flexural vibration analysis of nonuniform double-beam

Research ArticleFlexural Vibration Analysis of Nonuniform Double-BeamSystem with General Boundary and Coupling Conditions

Lujun Chen12 Deshui Xu 3 Jingtao Du 3 and Chengwen Zhong 1

1School of Aeronautics Northwestern Polytechnical University Xirsquoan 710072 China2China Aerodynamics Research and Development Center Mianyang 621000 China3College of Power and Energy Engineering Harbin Engineering University Harbin 150001 China

Correspondence should be addressed to Deshui Xu xudeshui2010163com

Received 18 May 2018 Revised 1 August 2018 Accepted 26 August 2018 Published 1 October 2018

Academic Editor Pedro Museros

Copyright copy 2018 Lujun Chen 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

In this paper an analytical modeling approach for the flexural vibration analysis of the nonuniform double-beam system isproposed via an improved Fourier series method in which both types of translational and rotational springs are introduced toaccount for the mechanical coupling on the interface as well as boundary restraints Energy formulation is employed for thedynamic description of the coupling system With the aim to treat the varying thickness across the beam in a unified pattern therelevant variables are all expanded into Fourier series Supplementary terms with the smoothed characteristics are introduced tothe standard Fourier series for the construction of displacement admissible function for each beam In conjunction with theRayleighndashRitz procedure the transverse modal characteristics of nonuniform double-beam system can be obtained by solvinga standard eigenvalue problem Instead of solving the certain value of nonideal boundary conditions the continuous springstiffnesses of the boundary conditions are considered and the rotational restrains are introduced in the coupling beam interfaceNumerical results are then presented to demonstrate the reliability of the current model and study the influence of variousparameters such as taper ratio boundary and coupling strength on the free vibration characteristics with the emphasis put on therotational restraining coefficients on the beam interface is work can provide an efficient modeling framework for the vibrationcharacteristics study of the complex double-beam system especially with arbitrary varying thickness and coupling stiffness

1 Introduction

emultiple-beam system has been extensively studied due toits wide application in various branches such as mechanicaland aeronautical engineering A good understanding on itsdynamic characteristics will be of great importance for theefficient design as well as vibration control of such complexsystem For this reason a lot of research attention has beendevoted to the vibration behavior of multiple-beam structureby many researchers in the past decades

For these multiple-beam structures mechanical couplingbetween each beam component is usually taken into accountthrough the introduction of translational spring across thebeam interface [1] Seelig and Hoppmann [2] formulated thedifferential equation of motion and obtained the solution forvibration analysis of the n elastically connected parallel beams

Rao [3] solved the differential governing equation for theflexural vibration of elastically connected parallel bars basedon the Timoshenko beam theory in which the effects of rotaryinertia and shear deformation are considered Li and Hua [4]employed the dynamic stiffness method to analyze the freevibration characteristics of a three-beam system with theelastic spring and dashpot on the coupling interface More-over Winkler elastic layers are also accoutered for in thevibration analysis of coupling interface of the multiple-beamsystem [5ndash7] Deng et al [8] also studied the double func-tionally graded Timoshenko beam system resting onWinklerndashPasternak elastic foundation using Hamiltonrsquosprinciple

In many occasions the nonuniform beam componentwill be encountered with the background of optimal designin which a better or more suitable distribution of mass and

HindawiShock and VibrationVolume 2018 Article ID 5103174 8 pageshttpsdoiorg10115520185103174

strength than the uniform beam is of great desire usvibration analysis of the nonuniform beam has been studiedcontinuously [9ndash11] Abrate [12] presented a method totransform the motion equation of a nonuniform beam intoits uniform counterpart Lee and Lee [13] developeda transfer-matrix method to investigate the free vibrationcharacteristics of a tapered BernoullindashEuler beam Besselfunctions are widely used in dealing with the nonuniformbeam system with different boundary conditions Rosa andAuciello [14] solved the nonuniform beam-governingequation and the elastic boundary condition bysubstituting the four flexibility coefficients of the con-structed functions Auciello and Ercolano [15] investigatedthe transverse vibration of a beam for which one part wastapered and the remaining part is the uniform thicknessTorabi et al [16] derived an exact closed-form solution forthe free vibration of EulerndashBernoulli conical and taperedbeams with any number of attached masses which aredescribed by Diracrsquos delta function Furthermore Chenand Pan Liu [17] employed the Bessel function and thenumerical assembly method (NAM) to investigate the vi-bration characteristics of a tapered beam with multiplearbitrarily placed rotational dampers

In the current studies on the vibration analysis of themultiple-beam system most of them are devoted to theuniform ones and just little exception can be found in theliterature for the vibration studies of nonuniform multiple-beam structure For example Mabie and Rogers [18 19]obtained an accurate solution for determining the first fivefrequencies of the cantilevered double-tampered beamsystem Takahashi and Yoshioka [20] utilized the transfer-matrix approach to analyze the vibration and stability ofa double-cantilever beam in which only translational springis used for the description of coupling strength between twobeams From the above literature review it can be found thatthe current study mainly considered the classical boundaryconditions and just one type of restraining spring on thebeam interface

From the practical point of view elastic boundary re-straint and other types of mechanical coupling strengthshould be consideredere is a clear gap in the literature onthis aspect In analysis of the real system the nearest idealboundary conditions such as clamped free and simplysupport is selected for the modeling However small de-viations from ideal conditions in real systems indeed occurFor example a beam connected at the ends to rigid supportsby pins is modeled using simply supported boundaryconditions which require deflections and moments to bezero But the hole and pin assembly may have small gapsandor friction which is called the nonideal boundary In thispaper the spring stiffnesses of the boundary conditions aredifferent and continuous so the elastic boundary is usedhere which includes the aforementioned nonideal boundaryMoreover when the beam is under axial force or in axialmoving the minimal rotations are emerged and the rotationrestrains are important For the double beam system whenthe similar rotation movements happen rotational couplingspring is introduced to restrain the relevant displacementClaeys et al [21] presents a direct comparison of measured

and predicted nonlinear vibrations of a clamped-clampedsteel beam with nonideal boundary conditions and theresults show that the nonideal boundary of transverse springand rotational spring have significant effect for the beam anda correct estimation is necessary to simulate the realstructurersquos characteristic

In this work motivated by the current limitation inliterature an efficient modeling approach for the vibrationanalysis of the nonuniform double-beam system withgeneral elastic boundary condition is proposed in whichboth the translational and rotational restraining springs aretaken into account to describe the dynamic interactionbetween each beam interface Energy principle is formu-lated for the analysis of the system motion equation withthe nonuniform thickness variation expanded into theFourier series in a unified pattern e transverse dis-placement admissible function is constructed as the su-perposition of the standard Fourier series and the boundarysmoothed auxillary terms In conjunction with theRayleighndashRitz procedure all the modal parameters can bederived by solving an eigenvalue matrix Numerical ex-amples are then given to demonstrate the reliability andeffectiveness of the current model Finally some con-cluding remarks are made

