Vibration Isolation Design and Transmissibility
Theory and Physics
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 $f_n$ — Widens the isolation region. Use softer mounts.
- However, too soft leads to large static deflection — Practical constraints.
- If operation passes through resonance, damping is necessary — Suppress peak with $\zeta$.
Selecting Vibration Isolation Mounts
| Mount | Spring Constant | Damping | Applications |
|---|---|---|---|
| Rubber Mount | Medium | Medium ($\zeta$ 5–15%) | 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 — $T < 1$ indicates isolation effect.
- Isolation region is $f > \sqrt{2} f_n$ — Lower $f_n$ increases effectiveness.
- Suppress resonance peak with damping — Appropriate setting of $\zeta$.
- Mount selection — Rubber, coil, air, wire rope.
- Calculate transmissibility with FEM — Spring elements + Harmonic Response Analysis.
The Golden Ratio of Vibration Isolation: Natural Frequency ≤ 1/3 of Excitation Frequency
The basic rule of vibration isolation design is "mount natural frequency fn ≤ excitation frequency f0 / √2 ≈ f0 × 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 ≤ 1Hz are standard, preventing image blur at magnifications of one million times.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, i.e., "mass × acceleration". Have you ever experienced being thrown forward when slamming on the brakes? That "feeling of being carried away" is precisely the inertial force. Heavier objects are harder to set in motion and harder to stop once moving. Buildings shake during earthquakes because the ground moves suddenly while the building's mass is "left behind". In static analysis, this term is set to zero, assuming "forces are applied slowly so acceleration can be ignored". It absolutely cannot be omitted for impact loads or vibration problems.
- Stiffness Term (Elastic Restoring Force): $Ku$ or $\nabla \cdot \sigma$. When you pull a spring, you feel a "force trying to return it", right? That's Hooke's law $F=kx$, the essence of the stiffness term. So, a question—if you pull an iron rod and a rubber band with the same force, which stretches more? Obviously the rubber. This "resistance to stretching" is Young's modulus $E$, which determines stiffness. A common misconception: "high stiffness = strong" is incorrect. Stiffness is "resistance to deformation", strength is "resistance to failure"—different concepts.
- External Force Term (Load Term): Body force $f_b$ (gravity, etc.) and surface force $f_s$ (pressure, contact force, etc.). Think of it this way—the weight of a truck on a bridge is a "force acting on the entire contents" (body force), the force of the tires pushing on the road is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A common pitfall here: getting the load direction wrong. Intending "tension" but applying "compression"—sounds like a joke, but it actually happens when coordinate systems are rotated in 3D space.
- Damping Term: Rayleigh damping $C\dot{u} = (\alpha M + \beta K)\dot{u}$. Try plucking a guitar string. Does the sound continue forever? No, it gradually fades. That's because vibration energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle—deliberately absorbing vibration energy to improve ride comfort. What if damping were zero? Buildings would keep swaying forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is crucial.
Assumptions and Applicability Limits
- Continuum assumption: Treats material as a continuous medium, ignoring microscopic heterogeneity.
- Small deformation assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, and stress-strain relationship is linear.
- Isotropic material (unless specified otherwise): Material properties are independent of direction (anisotropic materials require separate tensor definition).
- Quasi-static assumption (for static analysis): Ignores inertial and damping forces, considering only equilibrium between external and internal forces.
- Non-applicable cases: For large deformation/large rotation problems, geometric nonlinearity is required. For nonlinear material behavior like plasticity or creep, constitutive law extension is needed.
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify loads and elastic modulus to MPa/N system. |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful of unit inconsistency when comparing with yield stress. |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the distinction between engineering strain and logarithmic strain (for large deformation). |
| Elastic Modulus $E$ | Pa | Steel: ~210 GPa, Aluminum: ~70 GPa. Note temperature dependence. |
| Density $\rho$ | kg/m³ | In mm system: tonne/mm³ (= 10⁻⁹ tonne/mm³ for steel). |
| Force $F$ | N (Newton) | Unify as N in mm system, N in m system. |
Numerical Methods and Implementation
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 — Spring constant at low frequency.
- Dynamic stiffness — Spring constant at high frequency (20–50% higher than static).
- Loss factor $\eta$ — 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 — Anisotropy is also possible.
- Dynamic stiffness of rubber is higher than static — Use DMA data.
- Transmissibility = Output/Input ratio — 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 0.5–1 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 2–10 Hz) in manufacturing buildings to 1/100 or less.
Linear Elements (1st Order Elements)
Linear interpolation between nodes. Low computational cost but lower stress accuracy. Beware of shear locking (mitigated with reduced integration or B-bar method).
Quadratic Elements (with Mid-side Nodes)
Can represent curved deformation. Stress accuracy improves significantly, but degrees of freedom increase by about 2–3 times. Recommended: when stress evaluation is critical.
Full Integration vs Reduced Integration
Full Integration: Risk of over-constraint (locking). Reduced Integration: Risk of hourglass modes (zero-energy modes). Choose appropriately for the application.
Adaptive Mesh
Automatic refinement based on error indicators (e.g., ZZ estimator). Efficiently improves accuracy in stress concentration areas. Includes h-method (element subdivision) and p-method (order increase).
Newton-Raphson Method
Standard method for nonlinear analysis. Updates tangent stiffness matrix each iteration. Quadratic convergence within convergence radius, but high computational cost.
Modified Newton-Raphson Method
Updates tangent stiffness matrix using initial value or every few iterations. Cost per iteration is low, but convergence speed is linear.
Convergence Criteria
Force residual norm: $||R|| / ||F_{ext}|| < \epsilon$ (typically $\epsilon = 10^{-3}$ to $10^{-6}$). Displacement increment norm: $||\Delta u|| / ||u|| < \epsilon$. Energy norm: $\Delta u \cdot R < \epsilon$
Load Increment Method
Instead of applying the full load at once, apply it in small increments. The arc-length method (Riks method) can trace beyond limit points on the load-displacement curve.
Analogy: Direct Method vs Iterative Method
The direct method is like "solving simultaneous equations accurately with pen and paper"—reliable but takes too long for large-scale problems. The iterative method is like "repeatedly guessing to approach the correct answer"—starts with a rough answer but improves accuracy with each iteration. It's the same principle as looking up a word in a dictionary: it's more efficient to open it at an estimated location and adjust forward/backward (iterative) than to search sequentially from the first page (direct).
Relationship Between Mesh Order and Accuracy
1st order elements are like "approximating a curve with a ruler"—represented by straight line segments, so accuracy is limited. 2nd order elements are like a "flexible curve"—can represent curved changes, dramatically improving accuracy even at the same mesh density. However, computational cost per element increases, so judge based on total cost-effectiveness.
Practical Guide
Practical Vibration Isolation Design
How do you proceed with vibration isolation design in practice?
STEP 1: Identify the Vibration Environment
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