Short Answer:

Transmissibility is the ratio of the amplitude of vibration transmitted to the supporting structure (or foundation) to the amplitude of the applied exciting force. It indicates how much of the vibration energy is passed through an isolation system.

In simple terms, transmissibility measures the effectiveness of vibration isolation. A high transmissibility means more vibration is transmitted, while a low transmissibility means the isolation system is effective. It depends on the frequency ratio and damping in the system, and is used to design machines, foundations, and isolation supports.

Detailed Explanation :

Transmissibility

Transmissibility is a very important concept in vibration control and isolation. It expresses how much of the vibration or force from a vibrating system is transmitted through the isolating medium to its base or surrounding structure. In mechanical systems, vibrations generated by machines, motors, or unbalanced components can cause noise, wear, and fatigue in connected parts. Therefore, engineers use transmissibility to evaluate the performance of vibration isolators such as springs, rubber mounts, and dampers.

Transmissibility is a dimensionless quantity and helps in determining whether a system is effectively isolating vibrations or transmitting them. The smaller the transmissibility value, the better the isolation performance.

1. Concept of Transmissibility

When a machine or system vibrates, the energy generated by the source can be either transmitted to its foundation or absorbed by the isolation system. The transmissibility (T) is defined as the ratio of the transmitted vibration amplitude () to the input or excitation amplitude ().

This ratio tells us how much of the vibration amplitude passes through the isolator.

• If : The vibration is amplified (poor isolation).
• If : The transmitted and input amplitudes are equal (no isolation).
• If : The vibration is reduced (effective isolation).

Thus, transmissibility is the most important parameter in designing isolation systems for machines and structures.

2. Mathematical Expression of Transmissibility

The transmissibility ratio can be derived using the mass-spring-damper system, which represents a typical vibration isolation setup.

For a system having:

• Mass =
• Damping coefficient =
• Stiffness =
• Forcing frequency =
• Natural frequency =

The transmissibility ratio is given by:

Where:

•  = damping ratio =
•  = frequency ratio =

This formula helps to determine how much vibration is transmitted for a given damping and excitation frequency.

3. Interpretation of Transmissibility

The value of transmissibility depends mainly on the frequency ratio (r) and the damping ratio (ζ).

1. When :
   • The transmissibility (T) is greater than 1.
   • The system amplifies vibration instead of isolating it.
   • In this region, the system is said to be in the amplification zone.
2. When :
   • Transmissibility equals 1.
   • There is no amplification or isolation — the transmitted vibration equals the input vibration.
3. When :
   • The transmissibility becomes less than 1.
   • The system effectively isolates vibrations, and less energy is transmitted to the base.
   • This is called the isolation zone.

Hence, for effective vibration isolation, the forcing frequency should be much higher than the natural frequency of the isolator.

4. Effect of Damping on Transmissibility

Damping plays a vital role in controlling transmissibility. Its effects are as follows:

1. At Low Frequencies (r < 1):
   • Increasing damping reduces vibration amplitude and improves stability.
2. At Resonance (r = 1):
   • Damping has the greatest effect. It reduces the sharpness of resonance peaks and prevents large vibration amplitudes.
3. At High Frequencies (r > √2):
   • Too much damping reduces isolation efficiency because some vibration energy is transmitted through the damping medium.
   • Hence, only optimum damping is desirable — not too low and not too high.

5. Practical Significance of Transmissibility

1. Design of Machine Foundations:
   Transmissibility helps engineers select proper materials and support systems (springs, pads, etc.) so that the vibrations from heavy machinery are not transmitted to the ground or nearby equipment.
2. Automobile Suspension Systems:
   The transmissibility concept is applied to design suspensions that isolate the vehicle body from road vibrations for comfort and safety.
3. Building and Bridge Design:
   Isolation pads and supports are designed using transmissibility analysis to reduce the transmission of vibrations from machinery or traffic loads.
4. Aerospace and Precision Equipment:
   Sensitive instruments like gyroscopes or sensors are mounted on isolation systems designed to achieve low transmissibility for accurate readings.
5. Noise Control:
   By reducing vibration transmission, transmissibility also helps minimize noise caused by structural vibrations in machines and vehicles.

6. Conditions for Effective Vibration Isolation

To achieve effective vibration isolation (T < 1):

• The excitation frequency () must be greater than √2 times the natural frequency ().
• The damping ratio () should be small enough to allow flexibility but large enough to prevent resonance.
• The isolating system should be properly tuned to the operating speed of the machine.

This balance ensures that most of the vibration energy is absorbed and very little is transmitted.

7. Graphical Representation

If we plot transmissibility (T) against the frequency ratio (r):

• The curve rises sharply at  (resonance condition).
• Beyond , T drops below 1, indicating good isolation.
• The slope and shape of the curve depend on the damping value.

This graphical relationship helps visualize how isolation improves as frequency ratio increases.

Conclusion:

Transmissibility is the ratio of transmitted vibration amplitude to input amplitude and is a key parameter for evaluating vibration isolation performance. It helps determine whether a system amplifies or isolates vibrations. Effective vibration isolation occurs when the excitation frequency is much greater than the natural frequency of the system (r > √2). By controlling stiffness, mass, and damping, engineers can design systems with low transmissibility, ensuring smooth operation, reduced noise, and protection of machinery and structures from vibration damage.