Vibration transmissibility is crucial in understanding how forces and motions are transferred through mechanical systems. It's the ratio of output to input amplitudes, influenced by frequency and damping. This concept helps engineers design effective isolation systems for various applications.

In forced vibrations, transmissibility determines whether vibrations are amplified or reduced. It peaks at resonance and decreases at higher frequencies. Damping plays a key role, affecting the maximum transmissibility and isolation effectiveness across different frequency ranges.

Vibration Transmissibility

Definition and Calculation

• Vibration transmissibility represents the ratio of output force or motion amplitude to input force or motion amplitude in a vibrating system
• Function of frequency ratio (excitation frequency to natural frequency) and damping ratio in single degree-of-freedom (SDOF) systems
• Derived from equation of motion and steady-state response to harmonic excitation for SDOF systems
• Transmissibility curve typically shows peak at resonance frequency and decreases at higher frequency ratios
• Amplification of input vibration occurs for frequency ratios below √2 (transmissibility > 1)
• Attenuation of input vibration occurs for frequency ratios above √2 (transmissibility < 1)
• Crossover frequency where transmissibility equals 1 occurs at frequency ratio of √2 for all damping ratios in SDOF systems
• Transmissibility equation for SDOF system: $T = \frac{\sqrt{1 + (2\zeta r)^2}}{(1 - r^2)^2 + (2\zeta r)^2}$
  • Where $r$ is the frequency ratio and $\zeta$ is the damping ratio

Characteristics and Applications

• Peak transmissibility value significantly influenced by damping ratio
  • Higher damping reduces maximum transmissibility at resonance
• Transmissibility curve flattens with increased damping
  • Reduces effectiveness of vibration isolation at high frequency ratios
• Excitation frequency relative to system's natural frequency determines vibration amplification or attenuation
• Approximation for lightly damped systems at resonance: $T_{max} \approx \frac{1}{2\zeta}$
• Trade-off in isolation system design
  • Increasing damping reduces transmissibility near resonance
  • Increases transmissibility at higher frequency ratios
• Maximum transmissibility frequency shifts slightly lower than natural frequency as damping increases
• Damping affects rate of transmissibility reduction at high frequency ratios
  • Higher damping leads to slower decrease in transmissibility
• Applications include automotive suspension systems, machinery mounts, and seismic isolation for buildings

Damping and Frequency Effects

Damping Influence

• Damping ratio significantly impacts peak transmissibility value
  • Higher damping reduces maximum transmissibility at resonance
• Transmissibility curve flattens with increased damping
  • Diminishes vibration isolation effectiveness at high frequency ratios
• Damping creates trade-off in isolation system design
  • Reduces transmissibility near resonance
  • Increases transmissibility at higher frequency ratios
• Maximum transmissibility frequency shifts slightly below natural frequency as damping increases
• Rate of transmissibility reduction at high frequency ratios affected by damping
  • Higher damping results in slower transmissibility decrease
• Critical damping ratio (ζ = 1) eliminates resonance peak entirely
• Overdamped systems (ζ > 1) exhibit monotonically decreasing transmissibility with frequency

Frequency Effects

• Excitation frequency relative to system's natural frequency determines vibration amplification or attenuation
• Frequency ratio (r) defined as ratio of excitation frequency to natural frequency
• Resonance occurs when frequency ratio approaches 1 (r ≈ 1)
  • Results in maximum transmissibility for underdamped systems
• Amplification region exists for frequency ratios below √2
  • Transmissibility exceeds 1, indicating vibration amplification
• Isolation region occurs for frequency ratios above √2
  • Transmissibility less than 1, indicating vibration attenuation
• Crossover frequency (where T = 1) always occurs at r = √2 for SDOF systems
• High-frequency asymptotic behavior: transmissibility decreases at a rate of 40 dB/decade (or 12 dB/octave) for undamped systems
• Frequency response function (FRF) relates to transmissibility: $T = |H(\omega)|$
  • Where $H(\omega)$ is the complex frequency response function

Vibration Isolation System Design

Design Principles

• Introduce compliance and damping elements to reduce vibration transmission from source to receiver
• Achieve frequency ratio significantly greater than √2 for effective vibration attenuation
• Select appropriate stiffness and damping characteristics to optimize isolation performance
• Use transmissibility concept to evaluate and compare effectiveness of different isolation system designs
• Consider practical constraints (static deflection, space limitations, environmental factors)
• Implement active vibration isolation systems for superior performance in certain applications
  • Utilize sensors and actuators to adapt to changing conditions
• Employ multi-stage isolation systems for greater vibration reduction, especially at high frequencies
• Design for both force and motion isolation depending on the application requirements

Design Considerations

• Static deflection: $\delta_{st} = \frac{g}{\omega_n^2}$
  • Where $g$ is gravitational acceleration and $\omega_n$ is natural frequency
• Natural frequency selection based on lowest excitation frequency to be isolated
• Damping ratio optimization to balance resonance control and high-frequency isolation
• Material selection for isolators (rubber, springs, air mounts) based on required stiffness and damping properties
• Temperature effects on isolator properties, especially for elastomeric materials
• Nonlinear behavior of isolators at large amplitudes or high frequencies
• Fatigue life and durability of isolation components
• Cost-effectiveness and ease of maintenance in industrial applications

Force vs Vibration Transmissibility

Force Transmissibility

• Ratio of force transmitted to support structure to excitation force applied to system
• Equation identical to motion transmissibility for SDOF systems: $T_f = \frac{F_{transmitted}}{F_{input}} = \frac{\sqrt{1 + (2\zeta r)^2}}{(1 - r^2)^2 + (2\zeta r)^2}$
• Crucial for analyzing dynamic loads transmitted to supporting structures or foundations
• Used in force excitation scenarios to calculate forces transmitted through isolation system to support
• Applications include machine foundation design and structural load analysis

Relationship to Motion Transmissibility

• Physical interpretation differs between force and motion transmissibility
• Relationship depends on whether system excited by force input or motion input
• For base excitation problems, motion transmissibility determines relative motion between mass and base
• In force excitation scenarios, force transmissibility calculates transmitted forces through isolation system
• Motion transmissibility: $T_d = \frac{X_{output}}{X_{input}}$
• Both force and motion transmissibility essential for comprehensive vibration analysis and control
• Example: Automotive engine mount design considers both force transmission to chassis and engine motion

Applications and Analysis

• Vibration isolation of sensitive equipment (optical tables, precision machinery)
• Seismic isolation of buildings and structures
• Vehicle suspension design for ride comfort and handling
• Analysis of vibration transmission in multi-body systems
• Prediction of structural responses to dynamic loads
• Optimization of isolation systems for both force and motion control
• Evaluation of vibration control strategies in complex mechanical systems