Calculator inputs
Use consistent SI units for force, mass, stiffness, damping, and frequency. The layout is single column overall, with a responsive three, two, and one column input grid.
Formula used
For a single degree of freedom system under harmonic force, the steady-state displacement amplitude is:
Where:
- X = vibration amplitude
- F0 = force amplitude
- m = mass
- k = stiffness
- c = viscous damping coefficient
- ω = forcing angular frequency
Supporting relationships used by the calculator:
How to use this calculator
- Enter the harmonic force amplitude in newtons.
- Provide the equivalent mass, spring stiffness, and damping coefficient.
- Enter the excitation frequency and choose Hz or rad/s.
- Select the displacement unit you want for the displayed amplitude.
- Set the sweep multiplier and graph points for the Plotly response curve.
- Click Calculate amplitude to see the result above the form.
- Review amplitude, damping ratio, magnification, phase angle, and the graph.
- Use the CSV or PDF buttons to export the displayed response summary.
Example data table
| Case | F0 (N) | m (kg) | k (N/m) | c (N·s/m) | f (Hz) | Amplitude (mm) | ζ | r | Phase (deg) |
|---|---|---|---|---|---|---|---|---|---|
| Example A | 100 | 8 | 12,000 | 90 | 3 | 10.737 | 0.145 | 0.487 | 10.50 |
| Example B | 150 | 12 | 18,000 | 220 | 5 | 16.206 | 0.237 | 0.811 | 48.31 |
| Example C | 250 | 20 | 30,000 | 400 | 7 | 12.741 | 0.258 | 1.136 | 116.28 |
Frequently asked questions
1. What does vibration amplitude mean here?
It is the steady-state peak displacement of the mass caused by a harmonic force. The calculator reports the response after transient effects have decayed.
2. Why does amplitude increase near resonance?
When forcing frequency approaches the natural frequency, the dynamic stiffness becomes small. That allows a much larger response, especially when damping is light.
3. How does damping change the result?
More damping reduces peak amplitude, broadens the response curve, and limits resonance severity. It also affects the phase angle between applied force and displacement.
4. Should I enter frequency in Hz or rad/s?
Use whichever value you already have. Select the correct unit first. The calculator converts internally and shows both frequency forms in the results.
5. What is the frequency ratio?
Frequency ratio is the operating angular frequency divided by natural angular frequency. It is a quick way to judge whether the system is below, near, or above resonance.
6. Why is phase angle useful?
Phase angle shows how much the displacement lags the applied harmonic force. It is important when comparing vibration timing, control response, and resonance behavior.
7. What happens with zero damping at resonance?
The ideal equation predicts unbounded amplitude. Real systems never behave perfectly like that, because structural damping, nonlinearities, and limits always appear in practice.
8. Which units must stay consistent?
Use newtons for force, kilograms for mass, newtons per meter for stiffness, newton-seconds per meter for damping, and hertz or rad/s for excitation frequency.