Abstract
The main aim of this paper is to contribute to the construction of Green’s functions for initial boundary value problems for fourth order partial differential equations. In this paper, we consider a transversely vibrating homogeneous semi-infinite beam with classical boundary conditions such as pinned, sliding, clamped or with a non-classical boundary conditions such as dampers. This problem is of important interest in the context of the foundation of exact solutions for semi-infinite beams with boundary damping. The Green’s functions are explicitly given by using the method of Laplace transforms. The analytical results are validated by references and numerical methods. It is shown how the general solution for a semi-infinite beam equation with boundary damping can be constructed by the Green’s function method, and how damping properties can be obtained.
Highlights
In engineering, many problems describing mechanical vibrations in elastic structures, such as for instance the vibrations of power transmission lines [13] and bridge cables [16], can be mathematically represented by initial-boundary-value problems for a wave or a beam equation
In order to suppress the undesired vibrations of the mechanical structures different kinds of dampers such as tuned mass dampers and oil dampers can be used at the boundary
We will consider the transverse vibrations of a onedimensional elastic Euler–Bernoulli beam which is infinitely long in one direction
Summary
Many problems describing mechanical vibrations in elastic structures, such as for instance the vibrations of power transmission lines [13] and bridge cables [16], can be mathematically represented by initial-boundary-value problems for a wave or a beam equation. The initial-boundary value problem for a semi-infinite clamped bar has already been solved to obtain its Green’s function by using the method of Laplace tranforms [21]. 4, three classical boundary conditions are considered and the Green’s functions for semi-infinite beams are represented by definite integrals. Numerical and asymptotic approximations of the roots of a characteristic equation for the beam-like problem on a finite domain will be calculated. It will be shown how boundary damping can be effectively used to suppress the amplitudes of oscillation.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
