A single and double mode approach to chaotic vibrations of ... · Abstract. The chaotic vibrations of a cylindrical shell of large deﬂection subjected to two-dimensional exertions

Shock and Vibration 11 (2004) 533–546 533IOS Press

A single and double mode approach tochaotic vibrations of a cylindrical shell withlarge deflection

Liming Daia,∗, Qiang Hanb and Mingzhe DongcaIndustrial Systems Engineering, University of Regina, 3737 Wascana Parkway, Regina, Saskatchewan, CanadaS4S0A2bDepartment of Mechanics, College of Traffic and Communications, South China University of Technology,Guangzhou, China 510640cPetroleum Systems Engineering, University of Regina, 3737 Wascana Parkway, Regina, Saskatchewan, CanadaS4S0A2

Received 13 October 2003

Revised 8 December 2003

Abstract. The chaotic vibrations of a cylindrical shell of large deflection subjected to two-dimensional exertions are studied inthe present research. The dynamic nonlinear governing equations of the cylindrical shell are derived on the basis of single anddouble mode models established. Two different types of nonlinear dynamic equations are obtained with varying dimensions andloading parameters. The criteria for chaos are determined via Melnikov function for the single mode model. The chaotic motionof the cylindrical shell is investigated and the comparison between the single and double mode models is carried out. Results ofthe research show that the single mode method usually used may lead to incorrect conclusions under certain conditions. Doublemode or higher order mode methods should be used in these cases.

Keywords: Galerkin principle, melnikov function, chaos, nonlinear dynamics, cylindrical shell, single and double mode methods,numerical analysis

1. Introduction

In recent years, chaos in nonlinear dynamic systems has aroused more and more interest in the field of theoreticaland experimental mechanics. Chaotic motion is regarded as a natural extension of the study object in nonlinearvibration. In solid mechanics, the chaotic behavior of buckled beams is studied by numerous researchers [1–4], andmotion of the beams has been well understood. Among the recent research, the periodic and chaotic behavior ofa viscoelastic nonlinear bar subjected to harmonic excitations was investigated by Suire et al. [5] on the based ofa dynamics model established with implementation of Galerkin principle. The periodic and chaotic response of aslender beam with an attached mass under vertical base excitation was also reported [6]. However, a significantly lessnumber of archival publications is available in investigating the chaotic properties of plates and shells. The forcedresponse of a nearly square plate, the nonlinear dynamics of a shallow arch, and the chaotic motion of a circular plateand a cylindrical shell are a few typical studies in mechanical and structural systems found in the research [7–9].Moreover, the single mode method is usually employed in the analysis of nonlinear dynamic systems. A typical

∗Corresponding author. E-mail: [email protected].

ISSN 1070-9622/04/$17.00 2004 – IOS Press and the authors. All rights reserved

534 L. Dai et al. / A single and double mode approach to chaotic vibrations of a cylindrical shell with large deflection

L

2Rox

xQxQ

rQ

Fig. 1. A pinned shell.

example can be found from the article by Moon [2]. This article significantly contributes to the chaotic response ofan elastic beam subjected to a periodic excitation with nonlinear boundary conditions and provided the criterion forchaos on the basis of a single mode model. In fact, most of the studies on nonlinear behavior of elastic elementsutilize the single mode method. Based on the current literature, the differences between the single and double orhigher mode methods have not been attended.

Cylindrical shells are widely used in civil, mechanical and petroleum engineering practices. However, a compre-hensive understanding of the nonlinear behavior of the shells under dynamical loading is far from being reached.Among the available publications, for instance, a systematical and thorough study on elastic cylindrical shells of largedeflection subjected to multi-axial exertions has not been found. The goal of the present research is to investigatethe nonlinear behavior of an elastic cylindrical shell under excitations in longitudinal and radial directions. Largedeflection of the shell is to be taken into consideration. Equations of motion for the cylindrical shell will be derivedwith both single and double modes. Nonlinear behavior of the shell, such as chaos, will be studied. The criteria forchaos of the cylindrical shell will be developed and the chaotic behavior of the transverse vibration of the shell willbe investigated through a numerical analysis by the P-T method [10]. Results generated by single and double modemodels will be compared and the differences of the two models will be identified and analyzed.

