4.4 Vibration transmissibility

Vibration Transmissibility

Vibration transmissibility quantifies how much of an input vibration (force or motion) gets passed through a mechanical system to the output side. It's the key metric for deciding whether a system amplifies or attenuates vibrations at a given frequency, and it drives nearly every decision in vibration isolation design.

Definition and Calculation

Transmissibility ($T$) is the ratio of the output amplitude to the input amplitude for a harmonically excited single-DOF system. It depends on two dimensionless parameters:

• Frequency ratio $r = \omega / \omega_n$, where $\omega$ is the excitation frequency and $\omega_n$ is the natural frequency
• Damping ratio $\zeta$

The transmissibility equation for an SDOF system is:

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

Note the square root in the denominator covers the entire expression. This formula comes from taking the magnitude of the steady-state response to harmonic excitation divided by the input magnitude.

Three frequency regions define the behavior of this curve:

• Amplification region ($r < \sqrt{2}$): $T > 1$, so the system makes vibrations worse
• Crossover point ($r = \sqrt{2}$): $T = 1$ regardless of damping. Every SDOF transmissibility curve passes through this point.
• Isolation region ($r > \sqrt{2}$): $T < 1$, so the system attenuates vibrations

The peak occurs near $r = 1$ (resonance), where transmissibility can become very large for lightly damped systems. A useful approximation at resonance for small $\zeta$:

$T_{max} \approx \frac{1}{2\zeta}$

So a system with $\zeta = 0.05$ would have a peak transmissibility of roughly 10, meaning the output amplitude is 10 times the input at resonance.

Damping and Frequency Effects

How Damping Shapes the Transmissibility Curve

Damping has a dual role, and this is one of the trickiest points in isolation design:

• Near resonance ($r \approx 1$): Higher damping reduces the peak transmissibility. This is good because it prevents dangerously large amplitudes during startup or shutdown when the system passes through resonance.
• In the isolation region ($r > \sqrt{2}$): Higher damping actually increases transmissibility. The curve doesn't drop off as steeply, so you lose isolation performance at high frequencies.

This trade-off is central to choosing a damping ratio. You want enough damping to survive resonance, but not so much that you compromise high-frequency isolation.

A few additional damping effects worth knowing:

• The frequency at which peak transmissibility occurs shifts slightly below $\omega_n$ as damping increases
• At critical damping ($\zeta = 1$), the resonance peak disappears entirely
• Overdamped systems ($\zeta > 1$) show a monotonically decreasing transmissibility curve with no peak at all

Frequency Ratio Effects

The frequency ratio $r$ is the single most important variable. At high frequency ratios, an undamped SDOF system's transmissibility rolls off at 40 dB/decade (equivalently, 12 dB/octave). This means doubling the excitation frequency reduces the transmitted amplitude by a factor of 4. Adding damping slows this rolloff rate, which is another way to see the high-frequency trade-off described above.

The frequency response function $H(\omega)$ of the system is directly related to transmissibility: $T = |H(\omega)|$, where $H(\omega)$ is the complex-valued transfer function between output and input.

Force vs. Motion Transmissibility

Force Transmissibility

Force transmissibility is the ratio of the force transmitted to the support structure to the applied excitation force:

$T_f = \frac{F_{transmitted}}{F_{input}} = \frac{\sqrt{1 + (2\zeta r)^2}}{\sqrt{(1 - r^2)^2 + (2\zeta r)^2}}$

This is used when a machine (like a motor or compressor) applies a harmonic force to a mass mounted on isolators, and you need to know how much force reaches the foundation.

Motion Transmissibility

Motion transmissibility applies to base excitation problems, where the support itself is vibrating and you want to know how much motion reaches the isolated mass:

$T_d = \frac{X_{output}}{X_{input}}$

For a standard SDOF system, the mathematical expression for $T_d$ is identical to $T_f$. However, the physical setup is different. In force transmissibility, the force is applied to the mass and you measure what reaches the base. In motion transmissibility, the base moves and you measure how much the mass responds.

Both types matter in real design. An automotive engine mount, for example, must limit force transmission from the engine to the chassis and limit how much road-induced base motion reaches the engine.

Vibration Isolation System Design

Design Principles

The goal of isolation is to operate well into the isolation region ($r \gg \sqrt{2}$). Here's the general design process:

1. Identify the lowest excitation frequency that needs to be isolated. This sets the target for your natural frequency.
2. Choose a natural frequency well below that excitation frequency so that $r > \sqrt{2}$ (and ideally $r > 3$ or more for significant attenuation).
3. Select a damping ratio that balances resonance protection against high-frequency isolation. Typical values for passive isolators fall in the range $\zeta = 0.05$ to $0.2$.
4. Verify static deflection is acceptable. Lower natural frequencies require softer springs, which means larger static deflections. The relationship is:

$\delta_{st} = \frac{g}{\omega_n^2}$

For example, a natural frequency of 5 Hz gives a static deflection of about 10 mm. Dropping to 2 Hz pushes that to roughly 62 mm, which may not be practical.

1. Evaluate the transmissibility curve across the full operating frequency range, including any transient passage through resonance.

Practical Considerations

• Isolator materials: Rubber mounts provide moderate stiffness and inherent damping. Metal springs offer lower damping and are better for high-frequency isolation. Air mounts (pneumatic isolators) can achieve very low natural frequencies.
• Temperature sensitivity: Elastomeric isolators can stiffen significantly in cold environments, shifting the natural frequency upward.
• Nonlinear behavior: At large amplitudes, many isolators exhibit nonlinear stiffness or damping, which changes the effective transmissibility.
• Multi-stage isolation: Stacking two isolation stages in series can achieve steeper rolloff (up to 80 dB/decade for two undamped stages), which is useful for sensitive equipment like optical tables.
• Active isolation: Systems using sensors and actuators can adapt in real time, offering better performance than passive isolators when excitation conditions change. These are more complex and expensive, but essential in some precision applications.

Common Applications

• Machine foundations (reducing transmitted forces from rotating equipment)
• Vehicle suspensions (isolating passengers from road vibrations)
• Seismic isolation of buildings (protecting structures from ground motion)
• Precision equipment mounts (optical tables, semiconductor fabrication tools)
• Aerospace payload isolation (protecting sensitive instruments during launch)