research article analytical solution for free vibration...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 470927 7 pageshttpdxdoiorg1011552013470927

Research ArticleAnalytical Solution for Free Vibration Analysis of Beam onElastic Foundation with Different Support Conditions

Baki Ozturk1 and Safa Bozkurt Coskun2

1 Department of Civil Engineering Faculty of Engineering Hacettepe University 06800 Ankara Turkey2Department of Civil Engineering Faculty of Engineering Kocaeli University 41380 Kocaeli Turkey

Correspondence should be addressed to Baki Ozturk bakiozturk1yahoocom

Received 15 December 2012 Accepted 17 April 2013

Academic Editor Mehmet Atay

Copyright copy 2013 B Ozturk and S B Coskun This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Analytical solutions for free vibration analyses of a beam on elastic foundation are obtained for different support conditions Theanalytical solutions are applied on three different axially loaded cases namely (1) one end clamped the other end simply supported(2) both ends clamped and (3) both ends simply supported cases Analytical solutions and frequency factors are evaluated fordifferent ratios of axial load N acting on the beam to Euler buckling load 119873

119903 The analytical solutions give results which are in

excellent agreement with the variational iteration method (VIM) and homotopy perturbation method (HPM) results available inthe literature for the particular problem considering all the cases provided in this study and the differential transform method(DTM) results available in the literature for the clamped-pinned case

1 Introduction

The free vibration equation of an axially loaded beam on elas-tic foundation is a fourth-order partial differential equationFor this particular engineering problem the analytical solu-tions will be implemented in this study Although the gov-erning equation seems to be a linear one finding the eigen-values for the free vibration analysis is still challenging Onemay not simply obtain the eigenvalues sequentially and theircorresponding eigen vectors even with a software

Free vibration equation of the beam on partially elasticfoundation including only bending moment effect was ana-lytically solved [1] while the eigenvalues for free vibration ofbeam-column systems on elastic foundation were obtainedusing a numerical approach [2] The separation of variablestechnique was used to obtain the free vibration circular fre-quencies of piles partially embedded in soils [3] In additiondifferential transform method (DTM) has been proposed tosolve eigenvalue problems for free and transverse vibrationproblems of a rotating twisted Timoshenko beam under axialloading [4 5] Furthermore the DTM was also used to findthe nondimensional natural frequencies of tapered cantilever

Bernoulli-Euler beam [6] Free vibration equations for oneend clamped and other end simply supported beam on elasticfoundation were solved by using the DTM for various axialloads acting on the beam [7]

Meanwhile both the variational iteration method (VIM)and homotopy perturbation method (HPM) were used tosolve the free vibration equations of beam on elastic founda-tion for support conditions of one end clamped and otherend simply supported both ends clamped and both endssimply supported considering various case studies [8ndash11]Thebeam on elastic foundation was investigated for these threedifferent support conditions considering various119873

119903values

Recently there have been also other studies which arehelpful to better understand dynamic behavior of both infi-nite beams resting on elastic foundation [12] and tapered col-umnwith pinned ends embedded inWinkler-Pasternak elas-tic foundation [13] In this study the analytical solutions andanalytical results for free vibration analysis of beam on elasticfoundation are providedThe analytical results are in excellentagreement with the results of the particular problem solvedusing the VIMmethod and the HPMmethod available in theliterature [8ndash10]

2 Mathematical Problems in Engineering

119873119873

119871119910

119909

119862119904119910(119909 119905) Elastic soil

(a)

119862119904119910(119909 119905)

119879(119909 119905)

119872(119909 119905)

119889119909

119873

119873

119872(119909 119905) +120597119872(119909 119905)

120597119909119889119909

119879(119909 119905) +120597119872(119909 119905)

120597119909119889119909

120597(119909 119905)

120597119909119889119909

120574

(b)

Figure 1 (a) beam on elastic foundation and (b) internal forces and deformations of the beam on elastic foundation

2 Problem Formulation

Beam on elastic foundation and internal forces and deforma-tions of differential segment of the beamhaving a length of dxare depicted in Figures 1(a) and 1(b) respectively The elasticfoundation is idealized by Winkler model Hence the rela-tionship between displacement function 119910(119909 119905) of the beamon elastic foundation and the distributed force 119902(119909 119905) appliedon the elastic foundation beneath the beam can be written by119902(119909 119905) = 119862

119904119910(119909 119905) In this equation119862

119904= 119862119900119887 where119862

119900is the

modulus of subgrade reaction while 119887 is the beam widthThe equilibrium equations of the internal forces acting

on differential beam segment dimensionless parameter 119911instead of position variable119909with 0 le 119911 le 119909119871 andneglectingthe second-order terms are used in order to define themotionequation of the beam on elastic foundation [3]