2. Governing equations

A pinned elastic cylindrical shell as shown in Fig. 1 will be studied. The cylindrical shell has diameter2R, thicknessh and lengthL and is subjected to uniformly distributed harmonic excitations,Q x andQr, in the longitudinal andradial directions respectively.

The excitationsQx andQr are expressed in the following form:

Qx =qx + qx0 cosωxt, Qr =qr + qr0 cosωrt (1)

The dynamic equation of the shell with large deflection can then be given in the following form with utilization ofvon Karman’s theory for large deflection of shells [11].

L(W, t) =D∇4W − 1R

∂2ϕ

∂x2− ∂2ϕ

∂x2· ∂

2W

∂y2+ 2 · ∂

2ϕ

∂x∂y· ∂

2W

∂x∂y− ∂2ϕ

∂y2− ∂2ϕ

∂y2· ∂

2W

∂x2+ ρh

∂2W

∂t2(2a)

+c∂2W

∂t2+ c

∂W

∂t−Qr = 0

∇4ϕ = Eh

[(∂2W

∂x∂y

)2

− ∂2W

∂x2· ∂

2W

∂y2− 1R

∂2W

∂x2

](2b)

where

D =Eh3

12(1 − µ2), (3a)

L. Dai et al. / A single and double mode approach to chaotic vibrations of a cylindrical shell with large deflection 535

∇4 =∂4

∂x4+ 2

∂4

∂x2∂y2+

∂4

∂y4(3b)

In Eqs (2a) and (2b),W denotes the radial displacement,ρ the density of the material,c the damping coefficient,µ the Poisson ratio of the material,E the elastic constant andϕ the stress function which gives the in-plane forcesas follows.

Nx =∂2ϕ

∂y2, Ny =

∂2ϕ

∂x2, Nxy = − ∂2ϕ

∂x∂y(4)

Following the Ritz method with two modes, one may have

W (x, y, t) = W1(t) sinπx

Lsin

πy

R+W2(t) sin

2πxL

sinπy

R= W1 sinαx sinβy +W2 sin 2αx sinβy (5)

where

α =π

L, β =

π

R(6)

Trigonometric mode function is widely employed in describing the motion of a shell or plate [12]. Selection of thetrigonometric mode function in the present research is based the consideration of the convenience of the functionsin theoretical analysis for the response of the shell under the uniform loadings.

Substitution of Eq. (5) into Eq. (2b) gives

∇4ϕ = Eh

[(W1αβ cosαx cosβy + 2W2αβ cos 2αx cosβy)2 − (−W1α

2 sinαx sinβy

−4W2α2 sin 2αx sinβy) · (−W1β

2 sinαx sin βy −W2β2 sin 2αx sinβy)

− 1R

(−W1α2 sinαx sinβ − 4W2α

2 sin 2αx sinβ)]

(7)

= Eh

[12α2β2W 2

1 (cos 2αx+ cos 2βy) +14α2β2W1W2(− cosαx+ 9 cos 3αx+ 9 cosαx cos 2βy

− cos 3αx cos 2βy)+2α2β2W 22 (cos 4αx+ cos 2βy)+

α2

R(W1 sinαx sin βy + 4W2 sin 2αx sinβy)

]

The stress functionϕ can be obtained as follows

ϕ = Eh

[12α2β2W 2

1 (A1 cos 2αx+A2 cos 2βy) +14α2β2W1W2(A3 cosαx+A4 cos 3αx

+A5 cosαx cos 2βy +A6 cos 3αx cos 2βy + 2α2β2W 22 (A7 cos 4αx+A8 cos 2βy) (8)

+α2

R(A9W1 sinαx sinβy +A10W2 sin 2αx sinβy)

]+y2

2Qx

whereA1 = 1

16α4 , A2 = 116β4 , A3 = − 1

α4 , A4 = 19α4 , A5 = 9

(α2+4β2)2

A6 = − 1(9α2+4β2)2 , A7 = 1

256α4 , A8 = 116β4 , A9 = 1

(α2+β2)2

A10 = 1(4α2+β2)2

(9)

Define the following shorthand notations:B1 = Eh

2 α2β2A1, B2 = Eh

2 α2β2A2, B3 = Eh

4 α2β2A3, B4 = Eh

4 α2β2A4

B5 = Eh4 α

2β2A5, B6 = Eh4 α

2β2A6, B7 = 2Ehα2β2A7, B8 = 2Ehα2β2A8

B9 = EhR α2A9, B10 = Eh

R α2A10

(10)

