Chaos, Solitons and Fractals 36 (2008) 1267–1272

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Nonlinear vibration analysis of tapered Timoshenko beams

Minmao Liao, Hongzhi Zhong *

Department of Civil Engineering, Tsinghua University, Beijing 100084, PR China

Accepted 28 July 2006

Communicated by Prof. Ji-Huan He

Abstract

The nonlinear flexural vibration analysis of tapered Timoshenko beams is conducted. The equations of motion fortapered Timoshenko beams are established in which the effects of nonlinear transverse deformation, nonlinear curva-ture as well as nonlinear axial deformation are taken into account. The nonlinear fundamental frequencies of taperedTimoshenko beams with two simply supported or clamped ends are presented.� 2006 Published by Elsevier Ltd.

1. Introduction

The nonlinear effects on slender structural elements are of increasing concern in the modern design process. As isknown, nonlinear vibrations of structural elements may be characterized by deterministic chaos and nonlinear bifurca-tion. The spatial chaos of an axially loaded infinitely long rod was addressed in [1,2]. A discrete element method wasproposed for nonlinear bifurcation analysis of elastic structures in [3]. The investigation of nonlinear flexural vibrationsof Timoshenko beams was initiated in [4], using the finite element method. Allowing for more nonlinear effects, the non-linear vibrations of Timoshenko beams were studied in [5]. With the method of multiscales, the nonlinear frequencies oflarge amplitude vibrations of simply supported Timoshenko beams were obtained in [6]. The influence of shear defor-mation on the buckling and spatial chaos of beams was investigated in [7]. The effects of boundary conditions on thenonlinear frequencies of Timoshenko beams were studied in [8,9], using the differential quadrature method.

As a preliminary effort for the analysis of the complex nonlinear behavior of Timoshenko beams, the nonlinear freeflexural vibrations of tapered Timoshenko beams are studied using the differential quadrature method in the presentwork. Linear variation of width or height is considered and the taper influence on the nonlinear frequencies of Timo-shenko beams is investigated.

0960-0779/$ - see front matter � 2006 Published by Elsevier Ltd.doi:10.1016/j.chaos.2006.07.055

* Corresponding author. Tel.: +86 10 62781891; fax: +86 10 62771132.E-mail address: hzz@tsinghua.edu.cn (H. Zhong).

1268 M. Liao, H. Zhong / Chaos, Solitons and Fractals 36 (2008) 1267–1272

2. Formulation

A Timoshenko beam with either tapered width or tapered height is shown in Fig. 1(a) and (b). Denote N as the axialforce, Q as the shear force, M as the bending moment. An infinitesimal element with all internal forces is illustrated inFig. 1(c), in which �qAwttdx represents the inertia force and �qIhttdx is the inertial bending moment. The subscript t orx denotes the partial differentiation with respect to time or spatial coordinate.

The nonlinear strain–displacement relations of Timoshenko beams are expressed as

Fig. 1infinite

e ¼ ux þ1

2w2

x ; ð1Þ

j ¼ hxffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ w2

x

p � hx 1� 1

2w2

x þ3

8w4

x

� �; ð2Þ

c ¼ tan�1 wx � h � wx �1

3w3

x � h; ð3Þ

where e is the axial strain, c is the shear strain, j is the bending curvature, h is the rotation of the cross-section, u is theaxial displacement and w is the deflection of the beam.

The strain energy U and the kinetic energy K for a tapered Timoshenko beam are given as

U ¼ 1

2

Z L

0

EAe2 þ EIj2 þ kGAc2� �

dx; ð4Þ

K ¼ 1

2

Z L

0

qAw2t þ qIh2

t

� �dx; ð5Þ

where E is the Young’s modulus, A is the cross-section area, I is the moment of inertia, k is the shear modification coef-ficient, G is the shear modulus, q is the mass density per unit volume and L is the length of the beam.

