Stochastic jump and bifurcation of a slender cantilever beam carrying a lumped mass under narrow-band principal parametric excitation

A detailed theoretical investigation into the single-mode approximate response of a slender cantilever beam carrying a lumped mass subjected to base narrow-band random excitation is presented for the first time. The method of multiple scales is used and the stochastic jump and bifurcation have been investigated for the principal parametric resonance of the system using the stationary joint probability. Results show that stochastic jump occurs mainly in the region of triple-valued solution. For the frequency-response domain, if the excitation central frequency is a variable and others keep constant, the basic phenomena imply that the higher the frequency, the more probable the jump from the stationary non-trivial branch to the stationary trivial one once the frequency exceeds a certain value. If the bandwidth is a variable and others keep constant, the basic phenomena indicate that the most probable motion is around the non-trivial branch when the bandwidth is smaller, whereas the most probable motion gradually approaches the trivial one when the bandwidth becomes higher. For the force-response domain, there is a region of excitation acceleration within which the joint probability density has two peaks: an outer flabellate peak and a central volcano peak. Results show that the outer flabellate peak decreases while the central volcano peak increases as the value of the excitation acceleration decreases.