Oscillations

A pendulum clock keeps time using simple harmonic motion.

A particle moves with simple harmonic motion 简谐运动 (SHM) when its acceleration 加速度 is:

• proportional to its displacement from a fixed equilibrium 平衡 point, and
• directed back towards that point — opposite in sign to the displacement.

The defining equation is

where $x$ is the displacement 位移 from equilibrium and $\omega$ is a positive constant, the angular frequency 角频率. The minus sign means "directed back towards equilibrium".

Many systems do this near a stable equilibrium: a mass on a spring 弹簧, a pendulum 单摆 (small swing), a floating block pushed down, the charge on a capacitor 电容器 in an LC circuit, atoms in a solid.

Acceleration always points back towards equilibrium, opposite to the displacement

Key terms

• displacement $x$ — distance from equilibrium at a moment (a vector along the line of motion).
• amplitude 振幅 $x_{0}$ — the largest displacement from equilibrium. Always positive.
• period 周期 $T$ — the time for one full oscillation.
• frequency 频率 $f$ — the number of oscillations per second; $f = 1/T$. Unit: Hz.
• angular frequency $\omega$ — $\omega = 2\pi/T = 2\pi f$. Unit: $\text{rad s}^{-1}$.
• phase difference 相位差 — the fraction of a cycle (in radians) by which one oscillation leads or lags another. A quarter-cycle apart is a phase difference of $\pi/2$.

So $T = 2\pi/\omega$ and $f = \omega/(2\pi)$ — given any one of $\omega$, $f$, $T$ you can find the others.

Worked example. A mass on a spring oscillates with SHM of amplitude $0.050\ \text{m}$ and frequency $2.5\ \text{Hz}$. Find its maximum acceleration.

The angular frequency is $\omega = 2\pi f = 2\pi \times 2.5 = 15.7\ \text{rad s}^{-1}$. The acceleration is largest at the extremes, where $|a| = \omega^{2}x_{0}$:

If the particle starts at $x = 0$ moving in the positive direction at $t = 0$, then

Differentiating once gives the velocity 速度:

where $v_{0} = x_{0} \omega$ is the maximum speed (as the particle passes through equilibrium).

Differentiating again gives the acceleration:

which is the SHM defining equation again. (If the particle instead starts at the extreme position $x = x_{0}$ at $t = 0$, use $x = x_{0} \cos(\omega t)$. Choose the one that fits the start conditions.)

Velocity in terms of displacement

A useful relation that does not use time:

• at equilibrium ($x = 0$): $v = \pm \omega x_{0}$ (maximum speed). Both signs, because the particle passes through equilibrium twice each cycle.
• at the extremes ($x = \pm x_{0}$): $v = 0$ (at rest for an instant).

Worked example. The same oscillation has amplitude $x_{0} = 0.050\ \text{m}$ and angular frequency $\omega = 15.7\ \text{rad s}^{-1}$. Find the speed when the displacement is $x = 0.030\ \text{m}$.

Graphs against time

For $x = x_{0}\sin\omega t$:

• $x$ vs $t$ — a sine curve, amplitude $x_{0}$, period $T = 2\pi/\omega$.
• $v$ vs $t$ — a cosine curve, leading $x$ by $\pi/2$, amplitude $\omega x_{0}$.
• $a$ vs $t$ — a negative sine curve, out of phase with $x$ by $\pi$ (180°), amplitude $\omega^{2} x_{0}$.

Displacement varies sinusoidally with time in simple harmonic motion

Velocity leads displacement by a quarter cycle; acceleration is exactly out of phase with displacement

Graph of $a$ against $x$

A straight line through the origin with negative gradient $-\omega^{2}$. So you can read $\omega$ from the graph: gradient $= -\omega^{2}$, so $\omega = \sqrt{|\text{gradient}|}$, then $T = 2\pi / \omega$. This is a common exam pattern.

The acceleration–displacement graph is a straight line through the origin with gradient $-\omega^{2}$

A simple harmonic oscillator keeps swapping energy between two forms:

• kinetic energy 动能 $E_{\text{K}} = \tfrac{1}{2} m v^{2}$.
• potential energy $E_{\text{P}}$ (elastic for a spring, gravitational for a pendulum).

With no damping 阻尼, the total energy is constant (this is conservation of energy 能量守恒).

