Complex Mode Analysis For Transverse Vibration Of Beams On Viscoelastic Foundation

Finite beams resting on foundations are widely used in engineering structures, such as roads, airport pavements, and railway engineering equipments, the vibration analysis of such a system has interested many researchers in science and engineering. As an important and basic parameter, natural frequency reflect structure characteristics, research on it helps to solve the problems of stucture resonance and fatigue damage to infrastructure. It is common engineering problems that transverse vibrations of foundation beams under moving load caused damage to infrastructure.In this dissertation, the complex mode analysis is applied to investigate transverse vibrations of finite beams on viscoelastic foundations, and the differential quadrature scheme is applied to verify the result. The explicit formulae for natural frequencies are derived. As a reference to structure design, it has important theoretical and engineering significance, and a broad application prospect. The dissertation are organized as following:The first chapter introduces the current research, and presents the aim, significance, main contents, and main innovations of this dissertation.In chapter 2, the free transverse vibrations of finite elastic Euler-Bernoulli beams on three-parameter viscoelastic Pasternak foundations are investigated. Complex frequency equations and modal function expressions are obtained with different boundary conditions. The effect of the stiffness, the viscoelastic coefficient and shear parameter of foundations on natural frequencies and modal functions are analyzed in numerical examples. The numerical results obtained with the differential quadrature confirm the results of the complex mode methods.In chapter 3, the complex mode analysis is applied to investigate transverse vibrations of finite Euler-Bernoulli beams on viscoelastic Winkler foundations, the explicit formulae for natural frequencies and modal functions are obtained. In numerical examples, for the finite Euler-Bernoulli beams on viscoelastic Winkler foundations, the exact solutions and numerical solutions of natural frequencies are compared, and are compared with numerical solutions of beams on viscoelastic Pasternak foundations.In chapter 4, the dynamic responses of elastic Euler-Bernoulli beams on viscoelastic Pasternak foundations subject to moving loads are investigated with the complex mode analysis method. The vibration equations of the beams are expressed in the state equations and decoupled into a set of ordinary differential equations, based on the orthogonality of the modal functions. In specific examples, the dynamical responses under two typical external excitations are given.In chapter 5, transverse free vibrations of finite elastic Timoshenko beams on viscoelastic Pasternak foundations are investigated, complex frequency equations and modal function expressions are presented with different boundary conditions. In numerical examples, the effect of the different parameters of beams and foundations on natural frequencies and modal functions are analyzed. Comparing the numerical results obtained by the differential quadrature with the results of the complex mode methods, two results are fit very well.In chapter 6, under simply supported boundary conditions, further analysis reveals that the frequency equations of free transverse vibration are explicit equations, the explicit formulae for natural frequencies and modal functions can be obtained. The free vibration equations of the Timoshenko beams on viscoelastic Pasternak foundations are expressed in the state equations, and decoupled into a set of ordinary differential equations, based on the orthogonality of the modal functions. For the specific example, the accurate solution of any order natural frequencies and modal functions can be calculated, avoiding calculation error and missing error from numerical ways.In chapter 7, the differential quadrature is applied to investigate transverse vibrations of elastic Euler–Bernoulli beams resting on nonlinear viscoelastic foundations. In numerical examples, the figures of the time history of a beam midpoint under the free motion and the dynamic responses under moving harmonic loads are given. The influences of nonlinearity, viscoelasticity and other system parameters on natural frequencies and dynamic responses are numerically studied.