120601119894V1(119911) + [120587

2

119873119903+

(1198981205962

minus 119862119878) 1198961198712

119860119866]120601119894119894

1(119911)

+

(119862119878minus 1198981205962

) 1198714

1198641198681206011(119911) sin (120596119905 + 120579) = 0

(1)

In (1) 1206011(z) is dimensionless displacement function of

the beam considering axial and shear forces t is the timevariable 120579 is the phase angle119873

119903= 1198731198712

(1205872

119864119868) is the ratio ofaxial load119873 acting on the beam to Euler buckling load 119898 isthe distributed mass of the beam 120596 is the beam circularfrequency 119896 is the shape factor for the shape of the beamsection considered L is the beam length A G E and I arecross-section area shear modulus elastic modulus andmoment of inertia of the beam and 120601

119894119894

1(119911) = 119889

2

1206011(119911)119889119905

2120601119894V1(119911) = 119889

4

1206011(119911)119889119905

4 respectively If the axial and shear forceeffects are neglected the dimensionless equation of motionfor the beam on elastic foundation becomes as follows [14]

120601119894V2(119911) +

(119862119878minus 1198981205962

) 1198714

1198641198681206012(119911) sin (120596119905 + 120579) = 0 (2)

In (2) 1206012(119911) is the dimensionless displacement function

of the beamneglecting axial and shear forcesDivision of bothsides of (1) and (2) by sin(120596119905+120579) gives the following equations

120601119894V1(119911) + [120587

2

119873119903+

(1198981205962

minus 119862119878) 1198961198712

119860119866]120601119894119894

1(119911)

+

(119862119878minus 1198981205962

) 1198714

1198641198681206011(119911) = 0

(3)

120601119894V2(119911) +

(119862119878minus 1198981205962

) 1198714

1198641198681206012(119911) = 0 (4)

3 Analytical Solutions of the Problem

The equation of motion in (3) can be rearranged as follows

120601119894V1(119911) + [120587

2

119873119903minus

119864119868119896

1198601198661198712(120582 minus 120574

4

)] 12060110158401015840

1(119911)

+ (120582 minus 1205744

) 1206011(119911) = 0

(5)

where 120582 is relative stiffness 120582 = 1198621198781198714

119864119868 and 120574 is frequencyfactor 120574 = 4

radic11989812059621198714119864119868In order to simplify the symbolic representation of solu-

tion process (5) is rewritten as

120601119894V1(119911) + 120585

112060110158401015840

1(119911) + 120585

21206011(119911) = 0 (6)

where

1205851= 1205872

119873119903minus

119864119868119896

1198601198661198712(120582 minus 120574

4

)

1205852= 120582 minus 120574

4

(7)

Assuming a solution of the form of 119890119903119911 characteristicequation for (6) becomes

1199034

+ 12058511199032

+ 1205852= 0 (8)

Mathematical Problems in Engineering 3

119873119873

119871

119862119904

(a)

119873119873

119871

119862119904

(b)

119873119873

119871

119862119904

(c)

Figure 2 (a) CS beam (b) CC beam and (c) SS beam

Equation (8) is a quadratic equation in terms of 1199032 and itssolution is

(1199032

)12

= minus1205851

2plusmn radic(

1205851

2)

2

minus 1205852 (9)

By taking the square root of (9) roots of (8) would beobtained as follows

1199041= radic(1199032)

1

1199042= minus radic(1199032)

1

1199043= radic(1199032)

2

1199044= minus radic(1199032)

2

(10)

Hence the solution finally takes the following form

1206011(119911) = 119862

11198901199041119911

+ 11986221198901199042119911

+ 11986231198901199043119911

+ 11986241198901199044119911

(11)

Coefficients 1198621 1198622 1198623 and 119862

4would be obtained by

inserting the boundary conditions for the beam In this studythree different support configurations for the beam will beconsidered as shown in Figure 2 These are SS for SimplySupported CC for Clamped-Clamped and CS for Clamped-Simply Supported beams The boundary conditions related tothese supports are given below

For a simple support

1206011

1003816100381610038161003816119911= 0 120601

10158401015840

1

10038161003816100381610038161003816119911= 0 (12)

For a clamped support

1206011

1003816100381610038161003816119911= 0 120601

1015840

1

10038161003816100381610038161003816119911= 0 (13)

Inserting the boundary conditions given in (12)-(13) into(11) for SS CC andCSbeams a nonlinear systemof equationsin the following form would be produced

