Almost sure stability of the trivial response of a slender cantilever beam subject to narrow-band random principal parametric resonances

The nonlinear integro-differential equations of motion for a slender cantilever beam subject to axial narrow-band random excitation are investigated. The method of multiple scales is used to determine a uniform first-order expansion of the solution of equations. According to solvability conditions, the nonlinear modulation equations for the principal parametric resonance are obtained. The almost sure stability of the trivial solution is taken into consideration. According to the corresponding FPK (Fokker-Planck-Kolmogrov) equation, the invariant measure, i.e., the stationary probability density function of the process is determined and the largest Lyapunov exponent lambda is quantificationally resolved, in which, the modified Bessel function of the first kind is introduced. Results show that the mesh surface of the largest Lyapunov exponent becomes flatter as the increase of the narrow-band random bandwidth gamma, which implies that the almost sure unstable areas will change with the increase of it. In fact, results show that the bottom of the isohypse curve of lambda = 0 rises, whereas its top is widened with the increase of gamma, which, indicates that the increase of gamma may facilitate the almost sure stability of the trivial response and stabilize the system for a lower acceleration oscillating amplitude a but intensify the instability of the trivial response for a higher one. Naturally, the numerical results are in full agreement with the conclusion of the aforementioned validity of stability or instability.