Damped Harmonic Oscillator

• $m\ddot x + b\dot x + kx = 0$

• $\ddot{x}+\gamma \dot{x}+\omega_0^2 x=0$

• $x(t)=A e^{-\gamma t / 2} \text { cos } \omega_0 t$

• $\gamma=\frac{b}{m}, \quad \omega_0^2=\frac{k}{m}$

$m \ddot{x} =-k x+F \cos \omega t$

$\ddot{x} +\omega_0^2 x=\frac{F}{m} \cos \omega t$

${r}-x(t)=(A \cos \omega \cdot t+B \sin \omega \cdot t)$

$+\frac{F}{m\left(\omega_0^2-\omega^2\right)} \cos \omega t$

Swing in the park.

Pendulum in clocks - you need a1 battery to keep it running.

$m \ddot{x}=-b \dot{x}-k x+F \cos \omega t \quad\left(\omega \neq \omega_0\right)$

$\ddot{x}+r \dot{x}+\omega_0^2 x=F \cos \omega t$

Equation for forced damped harmonic oscillator.

Recall from forced undamped oscillator.

Motion was a combination of two diffrent frequences $\rightarrow \omega_0$ and $\omega$

$\ddot x + \omega_0^2x = \frac{F}{m} \cos\omega t$

$x(t)=\left(A \cos \omega_2 t+B \sin \omega t\right) +\frac{F}{m\left(\omega_1^2-\omega^2\right)} \cos \omega t$

$\ddot{x}+\gamma \dot{x}+\omega_0^2 x=\frac{F}{m} \cos at$

The solution is a combination (superposition) of two different solution

$x(t)=A e^{-\frac{\gamma}{2} t} \cos \omega_0 t+B e^{-\frac{\gamma}{2} t} \ sin \omega_0 t$

$x_{p}$ = depends on the applied force

As t become large, the solution

A $e^{-\gamma / 2 t} \cos\omega_0 t$ + $Be^{-\gamma / 2 t} \sin\omega_0 t \rightarrow 0$

We are left with steady state solution $\rightarrow$ $x_{\rho}(l)= x(t)$

Steady State Solution for a Forced Damped Harmonic Oscillator.

$A e^{-\frac{\gamma}{2} t} \cos\omega t$ + $B e^{-\frac{\gamma}{2} t} \sin\omega t \rightarrow 0$

Transient solution have died down.

We are left with only the steady state solution motion.

• $\ddot x + \gamma \dot x + \omega+0^2x = \frac{F}{m}\cos\omega t$

Recall in undamped oscillator

• $\ddot x+\omega_0^2 x = \frac{F}{m} \cos \omega t$

We had oscillator a solution x(t) = A $\cos \omega t$

• $\ddot{x}+\gamma \dot{x}+\omega_0^2 x=\frac{F}{m} \cos \omega t$

We want to try the solution of the form x(t) = A $\cos \omega t$

Substitute this is the equation of motion

• $\dot x(t) = - \omega A sin \omega t$

• $\ddot x (t) = - \omega A cos\omega t$

• $- \omega^2 A cos\omega t + \gamma (-\omega A) sin\omega t +\omega_0^2 cos\omega t=\frac{F}{m} cos\omega t$

LHS contains both $\cos\omega t$ and $\sin\omega t$.

RHS contains only $\cos\omega t$.

x(t) = $A \cos\omega t + B \sin\omega t$

$\dot x(t) = - \omega A \sin\omega t + \omega B \cos\omega t$

$\ddot x (t) = - \omega^2 A\cos\omega t - \omega^2 B \sin\omega t$

$(- \omega^2 A + \gamma \omega B + \omega_0^2 A) \cos\omega t$

$+ (- \omega^2 B - \gamma \omega A + \omega_0^2 B) \sin\omega t$

$=\frac{F}{m}\cos\omega t$

$-\omega^2 A + \gamma\omega B + \omega_0^2 A=\frac{A}{m}$

$-\omega^2 B - \gamma\omega A + \omega_0^2 B =0$

Two equation and two unknowns.