[119870 (120574)] 119862 = 0 (14)

where 119862119879 = 1198621119862211986231198624 and includes the coefficients

of (11) Coefficient matrix [119870(120574)] is dependent on frequencyfactor 120574 and equating the determinant to zero gives thefrequency factors for vibration modes of the correspondingbeam

det [119870 (120574)] = 0 (15)

Positive real roots of this equation are free vibrationfrequencies of the beams on elastic foundations shown inFigure 2

4 Numerical Study

Analytical solution for free vibration of a one end clampedand one end pinned beam on elastic foundation was previ-ously given by Catal [7] with additional analyses by DTMHowever Catal [7] gives analytical solutions for only CSbeam and in this study it is observed that these results fail tocoincide with the exact results

Previously both the variational iteration method (VIM)and homotopy perturbation method (HPM) were used tosolve the free vibration equations of beam on elastic foun-dation for support conditions of one end clamped and otherend simply supported both ends clamped and both ends sim-ply supported considering various case studies [8ndash10] In thisstudy analytical solutions for three cases are provided whichare shown in Figure 2 Each case was previously analyzed bythe use of both VIM and HPMwith the previously explainedprocedure [8ndash10] Numerical values in this study are chosenas the same used in Catal [7] Hence an IPB 500 steel profileresting on a Winkler foundation having a modulus of sub-grade reaction of 50000 kNm2 is considered Other numeri-cal values are as follows

119868 = 1072 lowast 10minus5m4 119860 = 239 lowast 10

minus2m2 119898 =

019 kN s2m 119896 = 3705119864 = 21 lowast 10

8 kNm2 119866 = 81 lowast 107 kNm2

Frequency factors 120574 = 4radic11989812059621198714119864119868 are calculated taking

bending moment shear and axial effects into considerationfor 119873

119903= 025 119873

119903= 05 and 119873

119903= 075 due to circular

frequencies of the beam


Table 1 Variation of frequency factor 120574 with relative stiffness 120582for a CS beam on elastic foundation (The analytical results of Catal[7] are provided below in bold characters)

1205741

1205742

1205743

119873119903= 025

120582 = 1281620011 397739569 482021485281574011 397658348 481918573

120582 = 10337458660 504748828 625843195337451696 504723072 625809765

120582 = 100405117292 609258233 783201825405107170 609248874 783184147

120582 = 1000587924217 732194493 934675005587935543 732217598 934584181

119873119903= 05

120582 = 1271457148 393377748 479443677271413994 393298292 479341412

120582 = 10326137662 499222355 622484874326133704 499199057 622451830

120582 = 100396066525 602920759 779040674396064973 602918911 779025650

120582 = 1000584614526 726848984 930219043584624672 726853895 930226231

119873119903= 075

120582 = 1259996785 388867219 476823873259956455 388787082 476721819

120582 = 10313469732 493509845 619072130313456678 493488312 619038296

120582 = 100386312198 596380562 774812917386301875 596338463 774796867

120582 = 1000581232280 721383882 925698856581241941 721407604 925701332

Table 2 Variation of frequency factor 120574 with relative stiffness 120582for a CC beam on elastic foundation

1205741

1205742

1205743

119873119903= 025

120582 = 1 339742898 432815705 516088024120582 = 10 405721417 552373153 668428571120582 = 100 466987707 668497843 836140368120582 = 1000 615541347 787417668 994633264

119873119903= 05

120582 = 1 333994125 429375653 514012431120582 = 10 399112373 548024811 665690121120582 = 100 460834095 663472888 832680496120582 = 1000 612479804 782851855 990820057

119873119903= 075

120582 = 1 327929840 425850809 511911832120582 = 10 392148631 543569919 662918805120582 = 100 454408497 658329012 829178517120582 = 1000 609362560 778201148 986962997

Variation of frequency factor 120574 is tabulated in Tables 1ndash3Table 1 includes comparison of analytical results of this study

Table 3 Variation of frequency factor 120574 with relative stiffness 120582for an SS beam on elastic foundation

1205741

1205742

1205741

119873119903= 025

120582 = 1 224318643 357344623 448319716120582 = 10 271977348 453601512 582884027120582 = 100 356133061 549391862 729761546120582 = 1000 571760639 682177513 875859570

119873119903= 05

120582 = 1 203688119 351272374 445090640120582 = 10 251190763 446032723 578712097120582 = 100 344330860 541052448 724679924120582 = 1000 568672681 675973740 870619682

119873119903= 075

120582 = 1 173715809 344867815 441789714120582 = 10 223396379 438057743 574447900120582 = 100 331172675 532308548 719489096120582 = 1000 565533583 669594299 865283438

for the CS beam with analytical results obtained by Catal [7]Tables 2 and 3 show the analytical results for both CC and SSbeams which are not solved by Catal [7]

