Short Answer:

Transmissibility is the ratio of the force transmitted to the foundation or supporting structure to the applied force that causes vibration in a system. It shows how much of the vibration is passed through a system to its base. A low transmissibility means the system effectively isolates vibration, while a high value means the vibration is transmitted more strongly.

Transmissibility is an important parameter in vibration isolation and control. It helps engineers design machines, vehicles, and structures that reduce vibration and noise transfer to their surroundings, improving performance and comfort.

Detailed Explanation :

Transmissibility

Transmissibility is a measure used to understand how vibrations are transferred from one part of a system to another, especially from a machine or vibrating source to its foundation or supporting structure. It is defined as the ratio of the transmitted force (or motion) to the applied exciting force (or motion). Mathematically, it can be expressed as:

where,
T = transmissibility ratio,
F_T = transmitted force or amplitude,
F_E = exciting or input force or amplitude.

This ratio indicates how effectively a vibration isolation system can reduce or transmit vibrations. If the value of transmissibility is less than one, the system isolates the vibration effectively; if it is greater than one, the vibration is amplified and transmitted more strongly to the base.

Transmissibility is particularly important in mechanical and structural engineering for analyzing vibration isolation systems such as machine mounts, vehicle suspensions, and building supports.

Principle of Transmissibility

When a machine or component vibrates, it creates periodic forces that act on its support or foundation. These forces can cause undesirable vibrations in nearby equipment or structures. To reduce this effect, vibration isolation systems are used. The effectiveness of such systems depends on the transmissibility ratio.

Transmissibility depends mainly on three factors:

Frequency Ratio (r): Ratio of the excitation frequency (ω) to the natural frequency (ωₙ) of the system, i.e., .
Damping Ratio (ζ): Resistance to vibration within the system.
Type of Excitation: Whether the input vibration is due to a steady rotating force, impact, or unbalanced motion.

At low frequencies (below the natural frequency), the system tends to move with the excitation, and transmissibility is high. When the excitation frequency equals the natural frequency (r = 1), resonance occurs, and transmissibility becomes maximum. Beyond this point, as frequency increases further, transmissibility decreases rapidly and becomes less than one, which means effective vibration isolation.

Mathematical Expression

The mathematical form of transmissibility for a damped vibration system is:

where,

T = transmissibility,
ζ = damping ratio,
r = frequency ratio (ω/ωₙ).

This equation shows that transmissibility depends on both the damping ratio and the frequency ratio. By adjusting these values, engineers can design systems that minimize vibration transmission.

Variation of Transmissibility

When r < √2:
The transmitted force is greater than the applied force (T > 1). In this region, vibration is amplified rather than isolated.
When r = √2:
Transmissibility equals one (T = 1). This is the boundary point between vibration amplification and isolation.
When r > √2:
Transmissibility becomes less than one (T < 1). In this region, vibration isolation occurs effectively, meaning less vibration is transmitted to the support or foundation.

Hence, for effective vibration isolation, the excitation frequency must be greater than √2 times the natural frequency of the system.

Importance of Transmissibility in Engineering

Transmissibility is a critical factor in designing systems that must operate smoothly without excessive vibration transmission. Its importance includes:

Vibration Control: Helps in designing isolators and dampers to reduce transmission of vibrations to surrounding parts.
Machine Protection: Reduces wear and tear caused by continuous vibration.
Noise Reduction: Lower transmissibility also reduces noise levels in mechanical systems.
Human Comfort: Used in vehicle design to reduce vibrations felt by passengers.
Structural Safety: Helps prevent vibration-induced damage to buildings, bridges, and machinery foundations.

By analyzing transmissibility, engineers can select suitable materials, damping devices, and mountings that achieve desired vibration isolation performance.

Practical Examples

Machine Foundation: Heavy machinery mounted on rubber or spring isolators uses transmissibility analysis to reduce vibrations transmitted to the ground.
Vehicle Suspension: Car suspensions are designed so that road vibrations are not transmitted to the vehicle body.
Buildings: Tall structures use damping and isolation systems to reduce vibration transmissibility during wind or seismic activity.
Electronic Equipment: Sensitive instruments are placed on vibration isolation tables to ensure accurate readings by minimizing vibration transmission.

These examples show that transmissibility has practical applications in almost every field of mechanical and civil engineering.

Conclusion:

Transmissibility is the ratio of transmitted force or motion to the applied excitation force in a vibrating system. It helps determine how much vibration passes from a source to its foundation or surroundings. A lower transmissibility means better vibration isolation and greater system stability. By controlling factors like natural frequency, damping ratio, and stiffness, engineers can design systems that reduce vibration transmission effectively, ensuring comfort, safety, and long service life of machinery and structures.