HCR - Definition of 'Impedance'

Impedance: $Z(s)$, such that:$F(s)=Z(s)\cdot \dot{X}(s)$For a given force the smaller the impedance ⇒ the higher is the velocity and viceversa.

---

• ~ Ex.: Mass-Spring Problem

$$\begin{array}{l}M\ddot{x}(t)+\beta \dot{x}(t)+kx(t)=F(t)\\ \mathcal{L}\left[\dot{x}(t)\right]=\dot{X}(s)\\ \mathcal{L}\left[\ddot{x}(t)\right]=s\dot{X}(s)\\ \mathcal{L}\left[x(t)\right]=\frac{1}{s}\dot{X}(s)\\ \\ \Rightarrow Ms\dot{X}(s)+\beta \dot{X}(s)+\frac{k}{s}\dot{X}(s)=F(s)\\ \\ \Rightarrow F(s)=\underset{Z(s)}{\underbrace{\left[Ms+\beta +\frac{k}{s}\right]}}\cdot \dot{X}(s)\end{array}$$

---

• Impedance:
  • For a given system the higher is the impedance, the lower will be its velocity:$Z(s):=\frac{F(s)}{\dot{X}(s)}$An higher impedance brings more stability to the system.
    It is the opposite of Compliance.