Driven Damped Harmonic Oscillator Calculator

Calculator Form

Mass m (kg)

Damping c (N·s/m)

Spring constant k (N/m)

Driving force F0 (N)

Driving angular frequency Ω (rad/s)

Driving phase φ (rad)

Initial displacement x(0) (m)

Initial velocity v(0) (m/s)

Evaluation time t (s)

Calculate Reset

Example Data Table

Sample setup uses m = 1 kg, c = 0.8 N·s/m, k = 25 N/m, and F0 = 10 N.

Formula Used

The calculator uses the standard forced vibration model:

m x'' + c x' + k x = F₀ cos(Ωt + φ)

Core Definitions

Natural angular frequency: ω₀ = √(k / m)

Damping factor: β = c / (2m)

Damping ratio: ζ = c / (2√km)

Steady-State Response

Amplitude: A = F₀ / √[(k - mΩ²)² + (cΩ)²]

Phase lag: δ = tan^-1(cΩ / (k - mΩ²))

Particular motion: x_s(t) = A cos(Ωt + φ - δ)

Transient Motion

Underdamped case: x_h(t) = e^-βt[C1 cos(ω_d t) + C2 sin(ω_d t)]

where ω_d = √(ω₀² - β²).

Critical case: x_h(t) = (C1 + C2 t)e^-βt

Overdamped case: x_h(t) = C1 e^r1t + C2 e^r2t

Total Motion

x(t) = x_h(t) + x_s(t)

Acceleration comes from the governing equation. Energy uses kinetic and spring terms.

How to Use This Calculator

1. Enter mass, damping coefficient, and spring constant.
2. Provide the external driving force amplitude and angular frequency.
3. Set the driving phase if your force does not start at zero phase.
4. Enter initial displacement and initial velocity for transient motion.
5. Choose the time where you want displacement, velocity, and acceleration evaluated.
6. Press calculate to place the result section above the form.
7. Use the CSV button to save values in spreadsheet form.
8. Use the PDF button to save a clean report.

Driven Damped Harmonic Oscillator Guide

What This Physics Calculator Measures

A driven damped harmonic oscillator is a mass spring system under an external periodic force. Real machines behave like this. So do many sensors, vehicle parts, and lab setups. This calculator estimates amplitude, phase lag, transient response, and total motion at a selected time.

Why Damping Changes Everything

Damping removes energy from motion. Small damping lets the system oscillate longer. Large damping kills motion quickly. The damping ratio tells you whether the system is underdamped, critically damped, or overdamped. That classification changes the transient formula and the way motion settles.

How Driving Frequency Affects Response

The driving frequency controls resonance behavior. When the forcing frequency moves close to the natural frequency, the amplitude can rise sharply. Damping limits that rise. The calculator shows this through steady state amplitude, phase lag, and the resonance estimate when that formula is valid.

Steady Motion and Transient Motion

Forced oscillation problems contain two parts. The transient part depends on initial displacement and initial velocity. It fades with time when damping exists. The steady part remains because the outside force keeps supplying energy. The total displacement is the sum of both parts.

How Students and Engineers Use These Outputs

Students can verify textbook examples fast. Engineers can inspect vibration sensitivity before deeper simulation. Phase lag helps describe how far the response trails the force. Instantaneous power helps explain energy transfer. Mechanical energy helps compare motion states at different times.

Reading the Results Correctly

A large amplitude does not always mean instability. It may only mean the chosen forcing frequency is near resonance. A high quality factor suggests a sharper peak. A low quality factor suggests broader, flatter response. Always keep units consistent and use angular frequency in radians per second.

FAQs