The Linear Stability of the Responses of Axially Moving Beams Supported by an Intermediate Spring

Abstract

In the present paper, the stability investigation of the linear responses of axially travelling beams supported by an intermediate linear spring is performed numerically by means of searching linear complex eigenvalues. The boundary conditions and the dynamic equation of motion are obtained by means of the Extended Principle of Hamilton for the two-span Euler-Bernoulli beam. Assuming the harmonic solution of the governing differential equations of motion, the dispersion relation is derived. To have a nontrivial solution in conjunction with the dispersion relation, the frequency equation is obtained by letting the determinant of the matrix representing the coefficients of the equations of the boundary conditions be equal to zero. The aim of this contribution is to compute the complex values of the natural frequencies from the nonlinear frequency equation by means of numerical methods for the beam supported by an intermediate spring. The real and imaginary parts of the complex eigenvalues versus the constant axial transport speed curves are depicted in order to show the linear instability. Depending on the signs of the real and imaginary parts of the complex eigenvalues, the boundaries of the instability regions were observed.