The graphical presentations of analytical solutions areprovided in Figures 3 4 and 5

5 Conclusion

In this study the analytical results for free vibration analysisof beam on elastic foundation are provided for three differentaxially loaded cases which are namely one end clamped theother end simply supported (CS beam) both ends clamped(CC beam) and both ends simply supported (SS beam) casesIn addition the analytical solution and the frequency factorsevaluated for different ratios of axial load 119873 acting on thebeam to Euler buckling load119873

119903are obtained In Figures 3 4

and 5 it is observed that for the same values of 120582 the largestvalues of 120574 are obtained for both ends clamped (CC) beamwhile the lowest values of 120574 are obtained for both ends pinned(SS) beam

For the CS Beam it is observed that analytical solutionsgiven by Catal [7] fail to coincide with the exact results pro-vided in this study Previous studies [8ndash10] employing boththe variational iteration method (VIM) and homotopy per-turbation method (HPM) to solve the free vibration equa-tions of beam on elastic foundation in this study converge tothe same results obtained in this study The reason for thatis the governing equation is a linear equation in nature andas expected analytical approximate methods converge to theexact solution In the study by Catal [7] the DTM convergesthe analytical results provided in the same study Presumablythe difference is due to the fact that Catal [7] used the lengthof the beam L in three-digit accuracy while in this study theexact values of length L are used in all calculations con-ducted


0123456789

10

1 10 100 1000120582

120574

(a)

0123456789

10

1 10 100 1000120582

120574

(b)

0123456789

10

1 10 100 1000

Mode 1Mode 2Mode 3

120582

120574

(c)

Figure 3 Variation of frequency factor 120574 with relative stiffness 120582 for a CS beam on elastic foundation (a) for119873119903= 025 (b) for119873

119903= 05

and (c) for119873119903= 075

0123456789

10

1 10 100 1000120582

120574

(a)

0123456789

10

1 10 1000100120582

120574

(b)

0123456789

10

1 10 100 1000

Mode 1Mode 2Mode 3

120582

120574

(c)

Figure 4 Variation of frequency factor 120574 with relative stiffness 120582 for a CC beam on elastic foundation (a) for119873119903= 025 (b) for119873

119903= 05

and (c) for119873119903= 075


0123456789

10

1 10 100 1000120582

120574

(a)

0123456789

10

1 10 100 1000120582

120574

(b)

0123456789

10

1 10 100 1000

Mode 1Mode 2Mode 3

120582

120574

(c)

Figure 5 Variation of frequency factor 120574 with relative stiffness 120582 for an SS beam on elastic foundation (a) for119873119903= 025 (b) for119873

119903= 05

and (c) for119873119903= 075

Accordingly this study provides the analytical results forfree vibration analysis of beam on elastic foundation for threedifferent axially loaded cases which are namely one endclamped the other end simply supported (CS beam) bothends clamped (CC beam) and both ends simply supported(SS beam) cases

References

[1] P F Doyle and M N Pavlovic ldquoVibration of beams on partialelastic foundationsrdquo Earthquake Engineering and StructuralDynamics vol 10 no 5 pp 663ndash674 1982

[2] H H West and M Mafi ldquoEigenvalues for beam-columns onelastic supportsrdquo Journal of Structural Engineering vol 110 no6 pp 1305ndash1320 1984

[3] H H Catal ldquoFree vibration of partially supported piles with theeffects of bending moment axial and shear forcerdquo EngineeringStructures vol 24 no 12 pp 1615ndash1622 2002

[4] C-K Chen and S-H Ho ldquoApplication of differential trans-formation to eigenvalue problemsrdquo Applied Mathematics andComputation vol 79 no 2-3 pp 173ndash188 1996

[5] C K Chen and S H Ho ldquoTransverse vibration of a rotatingtwisted Timoshenko beams under axial loading using differen-tial transformrdquo International Journal ofMechanical Sciences vol41 no 11 pp 1339ndash1356 1999

[6] O Ozdemir and M O Kaya ldquoFlapwise bending vibration anal-ysis of a rotating tapered cantilever Bernoulli-Euler beam bydifferential transformmethodrdquo Journal of Sound and Vibrationvol 289 no 1-2 pp 413ndash420 2006

[7] S Catal ldquoSolution of free vibration equations of beam onelastic soil by using differential transform methodrdquo AppliedMathematical Modelling vol 32 no 9 pp 1744ndash1757 2008

[8] B Ozturk ldquoFree vibration analysis of beam on elastic founda-tion by the variational iteration methodrdquo International Journalof Nonlinear Sciences and Numerical Simulation vol 10 no 10pp 1255ndash1262 2009

