Storage of Rotational Kinetic Potential Energy

Newton’s second law indicates that time derivative of a mass’ angular momentum is the net torque on that object. Taking the inertia, $J$, to be constant, the product of inertia and torque is obtained.

$$ T=\frac{dL}{dt}=J\frac{d\omega}{dt} $$

Given that power is the product of torque and angular velocity, we substitute the above and integrate to obtain an expression for rotational kinetic potential energy.

$$ \frac{d\boldsymbol{\large\varepsilon}_K}{dt}=T\omega\Rarr\int d\boldsymbol{\large\varepsilon}_K=\int T\omega~dt=\int J\omega ~d\omega\Rarr\boldsymbol{\large\varepsilon}_K=\frac 12J\omega^2 $$

This derivation draws our attention to the fact that an instantaneous jump in angular velocity is impossible because it would require the application of infinite power.

Storage of Spring Potential Energy

An ideal rotational spring generates a torque proportional to its angular deflection from equilibrium position. Thus, the rate of change of torque is proportional to the rate of change of angular deflection.

$$ T_s=K\theta\Rarr\frac{dT_s}{dt}\frac 1K=\omega $$

Again we apply conservation of energy, then substitute the above expression for angular velocity, and integrate to obtain an expression for rotational spring potential energy. Substituting our initial definition further renders the normal expression.

$$ \frac{d\boldsymbol{\large\varepsilon}_S}{dt}=T_s\omega\Rarr\int d\boldsymbol{\large\varepsilon}_S=\int T_s\omega~dt=\frac 1K\int T_s~dT_s\Rarr\boldsymbol{\large\varepsilon}_S=\frac 1{2K}T_k^2=\frac 12K\theta^2 $$

This derivation draws our attention to the fact that an instantaneous jump in spring torque is impossible because it would require the application of infinite power and an infinite speed of angular deflection.

Dissipation of Energy by Damping

A rotational damper provides a resistive torque related to angular velocity, and in the ideal case, this relationship is linear.

$$ T_{NLd}=f(\omega)~~~~T_{Ld}=b\omega $$

While a rotational inertia may store rotational kinetic potential energy, and a rotational spring may store rotational spring potential energy, a rotational damper only dissipates energy.