Using the Hamilton’s principle

dZ t2

t1

ðK � UÞdt ¼ 0 ð6Þ

w

w

x

x

O

O

a

b

c

h

bLL

dx

h0

b

hL

dx

L

N

M

Q

M+dM

1tan ( )xw-

b0

Q+dQN

ttI dx-

ttAw dx -

. (a) A tapered Timoshenko beam with varying width, (b) a tapered Timoshenko beam with varying height and (c) ansimal element with all internal forces.

M. Liao, H. Zhong / Chaos, Solitons and Fractals 36 (2008) 1267–1272 1269

the equations of motion are obtained. With the immovable condition at the two ends of the beam,

uð0; tÞ ¼ uðL; tÞ ¼ 0; ð7Þ

the axial force is found independent of spatial variable and given as

NðtÞ ¼ E2

Z L

0

w2x dx�Z L

0

1

Adx: ð8Þ

Under the concurrent reversal assumption adopted by some researchers [9,10], which says that the maximum amplitudeis reached at the motion reversal point of every point of the beam, the equations of motion will be simplified into aneigenvalue problem. Mathematically, the assumption prescribes that at the reversal time there exist

o2wot2¼ �x2w;

owot¼ 0;

o2h

ot2¼ �x2h;

ohot¼ 0; ð9Þ

where x is the nonlinear frequency.

2.1. Beams with varying width

For a tapered Timoshenko beam with linearly varying width, the cross-section area and the moment of inertia areexpressed as

A ¼ A0ð1þ ax=LÞ; I ¼ I0ð1þ ax=LÞ; ð10Þ

with

a ¼ ðbL � b0Þ=b0; ð11Þ

where A0, I0 and b0 are the cross-section area, the moment of inertia and the beam width at the left end x = 0, respec-tively. bL is the beam width at the right end x = L.

The following dimensionless parameters are introduced for convenience:

n ¼ xL; w ¼ av; U ¼ L

ah; r0 ¼

ffiffiffiffiffiI0

A0

r; k ¼ E

kG; g ¼ I0

A0L2; X ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqA0L4x2

EI0

s: ð12Þ

The dimensionless governing equations are obtained as

1

kgðUn � vnnÞ �

að1þ anÞkg

ðvn � UÞ þ 4a3ð1þ anÞk

ar0

� �2

v3n �

að1þ anÞk

ar0

� �2

Uv2n �

1

2

ar0

� �2

� að1þ anÞ lnð1þ aÞ

Z 1

0

v2n dn

� �vnn þ g

ar0

� �2

U2nvnn �

2a1þ an

g2 ar0

� �4

U2nv3

n þ 2gar0

� �2

UnUnnvn

þ a1þ an

gar0

� �2

U2nvn � 6g2 a

r0

� �4

U2nv2

nvnn � 4g2 ar0

� �4

UnUnnv3n �

1

kar0

� �2

Unv2n þ 2Uvnvnn � 4v2

nvnn

� �� X2v ¼ 0;

ð13aÞ

�að1þ anÞg Un �

1

gUnn �

a1þ an

ar0

� �4

gUnv4n þ

1

kg2ðU� vnÞ þ

ar0

� �2

v2nUnn þ 2

ar0

� �2

vnvnnUn

þ a1þ an

ar0

� �2

Unv2n � g

ar0

� �4

v4nUnn � 4g

ar0

� �4

v3nvnnUn þ

1

3kgar0

� �2

v3n � X2U ¼ 0: ð13bÞ

2.2. Beams with varying height

For a tapered Timoshenko beam with linearly varying height, the cross-section area and the moment of inertia are

A ¼ A0ð1þ bx=LÞ; I ¼ I0ð1þ bx=LÞ3; ð14Þ

with

b ¼ ðhL � h0Þ=h0; ð15Þ

where h0 and hL are the height of the beam at the left end x = 0 and the right end x = L, respectively.

