The Electrical-Mechanical Analogy — Question 3

Question 3

A scaled circuit analogue can run faster than the mechanical motion

Let a mechanical solution x(s) satisfy $m x_{ss}+b x_s+kx=F(s)$. Build an electrical analogue with $q(t)=a x(\gamma t)$, where a has units C/m and gamma is dimensionless. Choose positive L.

1. Derive R and C so the circuit equation is a constant multiple of the mechanical equation.

2. Find the required driving voltage in terms of F.

Question 3 - Solution

A scaled circuit analogue can run faster than the mechanical motion

(a) Method and calculation

$$\dot q=a\gamma x_s,\quad\ddot q=a\gamma^2x_{ss}.$$$$\boxed{R=L\gamma b/m,\quad C=m/(L\gamma^2k)}.$$

(b) Interpretation and further calculation

$$L\ddot q+R\dot q+q/C=\frac{La\gamma^2}{m}(mx_{ss}+bx_s+kx).$$$$\boxed{v(t)=\frac{La\gamma^2}{m}F(\gamma t)}.$$

Physical check

The time scale changes both damping and stiffness ratios; matching just the natural frequency is insufficient for a damped analogue.