Forced Vibration Calculator — Resonance & Damping
Calculate the steady-state response of a damped spring-mass system to harmonic forcing: amplitude, phase lag, natural frequency, damping ratio and resonance.
Oscillatory vibration calculator
Displacement, velocity, acceleration
- Envelope
- x(t)
- v(t)
- a(t)
Gain and phase vs frequency ratio
- Magnitude ratio
- Phase (deg)
x versus v orbit
Mechanical energy and damping dissipation
- Mechanical energy
- Damping dissipation
Isometric spring-mass-damper
Gain over frequency ratio and damping ratio
Numerical checks
A clear path from inputs to results.
See how a single spring-mass-damper system responds when you push it with a steady oscillating force: how big the steady vibration gets, how far it lags the force, and how sharply it amplifies near resonance. It is the textbook model behind machine mounts, vibration isolators, and any structure driven at a fixed frequency.
How to use
- Enter the system properties: mass m, spring stiffness k, and viscous damping coefficient c.
- Enter the harmonic forcing: the force amplitude F₀ and the drive frequency (in Hz), which the tool converts to the angular drive frequency ω = 2πf.
- Optionally set the initial displacement x₀ and initial velocity v₀ to see the transient that decays before the steady state settles.
- Read the natural frequency ωₙ = √(k/m) and damping ratio ζ = c/(2mωₙ), which fix the system’s character.
- Read the steady-state amplitude X and phase lag φ, and compare the drive frequency to ωₙ through the frequency ratio r = ω/ωₙ.
- Check the dynamic magnification factor M and quality factor Q to judge how close you are to resonance and how strongly the force is amplified.
The formula
Reference: Steady-state response of a damped SDOF oscillator to harmonic forcing; Rao, Mechanical Vibrations; MIT OCW 2.003/18.03 forced-vibration notes.
Worked example
Machine on a spring mount driven near resonance
- Mass m = 1 kg; stiffness k = 100 N/m; damping c = 1.4 N·s/m; force amplitude F₀ = 20 N; drive frequency 1.4 Hz.
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The natural frequency is ωₙ = √(k/m) = √100 = 10 rad/s and the damping ratio is ζ = c/(2mωₙ) = 1.4/(2·1·10) = 0.07. The drive frequency 1.4 Hz is ω = 2π×1.4 = 8.7965 rad/s, so the frequency ratio is r = ω/ωₙ = 0.8797 — just below resonance. The dynamic stiffness terms are k − mω² = 100 − 77.38 = 22.62 N/m and cω = 1.4×8.7965 = 12.32 N·s⁻¹. The steady amplitude is X = 20 / √(22.62² + 12.32²) = 20 / 25.76 = 0.7765 m, the phase lag is φ = atan2(12.32, 22.62) = 0.499 rad = 28.56°, and the magnification factor is M = 1/√((1−r²)² + (2ζr)²) = 3.882.
Steady-state amplitude X ≈ 0.776 m, lagging the force by φ ≈ 28.6°; the response is magnified about 3.88× over the static deflection F₀/k = 0.20 m. The quality factor is Q = 1/(2ζ) ≈ 7.14, and the peak magnification at resonance would reach about 7.16.
Interpreting the result
The amplitude peaks when the drive frequency approaches the natural frequency; with light damping (ζ = 0.07 here) that peak is tall and narrow, so a small change in drive frequency sharply changes the response. The phase lag passes through 90° exactly at resonance. The quality factor Q ≈ 1/(2ζ) summarises the sharpness of the peak: Q ≈ 7.14 means the system amplifies forces near ωₙ by roughly seven times. To cut vibration, move the drive frequency away from ωₙ or add damping to flatten the peak.
Assumptions & limitations
- The system is linear and has a single degree of freedom: one mass, one linear spring, and one viscous (velocity-proportional) damper.
- The forcing is a pure sinusoid of constant amplitude and frequency; the reported amplitude and phase are the steady-state response after the transient has decayed.
- Damping is subcritical (ζ < 1), so the free response oscillates while decaying; the steady-state formula itself holds for any positive damping.
- Stiffness and damping are constant (no amplitude-dependent or frequency-dependent behaviour), and deflections are small enough for Hooke’s law to apply.
- Gravity and any static preload only shift the equilibrium position; they do not affect the vibration amplitude about that equilibrium.
Frequently asked questions
What is resonance in this model?
Resonance is when the drive frequency approaches the natural frequency ωₙ = √(k/m), making the frequency ratio r approach 1. The dynamic magnification factor M then reaches its maximum and the amplitude is largest. In the worked example r ≈ 0.88, just below resonance, which already magnifies the static deflection by about 3.88 times.
What does the quality factor Q mean?
Q = 1/(2ζ) measures how sharply the system amplifies at resonance and how slowly free vibrations decay. High Q (light damping) means a tall, narrow resonance peak and long ring-down; low Q (heavy damping) means a broad, flat response. Here ζ = 0.07 gives Q ≈ 7.14.
What is the difference between ωₙ and the damped frequency?
The undamped natural frequency is ωₙ = √(k/m); the damped free-vibration frequency is ω_d = ωₙ√(1−ζ²), slightly lower. For light damping the difference is tiny — at ζ = 0.07, ω_d is about 99.75% of ωₙ — which is why resonance is still essentially at ωₙ.
Why does the response lag the force?
A damped system cannot respond instantly, so the displacement reaches its peak slightly after the force does. The lag φ = atan2(cω, k − mω²) grows from near 0° well below resonance to 90° exactly at resonance and toward 180° well above it. The example lags by about 28.6°.
How do I reduce the vibration amplitude?
Three levers: shift the natural frequency away from the drive frequency by changing mass or stiffness, add damping to lower the magnification factor near resonance, or reduce the forcing amplitude F₀. Isolation is most effective when the drive frequency is well above √2·ωₙ, where the system transmits less than the applied force.
Methodology last reviewed 2026-10-06. Results are informational and not financial or engineering advice — see our disclaimer.