## Forced Vibrations

Forced oscillations occur when a system is subject to an external periodic force
$f(t)$
.
Simple harmonic oscillations in one dimension, with no damping and no external force the equation
$m \frac{d^2x}{dt^2}+ k x= A$
. If a resistive force proportional to speed is present, the equation becomes
$m \frac{d^2x}{dt^2}+r \frac{dx}{dt}+ k x= A$
. If now a forcing term
$f(t)$
is present, the equation becomes
$m \frac{d^2x}{dt^2}+r \frac{dx}{dt}+ k x= f(t)$
.
The solution of this equation is the sum of two terms.
1. The complementary solution
$x_c$
of the homogeneous equation
$m\frac{d^2x}{dt^2}+r \frac{dx}{dt}+ k x=0$
.
Assume a solution of the form
$x_c=Ae^{\lambda t}$
then
$\frac{dx}{dt}=A \lambda e^{\lambda t}, \; \frac{d^2x}{dt^2}=\lambda^2 Ae^{\lambda t}$
so the equation becomes
$m \lambda^2 Ae^{\lambda t}+r A \lambda e^{\lambda t}+ k A e^{\lambda t}= 0$
.
Divide by
$e^{\lambda t} \neq 0$
.
$m \lambda^2 +r \lambda + k = 0 \rightarrow \lambda = \frac{-r \pm \sqrt{r^2-4mk}}{2m}$
.
If
$r^2-4mk \gt 0$
then
$\lambda_1, \; \lambda_2$
are real and distinct, and
$x_p=Ae^{{\frac{-r + \sqrt{r^2-4mk}}{2m}}t}+Be^{{\frac{-r - \sqrt{r^2-4mk}}{2m}}t}$
.
If
$r^2-4mk \lt 0$
then
$\lambda_1, \; \lambda_2$
are complex and distinct, and
$x_p=e^{- \frac{r}{2m}t}(Acos (\frac{\sqrt{4mk-r^2}}{2m}t) + Bsin (\frac{\sqrt{4mk-r^2}}{2m}t) )$
If
$r^2-4mk = 0$
then
$\lambda_1, \; \lambda_2$
are equal, and
$x_p=e^{- \frac{r}{2m}} (A+Bt )$
.
Choose the particular solution
$y_p$
from this table, then
$x=x_c+x_p$
..
The constants can be found from suitable initial conditions. If there is no resistive force then put
$r=0$
in the above solutions.