Explanation on the importance of narrow-band random excitation characters in the response of a cantilever beam

A detailed theoretical investigation into the first- and second-mode response of a parametrically excited slender cantilever beam, where, the narrow-band random excitation characters are taken into consideration, is presented. The method of multiple scales is used to determine a uniform first-order expansion of the solution of the nonlinear integro-differential equations. The first-order moment frequency- and force–response data (curves) of a specimen beam tested by other investigators are obtained. Further comparisons have been made and results show that whether the first-order moment frequency–response data (curves) or the first-order moment force–response data (curves) of the first two modes are all in agreement with other investigators’ experimental results. Furthermore, the stochastic jump and bifurcation have been investigated for the first modal parametric principal resonance by using the stationary joint probability of amplitude and phase. Results show that stochastic jump occurs mainly in the region of triple-valued solution. For the frequency–response domain, if the bandwidth γ is a variable and others keep constant, the basic phenomena indicate that the most probable motion is around the higher branch when the bandwidth is smaller, whereas the most probable motion gradually approaches the lower one when the bandwidth becomes higher; if the excitation central frequency f is a variable and others keep constant, the basic phenomena imply that the higher is f, the more probable is the jump from the higher branch to the lower one once f exceeds an certain value. For the force–response domain, there is a region of excitation acceleration a within which the joint probability density has two peaks: an upper peak and a lower peak. Results show that the upper peak decreases while the lower peak increases as the value of a decreases.