A single and double mode approach to chaotic vibrations of a cylindrical shell with large deflection

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


Introduction
In recent years, chaos in nonlinear dynamic systems has aroused more and more interest in the field of theoretical and experimental mechanics.Chaotic motion is regarded as a natural extension of the study object in nonlinear vibration.In solid mechanics, the chaotic behavior of buckled beams is studied by numerous researchers [1][2][3][4], and motion of the beams has been well understood.Among the recent research, the periodic and chaotic behavior of a viscoelastic nonlinear bar subjected to harmonic excitations was investigated by Suire et al. [5] on the based of a dynamics model established with implementation of Galerkin principle.The periodic and chaotic response of a slender beam with an attached mass under vertical base excitation was also reported [6].However, a significantly less number of archival publications is available in investigating the chaotic properties of plates and shells.The forced response of a nearly square plate, the nonlinear dynamics of a shallow arch, and the chaotic motion of a circular plate and a cylindrical shell are a few typical studies in mechanical and structural systems found in the research [7][8][9].Moreover, the single mode method is usually employed in the analysis of nonlinear dynamic systems.A typical example can be found from the article by Moon [2].This article significantly contributes to the chaotic response of an elastic beam subjected to a periodic excitation with nonlinear boundary conditions and provided the criterion for chaos on the basis of a single mode model.In fact, most of the studies on nonlinear behavior of elastic elements utilize the single mode method.Based on the current literature, the differences between the single and double or higher mode methods have not been attended.
Cylindrical shells are widely used in civil, mechanical and petroleum engineering practices.However, a comprehensive 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 large deflection subjected to multi-axial exertions has not been found.The goal of the present research is to investigate the nonlinear behavior of an elastic cylindrical shell under excitations in longitudinal and radial directions.Large deflection of the shell is to be taken into consideration.Equations of motion for the cylindrical shell will be derived with both single and double modes.Nonlinear behavior of the shell, such as chaos, will be studied.The criteria for chaos of the cylindrical shell will be developed and the chaotic behavior of the transverse vibration of the shell will be investigated through a numerical analysis by the P-T method [10].Results generated by single and double mode models will be compared and the differences of the two models will be identified and analyzed.

Governing equations
A pinned elastic cylindrical shell as shown in Fig. 1 will be studied.The cylindrical shell has diameter 2R, thickness h and length L and is subjected to uniformly distributed harmonic excitations, Q x and Q r , in the longitudinal and radial directions respectively.
The excitations Q x and Q r are expressed in the following form: The dynamic equation of the shell with large deflection can then be given in the following form with utilization of von Karman's theory for large deflection of shells [11]. where 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 and ϕ the stress function which gives the in-plane forces as follows.
Following the Ritz method with two modes, one may have where Trigonometric mode function is widely employed in describing the motion of a shell or plate [12].Selection of the trigonometric mode function in the present research is based the consideration of the convenience of the functions in theoretical analysis for the response of the shell under the uniform loadings.
Substitution of Eq. ( 5) into Eq.(2b) gives The stress function ϕ can be obtained as follows Define the following shorthand notations: The stress function ε can be rewritten as Substituting Eqs ( 5) and ( 11) into Eq.(2a), one may have the following nonlinear modal equations can be obtained.
where 14) can be written as