2 Theoretical Formulations

21 Model Description Consider an elastically connectednonuniform double-beam system as illustrated in Figure 1e beam member with arbitrary boundary conditions iscoupled with other through the elastic restraining springs onthe coupling interfaces Such mechanical interaction isrepresented using two types of springs namely translationaland rotational one Any boundary condition can be easilyobtained by setting the relevant restraining spring stiff-nesses For the beam vibration there are two freedom de-grees at each field point namely translation and rotationen a full-coupling restraint should include the trans-lational and rotational spring distributions accordingly Inthis way the familiar Winkler-type of elastic interface can bereadily derived by setting the coefficients of rotationalsprings into zero

22 Double-Beam SystemDynamics and Its Solution For thecoupled beam structure as shown in Figure 1 it will bedescribed from the viewpoint of energy Although thedouble-beam system is considered here the current mod-eling framework is suitable for the general N beam systemWithout losing the generality the subsequent formulationwill be given for such N-beam coupling system Lagrangianfunction is used for the description of system dynamicbehavior which can be written as

L VminusT 1113944N

i1Vi + Vcoupling minus 1113944

N

i1Ti (1)

where L is the system Lagrangian V and T are the totalpotential energy and kinetic energy Vi is the potentialenergy associated with the ith beam member Vcoupling is

2 Shock and Vibration

the potential energy stored in the interface couplingsprings and Ti is the kinetic energy due to the ith vibratingbeam

For the ith beam member with elastically restrainededges its potential energy Vi is

Vi 12intL

0EiIi(x)

z2wi(x)zx2

[ ]2

dx +12Ki

T0w2i (x)

∣∣∣∣∣x0

+12Ki

R0zwi(x)zx

[ ]2∣∣∣∣∣∣∣∣∣x0

+12Ki

TLw2i (x)

∣∣∣∣∣xL

+12Ki

RLzwi(x)zx

[ ]2∣∣∣∣∣∣∣∣∣xL

(2)

where w(x) is the transverse vibration displacement eldfunction KT0 and KR0 are respectively the stiness co-ecients for the translational and rotational springs at theend x 0 and similar meaning can be deduced for the rightend of x L e subscript i means that this variable isassociated with the ith beammember Ii(x) is the moment ofinertia of the nonuniform beam

e total kinetic energy of the ith beam structure is

Ti 12intL

0ρiSi(x)

zwi(x t)zt

[ ]2

dx 12ω2 int

L

0ρiSi(x)w

2i (x) dx

(3)

where ω is the radian frequency and ρi and Si(x) are re-spectively the mass density and cross section area of the ithbeam member

e coupling potential energy between the interfaces canbe written as

Vcoupling 12intL

0kc wi minuswi+1( )2 +Kc

zwizxminuszwi+1zx

( )2

dx

(4)

in which kc and Kc are respectively the translational androtational coupling spring stinesses distributed across theinterface between the ith and i + 1st beam member

In this work in order to treat the cross-section areavariation in a most general unied pattern variables such asarbitrary inertia Ii(x) and cross-section area Si(x) associ-ated with the nonuniform thickness variation are expandedinto Fourier cosine series with the corresponding expansioncoecients dened as follows

Si(x) S0f1(x) S0c02+ suminfin

m1cm cos λmx

S0 suminfin

m0cm cos λmx

(5)

cm

1LintL

0f1(x)dx m 0

2LintL

0f1(x) cos

mπLxdx mne 0

(6)

and

Ii(x) I0f2(x) I0d02+ suminfin

m1dm cos λmx

I0 suminfin

m0dm cos λmx

(7)

dm

1LintL

0f2(x)dx m 0

2LintL

0f2(x) cos

mπLxdx mne 0

(8)

in which λm mπL and m is the Fourier series termFor various kinds of cross sections the dierence be-

tween them is merely the Fourier coecients and the itemswhich can be obtained easily through the Fourier trans-formation For the traditional uniform beam Si(x) andIi(x) will be constant across the beam length

Once the system Lagrangian is obtained the other thingis to construct the appropriate admissible function Dif-ferential continuity of the constructed function has signif-icant eect on the nal convergence and accuracy Here animproved Fourier series method is employed for this pur-pose in which the additional functions are introduced to thestandard Fourier series to remove all the discontinuitiesassociated with the spatial dierentiation of the displace-ment eld functions For each beam member its exuralvibrating displacement function is expanded as [22]

w(x) suminfin

n0an cos

nπLx( ) + b1ζ1(x)

+ b2ζ2(x) + b3ζ3(x) + b4ζ4(x)

(9)

in which

ζ1(x) 9L4π

sinπx2L( )minus

L

12πsin

3πx2L

( )

ζ2(x) minus9L4π

cosπx2L( )minus

L

12πcos

3πx2L

( )

ζ3(x) L3

π3 sinπx2L( )minus

L3

3π3sin

3πx2L

( )

ζ4(x) minusL3

π3cos

πx2L( )minus

L3

3π3cos

3πx2L

( )

(10)

x2

K1R0 K1RLK1TLK1T0

K2R0 K2RL

K2T0 K2TL

Beam 1

Kc

Beam 2

kc

x1

Figure 1 Nonuniform double-beam system with arbitraryboundary conditions in which both the translational and rotationalcoupling eects are taken into account

Shock and Vibration 3

It can be easily proven that the current constructedtrigonometric function can satisfy the displacement and itshigher-order differentiation continuity requirement in theinterval (0 L) It should be pointed out that the choice of thesupplementary functions is not unique while the appro-priate form will be helpful for simplifying the subsequentmathematical formulations

Substituting the admissible function Equation (9) intothe elastically connected double-beam system LagrangianEquations (1)ndash(4) minimizing it with respect to all theunknown Fourier series coefficients and truncating theFourier series into finite number n N one will obtain thesystem characteristic equation in the matrix form

Kminusω2M1113872 1113873A 0 (11)

where K and M are the stiffness and mass matrices for theelastically connected double-beam system respectively andA is the unknown Fourier series coefficient vector Bysolving such standard eigenvalue problem all the modalparameters can be easily obtained

3 Numerical Examples and Discussions

In this section the aforementioned modeling framework isprogrammed in the MATLAB environment Several nu-merical results of different kinds of cross sections withvarious boundary conditions will be presented to demon-strate the effectiveness and advantage of the proposedmodel As no research has been published about the freevibration of double-beam system with uniform cross sec-tions the results of the current method will be the firstcompared with those from other relevant literatures tovalidate the correctness In current solution framework theelastic boundary conditions are easily obtained by setting therestraining stiffness coefficient into various values accord-ingly Similarly variation of the arbitrary inertia Ii(x) andcross-section area Si(x) can be easily handled using the sameMATLAB program by just changing the mapping param-eters of the nonuniform profile in Fourier space In thefollowing analysis the nondimensional frequency parameterwill be used with their definitions as Ω ωL2

