Vibration Isolation Design and Transmissibility
Vibration Isolation Design and Transmissibility: Theoretical Foundations
What is Vibration Isolation?
Professor, does "vibration isolation" mean stopping vibration?
It's not about "stopping" vibration, but about reducing its transmission. A spring (vibration isolation rubber, vibration isolation mount) is placed between the vibration source and the object to be isolated to suppress the transmission of vibration.
Transmissibility
Transmissibility $T$ is the ratio of output to input:
Where $r = \omega / \omega_n$ (frequency ratio), $\zeta$ is the damping ratio.
At $r = 1$ (resonance), transmissibility peaks, and for $r > \sqrt{2}$, $T < 1$ (vibration isolation effect), right?
Perfect. The region $r > \sqrt{2}$ (i.e., $f > \sqrt{2} f_n$) is the isolation region. Here, the output becomes smaller than the input.
Key design points:
- Lower $f_n$ โ Widens the isolation region. Use softer mounts.
- However, too soft leads to large static deflection โ Practical constraints.
- If operation passes through resonance, damping is necessary โ Suppress peak with $\zeta$.
Selecting Vibration Isolation Mounts
| Mount | Spring Constant | Damping | Applications |
|---|---|---|---|
| Rubber Mount | Medium | Medium ($\zeta$ 5โ15%) | Engine mounts, equipment mounts |
| Coil Spring | Low | Low ($\zeta$ < 1%) | Precision equipment isolation |
| Air Spring | Very Low | Low | Semiconductor manufacturing equipment |
| Wire Rope Mount | Medium | Medium (friction damping) | Military equipment |
Air springs for precision equipment... So $f_n$ can be lowered to around 0.5 Hz, right?
Air springs have $f_n = 0.5 \sim 2$ Hz. They can block almost all external vibration. Air springs are standard for semiconductor exposure equipment and laser equipment.
Vibration Isolation Design in FEM
Vibration isolation design in FEM:
1. Build an FEM model of the equipment + mounts + foundation.
2. Model mounts as spring elements (+ dampers).
3. Apply input vibration to the foundation (frequency response or time history).
4. Calculate the equipment's response (displacement, acceleration).
5. Plot transmissibility $T = |X_{out}| / |X_{in}|$.
6. Confirm $T < T_{target}$.
Summary
Let me organize vibration isolation design and transmissibility.
Key points:
- Transmissibility $T$ is the central design metric โ $T < 1$ indicates isolation effect.
- Isolation region is $f > \sqrt{2} f_n$ โ Lower $f_n$ increases effectiveness.
- Suppress resonance peak with damping โ Appropriate setting of $\zeta$.
- Mount selection โ Rubber, coil, air, wire rope.
- Calculate transmissibility with FEM โ Spring elements + Harmonic Response Analysis.
The Golden Ratio of Vibration Isolation: Natural Frequency โค 1/3 of Excitation Frequency
The basic rule of vibration isolation design is "mount natural frequency fn โค excitation frequency f0 / โ2 โ f0 ร 0.7 or lower," and lowering fn to f0/3 (the rule of thirds) reduces transmissibility to 1/8 or less. This rule originates from the transmissibility curve shown by J.P. Den Hartog in his 1934 book 'Mechanical Vibrations'. For electron microscope (SEM/TEM) installation, ultra-low stiffness air mounts with fn โค 1Hz are standard, preventing image blur at magnifications of one million times.
Computational Methods for Vibration Isolation Design and Transmissibility
Modeling Vibration Isolation Mounts in FEM
How do you model vibration isolation mounts in FEM?
Represented by spring element + damper (viscous element) in parallel. Set spring constant and damping coefficient for 3 directions.
Nastran
```
CBUSH, 100, 200, 1000, 2000 $ Bush element
PBUSH, 200, K, 1000., 1000., 5000. $ kx, ky, kz
, B, 10., 10., 50. $ cx, cy, cz
```
Abaqus
```
*CONNECTOR SECTION, BEHAVIOR=mount
BUSHING,
*CONNECTOR BEHAVIOR, NAME=mount
*CONNECTOR ELASTICITY
1000., 1000., 5000.
*CONNECTOR DAMPING
10., 10., 50.
```
Nonlinear Characteristics of Rubber Mounts
Rubber mounts have frequency-dependent stiffness and damping (viscoelastic properties).
- Static stiffness โ Spring constant at low frequency.
- Dynamic stiffness โ Spring constant at high frequency (20โ50% higher than static).
- Loss factor $\eta$ โ Frequency-dependent damping.
Dynamic stiffness is higher than static stiffness?
Because rubber is a viscoelastic material, it hardens as vibration frequency increases. Using the spring constant obtained from static tests directly for dynamic analysis overestimates the isolation effect. Frequency-dependent properties should be measured via dynamic tests (DMA: Dynamic Mechanical Analysis).
Calculating Transmissibility
```
$ Transmissibility = Output point acceleration / Input point acceleration
T(f) = |a_output(f)| / |a_input(f)|
```
In FEM frequency response analysis, output the acceleration at input and output points, then take the ratio.
Summary
Let me organize the numerical methods for vibration isolation design.
Key points:
- Represent mounts with CBUSH (Nastran) / CONNECTOR (Abaqus)
- Set spring constant + damping for 3 directions โ Anisotropy is also possible.
- Dynamic stiffness of rubber is higher than static โ Use DMA data.
- Transmissibility = Output/Input ratio โ Calculated from frequency response analysis.
Air Spring Natural Frequency Changes with Pipe Length
The natural frequency of an air spring (pneumatic spring) is proportional to the -1/2 power of the enclosed air volume V, so increasing the volume with a supplementary tank can lower the natural frequency (down to about 0.5โ1 Hz). Since it's difficult to achieve a natural frequency below 3 Hz with coil springs alone, semiconductor manufacturing equipment (e.g., ASML TWINSCAN) universally uses air springs. They attenuate floor vibrations (mainly 2โ10 Hz) in manufacturing buildings to 1/100 or less.
Vibration Isolation Design and Transmissibility in Practice
Practical Vibration Isolation Design
How do you proceed with vibration isolation design in practice?
STEP 1: Identify the Vibration Environment
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