Closed-form solution for response of linear systems subjected to periodic non-harmonic excitation

Abstract

A scheme is proposed for linking together into a single equation the equations and inequalities that make up a piecewise representation of a non-harmonic, periodic, continuous or discontinuous function. A method using this scheme is proposed in order to obtain, in closed form, the displacement response of linear vibration problems with piecewise-continuous forcing functions. Since the solution is exact, so are the derivatives, i.e. the velocity and acceleration responses. An example is presented.

Exact, Single Equation, Closed-Form Solution of Vibrating Systems With Piecewise Linear Springs

“…The periodizer function, proposed in reference [2], is a piecewise linear, periodic (sawtooth) function of the independent variable:…”

“…Furthermore, a &&periodizer function'' was introduced in reference [2], which together with the uni"cation scheme of reference [1] was used in the linear vibration problem with non-harmonic or discontinuous periodic excitation. This allows the steady state displacement response to be obtained directly, in closed form, without resorting to either Fourier series or the Laplace, or any other, transform.…”

“…Pursuing further the ideas of references [1,2], in this paper a single equation is used to represent the force}displacement relation of a piecewise linear spring. This leads to the analytical, single equation, phase-plane solution of the free vibrations of a system made up of such a spring and a mass.…”

Exact, no-transform solution of multi-degree-of-freedom vibrations with non-harmonic, periodic excitation

“…The periodizer function introduced in reference [2] converts any function, composite or not, into a periodic function by simply repeating any chosen interval of the function in question. Thus, the interval from tˆ0 to tˆT of the function f (t) is converted into a periodic function by simply replacing t by the periodizer function, p(t), and expressing the resulting function,…”

“…This was accomplished without resorting to the Laplace or any other transform, thus avoiding the concomitant complexities. However, reference [2] is not speci cally addressed to the multidegree-of-freedom problem and it contains only a single-degree-offreedom problem to illustrate the procedure. The main purpose of this paper is to point out a few generalities regarding the methodology as well as how it may be applied to problems with any nite number of degrees of freedom.…”