ρSEI1113968

31 Double-Beam System with Uniform Cross SectionHere by setting the relevant parameters to constant inEquations (5) and (7) several vibration results can be ob-tained for a uniform double-beam system which has beeninvestigated in some former studies With the comparisonpurpose modal parameters are kept the same as those usedin Ref [23] namely E1I1 4 times 106 Nm2 E2I2 2 times E1I1ρ1A1 100 kgm ρ2A2 2 times ρ1A1 L 10 m From thecomparison tabulated in Table 1 it can be observed that thecurrent results can agree well with those from other ap-proaches in which the parameters of beam 1 are used for thecalculation of nondimensional frequency Ω Comparing theresults of Tables 1 and 2 it can be found that the variation ofcoupling stiffness will not affect the odd-mode frequency inthe uniform double-beam system

32 Single Nonuniform Beam with Different Cross-SectionParameters In order to validate the proposed method forthe case of variable cross sections the coupling trans-lational spring kc and rotational spring Kc are set to zerowhich transfer the double-beam system to two single-beams with the same vibration characteristics e poly-nomial function of inertia Ii(x) and cross-section areaSi(x) is as follows

Si(x) S0 aminus bx

L1113874 1113875

r

(12)

and

Ii(x) I0 aminus bx

L1113874 1113875

r+2 (13)

where b is the taper ratio of the cross-section areaTable 3 shows the results of one single nonuniform beam

with clamped-free (C-F) boundary condition in which thedata taken form Ref [24] are also presented to validate thecorrectness of the present method in dealing with the dif-ferent cross sections With the increase of the taper ratio bthe frequency will raise which represents the decrease of thebeamrsquos stiffness

Moreover a single beamwith three-step changes in crosssection is analyzed e geometrical dimensions and ma-terial properties are LR 1m mR 1 kgm EIR 1Nm2μ1 10 μ2 08 μ3 065 μ4 025 L1 025LR L2 03LRL3 025LR and L4 02LR Table 4 lists the first four fre-quency parameters for various combinations of boundaryconditions e results can agree well with those from thereference approaches

33 Nonuniform Double-Beam System with Uniform CrossSection of Beam 2 e nonuniform double beam system isconsidered with C-F boundary condition for both beams inwhich the upper beamrsquos cross-section function is the same asdefined in Equations (12) and (13) in Section 32 e lowerbeamrsquos cross section is uniform with the same parameter ofthe upper beam at x 0 Other material parameters are thesame for the two beams Table 5 lists the results of differentcoupling stiffness boundaries When the coupling trans-lational spring kc and rotational spring Kc are both zero thefrequencies are from the two single beams in which the odd-mode results of the lower uniform beam is larger than theeven-mode results of the upper taper beam because thenonuniform cross sectionmakes the beamrsquos stiffness smallerAfter the coupling stiffness kc and Kc becomes bigger thefrequencies for both beams becomes larger As the value ofkc and Kc are both 107 the frequencies are similar as thesingle beam in which the value is bigger than odd- and even-mode results

e influence of different coupling springs kc and Kc onthe fundamental frequency of the double-beam system ispresented in Figure 2 which can illustrate that the influenceof rotational spring Kc is obvious than that of translationalspring kc which means the rotational spring cannot be ig-nored in the double-beam system


34 Nonuniform Double-Beam System with Elastic BoundaryCondition Finally the nonuniform double-beam system isconsidered in which both beams are tapered beams asdefined in Section 32 Here the boundary condition isclamped at x 0 and elastic at x L e influence of elasticboundary stiffness K2 (K2 KTL KRL) at x L of bothbeams and coupling spring KC (KC kc Kc) are plotted in

Figure 3 As discussed in Section 33 the rotational springcannot be ignored and the translational and rotationalsprings of the coupling spring are set at the same value FromFigure 3 when the elastic boundary stiffness increases thefrequencies will become bigger and there will be a sensitivearea in which the influence to the fundamental frequency isuniform Otherwise the fundamental frequencies will not

Table 2 e first six nondimensional frequency Ω for the uniform double-beams with different boundary conditions (kc 0Kc 1 times 105 Nm2)

Boundary conditions Nondimensional frequency Ωi

Beam 1 Beam 2 1 2 3 4 5 6

S-S S-S 98683 115923 394576 412919 887207 90593098649a 121168 394240 418523 885772 910371

C-C C-C 223008 233205 614048 628443 1201720 1218760223421a 236691 614904 633157 1202885 1222928

C-F C-F 35113 53243 219985 245854 615643 63893935148a 57683 220091 253841 615438 645943

S-S C-F 42204 109595 229299 406538 623415 89975548060a 108866 238391 405768 631366 897648

S-S C-C 110137 226417 406693 618348 899830 1205744109830a 230436 406083 624392 897773 1213189

C-C C-F 42334 222624 236115 616101 631334 120541247805a 223691 243769 616224 638560 1204110

aResults from Ref [23]

Table 3 Nondimensional natural frequencies for a tapered nonuniform beam with clamped-free (C-F) boundary condition and differentcross-section parameters

bc 1 c 2

Ω1 Ω2 Ω3 Ω1 Ω2 Ω3

02 Current 360566 206030 561232 385241 210389 565570Ref [24] 360827 206210 561923 385511 210568 566303

04 Current 373471 190993 502941 431621 200358 512654Ref [24] 373708 191138 503537 431878 200500 513346

06 Current 393214 174767 439735 500654 190544 456680Ref [24] 393428 174878 440248 500904 190649 457384

08 Current 429053 157350 368378 619373 183793 397350Ref [24] 429249 157427 368846 619639 183855 398336

Table 1 e first six nondimensional frequency Ω for the uniform double-beams with different boundary conditions (kc 1 times 105 Nm2Kc 0)


Beam 1 Beam 2 1 2 3 4 5 6

S-S S-S 98683 217345 394577 439572 887208 90830998696a 217350 394784 439721 888265 909128

C-C C-C 223011 295464 614070 644416 1201731 1219079223733a 295900 616728 646416 1209034 1224444

C-F C-F 35113 196809 219986 293130 615645 64564335160a 196815 220345 293346 616972 646649

S-S C-F 87062 185912 257651 424830 626190 901489103911b 202322 284861 424398 636832 899532

S-S C-C 162054 265768 423472 624802 901361 1205933151683b 294314 421633 636958 899281 1213566

C-C C-F 108006 229898 291021 617083 643423 1205406108090a 230286 291356 618652 644954 1209652

aResults from Ref [23] bResults from FEM with 500 elements


change when there is an increase in the coupling stiness inFigure 3(a) which illustrates the same characteristic as inSection 31 for the uniform double-beam system In Fig-ure 3(b) the second mode frequency will increase alongwith the increase of elastic boundary and coupling springWith the increase of coupling spring KC there is a rapidchange when the coupling spring KC is around 105Nm2which means the inuence of coupling spring on thefrequency is more obvious than that of elastic boundary

In order to analyze the coupling inuence of elasticboundary and taper ratio of the nonuniform beam thefundamental frequency of the same structure is shown inTable 6 and the corresponding results are plotted in Fig-ure 4 e sensitive area of the inuence of elastic boundarywill move when the taper ratio changes At the left side of thesensitive which means the elastic boundary stiness issmaller the fundamental frequency will increase along withthe raise of the taper ratio and the contrary tendency willoccur when on the right side In the sensitive area each linein Figure 4 will intersect with each other which means thefundamental frequency are equal at the certain value ofelastic boundary condition is phenomenon implies thatwhen the change of elastic boundary is dicult to achieve atone can get the same result by changing the taper ratios ofthe beams

