Nonlinear Vibration of A Cantilever With A Derjaguin–müller–toporov Contact End

In this paper, the principal resonance is investigated for a cantilever with a contact end. The cantilever is modeled as an Euler–Bernoulli beam, and the contact is modeled by the Derjaguin–Müller–Toporov theory. The problem is formulated as a linear nonautonomous partial-differential equation with a nonlinear autonomous boundary condition. The method of multiple scales is applied to determine the steady-state response. The equation of response curves is derived from the solvability condition of eliminating secular terms. The stability of steady-state responses is analyzed by using the Lyapunov-linearized stability theory. Numerical examples are presented to highlight the effects of the excitation amplitude, the damping coefficient, and the coefficients related to the contact.