Maximum and minimum

• at equilibrium ($x = 0$): $v$ is largest, so $E_{\text{K}}$ is largest and $E_{\text{P}}$ is smallest (zero, by choice).
• at the extremes ($x = \pm x_{0}$): $v = 0$, so $E_{\text{K}} = 0$ and $E_{\text{P}}$ is largest.

Total energy

Using $v_{\text{max}} = \omega x_{0}$:

Two key facts: the total energy is proportional to the square of the amplitude (doubling $x_{0}$ gives four times the energy), and to $\omega^{2}$.

Energy against displacement

Using $v^{2} = \omega^{2}(x_{0}^{2} - x^{2})$:

So $E_{\text{K}}$ is a downward parabola (peak at $x = 0$, zero at $x = \pm x_{0}$) and $E_{\text{P}}$ is an upward parabola (zero at $x = 0$, largest at $x = \pm x_{0}$). Their sum is constant.

Kinetic and potential energy swap over a cycle while the total energy stays constant

Worked example. A $0.20\ \text{kg}$ mass oscillates with amplitude $x_{0} = 0.050\ \text{m}$ and angular frequency $\omega = 15.7\ \text{rad s}^{-1}$. Find the total energy of the oscillation.

The total energy equals the maximum kinetic energy, as the mass passes through $x = 0$ at $v_{\text{max}} = \omega x_{0}$:

This stays constant, swapping between kinetic and potential form twice each cycle.

A resistive force (friction 摩擦力, drag 阻力, air resistance 空气阻力) causes damping — the amplitude shrinks over time as energy is lost as heat. Three named cases:

Light damping

The amplitude shrinks slowly over many cycles (a light damping 轻阻尼 case). The system still oscillates near its natural frequency, but each cycle is smaller than the last. A car's suspension is light-to-medium damped, so bumps die away but the ride stays smooth.

In light damping the amplitude dies away slowly over many cycles

Critical damping

The least damping that brings the system back to equilibrium without overshooting and without oscillating — a critical damping 临界阻尼 case. It returns in the shortest time. A galvanometer 检流计 or analogue voltmeter 电压表 is critically damped so the needle settles quickly.

Heavy damping

So much resistance that the system returns slowly, with no oscillation, but more slowly than the critical case — a heavy damping 过阻尼 case. A door with a strong closer is heavily damped.

On a displacement–time graph: light damping is a wave whose size dies away smoothly; critical damping returns quickly with no overshoot; heavy damping returns slowly.

Critical damping returns to equilibrium fastest without overshoot; overdamping returns more slowly

A plucked guitar string vibrates at its resonant frequencies.

A forced oscillation 受迫振动 is driven by an outside periodic force at a frequency $f_{\text{d}}$ chosen by the experimenter. The system then oscillates at this driving frequency 驱动频率 $f_{\text{d}}$, not at its own natural frequency. A plot of amplitude against $f_{\text{d}}$ is a resonance curve 共振曲线 with a peak.

Resonance

Resonance 共振 happens when the driving frequency equals the system's natural frequency 固有频率 $f_{0}$. At resonance the amplitude is largest and the energy transfer from the driver is most efficient.

The amplitude of a forced oscillation peaks at resonance, when the driving frequency equals the natural frequency

Resonance can destroy. In 1940 the wind pushed the Tacoma Narrows Bridge close to its natural frequency; with little damping the twisting grew and grew until the deck ripped apart. Engineers now design bridges and buildings so their natural frequencies avoid such driving forces

Examples:

• a swing pushed at the right rate builds up a large amplitude.
• a wine glass broken by a sound at its natural ringing frequency.
• a building shaken by an earthquake whose frequency matches a natural frequency — engineers design buildings so their natural frequencies avoid the main earthquake range.

The peak's shape depends on damping: lighter damping → a sharper, higher peak; heavier damping → a broader, lower peak, shifted slightly to lower frequency.

• The SHM condition is $a = -\omega^2 x$ (acceleration proportional to displacement, directed back to equilibrium).
• Learn $x = x_0\sin\omega t$ (or cos), $v_{max} = \omega x_0$, $a_{max} = \omega^2 x_0$; KE and PE interchange with the total energy constant.
• Velocity is zero at the extremes and maximum at the centre.
• Resonance occurs when the driving frequency equals the natural frequency; damping lowers and broadens the peak.