Research Of Transverse Vibration Of An Axially Moving Viscoelastic Sandwich Beam

The viscoelastic damped sandwich structures are usually composed of two thin constrainted layers with high stiffness and a thick soft core with large damping. It is widely used to reduce vibration and noise in high speed train's carriage, aircraft envelope,spacecraft deployment accessories, missile in high speed flight and etc, as this material has the advantages of light in weight, fatigue resistance, sound absorption and thermal insulation.The previous researches on axially moving material indicate that the axially moving will induce transverse vibration and the strength of vibration is closely related to the axially moving velocity. The system will be unstable when the velocity reaches or exceeds the critical value. The researches of the effect of axially moving on the dynamical characteristic,such as natural frequencies, loss factors, bifurcations and chaos of the viscoelastic damping sandwich structure are rare. This dissertation simplifies the model of viscoelastic damping sandwich structure traveling at high speed to the model of axially moving viscoelasticsandwich beam. The influences of axially moving velocity,viscoelastic coefficient,initial tension, the thickness of soft core, the frequency and amplitude of external excitation on the stability, the characteristic of vibration and nonlinear dynamical behavior of the axially moving viscoelastic sandwich beam are investigated.In Chapter 1, the background and significance of this work are presented. From the aspects of sandwich structure, axially moving structure and axially moving sandwich structure, a review on the results and current situation of researches and the main problem in these studies are illustrated. A brief statement on the research plan of this dissertation is given.By comprehensively considering the effects of axially moving, initial tension,geometric nonlinearity and structural damping, the governing equation of the axially moving viscoelastic sandwich beam which is incompressible in the transverse direction is established by using the principle of D'Alembert. The detailed deduced process as well as the specific expressions are given in Chapter 2. The accuracy and usability of this model are verified by the results of finite element and previous models. In this Chapter, as the core layer is composed of viscoelastic material, Kelvin-Viogt differential constitute model is adopted here and material derivative is employed, rather than using just the partial derivative of time.Secondly, according to the shortcoming of traditional sandwich beam theory that the sandwich beam is incompressible in the thickness direction, a new sandwich beam theory is proposed by introducing independent variables in terms of the displacements of top constrainted layer, middle plane of soft core and bottom constrainted layer. The displacement of soft core is approximately equal to a quadratic polynomial to be determined in the thickness direction. Using continuity conditions along the constrainted layer and soft core,transverse displacement of soft core is obtained. Normal strain and shearing strain of soft core in the thickness direction are also got. Based on the principal of Hamilton, the governing equation of axially moving viscoelastic soft sandwich beam is established and it is noticed that the incompressible model of sandwich beam is the special form of soft sandwich beam.In Chapter 3, the dynamical characteristic of an axially moving viscoelastic sandwich beam with simple supported boundary conditions is studied by using the method of Galerkin truncation and complex modal analysis. The influences of the ratio of core layer, axially moving velocity, viscoelastic coefficient and axial tension on the natural frequencies and critical velocity are discussed. The effect of axially moving velocity on the asymmetry of complex eigenfunctions is investigated as well.In Chapter 4, the method of Galerkin truncation is used to solve the governing equation of an axially moving viscoelastic soft sandwich beam. The partial differential equations are discretized into ordinary differential equations, natural frequencies and eigenvectors are obtained by solving eigenvalue problem, and the modal shapes, the modal functions, the response of free vibration are also obtained. The effects of axially moving velocity and the thickness of soft core on frequencies and critical velocity are investigated. The result also shows that the incompressible model of sandwich beam is the special form of soft sandwich beam, and the soft sandwich beam has some properties that incompressible sandwich beam doesn't have.In Chapter 5, the transverse forced vibration of an axially moving soft sandwich beam is investigated using the method of Green's function and the exact solution in closed form is obtained and is compared with the results got by the method of finite element. The results indicate that the exact solution obtained by the method of Green's function has a higher accuracy.Non-linearly parametric resonance of an axially moving viscoelastic sandwich beam is investigated in Chapter 6. The beam is moving with a time-dependent velocity, namely a harmonically varied velocity about the mean velocity. Based on the governing equation established in Chapter 2,the partial differential equation is discretized into nonlinear ordinary differential equations via the method of Galerkin truncation and then the steady-state response are obtained using the method of multiple scales, an approximate analytical method. The amplitude of periodic vibration of the system is got by eliminating secular terms. Additionally, the stability conditions of trivial and nontrivial solutions are analyzed using the Routh-Hurwitz criterion. Eventually, the effects of various parameters such as the ratio of core layer, mean velocity, the amplitude of axially moving velocity fluctuation and viscoelastic coefficient on response curves and unstable regions are investigated. The analytical solutions obtained by the method of multiple scales are compared with those obtained by numerically integrating the ordinary differential equations by the method of fourth order Runge-Kutta and the results are found to be in good agreement.The nonlinear dynamical behavior and stability of forced vibration of an axially moving viscoelastic sandwich beam are studied in Chapter 7. The method which we used in this chapter is the similar as in Chapter 6. The approximate solutions are obtained. The stability and bifurcation are studied in detail when the low-frequency primary resonance and high-frequency primary resonance occurring, and the influence of various parameters on response curves and unstable regions are also researched.Non-linearly parametric resonances of an axially accelerating moving viscoelastic sandwich beam with time-dependent tension are investigated in Chapter 8. The system is subjected to a time varied velocity and a harmonic changed axial tension, namely the governing equation contains linear parametric terms and two frequencies, one is the frequency of axially moving velocity and the other one is the frequency of axial tension. The method of multiple scales is applied directly to the governing equation. The amplitude of steady-state response is obtained by eliminating secular terms. The stability and bifurcation are studied under two different resonance conditions, simultaneous resonance and subharmonic resonance. The influences of various parameters such as the amplitude of axial tension fluctuation, the phase angle between the two frequencies on response curves and unstable regions are investigated. With the help of numerical results, it has been shown that the vibration of the sandwich beam can be significantly controlled by selecting appropriate parameters.Finally, the research contents and the results of this dissertation are summed up, and a brief plan for future researches is given.