4 Conclusion

In this paper an ecient modeling approach for the vi-bration analysis of nonuniform double-beam system withgeneral boundary condition is established in which the full-

coupling on the common interface are taken into account byintroducing both the translational and rotational restrainingsprings across the beams e elastic boundary condition ofthe beams and various rotational restrains in the beamrsquosinterface can be studied easily in the current modeling Inorder to treat the nonuniform beam prole in the mostgeneral pattern the arbitrary thickness variation functionsare all expanded into Fourier series Energy formulation isemployed for the description of double-beam dynamics

Table 5 Nondimensional natural frequencies for the nonuniform double-beam system under dierent coupling spring stiness with c 02and n 2 of beam 2

Ω kc Kc 0 (Nm2) kc 0 Kc 104 (Nm2) kc 104 Kc 0 (Nm2) kc Kc 104 (Nm2) kc Kc 107 (Nm2)1 35132 35972 36543 36545 366412 38524 41375 86261 87880 2164243 210389 212602 214288 214751 5957034 220129 222475 229277 233122 11616075 565570 567903 568261 570392 19132096 616177 617810 618310 620116 24461187 1096314 1098539 1097742 1099922 26721768 1206885 1208388 1207933 1209473 28530059 1800414 1802551 1801284 1803406 313157110 1993790 1995227 1994420 1995869 3792597

Table 4 e rst four dimensionless frequencies Ω for the three-stepped beams

BC Ω1 Ω2 Ω3 Ω4

SS-SS 3097 6184 9343 126053096a 6184 9343 12605

SS-F 4313 7331 10240 132974313a 7331 10240 13297

C-C 4536 7663 10808 140684541a 7660 10809 14064

C-F 2287 5133 8083 109782285a 5133 8083 10978

aResults from [25]

2 25 3 35 4 45 52

25

3

35

4

45

5

log10(kc)

log10(K c)

356

358

36

362

364

366

Figure 2 e inuence of the coupling translational spring kc androtational spring Kc to the nondimensional fundamental fre-quencies for the nonuniform double-beam system with c 02 andr 2


with the admissible function constructed as the superpo-sition of Fourier series and boundary smoothed auxiliaryterms In conjunction with the RayleighndashRitz procedure allthe modal parameters can be derived by solving a standardeigenvalue problem

Numerical examples are then presented to demonstratethe correctness and reliability of the current model forpredicting the modal frequencies of the uniform double-beam and nonuniform beam structures Based on the modelestablished the inuence of boundary condition and cou-pling strength on the modal characteristics of the non-uniform beam is investigated and addressed e resultsshow that the rotational coupling stiness can also play animportant role in aecting the double-beam system whichhas received little research attention in the existing literatureIt can be also found that the variation of the thickness prolecan be utilized to adjust the system modal parameters whenthe change of boundary andor coupling conditions may bedicult to perform Although just simulation is imple-mented for the double-beam system this approach can bevery easy for handling the multiple-beam structure of anynumber is work can provide an ecient modeling ap-proach for the dynamic study of the multiple-beam systemwith complex boundary coupling and thickness variationconditions

Data Availability

All data generated or analyzed during this study are includedin this published article

Conflicts of Interest

e authors declare that they have no conicts of interest

Acknowledgments

is work was supported by the Fok Ying Tung EducationFoundation (Grant no 161049)

c = 0c = 02c = 04

c = 06c = 08

0

5

10

15

20

25

Ω

1 2 3 4 5 6 7 8 90log10(K2)

Figure 4 Nondimensional fundamental frequencies of the non-uniform double-beam system with r 2 and kc Kc 100 Nm2

02

46

810

0

5

100

5

10

15

20

25

log10(K2)log10(KC)

Ω

(a)

log10(K2) 0

5

10

0

5

100

20

40

60

log10(KC)

Ω

(b)

Figure 3 Nondimensional fundamental and second mode frequencies of the nonuniform double-beam system with c 02 and r 2 (a)Mode 1 (b) mode 2

Table 6 Nondimensional fundamental frequencies for the non-uniform double-beam system under elastic boundary with r 2and kc sc 100 Nm2

K2 (Nm2)c

0 02 04 06 080 35133 38526 43165 50070 6194710 35146 38544 43191 50113 62036102 35276 38720 43448 50537 62906103 36548 40431 45915 54483 70016104 47161 54056 63962 78291 95231105 94720 105052 114273 120812 117939106 157713 158984 156945 145994 123139107 205987 192828 174287 151154 123731108 221393 199912 176770 151730 123791109 223299 200700 177029 151788 123797


References

[1] Z Oniszczuk ldquoFree transverse vibrations of an elasticallyconnected complex beamndashstring systemrdquo Journal of Soundand Vibration vol 254 no 4 pp 703ndash715 2002

[2] J M Seelig and W H Hoppmann ldquoNormal mode vibrationsof systems of elastically connected parallel barsrdquo Journal of theAcoustical Society of America vol 36 no 1 pp 93ndash99 1964

[3] S S Rao ldquoNatural vibrations of systems of elastically con-nected Timoshenko beamsrdquo Journal of the Acoustical Societyof America vol 55 no 6 pp 1232ndash1237 1974

[4] J Li andH X Hua ldquoDynamic stiffness vibration analysis of anelastically connected three-beam systemrdquo Applied Acousticsvol 69 no 7 pp 591ndash600 2008

[5] Z Oniszczuk ldquoFree transverse vibrations of elastically con-nected simply supported double-beam complex systemrdquoJournal of Sound and Vibration vol 232 no 2 pp 387ndash4032000

[6] S G Kelly and S Srinivas ldquoFree vibrations of elasticallyconnected stretched beamsrdquo Journal of Sound and Vibrationvol 326 no 3ndash5 pp 883ndash893 2009

[7] M A De Rosa and M Lippiello ldquoNon-classical boundaryconditions and DQM for double-beamsrdquoMechanics ResearchCommunications vol 34 no 7-8 pp 538ndash544 2007

[8] H Deng W Cheng and S G Zhao ldquoVibration and bucklinganalysis of double-functionally graded Timoshenko beamsystem on Winkler-Pasternak elastic foundationrdquo CompositeStructures vol 160 pp 152ndash168 2017

[9] J R Banerjee and F W Williams ldquoExact BernoullindashEulerdynamic stiffness matrix for a range of tapered beamsrdquo In-ternational Journal for Numerical Methods in Engineeringvol 21 no 12 pp 2289ndash2302 1985

[10] E M Cem M Aydogdu and V Taskin ldquoVibration ofa variable cross-section beamrdquo Mechanics Research Com-munications vol 34 no 1 pp 78ndash84 2007

[11] S Lenci F Clementi and C E N Mazzilli ldquoSimple formulasfor the natural frequencies of non-uniform cables and beamsrdquoInternational Journal of Mechanical Sciences vol 77pp 155ndash163 2013

