MECHANICAL and ELECTRICAL VIBRATIONS

Equation of Motion for Mass-Spring System

$$mu''(t) + \gamma u'(t) + ku(t) = F(t),$$

where u is displacement, m is mass, $\gamma$ is damping or resistance factor, k is spring constant and F is external force.

Cases

• Undamped, no external force:
  natural frequency $\omega_0 = \sqrt{\frac{k}{m}}$, period $T = \frac{2\pi}{\omega_0}$.
  Solutions have the form
  
  $$u(t) = A\cos(\omega_0 t) + B\sin(\omega_0 t),$$
  
  
  which can be rewritten as
  
  $$u(t) = R\cos(\omega_0 t - \delta);$$
  
  
  phase $\delta = \tan^{-1}(B/A)$, amplitude $R = \sqrt{A^2 +B^2}$.
• Damped, no external force: characteristic equation roots
  $r_{1,2} =\frac{\gamma}{2m}(-1\pm\sqrt{1-\frac{4km}{\gamma^2}})$.
  Solutions have the form
  
  $$\begin{eqnarray*}u(t)=&Ae^{r_1t} + Be^{r_2t}&\mbox{\ if\ } \gamma^2-4km > 0,\\ ... ...rac{\gamma t}{2m}}cos(\mu t-\delta)&\mbox{\ if\ }\gamma^2-4km<0, \end{eqnarray*}$$
  
  where $\mu = \sqrt{4km-\gamma^2}/(2m)$.

Electric Circuits

LQ''(t) + R Q'(t) + Q(t)/C = E(t)

where Q is capacitor charge, L is inductance, R is resistance, C is capacitance and E is impressed voltage.

FORCED VIBRATIONS

Spring-Mass System:

consider

$$mu''(t) + \gamma u'(t) + ku(t) = F_0 \cos(\omega t)$$

with mass m, damping $\gamma$ and spring constant k.

Cases

• Undamped: let $\omega_0 = \sqrt{k/m}$.
  i) If $\omega_0\neq\omega$, solutions have the form
  
  $$u(t) = A\cos(\omega_0 t) + B\sin(\omega_0 t) + \frac{F_0}{m(\omega_0^2-\omega^2)}\cos(\omega t).$$
  
  
  If u(0) = u'(0) = 0,
  
  $$u(t) = \frac{2F_0}{m(\omega_0^2-\omega^2)}\sin((\omega_0-\omega) \frac{t}{2}) \sin((\omega_0+\omega) \frac{t}{2}).$$
  
  
  ii) If $\omega_0=\omega$ (resonance), solutions have the form
  
  $$u(t) = A\cos(\omega_0 t) + B\sin(\omega_0 t) + \frac{tF_0}{2m\omega_0}\sin(\omega_0 t).$$
  
  

• Damped: let $r_{1,2} =\frac{\gamma}{2m}(-1\pm\sqrt{1-\frac{4km}{\gamma^2}})$
  and $\mu = \sqrt{4km-\gamma^2}/(2m)$; solutions are
  
  $$\begin{eqnarray*}u(t)=&Ae^{r_1t} + Be^{r_2t} + U(t)&\mbox{\ if\ } \gamma^2-4km >... ...t}{2m}}R\cos(\mu t - \delta)+U(t)& \mbox{\ if\ }\gamma^2-4km<0. \end{eqnarray*}$$
  
  Let $\tan(\delta) = \frac{\gamma\omega}{m(\omega_0^2-\omega^2)}$; steady-state solution is
  
  $$U(t) = \frac{F_0}{\sqrt{m^2(\omega_0^2-\omega^2)^2+\gamma^2\omega^2}} \cos(\omega t - \delta).$$