Forced Oscillations, Beats, and Resonance

Definition Let $t$ be time and let primes denote derivatives with respect to $t$. A forced oscillation is a solution of

where $m > 0$, $k > 0$ and $c \ge 0$ are the mass, stiffness and damping coefficient of the free model, $F_0 > 0$ is the force amplitude in newtons and $\omega > 0$ is the driving angular frequency in radians per second, so the force completes $\omega/(2\pi)$ cycles per second. Every solution has the form $x = x_h + x_p$, where $x_p$ is any one particular solution and $x_h$ is a free solution of the homogeneous equation, chosen to match the initial data. Without damping, and with $\omega \ne \omega_0 = \sqrt{k/m}$, one particular solution is

Definition The natural angular frequency of the oscillator is

measured in radians per second. It is the angular frequency of free undamped motion $x = A\cos\omega_0 t + B\sin\omega_0 t$, whose period is $2\pi/\omega_0$ and whose ordinary frequency is $\omega_0/(2\pi)$ cycles per second. It depends only on $m$ and $k$. It is different from the driving angular frequency $\omega$, which is set from outside, and from the damped angular frequency $\omega_d = \sqrt{\omega_0^2 - (c/(2m))^2}$ of an underdamped system, which is smaller.

Definition Beats are the slow periodic swelling and fading of an oscillation that is the sum of two oscillations with nearby frequencies. In an undamped forced oscillator ($c = 0$, $\omega \ne \omega_0$) started from rest, $x(0) = 0$ and $x'(0) = 0$, the response is $x = K(\cos\omega t - \cos\omega_0 t)$ with $K = F_0/(m(\omega_0^2 - \omega^2))$. The difference of cosines identity turns it into a product,

When $\omega$ is close to $\omega_0$, the second factor oscillates fast and the first changes slowly, so the slow factor sets the size of the fast oscillation. The magnitude envelope $2\lvert K \rvert \lvert \sin((\omega_0 - \omega)t/2) \rvert$ repeats with the beat period

while the signed slow factor takes twice as long to repeat. For every fixed $\omega \ne \omega_0$ the response stays bounded, with $\lvert x \rvert \le 2\lvert K \rvert$.

Definition Undamped resonance is the case $c = 0$ and $\omega = \omega_0$. The usual trial $a\cos\omega_0 t + b\sin\omega_0 t$ solves the homogeneous equation, so substituting it gives $0$ and never the force. Multiplying the trial by $t$, as the modification rule of undetermined coefficients prescribes, gives the particular solution

It also satisfies $x_p(0) = 0$ and $x_p'(0) = 0$, so it is the response from rest. Every other solution adds a bounded free oscillation to it, so every solution is unbounded as $t \to \infty$.

Definition Secular growth is growth of an oscillation’s envelope in proportion to a power of $t$, here to $t$ itself. For $t \ge 0$ the resonant response from rest satisfies

with equality whenever $\lvert \sin\omega_0 t \rvert = 1$, so its envelope is a pair of straight lines through the origin with slopes $\pm F_0/(2m\omega_0)$. Unlike beats, which stay below a fixed bound, secular growth passes every bound. A real spring eventually leaves its linear range, and any damping $c > 0$ keeps the amplitude bounded.

Definition Let $m$, $k$ and $c$ be positive. Undetermined coefficients give exactly one particular solution of the form $a\cos\omega t + b\sin\omega t$, and its amplitude $\sqrt{a^2 + b^2}$ is

where $D(\omega) = (k - m\omega^2)^2 + c^2\omega^2$. Since $c > 0$, the two squares never vanish together, so $A(\omega)$ is finite at every driving frequency. With $\Omega^2 = k/m - c^2/(2m^2)$, a number that may be negative, differentiating gives

So the amplitude has a maximum at a positive frequency if and only if $c^2 < 2mk$, and the peak is then at

with largest amplitude $F_0/(c\,\omega_d)$. If $c^2 \ge 2mk$, the amplitude decreases for all $\omega > 0$ and stays below its zero-frequency value $A(0) = F_0/k$, the static deflection. The condition $c^2 < 2mk$ is stronger than underdamping, which asks only $c^2 < 4mk$, and a peak, when it exists, lies below both frequencies, $\omega_{\text{peak}} < \omega_d < \omega_0$. Some books call this peak practical resonance.