdquo Journal of Sound and Vibration vol 375pp 289ndash307 2016

[9] D Zhu H Chen X Kong and W Zhang ldquoA hybrid finiteelement-energy finite element method for mid-frequencyvibrations of built-up structures under multi-distributedloadingsrdquo Journal of Sound amp Vibration vol 333 no 22pp 5723ndash5745 2014

[10] M Sadeghmanesh H Haddadpour M T Abadi andH M Navazi ldquoA method for selection of structural theories

10 Mathematical Problems in Engineering

for low to high frequency vibration analysesrdquo EuropeanJournal of Mechanics-A Solids vol 75 pp 27ndash40 2019

[11] W Zhang AWang N Vlahopoulos and KWu ldquoA vibrationanalysis of stiffened plates under heavy fluid loading by anenergy finite element analysis formulationrdquo Finite Elements inAnalysis and Design vol 41 no 11-12 pp 1056ndash1078 2005

[12] J-B Han S-Y Hong and J-H Song ldquoEnergy flow model forthin plate considering fluid loading with mean flowrdquo Journalof Sound and Vibration vol 331 no 24 pp 5326ndash5346 2012

[13] H-W Kwon S-Y Hong D-K Oh et al ldquoExperimental studyon the energy flow analysis of underwater vibration for thereinforced cylindrical structurerdquo Journal of Mechanical Sci-ence and Technology vol 28 no 9 pp 3405ndash3410 2014

[14] S Wang and R J Bernhard ldquoPrediction of averaged energyfor moderately damped systems with strong couplingrdquoJournal of Sound and Vibration vol 319 no 1-2 pp 426ndash4442009

[15] V S Pereira and J M C Dos Santos ldquoCoupled plate energymodels at mid- and high-frequency vibrationsrdquo Computers ampStructures vol 134 pp 48ndash61 2014

[16] H-W Kwon S-Y Hong and J-H Song ldquoVibrational energyflow analysis of coupled cylindrical thin shell structuresrdquoJournal of Mechanical Science and Technology vol 30 no 9pp 4049ndash4062 2016

[17] Y-H Park and S-Y Hong ldquoHybrid power flow analysis usingcoupling loss factor of SEA for low-damping system-part Iformulation of 1-D and 2-D casesrdquo Journal of Sound andVibration vol 299 no 3 pp 484ndash503 2007

[18] Z Liu J Niu and X Gao ldquoAn improved approach for analysisof coupled structures in Energy Finite Element Analysis usingthe coupling loss factorrdquo Computers amp Structures vol 210pp 69ndash86 2018

[19] H-W Kwon S-Y Hong and J-H Song ldquoEnergy flowmodels for underwater radiation noise prediction in medium-to-high-frequency rangesrdquo Proceedings of the Institution ofMechanical Engineers Part M Journal of Engineering for theMaritime Environment vol 230 no 2 pp 404ndash416 2016

[20] X Zheng W Dai Y Qiu and Z Hao ldquoPrediction and energycontribution analysis of interior noise in a high-speed trainbased on modified energy finite element analysisrdquoMechanicalSystems and Signal Processing vol 126 pp 439ndash457 2019

[21] W Zhang H Chen D Zhu and X Kong ldquo)e thermal effectson high-frequency vibration of beams using energy flowanalysisrdquo Journal of Sound and Vibration vol 333 no 9pp 2588ndash2600 2014

[22] W Di X Miaoxia and L Yueming ldquoHigh-frequency dy-namic analysis of plates in thermal environments based onenergy finite elementmethodrdquo Shock and Vibration vol 2015pp 1ndash14 2015

[23] Z Chen Z Yang N Guo and G Zhang ldquoAn energy finiteelement method for high frequency vibration analysis ofbeams with axial forcerdquo Applied Mathematical Modellingvol 61 no 5 pp 521ndash539 2018

[24] P M Morse and K U Indard Ceoretical AcousticsPrinceton University Press Princeton NJ USA 1986

[25] W Weaver S P Timoshenko and D H Young VibrationProblems in Engineering John Wiley amp Sons Hoboken NJUSA 1990

[26] M N Ichchou A Le Bot and L Jezequel ldquoEnergy models ofone-dimensional multi-propagative systemsrdquo Journal ofSound and Vibration vol 201 no 5 pp 535ndash554 1997

