Calculator
Formula used
A damped, driven single‑degree‑of‑freedom oscillator satisfies m ẍ + c ẋ + k x = F0cos(ωt). The steady‑state displacement can be written as x(t)=Xcos(ωt−φ).
| Amplitude | X = F0 / √[(k − mω²)² + (cω)²] |
|---|---|
| Phase lag | φ = atan2(cω, k − mω²) |
| Natural frequency | ω0 = √(k/m) and f0=ω0/(2π) |
| Damping ratio | ζ = c / (2√(km)) with ccrit=2√(km) |
| Static deflection | Xstatic = F0/k and magnification M=X/Xstatic |
Resonance peak (light damping): ωr≈ω0√(1−2ζ²), valid when 0<ζ<1/√2.
How to use this calculator
- Enter force amplitude, mass, stiffness, and damping with units.
- Select frequency input mode: Hz or angular frequency.
- Provide the driving value and choose the matching unit.
- Press Calculate to view results above the form.
- Use CSV or PDF buttons to export results.
Example data table
| F0 (N) | m (kg) | k (N/m) | c (N·s/m) | f (Hz) | Amplitude X (m) |
|---|---|---|---|---|---|
| 10 | 2 | 2000 | 15 | 5 | 0.00610 |
| 15 | 1.5 | 3000 | 10 | 10 | 0.00524 |
| 20 | 4 | 5000 | 40 | 8 | 0.00417 |
1) What the calculator solves
This tool evaluates the steady‑state displacement amplitude of a single‑degree‑of‑freedom mass–spring–damper under harmonic forcing. You enter force amplitude, mass, stiffness, damping, and the driving frequency, then receive displacement amplitude, magnification factor, and phase lag in a consistent SI output set.
2) Core response data you get
The primary output is X in meters, computed from the classic frequency response function. The calculator also reports Xstatic=F0/k and magnification M=X/Xstatic, which is useful when comparing designs at different load levels.
3) Frequency ratio and resonance
The frequency ratio r=ω/ω0 indicates proximity to resonance. For light damping, the peak typically occurs near r≈1. For example, with ζ=0.05, peak magnification can exceed 10, while ζ=0.20 often reduces the peak below about 3.
4) Damping ratio interpretation
Damping is expressed as ζ=c/ccrit, where ccrit=2√(km). Values ζ<1 describe under‑damped systems that oscillate. e appreciably; ζ≈0.02–0.10 is common in lightly damped structures, while ζ>0.30 is typical of highly damped mounts and isolators.
5) Phase lag as diagnostic data
The phase angle φ helps validate sensor timing and control logic. As r increases, φ transitions from near 0° (stiffness‑controlled) toward 180° (mass‑controlled). Around resonance, φ is close to 90° for moderate damping, which is often seen in shaker tests.
6) Units and practical checks
Inputs accept common engineering units and are converted internally to SI. A quick check is dimensional: stiffness in N/m and force in N should produce displacement in meters. If results appear too large, verify stiffness and damping units, especially N/cm versus N/m conversions.
7) Using results for design decisions
Compare amplitude against allowable displacement, fatigue limits, or clearance constraints\.\ If M is high near operating speed, consider increasing damping, shifting ω0 by changing k or m,\ or reducing excitation force\. Small changes around r≈1 can produce large response differences\. Compare operating points across a sweep of frequency to spot risk bands\. For isolation problems, you typically want r>√2 so the transmitted motion drops, then choose ζ to balance peak control with high‑frequency attenuation\. Document the worst‑case X at expected tolerances in k and c\.
8) When this model applies
The equations assume linear stiffness, viscous damping, and a single dominant mode. For non‑linear springs, friction damping, or multiple closely spaced modes, use this calculator as a first estimate, then validate with testing or a multi‑DOF simulation.
FAQs
1) What is forced vibration amplitude?
The forced vibration amplitude is the steady‑state displacement magnitude produced by a periodic driving force after transient effects decay, for a given frequency and damping.
2) What does the magnification factor mean?
Magnification M compares dynamic amplitude to static deflection under the same force. M>1 indicates dynamic amplification; near resonance, M can become very large when damping is small.
3) Why does damping reduce resonance peaks?
Damping dissipates vibration energy each cycle. Higher damping increases the response denominator, flattening the frequency response and reducing the peak amplitude near the natural frequency.
4) Can I input frequency in rad/s?
Yes. Select angular frequency mode and enter ω directly in rad/s. In Hz mode, the calculator converts using ω=2πf.
5) What range of damping ratio is typical?
Lightly damped metal structures often fall around ζ=0.01–0.05. Rubber mounts and viscoelastic isolators can reach ζ=0.10–0.30 or higher, depending on temperature and preload.
6) How is phase lag used in practice?
Phase lag helps align force and displacement measurements and supports control tuning. Near resonance, displacement lags force by roughly 90° for many under‑damped systems.
7) What if my system has multiple modes?
If multiple modes contribute significantly, a single‑mode SDOF model may under‑ or over‑predict response. Use this output as a quick estimate, then refine with modal superposition or simulation.