Continuous Systems

Abstract

Let us consider the system in which the masses, interconnected by a weightless beam, interact with nonlinearly elastic supports distributed equidistantly along the length (Fig. 3.1). The corresponding equation of the free motion in the absence of friction is written in the form

$$ m{\rm{ }}\sum\limits_{j = -\infty }^\infty {{{{\partial ^2}w} \over {\partial {t^2}}}\delta \left( {x -jl} \right)} + EJ{{{\partial ^4}w} \over {\partial {x^4}}} -S{{{\partial ^2}w} \over {\partial {x^2}}} + \sum\limits_{j = -\infty }^\infty {q\left( w \right)} \delta \left( {x -jl} \right) = 0; $$

((3.1.1))

here m is the magnitude of each concentrated mass, w[x, t) is the transversal displacement, l is the spacing between supports (masses), δ is the Dirac delta function, q(w) = aw + bw³, and S is the stretching force.