Vibration Isolation and Transmissibility

Why it matters

Vibration isolation and transmissibility are crucial in mechanical systems to prevent damage and ensure longevity. They help in reducing noise and improving the comfort and safety of machinery and structures.

Key ideas

• Vibration Isolation: The process of reducing the transmission of vibrations from a vibrating source to its surroundings. It is essential in protecting sensitive equipment and improving the performance of mechanical systems.
• Transmissibility: The ratio of the amplitude of the forced vibration to the amplitude of the base excitation. It indicates how much of the vibration is transmitted through the system.
• Natural Frequency: The frequency at which a system tends to oscillate in the absence of any driving or damping force. It is a critical factor in designing vibration isolation systems.
• Damping: The process of reducing the amplitude of vibrations. Damping materials and techniques are used to absorb energy and reduce transmissibility.
• Isolation Efficiency: A measure of how effectively a system isolates vibrations. It depends on the frequency ratio and damping ratio.

Specify the transmitted quantity

For a linear mass–spring–damper with natural frequency f_n = √(k/m)/(2π), let r = f/f_n. Under harmonic base displacement y(t), the steady absolute displacement transmissibility is:

T = X/Y = √[1 + (2ζr)²] / √[(1 − r²)² + (2ζr)²].

The same expression gives transmitted-force amplitude divided by applied harmonic force amplitude for force excitation on a fixed base. The expression with numerator 1 instead is the displacement magnification X/(F₀/k) for force excitation; it is a different quantity.

For this model, isolation T < 1 requires r > √2. Damping reduces the resonance peak but increases T in the isolation region for a fixed frequency ratio. More damping is therefore not always better for high-frequency isolation.

Worked example

A 50 kg mass on a spring of stiffness 2000 N/m has damping ratio 0.1. Its base moves sinusoidally at 5 Hz. Find X/Y in steady state.

f_n = √(2000/50)/(2π) = 1.006584 Hz.

r = 5/1.006584 = 4.967294.

T = √[1 + (0.2 × 4.967294)²] / √[(1 − 4.967294²)² + (0.2 × 4.967294)²] = 0.05949.

Thus the mass's absolute displacement amplitude is about 5.95% of the base amplitude. This calculation assumes linear stiffness, viscous damping, and steady harmonic excitation.

Common mistakes

• Confusing natural frequency with the frequency of excitation.
• Ignoring the damping ratio in calculations, leading to incorrect transmissibility values.
• Misapplying the formula for transmissibility by not squaring terms correctly.

For GATE ME

Questions often involve calculating natural frequency, transmissibility, and analyzing the effect of damping. Practice problems on determining the isolation efficiency and designing systems for specific vibration isolation requirements.

Quick check

1. What is the purpose of vibration isolation?
2. Define transmissibility in the context of vibrations.
3. How does damping affect transmissibility?

Answers: 1. To reduce the transmission of vibrations. 2. The ratio of the amplitude of forced vibration to base excitation. 3. Damping reduces the resonance peak but can increase transmissibility in the isolation region.