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Calcimator

Vibration Analysis Calculator

Analyze single-degree-of-freedom vibration systems. Calculate natural frequency, damping ratio, transmissibility, and amplitude magnification.

About this calculator

This calculator models a classic single-degree-of-freedom (SDOF) mass-spring-damper system, the standard first analysis for vibration isolation and machinery mounting. Natural Frequency follows fn = (1/2π)√(k/m), where Spring Stiffness (k) sets it directly and Supported Weight (converted to mass via m = W/g) sets it inversely -- a stiffer mount or a lighter load both raise the frequency at which the system would freely vibrate. Neither Damping Coefficient (c) nor Excitation Frequency changes Natural Frequency at all, since it describes the UNDAMPED system's own resonant behavior, independent of how hard or how fast you're driving it. Damping Ratio (ζ) is c/(2√(km)), rising directly with Damping Coefficient (c) and independent of Excitation Frequency -- it is a property of the mount, not the forcing.

Frequency Ratio (r) is simply Excitation Frequency divided by Natural Frequency, the single number that determines whether the system is being driven below resonance (r < 1), near resonance (r ≈ 1, where response is largest and most sensitive to damping), or well above resonance (r >> 1). Transmissibility and Amplitude Magnification are both standard forced-vibration formulas built from Frequency Ratio (r) and Damping Ratio (ζ); NEITHER behaves monotonically as Excitation Frequency changes across a wide range -- both typically rise toward a peak near resonance (r ≈ 1) and fall away on either side, so "higher Excitation Frequency" does not mean "more transmitted vibration" in general, only within specific ranges relative to Natural Frequency. A Transmissibility below 1.0 means the mount is successfully isolating vibration; above 1.0 means the mount is amplifying it, which typically happens for excitation near or below resonance in lightly damped systems.

Inputs

lb/in
lb
lb·s/in
Hz

Results

Natural Frequency

31.29 Hz

Transmissibility

8.11

Amplitude Magnification

8.07

Damping Ratio (ζ)0.05
Frequency Ratio (r)0.96
How to Use This Calculator
  1. Enter the Spring Stiffness (k) in lb/in — for isolator pads or mounts, the stiffness is listed in the product datasheet. Add values in parallel.
  2. Enter the Supported Weight in lb — the total weight of the equipment or vibrating mass.
  3. Enter the Damping Coefficient (c) in lb·s/in — for steel structures ≈ 1–5% critical damping; rubber mounts ≈ 5–15%.
  4. Enter the Excitation Frequency in Hz — for rotating machinery use f = RPM / 60.
  5. Review the Natural Frequency in Hz — the excitation frequency should not be within 20% of the natural frequency to avoid resonance.
  6. Check Transmissibility — a value below 1.0 means isolation is effective; above 1.0 means vibration is being amplified.

How the result changes with Supported Weight

Supported WeightNatural FrequencyTransmissibilityAmplitude Magnification
544.24 Hz1.831.82
7.536.13 Hz3.13.08
1525.54 Hz2.572.56
2519.79 Hz0.770.77

What each input means

Spring Stiffness (k)
Total spring stiffness of the support system. For multiple springs in parallel, add stiffness values.
Supported Weight
Weight of the vibrating mass (equipment, rotor, etc.). Converted internally to mass using g = 386.4 in/s².
Damping Coefficient (c)
Viscous damping coefficient. Steel structures ≈ 1-5% critical; rubber mounts ≈ 5-15% critical.
Excitation Frequency
Frequency of the forcing vibration. For rotating machinery: f = RPM/60.

How this is calculated

Worked example, using the default values

  1. Identify Input Parameters
    4 parameters
    Spring Stiffness (k) = 1000, Supported Weight = 10, Damping Coefficient (c) = 0.5, Excitation Frequency = 30 = 4 input(s) provided
  2. Calculate Natural Frequency
    Natural Frequency
    31.29 = 31.29
  3. Calculate Transmissibility
    Transmissibility
    8.105 = 8.105
  4. Calculate Amplitude Magnification
    Amplitude Magnification
    8.069 = 8.069
  5. Calculate Damping Ratio
    Damping Ratio
    0.0491 = 0.0491
  6. Calculate Frequency Ratio
    Frequency Ratio = r
    0.959 = 0.959

Engine last updated . Checked against 2 independently-derived tests — how we verify calculators. Built by Paul Gunder, a software engineer, not a licensed financial, medical, or legal professional.

Frequently Asked Questions

Why doesn't Excitation Frequency change Natural Frequency?

Natural Frequency describes how the mass-spring system would vibrate on its OWN if displaced and released, a property set entirely by Spring Stiffness (k) and Supported Weight -- it exists independently of any external forcing. Excitation Frequency instead describes how hard you're DRIVING the system from outside; the ratio between the two (Frequency Ratio) is what determines the system's response, not Natural Frequency itself.

Why doesn't Excitation Frequency change Damping Ratio either?

Damping Ratio (ζ = c/(2√(km))) is a fixed property of the physical mount -- Damping Coefficient (c), Spring Stiffness (k), and Supported Weight alone -- describing how quickly the system dissipates energy on its own. It doesn't depend on how fast an external force is oscillating; Frequency Ratio is the input that captures the relationship between the forcing and the mount's own natural behavior.

Does raising Excitation Frequency always increase Transmissibility?

No -- Transmissibility typically RISES toward a peak somewhere near Frequency Ratio ≈ 1 (excitation near resonance) and then FALLS again as Excitation Frequency climbs well past Natural Frequency, especially in lightly damped systems. This is why vibration isolators are deliberately designed with a low natural frequency relative to the excitation they need to isolate -- operating well above resonance (high Frequency Ratio) is where Transmissibility drops below 1.0 and true isolation begins.

What does a Frequency Ratio (r) close to 1.0 mean for my design?

It means Excitation Frequency is close to Natural Frequency -- the system is operating near resonance, where Transmissibility and Amplitude Magnification are both largest and most sensitive to small changes in Damping Ratio. This is generally the condition to avoid in a mount or isolator design; the standard howToUse guidance here is to keep Excitation Frequency at least 20% away from Natural Frequency.

How does adding damping change the picture near resonance?

Higher Damping Ratio (from a larger Damping Coefficient relative to Spring Stiffness and Supported Weight) reduces the height of the Transmissibility and Amplitude Magnification peak near Frequency Ratio ≈ 1, trading a lower resonance spike for typically worse isolation performance well above resonance -- a classic vibration-isolation design trade-off between suppressing the resonance peak and maximizing high-frequency isolation.

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