[9] B Ozturk and S B Coskun ldquoThe homotopy perturbationmethod for free vibration analysis of beamon elastic foundationStructural Engineering and Mechanicsrdquo An International Jour-nal vol 37 no 4 2011

[10] B Ozturk S B Coskun M Z Koc andM T Atay ldquoHomotopyperturbation method for free vibration analysis of beams onelastic foundationsrdquo IOP Conference Series vol 10 no 1 2010

[11] S B Coskun M T Atay and B Ozturk ldquoTransverse vibrationanalysis of euler-bernoulli beams using analytical approximatetechniquesrdquo inAdvances in Vibration Analysis Research chapter1 pp 1ndash22 InTech Vienna Austria 2011

[12] I G Raftoyiannis T P Avraam andG TMichaltsos ldquoDynamicbehavior of infinite beams resting on elastic foundationunder the action of moving loads Structural Engineering andMechanicsrdquo An International Journal vol 35 no 3 2010


[13] O Civalek and B Ozturk ldquoFree vibration analysis of taperedbeam-column with pinned ends embedded in Winkler-Paster-nak elastic foundationrdquo Geomechanics and Engineering vol 2no 1 pp 45ndash56 2010

[14] J Tuma and F Cheng Theory and Problems of Dynamic Struc-tural Analysis Schaumrsquos Outline Series McGraw-Hill NewYork NY USA 1983

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Stochastic AnalysisInternational Journal of


119873119873

119871119910

119909

119862119904119910(119909 119905) Elastic soil

(a)

119862119904119910(119909 119905)

119879(119909 119905)

119872(119909 119905)

119889119909

119873

119873

119872(119909 119905) +120597119872(119909 119905)

120597119909119889119909

119879(119909 119905) +120597119872(119909 119905)

120597119909119889119909

120597(119909 119905)

120597119909119889119909

120574

(b)

Figure 1 (a) beam on elastic foundation and (b) internal forces and deformations of the beam on elastic foundation

2 Problem Formulation

Beam on elastic foundation and internal forces and deforma-tions of differential segment of the beamhaving a length of dxare depicted in Figures 1(a) and 1(b) respectively The elasticfoundation is idealized by Winkler model Hence the rela-tionship between displacement function 119910(119909 119905) of the beamon elastic foundation and the distributed force 119902(119909 119905) appliedon the elastic foundation beneath the beam can be written by119902(119909 119905) = 119862

119904119910(119909 119905) In this equation119862

119904= 119862119900119887 where119862

119900is the

modulus of subgrade reaction while 119887 is the beam widthThe equilibrium equations of the internal forces acting

on differential beam segment dimensionless parameter 119911instead of position variable119909with 0 le 119911 le 119909119871 andneglectingthe second-order terms are used in order to define themotionequation of the beam on elastic foundation [3]

120601119894V1(119911) + [120587

2

119873119903+

(1198981205962

minus 119862119878) 1198961198712

119860119866]120601119894119894

1(119911)

+

(119862119878minus 1198981205962

) 1198714

1198641198681206011(119911) sin (120596119905 + 120579) = 0

(1)

In (1) 1206011(z) is dimensionless displacement function of

the beam considering axial and shear forces t is the timevariable 120579 is the phase angle119873

119903= 1198731198712

(1205872

119864119868) is the ratio ofaxial load119873 acting on the beam to Euler buckling load 119898 isthe distributed mass of the beam 120596 is the beam circularfrequency 119896 is the shape factor for the shape of the beamsection considered L is the beam length A G E and I arecross-section area shear modulus elastic modulus andmoment of inertia of the beam and 120601

119894119894

1(119911) = 119889

2

1206011(119911)119889119905

2120601119894V1(119911) = 119889

4

1206011(119911)119889119905

4 respectively If the axial and shear forceeffects are neglected the dimensionless equation of motionfor the beam on elastic foundation becomes as follows [14]

120601119894V2(119911) +

(119862119878minus 1198981205962

) 1198714

1198641198681206012(119911) sin (120596119905 + 120579) = 0 (2)

In (2) 1206012(119911) is the dimensionless displacement function

of the beamneglecting axial and shear forcesDivision of bothsides of (1) and (2) by sin(120596119905+120579) gives the following equations

120601119894V1(119911) + [120587

2

119873119903+

(1198981205962

minus 119862119878) 1198961198712

119860119866]120601119894119894

1(119911)

+

(119862119878minus 1198981205962

) 1198714

1198641198681206011(119911) = 0

(3)

120601119894V2(119911) +

(119862119878minus 1198981205962

) 1198714

1198641198681206012(119911) = 0 (4)

