Finite elements for a beam system with nonlinear contact under periodic excitation

Abstract

Solar arrays are structures which are connected to satellites; during launch, they are in a folded position and submitted to high vibrations. In order to save mass, the flexibility of the panels is not negligible and they may strike each other; this may damage the structure. To prevent this, rubber snubbers are mounted at well chosen points of the structure; a prestress is applied to the snubber; but it is quite difficult to check the amount of prestress and the snubber may act only on one side; they will be modeled as one sided springs (see figure 2). In this article, some analysis for responses (displacements) in both time and frequency domains for a clamped-clamped Euler-Bernoulli beam model with a spring are presented. This spring can be unilateral or bilateral fixed at a point. The mounting (beam +spring) is fixed on a rigid support which has a sinusoidal motion of constant frequency. The system is also studied in the frequency domain by sweeping frequencies between two fixed values, in order to save the maximum of displacements corresponding to each frequency. Numerical results are compared with exact solutions in particular cases which already exist in the literature. On the other hand, a numerical and theoretical investigation of nonlinear normal mode (NNM) can be a new method to describe nonlinear behaviors, this work is in progress.

Asymptotic expansions of vibrations with small unilateral contact

“…For our first simple example, we can compute explicitly α 2 . After lengthy and tedious computations involving numerical series, we obtain α 2 = −3(4ω 0 ) −2 , thus ω 2 = −(2ω 0 ) −3 as we have yet obtained in (2). More generally, we have:…”

The method of strained coordinates for vibrations with weak unilateral springs

“…The numerical expansions of the previous subsection gives valid results for ε small enough; in many practical cases such as [8], ε may be quite large; in this case, it is natural to try to solve numerically the following equations with respect to the period T and the initial condition X(0). In other words, we look for a periodic solution of prescribed energy; this last condition is to ensure to obtain an isolated local solution: the previous expansions show that in general, the period of the solution depends on its amplitude prescribed here by its energy.…”