The stress functionε can be rewritten as


ϕ = W 21 (B1 cos 2αx+B2 cos 2βy) +W1W2(B3 cosαx+B4 cos 3αx+B5 cosαx cos 2βy

+B6 cos 3αx cos 2βy) +W 22 (B7 cos 4αx+B8 cos 2βy) +B9W1 sinαx sinβy (11)

+B10W2 sin 2αx sinβy +y2

2Qx

Substituting Eqs (5) and (11) into Eq. (2a), one may have

L(W ) =D[W1(α2 + β2)2 sinαx sin βy +W2(4α2 + β2)2 sin 2αx sinβy] − [W 21 (−B1 · 4α2 · cos 2αx)

+W1W2(−B3α2 cosαx−B4 · 9α2 cos 3αx−B5α

2 cosαx cos 2βy −B6 · 9α2 cos 3αx cos 2βy)

+W 22 · (−B7 · 16α2 cos 4αx) −W1B9 · α2 sinαx sinβy −W2 ·B10 · 4α2 sin 2αx sinβ]

·[

1R

+ (−W1β2 sinαx · sinβy −W1β

2 sin 2αx sinβy)]

+ 2[W1W2(B5 · 2αβ sinαx sin 2βy

+B6 · 6αβ sin 3βx sin 2βy) +W1B9 · αβ cosαx cosβy +W2 ·B10 · 2αβ cos 2αx cosβy](12)

· [W1αβ cosαx cos βy +W22αβ cos 2αx cos β] − [W 21 (−B2 · 4β2 cos 2βy)

+W1W2(−B5 · 4β2 cosαx cos 2βy −B6 · 4β2 cos 3βx cos 2βy) +W 22 (−B8 · 4β2 cos 2βy)

−W1B9 sinαx sin βy −W2B10β2 sin 2αx sinβy +Qx] · [−W1α

2 sinαx sin βy

−W2 · 4α2 sin 2αx sinβy] + ρh(W1 sinαx sin βy + W2 sin 2αx sinβy)

+c(W1 sinαx sin βy + W2 sin 2αx sinβy) −Qr = 0

Employing the Galerkin principle{∫∫L(W ) sinαx sin βydxdy = 0∫∫L(W ) sin 2αx sinβydxdy = 0, (13)

the following nonlinear modal equations can be obtained.{W1 +m1W1 +m2W

31 +m3W

21 +m4W1W

22 +m5W1 +m6W1 +m7W

22 +m8 = 0

W2 + n1W2 + n2W32 + n3W2W

21 + n4W2W1 + n5W2 + n6W2 = 0

(14)

where

m1 = cρh , m2 = 2α2β2

ρh (B1 +B2), m3 = − 32α2

3π2ρhR (2B1 + β2B9R)

m4 = α2β2

4ρh (18B4 −B6 − 2B3 + 9B5 + 8B8), m5 = 1ρhR (α2B9 +D(α2 + β2)2)

m6 = α2Qx

ρh , m7 = − 12815

α2

π2ρhR (2B7 + 3β2B10R), m8 = − 16Qr

π2ρh

n1 = cρh , n2 = 8α2β2

ρh (B7 +B8), n3 = α2β2

4ρh (32B2 + 18B4 −B6 + 9B5 − 2B3)

n4 = − α2

180π2ρhR (−3456B6 + 4608β2B9R+ 640B5 + 10368B4 + 4608β2B10R− 1920B3)

n5 = 1ρhR (Dβ4R+ 16Dα4R+ 8Dα2β2R+ 4α2B10), n6 = 4α2Qx

ρh

(15)

LetW1 = c1x1,W2 = c2x2, t = c0τ,m5c20 = 1,m2c

20c

21 = 1, n2c

20c

22 = 1, Eq. (14)

can be written as{x1 + m1√

m5x1 + x3

1 +√

m3√m2m5

x21 + m4

n2x1x

22 + x1 + m6

m5x1 + m7

n2

√m2m5x2

2 + m8m5

√m2m5

= 0

x2 + n1√m5x2 + x3

2 + n3m2x2x

21 + n4√

m2m5x2x1 + n5

m5x2 + n6

m5x2 = 0

(16)


