Loading Pyodide...

Vibration Isolation Designer

Size and validate single-degree-of-freedom isolators using base or force excitation models.

Tool Purpose & README

Loading README...

Inputs

Expert Mode Show all parameters

Analyze an existing isolator and check transmissibility at the excitation frequency.

Core Inputs
?
?
?
?
?
?
?

Advanced Inputs

Damping

By default, plots show curves for a range of typical damping ratios (zeta). Enable specific zeta to calculate for a known value.

Response Amplitudes
?
Design Checks
?

Key Results

Enter inputs and press Calculate to see results.

Transmissibility Derivation
1. Transmissibility Equation
$$ T = \frac{\sqrt{1 + (2\zeta r)^2}}{\sqrt{(1-r^2)^2 + (2\zeta r)^2}} $$
2. Substituted Values
Loading...
Natural Frequency Derivation
1. Natural Frequency Equation
$$ f_n = \frac{1}{2\pi}\sqrt{\frac{k_{total}}{m}} $$
2. Substituted Values
Loading...
Frequency Ratio Derivation
1. Frequency Ratio Definition
$$ r = \frac{f_{exc}}{f_n} $$
2. Substituted Values
Loading...
Phase Angle Derivation
1. Phase Angle Equation
$$ \phi = \arctan\left(\frac{2\zeta r}{1 - r^2}\right) $$
2. Physical Interpretation

The phase angle describes the time lag between excitation and response:
• At low frequencies (r << 1): φ ≈ 0° (in-phase)
• At resonance (r = 1): φ = 90° (quadrature)
• At high frequencies (r >> 1): φ ≈ 180° (anti-phase)

3. Substituted Values
Loading...

Transmissibility vs Frequency Ratio

What this shows: Transmissibility as a function of frequency ratio r = f_exc / f_n. Key insight: The red dashed line marks resonance (r=1) where amplification is maximum. The green dashed line marks the isolation threshold (r=sqrt(2)) where isolation begins. Your operating point is shown as an orange marker.

Phase Angle vs Frequency Ratio

What this shows: The phase lag between excitation and response as a function of frequency ratio. Key insight: At low frequencies the response is in-phase with excitation (0°). At resonance (r=1), the phase is exactly 90°. At high frequencies, the response is nearly 180° out of phase (anti-phase). The transition is sharper with lower damping.

Transmissibility vs Damping Ratio

What this shows: How transmissibility changes with damping ratio at your current frequency ratio. Key insight: Higher damping reduces resonance peaks but can increase transmissibility in the isolation region (r > sqrt(2)).
Contents
  1. Overview
  2. Key Equations
  3. Interpreting Results
  4. Limitations
  5. References

1. Overview

This tool analyzes and designs single-degree-of-freedom (SDOF) vibration isolators. Vibration isolation is achieved by mounting equipment on compliant elements (springs/elastomers) that reduce the transmission of vibration between the equipment and its base.

Key Principle: Isolation occurs when the excitation frequency exceeds sqrt(2) times the natural frequency of the isolated system. Below this threshold, the isolator amplifies vibration.

2. Key Equations

Natural Frequency $$ f_n = \frac{1}{2\pi}\sqrt{\frac{k_{total}}{m}} $$
  • k_total: Total stiffness of all mounts in parallel (N/m)
  • m: Supported mass (kg)
Transmissibility (Base or Force Excitation) $$ T = \frac{\sqrt{1 + (2\zeta r)^2}}{\sqrt{(1-r^2)^2 + (2\zeta r)^2}} $$
  • zeta: Damping ratio (dimensionless)
  • r: Frequency ratio = f_exc / f_n
  • For base excitation: T = X/Y (absolute/base displacement)
  • For force excitation: T = F_T/F_0 (transmitted/applied force)
Isolation Threshold $$ r > \sqrt{2} \approx 1.414 $$
  • When r > sqrt(2), transmissibility T < 1 (isolation achieved)
  • When r = 1, resonance occurs (maximum amplification)

3. Interpreting Results

Isolation Region

r > sqrt(2): The system is in the isolation region. Transmissibility is less than 1, meaning vibration is reduced. Higher r values provide better isolation.
1 < r < sqrt(2): The system is past resonance but not yet isolating. Transmissibility may still be greater than 1.
r near 1: Resonance region - avoid operating here. Very high amplification occurs, especially with low damping.

Effect of Damping

Damping has a complex effect on isolation. High damping reduces the resonance peak, which is beneficial if the system must pass through resonance during startup/shutdown. However, high damping also reduces isolation effectiveness in the high-frequency region.

4. Limitations

  • This tool assumes linear, viscous damping and small displacements
  • Real isolators may have nonlinear stiffness and hysteretic damping
  • Multi-degree-of-freedom effects (rocking, bouncing) are not modeled
  • Temperature and frequency dependence of material properties are not included
  • Shock isolation requires separate analysis (transient response)

5. References

  • Inman, D. J., Engineering Vibration, 4th ed., Pearson, 2014.
  • Rao, S. S., Mechanical Vibrations, 6th ed., Pearson, 2017.
  • Harris, C. M. & Piersol, A. G., Harris' Shock and Vibration Handbook, 5th ed., McGraw-Hill, 2002.