Input Parameters
Model: single-degree-of-freedom mass–spring–damper. Choose forced for steady-state, or free for transient decay.
Example Data Table
Use these sample inputs to test typical vibration cases.
| Case | m (kg) | k (N/m) | c (N·s/m) | F0 (N) | f (Hz) | x0 (m) |
|---|---|---|---|---|---|---|
| Light damping, near resonance | 2.50 | 1200 | 3.0 | 20 | 3.40 | 0.010 |
| Moderate damping, off resonance | 5.00 | 2500 | 18.0 | 30 | 2.00 | 0.005 |
| Free decay validation | 1.20 | 900 | 4.0 | 0 | 0 | 0.015 |
Formula Used
This tool uses standard single-degree-of-freedom vibration relationships.
- Equation of motion: m ẍ + c ẋ + k x = F0 cos(ωt)
- Natural frequency: ωn = √(k/m), and fn = ωn/(2π)
- Critical damping: cc = 2 √(km)
- Damping ratio: ζ = c/cc
- Damped frequency (underdamped): ωd = ωn √(1 − ζ²)
- Static deflection: xstatic = F0/k
- Frequency ratio: r = ω/ωn
- Magnification factor: M = 1 / √((1 − r²)² + (2ζr)²)
- Steady displacement amplitude: X = xstatic · M
- Phase angle: φ = atan2(2ζr, 1 − r²)
- Forced response at time: x(t) = X cos(ωt − φ)
- Free underdamped response: x(t) = e^(−ζωn t)[x0 cos(ωd t) + (v0+ζωn x0)/ωd · sin(ωd t)]
Notes: Assumes linear stiffness, viscous damping, and small oscillations. For multi-degree systems, compute modal parameters and repeat per mode.
How to Use This Calculator
- Enter mass, stiffness, and damping from your model.
- Select Forced steady-state to evaluate harmonic forcing.
- Provide F0 and f, then choose a time t.
- Select Free response to study decay with x0 and v0.
- Press Calculate to show results above the form.
- Use Download CSV or Download PDF for reports.
Professional Notes on Vibration Analysis
An applied reference for interpreting the calculator outputs in design and testing workflows.
1) Why single-degree-of-freedom modeling matters
Many machines and structures have one dominant mode near a critical operating speed. Approximating the motion with an equivalent mass, stiffness, and damping captures the peak response trend while keeping inputs measurable. Use this tool when one mode drives risk.
2) Natural frequency as a design target
Natural frequency depends on stiffness-to-mass ratio. Raising stiffness or reducing mass increases fn. For rotating equipment, separating operating frequency from fn by at least 20–30% reduces amplification in many practical cases.
3) Damping ratio and energy loss
Damping ratio ζ compares actual damping to critical damping. Small values (for example 0.01–0.05) are common in lightly damped metal assemblies and can produce large magnification near resonance. Adding isolators or tuned damping increases stability margins.
4) Magnification factor and resonance peak
The magnification factor M indicates how much dynamic displacement exceeds static deflection F0/k. At resonance, M approaches approximately 1/(2ζ) for light damping, which links directly to the reported quality factor.
5) Phase angle for diagnostics
Phase angle shifts from near 0° at low frequency to near 180° at high frequency. Around resonance, phase is close to 90°. In field data, this transition helps distinguish unbalance, misalignment, and stiffness changes when combined with amplitude trends.
6) Velocity and acceleration outputs
Displacement highlights compliance, velocity relates to fatigue and bearing heating, and acceleration correlates with shock and noise. The tool computes steady-state velocity amplitude ωX and acceleration amplitude ω²X, useful for comparing sensor units and limits.
7) Free decay for validation and identification
Free response uses initial conditions to estimate transient behavior. A measured ring-down can be used to back-calculate damping by fitting exponential decay. Matching decay rate and oscillation period improves confidence that your equivalent parameters represent the physical system.
8) Practical workflow and reporting
Start with measured or estimated m, k, and c, then sweep excitation frequency to map response. Export results for design reviews, maintenance logs, and compliance audits. Document assumptions, boundary conditions, and units to keep comparisons consistent. For acceptance, many teams set alert thresholds on velocity and track trends over time. When instrumenting, note sensor bandwidth and mounting stiffness; poor mounting can under-report acceleration. Record operating speed, temperature, and load so exported results remain comparable. in commissioning and condition monitoring programs.
FAQs
1) What does the frequency ratio r tell me?
r compares excitation to natural frequency. Values near 1 indicate resonance risk, while values far below or above 1 typically reduce displacement amplification for the same forcing level.
2) Why can small damping cause very large amplitudes?
Near resonance, stored energy grows each cycle. With low damping, little energy is dissipated, so the magnification factor rises sharply, often approximating 1/(2ζ) for light damping.
3) Should I use forced or free mode?
Use forced mode for steady response to harmonic input and phase. Use free mode to study decay from initial conditions, such as ring-down testing or startup transients.
4) How accurate is the resonance frequency estimate?
The resonance estimate assumes linear viscous damping and an SDOF model. It is most reliable for lightly damped systems where one dominant mode governs the response peak.
5) What units should I use?
Use consistent SI units: kg, N/m, N·s/m, N, Hz, and meters. If you use other units, convert inputs first to avoid scaling errors in frequency and amplitude results.
6) Why does phase approach 180 degrees at high frequency?
At high excitation frequency, inertia dominates. Displacement tends to oppose the forcing direction, producing a phase lag near 180°, while low frequency behavior follows stiffness with small lag.
7) Can I use this for multi-degree systems?
Yes, by applying it per mode using modal mass, stiffness, and damping. Combine modal responses using appropriate superposition methods and consider mode shapes for spatial predictions.
Use this tool to understand vibration behavior very quickly.