Harmonic Motion

Definition The equilibrium position of an object that can move back and forth along a line is a position where it can stay at rest, because the forces on it balance. For a weight hanging on a spring it is the point where the upward pull of the stretched spring exactly balances the downward pull of gravity. In a model of the motion, positions are measured from this point, so the equilibrium position is displacement $0$.

Definition Choose a positive direction along the line of motion. The displacement of the object at time $t$, written $d(t)$, is its signed distance from the equilibrium position, positive on the chosen side and negative on the other. If positions are measured as $y(t)$ on a coordinate line that points in the chosen positive direction, and the equilibrium position is $y = D$, then

The distance from equilibrium is $\lvert d(t) \rvert$.

Definition An object moves in simple harmonic motion if its displacement from equilibrium at time $t$ is

where $a$, $\omega$ and $\varphi$ are real constants with $a \ne 0$ and $\omega > 0$, and the angle $\omega t + \varphi$ is in radians. The most common cases are $d(t) = a\sin(\omega t)$ and $d(t) = a\cos(\omega t)$, which is the case $\varphi = \frac{\pi}{2}$ because $\cos x = \sin\left(x + \frac{\pi}{2}\right)$ for every real $x$. Physics shows that an object moves this way when the force pulling it back toward equilibrium is proportional to its displacement and nothing slows it down, as for a weight on an ideal spring without friction.

Definition The amplitude of the motion $d(t) = a\sin(\omega t + \varphi)$ is $\lvert a \rvert$. For every $t$

and both bounds are reached, so the amplitude is the largest distance from equilibrium. In simple harmonic motion the amplitude has no effect on the period.

Definition The period of $d(t) = a\sin(\omega t + \varphi)$ is the time $T$ of one full cycle, and the frequency $f$ is the number of cycles per unit of time,

The period has this value because the angle $\omega t + \varphi$ grows by $2\pi$ exactly when $t$ grows by $\frac{2\pi}{\omega}$. The constant $\omega$ is the angular frequency, in radians per unit of time. With $t$ in seconds, frequency is measured in cycles per second, also called hertz. For a weight of mass $m$ on an ideal spring with spring constant $k$, physics gives

Definition In $d(t) = a\sin(\omega t + \varphi)$ the angle $\omega t + \varphi$ is the phase of the motion at time $t$, and the constant $\varphi$ is the phase constant, the phase at $t = 0$. Adding a multiple of $2\pi$ to $\varphi$ does not change the motion, so the phase constant is determined only up to such a multiple. Writing

shows that the graph of $d$ is the graph of $a\sin(\omega t)$ shifted by $\frac{\lvert \varphi \rvert}{\omega}$ time units, to the left when $\varphi > 0$ and to the right when $\varphi < 0$. So the phase shift, the number $C$ in $A\sin\bigl(B(t - C)\bigr) + D$, is $C = -\frac{\varphi}{\omega}$, which is a different number from the phase constant $\varphi$. Two motions with the same $\omega$ and with $a > 0$ in both are in phase when their phase constants differ by a multiple of $2\pi$, so they reach their highest points at the same moments. They are opposite in phase when their phase constants differ by $\pi$ plus a multiple of $2\pi$, since $\sin(x + \pi) = -\sin x$.

Definition A sinusoidal model of a quantity that oscillates around a fixed level is a function

where $A$, $B$, $C$ and $D$ are real constants with $A \ne 0$ and $B > 0$, and $t$ is time. When $y(t)$ is the position of an object in simple harmonic motion, the midline $D$ is the equilibrium position, $\lvert A \rvert$ is the amplitude, $B$ is the angular frequency $\omega$ and $y(t) - D$ is the displacement. To build a model, read off the equilibrium, the amplitude and the period, as in amplitude, period, and phase shift. Then choose sine or cosine, the sign of $A$ and the shift $C$ so that the model starts in the right place.

Definition In damped harmonic motion, friction makes the swings shrink over time. A standard model is

with constants $a > 0$, $c > 0$ and $\omega > 0$. Since $\lvert \cos(\omega t) \rvert \le 1$ and $e^{-ct} > 0$,

so the graph stays between the exponential curves $y = a e^{-ct}$ and $y = -a e^{-ct}$, called the envelope. It touches the upper curve where $\cos(\omega t) = 1$ and the lower curve where $\cos(\omega t) = -1$, and it passes through equilibrium where $\cos(\omega t) = 0$, once every $\frac{\pi}{\omega}$ time units. Over each time span of length $\frac{2\pi}{\omega}$ the envelope shrinks by the factor $e^{-2\pi c / \omega}$. Setting $c = 0$ gives back simple harmonic motion.