Study On The Dynamical Responses Of Beam Structures Using Green’s Functions

Beam structures have been extensively employed in many engineering applications as basic structures, such as aircraft engines, space vehicles and buildings. When the mass and shear centers of the beam sections are separated by a distance, bending-torsion coupled vibration of the beam is caused. The vibration characteristics of composite structures should be considered when the beams are connected to each other. And the external loadings of the beam structures are complex for its various environment. Hence it is important for the research and design of the engineering beams to investigate the forced vibration responses of the typical beam structures.In this paper, the bending-torsional vibration of a Timoshenko beam and the nonlinear vibrations of a cable-stayed beam are analyzed. The forced dynamical responses of these basic units of beam structures are studied, which provide the theoretical bases and approaches for engineering application.A method based on Green’s functions is proposed for the analysis of the steady-state dynamic response of bending-torsion coupled Timoshenko beam subjected to distributed and/or concentrated loadings. Damping effects on the bending and torsional directions are taken into account in the vibration equations. The elastic boundary conditions with bending-torsion coupling and damping effects are derived. The Laplace transform technology is employed to work out the Green’s functions for the beam with arbitrary boundary conditions. The Green’s functions are obtained for the beam subject to external lateral force and external torque, respectively. The direct expressions of the steady-state responses with various loadings are obtained by using the superposition principle. The present Green’s functions for the Timoshenko beam can be reduced to those for Rayleigh and Euler-Bernoulli beams by setting the values of shear rigidity and rotational inertia. The influences of external loading frequency and eccentricity on Green’s functions of bending-torsion coupled Timoshenko beam are investigated in terms of the numerical results for both simply supported and cantilever beams. And the natrual frequencies of system are obtained. Moreover, the symmetric property of the Green’s functions and the damping effects on the amplitude of Green’s functions of the beam are discussed particularly.A method based on Green’s functions is proposed to study the nonlinear interactions between beam and cable steady-state dynamics in stayed-systems subjected to distributed and/or concentrated loadings. The initial sag of cable is considered and the quadratic and cubic nonlinearities both in the cable equation and at the boundary conditions are presented of a simple cable-stayed beam. The nonlinear governing equations are solved using perturbation method and the Laplace transform technology is employed to work out the Green’s functions for the beam and cable with boundary conditions. The Green’s functions are obtained for the beam and cable subject to external forces which act on beam and cable, respectively. And the closed-formed solutions are presented in an integral form based on the presented Green’s functions. The difference of Green’s functions between the linear and nonlinear systems is caused by the change of boundary conditions. The present Green’s functions for the cable-stayed beam can be extended to those for cable-stayed bridge by inducting boundary conditions of connections between elements. In order to demonstrate the validity of the solutions proposed, results obtained for special case is given for a comparison with those given in the literature and they agree with each other exactly. The numerical results show that the more complex amplitudes of the Green’s functions and vibrations can be conducted by loadings with high frequencies. And the symmetric property of the Green’s functions is destroyed with the existence of dynamic strain. Also, the vibrations of the structures are restrained by damping effects. Moreover, the influ ences of nonlinear terms on the amplitude of Green’s functions and vibration are investigated.