[27] Y Lase M N Ichchou and L Jezequel ldquoEnergy flow analysisof bars and beams theoretical formulationsrdquo Journal of Soundand Vibration vol 192 no 1 pp 281ndash305 1996

[28] D J Nefske and S H Sung ldquoPower flow finite elementanalysis of dynamic systems basic theory and application tobeamsrdquo Journal of Vibration and Acoustics vol 111 no 1pp 94ndash100 1989

Mathematical Problems in Engineering 11

)e general solution of equation (A1) is shown inequation (7) whereas the frequency equation with bothsimply supported ends is derived as

sin kiL 0 (B2)

)e vibration mode of the beam is given by

ϕi(x) sin kix (B3)

B2 Beam Fixed at Both Ends At a fixed end the transversedisplacement and the slope of the displacement are zeroHence the boundary conditions are given by

w(0) 0

zw(0)

zx 0

w(L) 0

zw(L)

zx 0

(B4)

By virtue of general solution in equation (7) andboundary conditions with both fixed ends the frequencyequation is obtained as

cos kiL cosh kiL minus 1 0 (B5)

)e vibration mode of the beam is expressed as

ϕi(x) cos kix minus cosh kix( 1113857 +cos kiL minus cosh kiL

sin kiL minus sinh kiL

middot sin kix minus sinh kix( 1113857

(B6)

B3BeamFixedatOneEndandFreeat theOther If the beamis fixed at x 0 and free at x L the transverse deflectionand its slope must be zero at x 0 and the bending momentand shear force must be zero at x L )us the boundaryconditions become

w(0) 0

zw(0)

zx 0

z3w(L)

zx3 0

z2w(L)

zx2 0

(B7)

According to the general solution in equation (7) andboundary conditions with one fixed end and the other freeend the frequency equation is derived as

cos kiL cosh kiL + 1 0 (B8)

)e vibration mode of the beam is written as

ϕi(x) cos kix minus cosh kix( 1113857 +cos kiL + cosh kiL

sin kiL + sinh kiL

middot sin kix minus sinh kix( 1113857

(B9)

Data Availability

)e data used to support the finding of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that there no conflicts of interest

Acknowledgments

)e presented work was supported by National NaturalScience Foundation of China (Grant no 51505096) andNatural Science Foundation of Heilongjiang Province ofChina (Grant no QC2016056)

References

[1] L Bot ldquoEntropy in sound and vibration towards a newparadigmrdquo Proceedings of the Royal Society A MathematicalPhysical and Engineering Science vol 473 no 2197 pp 1ndash92017

[2] A Acri ldquoA literature review on energy flow analysis of vi-brating structuresrdquo ASRO Journal of Applied Mechanicsvol 1 no 1 pp 12ndash15 2017

[3] J C Wohlever and R J Bernhard ldquoMechanical energy flowmodels of rods and beamsrdquo Journal of Sound and Vibrationvol 153 no 1 pp 1ndash19 1992

[4] Z Chen Z Yang Y Wang and X Wang ldquoA high-frequencyshock response analysis method based on energy finite ele-ment method and virtual mode synthesis and simulationrdquoActa Aeronautica et Astronautica Sinica vol 39 no 08pp 182ndash191 2018

[5] Z Liu and J Niu ldquoVibrational energy flow model for func-tionally graded beamsrdquo Composite Structures vol 186pp 17ndash28 2018

[6] X Kong H Chen D Zhu and W Zhang ldquoStudy on thevalidity region of energy finite element analysisrdquo Journal ofSound and Vibration vol 333 no 9 pp 2601ndash2616 2014

[7] A Nokhbatolfoghahai H Navazi Y Ghobaad andH Haddadpour ldquoExperimental measurement of energydensity of a vibrating beamrdquo Journal of Vibration and Controlvol 24 no 24 pp 5735ndash5746 2018

[8] H M Navazi A Nokhbatolfoghahaei Y Ghobad andH Haddadpour ldquoExperimental measurement of energydensity in a vibrating plate and comparison with energy finiteelement analysisrdquo Journal of Sound and Vibration vol 375pp 289ndash307 2016

[9] D Zhu H Chen X Kong and W Zhang ldquoA hybrid finiteelement-energy finite element method for mid-frequencyvibrations of built-up structures under multi-distributedloadingsrdquo Journal of Sound amp Vibration vol 333 no 22pp 5723ndash5745 2014

