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Vibration Isolator Simulator — Transmissibility TR & Frequency Ratio

Quick answer
The displacement transmissibility of a vibration isolator is TR = √[(1+(2ζr)²)/((1−r²)² + (2ζr)²)], with frequency ratio r = ω/ω_n, damping ratio ζ, and ω_n = √(k/m). Isolation only occurs for r > √2 ≈ 1.414; below that, TR ≥ 1 and the vibration is amplified instead.

Real-time displacement transmissibility from base to equipment. Vary mass, stiffness, damping, and forcing frequency to watch the TR-r curve cross the r=√2 boundary into the true isolation region.

Parameters
Mass m
kg
Stiffness k
N/m
Damping ratio ζ
—
Forcing freq ω
rad/s

Isolation is effective when the frequency ratio r=ω/ω_n exceeds √2≈1.414 and TR<1.

While paused, moving a slider updates the results immediately.

Results

Bottom = excited base (input wave) / Top = machine on isolation mounts (transmitted wave) / Right = TR–r curve and current operating point (red). Resonance occurs at r=1 (red dashed line), and the isolation threshold is r=√2 (green line).

TR-r diagram (curves by damping ratio)
—
Natural freq f_n
—
Frequency ratio r=ω/ω_n
—
Transmissibility TR
—
Isolation efficiency IE=1-TR
Theory & Key Formulas

Horizontal axis = frequency ratio r (log) / vertical axis = transmissibility TR (log) / green region = effective isolation region (r>√2 and TR<1) / red dot = current point

Theory & Key Formulas

The ratio of motion transmitted from a sinusoidally vibrating base to equipment through a spring and dashpot is called the displacement transmissibility TR. It is derived from the frequency response of a 1-DOF viscously damped system.

Natural angular frequency ω_n and natural frequency f_n. k is the spring constant and m is the mass:

$$\omega_n = \sqrt{k/m},\qquad f_n = \frac{\omega_n}{2\pi}$$

Frequency ratio r and damping ratio ζ. c is the viscous damping coefficient:

$$r = \frac{\omega}{\omega_n},\qquad \zeta = \frac{c}{2\sqrt{km}}$$

Displacement transmissibility TR and isolation efficiency IE:

$$TR = \sqrt{\frac{1+(2\zeta r)^2}{(1-r^2)^2 + (2\zeta r)^2}},\qquad IE = 1 - TR$$

r=1 is the resonance point, where TR is maximum. Isolation occurs only when r>√2≈1.414; below this value, TR≥1 and vibration is amplified rather than isolated.

What is the Vibration Isolator Simulator?

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So "vibration isolation" basically means putting a rubber pad under the machine, right? Does that really stop vibration?
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Roughly yes, but it doesn't "stop" vibration — it lowers the fraction that gets through. We call that fraction transmissibility, TR. TR = 0.13 means only 13 % is transmitted, i.e. 87 % is isolated. Try the default values above (m = 100 kg, k = 100 000 N/m) — you should see TR ≈ 13 %.
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OK. When I sweep ω the whole curve shifts. Why the red vertical line near the middle, around r = 1?
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That's resonance — the forcing frequency ω equals the system's natural frequency ω_n, so vibration is amplified the most. In $TR=\sqrt{(1+(2\zeta r)^2)/((1-r^2)^2+(2\zeta r)^2)}$ the denominator term $(1-r^2)^2$ goes to zero there. With zeta near zero, TR blows up. Try setting zeta near 0 and r = 1 in the tool — you'll get a huge number.
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So bigger damping is always better? With zeta = 1.0 the peak almost disappears.
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Here's the catch: more damping raises TR in the isolation region (r > sqrt(2)). The curves for different zeta meet around r = sqrt(2) and then cross. Real machines have to pass through resonance during start-up and shut-down, so you want some damping, but not too much in steady-state. Engineers typically compromise around zeta = 0.05–0.15.
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The green vertical line at r = sqrt(2) ≈ 1.414 is the "isolation boundary"? Nothing works below it?
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Exactly — the iron rule of vibration isolation. For r < sqrt(2), TR ≥ 1, so you amplify rather than isolate. A common practical target is to set the natural frequency below half of the forcing frequency (r ≥ 2). For a 50 Hz motor, that means f_n ≤ 25 Hz. Softer springs lower f_n, which is the basic strategy.

Frequently Asked Questions

Rubber mounts are compact and cheap with damping ratios around 0.05–0.10, widely used for HVAC units and small machines. Steel coil springs have nearly zero damping and reach very low frequencies but resonate violently unless paired with a damper. Air springs lower the natural frequency to about 1 Hz, suitable for precision equipment and vehicles. Pick whichever lets you set the natural frequency below the forcing frequency by a factor of sqrt(2).
No. Larger damping suppresses the peak at resonance (r near 1), but in the isolation region (r > sqrt(2)) it actually raises the transmissibility TR and hurts isolation. The curves cross near r = sqrt(2). Since real machines pass through resonance during start-up and shut-down, designers compromise around zeta = 0.05–0.15 to balance peak suppression with high-frequency isolation.
If the forcing frequency (operating rpm or excitation source) is close to the machine's natural frequency, resonance amplifies vibration and causes equipment damage, noise and disturbance to neighbours. At the planning stage you compute omega_n = sqrt(k/m) and design the mount stiffness k so f_n stays below about 0.7x the operating frequency. This is the first step toward satisfying r > sqrt(2).
Yes. Base-isolated buildings deliberately lengthen the natural period (lower omega_n) well beyond the dominant period of seismic ground motion, which reduces transmissibility. Laminated rubber bearings or sliding bearings stretch the period to 3–5 seconds. Oil or lead dampers are added because insufficient damping lets the isolation layer drift too far. The design philosophy is essentially the same as mechanical vibration isolation.

