Vibration Isolation Example
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## 1 Introduction to Vibration Isolation

The dynamic forces of satellites and machinery are often large by comparison to the desired performance. When this happens passive vibration isolation is implemented.

A passive isolator is employed. The employed isolator will typically be well modeled by second order system with a small damping (ζ). The lower the natural frequency of the isolator the harder it is to realize or implement.

## 2 Vibration Isolation: Transmitted Force[1]

The force transmitted to the foundation is the sum of the spring force and the damper force. Thehe transmitted force is

 $LaTeX: F_{T}=\sqrt{\left \[ \left(kX \right)^2 + \left(c\nu X \right)^2 \right ]}$

The transmissibility is

 $LaTeX: T_{R}=\frac{F_{T}}{F}=\frac{X\sqrt{k^2 + \left(c\nu \right)^2}}{F}$

 $LaTeX: X=\frac{\frac{F}{k}}{\sqrt{\left( \left \{ 1-\left(\frac{\nu}{\omega}\right)^2 \right \} ^2 + \left \{ 2\zeta \frac{\nu}{\omega} \right \} ^2 \right)}}$

 $LaTeX: T_{R}=\frac{\sqrt{1+\left( 2\zeta \frac{\nu}{\omega} \right)^2}}{\sqrt{\left( \left \{ 1-\left( \frac{\nu}{\omega} \right)^2 \right \} ^2 + \left \{ 2\zeta \frac{\nu}{\omega} \right \} ^2 \right)}}$

For hysteretic damping the transmissibility is

 $LaTeX: T_{R}=\frac{\sqrt{1+\eta^2}}{\sqrt{\left( \left \{ 1-\left( \frac{\nu}{\omega} \right)^2 \right \} ^2 + \eta^2 \right)}}$

where

$LaTeX: \eta=\frac{c\nu}{k}=2\zeta \frac{\nu}{\omega}$

### 2.1 Notes on Vibration Isolation

Vibration imposed on the system structure will be transmitted throughout the structure. This transmitted vibration can find its way to systems which are susceptible to these vibrations. Certain system components will resonate and this will lead to large noise values.

As mentioned before, the ideal passive isolator would have a low natural frequency in order to maximize vibration isolation. Realizing these low natural frequency isolators is difficult. In some instances the passive isolator will be too soft and the system will not be stable.

There are a couple of design options for improving the stability of the very soft vibration isolation. The system can be attached (rigidly) to additional masses so that stiffer isolators can be used. The center of mass can also be lowered to improve stability. Lastly, snubbers can be added to the system. Snubbers control large amplitude vibration and provide little damping for small amplitude motion.

#### 2.1.1 Notes on Vibration Isolation: Single Degree of Freedom vs. 6 Degrees

Proper design of a passive isolator for a single degree of freedom (DOF) is often fairly simple. The structure will pass vibrations at various amplitudes and frequencies. Components in the system will be susceptible to certain frequencies. A passive isolator will be designed to attenuate the vibrations above a certain frequency. So if you have structural vibrations at 100 and 1000 Hz that will make system components ring then design a passive isolator with a natural frequency of around 10 Hz.

The cost of this design approach is that these isolators can introduce a new natural frequency to the system in a different axis. The passive isolator(s) designed must account for all 6 DOFs (3 translational, 3 rotational) and this is much more complicated. Expect to go through lots of iteration cycles in order to achieve acceptable vibration isolation in all 6 DOFs.

## 3 Vibration Isolation Guidelines[2]

• The isolating system must have a natural frequency less than $LaTeX: \frac{1}{\sqrt{2}}$ times the disturbing frequency.
• The forcing frequency, ff, is usually known. The natural frequency can be found from
 $LaTeX: f=\frac{1}{2\pi}\sqrt{\frac{g}{\delta_{st}}}$ Natural Frequency

where

g is gravity
$LaTeX: \delta_{st}=\frac{M}{k}$
where
M is the equipment weight
k is the spring stiffness described is Forced Vibration in Second Order Systems
• The isolation efficiency is
 $LaTeX: \eta=1-\frac{1}{\left( \frac{f_{f}}{f} \right) ^2 -1}$

• Damping is only useful when
$LaTeX: \frac{f_{f}}{f} > \sqrt{2}$

## 4 References

• Leigh, J. R. 2004 Control Theory. ISBN 0863413390
• Lindeburg, M., (1994). Mechanical Engineering Reference Manual. Belmont: Professional Publications. ISBN 0-912045-72-8

### 4.1 Notes

1. Leigh, pp. 56-62
2. Lindeburg, pp. 16-15 to 16-16