3 Analytical Solutions of the Problem

The equation of motion in (3) can be rearranged as follows

120601119894V1(119911) + [120587

2

119873119903minus

119864119868119896

1198601198661198712(120582 minus 120574

4

)] 12060110158401015840

1(119911)

+ (120582 minus 1205744

) 1206011(119911) = 0

(5)

where 120582 is relative stiffness 120582 = 1198621198781198714

119864119868 and 120574 is frequencyfactor 120574 = 4

radic11989812059621198714119864119868In order to simplify the symbolic representation of solu-

tion process (5) is rewritten as

120601119894V1(119911) + 120585

112060110158401015840

1(119911) + 120585

21206011(119911) = 0 (6)

where

1205851= 1205872

119873119903minus

119864119868119896

1198601198661198712(120582 minus 120574

4

)

1205852= 120582 minus 120574

4

(7)

Assuming a solution of the form of 119890119903119911 characteristicequation for (6) becomes

1199034

+ 12058511199032

+ 1205852= 0 (8)


119873119873

119871

119862119904

(a)

119873119873

119871

119862119904

(b)

119873119873

119871

119862119904

(c)



(1199032

)12

= minus1205851

2plusmn radic(

1205851

2)

2

minus 1205852 (9)


1199041= radic(1199032)

1

1199042= minus radic(1199032)

1

1199043= radic(1199032)

2

1199044= minus radic(1199032)

2

(10)


1206011(119911) = 119862

11198901199041119911

+ 11986221198901199042119911

+ 11986231198901199043119911

+ 11986241198901199044119911

(11)





1206011

1003816100381610038161003816119911= 0 120601

10158401015840

1

10038161003816100381610038161003816119911= 0 (12)


1206011

1003816100381610038161003816119911= 0 120601

1015840

1

10038161003816100381610038161003816119911= 0 (13)


[119870 (120574)] 119862 = 0 (14)



det [119870 (120574)] = 0 (15)


4 Numerical Study




minus2m2 119898 =

019 kN s2m 119896 = 3705119864 = 21 lowast 10

8 kNm2 119866 = 81 lowast 107 kNm2



119903= 025 119873

119903= 05 and 119873





1205741

1205742

1205743

119873119903= 025

120582 = 1281620011 397739569 482021485281574011 397658348 481918573

120582 = 10337458660 504748828 625843195337451696 504723072 625809765

120582 = 100405117292 609258233 783201825405107170 609248874 783184147

120582 = 1000587924217 732194493 934675005587935543 732217598 934584181

119873119903= 05

120582 = 1271457148 393377748 479443677271413994 393298292 479341412

120582 = 10326137662 499222355 622484874326133704 499199057 622451830

120582 = 100396066525 602920759 779040674396064973 602918911 779025650

120582 = 1000584614526 726848984 930219043584624672 726853895 930226231

119873119903= 075

120582 = 1259996785 388867219 476823873259956455 388787082 476721819

120582 = 10313469732 493509845 619072130313456678 493488312 619038296

120582 = 100386312198 596380562 774812917386301875 596338463 774796867

120582 = 1000581232280 721383882 925698856581241941 721407604 925701332


1205741

1205742

1205743

119873119903= 025

120582 = 1 339742898 432815705 516088024120582 = 10 405721417 552373153 668428571120582 = 100 466987707 668497843 836140368120582 = 1000 615541347 787417668 994633264

119873119903= 05

120582 = 1 333994125 429375653 514012431120582 = 10 399112373 548024811 665690121120582 = 100 460834095 663472888 832680496120582 = 1000 612479804 782851855 990820057

119873119903= 075

120582 = 1 327929840 425850809 511911832120582 = 10 392148631 543569919 662918805120582 = 100 454408497 658329012 829178517120582 = 1000 609362560 778201148 986962997



1205741

1205742

1205741

119873119903= 025

120582 = 1 224318643 357344623 448319716120582 = 10 271977348 453601512 582884027120582 = 100 356133061 549391862 729761546120582 = 1000 571760639 682177513 875859570

119873119903= 05

120582 = 1 203688119 351272374 445090640120582 = 10 251190763 446032723 578712097120582 = 100 344330860 541052448 724679924120582 = 1000 568672681 675973740 870619682

119873119903= 075

120582 = 1 173715809 344867815 441789714120582 = 10 223396379 438057743 574447900120582 = 100 331172675 532308548 719489096120582 = 1000 565533583 669594299 865283438



5 Conclusion






0123456789

10

1 10 100 1000120582

120574

(a)

0123456789

10

1 10 100 1000120582

120574

(b)