3. Melnikov function for the single mode motion

From Eq. (16), the following nonlinear dynamic equation can be constructed if only one mode is considered.

x+m1√m5

x+ x3 +m3√m2m5

x2 + x+m6

m5x+

m8

m5

√m2

m5= 0 (17)

Two cases are studied using the following common parameters.L = 0.5 m, R = 0.1 m, h = 0.002 m, α = π

L m−1, β = πR m−1

E = 69.7 × 109Pa, ρ = 2.78 × 103 kg/m3, µ = 0.3, c = 1.67 × 107 kg · (m2 · s)−1

qx = 0, qr0 = 0, qr = g × 1.56 × 1010 N/m2(18)

In the first case,qx0 = −106 N/m, and Eq. (17) takes the following form

x+ 0.5x− 3.8x2 + x3 = g∗ cosωτ − 0.82x (19)

which in general can be expressed as

x+ x+ αx2 + βx3 = ε(g cosωt− µx) (20)

whereα, β, andε are system parameters.In the second case,qx0 = −4 × 106 N/m, and Eq. (17) reads

x− x− 3.8x2 + x3 = g∗ cosωτ − 0.82x (21)

The corresponding general form of this system is

x− x+ λ2x2 + λ3x

3 = ε(g cosωt− µx) (22)

a) The Melnikov function of Eq. (20)Whenε = 0, the corresponding unperturbed system is{

x = yy = −x− αx2 − βx3 (23)

Its Hamilton function can be expressed as

H(x, y) =12y2 +

12x2 +

α

3x3 +

β4

4x4 (24)

The three fixed points are (0,0) and((−α ±√α2 − 4β)/2β, 0). For the homoclinic orbit of the system,

x0(t), y0(t))T , the Melnikov function [13] is defined as follows.

M(t0) =∫ +∞

−∞y0(t){−µy0(t) + g cos[ω(t+ t0)]}dt = −µA+ gB(t0) (25)

where

A =∫ +∞

−∞y20(t)dt (26)

B(t0) =∫ +∞

−∞y0(t) cos[ω(t+ t0)]dt =

√B2

1 +B22 cos[ω(t+ τ0)] (27)

B1 =∫ +∞

−∞y0(t) cosωtdt, B2 =

∫ +∞

−∞y0(t) sinωtdt, τ0 = arctan(B2/B1). (28)

One may therefore obtain

M(t0) = −µA+ g√B2

1 +B22 cos[ω(t0 + τ0)] (29)


The homoclinic orbitx0(t), y0(t))T can be determined by using the following equation

H(x0, y0) =12y2 +

12x2 +

α

3x3 +

β

4x4 (30)

where(x0, y0)T is the saddle point, and we have

dt/dx = ±N−1 (31)

where

N =

√2H − x2 − 2α

3x3 − β

2x4, H = H(x0, y0). (32)

With these equations,A,B1 andB2 can be determined as

|A| =∣∣∣∣∫ +∞

−∞y20(t)dt

∣∣∣∣ = 2

∣∣∣∣∣∫ x(+∞)

x(0)

Ndx

∣∣∣∣∣ , |B1| =∣∣∣∣∫ +∞

−∞y0(t) cosωtdt

∣∣∣∣ = 0 (33)

|B2| =∣∣∣∣∫ +∞

−∞y0(t) sinωtdt

∣∣∣∣ = 2

∣∣∣∣∣∫ x(+∞)

x(0)

sin(ω∫ x

x(0)

N−1ds)dx

∣∣∣∣∣ (34)

When the Melnikov function has simple zero points, the stable and unstable manifolds intersect. The Poincaremap has a horseshoes; there therefore exists a strange constant set [14]. As such, it is possible for the dissipativesystem to enter chaos. According to the Melnikov method, the criterion for chaos can be determined as

|g/µ| > A/√B2

1 + B22 =

∣∣∣∫ x(+∞)

x(0) Ndx∣∣∣∣∣∣∫ x(+∞)

x(0)sin(ω

∫ x

x(0)N−1ds)dx

∣∣∣ (35)

b) The Melnikov function for Eq. (22)Equation (22) can be rewritten in the following form{

x = yy = x− λ2x

2 − λ3x3 + ε(g cosωt− µx) (36)

Its unperturbed system is

x− x+ λ2x2 + λ3x

3 = 0. (37)

