\\[ X(s) = \underbrace{\frac{cs + k}{ms^2 + cs + k}}\_{T\_{wx}(s)} W(s) + \underbrace{\frac{1}{ms^2 + cs + k}}\_{T\_{Fx}(s)} F(s) \\]
- \\(T\_{wx}(s)\\) is called the **transmissibility** of the isolator. It characterize the way seismic vibrations \\(w\\) are transmitted to the equipment.
- \\(T\_{Fx}(s)\\) is called the **compliance**. It characterize the capacity of disturbing forces \\(F\\) to create motion \\(x\\) of the equipment.
In order to minimize the vibrations of a sensitive equipment, a general objective to design a good isolator is to minimize both \\(\abs{T\_{wx}}\\) and \\(\abs{T\_{Fx}}\\) in the frequency range of interest.
To decrease the amplitude of the overshoot at the resonance frequency, **damping** can be increased.
The price to pay is degradation of the isolation at high frequency (the roll off becomes \\(-1\\) instead of \\(-2\\)).
**First Trade-off**: Trade-off between damping and isolation.
To improve the transmissibility, the resonance frequency can be decreased.
However, the systems becomes more sensitive to external force \\(F\\) applied on the equipment.
**Second trade-off**: Trade-off between isolation and robustness to external force
### Active Isolation {#active-isolation}
We apply a feedback control.
The general expression of the force delivered by the actuator is \\(f = g\_a \ddot{x} + g\_v \dot{x} + g\_p x\\). \\(g\_a\\), \\(g\_v\\) and \\(g\_p\\) are constant gains.