efficient model order reduction of structural dynamic

Research ArticleEfficient Model Order Reduction of Structural DynamicSystems with Local Nonlinearities under Periodic Motion

M. Mohammadali and H. Ahmadian

Center of Excellence in Experimental Solid Mechanics and Dynamics, School of Mechanical Engineering,Iran University of Science and Technology, Narmak, Tehran 16844, Iran

Correspondence should be addressed to M. Mohammadali; [email protected]

Received 20 October 2012; Accepted 19 November 2012; Published 9 June 2014

Academic Editor: Hamid Mehdigholi

Copyright © 2014 M. Mohammadali and H. Ahmadian. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

In many nonlinear structural systems, compared with the local regions with induced nonlinear effects, the main portions of thestructures are linear. An exact condensation technique based on the harmonic balance method (HBM) in conjunction with themodal expansion technique is employed to convert the motion equations of such a system to a set of nonlinear algebraic equationsthat are considerably small and adequately accurate to determine periodic responses. To demonstrate the capability of the suggestedmethod, few case studies consisting of discrete systems with weak and essential nonlinearities are studied, and the results arecompared to other methodologies results.

1. Introduction

Compared with the local regions with induced nonlineareffects, the main portions of the structural systems are linear.Typical engineering examples of these localized nonlinearsources are friction and vibroimpact in joints, local buckling,crack, nonlinear isolator, dead zone (gap), and squeeze filmdampers in mechanical systems. Consequently, the dynamicresponse of such a system is nonlinear. A dynamic model forthis structural system is useful for detailed study, predictionof response, identification of unknown parameters, or healthmonitoring of the overall system. Therefore, much time andeffort are consumed in the development of this dynamicmodel that includes desired effects. Usually, the dynamicproperties of the structure components are known (analysedor tested) as separate components, and they are used fordevelopment of the dynamic model for the total assembledsystem.

Anynonlinear response analysis involves significant com-putational effort, especially if the full dynamic model isused and a set of performance characteristics related to tem-perature, preload, deflection, and so forth characterize thenonlinear elements. Although there is significant motivationto develop a set of model order reduction (MOR) techniques

to project the system model to a condensed space andconsiderably reduce time of nonlinear response computation,accuracy of these MOR techniques and solution algorithmsare still issues. For reduction of nonlinear discrete modelsone can apply linear MOR techniques [1] such as Guyanreduction [2, 3], improved reduced system (IRS) [4], iter-ated improved reduced system (IIRS) [5], system equivalentreduction expansion process (SEREP), [6] or componentmode synthesis (CMS) [4, 7].

Other reduction methods that can be applied to reducecontinuous and discrete models size are commonly usedlinear normal modes (LNMs) of linearized system andnonlinear normal modes (NNMs). Generation of NNMs fora small system requires considerable effort [8, 9], especiallyin the presence of internal resonance [10] and/or externalexcitation [11]. In the case of a large system, it is possible touse other reduction techniques and then generate NNMs ofthe reduced order model [12].

Reduction techniques based on linear MOR techniques,in best, will lead to an acceptable answer for weakly nonlinearsystem, andby growing the nonlinearity their accuracywill belost due to their linear nature. Local equivalent linear stiffnessmethod (LELMS) was proposed [13] to reduce the nonlineardiscrete models size. In this method, an equivalent linear

Hindawi Publishing CorporationShock and VibrationVolume 2014, Article ID 152145, 5 pageshttp://dx.doi.org/10.1155/2014/152145

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2 Shock and Vibration

system of the nonlinear system is iteratively calculated whosemode shapes are used to reduce the system size.

There aremany different algorithms for solving nonlinearsystems. A semianalytical method that is generally valid overa much larger domain compared to the other methodologiesis the harmonic balance method (HBM) [14]. The HBMcan solve many types of nonlinear problems in the form ofperiodic response. The HBM is used for handling nonlinearproblems consisting of forced excited, strongly nonlinearresponse, chaotic behaviour, and internal resonance and hasa great potential for identification and health monitoring.Furthermore, this technique is applied to estimate NNMs ofnonlinear systems [8].HBMexpands the periodic response ofthe nonlinear system in the form of truncated Fourier series,whose coefficients are estimated by solving a set of nonlinearalgebraic equations. In the absence of internal resonancein many cases, the solution of one term harmonic balancehas good correlation with the recorded experimental results[15]. Typically, more accurate approximation to the solutionobtained by using a Fourier series with a higher number ofterms and, in general, the amplitudes of the higher termsgradually decrease. Consequently, in experiment if the higherterm’s amplitudes are small, then inclusion of measurementnoise will result in the difference between these terms in theanalysis and experiment.