0123456789

10

1 10 100 1000

Mode 1Mode 2Mode 3

120582

120574

(c)


119903= 05

and (c) for119873119903= 075

0123456789

10

1 10 100 1000120582

120574

(a)

0123456789

10

1 10 1000100120582

120574

(b)

0123456789

10

1 10 100 1000

Mode 1Mode 2Mode 3

120582

120574

(c)


119903= 05

and (c) for119873119903= 075


0123456789

10

1 10 100 1000120582

120574

(a)

0123456789

10

1 10 100 1000120582

120574

(b)

0123456789

10

1 10 100 1000

Mode 1Mode 2Mode 3

120582

120574

(c)


119903= 05

and (c) for119873119903= 075


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119873119873

119871

119862119904

(a)

119873119873

119871

119862119904

(b)

119873119873

119871

119862119904

(c)



(1199032

)12

= minus1205851

2plusmn radic(

1205851

2)

2

minus 1205852 (9)


1199041= radic(1199032)

1

1199042= minus radic(1199032)

1

1199043= radic(1199032)

2

1199044= minus radic(1199032)

2

(10)


1206011(119911) = 119862

11198901199041119911

+ 11986221198901199042119911

+ 11986231198901199043119911

+ 11986241198901199044119911

(11)





1206011

1003816100381610038161003816119911= 0 120601

10158401015840

1

10038161003816100381610038161003816119911= 0 (12)


1206011

1003816100381610038161003816119911= 0 120601

1015840

1

10038161003816100381610038161003816119911= 0 (13)


[119870 (120574)] 119862 = 0 (14)



det [119870 (120574)] = 0 (15)


4 Numerical Study




minus2m2 119898 =

019 kN s2m 119896 = 3705119864 = 21 lowast 10

8 kNm2 119866 = 81 lowast 107 kNm2



119903= 025 119873

119903= 05 and 119873





1205741

1205742

1205743

119873119903= 025

120582 = 1281620011 397739569 482021485281574011 397658348 481918573

120582 = 10337458660 504748828 625843195337451696 504723072 625809765

120582 = 100405117292 609258233 783201825405107170 609248874 783184147

120582 = 1000587924217 732194493 934675005587935543 732217598 934584181

119873119903= 05

120582 = 1271457148 393377748 479443677271413994 393298292 479341412

120582 = 10326137662 499222355 622484874326133704 499199057 622451830

120582 = 100396066525 602920759 779040674396064973 602918911 779025650

120582 = 1000584614526 726848984 930219043584624672 726853895 930226231

119873119903= 075

120582 = 1259996785 388867219 476823873259956455 388787082 476721819

120582 = 10313469732 493509845 619072130313456678 493488312 619038296

120582 = 100386312198 596380562 774812917386301875 596338463 774796867

120582 = 1000581232280 721383882 925698856581241941 721407604 925701332


1205741

1205742

1205743

119873119903= 025

120582 = 1 339742898 432815705 516088024120582 = 10 405721417 552373153 668428571120582 = 100 466987707 668497843 836140368120582 = 1000 615541347 787417668 994633264

119873119903= 05

120582 = 1 333994125 429375653 514012431120582 = 10 399112373 548024811 665690121120582 = 100 460834095 663472888 832680496120582 = 1000 612479804 782851855 990820057

119873119903= 075

120582 = 1 327929840 425850809 511911832120582 = 10 392148631 543569919 662918805120582 = 100 454408497 658329012 829178517120582 = 1000 609362560 778201148 986962997



1205741

1205742

1205741

119873119903= 025

120582 = 1 224318643 357344623 448319716120582 = 10 271977348 453601512 582884027120582 = 100 356133061 549391862 729761546120582 = 1000 571760639 682177513 875859570

119873119903= 05

120582 = 1 203688119 351272374 445090640120582 = 10 251190763 446032723 578712097120582 = 100 344330860 541052448 724679924120582 = 1000 568672681 675973740 870619682

119873119903= 075

120582 = 1 173715809 344867815 441789714120582 = 10 223396379 438057743 574447900120582 = 100 331172675 532308548 719489096120582 = 1000 565533583 669594299 865283438



5 Conclusion






0123456789

10

1 10 100 1000120582

120574

(a)

0123456789

10

1 10 100 1000120582

120574

(b)

0123456789

10

1 10 100 1000

Mode 1Mode 2Mode 3

120582

120574

(c)


119903= 05

and (c) for119873119903= 075

0123456789

10

1 10 100 1000120582

120574

(a)

0123456789

10

1 10 1000100120582

120574

(b)

0123456789

10

1 10 100 1000

Mode 1Mode 2Mode 3

120582

120574

(c)