• $B=\left(\frac{\gamma \omega}{\omega_0^2-\omega^2}\right) A$

$A=\frac{\left(\omega_0^2-\omega^2\right)}{\left(\omega_0^2-\omega^2\right)^2+\gamma^2 \omega^2}\left(\frac{F}{m}\right)$

$B=\frac{\gamma \omega}{\left(\omega_0^2-\omega^2\right)} A$

$=\frac{\gamma \omega}{\left(\omega_0^2-\omega^2\right)^2+\gamma^2 \omega^2}\left(\frac{F}{m}\right)$

Steady - state solution

• $x(t)=\left[\frac{\left(\omega_0^2-\omega^2\right)}{\left(\omega_0^2-\omega^2\right)^2+\gamma^2 \omega^2} \text { as } \omega t\right.+\left.\frac{\gamma \omega}{\left(\omega^2-\omega y+\gamma^2 \omega\right.} \sin \omega t\right] \frac{F}{m}$

$x(t)= \frac{\left(\omega_0^2-\omega^2\right)}{\left(\omega_0^2-\omega^2+\gamma^2 \omega^2\right.}\left(\frac{F}{m}\right) \cos \omega t$

$+\frac{\gamma \omega}{\left(\omega_0^2-\omega^2\right)+\gamma^2 \omega^2}\left(\frac{F}{m}\right) \sin \omega t$

$x (t)$ is periodic with the same frequency.

i.e, $\omega$ as the frequency of the applied force.

It contains both $\cos\omega t$ and $\sin\omega t$ terms.

Amplitude of Motion (The Phase of Motion)

• $x(t)= \frac{\left(\omega_0^2-\omega^2\right)}{\left(\omega_0^2-\omega^2\right)^2+\gamma^2 \omega^2}\left(\frac{E}{m}\right) \cos \omega t$

• $+\frac{\gamma \omega}{\left(\omega_0^2-\omega^2\right)^2+\gamma^2 \omega^2}\left(\frac{F}{m}\right) \sin \omega t$

• $\frac{\omega_0^2-\omega^2}{\left(\omega_0^2-\omega^2\right)^2+ \gamma^2 \omega^2}$ =

• $\frac{1}{\sqrt{\left(\omega_0^2-\omega^2\right)^2+\gamma^2 \omega^2}} \left (\frac{\omega_0^2-\omega^2}{\sqrt{\left(\omega_0^2-\omega^2\right)^2+\gamma^2\omega^2}}\right)$

• $\frac{\gamma\omega}{\left(\omega^2-\omega)^2+\gamma^2 \omega^2\right.}$=

$\frac{1}{\sqrt{(\omega_0^2-\omega^2)^2}+\gamma^2\omega^2} \left( \frac{\gamma\omega}{\sqrt{\left(\omega_0^2-\omega^2\right)^2+\gamma^2\omega^2}}\right)$

$\left|c_1\right| \leqslant 1 \left|c_2\right| \leqslant 1 \quad c_1^2+c_2^2 = 1$

$x(t)=\frac{F}{m \sqrt{\left(\omega_0^2-\omega_0\right)^2+\gamma^2 \omega^2}} \cos (\omega t-\phi)$

$c_1 = \cos \phi=\frac{\omega_0^2-\omega^2}{\sqrt{\left(\omega_0^2-\omega^2\right)+\gamma^2 \omega^2}}$

$c_2 = \sin \phi=\frac{\gamma \omega}{\sqrt{(\omega_0^2-\omega^2)^2+\gamma^2 a^2}}$

$\tan \phi=\frac{\gamma \omega}{\left(\omega _0^2-\omega^2\right)}$

Amplitude $\frac{F}{m \sqrt{\left(\omega_2^2-\omega^2\right)+\gamma^2 \omega^2}}$ , $\phi=-\frac{\gamma \omega}{\left(\omega_1^2-\omega^2\right)}$

Same frequency on the applied force

It there is reasonance?

