Forced response of low-frequency pendulum mechanism

• Strongly nonlinear pendulum mechanism
• Jacobi elliptic loading function
• Exact analytical solution for forced steady-state oscillation response
• Closed-form amplitude–frequency relation
• Passive mass damper for large engineering structures

A strongly nonlinear pendulum mechanism is considered in which the restoring force is approximately a cubic function of the displacement variable. Its free oscillation frequency is approximately proportional to the amplitude of oscillation and distinctly lower than that of a simple pendulum. The mechanism has therefore been named infra-pendulum. The forced undamped oscillation response of the mechanism to non-harmonic periodic loading is studied under the assumption of small displacements. The loading function is derived from the free oscillation response whose time course follows a Jacobi elliptic function. It is chosen such that exact analytical solutions are obtained for the steady-state response and the amplitude–frequency relation. The equation describing the amplitude–frequency relation is a cubic polynomial equation. Its solutions are presented. The general approach of using non-harmonic periodic loading functions is transferable to other types of nonlinear oscillators.

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