119903= 05

and (c) for119873119903= 075


0123456789

10

1 10 100 1000120582

120574

(a)

0123456789

10

1 10 100 1000120582

120574

(b)

0123456789

10

1 10 100 1000

Mode 1Mode 2Mode 3

120582

120574

(c)


119903= 05

and (c) for119873119903= 075


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1205741

1205742

1205743

119873119903= 025

120582 = 1281620011 397739569 482021485281574011 397658348 481918573

120582 = 10337458660 504748828 625843195337451696 504723072 625809765

120582 = 100405117292 609258233 783201825405107170 609248874 783184147

120582 = 1000587924217 732194493 934675005587935543 732217598 934584181

119873119903= 05

120582 = 1271457148 393377748 479443677271413994 393298292 479341412

120582 = 10326137662 499222355 622484874326133704 499199057 622451830

120582 = 100396066525 602920759 779040674396064973 602918911 779025650

120582 = 1000584614526 726848984 930219043584624672 726853895 930226231

119873119903= 075

120582 = 1259996785 388867219 476823873259956455 388787082 476721819

120582 = 10313469732 493509845 619072130313456678 493488312 619038296

120582 = 100386312198 596380562 774812917386301875 596338463 774796867

120582 = 1000581232280 721383882 925698856581241941 721407604 925701332


1205741

1205742

1205743

119873119903= 025

120582 = 1 339742898 432815705 516088024120582 = 10 405721417 552373153 668428571120582 = 100 466987707 668497843 836140368120582 = 1000 615541347 787417668 994633264

119873119903= 05

120582 = 1 333994125 429375653 514012431120582 = 10 399112373 548024811 665690121120582 = 100 460834095 663472888 832680496120582 = 1000 612479804 782851855 990820057

119873119903= 075

120582 = 1 327929840 425850809 511911832120582 = 10 392148631 543569919 662918805120582 = 100 454408497 658329012 829178517120582 = 1000 609362560 778201148 986962997



1205741

1205742

1205741

119873119903= 025

120582 = 1 224318643 357344623 448319716120582 = 10 271977348 453601512 582884027120582 = 100 356133061 549391862 729761546120582 = 1000 571760639 682177513 875859570

119873119903= 05

120582 = 1 203688119 351272374 445090640120582 = 10 251190763 446032723 578712097120582 = 100 344330860 541052448 724679924120582 = 1000 568672681 675973740 870619682

119873119903= 075

120582 = 1 173715809 344867815 441789714120582 = 10 223396379 438057743 574447900120582 = 100 331172675 532308548 719489096120582 = 1000 565533583 669594299 865283438



5 Conclusion






0123456789

10

1 10 100 1000120582

120574

(a)

0123456789

10

1 10 100 1000120582

120574

(b)

0123456789

10

1 10 100 1000

Mode 1Mode 2Mode 3

120582

120574

(c)


119903= 05

and (c) for119873119903= 075

0123456789

10

1 10 100 1000120582

120574

(a)

0123456789

10

1 10 1000100120582

120574

(b)

0123456789

10

1 10 100 1000

Mode 1Mode 2Mode 3

120582

120574

(c)


119903= 05

and (c) for119873119903= 075


0123456789

10

1 10 100 1000120582

120574

(a)

0123456789

10

1 10 100 1000120582

120574

(b)

0123456789

10

1 10 100 1000

Mode 1Mode 2Mode 3

120582

120574

(c)


119903= 05

and (c) for119873119903= 075


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0123456789

10

1 10 100 1000120582

120574

(a)

0123456789

10

1 10 100 1000120582

120574

(b)

0123456789

10

1 10 100 1000

Mode 1Mode 2Mode 3

120582

120574

(c)


119903= 05

and (c) for119873119903= 075

0123456789

10

1 10 100 1000120582

120574

(a)

0123456789

10

1 10 1000100120582

120574

(b)

0123456789

10

1 10 100 1000

Mode 1Mode 2Mode 3

120582

120574

(c)


119903= 05

and (c) for119873119903= 075


0123456789

10

1 10 100 1000120582

120574

(a)

0123456789

10

1 10 100 1000120582

120574

(b)

0123456789

10

1 10 100 1000

Mode 1Mode 2Mode 3

120582

120574

(c)


119903= 05

and (c) for119873119903= 075


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0123456789

10

1 10 100 1000120582

120574

(a)

0123456789

10

1 10 100 1000120582

120574

(b)

0123456789

10

1 10 100 1000

Mode 1Mode 2Mode 3

120582

120574

(c)


119903= 05

and (c) for119873119903= 075


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article analytical solution for free vibration...

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