[12] S Abrate ldquoVibration of non-uniform rods and beamsrdquoJournal of Sound and Vibration vol 185 no 4 pp 703ndash7161995

[13] J W Lee and J Y Lee ldquoFree vibration analysis using thetransfer-matrix method on a tapered beamrdquo Computers ampStructures vol 164 pp 75ndash82 2016

[14] M A Rosa and N M Auciello ldquoFree vibrations of taperedbeams with flexible endsrdquo Computers amp Structures vol 60no 2 pp 197ndash202 1996

[15] N M Auciello and A Ercolano ldquoExact solution for thetransverse vibration of a beam a part of which is a taper beamand other part is a uniform beamrdquo International Journal ofSolids and Structures vol 34 no 17 pp 2115ndash2129 1997

[16] K Torabi H Afshari M Sadeghi and H Toghian ldquoExactclosed-form solution for vibration analysis of truncatedconical and tapered beams carrying multiple concentratedmassesrdquo Journal of Solid Mechanics vol 9 no 4 pp 760ndash7822017

[17] Y Chen and L Pan Liu ldquoA complex-valued solution of freevibration for tapered beams with any number of rotationaldampersrdquo Applied Mathematics amp Information Sciencesvol 6 no 3 pp 749ndash758 2012

[18] H H Mabie and C B Rogers ldquoTransverse vibrations ofdouble-tapered cantilever beamsrdquo Journal of the AcousticalSociety of America vol 51 no 5B pp 1771ndash1774 1972

[19] H H Mabie and C B Rogers ldquoTransverse vibrations ofdouble-tapered cantilever beams with end support and withend massrdquo Journal of the Acoustical Society of Americavol 55 no 5 pp 986ndash991 1974

[20] I Takahashi and T Yoshioka ldquoVibration and stability ofa non-uniform double-beam subjected to follower forcesrdquoComputers amp Structures vol 59 no 6 pp 1033ndash1038 1996

[21] M Claeys J J Sinou J P Lambelin and B Alcoverro ldquoMulti-harmonic measurements and numerical simulations ofnonlinear vibrations of a beam with non-ideal boundaryconditionsrdquo Communications in Nonlinear Science and Nu-merical Simulation vol 19 no 12 pp 4196ndash4212 2014

[22] W L Li X F Zhang J T Du and Z G Liu ldquoAn exact seriessolution for the transverse vibration of rectangular plates withgeneral elastic boundary supportsrdquo Journal of Sound andVibration vol 321 no 1-2 pp 254ndash269 2009

[23] Q B Mao ldquoFree vibration analysis of elastically connectedmultiple-beams by using the Adomian modified de-composition methodrdquo Journal of Sound and Vibrationvol 331 no 11 pp 2532ndash2542 2012

[24] J R Banerjee H Su and D R Jackson ldquoFree vibration ofrotating tapered beams using the dynamic stiffness methodrdquoJournal of Sound and Vibration vol 298 no 4-5 pp 1034ndash1054 2006

[25] X Wang and Y Wang ldquoFree vibration analysis of multiple-stepped beams by the differential quadrature elementmethodrdquo Applied Mathematics and Computation vol 219no 11 pp 5802ndash5810 2013


International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

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Active and Passive Electronic Components

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

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Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

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Journal of



Journal ofEngineeringVolume 2018

SensorsJournal of



RotatingMachinery


Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018


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Hindawi

wwwhindawicom Volume 2018

Advances in

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Submit your manuscripts atwwwhindawicom

strength than the uniform beam is of great desire usvibration analysis of the nonuniform beam has been studiedcontinuously [9ndash11] Abrate [12] presented a method totransform the motion equation of a nonuniform beam intoits uniform counterpart Lee and Lee [13] developeda transfer-matrix method to investigate the free vibrationcharacteristics of a tapered BernoullindashEuler beam Besselfunctions are widely used in dealing with the nonuniformbeam system with different boundary conditions Rosa andAuciello [14] solved the nonuniform beam-governingequation and the elastic boundary condition bysubstituting the four flexibility coefficients of the con-structed functions Auciello and Ercolano [15] investigatedthe transverse vibration of a beam for which one part wastapered and the remaining part is the uniform thicknessTorabi et al [16] derived an exact closed-form solution forthe free vibration of EulerndashBernoulli conical and taperedbeams with any number of attached masses which aredescribed by Diracrsquos delta function Furthermore Chenand Pan Liu [17] employed the Bessel function and thenumerical assembly method (NAM) to investigate the vi-bration characteristics of a tapered beam with multiplearbitrarily placed rotational dampers

In the current studies on the vibration analysis of themultiple-beam system most of them are devoted to theuniform ones and just little exception can be found in theliterature for the vibration studies of nonuniform multiple-beam structure For example Mabie and Rogers [18 19]obtained an accurate solution for determining the first fivefrequencies of the cantilevered double-tampered beamsystem Takahashi and Yoshioka [20] utilized the transfer-matrix approach to analyze the vibration and stability ofa double-cantilever beam in which only translational springis used for the description of coupling strength between twobeams From the above literature review it can be found thatthe current study mainly considered the classical boundaryconditions and just one type of restraining spring on thebeam interface

From the practical point of view elastic boundary re-straint and other types of mechanical coupling strengthshould be consideredere is a clear gap in the literature onthis aspect In analysis of the real system the nearest idealboundary conditions such as clamped free and simplysupport is selected for the modeling However small de-viations from ideal conditions in real systems indeed occurFor example a beam connected at the ends to rigid supportsby pins is modeled using simply supported boundaryconditions which require deflections and moments to bezero But the hole and pin assembly may have small gapsandor friction which is called the nonideal boundary In thispaper the spring stiffnesses of the boundary conditions aredifferent and continuous so the elastic boundary is usedhere which includes the aforementioned nonideal boundaryMoreover when the beam is under axial force or in axialmoving the minimal rotations are emerged and the rotationrestrains are important For the double beam system whenthe similar rotation movements happen rotational couplingspring is introduced to restrain the relevant displacementClaeys et al [21] presents a direct comparison of measured

and predicted nonlinear vibrations of a clamped-clampedsteel beam with nonideal boundary conditions and theresults show that the nonideal boundary of transverse springand rotational spring have significant effect for the beam anda correct estimation is necessary to simulate the realstructurersquos characteristic

In this work motivated by the current limitation inliterature an efficient modeling approach for the vibrationanalysis of the nonuniform double-beam system withgeneral elastic boundary condition is proposed in whichboth the translational and rotational restraining springs aretaken into account to describe the dynamic interactionbetween each beam interface Energy principle is formu-lated for the analysis of the system motion equation withthe nonuniform thickness variation expanded into theFourier series in a unified pattern e transverse dis-placement admissible function is constructed as the su-perposition of the standard Fourier series and the boundarysmoothed auxillary terms In conjunction with theRayleighndashRitz procedure all the modal parameters can bederived by solving an eigenvalue matrix Numerical ex-amples are then given to demonstrate the reliability andeffectiveness of the current model Finally some con-cluding remarks are made