[10] M Sadeghmanesh H Haddadpour M T Abadi andH M Navazi ldquoA method for selection of structural theories

10 Mathematical Problems in Engineering

for low to high frequency vibration analysesrdquo EuropeanJournal of Mechanics-A Solids vol 75 pp 27ndash40 2019

[11] W Zhang AWang N Vlahopoulos and KWu ldquoA vibrationanalysis of stiffened plates under heavy fluid loading by anenergy finite element analysis formulationrdquo Finite Elements inAnalysis and Design vol 41 no 11-12 pp 1056ndash1078 2005

[12] J-B Han S-Y Hong and J-H Song ldquoEnergy flow model forthin plate considering fluid loading with mean flowrdquo Journalof Sound and Vibration vol 331 no 24 pp 5326ndash5346 2012

[13] H-W Kwon S-Y Hong D-K Oh et al ldquoExperimental studyon the energy flow analysis of underwater vibration for thereinforced cylindrical structurerdquo Journal of Mechanical Sci-ence and Technology vol 28 no 9 pp 3405ndash3410 2014

[14] S Wang and R J Bernhard ldquoPrediction of averaged energyfor moderately damped systems with strong couplingrdquoJournal of Sound and Vibration vol 319 no 1-2 pp 426ndash4442009

[15] V S Pereira and J M C Dos Santos ldquoCoupled plate energymodels at mid- and high-frequency vibrationsrdquo Computers ampStructures vol 134 pp 48ndash61 2014

[16] H-W Kwon S-Y Hong and J-H Song ldquoVibrational energyflow analysis of coupled cylindrical thin shell structuresrdquoJournal of Mechanical Science and Technology vol 30 no 9pp 4049ndash4062 2016

[17] Y-H Park and S-Y Hong ldquoHybrid power flow analysis usingcoupling loss factor of SEA for low-damping system-part Iformulation of 1-D and 2-D casesrdquo Journal of Sound andVibration vol 299 no 3 pp 484ndash503 2007

[18] Z Liu J Niu and X Gao ldquoAn improved approach for analysisof coupled structures in Energy Finite Element Analysis usingthe coupling loss factorrdquo Computers amp Structures vol 210pp 69ndash86 2018

[19] H-W Kwon S-Y Hong and J-H Song ldquoEnergy flowmodels for underwater radiation noise prediction in medium-to-high-frequency rangesrdquo Proceedings of the Institution ofMechanical Engineers Part M Journal of Engineering for theMaritime Environment vol 230 no 2 pp 404ndash416 2016

[20] X Zheng W Dai Y Qiu and Z Hao ldquoPrediction and energycontribution analysis of interior noise in a high-speed trainbased on modified energy finite element analysisrdquoMechanicalSystems and Signal Processing vol 126 pp 439ndash457 2019

[21] W Zhang H Chen D Zhu and X Kong ldquo)e thermal effectson high-frequency vibration of beams using energy flowanalysisrdquo Journal of Sound and Vibration vol 333 no 9pp 2588ndash2600 2014

[22] W Di X Miaoxia and L Yueming ldquoHigh-frequency dy-namic analysis of plates in thermal environments based onenergy finite elementmethodrdquo Shock and Vibration vol 2015pp 1ndash14 2015

[23] Z Chen Z Yang N Guo and G Zhang ldquoAn energy finiteelement method for high frequency vibration analysis ofbeams with axial forcerdquo Applied Mathematical Modellingvol 61 no 5 pp 521ndash539 2018

[24] P M Morse and K U Indard Ceoretical AcousticsPrinceton University Press Princeton NJ USA 1986

[25] W Weaver S P Timoshenko and D H Young VibrationProblems in Engineering John Wiley amp Sons Hoboken NJUSA 1990

[26] M N Ichchou A Le Bot and L Jezequel ldquoEnergy models ofone-dimensional multi-propagative systemsrdquo Journal ofSound and Vibration vol 201 no 5 pp 535ndash554 1997

[27] Y Lase M N Ichchou and L Jezequel ldquoEnergy flow analysisof bars and beams theoretical formulationsrdquo Journal of Soundand Vibration vol 192 no 1 pp 281ndash305 1996

[28] D J Nefske and S H Sung ldquoPower flow finite elementanalysis of dynamic systems basic theory and application tobeamsrdquo Journal of Vibration and Acoustics vol 111 no 1pp 94ndash100 1989

