At the steady state, the average stored energy is given by $$E = \dfrac{1}{2} m\left\langle (x')^2\right\rangle + \dfrac{1}{2} m{\omega_0}^2 \left\langle x^2 \right\rangle$$
They become equal when the driving frequency $\omega = \omega_0$.
Here is the qualitative explanation given by Frank S. Crawford in his book:
If $\omega$ is large compared with $\omega_0$, the velocity of $m$ gets reversed before it has a chance to acquire large displacement & hence it can acquire a large potential energy.
On the other hand, if $\omega$ is small compared with $\omega_0$, the velocity never gets very large, & then the time_average potential energy dominates.
I'm having some problem with the explanation. Suppose if I give initially a certain velocity to the oscillator, the restoring force works against it & if the spring-constant is more, it only takes less time to bring the oscillator to rest converting the kinetic energy completely to potential energy in a lesser time than the time required if the spring-constant were smaller than $\omega$ . So, I really couldn't understand the first explanation. Same is with the second also. I am failing to visualise what he is saying. Please help me explain his words
Best Answer
ACuriousMind is right. Assuming the system contains a dissipative component, for any fixed driving frequency, when the rate of energy input from the driving source equals the rate at which energy is lost from any dissipative elements in the system, a dynamic equilibrium is reached in the system. At dynamic equilibrium the sum of the average potential and average kinetic energy are a constant.
What Crawford is trying to illustrate is what particular form of energy is dominant below and above the natural (resonant) frequency of the system. Below the resonant frequency kinetic energy is dominant and above the resonant frequency potential energy is dominant. And at the resonant frequency each form is equal.
Note that prefaced "For any fixed driving frequency ..." above. The total energy at dynamic equilibrium is a constant, but only at a fixed frequency. The total energy is a maximum at the resonant frequency. At the resonant frequency, the system has its maximum capacity for energy.
Now if there is no energy dissipation or the rate of dissipation is less than the rate of energy input - there is danger of the system 'blowing up'. But in real physical systems there is always nonlinearity. The nonlinearity may provide a safe release for energy once some limiting factor is reached, or in other cases (like the Tacoma Narrows Bridge) reach a point of disaster.