2 Theoretical Formulations

21 Model Description Consider an elastically connectednonuniform double-beam system as illustrated in Figure 1e beam member with arbitrary boundary conditions iscoupled with other through the elastic restraining springs onthe coupling interfaces Such mechanical interaction isrepresented using two types of springs namely translationaland rotational one Any boundary condition can be easilyobtained by setting the relevant restraining spring stiff-nesses For the beam vibration there are two freedom de-grees at each field point namely translation and rotationen a full-coupling restraint should include the trans-lational and rotational spring distributions accordingly Inthis way the familiar Winkler-type of elastic interface can bereadily derived by setting the coefficients of rotationalsprings into zero

22 Double-Beam SystemDynamics and Its Solution For thecoupled beam structure as shown in Figure 1 it will bedescribed from the viewpoint of energy Although thedouble-beam system is considered here the current mod-eling framework is suitable for the general N beam systemWithout losing the generality the subsequent formulationwill be given for such N-beam coupling system Lagrangianfunction is used for the description of system dynamicbehavior which can be written as

L VminusT 1113944N

i1Vi + Vcoupling minus 1113944

N

i1Ti (1)

where L is the system Lagrangian V and T are the totalpotential energy and kinetic energy Vi is the potentialenergy associated with the ith beam member Vcoupling is




Vi 12intL

0EiIi(x)

z2wi(x)zx2

[ ]2

dx +12Ki

T0w2i (x)

∣∣∣∣∣x0

+12Ki

R0zwi(x)zx

[ ]2∣∣∣∣∣∣∣∣∣x0

+12Ki

TLw2i (x)

∣∣∣∣∣xL

+12Ki

RLzwi(x)zx

[ ]2∣∣∣∣∣∣∣∣∣xL

(2)



Ti 12intL

0ρiSi(x)

zwi(x t)zt

[ ]2

dx 12ω2 int

L

0ρiSi(x)w

2i (x) dx

(3)



Vcoupling 12intL


zwizxminuszwi+1zx

( )2

dx

(4)




m1cm cos λmx

S0 suminfin

m0cm cos λmx

(5)

cm

1LintL

0f1(x)dx m 0

2LintL

0f1(x) cos

mπLxdx mne 0

(6)

and


m1dm cos λmx

I0 suminfin

m0dm cos λmx

(7)

dm

1LintL

0f2(x)dx m 0

2LintL

0f2(x) cos

mπLxdx mne 0

(8)




w(x) suminfin

n0an cos

nπLx( ) + b1ζ1(x)

+ b2ζ2(x) + b3ζ3(x) + b4ζ4(x)

(9)

in which

ζ1(x) 9L4π

sinπx2L( )minus

L

12πsin

3πx2L

( )

ζ2(x) minus9L4π

cosπx2L( )minus

L

12πcos

3πx2L

( )

ζ3(x) L3


L3

3π3sin

3πx2L

( )

ζ4(x) minusL3

π3cos

πx2L( )minus

L3

3π3cos

3πx2L

( )

(10)

x2

K1R0 K1RLK1TLK1T0

K2R0 K2RL

K2T0 K2TL

Beam 1

Kc

Beam 2

kc

x1





Kminusω2M1113872 1113873A 0 (11)




ρSEI1113968



Si(x) S0 aminus bx

L1113874 1113875

r

(12)

and

Ii(x) I0 aminus bx

L1113874 1113875

r+2 (13)











Beam 1 Beam 2 1 2 3 4 5 6

S-S S-S 98683 115923 394576 412919 887207 90593098649a 121168 394240 418523 885772 910371

C-C C-C 223008 233205 614048 628443 1201720 1218760223421a 236691 614904 633157 1202885 1222928

C-F C-F 35113 53243 219985 245854 615643 63893935148a 57683 220091 253841 615438 645943

S-S C-F 42204 109595 229299 406538 623415 89975548060a 108866 238391 405768 631366 897648

S-S C-C 110137 226417 406693 618348 899830 1205744109830a 230436 406083 624392 897773 1213189

C-C C-F 42334 222624 236115 616101 631334 120541247805a 223691 243769 616224 638560 1204110



bc 1 c 2

Ω1 Ω2 Ω3 Ω1 Ω2 Ω3

02 Current 360566 206030 561232 385241 210389 565570Ref [24] 360827 206210 561923 385511 210568 566303

04 Current 373471 190993 502941 431621 200358 512654Ref [24] 373708 191138 503537 431878 200500 513346

06 Current 393214 174767 439735 500654 190544 456680Ref [24] 393428 174878 440248 500904 190649 457384

08 Current 429053 157350 368378 619373 183793 397350Ref [24] 429249 157427 368846 619639 183855 398336



Beam 1 Beam 2 1 2 3 4 5 6

S-S S-S 98683 217345 394577 439572 887208 90830998696a 217350 394784 439721 888265 909128

C-C C-C 223011 295464 614070 644416 1201731 1219079223733a 295900 616728 646416 1209034 1224444

C-F C-F 35113 196809 219986 293130 615645 64564335160a 196815 220345 293346 616972 646649

S-S C-F 87062 185912 257651 424830 626190 901489103911b 202322 284861 424398 636832 899532

S-S C-C 162054 265768 423472 624802 901361 1205933151683b 294314 421633 636958 899281 1213566

C-C C-F 108006 229898 291021 617083 643423 1205406108090a 230286 291356 618652 644954 1209652





4 Conclusion






BC Ω1 Ω2 Ω3 Ω4

SS-SS 3097 6184 9343 126053096a 6184 9343 12605

SS-F 4313 7331 10240 132974313a 7331 10240 13297

C-C 4536 7663 10808 140684541a 7660 10809 14064

C-F 2287 5133 8083 109782285a 5133 8083 10978

aResults from [25]

2 25 3 35 4 45 52

25

3

35

4

45

5

log10(kc)

log10(K c)

356

358

36

362

364

366





Data Availability




Acknowledgments


c = 0c = 02c = 04

c = 06c = 08

0

5

10

15

20

25

Ω

1 2 3 4 5 6 7 8 90log10(K2)


02

46

810

0

5

100

5

10

15

20

25

log10(K2)log10(KC)

Ω

(a)

log10(K2) 0

5

10

0

5

100

20

40

60

log10(KC)

Ω

(b)



K2 (Nm2)c

0 02 04 06 080 35133 38526 43165 50070 6194710 35146 38544 43191 50113 62036102 35276 38720 43448 50537 62906103 36548 40431 45915 54483 70016104 47161 54056 63962 78291 95231105 94720 105052 114273 120812 117939106 157713 158984 156945 145994 123139107 205987 192828 174287 151154 123731108 221393 199912 176770 151730 123791109 223299 200700 177029 151788 123797


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




Vi 12intL

0EiIi(x)

z2wi(x)zx2

[ ]2

dx +12Ki

T0w2i (x)

∣∣∣∣∣x0

+12Ki

R0zwi(x)zx

[ ]2∣∣∣∣∣∣∣∣∣x0

+12Ki

TLw2i (x)

∣∣∣∣∣xL

+12Ki

RLzwi(x)zx

[ ]2∣∣∣∣∣∣∣∣∣xL

(2)