This has the first integration

12x2 − 1

2x2 +

13λ2x

3 +14λ3x

4 = H (38)

With this expression, different values ofH indicate different curves in the corresponding phase portraits. TheHvalues are determined with the initial conditions. The three fixed points areO,A andB, whereO is a hyperbolic-typefixed point and the others are stable fixed points, such that

O(0, 0), A

(−−λ2 −

√λ2

2 + 4λ3

2λ3, 0

), B

(−λ2 +

√λ2

2 + 4λ3

2λ3, 0

)

Now let us find the homoclinic motion; withH = 0. In this case

x = ±√x∗2 − 2

3λ2x∗3 − 1

2λ3x∗4 (39)

The homoclinic orbit takes the following form.∣∣∣∣∣∣√

1 − 23λ2x∗ − 1

2λ3x∗2 − 1

x∗− λ2

3

∣∣∣∣∣∣ = C2e±√

t (40)


µg

Zone of chaos )(ω

µR

g=

ω

Fig. 2. Sketch of chaos criterion.

The corresponding Melnikov function is

M(τ) =∫ +∞

−∞x∗ · ψ[x∗, x∗, ω(t+ τ)]dt (41)

where

Ψ = Ψ(x, x, ωt) = g cosωt− µx (42)

therefore

M(τ) = −µ∫ +∞

−∞x∗2dt+ g

∫ +∞

−∞x∗ cosω(t+ τ)dt. (43)

According to the residual law, the Melnikov function can be expressed as

M(τ)=− 4µ3λ3

[1+

λ22

3λ3+λ2 · 2λ2

2+9λ3

9λ3

√2λ3

arcsin√

2λ2√2λ2

2+9λ3

]+4πgω · sinω(ξ + τ)(chωη1−chωη2)√

2λ3(ch2πω − 1)(44)

where

ξ = ln[

13C2

√2λ2

2+9λ3

2

]

η1 = π + arctan(

3λ2

√λ32

),

η2 = π − arctan(

3λ2

√λ32

) (45)

whereC2 is a constant and can be determined by the initial conditions. When the Melnikov function has simple zeropoints, the nonlinear system may lead to a Smale horseshoe type of chaos. That implies that the criterion of chaosin this case should take the following form:

g

µ� 2(ch2πω − 1)

3√

2ω√λ3(chωη1 − chωη2)

[1 +

λ22

3λ3+ λ2 · 2λ2

2 + 9λ3

9λ3

√2λ3

arcsin√

2λ2√2λ2

2 + 9λ3

]= R(ω), (46)

This criterion is graphically demonstrated in Fig. 2.As illustrated in Fig. 2 that chaos occurs when the value ofg/µ is greater thanR(ω) expressed in Eq. (46).

4. Numerical simulations

According to the nonlinear dynamic Eqs (16) and (17), the following numerical computations are carried out bythe P-T method [10]. The parameters used are those as indicated in Eq. (18). The initial conditions of the numericalsimulations for the single mode model arex = 0.0, andx = 0.0 ast = 0 For the double mode model, the initialconditions arex1 = 0.0, x1 = 0.0, x2 = 0.00001, x2 = 0.0 ast = 0


6 7 8-1 8

-1 7

-1 6

-1 5 0 15- 1 0 0

0

1 0 0

3 9 0 3 9 5 4 0 0-1 5

0

15

Poincare map phase portrait time history

(a) single mode model x

6 7 8- 1 8

- 1 7

- 1 6

- 1 5 0 1 5- 1 0 0

0

1 0 0

3 9 0 3 9 5 4 0 0- 1 5

0

1 5


(b) double mode model 1x

-1 0 1-1

0

1

-2 0 2-1 5

0

15

3 8 0 3 9 0 4 0 0-2

0

2


(The vertical axis is amplified by 6510 times)

(c) double mode model 2x

Fig. 3. Comparison between single and double mode models (ω = π, g = 610).

