© 1970 by Oxford University Press
THE TRANSIENT AND STEADY-STATE RESPONSE OF THE INFINITE BERNOULLI-EULER BEAM WITH DAMPING AND AN ELASTIC FOUNDATION
(
School of Engineering Science and Mechanics and School of Applied Mathematics, Georgia Institute of Technology Atlanta
School of Engineering Science and Mechanics, Georgia Institute of Technology Atlanta
)
The closed-form analytical solution is obtained for the dynamic response of the infinite Bernoulli-Euler beam with arbitrary initial conditions and subjected to an arbitrary load. The simultaneous effects of damping, an elastic foundation and a constant axial load are considered. The values of the parameters describing these effects are unrestricted; in particular, it is not necessary to restrict the damping coefficient to small values as has been the case in previous papers. The character of the solution changes radically for certain combinations of these parameters; thus, it is possible to define critical, subcritical, and super-critical damping analogous to a simple, damped spring-mass system. It is shown, in addition, that the steady-state solutions dealt with in several papers may be obtained by a formal passage to the limit as time tends to infinity in the transient solution. It must be emphasized that the solution given here contains all cited previous solutions as special cases, and that the transient solution of the infinite Bernoulli-Euler beam, even when only an elastic foundation was included, has not been presented in the literature. In particular, the general solution may be employed to obtain the dynamic response due to arbitrarily moving, time-dependent loads, where previously even the simple case of a constant load moving with constant velocity presented serious difficulties.