Driven Damped Harmonic Oscillator Calculator

Driven Damped Harmonic Oscillator Calculator

Mass m

Damping coefficient c

Spring constant k

Force amplitude F₀

Drive frequency f (Hz)

Time point t (s)

Initial displacement x(0)

Initial velocity v(0)

Sweep minimum frequency (Hz)

Sweep maximum frequency (Hz)

Sweep points

Calculate Reset

Example Data Table

Formula Used

The governing equation is m x'' + c x' + k x = F₀ cos(ωt).

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

Natural frequency: f_n = ω_n / 2π

Critical damping: c_c = 2√(km)

Damping ratio: ζ = c / c_c

Steady-state amplitude: X = F₀ / √[(k - mω²)² + (cω)²]

Phase lag: φ = tan^-1(cω / (k - mω²))

Velocity amplitude: V = ωX

Acceleration amplitude: A = ω²X

Average absorbed power: P = 0.5 c (ωX)²

The time-response block adds the transient solution from x(0) and v(0) to the steady forced motion.

How to Use This Calculator

1. Enter mass, damping coefficient, spring constant, and force amplitude.
2. Enter the driving frequency in hertz.
3. Add initial displacement, initial velocity, and a time point.
4. Set the minimum frequency, maximum frequency, and sweep points.
5. Press Calculate to show results above the form.
6. Review the summary metrics and the sweep table.
7. Use the CSV button for spreadsheet work.
8. Use the PDF button for a compact report file.

Driven Damped Harmonic Oscillator Guide

Why this model matters

Driven damped harmonic motion appears in many real systems. It describes a mass, spring, damping element, and repeating force. Engineers use it for suspension design, vibration isolation, machine tuning, and resonance checks. Physicists use it to study energy transfer and forced response.

What the calculator tells you

A useful calculator turns the model into practical numbers. It shows amplitude, phase lag, damping ratio, and average absorbed power. These values explain how a system reacts at one driving frequency. They also reveal whether the response is safe, efficient, or close to resonance.

Core physics behind the output

The core equation is m x'' + c x' + k x = F₀ cos(ωt). Mass controls inertia. The spring constant controls restoring force. The damping coefficient removes energy. The driving force keeps energy entering the system. Together they define the final steady response and the short transient motion.

Frequency and damping effects

Natural frequency sets the free oscillation scale. Damping ratio shows how strongly motion dies out. A small damping ratio allows a sharp resonance peak. A larger ratio spreads the response and lowers the peak. Phase also changes with frequency. Low frequencies stay almost in phase. High frequencies move closer to opposite phase.

Most useful design outputs

The most useful output is steady-state amplitude. It tells you how far the system moves under repeated loading. Velocity amplitude matters for wear and energy loss. Acceleration amplitude matters for force transmission and comfort studies. Average power helps estimate how much energy the damper absorbs.

Why time response helps

Time response is also important. Initial displacement and initial velocity create a transient term. That transient fades when damping is present. The forced term remains and dominates long-run motion. This helps compare startup behavior with long-run operating behavior.

Why a sweep is valuable

A frequency sweep adds more insight. It shows how amplitude and phase change across a range. You can spot resonance zones, safe operating windows, and inefficient settings. This is valuable in laboratories, rotating equipment, loudspeaker systems, and structural vibration studies.

How to interpret the ratio

You can also compare drive frequency with natural frequency. That ratio quickly shows whether the machine runs below, near, or above resonance. Small changes near resonance can produce large response shifts.

Use this calculator when you need a quick but detailed forced vibration check. It combines formulas, interpretation, export tools, and example data in one place.

FAQs

1) Can this model represent real machines?

Yes. The equation models any forced linear oscillator with viscous damping. It fits springs, suspensions, mounts, and many vibration test setups when stiffness and damping stay approximately constant.

2) What does amplitude mean here?

Amplitude is the steady displacement size at the chosen drive frequency. Phase lag shows how far the motion shifts behind the driving force in time.

3) What is resonance in this calculator?

Resonance happens when the driving frequency approaches the response peak. Low damping makes the peak sharper and usually raises amplitude.

4) How does damping change the response?

Higher damping lowers the peak amplitude, increases energy loss, and reduces resonance sharpness. It also changes phase more smoothly across the frequency range.

5) Is the resonance frequency always the natural frequency?

Not always. The exact peak shifts below the undamped natural frequency when damping is present. With enough damping, a clear amplitude peak may disappear.

6) What is the quality factor?

Quality factor measures how lightly damped the system is. A higher value means narrower resonance and lower energy loss per cycle.

7) Why do initial conditions matter?

Initial conditions affect the transient response and short-term motion. They do not change the final steady-state amplitude and phase once the transient decays.

8) When should I use the export buttons?

Use the CSV file for spreadsheets and data review. Use the PDF file for reports, sharing, or quick printable result snapshots.