Physical pendulum

A uniform circular disc of radius $a$ and mass $m$, pivoted on a perpendicular axis through its rim, has moment of inertia $I_A=3m{a}^2/2$ by the parallel axis theorem. Its hanging small oscillations have period $2\pi \sqrt{3a/(2g)}$. Measured from the upright position, the falling equation is $\ddot{\theta }=2g\sin \theta /(3a)$. Exact upright rest remains an unstable equilibrium; the energy level of that state describes a limiting separatrix after perturbation, with ${\dot{\theta }}^2=4g(1-\cos \theta )/(3a)$. The center acceleration and pivot reaction follow by resolving radial and tangential components.

In nondimensional time the quadratic damping removes energy from the pendulum at rate $-c|\omega {|}^3$. Downward resting positions are nonhyperbolic asymptotically stable spirals, even though the linearized equation has a centre. Upward resting positions remain saddle equilibria. Each fixed-sign branch of $\omega$ has a linear equation for ${\omega }^2$ as a function of angle.

For equal hoop and bead masses, with the hoop suspended from its circumference, the quadratic mass and stiffness matrices are proportional toThe dimensionless squared frequencies are the roots of $2c^2-5c+2$, namely $c=1/2$ and $c=2$.