HBM is improved in different ways because of its capa-bility. Incremental harmonic balance (IHB) [16] suggests anew formulation based on HBM for achieving frequencyresponses of nonlinear systems. With the aim of reducingthe computational cost and preserving the accuracy, adaptiveHBM (AHBM) [17] has proposed selection algorithms for thenumber of Fourier series terms. Recently, a new exact reduc-tion technique [18, 19] was proposed for discrete systems thatcondense the problem into the nonlinear DOFs only.

The aim of this work is to study the technique suggestedby KIM [18, 19] and extend this by expanding the nonlinearDOFsmotions by natural modes of the condensed system. Todemonstrate the capability of the suggested method, few casestudies consisting of a discrete systemwithweak and essentialnonlinearity are studied and the results are compared.

2. Formulation

The general form of a nonlinear dynamic system with 𝑁

degrees of freedom (DOFs) is shown below:

Mq + Cq + Kq + fnl = f𝐸, (1)

where q is the field deformation vector,M,C, andK aremass,damping, and stiffness matrices, fnl is nonlinear force, andf𝐸is periodic excitation force. q, fnl, and f

𝐸are function of

time which time argument, 𝑡, is dropped for convenience. Ifthe steady-state solution of above equation with the period of𝑇 = 2𝜋/𝜔 is requested, the deformation field can be expandedwith the aid of HBM in the form of

q =𝑚

∑

𝑗=0

(𝜓𝑗cos (𝑗𝜔𝑡) + 𝜑

𝑗sin (𝑗𝜔𝑡)) , (2)

where 𝜓𝑗and 𝜑

𝑗are vectors of vibration amplitude at the

frequency of 𝑗𝜔. By substituting (2) into (1) and usingGalerkin method, the equation of motion is decoupled in theform below:

−𝑗2𝜔2M𝜓𝑗+𝑗𝜔C𝜑

𝑗+K𝜓𝑗+ fnl,𝑐𝑗=f𝐸,𝑐𝑗 ,

− 𝑗2𝜔2M𝜑𝑗−𝑗𝜔C𝜓

𝑗+K𝜑𝑗+fnl,𝑠𝑗=f𝐸,𝑠𝑗 ,

𝑗 = 0, 1, 2, . . . , 𝑚,

{f𝑋,𝑐𝑗,f𝑋,𝑠𝑗

} = (1 + sgn (𝑗))

×𝜔

2𝜋∫

2𝜋/𝜔

0

{f𝑋cos (𝑗𝜔𝑡) , f

𝑋sin (𝑗𝜔𝑡)} 𝑑𝑡,

(3)

where 𝑋 is replaced by arbitrary subscripts of force vec-tors. Equation (3) expresses one static equilibrium and 2𝑚dynamic equations. By using imaginary unit 𝜄 = √−1, (3) canbe put together in the form

(−𝑗2𝜔2M−𝜄𝑗𝜔C+K)𝜒

𝑗+ fnl𝑗=f𝐸𝑗

𝑗 = 0, 1, 2, . . . , 𝑚,

(4)

where

𝜒𝑗= 𝜓𝑗+ 𝜄𝜑𝑗,

f𝑋𝑗

= f𝑋,𝑐𝑗+ 𝜄f𝑋,𝑠𝑗

.

(5)

In the case of localized nonlinearity, it is possible to writenonlinear force vector in the form

f𝑇nl = {𝑓nl1, . . . , 𝑓nl𝑝, 0, 0, . . . , 0} ,

fnl = {fnl𝛼0 } .

(6)

Furthermore, the deformation field can be divided intotwo vectors with subscripts 𝛼 and 𝛽, which stand for activeor master coordinates and deleted or slave coordinates,respectively. Active coordinates are experiencing nonlinearforces and deleted coordinates are only subjected to linearforces. Also, f

𝐸vector and M, C, and K matrices can be

divided appropriately with subscripts 𝛼 and 𝛽. Therefore, themotion equation in (4) can be written in the form

(−𝑗2𝜔2[M𝛼𝛼

M𝛼𝛽

M𝛽𝛼

M𝛽𝛽

]−𝜄𝑗𝜔 [C𝛼𝛼

C𝛼𝛽

C𝛽𝛼

C𝛽𝛽

]

+ [K𝛼𝛼

K𝛼𝛽

K𝛽𝛼

K𝛽𝛽

]){

𝜒𝛼𝑗

𝜒𝛽𝑗

} + {fnl𝛼0 }

={f𝐸𝛼

f𝐸𝛽

} , 𝑗 = 0, 1, 2, . . . , 𝑚.

(7)

Shock and Vibration 3

k1 k2 k3 k4 k5 k6 k7 k8 k9 k10 k11 k12

c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 c11 c12m1 m2 m3 m4 m5 m6 m7 m8 m9 m10 m11

knl8 knl9knl10 knl11

f8

knl12

Figure 1: The discrete system model.

