Simple Harmonic Motion Simulator

📐 Governing Equation of Motion

mẍ + bẋ + kx = F₀ sin(ωt + φ)

mMass of the ball (kg)

bDamping factor (Ns/m)

kSpring constant (N/m)

x₀Initial displacement (m)

v₀Initial velocity (m/s)

ωDriving frequency (rad/s)

F₀Driving amplitude (N)

φPhase angle (rad)

🔬 Types of Oscillators

Free Oscillator

No damping (b = 0) and no driving force (F₀ = 0). The mass oscillates indefinitely at its natural frequency ω₀ = √(k/m).

Condition: b = 0, F₀ = 0

e.g. Ideal pendulum, LC circuit

Damped Oscillator

Energy is lost to friction/resistance. Amplitude decreases exponentially: x(t) = Ae^−γtcos(ωt + φ)

Under: b < 2√(km) Critical: b = 2√(km) Over: b > 2√(km)

e.g. Shock absorbers, pendulum in air

Forced Oscillator

An external periodic force drives the system. At resonance (ω = ω₀), amplitude peaks dramatically.

Condition: F₀ ≠ 0

e.g. Pushed swing, resonating wineglass

🔄 Simulation Process Flow

1

Select Oscillator Type

Choose Manual, Under, Critical, or Over damping using the radio buttons in the control panel.

▼

2

Set Parameters

Adjust sliders for b (damping), v₀ (initial velocity), ω (driving frequency), and F₀ (driving force). Initial displacement x₀ appears after pressing Play.

▼

3

Run the Simulation

Press the ▶ Play/Pause button to start. Watch the spring-mass system animate on the left canvas and the displacement (red) / velocity (blue) traces appear on the graph in real time.

▼

4

Toggle Velocity Trace

Check the V checkbox to overlay the velocity curve (blue) on the displacement graph. This reveals the 90° phase difference between x(t) and v(t).

▼

5

Observe & Analyse

Compare how the amplitude envelope changes across damping types. Increase ω towards ω₀ to witness resonance — the maximum amplitude spike.

▼

6

Reset & Repeat

Use Reset Time to clear the graph trace, or Reset Simulation to restore all parameters to their default values. Try a new configuration!

🎮 Controls Reference

ControlActionEffect on Simulation

▶ / ⏸ Play/Pause Start or pause Begins/halts the animation and graph plotting

⏹ Reset Time Clear traces Erases graph history; keeps current parameters

🔄 Reset Simulation Full reset Restores all parameters and clears the canvas

b slider Damping factor 0 → free; 2.8 → critical; >2.8 → over-damped

v₀ slider Initial velocity Sets velocity at t = 0; affects starting momentum

ω slider Driving frequency Approach ω₀ ≈ 1 rad/s for resonance peak

F₀ slider Driving amplitude 0 → no external force (free/damped modes)

V checkbox Velocity trace Overlays blue v(t) curve on the graph

💡 Quick Tips

Find resonance: Set b to a small value (~0.5), set F₀ > 0, then slowly increase ω from 0 to 2 rad/s — watch the amplitude explode near ω = 1.
Critical vs Over: Select "Critical" mode and compare the return-to-equilibrium speed with "Over" — critical is always faster.
Phase difference: Enable the V checkbox; notice displacement and velocity curves are always exactly 90° apart in undamped SHM.
Energy loss: In "Under" mode, each successive peak is smaller by a factor of e^−γT where T is the period.

Types of Oscillators — Simulation Guide