Oscillations, Damping, and Resonance

Definition A mass $m > 0$ hangs on a spring with spring constant $k > 0$. Measure the displacement $x(t)$ from the equilibrium position, with downward positive. Hooke’s law says the spring pulls with force $-k$ times its stretch. At equilibrium the spring is stretched by $\ell$ with $k\ell = mg$, so gravity and the spring balance. With displacement $x$ the stretch is $\ell + x$, and the net of gravity and spring is $mg - k(\ell + x) = -kx$. Adding a damping force $-c\,x'$ with $c \ge 0$ and an applied force $F(t)$, Newton’s second law gives

Definition When $c = 0$ and $F = 0$, the equation $m x'' + k x = 0$ has the general solution

With $R = \sqrt{c_1^2 + c_2^2}$ and an angle $\delta$ such that $c_1 = R\cos\delta$ and $c_2 = R\sin\delta$, this is $x = R\cos(\omega_0 t - \delta)$. The number $R$ is the amplitude, $\omega_0$ is the natural angular frequency in radians per second, and the period is $2\pi/\omega_0$.

Definition With damping constant $c > 0$ and no applied force, the characteristic equation $m r^2 + c r + k = 0$ has roots

Since $m$, $c$ and $k$ are positive, both roots have negative real part, so every solution tends to $0$ as $t \to \infty$. The sign of $c^2 - 4mk$ sorts the motion into three cases, treated next.

Definition If $0 < c^2 < 4mk$, the roots are $-\frac{c}{2m} \pm i\mu$ with $\mu = \frac{\sqrt{4mk - c^2}}{2m} > 0$, and

The mass crosses equilibrium infinitely often, at time gaps of $\pi/\mu$, and the oscillation is squeezed by the envelope $\pm R\,e^{-ct/(2m)}$. The number $2\pi/\mu$ is called the quasi period.

Definition If $c^2 = 4mk$, the characteristic equation has the double root $-\frac{c}{2m}$, and

Unless $x$ is identically zero, the factor $c_1 + c_2 t$ has at most one zero, so the mass passes through equilibrium at most once.

Definition If $c^2 > 4mk$, the roots $r_1$ and $r_2$ are real, distinct and negative, and

As with critical damping, a solution that is not identically zero passes through equilibrium at most once, since $c_1 e^{r_1 t} = -c_2 e^{r_2 t}$ has at most one solution $t$. Order the roots so that $r_2 < r_1 < 0$. For fixed $m$ and $k$, the slower root satisfies

So every solution with $c_1 \ne 0$ decays like $e^{r_1 t}$, more slowly in the long run than every nonzero critically damped solution, which decays like $e^{-\omega_0 t}$ or $t\,e^{-\omega_0 t}$. Initial data that make $c_1 = 0$ excite only the fast root and decay faster.

Definition A periodic applied force $F(t) = F_0 \cos \omega t$, with $F_0 \ne 0$ and $\omega > 0$, gives $m x'' + c x' + k x = F_0 \cos\omega t$. In the undamped case $c = 0$ with $\omega \ne \omega_0$, undetermined coefficients gives the particular solution

and the general solution is $x_p$ plus simple harmonic motion at frequency $\omega_0$.

Definition Pure resonance happens when $c = 0$ and $\omega = \omega_0$. Then $\cos\omega_0 t$ solves the homogeneous equation, the trial needs a factor $t$, and

which is unbounded as $t \to \infty$. When $c > 0$ every solution stays bounded, and for $F_0 > 0$ the steady periodic solution has amplitude

If $c^2 < 2mk$, then on $\omega > 0$ the amplitude $A(\omega)$ has its largest value at

This peak is called practical resonance, and its frequency $\omega_r$ is less than $\omega_0$. If $c^2 \ge 2mk$, then $A(\omega)$ strictly decreases for $\omega > 0$ and there is no peak. Both facts follow by writing $u = \omega^2$ and differentiating $(k - mu)^2 + c^2 u$, which gives $2m^2 u - 2mk + c^2$.

Definition A series circuit with inductance $L > 0$ henries, resistance $R \ge 0$ ohms, capacitance $C > 0$ farads and applied voltage $E(t)$ volts satisfies, for the charge $q(t)$ in coulombs on the capacitor,

and the current is $I = q'$ amperes. It is the spring-mass equation with $L$ for $m$, $R$ for $c$ and $1/C$ for $k$, so every result above carries over.

Definition Suppose $m, c, k > 0$ and $F(t) = F_0\cos\omega t$. Every solution is $x = x_c + x_p$, where $x_c$ solves the homogeneous equation and tends to $0$ as $t \to \infty$. The part $x_c$ is the transient. The periodic particular solution

is the steady periodic response. Here $F_0 > 0$, $A(\omega)$ is the amplitude from the resonance section, and the phase lag $\gamma$ is the angle in $(0, \pi)$ with

So the response lags the force by less than a quarter period when $\omega < \omega_0$, by exactly $\pi/2$ when $\omega = \omega_0$, and by more when $\omega > \omega_0$. The steady response does not depend on the initial conditions.