Stability and Hopf bifurcation analysis of axially narrow-band random oscillating flexible beams

According to the nonlinear dynamic equations of motions for flexible beams excited axially by narrow-band random processes, a set of nonlinear modulation equations for the combination parametric resonance between two natural modes is established based on the method of multiple scales and Cartesian transformation. Then the case of harmonic parametric excitation is taken into consideration. The equations of approximate transition curves that separate stable solutions from the unstable trivial ones, which belong to unstable foci, are derived. The type of Hopf bifurcation is determined in the vicinity of the bifurcation point via the center manifold theorem and the corresponding limit cycle is numerically found. Finally, the case of narrow-band random parametric excitation is focused on. To determine the almost sure stability of the trivial response of the system, the numerical results for the largest Lyapunov exponent are obtained, which show that with the increase of narrow-band width γ, the unstable region will be widened. Furthermore, the increase of γ results in a fact that the limit cycle gradually becomes a diffused one, i.e. the width of the limit cycle is increased to some extent.