Does the amplitude go to a very large value for center applied frequency

• $A=\frac{F}{m \sqrt{\left(\omega_0^2-\omega^2\right)^2+\gamma^2 \omega^2}}$

$\frac{d A}{d \omega}=0$

$\frac{d A}{d \omega}=-\frac{F}{m} \quad \frac{1}{2} \frac{1}{\left.\left(\omega _ 0 ^2-\omega^2\right)^2+\gamma^2 \omega^2\right)^{3 / 2}} \times \frac{\left(2\left(\omega_0^2-\omega^2\right) \times-2 \omega+2 \gamma^2 \omega\right)}{1} \ = 0$

$2 \left(\omega_0^2-\omega^2\right)(- 2\omega) + 2\gamma^2\omega=0$

$- 2\left(\omega_0^2-\omega^2\right)= -\gamma^2$

or $\omega_0^2-\omega^2= \frac{\gamma^2}{2}$

$\omega^2=\omega_0^2-\frac{\gamma^2}{2}$

$\omega=\omega_0\left(1-\frac{\gamma^2}{2\omega_0^2}\right)^{\frac{1}{2}}$

For $\gamma << \omega_0$

$\omega=\omega_0\left(1-\frac{\gamma^2}{4\omega_0^2}\right)$

$=\omega_0-\frac{\gamma^2}{t\omega_0} \simeq\omega_0$

$A \left(\omega\right)$ = $\frac{c}{\sqrt{\left(\omega_0^2-\omega^2\right)^2+\gamma^2\omega^2}}$

At frequency $\omega=\omega_0$, the amplitude of steady state motion becomes large.

Power supplied by the force = Fv

• $x(t) = A \cos\left(\omega t -\phi\right)$

• $v(t) = -\omega A \sin \left(\omega t - \phi\right)$

Power = Fv = $- \omega A F \cos\omega t . sin\left(\omega t -\phi\right)$

• = $- \omega A F \left(\cos \omega t \sin\omega t \cos\phi - cos^2 \omega t \sin\phi\right)$

Power = $-\omega AF \left(\cos\omega t \sin\omega t \cos\phi - cos^2 \omega t \sin\phi\right)$

$\frac{\text{Average power}}{Cycle} = \frac{1}{T} \int_0^T \operatorname{Power}(t) d t$

= $-\omega A F\left[\frac{1}{T} \int_0^T \cos \omega t \sin \omega t d t \cdot \cos \phi-\frac{1}{T} \int_0^T \cos ^2 \omega t \quad \sin \phi d t\right]$

$T =\frac{2\pi}{\omega}$

$\int_0^T \cos \omega t \sin \omega t d t= 0$

$\frac{1}{T} \int \cos ^2 \omega t d t=\frac{1}{2}$

Average power = $\frac{\omega AF}{2} \sin\phi$

$\sin \phi$ = Power factor

This will be maximum value $sin\phi$ is maximum.

Average power= $\frac{\omega F^2}{m \sqrt{\left(\omega^2-\omega_0^2\right)+x^2 \gamma^2}} \sin \phi$

$\left.\operatorname{sin} \phi\right|_{\text {maximum }}=1 \Rightarrow \phi=\frac{\pi}{2} \Rightarrow \cos \phi=0$

$\frac{\omega_0^2-\omega^2}{m \sqrt{\left(\omega^2-\omega_0^2\right)+x^2 \gamma^2}} = 0 \rightarrow \omega_0^2 = \omega^2$

Average power supplied by us is maximum at resonance.

• $\rightarrow$ In steady state motion : the power supplied = power dissipated.

• = At resonance power diss by the system is maximum

$\sin\phi$ = Power Forced Oscillation LCR circuit.

(1) Periodic Motion

(ii) Simple Harmonic Motion

$m \ddot{x}=-k x$

$x(t)=A \cos \omega t+B \sin \omega t, \quad \omega=\sqrt{\frac{k}{m}}$

Resonance in forced undamped harmonic oscillation

Damping into the system

Friction

Velocity Dependent Damped

• $x(t)= A e^{-\gamma / 2 t} \cos t$

• $+B e^{-\gamma / 2} \sin \omega t$

Forced damped harmonic oscillation