When does vibration isolation work?

Isolation is achieved only when the frequency ratio r = ω/ωn exceeds √2 (≈1.41). Below this threshold, vibration is amplified rather than attenuated. The design goal is to choose a soft mount so that r > √2 at the operating frequency.

How does damping ratio affect isolation performance?

Higher damping reduces the resonance peak at r ≈ 1, making the system safer during start-up. However, at r >> √2, more damping slightly degrades high-frequency isolation. Precision instruments often use very low damping; machinery subject to shock loads needs moderate damping.

What is static deflection and why does it matter?

Static deflection δ_st = g/ωn² is the sag of the mount under the machine's weight. Softer mounts (lower ωn, better isolation) deflect more. At 5 Hz natural frequency, δ_st ≈ 10 mm; at 1 Hz, δ_st ≈ 250 mm. Space constraints may limit how soft the mount can be.

How do multiple mounts affect natural frequency?

N mounts in parallel multiply the total stiffness by N, raising the natural frequency by √N. To maintain the same isolation performance with more mounts, each individual mount must be correspondingly softer.

Advanced Vibration Isolation Mount Design Calculator

Presets

Machine Parameters

Machine mass m (kg)

Operating speed N (RPM)

rpm

Disturbing freq. f (Hz)

Mount Parameters

Mount stiffness k (kN/m)

kN/m

Damping ratio ζ

Number of mounts n

Results

Calculating...

While paused, move the sliders to update the result instantly.

Two-Stage Isolation — Live Animation

A harmonic force F₀ shakes the machine m₁; vibration passes through the primary mount to the intermediate mass m₂, then through the secondary mount to the base. Two stages give a much steeper high-frequency roll-off (ideal −24 dB/oct vs −12 dB/oct for a single stage), greatly reducing the force transmitted to the floor. The inset plot compares T₁ (single-stage) with T₂ (two-stage).

Live Results

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Natural freq. fn (Hz)

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Freq. ratio r

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Single-stage T₁

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Two-stage T₂

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Isolation eff. (2-stage, %)

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High-freq roll-off

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Static deflect. (mm)

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Transmitted force (N)

Transmissibility T vs Frequency Ratio r — Red Region: Resonance Danger Zone (r < √2)

Theory & Key Formulas

Transmissibility: $T = \sqrt{\dfrac{1+(2\zeta r)^2}{(1-r^2)^2+(2\zeta r)^2}}$

$r = \omega/\omega_n,\quad \omega_n = \sqrt{k_{eq}/m}$

Isolation eff.: $IE = (1-T)\times 100\%$

Static deflection: $\delta_{st} = g/\omega_n^2$

What is Vibration Isolation?

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What exactly is "vibration transmissibility"? I see it's the main output of this simulator.

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Basically, it's the ratio of the vibration force transmitted through the mounts to the foundation, compared to the force generated by the machine. A transmissibility (T) of 1 means all the shaking gets through. If T is 0.2, only 20% of the vibration is transmitted—that's good isolation! Try moving the "Mount stiffness k" slider down in the simulator. You'll see T drop when the mounts get softer.

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Wait, really? So softer mounts are always better? But the graph shows a huge peak at one point...

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Great observation! That peak is the resonance danger zone. If the machine's operating speed matches the natural frequency of the system ($r = \omega/\omega_n \approx 1$), vibration is amplified dramatically. For instance, a pump starting up must pass through this speed quickly. The design trick is to ensure your normal operating point is to the right of that peak, where $r > \sqrt{2}$. Adjust the "Operating speed N" and see the vertical line move across the graph.

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So what's the damping ratio for? If soft mounts are good for isolation, why would I want more damping?

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In practice, damping is your safety net. It controls that dangerous resonance peak. A common case is a washing machine during the spin cycle—it goes through resonance. High damping (try setting ζ to 0.5 in the simulator) squashes the peak, preventing violent shaking. But notice, at high operating speeds (far right of the graph), more damping actually worsens isolation slightly. It's a trade-off between safety during startup and performance at full speed.

Physical Model & Key Equations

The core model is a single-degree-of-freedom system: a mass (the machine) supported on springs and dampers (the mounts). The key parameter is the frequency ratio $r$, which compares the disturbance frequency to the system's natural frequency.

$$ r = \frac{\omega}{\omega_n}, \quad \omega_n = \sqrt{\frac{k_{eq}}{m}}$$

Here, $\omega = 2\pi f$ is the disturbing angular frequency (from the "Disturbing freq. f" input), $\omega_n$ is the natural angular frequency, $k_{eq}= n \times k$ is the total stiffness from all mounts, and $m$ is the machine mass.

Transmissibility $T$ predicts how much vibration force passes through the mounts to the foundation. It depends critically on $r$ and the damping ratio $\zeta$.

$$ T = \sqrt{\dfrac{1+(2\zeta r)^2}{(1-r^2)^2+(2\zeta r)^2}}$$

When $r \lt \sqrt{2}$, $T \gt 1$ (amplification occurs). When $r \gt \sqrt{2}$, $T \lt 1$ (isolation is achieved). The isolation efficiency is simply $IE = (1-T) \times 100\%$, which you see calculated live in the simulator.

Real-World Applications

HVAC Systems on Building Roofs: Large air handling units and chillers operate at constant speeds (e.g., 1800 RPM). Engineers use this exact calculation to design rubber or spring isolators that shift the system's natural frequency well below 25 Hz, ensuring $r \gt \sqrt{2}$ and preventing structure-borne noise throughout the building.

Precision Manufacturing Equipment: CNC machines and semiconductor lithography tools must be isolated from floor vibrations to maintain micron-level accuracy. Here, very soft air springs or active mounts are used to achieve an extremely low $\omega_n$, providing high isolation efficiency ($IE \gt 95\%$) against typical floor vibration frequencies around 10-30 Hz.

Vehicle Engine Mounts: Mounts must support the engine's weight (static deflection) while isolating high-frequency combustion vibrations and managing the large shake forces when the engine idles near resonance. The damping ratio is carefully chosen to control the lurch felt at stoplights without compromising high-speed isolation.

Data Center Server Racks: In earthquake-prone areas, server racks use isolation mounts with high damping. The design ensures isolation from typical building vibrations during normal operation but locks down or provides damping to limit resonant amplification during the low-frequency, high-amplitude shaking of a seismic event.

Common Misconceptions and Points to Note

First, the assumption that a mount with higher stiffness is safer is dangerous. While it's true that static deflection may be smaller, the natural frequency increases. This results in no isolation effect for low-frequency vibrations (e.g., from low-speed pumps or fans), essentially creating a "rigidly fixed" state. For example, using a high-stiffness mount on a machine running at 1200 rpm (20 Hz) could leave the operating frequency on the left side of the resonance danger zone on the graph, potentially causing the transmissibility to exceed 1.

Next, do not overestimate the meaning of "90% isolation efficiency." This value is based on theoretical transmissibility, and the actual effect is heavily influenced by factors like the stiffness of the installation surface and internal unbalanced masses within the machine. The simulator uses an idealized "single-degree-of-freedom" model. In reality, you often need to consider the "six-degree-of-freedom" behavior where the machine rocks in various directions. Even if the tool shows good numbers, on-site measurement is essential.

Finally, beware of overlooking static deflection. Even if calculations show no issue, pipes or electrical cables may fail to accommodate the settling of the mounts, leading to accidents like tearing. Also, when supporting a machine with multiple mounts, uneven load distribution due to stiffness variations or surface irregularities can prevent the intended isolation performance. In design, it's practical wisdom to include a safety margin of at least 20%.