1270 M. Liao, H. Zhong / Chaos, Solitons and Fractals 36 (2008) 1267–1272

In a manner similar to the beams with varying width, the dimensionless governing equations are written as

TableVariat

a/r0

0.100.200.400.600.801.001.201.50

TableVariat

a/r0

0.100.200.400.600.801.001.201.50

1

kgðUn � vnnÞ �

bð1þ bnÞkg

ðvn � UÞ þ 4b3ð1þ bnÞk

ar0

� �2

v3n �

bð1þ bnÞk

ar0

� �2

Uv2n

� 1

2

ar0

� �2 bð1þ bnÞ lnð1þ bÞ

Z 1

0

v2n dn

� �vnn þ g

ar0

� �2

ð1þ bnÞ2U2nvnn

� 6bð1þ bnÞg2 ar0

� �4

U2nv3

n þ 2gar0

� �2

ð1þ bnÞ2UnUnnvn þ 3bð1þ bnÞg ar0

� �2

U2nvn � 6g2 a

r0

� �4

ð1þ bnÞ2U2nv2

nvnn

� 4g2 ar0

� �4

ð1þ bnÞ2UnUnnv3n �

1

kar0

� �2

ðUnv2n þ 2Uvnvnn � 4v2

nvnnÞ � X2v ¼ 0; ð16aÞ

�3bð1þ bnÞg Un �

1

gUnn �

3b1þ bn

ar0

� �4

gUnv4n þ

1

kg2ð1þ bnÞ2ðU� vnÞ þ

ar0

� �2

v2nUnn þ 2

ar0

� �2

vnvnnUn

þ 3b1þ bn

ar0

� �2

Unv2n � g

ar0

� �4

v4nUnn � 4g

ar0

� �4

v3nvnnUn þ

1

3kg 1þ bnð Þ2ar0

� �2

v3n � X2U ¼ 0: ð16bÞ

The motion equations (13) and (16) for tapered beams will be simplified into those for uniform beams given in [9] whenthe taper vanishes. Applying the differential quadrature rule to the governing equations, the nonlinear fundamental fre-quencies can be extracted from the resultant algebraic equations. The details can be referred to [8].

3. Results and discussion

In all computations, the Poisson’s ratio is taken as t = 0.30 and the shear modification coefficient is k = 5/6. Tables 1and 2 give the variation of the nonlinear frequency to linear frequency ratio (X/XL) of beams with two simply supportedends for slenderness ratio L/r0 = 20 (short beams) and L/r0 = 100 (slender beams), respectively. Tables 3 and 4 show thevariation of nonlinear frequency to linear frequency ratio for beams with two clamped ends for slenderness ratio

1ion of frequency ratio X/XL for simply supported tapered Timoshenko beams with slenderness ratio L/r0 = 20

Width taper Height taper

a = 0.25 a = 0.50 a = 0.75 a = 1.00 b = 0.25 b = 0.50 b = 0.75 b = 1.00

1.0012 1.0012 1.0012 1.0011 1.0009 1.0007 1.0006 1.00051.0047 1.0046 1.0046 1.0046 1.0037 1.0030 1.0025 1.00211.0185 1.0184 1.0183 1.0182 1.0146 1.0119 1.0100 1.00851.0411 1.0409 1.0407 1.0405 1.0325 1.0266 1.0224 1.01901.0719 1.0716 1.0713 1.0709 1.0570 1.0467 1.0394 1.03361.1102 1.1097 1.1092 1.1086 1.0878 1.0721 1.0608 1.05191.1551 1.1545 1.1537 1.1528 1.1239 1.1022 1.0864 1.07401.2333 1.2323 1.2311 1.2297 1.1878 1.1552 1.1315 1.1131

2ion of frequency ratio X/XL for simply supported tapered Timoshenko beams with slenderness ratio L/r0 = 100

Width taper Height taper

a = 0.25 a = 0.50 a = 0.75 a = 1.00 b = 0.25 b = 0.50 b = 0.75 b = 1.00

1.0012 1.0012 1.0012 1.0012 1.0010 1.0008 1.0007 1.00061.0050 1.0049 1.0049 1.0049 1.0040 1.0033 1.0028 1.00251.0197 1.0196 1.0195 1.0194 1.0158 1.0132 1.0113 1.00981.0438 1.0437 1.0434 1.0432 1.0353 1.0294 1.0252 1.02191.0767 1.0764 1.0760 1.0755 1.0619 1.0516 1.0444 1.03871.1175 1.1170 1.1164 1.1157 1.0950 1.0795 1.0685 1.05971.1654 1.1647 1.1638 1.1628 1.1344 1.1127 1.0972 1.08491.2488 1.2477 1.2464 1.2448 1.2033 1.1712 1.1479 1.1296