Ti 12intL

0ρiSi(x)

zwi(x t)zt

[ ]2

dx 12ω2 int

L

0ρiSi(x)w

2i (x) dx

(3)



Vcoupling 12intL


zwizxminuszwi+1zx

( )2

dx

(4)




m1cm cos λmx

S0 suminfin

m0cm cos λmx

(5)

cm

1LintL

0f1(x)dx m 0

2LintL

0f1(x) cos

mπLxdx mne 0

(6)

and


m1dm cos λmx

I0 suminfin

m0dm cos λmx

(7)

dm

1LintL

0f2(x)dx m 0

2LintL

0f2(x) cos

mπLxdx mne 0

(8)




w(x) suminfin

n0an cos

nπLx( ) + b1ζ1(x)

+ b2ζ2(x) + b3ζ3(x) + b4ζ4(x)

(9)

in which

ζ1(x) 9L4π

sinπx2L( )minus

L

12πsin

3πx2L

( )

ζ2(x) minus9L4π

cosπx2L( )minus

L

12πcos

3πx2L

( )

ζ3(x) L3


L3

3π3sin

3πx2L

( )

ζ4(x) minusL3

π3cos

πx2L( )minus

L3

3π3cos

3πx2L

( )

(10)

x2

K1R0 K1RLK1TLK1T0

K2R0 K2RL

K2T0 K2TL

Beam 1

Kc

Beam 2

kc

x1





Kminusω2M1113872 1113873A 0 (11)




ρSEI1113968



Si(x) S0 aminus bx

L1113874 1113875

r

(12)

and

Ii(x) I0 aminus bx

L1113874 1113875

r+2 (13)











Beam 1 Beam 2 1 2 3 4 5 6

S-S S-S 98683 115923 394576 412919 887207 90593098649a 121168 394240 418523 885772 910371

C-C C-C 223008 233205 614048 628443 1201720 1218760223421a 236691 614904 633157 1202885 1222928

C-F C-F 35113 53243 219985 245854 615643 63893935148a 57683 220091 253841 615438 645943

S-S C-F 42204 109595 229299 406538 623415 89975548060a 108866 238391 405768 631366 897648

S-S C-C 110137 226417 406693 618348 899830 1205744109830a 230436 406083 624392 897773 1213189

C-C C-F 42334 222624 236115 616101 631334 120541247805a 223691 243769 616224 638560 1204110



bc 1 c 2

Ω1 Ω2 Ω3 Ω1 Ω2 Ω3

02 Current 360566 206030 561232 385241 210389 565570Ref [24] 360827 206210 561923 385511 210568 566303

04 Current 373471 190993 502941 431621 200358 512654Ref [24] 373708 191138 503537 431878 200500 513346

06 Current 393214 174767 439735 500654 190544 456680Ref [24] 393428 174878 440248 500904 190649 457384

08 Current 429053 157350 368378 619373 183793 397350Ref [24] 429249 157427 368846 619639 183855 398336



Beam 1 Beam 2 1 2 3 4 5 6

S-S S-S 98683 217345 394577 439572 887208 90830998696a 217350 394784 439721 888265 909128

C-C C-C 223011 295464 614070 644416 1201731 1219079223733a 295900 616728 646416 1209034 1224444

C-F C-F 35113 196809 219986 293130 615645 64564335160a 196815 220345 293346 616972 646649

S-S C-F 87062 185912 257651 424830 626190 901489103911b 202322 284861 424398 636832 899532

S-S C-C 162054 265768 423472 624802 901361 1205933151683b 294314 421633 636958 899281 1213566

C-C C-F 108006 229898 291021 617083 643423 1205406108090a 230286 291356 618652 644954 1209652





4 Conclusion






BC Ω1 Ω2 Ω3 Ω4

SS-SS 3097 6184 9343 126053096a 6184 9343 12605

SS-F 4313 7331 10240 132974313a 7331 10240 13297

C-C 4536 7663 10808 140684541a 7660 10809 14064

C-F 2287 5133 8083 109782285a 5133 8083 10978

aResults from [25]

2 25 3 35 4 45 52

25

3

35

4

45

5

log10(kc)

log10(K c)

356

358

36

362

364

366





Data Availability




Acknowledgments


c = 0c = 02c = 04

c = 06c = 08

0

5

10

15

20

25

Ω

1 2 3 4 5 6 7 8 90log10(K2)


02

46

810

0

5

100

5

10

15

20

25

log10(K2)log10(KC)

Ω

(a)

log10(K2) 0

5

10

0

5

100

20

40

60

log10(KC)

Ω

(b)



K2 (Nm2)c

0 02 04 06 080 35133 38526 43165 50070 6194710 35146 38544 43191 50113 62036102 35276 38720 43448 50537 62906103 36548 40431 45915 54483 70016104 47161 54056 63962 78291 95231105 94720 105052 114273 120812 117939106 157713 158984 156945 145994 123139107 205987 192828 174287 151154 123731108 221393 199912 176770 151730 123791109 223299 200700 177029 151788 123797


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




Kminusω2M1113872 1113873A 0 (11)




ρSEI1113968



Si(x) S0 aminus bx

L1113874 1113875

r

(12)

and

Ii(x) I0 aminus bx

L1113874 1113875

r+2 (13)











Beam 1 Beam 2 1 2 3 4 5 6

S-S S-S 98683 115923 394576 412919 887207 90593098649a 121168 394240 418523 885772 910371

C-C C-C 223008 233205 614048 628443 1201720 1218760223421a 236691 614904 633157 1202885 1222928

C-F C-F 35113 53243 219985 245854 615643 63893935148a 57683 220091 253841 615438 645943

S-S C-F 42204 109595 229299 406538 623415 89975548060a 108866 238391 405768 631366 897648

S-S C-C 110137 226417 406693 618348 899830 1205744109830a 230436 406083 624392 897773 1213189

C-C C-F 42334 222624 236115 616101 631334 120541247805a 223691 243769 616224 638560 1204110



bc 1 c 2

Ω1 Ω2 Ω3 Ω1 Ω2 Ω3

02 Current 360566 206030 561232 385241 210389 565570Ref [24] 360827 206210 561923 385511 210568 566303

04 Current 373471 190993 502941 431621 200358 512654Ref [24] 373708 191138 503537 431878 200500 513346

06 Current 393214 174767 439735 500654 190544 456680Ref [24] 393428 174878 440248 500904 190649 457384

08 Current 429053 157350 368378 619373 183793 397350Ref [24] 429249 157427 368846 619639 183855 398336



Beam 1 Beam 2 1 2 3 4 5 6

S-S S-S 98683 217345 394577 439572 887208 90830998696a 217350 394784 439721 888265 909128

C-C C-C 223011 295464 614070 644416 1201731 1219079223733a 295900 616728 646416 1209034 1224444

C-F C-F 35113 196809 219986 293130 615645 64564335160a 196815 220345 293346 616972 646649

S-S C-F 87062 185912 257651 424830 626190 901489103911b 202322 284861 424398 636832 899532

S-S C-C 162054 265768 423472 624802 901361 1205933151683b 294314 421633 636958 899281 1213566