Melnikov function for the single mode motion
From Eq. ( 16), the following nonlinear dynamic equation can be constructed if only one mode is considered.
Two cases are studied using the following common parameters.
In the first case, q x0 = −10 6 N/m, and Eq. ( 17) takes the following form which in general can be expressed as where α, β, and ε are system parameters.In the second case, q x0 = −4 × 10 6 N/m, and Eq. ( 17) reads The corresponding general form of this system is a) The Melnikov function of Eq. ( 20) When ε = 0, the corresponding unperturbed system is Its Hamilton function can be expressed as The three fixed points are (0,0) and ((−α ± α 2 − 4β)/2β, 0).For the homoclinic orbit of the system, x 0 (t), y 0 (t)) T , the Melnikov function [13] is defined as follows. where One may therefore obtain The homoclinic orbit x 0 (t), y 0 (t)) T can be determined by using the following equation where (x 0 , y 0 ) T is the saddle point, and we have where With these equations, A, B 1 and B 2 can be determined as When the Melnikov function has simple zero points, the stable and unstable manifolds intersect.The Poincare map has a horseshoes; there therefore exists a strange constant set [14].As such, it is possible for the dissipative system to enter chaos.According to the Melnikov method, the criterion for chaos can be determined as b) The Melnikov function for Eq. ( 22) Equation ( 22) can be rewritten in the following form Its unperturbed system is This has the first integration With this expression, different values of H indicate different curves in the corresponding phase portraits.The H values are determined with the initial conditions.The three fixed points are O, A and B, where O is a hyperbolic-type fixed point and the others are stable fixed points, such that Now let us find the homoclinic motion; with H = 0.In this case The homoclinic orbit takes the following form.
The corresponding Melnikov function is where According to the residual law, the Melnikov function can be expressed as where where C 2 is a constant and can be determined by the initial conditions.When the Melnikov function has simple zero points, the nonlinear system may lead to a Smale horseshoe type of chaos.That implies that the criterion of chaos in this case should take the following form: This criterion is graphically demonstrated in Fig. 2.
As illustrated in Fig. 2 that chaos occurs when the value of g/µ is greater than R(ω) expressed in Eq. (46).

Numerical simulations
According to the nonlinear dynamic Eqs ( 16) and (17), the following numerical computations are carried out by the P-T method [10].The parameters used are those as indicated in Eq. ( 18).The initial conditions of the numerical simulations for the single mode model are x = 0.0, and ẋ = 0.0 as t = 0 For the double mode model, the initial conditions are x 1 = 0.0, ẋ1 = 0.0, x 2 = 0.00001, ẋ2 = 0.0 as t = 0 The displacement modes used in this paper based on the single and double mode models can be expressed as follows.
w(x, y, t) = x(t) sin πx L sin πy R , for the single mode model (47) , for the double mode model (48) If the single mode model is tenable, the following expression should be true.
As can be seen from Fig. 3 to Fig. 5, the results obtained from the single and double mode models are completely identical when, ω = π, g = 610, ω = π, g = 1400 and ω = π, g = 1400 if the other parameters are kept as the same.This is to say; in these cases the single mode model analysis is sufficient and correct.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) << 1 is not satisfied; whereas the corresponding results created by the single mode model indicate periodic or quasiperiodic behavior as shown in the figures.In other words, the single mode method, which is widely used in the dynamic analysis, may lead to incorrect conclusions in these cases.It is therefore clear that the single mode method has limits in analyzing the elastic structure's nonlinear response.For the cases as indicated above, double mode or higher order mode method should be used for reliable results.

Concluding remarks
The characteristics of the nonlinear transverse vibration of an elastic cylindrical shell with large deflection and under uniform harmonic excitations are investigated in the present research based on the single and double mode models.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.Chaotic behavior of the cylindrical shell is evident as found in the present research.The criteria of chaos are determined for the motion of the cylindrical shell with the Melnikov function for the single mode model.Differences between the single and double mode models are apparent and a comparison between the results generated by the single and double mode models is carried out using numerical computations via the P-T method.Based on the theoretical and numerical analyses of the present research, it can also be stated that W 2 (ξ, t) = T 1 (t)W * 1 (ξ) + T 2 (t)W * 2 (ξ) of the double mode model is close to W 1 (ξ, t) = T (t)W * 1 (ξ) of the single mode model, provided that x2(t)−x1(t) x1(t) << 1 is complied.However, w 2 (ξ, t) is a better approximation solution over W 1 (ξ, t), when the conditions of << 1 is not maintained, the single mode method, which is conventionally used in the literature for analyzing the nonlinear behavior of an elastic system, may lead to incorrect conclusions and is therefore no longer reliable.The double mode or higher order mode models should then be employed in these cases.
<< 1 are satisfied.In this case, if T (t) is chaotic, T 1 (t) is also chaotic correspondingly, and vice versa.When the condition