5.5 - Oscillations

Displacement $x$ is measured from equilibrium. The minus sign means force and acceleration always point back towards equilibrium; proportionality means their magnitudes grow linearly with distance from equilibrium.

For a mass on an ideal spring, extension about the equilibrium position produces restoring force $-kx$. A simple pendulum is approximately SHM only at small angles, where the restoring component is approximately proportional to displacement.

The cosine form chooses $t=0$ at positive maximum displacement. A different initial condition requires a phase constant, but the relationships between $x$, $v$ and $a$ remain: velocity is one quarter-cycle out of phase with displacement, and acceleration is in antiphase with displacement.

For $A=0.040\,\mathrm{m}$ and $f=2.0\,\mathrm{Hz}$, $\omega=2\pi f=12.6\,\mathrm{rad\,s^{-1}}$, so $v_{max}=A\omega=0.50\,\mathrm{m\,s^{-1}}$ and $a_{max}=A\omega^2=6.3\,\mathrm{m\,s^{-2}}$.

A $0.20\,\mathrm{kg}$ mass on a $50\,\mathrm{N\,m^{-1}}$ spring has $T=2\pi\sqrt{0.20/50}=0.40\,\mathrm{s}$. Quadrupling the mass doubles the period because $T\propto\sqrt m$.

Use the oscillating mass that moves with the spring. Pendulum length is from pivot to the bob's centre of mass, and the pendulum equation assumes small angular displacement, negligible air resistance and a light inextensible support.

For SHM, displacement varies sinusoidally between $+A$ and $-A$ about the equilibrium line $x=0$. The period $T$ is the time between equivalent points moving in the same direction; amplitude is the maximum distance from equilibrium.

Fix the axes and units, mark $\pm A$, and place repeated peaks exactly one period apart. If initial displacement is $+A$ with release from rest, begin at a maximum and follow a cosine curve; other initial states shift the phase without changing $A$ or $T$.

SHM velocity is sinusoidal with the same period as displacement and amplitude $v_{max}=A\omega$. The gradient of a velocity-time graph is instantaneous acceleration.

If $x=A\cos\omega t$, then $v=-A\omega\sin\omega t$. Velocity therefore reaches each corresponding feature one quarter-period after displacement, and its sign records direction. Draw peaks separated by $T$ and label the vertical scale using $A\omega$.

Resonance occurs when a periodic driving force acts at the natural frequency, or very close to it in a damped real system. The driver then transfers energy efficiently on successive cycles, producing the largest steady oscillation amplitude.

Away from the natural frequency, the driver and oscillator do not stay in the most effective timing relationship, so less net energy is transferred and the response amplitude is smaller.

Attach the same spring to a vibration generator and add a known mass. Drive it sinusoidally, vary frequency in small steps, and record the resonant frequency at the maximum steady amplitude. Repeat the frequency sweep from both directions and average the peak value.

Calibration can absorb a small constant effective contribution from the moving spring or holder: use the fitted relationship rather than forcing a line through the origin without evidence. Repeat the unknown measurement and propagate the frequency reading uncertainty through the graph.

For undamped SHM, $E=\tfrac12mv^2+\tfrac12kx^2=\tfrac12kA^2$ is constant. A pendulum similarly exchanges gravitational potential and kinetic energy while total mechanical energy remains constant.

With damping, resistive forces do negative work. Mechanical energy is transferred to thermal energy, sound or deformation in the surroundings, so the maximum energy and amplitude decrease on successive cycles even though total energy of the larger closed system is conserved.

A sequence of impulses can maintain a forced oscillation when each arrives at the same effective phase, adding energy repeatedly. The drive need not move sinusoidally, but its timing must contain the relevant periodic component.

For a fixed driving-force amplitude, the steady response is small far below the natural frequency, rises as the driving frequency approaches it, reaches a maximum near resonance, and falls again above it.

At steady amplitude, energy supplied by the driver each cycle equals energy dissipated. Near resonance, the timing permits the greatest energy transfer; increasing damping removes more mechanical energy per cycle, so a smaller amplitude is enough to balance the input.

Damping forces oppose motion and do work on the surroundings. Fluid resistance, friction or a dashpot converts organised oscillation energy into thermal energy, so each cycle returns less mechanical energy and the amplitude falls.

A ductile component designed to yield during strong structural motion absorbs work over its force-extension cycle. Together with viscous dampers, this reduces energy available for the next swing and limits damaging amplitudes.