C-C C-F 108006 229898 291021 617083 643423 1205406108090a 230286 291356 618652 644954 1209652





4 Conclusion






BC Ω1 Ω2 Ω3 Ω4

SS-SS 3097 6184 9343 126053096a 6184 9343 12605

SS-F 4313 7331 10240 132974313a 7331 10240 13297

C-C 4536 7663 10808 140684541a 7660 10809 14064

C-F 2287 5133 8083 109782285a 5133 8083 10978

aResults from [25]

2 25 3 35 4 45 52

25

3

35

4

45

5

log10(kc)

log10(K c)

356

358

36

362

364

366





Data Availability




Acknowledgments


c = 0c = 02c = 04

c = 06c = 08

0

5

10

15

20

25

Ω

1 2 3 4 5 6 7 8 90log10(K2)


02

46

810

0

5

100

5

10

15

20

25

log10(K2)log10(KC)

Ω

(a)

log10(K2) 0

5

10

0

5

100

20

40

60

log10(KC)

Ω

(b)



K2 (Nm2)c

0 02 04 06 080 35133 38526 43165 50070 6194710 35146 38544 43191 50113 62036102 35276 38720 43448 50537 62906103 36548 40431 45915 54483 70016104 47161 54056 63962 78291 95231105 94720 105052 114273 120812 117939106 157713 158984 156945 145994 123139107 205987 192828 174287 151154 123731108 221393 199912 176770 151730 123791109 223299 200700 177029 151788 123797


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






Beam 1 Beam 2 1 2 3 4 5 6

S-S S-S 98683 115923 394576 412919 887207 90593098649a 121168 394240 418523 885772 910371

C-C C-C 223008 233205 614048 628443 1201720 1218760223421a 236691 614904 633157 1202885 1222928

C-F C-F 35113 53243 219985 245854 615643 63893935148a 57683 220091 253841 615438 645943

S-S C-F 42204 109595 229299 406538 623415 89975548060a 108866 238391 405768 631366 897648

S-S C-C 110137 226417 406693 618348 899830 1205744109830a 230436 406083 624392 897773 1213189

C-C C-F 42334 222624 236115 616101 631334 120541247805a 223691 243769 616224 638560 1204110



bc 1 c 2

Ω1 Ω2 Ω3 Ω1 Ω2 Ω3

02 Current 360566 206030 561232 385241 210389 565570Ref [24] 360827 206210 561923 385511 210568 566303

04 Current 373471 190993 502941 431621 200358 512654Ref [24] 373708 191138 503537 431878 200500 513346

06 Current 393214 174767 439735 500654 190544 456680Ref [24] 393428 174878 440248 500904 190649 457384

08 Current 429053 157350 368378 619373 183793 397350Ref [24] 429249 157427 368846 619639 183855 398336



Beam 1 Beam 2 1 2 3 4 5 6

S-S S-S 98683 217345 394577 439572 887208 90830998696a 217350 394784 439721 888265 909128

C-C C-C 223011 295464 614070 644416 1201731 1219079223733a 295900 616728 646416 1209034 1224444

C-F C-F 35113 196809 219986 293130 615645 64564335160a 196815 220345 293346 616972 646649

S-S C-F 87062 185912 257651 424830 626190 901489103911b 202322 284861 424398 636832 899532

S-S C-C 162054 265768 423472 624802 901361 1205933151683b 294314 421633 636958 899281 1213566

C-C C-F 108006 229898 291021 617083 643423 1205406108090a 230286 291356 618652 644954 1209652





4 Conclusion






BC Ω1 Ω2 Ω3 Ω4

SS-SS 3097 6184 9343 126053096a 6184 9343 12605

SS-F 4313 7331 10240 132974313a 7331 10240 13297

C-C 4536 7663 10808 140684541a 7660 10809 14064

C-F 2287 5133 8083 109782285a 5133 8083 10978

aResults from [25]

2 25 3 35 4 45 52

25

3

35

4

45

5

log10(kc)

log10(K c)

356

358

36

362

364

366





Data Availability




Acknowledgments


c = 0c = 02c = 04

c = 06c = 08

0

5

10

15

20

25

Ω

1 2 3 4 5 6 7 8 90log10(K2)


02

46

810

0

5

100

5

10

15

20

25

log10(K2)log10(KC)

Ω

(a)

log10(K2) 0

5

10

0

5

100

20

40

60

log10(KC)

Ω

(b)



K2 (Nm2)c

0 02 04 06 080 35133 38526 43165 50070 6194710 35146 38544 43191 50113 62036102 35276 38720 43448 50537 62906103 36548 40431 45915 54483 70016104 47161 54056 63962 78291 95231105 94720 105052 114273 120812 117939106 157713 158984 156945 145994 123139107 205987 192828 174287 151154 123731108 221393 199912 176770 151730 123791109 223299 200700 177029 151788 123797


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




4 Conclusion






BC Ω1 Ω2 Ω3 Ω4

SS-SS 3097 6184 9343 126053096a 6184 9343 12605

SS-F 4313 7331 10240 132974313a 7331 10240 13297

C-C 4536 7663 10808 140684541a 7660 10809 14064

C-F 2287 5133 8083 109782285a 5133 8083 10978

aResults from [25]

2 25 3 35 4 45 52

25

3

35

4

45

5

log10(kc)

log10(K c)

356

358

36

362

364

366





Data Availability




Acknowledgments


c = 0c = 02c = 04

c = 06c = 08

0

5

10

15

20

25

Ω

1 2 3 4 5 6 7 8 90log10(K2)


02

46

810

0

5

100

5

10

15

20

25

log10(K2)log10(KC)

Ω

(a)

log10(K2) 0

5

10

0

5

100

20

40

60

log10(KC)

Ω

(b)



K2 (Nm2)c

0 02 04 06 080 35133 38526 43165 50070 6194710 35146 38544 43191 50113 62036102 35276 38720 43448 50537 62906103 36548 40431 45915 54483 70016104 47161 54056 63962 78291 95231105 94720 105052 114273 120812 117939106 157713 158984 156945 145994 123139107 205987 192828 174287 151154 123731108 221393 199912 176770 151730 123791109 223299 200700 177029 151788 123797


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




Data Availability




Acknowledgments


c = 0c = 02c = 04

c = 06c = 08

0

5

10

15

20

25

Ω

1 2 3 4 5 6 7 8 90log10(K2)


02

46

810

0

5

100

5

10

15

20

25

log10(K2)log10(KC)

Ω

(a)

log10(K2) 0

5

10

0

5

100

20

40

60

log10(KC)

Ω

(b)



K2 (Nm2)c

0 02 04 06 080 35133 38526 43165 50070 6194710 35146 38544 43191 50113 62036102 35276 38720 43448 50537 62906103 36548 40431 45915 54483 70016104 47161 54056 63962 78291 95231105 94720 105052 114273 120812 117939106 157713 158984 156945 145994 123139107 205987 192828 174287 151154 123731108 221393 199912 176770 151730 123791109 223299 200700 177029 151788 123797


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


flexural vibration analysis of nonuniform double-beam

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