A low-frequency pendulum mechanism

• Free oscillation response described by nonlinear differential equation
• Solution leads to Jacobi elliptic function for displacement curve
• Closed-form expression for natural frequency of oscillation
• Natural frequency proportional to amplitude of oscillation
• Usable as tuned mass damper for large engineering structures

A pendulum mechanism is presented whose natural frequency of oscillation is distinctly lower than that of a conventional pendulum of comparable size. Furthermore, its natural frequency is approximately proportional to its amplitude of oscillation. The mechanism can thus be tuned to extremely low frequencies by using small amplitudes. The undamped free oscillation response of the mechanism is studied. The derivation of the equation of motion is outlined for both large and, after neglecting higher order terms, small displacements. In both cases, a second-order nonlinear differential equation results. When higher order terms are neglected, the equation of motion is of simple form and can be solved symbolically in terms of a Jacobi elliptic function. Based on this solution, a closed-form expression for the natural frequency is derived and the characteristics of the free oscillation response are discussed.