The displacement modes used in this paper based on the single and double mode models can be expressed asfollows.

w(x, y, t) = x(t) sinπx

Lsin

πy

R, for the single mode model (47)

W ∗(x, y, t) = x1(t) sinπx

Lsin

πy

R+ x2(t) sin

2πxL

sinπy

R, for the double mode model (48)


5 10 15- 1 0 0

0

1 0 0

-2 0 0 20- 2 0 0

0

2 0 0

3 8 0 3 9 0 4 0 0-2 0

0

20



5 10 15- 1 0 0

0

1 0 0

-2 0 0 20- 2 0 0

0

2 0 0

3 8 0 3 9 0 4 0 0-2 0

0

20



-1 0 1-1

0

1

-1 0 1-1 0

0

10

3 8 0 3 9 0 4 0 0-0.6

0

0.6





If the single mode model is tenable, the following expression should be true.

W (x, y, t) ≈W ∗(x, y, t), x(t) ≈ x1(t),∣∣∣∣x2(t) − x1(t)

x1(t)

∣∣∣∣ << 1 (49)

As can be seen from Fig. 3 to Fig. 5, the results obtained from the single and double mode models are completelyidentical when,ω = π, g = 610, ω = π, g = 1400 andω = π, g = 1400 if the other parameters are kept as thesame. This is to say; in these cases the single mode model analysis is sufficient and correct.


8 12 160

25

50

-2 0 0 20- 2 0 0

0

2 0 0

3 5 0 3 5 5 3 6 0-2 0

0

20



8 12 160

25

50

-2 0 0 20- 2 0 0

0

2 0 0

3 5 0 3 5 5 3 6 0-2 0

0

20



-1 0 1-1

0

1

-2 0 2-3 0

0

30

3 4 0 3 5 5 3 7 0-2

0

2




Fig. 5. Comparison between single and double mode models (ω = 1.1π, g = 1400).

As illustrated in Fig. 6, 7 and 8, the single and double mode models generate completely different results. Chaotic

behavior of the motion is identified by the results of the double mode model for the cases in which∣∣∣x2(t)−x1(t)

x1(t)

∣∣∣ << 1is not satisfied; whereas the corresponding results created by the single mode model indicate periodic or quasiperiodicbehavior as shown in the figures. In other words, the single mode method, which is widely used in the dynamicanalysis, may lead to incorrect conclusions in these cases. It is therefore clear that the single mode method has limitsin analyzing the elastic structure’s nonlinear response. For the cases as indicated above, double mode or higher ordermode method should be used for reliable results.


8 9 10 110

25

50

-1 5 0 15-5 0

0

50

1 9 8 0 1 9 9 0 2 0 0 0-1 5

0

15


(a) single mode x

-2 0 0 20- 1 0 0

0

1 0 0

-1 0 0 10- 1 0 0

0

1 0 0

1 9 8 0 1 9 9 0 2 0 0 0-1 0

0

10


(b) double mode 1x

-2 0 0 20- 1 0 0

0

1 0 0

-2 0 0 20- 1 5 0

0

1 5 0

1 9 8 0 1 9 9 0 2 0 0 0-2 0

0

20


(c) double mode 2x


5. Concluding remarks

The characteristics of the nonlinear transverse vibration of an elastic cylindrical shell with large deflection andunder uniform harmonic excitations are investigated in the present research based on the single and double modemodels. In the current literature, systematical studies on such a shell exerted by multiple loadings are not found.Dynamic nonlinear modal equations for the motion of the cylindrical shell are derived on the basis of the models.Two types of nonlinear dynamic equations are obtained with a variety of system and loading parameters. Chaoticbehavior of the cylindrical shell is evident as found in the present research. The criteria of chaos are determinedfor the motion of the cylindrical shell with the Melnikov function for the single mode model. Differences betweenthe single and double mode models are apparent and a comparison between the results generated by the single anddouble mode models is carried out using numerical computations via the P-T method.


6 6.5 7-2 2

-2 1

-2 0

-1 5 0 15- 1 0 0

0

1 0 0

1 9 9 0 1 9 9 5 2 0 0 0-1 5

0

15


(a) single mode x

-2 0 0 20- 1 5 0

0

1 5 0

-1 5 0 15- 1 0 0

0

1 0 0

1 9 8 0 1 9 9 0 2 0 0 0-2 0

0

20


(b) double mode 1x

-2 0 0 20- 2 0 0

0

2 0 0

-2 0 0 20- 2 0 0

0

2 0 0

1 9 8 0-2 0

0

20


(c) double mode 2x

1 9 9 0 2 0 0 0


Based on the theoretical and numerical analyses of the present research, it can also be stated thatW 2(ξ, t) =T1(t)W ∗