1st harmonic3rd harmonic

5th harmonic7th harmonic

0

2

4

6

8

10

12

Peak

ampl

itude

1 1.1 1.2 1.3

𝜔∗

𝜔∗ = 1.21

(a)

1st harmonic3rd harmonic

5th harmonic7th harmonic

0

1

2

3

4

5

6

Peak

ampl

itude

1 1.1 1.2 1.3 1.4 1.5 1.6

𝜔∗

𝜔∗ = 1.36

(b)

Figure 2: The response peak amplitudes of each harmonic at 8th DOF: (a) weakly nonlinear system; (b) essentially nonlinear system.

In above equation, one can eliminate 𝜒𝛽𝑗

from the firstrow with the aid of the second row; this will result in exactreduced form of motion equation

(−𝑗2𝜔2 M𝛼𝛼−𝜄𝑗𝜔C

𝛼𝛼+K𝛼𝛼−K𝑀𝑗)𝜒𝛼𝑗+ fnl𝛼 𝑗

= f𝐸𝛼𝑗−f𝐸𝑇𝛽𝑗

,

A𝑗= (−𝑗

2𝜔2M𝛼𝛽−𝜄𝑗𝜔C

𝛼𝛽+K𝛼𝛽)

× (−𝑗2𝜔2M𝛽𝛽−𝜄𝑗𝜔C

𝛽𝛽+K𝛽𝛽)−1

K𝑀𝑗

= A𝑗(−𝑗2𝜔2M𝛽𝛼−𝜄𝑗𝜔C

𝛽𝛼+K𝛽𝛼) ,

f𝐸𝑇𝛽𝑗

=A𝑗f𝐸𝛽𝑗.

(8)

The deformation field vectors, 𝜒𝛼𝑗, can be expanded by

one of its basis. Orthogonal bases which consist of arrangedvectors with minimum energy required for excitation will besuitable bases to approximate the deformation field vectorsand reduce the size of condense system. To find such bases,

K𝑀𝑗

can be considered as a constant stiffness matrix, whosedependence on frequency, 𝑗𝜔, is neglected; then

(−𝜆2

𝑗𝑘M𝛼𝛼−𝜄𝜆𝑗𝑘C𝛼𝛼+K𝛼𝛼−K𝑀𝑗)𝜙𝑗𝑘= 0 (9)

is the eigenvalue problem of the linearized condensed system,where 𝜙

𝑗𝑘is the 𝑘th mass normalized, maximum real right

eigenvector and 𝜆𝑗𝑘is its companion eigenvalue. The 𝜙

𝑗𝑘are

orthogonal basis of 𝜒𝛼𝑗

and by increasing the 𝑘 index therequired energy for excitation 𝜙

𝑗𝑘in linearized condensed

system will grow. Considering that the motion of the non-linear condensed system is close to linearized system, thedeformation field in (8) can be approximated by few vectors ofthese orthogonal bases. Therefore, the deformation field canbe written in the form

𝜒𝛼𝑗≅ Φ𝑗(𝜂𝑐𝑗+ 𝜄𝜂𝑠𝑗) , (10)

where 𝜂𝑐𝑗and 𝜂

𝑠𝑗are unknown real vectors and Φ

𝑗is the

modal matrix in which collaborated mode shapes, 𝜙𝑗𝑘, are

arranged in column fashion. After substituting (10) into (8)

4 Shock and Vibration

10th DOF HBM11th DOF HBM7th DOF proposed method8th DOF proposed method9th DOF proposed method10th DOF proposed method11th DOF proposed method

0

5

10

15

0

7th DOF HBM8th DOF HBM9th DOF HBM

7th DOF numerical method8th DOF numerical method9th DOF numerical method10th DOF numerical method11th DOF numerical method

Disp

lace

men

t

Time

−5

−10

−15T/4 T/2 3T/4 T

(a)

0

2

4

6

8

Disp

lace

men

t

−2

−4

−6

−80

TimeT/4 T/2 3T/4 T

10th DOF HBM11th DOF HBM7th DOF proposed method8th DOF proposed method9th DOF proposed method10th DOF proposed method11th DOF proposed method

7th DOF HBM8th DOF HBM9th DOF HBM

7th DOF numerical method8th DOF numerical method9th DOF numerical method10th DOF numerical method11th DOF numerical method

(b)

Figure 3: Time response of nonlinear systems at their resonance: (a) weakly nonlinear system 𝜔∗= 1.21; (b) essentially nonlinear system

𝜔∗= 1.36.

