Understanding Transmissibility at Resonance

Transmissibility (TR) is a key concept in vibration analysis. It measures how effectively vibrations are transmitted through a system. Specifically, it's the ratio of the amplitude of the output (force transmitted or motion) to the amplitude of the input (exciting force or motion).

Transmissibility Formula

For a viscously damped single-degree-of-freedom system, the transmissibility (TR) is given by the formula:

$$TR = \frac{\sqrt{1 + (2\zeta)^2}}{\sqrt{(1 - (\omega/\omega_n)^2)^2 + (2\zeta(\omega/\omega_n))^2}}$$

Where:

• $TR$ is the transmissibility.
• $\zeta$ is the damping ratio (a measure of how quickly vibrations decay).
• $\omega$ is the frequency of the external excitation (in rad/s).
• $\omega_n$ is the natural frequency of the system (in rad/s).

Analyzing the Condition: $\omega/\omega_n = 1$

The condition $\omega/\omega_n = 1$ signifies that the system is excited at its natural frequency. This condition is known as resonance.

Let's substitute $\omega/\omega_n = 1$ into the transmissibility formula:

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

Simplifying the expression:

$$TR = \frac{\sqrt{1 + 4\zeta^2}}{\sqrt{(1 - 1)^2 + (2\zeta)^2}}$$

$$TR = \frac{\sqrt{1 + 4\zeta^2}}{\sqrt{0^2 + (2\zeta)^2}}$$

$$TR = \frac{\sqrt{1 + 4\zeta^2}}{\sqrt{4\zeta^2}}$$

$$TR = \frac{\sqrt{1 + 4\zeta^2}}{2\zeta}$$

Transmissibility Behavior at Resonance

Now, consider the behavior of the transmissibility as the damping ratio $\zeta$ changes, particularly when it is very small (approaching zero), which is common in many physical systems.

• Numerator: As $\zeta \to 0$, the term $4\zeta^2$ approaches 0. So, the numerator $\sqrt{1 + 4\zeta^2}$ approaches $\sqrt{1 + 0} = \sqrt{1} = 1$.
• Denominator: As $\zeta \to 0$, the denominator $2\zeta$ also approaches 0.

When the numerator approaches 1 and the denominator approaches 0, the fraction $\frac{1}{0}$ tends towards infinity.

Therefore, when $\omega/\omega_n = 1$ (resonance), the transmissibility:

$$TR \to \infty$$

This means that at resonance, with very little damping, the amplitude of the transmitted force or motion can become extremely large, potentially leading to system failure.

Conclusion

When the excitation frequency ($\omega$) equals the natural frequency ($\omega_n$), or $\omega/\omega_n = 1$, the transmissibility approaches infinity (or proportionality, denoted by $\prop$), especially in systems with low damping.