Real-world Applications

HVAC and chiller foundations: Rooftop air handling units and chillers transmit compressor vibration to lower floors and harm occupant comfort. Against a typical 1500 rpm (25 Hz) running speed, designers use rubber or spring mounts to drop the natural frequency to 5–10 Hz, putting the system in the r = 2.5–5 isolation region. Trying f_n = 5 Hz with ω = 25 Hz × 2π in the simulator confirms isolation efficiency above 90 %.

Vibration isolation tables for precision instruments: Electron microscopes, semiconductor lithography tools and optical benches are degraded by micron-level floor vibration. Pneumatic isolation tables drop the natural frequency to 1–3 Hz and isolate floor vibration above 5 Hz by more than 90 %. Advanced units add active control that injects an out-of-phase counter-force to remain effective even at low frequencies.

Automotive suspensions: Against road irregularities (effectively 5–20 Hz), vehicle bodies are designed with natural frequencies of 1–1.5 Hz. That ratio is the essence of ride comfort, set by a delicate balance of spring rate, mass and damping. Sports cars raise the natural frequency for sharper road tracking; luxury cars lower it for comfort — a textbook trade-off.

Base-isolated and damped buildings: Base-isolated buildings stretch the natural period to 3–5 seconds with laminated rubber bearings, moving it away from the dominant seismic periods of 0.5–2 seconds and cutting transmissibility. Supplemental dampers (oil, viscoelastic) add damping to control large displacements at resonance. The whole building is approximated as an SDOF system, conceptually identical to machine isolation.

Common Misunderstandings and Cautions

The most common mistake is to assume that softer springs are always better for isolation. Lowering the natural frequency does push r up and improve isolation, but too soft a spring causes large static deflection, equipment tilt and violent start-up motion. In practice mounts are sized to keep static deflection between roughly 10 and 50 mm, balancing isolation performance against stability. Reducing k to 100 in the simulator yields an absurd f_n of 0.16 Hz, which is not realistic.

Another frequent error is treating the resonance point r = 1 as something you can simply "avoid". Real machines always pass through resonance during start-up and shut-down. A 60 Hz motor sweeps continuously from 0 Hz to 60 Hz during the first few seconds, so r = 1 is crossed every time. With insufficient damping the peak amplitude is huge and damages the equipment. Setting zeta = 0 and r = 1 in the simulator shows TR diverging. Designs must consider transient response, not only steady operation.

Finally, note that this simulator covers displacement transmissibility, which coincidentally matches force transmissibility in an SDOF system. The ratio of equipment displacement to base displacement, and the ratio of force transmitted to the base from machine excitation, are both TR = sqrt((1+(2ζr)²)/((1-r²)²+(2ζr)²)). However acceleration transmissibility and 2-DoF systems (e.g. dynamic absorbers) follow different formulas. Complex assemblies need finite element analysis to capture multiple modes accurately.

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How to Use

  1. Set mass M (kg) using the slider; typical values range 50–500 kg for machinery mounts.
  2. Adjust stiffness K (N/m) with the input field; stiffer isolators (higher K) shift natural frequency upward.
  3. Input damping ratio ζ (zeta) via the input field between 0.01–0.3; rubber mounts typically exhibit ζ ≈ 0.05–0.15.
  4. Vary excitation frequency ω (rad/s) using the input field to sweep across resonance and observe transmissibility TR response.
  5. Read natural frequency f_n, frequency ratio r, transmissibility TR, and isolation efficiency IE in real time.

Worked Example

Compressor isolation: M=200 kg, K=8000 N/m yields f_n≈3.2 Hz. At excitation ω=20 rad/s (r≈2.24), transmissibility TR≈0.22, giving isolation efficiency IE≈78%. Increasing damping to ζ=0.10 reduces resonant peak amplification from 5× to 2.5× near f_n, though transmissibility at high frequencies increases slightly. For r>√2≈1.41, isolation begins; r=3.0 achieves TR<0.12 with minimal damping.

Practical Notes

  1. Select K such that natural frequency stays 2–4 Hz below machinery operating frequency to ensure r>1.5 and TR<0.3 in steady state.
  2. Damping reduces resonance overshoot but increases transmissibility at high frequencies; balance via ζ≈0.05–0.08 for optimal broadband performance on reciprocating equipment.
  3. Isolator stiffness degrades 10–20% over 5 years in elastomer mounts; design with K_initial 15% higher to maintain isolation margin.
  4. Critical frequency r=√2 marks the isolation threshold; below this, vibration amplification occurs and isolation fails.