Table 4Variation of frequency ratio X/XL for clamped tapered Timoshenko beams with slenderness ratio L/r0 = 100

a/r0 Width taper Heigh taper

a = 0.25 a = 0.50 a = 0.75 a = 1.00 b = 0.25 b = 0.50 b = 0.75 b = 1.00

0.10 1.0003 1.0003 1.0003 1.0003 1.0002 1.0002 1.0002 1.00010.20 1.0012 1.0012 1.0012 1.0012 1.0009 1.0008 1.0007 1.00060.40 1.0047 1.0047 1.0047 1.0047 1.0038 1.0031 1.0027 1.00230.60 1.0106 1.0106 1.0106 1.0105 1.0085 1.0070 1.0060 1.00520.80 1.0187 1.0187 1.0187 1.0186 1.0150 1.0124 1.0106 1.00921.00 1.0291 1.0290 1.0290 1.0289 1.0233 1.0193 1.0165 1.01441.20 1.0416 1.0415 1.0414 1.0414 1.0334 1.0278 1.0237 1.02061.50 1.0641 1.0640 1.0639 1.0638 1.0517 1.0430 1.0367 1.0320

Table 3Variation of frequency ratio X/XL for clamped tapered Timoshenko beams with slenderness ratio L/r0 = 20

a/r0 Width taper Height taper

a = 0.25 a = 0.50 a = 0.75 a = 1.00 b = 0.25 b = 0.50 b = 0.75 b = 1.00

0.10 1.0002 1.0002 1.0002 1.0002 1.0002 1.0001 1.0001 1.00010.20 1.0009 1.0009 1.0009 1.0008 1.0006 1.0005 1.0004 1.00030.40 1.0034 1.0034 1.0034 1.0034 1.0025 1.0019 1.0015 1.00120.60 1.0077 1.0077 1.0076 1.0076 1.0057 1.0043 1.0034 1.00260.80 1.0135 1.0135 1.0135 1.0134 1.0100 1.0076 1.0060 1.00471.00 1.0210 1.0210 1.0209 1.0208 1.0156 1.0119 1.0093 1.00731.20 1.0300 1.0299 1.0299 1.0297 1.0223 1.0170 1.0133 1.01051.50 1.0460 1.0460 1.0459 1.0457 1.0344 1.0263 1.0206 1.0164

M. Liao, H. Zhong / Chaos, Solitons and Fractals 36 (2008) 1267–1272 1271

L/r0 = 20 and L/r0 = 100, respectively. It is seen that the frequency ratio increases with the increase of amplitude ratioand the increase of slenderness ratio but decreases with the increase of either width taper or height taper. In comparisonwith beams with simply supported ends, the frequency ratio of beams with clamped ends increases relatively slowly inall cases. It is observed that the change of height taper has more significant influence on the frequency ratio than thechange of the width taper. This is attributed to the cubic relation of the moment of inertia with the height of beam cross-section contrasting with the linear relation of the moment of inertia with the width.

4. Conclusion

The nonlinear flexural vibrations of tapered Timoshenko beams with two simply supported or clamped ends have beenstudied using the differential quadrature method. The governing equations of motion have been established. The taperinfluence on the nonlinear fundamental frequencies has been investigated. The effect of height taper change on the non-linear fundamental frequency is found larger than that of width taper change. Generally, the nonlinear effects on the fre-quencies of beams with two clamped ends are less significant than those on the frequencies of simply supported beams.

Acknowledgement

The present work was undertaken under the financial support of the Tsinghua Fundamental Research FoundationJCxx2005069.

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