Mathematical Problems in Engineering 11

for low to high frequency vibration analysesrdquo EuropeanJournal of Mechanics-A Solids vol 75 pp 27ndash40 2019

[11] W Zhang AWang N Vlahopoulos and KWu ldquoA vibrationanalysis of stiffened plates under heavy fluid loading by anenergy finite element analysis formulationrdquo Finite Elements inAnalysis and Design vol 41 no 11-12 pp 1056ndash1078 2005

[12] J-B Han S-Y Hong and J-H Song ldquoEnergy flow model forthin plate considering fluid loading with mean flowrdquo Journalof Sound and Vibration vol 331 no 24 pp 5326ndash5346 2012

[13] H-W Kwon S-Y Hong D-K Oh et al ldquoExperimental studyon the energy flow analysis of underwater vibration for thereinforced cylindrical structurerdquo Journal of Mechanical Sci-ence and Technology vol 28 no 9 pp 3405ndash3410 2014

[14] S Wang and R J Bernhard ldquoPrediction of averaged energyfor moderately damped systems with strong couplingrdquoJournal of Sound and Vibration vol 319 no 1-2 pp 426ndash4442009

[15] V S Pereira and J M C Dos Santos ldquoCoupled plate energymodels at mid- and high-frequency vibrationsrdquo Computers ampStructures vol 134 pp 48ndash61 2014

[16] H-W Kwon S-Y Hong and J-H Song ldquoVibrational energyflow analysis of coupled cylindrical thin shell structuresrdquoJournal of Mechanical Science and Technology vol 30 no 9pp 4049ndash4062 2016

[17] Y-H Park and S-Y Hong ldquoHybrid power flow analysis usingcoupling loss factor of SEA for low-damping system-part Iformulation of 1-D and 2-D casesrdquo Journal of Sound andVibration vol 299 no 3 pp 484ndash503 2007

[18] Z Liu J Niu and X Gao ldquoAn improved approach for analysisof coupled structures in Energy Finite Element Analysis usingthe coupling loss factorrdquo Computers amp Structures vol 210pp 69ndash86 2018

[19] H-W Kwon S-Y Hong and J-H Song ldquoEnergy flowmodels for underwater radiation noise prediction in medium-to-high-frequency rangesrdquo Proceedings of the Institution ofMechanical Engineers Part M Journal of Engineering for theMaritime Environment vol 230 no 2 pp 404ndash416 2016

[20] X Zheng W Dai Y Qiu and Z Hao ldquoPrediction and energycontribution analysis of interior noise in a high-speed trainbased on modified energy finite element analysisrdquoMechanicalSystems and Signal Processing vol 126 pp 439ndash457 2019

[21] W Zhang H Chen D Zhu and X Kong ldquo)e thermal effectson high-frequency vibration of beams using energy flowanalysisrdquo Journal of Sound and Vibration vol 333 no 9pp 2588ndash2600 2014

[22] W Di X Miaoxia and L Yueming ldquoHigh-frequency dy-namic analysis of plates in thermal environments based onenergy finite elementmethodrdquo Shock and Vibration vol 2015pp 1ndash14 2015

[23] Z Chen Z Yang N Guo and G Zhang ldquoAn energy finiteelement method for high frequency vibration analysis ofbeams with axial forcerdquo Applied Mathematical Modellingvol 61 no 5 pp 521ndash539 2018

[24] P M Morse and K U Indard Ceoretical AcousticsPrinceton University Press Princeton NJ USA 1986

[25] W Weaver S P Timoshenko and D H Young VibrationProblems in Engineering John Wiley amp Sons Hoboken NJUSA 1990

[26] M N Ichchou A Le Bot and L Jezequel ldquoEnergy models ofone-dimensional multi-propagative systemsrdquo Journal ofSound and Vibration vol 201 no 5 pp 535ndash554 1997

[27] Y Lase M N Ichchou and L Jezequel ldquoEnergy flow analysisof bars and beams theoretical formulationsrdquo Journal of Soundand Vibration vol 192 no 1 pp 281ndash305 1996

[28] D J Nefske and S H Sung ldquoPower flow finite elementanalysis of dynamic systems basic theory and application tobeamsrdquo Journal of Vibration and Acoustics vol 111 no 1pp 94ndash100 1989

Mathematical Problems in Engineering 11