Short Answer:

The transmissibility ratio is the ratio of the force transmitted to the foundation or support to the applied external force acting on a vibrating system. It indicates how much of the vibration force is passed through the system to its base or surroundings.

In simple words, the transmissibility ratio shows the effectiveness of a vibration isolation system. A lower transmissibility ratio means less vibration is transmitted, which is desirable. It helps engineers design systems that minimize vibration transmission to reduce noise, wear, and structural damage.

Detailed Explanation :

Transmissibility Ratio

The transmissibility ratio is an important parameter in the study of vibration control and isolation. It measures how much of the vibratory force or motion from a machine or vibrating system is transmitted to its foundation or support.

When a machine or system vibrates due to an unbalanced or periodic external force, some part of this vibration energy is absorbed by the system itself, while the remaining part is transmitted to the base or surroundings. The transmissibility ratio (TR) helps in understanding how effectively the system isolates these vibrations.

It is particularly important in designing vibration isolators, machine foundations, and mounting systems, where the goal is to reduce the amount of vibration transmitted from the machine to its surroundings.

Definition

The transmissibility ratio (TR) is defined as:

It is also known as the force transmissibility or vibration transmissibility.

In mathematical form, for a damped system under harmonic excitation, it is expressed as:

Where:

•  = damping ratio,
•  = frequency ratio (ratio of excitation frequency to natural frequency),
•  = frequency of external excitation,
•  = natural frequency of the system.

Physical Meaning of Transmissibility Ratio

The transmissibility ratio tells how efficiently the system isolates vibrations.

• If TR > 1: The transmitted force is greater than the applied force. This means amplification of vibration, which usually happens when the system operates below its natural frequency.
• If TR = 1: The transmitted and applied forces are equal. No isolation occurs.
• If TR < 1: The transmitted force is less than the applied force. This is the desired condition, indicating effective vibration isolation.

Therefore, to achieve proper vibration isolation, the frequency of excitation () must be greater than √2 times the natural frequency ().

Derivation of Transmissibility Ratio

Consider a single-degree-of-freedom (SDOF) system with a mass , spring stiffness , and damping coefficient , subjected to a harmonic external force .

The equation of motion of the system is:

The transmitted force to the foundation is the vector sum of the spring force and the damping force:

Where  is the amplitude of the vibration.

The amplitude of steady-state vibration  is given by:

Substituting this into the above equation gives:

This is the transmissibility ratio equation used in vibration isolation analysis.

Effect of Frequency Ratio (r) on Transmissibility Ratio

1. When r < 1 (Low Frequency Region):
   The transmissibility ratio (TR) is greater than 1, meaning the transmitted force is amplified. In this case, vibration isolation is not effective because the system moves almost in phase with the external force.
2. When r = 1 (At Resonance):
   The transmissibility ratio reaches its maximum value. The system vibrates with maximum amplitude, and the transmitted force is very high. This is an undesirable condition that can lead to system failure.
3. When r > √2 (High Frequency Region):
   The transmissibility ratio becomes less than 1. This means the transmitted force is smaller than the applied force, and vibration isolation becomes effective.

Hence, for proper vibration isolation, the excitation frequency should be much greater than the natural frequency of the system.

Effect of Damping on Transmissibility Ratio

Damping plays a crucial role in controlling the transmissibility ratio:

• At Low Frequency (r < 1):
  Increasing damping reduces the amplitude of vibration and slightly reduces transmissibility.
• At Resonance (r = 1):
  Damping significantly reduces the peak transmissibility and prevents excessive vibration amplification.
• At High Frequency (r > √2):
  Excessive damping can slightly increase the transmissibility ratio. Therefore, an optimum level of damping is desired for the best performance.

Thus, damping should be chosen carefully — too little damping causes large vibrations at resonance, while too much damping reduces isolation efficiency at high frequencies.

Importance of Transmissibility Ratio

The transmissibility ratio is a key parameter in vibration analysis and isolation system design. Its importance includes:

1. Vibration Isolation:
   It helps determine how much vibration from a machine is transmitted to its foundation or surroundings.
2. Machine Safety:
   Prevents damage due to excessive vibration transmission, especially near resonance.
3. Comfort and Noise Reduction:
   Reduces noise and provides comfort in vehicles, buildings, and machinery by isolating vibration.
4. Design of Mounts and Dampers:
   Helps in selecting proper spring stiffness and damping levels for isolators.
5. Performance Improvement:
   Ensures smoother and quieter operation of rotating machines and dynamic systems.

Practical Applications of Transmissibility Ratio

1. Automobile Suspensions:
   Reduces vibration transmitted from the road to the vehicle body.
2. Machine Foundations:
   Used to design vibration isolators that protect buildings from dynamic loads.
3. Aerospace Systems:
   Controls vibration transmission in aircraft structures and equipment.
4. Electronic Devices:
   Isolates sensitive instruments from external vibrations.
5. Industrial Machines:
   Ensures stability and accuracy of machines like compressors, pumps, and turbines.

Conclusion

In conclusion, the transmissibility ratio is the ratio of the transmitted force to the applied external force in a vibrating system. It helps in determining how much vibration energy passes through the system to its foundation. The ratio depends on the frequency ratio and damping of the system. Effective vibration isolation occurs when the transmissibility ratio is less than one, which happens at frequencies higher than √2 times the natural frequency. Understanding and controlling the transmissibility ratio is essential for designing efficient and safe vibration isolation systems in mechanical and structural applications.