Study On Internal Resonance And Bifurcations Of Thin Circular Cylindrical Shells

Thin circular cylindrical shells are widely used in many branches of industry, including civil engineering, aerospace engineering, mechanical engineering, ocean engineering, and automobile engineering. In actual applications, these structures are commonly subjected to various dynamic loads whose size, direction, and acting points change with time quickly. So the design and study of these structures always need dynamic analysis, which has become indispensable fundamental research for the development of engineering technology, and has important scientific value. Nowadays, the advances in composite manufacturing methods have contributed to the increased usage of laminated composite materials in many modern applications. When compared to traditional metallic materials, laminated composites offer advantages such as higher strength-to-weight and stiffness-to-weight ratios, improved chemical and environmental resistance. Thus laminated composite circular cylindrical shells now constitute a large percentage of engineering structures.When circular cylindrical shells vibrate, each point on the shells will have displacements which are represented by u, v and w in cylindrical coordinates. These three displacement components are all the functions of space coordinates x,6, r and time t. In general, it is very difficult to find the closed form solution for the nonlinear vibrational problems. In this paper, the partial differential equations are discretized by Galerkin’s method. Some dynamic behaviour including periodic vibration, stability, resonant response and bifurcations for transverse vibration of laminated composite circular cylindrical shells are researched by using approximate analytical methods. The main work is as follows:Nonlinear dynamic equation for the transverse vibration of laminated circular cylindrical shells is obtained by using Donnell’s simplified shell theory. Nonlinearities due to large-amplitude shell motion are considered, with account taken of the effects of viscous structure damping and dynamic Young’s modulus. The system is discretized by Galerkin’s method while a model involving two neighboring axial modes is adopted. The method of averaging is developed to study the nonlinear frequency-responses of the multi-degrees-of-freedom system, and the results show internal resonance characteristics. The effects of amplitude of force and small parameters on the nonlinear frequency-responses are discussed, and the stability of steady-state solutions is also analyzed in detail.The mechanism of internal resonance of clamped-free laminated composite circular cylindrical shells containing inviscid, incompressible quiescent fluid is studied. Based on linear potential flow theory, Donnell’s simplified shell theory and classical laminated shell theory, nonlinear dynamic equation is derived, in which the effect of fluid-structure interaction is considered. Linear potential flow theory is applied to describe the fluid-structure interaction. In fact, the amplitude of shell displacements remains small enough for linear fluid mechanics to be adequate. Experimental investigation of free vibration of the shell is carried out. It can be found the experimental results agree with theoretical values obtained by using dynamic Young’s modulus very well. The method of averaging is developed to study the nonlinear frequency-responses of the multi-degrees-of-freedom system with two neighboring axial modes participation, and the results show1:1internal resonance characteristics. The mechanism of internal resonance of the system is analyzed in detail. The stability of steady-state solutions is analyzed by using Lyapunov’s first approximation theory, and the effects of amplitude of force and phase angle on the complex dynamic behaviour are also discussed.Nonlinear vibrational equation for the transverse vibration of axially moving laminated circular cylindrical shells is obtained by using Donnell’s simplified shell theory, in which the effect of dynamic Young’s modulus is considered. The method of averaging is used to study the nonlinear frequency-responses of the multi-degrees-of-freedom system with two neighboring axial modes participation. Results show that, the frequency-responses curves of axially moving laminated circular cylindrical shells show hardening spring characteristics. Due to the frequencies of the two modes selected are very near, response values jump from one to the other between two stable frequency-response curves continuously. Energy of the system transfers between the two modes. So it exists internal resonance in the system.Without introducing Airy stress function, Nonlinear vibrational equation of axially moving laminated composite circular cylindrical shells is obtained by using the nonlinear Donnell’s shallow shell theory. Though the form of the equation is a bit complicated, the solving difficulty of this problem is greatly reduced because of the absence of Airy stress function. Galerkin method is used to discretize the basic differential equation and ordinary differential equations are gained. The method of harmonic balance is applied to analyze the system, and the stability of steady-state solutions is analyzed by using Lyapunov’s first approximation theory. The effects of different physical parameters on the internal resonance response of axially moving laminated circular cylindrical shells are researched in detail.Bifurcations of a cantilever cylindrical shell under a concentrated harmonic excitation moving in a concentric circular path and that of a simply-supported laminated composite circular cylindrical shell with geometric large-amplitude vibrations are investigated, respectively. Results show that there exists traveling wave vibration for the steel shell. When principal resonance occurs, bifurcation set of the system is divided into two parts, namely forward and backward waves bifurcation sets of the principal resonance, respectively; when the second order resonance occurs, bifurcation set of the system is also divided into two parts, namely forward and backward waves bifurcation sets of the second order resonance, respectively. Principal resonance is not coupled with the second order resonance, and they are decoupled in the averaged equations. Besides, the bifurcation phenomenon of laminated composite circular cylindrical shells with1:1internal resonance is quite different from systems without internal resonance:In each bifurcation area of the system without internal resonance, there is only phase trajectories for one mode, but in each bifurcation area of the system with1:1internal resonance, phase trajectories for the first-order and second-order modes both appear, which show the first-order and second-order modes are both excited and they are coupled with each other; the bifurcation plane of the system with1:1internal resonance is divided by several curves, showing the nonlinear vibrational characteristics of the system with1:1 internal resonance is more sensitive to the system parameters; phase trajectories for the system with1:1internal resonance appear many sorts and numbers singular points. Its bifurcation phenomenon is more complex, and its bifurcation forms are more varied, which show the modes coupling characteristics and complex dynamic behavior of internal resonance systems.