Viscous Damped System: Introduction to Force Transmissibility

When analyzing mechanical vibrations, especially in a single degree of freedom viscous damped system, understanding force transmissibility is crucial. Force transmissibility refers to the ratio of the force transmitted to the foundation or support to the exciting force applied to the system. It helps us determine how much of the external vibration force is actually transmitted through the system. This concept is fundamental in designing systems for effective vibration isolation.

Frequency Ratio and Force Transmissibility Formula

The behavior of force transmissibility depends significantly on the frequency ratio and the damping ratio of the system. The frequency ratio (\(r\)) is defined as the ratio of the excitation frequency (\(\omega\)) to the natural frequency of the system (\(\omega_n\)). The damping ratio (\(\zeta\)) indicates the level of damping present in the system.

For a single degree of freedom viscous damped system, the force transmissibility (\(T_F\)) is given by the formula:

Where:

• \(T_F\) is the force transmissibility
• \(\zeta\) is the damping ratio
• \(r = \frac{\omega}{\omega_n}\) is the frequency ratio
• \(\omega\) is the excitation frequency
• \(\omega_n\) is the natural frequency of the system

Transmitted Force Behavior at the Critical Frequency Ratio

Let's examine the force transmissibility at a specific frequency ratio, \(r = \sqrt{2}\). This point is particularly important because its behavior is independent of the damping ratio (\(\zeta\)).

Substituting \(r = \sqrt{2}\) into the transmissibility formula:

This calculation shows that when the frequency ratio \(r = \sqrt{2}\), the force transmissibility \(T_F\) is equal to 1. This means the transmitted force is exactly equal to the exciting force, regardless of the damping present in the system. This point is often referred to as the "isolation point" or "crossover point" because it marks the transition where isolation starts to become effective.

Exciting Force Relationship when Frequency Ratio is Greater than √2

Now, let's consider the scenario where the frequency ratio is greater than \(\sqrt{2}\), i.e., \(r > \sqrt{2}\). This is a critical range for achieving effective vibration isolation in a viscous damped system.

When \(r > \sqrt{2}\), the term \((1 - r^2)^2\) in the denominator of the transmissibility formula becomes significantly larger as \(r\) increases. As \(r\) increases beyond \(\sqrt{2}\), the value of \((1 - r^2)\) becomes increasingly negative, but because it's squared, the denominator becomes very large.

Consider the general behavior:

• At \(r = 0\), \(T_F = 1\).
• For \(0 < r < \sqrt{2}\), particularly around resonance (\(r=1\)), \(T_F\) can be greater than 1, meaning the transmitted force can be amplified.
• At \(r = \sqrt{2}\), \(T_F = 1\).
• For \(r > \sqrt{2}\), the denominator term \((1 - r^2)^2\) grows rapidly, leading to the overall value of \(T_F\) becoming less than 1. The damping term \((2\zeta r)^2\) also contributes, but the \((1-r^2)^2\) term dominates for large \(r\).

Therefore, if the frequency ratio \(r\) is greater than \(\sqrt{2}\), the force transmissibility \(T_F\) will be less than 1 (\(T_F < 1\)).

This implies that:

Which leads to:

In practical terms, this condition means that the system effectively isolates the vibrations, and the force transmitted to the foundation or supporting structure is less than the force applied by the excitation. This is the desired outcome for vibration isolation applications.

Based on the analysis, when the frequency ratio is greater than \(\sqrt{2}\), the transmitted force is less than the exciting force.