and using Galerkinmethod themotion equation will becomein the form

Θ𝑗(𝜂𝑐𝑗+ 𝜄𝜂𝑠𝑗) + ⟨fnl𝛼 𝑗,Φ𝑗⟩=⟨f𝐸𝛼 𝑗−f𝐸𝑇𝛽𝑗 ,Φ𝑗⟩ ,

Θ𝑗= ⟨(−𝑗

2𝜔2M𝛼𝛼−𝜄𝑗𝜔C

𝛼𝛼+K𝛼𝛼−K𝑀𝑗)Φ𝑗,Φ𝑗⟩ ,

⟨a, b⟩ = ba,

(11)

where ∗ is transpose operator. The real and imaginary partof above equation will result in two different nonlinearequations. Therefore,

[Re (Θ

𝑗) − Im (Θ

𝑗)

Im (Θ𝑗) Re (Θ

𝑗)]{𝜂𝑐𝑗

𝜂𝑠𝑗

} +{

{

{

Re (⟨fnl𝛼𝑗 ,Φ𝑗⟩)

Im (⟨fnl𝛼𝑗 ,Φ𝑗⟩)}

}

}

={

{

{

Re (⟨f𝐸𝛼𝑗−f𝐸𝑇𝛽𝑗

,Φ𝑗⟩)

Im (⟨f𝐸𝛼𝑗−f𝐸𝑇𝛽𝑗

,Φ𝑗⟩)

}

}

}

.

(12)

By solving (12) with an arbitrarymethod the final solutionof motion will be achieved.

3. Numerical Results

In what follows, to prove the power of the proposed reductiontechnique discrete vibrating systems are studied with weakand essential nonlinearities of the cubic form. These systems

have 11 DOFs in which 5 of its DOFs are connected togetherwith nonlinear stiffness (Figure 1).

The steady-state responses of the system under periodicexcitation are determined by the numerical method, HBM,and proposed method. The resultant nonlinear algebraicequations of HBM and proposed method are solved byNewton-Raphson algorithm. The computational complexityof thismethod at each step can be approximated by the squareof the number of equations.

The motion of these systems is governed by

𝑚��𝑖+ 𝑐 (2��

𝑖− ��𝑖+1

− ��𝑖−1

) + 𝑘 (2𝑥𝑖− 𝑥𝑖+1

− 𝑥𝑖−1)

+ 𝑘nl𝑖(𝑥𝑖 − 𝑥𝑖−1)3+ 𝑘nl𝑖+1(𝑥𝑖 − 𝑥𝑖+1)

3= 𝑓𝑖,

(13)

where 𝑖 = 1, 2, . . . , 11,𝑚 = 1, 𝑐 = 1/10, 𝑘 = 1, and

𝑓𝑖= {

0 𝑖 = 8

cos (𝜔𝑡) 𝑖 = 8.(14)

For the case of weak and essential nonlinearities 𝑘nl𝑖 is takenas

{

{

{

0 𝑖 < 7

1

5𝑖 ≥ 7,

{0 𝑖 < 7

2 𝑖 ≥ 7,(15)

respectively. 𝜔1= 0.26Hz, real part of the first resonance

frequency of linearized system, is used to normalize theexcitation frequency 𝜔∗ = 𝜔/𝜔

1. Peak amplitudes of each

Shock and Vibration 5

harmonic at 8th DOF are shown in Figure 2 for differentfrequencies.

The steady-state responses of the systems at their reso-nances are approximated by first 7 harmonies.

Thenonlinear systemswith 11DOFs are reduced to 5mas-ter DOFs and solved using HBM, which causes around 80%decrease in computational cost. By applying the proposedmethod to the condensed system and expanding the responseusing its first three mode shapes, another 64% computationalcost reduction is achieved due to reduction of systems sizefrom 5 DOFs to 3 DOFs. The results are shown in Figure 3.

In both cases, there are negligible differences betweennumerical method and the HBMdue to harmonic truncationorder; therefore, this error is present in the proposed methodresults.

Compared to the results of the HBM and numericalmethod, phase of the DOFs vibration computed by theproposed method is accurate in both nonlinear systems.However, the difference between the calculated amplitudeswith the proposed and other methods is increased withgrowing the nonlinearity, but the captured response is stillacceptable. These differences are due to reduction of exactcondensed system size, from 5 DOFs to 3 DOFs, which leadsto a stiffer system.

4. Conclusion

In this paper, a new condensation technique is suggested fordiscrete systems by the combination of exact condensationtechniques based on HBM and modal expansion techniqueto reduce the computational cost and preserve the accuracy.The proposed method is used to reduce the size of two non-linear systems with weak and essential nonlinearity. At last,the predicted periodic responses are compared with othermethodologies results. The correlation between the results ofthe proposed method and other techniques is appropriate.Using the suggested technique, the reduced form of thesystem is independent of system nonlinearities; therefore,this technique has a great potential for the identification ofsystem nonlinearities. The main drawback of the suggestedtechnique is its frequency dependency.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

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