Oscillations — SHM, Damped & Forced Oscillations

You are handed a mass on a spring, a swinging pendulum, or an RLC circuit and asked how it behaves over time. Rather than reaching for a formula immediately, the reliable move is to classify the system first. Once you know which of three categories it falls into, the right tools follow.

The three categories are ideal simple harmonic motion, damped oscillation, and forced oscillation. A short procedure sorts any oscillating system into one of them.

When This Approach Applies

Use this classification whenever a system repeats its motion around an equilibrium position. An oscillating system always has two ingredients: an equilibrium position and a restoring effect that pushes the system back after it is displaced. Once it passes back through equilibrium, inertia carries it past center, so the motion repeats.

If those two ingredients are present, the procedure below tells you which model to use.

The Procedure, Step By Step

1. Find the equilibrium

Identify the position where the net restoring effect is zero and the system would rest if undisturbed. Everything else is measured relative to this point.

2. Check the restoring behavior

If the restoring force or torque is proportional to displacement, the ideal motion can be treated as SHM. For a spring that condition is

$$F = -kx$$

The minus sign matters: it shows the force points back toward equilibrium. In SHM the acceleration obeys $a = -\omega^2 x$, which produces sinusoidal motion with

$$\omega = \sqrt{\frac{k}{m}}, \qquad T = 2\pi \sqrt{\frac{m}{k}}$$

A stiffer spring pulls harder, so the oscillation is faster; a larger mass resists acceleration, so it is slower.

3. Check for energy loss

If friction, drag, or electrical resistance matters, the oscillation is damped and the amplitude decreases with time. For light damping the motion still looks roughly periodic; for heavy damping the system may return to equilibrium without completing repeated swings.

4. Check for external driving

If a periodic force keeps acting, the motion is forced and the response depends strongly on the driving frequency. When the driving frequency is far from the natural frequency the response stays modest; when it is close, and damping is light, the amplitude can grow much larger. That large-response region is resonance.

Full Worked Example: One Spring, Three Categories

Take a mass of $0.50\ \mathrm{kg}$ on an ideal spring with $k = 200\ \mathrm{N/m}$, and run the steps.

Step 2 gives the angular frequency:

$$\omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{200}{0.50}} = \sqrt{400} = 20\ \mathrm{rad/s}$$

and the period and natural frequency:

$$T = \frac{2\pi}{\omega} = \frac{2\pi}{20} \approx 0.314\ \mathrm{s}, \qquad f = \frac{\omega}{2\pi} = \frac{20}{2\pi} \approx 3.18\ \mathrm{Hz}$$

So in the ideal SHM model the system completes one oscillation in about $0.314$ seconds, or roughly $3.18$ oscillations per second. Now run steps 3 and 4 on the same system:

• Release it with no friction: the motion is ideal SHM.
• Add air resistance or internal friction: the amplitude decays, so the motion is damped.
• Push it periodically with a motor: the motion is forced, and if the driving rate is near $3.18\ \mathrm{Hz}$ with light damping, the response can grow far beyond its off-resonance size.

This one example organizes the whole topic: SHM sets the ideal timing, damping explains why real motion fades, and forcing explains how an outside input sustains or enlarges it.

Where Students Get Stuck, And How To Check

• Calling every oscillation SHM. Back-and-forth motion is not enough; verify the restoring effect is proportional to displacement (step 2) before using the SHM formulas.
• Thinking damping only shrinks amplitude. The shrinking amplitude is the visible signature, but damping also changes the detailed motion, so do not treat it as cosmetic.
• Assuming forced motion grows without limit. Real systems have damping, which caps the steady response. If your prediction blows up, you have probably dropped the damping check.
• Pinning resonance to the exact natural frequency. For light damping the strongest response is near the natural frequency, and the exact peak depends on damping and on whether you track displacement, velocity, or power.

Where Oscillations Show Up

The same three categories appear in car suspensions, pendulum clocks, guitar strings, sound and vibration, molecular motion, sensors, and RLC circuits. They all share one language: natural frequency, damping, and driving.

Practice the Classification

Take the same spring and double the mass to $1.0\ \mathrm{kg}$. Rerun the steps to recompute $T$ and the natural frequency, then ask what a driving force near the new natural frequency would do. For extra range, run the same four steps on a pendulum or an RLC circuit and watch the procedure stay identical.