1 (ξ) +T2(t)W ∗2 (ξ) of the double mode model is close toW1(ξ, t) = T (t)W ∗

1 (ξ) of the single mode model,

provided that∣∣∣x2(t)−x1(t)

x1(t)

∣∣∣ << 1 is complied. However,w2(ξ, t) is a better approximation solution overW1(ξ, t),

when the conditions of∣∣∣ T (t)T1(t)

∣∣∣ ≈ 1 and∣∣∣T1(t)W

∗2 (ξ)−T1(t)W∗

1 (ξ)T1(t)W∗

1 (ξ)

∣∣∣ << 1 are satisfied. In this case, ifT (t) is chaotic,

T1(t) is also chaotic correspondingly, and vice versa. When the condition∣∣∣T2(t)W∗

2 (ξ)−T1(t)W∗1 (ξ)

T1(t)W∗1 (ξ)

∣∣∣ << 1 is not

maintained, the single mode method, which is conventionally used in the literature for analyzing the nonlinearbehavior of an elastic system, may lead to incorrect conclusions and is therefore no longer reliable. The doublemode or higher order mode models should then be employed in these cases.


0 10 20-5 0

0

50

-2 0 0 20- 2 0 0

0

2 0 0

1 9 8 0 1 9 9 0 2 0 0 0-2 0

0

20


(a) single mode x

0 10 20- 2 0 0

0

2 0 0

-2 0 0 20- 2 0 0

0

2 0 0

1 9 8 0 1 9 9 0 2 0 0 0-2 0

0

20


(b) double mode 1x

-1 0 0 10- 1 0 0

0

1 0 0

-1 0 0 10- 2 0 0

0

2 0 0

1 9 8 0 1 9 9 0 2 0 0 0-1 0

0

10


(c) double mode 2x


Acknowledgement

This research is supported by NSERC, CFI and NNSFC.

References

[1] P. Holms and J. Marsden, A Partial Differential Equation with Infinitely Many Periodic Orbits: Chaotic Oscillation of a Forced Beam,Arch. Rat. Mech. and Analysis 2 (1981), 135–165.

[2] F.C. Moon and S.W. Shaw, Chaotic Vibration of A Beam with Nonlinear Boundary Conditions,Non-linear Mech 18 (1983), 230–240.[3] P.D. Baran, Mathematical Models Used in Studying the Chaotic Vibration of Buckled Beam,Mechanics Research Communications 21

(1994), 189–196.


[4] V. Keragiozov and D. Keoagiozova, Chaotic Phenomena in the Dynamic Buckling of an Elastic-Plastic Column Under an Impact,NonlinearDynamics 13 (1995), 1–16.

[5] G. Suire and G. Cederbaum,Periodic and Chaotic Behavior of Viscoelastic Nonlinear (Elastica) Bars under Harmonic Excitations 37(1995), 753–772.

[6] S. Dwivedy and R. Kar, Dynamics of a Slender Beam with an Attached Mass under Combination Parametric and Internal Resonances,Part II: Periodic and Chaotic Responses,J. Sound Vib. 222 (1999), 281–305.

[7] X.L. Yang and P.R. Sethna, Nonlinear Forced Vibrations of a Nearly Square Plate-Antisymmetric Case,J. Sound Vib. 155 (1992), 413–441.[8] W. Tien, N. Namachchivaya and N. Malhotra, Non-Linear Dynamics of a Shallow Arch Under Periodic Excitation-II.1:1 Internal Resonance,

Int. J. Non-Linear Mech. 29 (1994), 367–385.[9] Q. Han, H.Y. Hu and G.T. Yang, A Study of Chaotic Motion in Elastic Cylindrical Shells,Eur. J. Mech. A/Solids 18 (1999), 351–360.

[10] L. Dai and M.C. Singh, A New Approach to Approximate and Numerical Solutions of Oscillatory Problems,J. Sound Vib. 263 (2003),535–548.

[11] T. von Karman, Festigkeits Probleme in Machinenbau,Encl. Der math. Wiss. 4 (1910), 348–351.[12] A.C. Ugural,Stress in Plates and Shells, McGraw-Hill, Boston, 1999.[13] S. Wiggins,Chaotic Transport in Dynamical Systems, Springer-Verlag, New York, 1992.[14] R. Temam,Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997.

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