\magnification 2200
\parindent 0pt
\parskip 13pt
\pageno=1
\hsize 7truein
\vsize 9.2truein
\def\u{\vskip -10pt}
\def\v{\vskip 3pt}
\def\w{\vskip 13pt}


3.8:  Mechanical and Electrical Vibrations

Trig background:

$cos (y \mp x) = cos(x \mp y) = cos(x) cos(y) \pm sin(x)sin(y)$

Let $A = Rcos(\delta)$,  $B = Rsin(\delta)$ in 


$Acos(\omega_0t) + B sin(\omega_0t)$
\u
= $Rcos(\delta)cos(\omega_0t) +  Rsin(\delta) sin(\omega_0t)$
\u
= $Rcos(\omega_0t - \delta)$

Amplitude = $R$
\u
frequency = $\omega_0$ (measured in radians per unit time).
\u
period = ${2\pi \over \omega_0}$
\u
phase (displacement) = $\delta$

\eject

Mechanical Vibrations:

\centerline{$mu''(t) + \gamma u'(t) + ku(t) = F_{external}, ~~m, \gamma, k \geq 0$} 

\centerline{ $mg - kL = 0$, ~~~~~$F_{viscous}(t) = \gamma u'(t)$}

$m$=  mass,
\u
$k$ = spring force proportionality constant,
\u
$\gamma$ = damping force proportionality constant

\u
g = 9.8 m/sec


\vskip 15 pt

Electrical Vibrations:
\vskip 4pt
\centerline{$L{dI(t) \over dt} + RI(t) + {1 \over C}Q(t) = E(t), ~~L, R, C \geq 0 {\hbox{ and 
}}
I = {dQ \over dt}$~~~~~~}

$L$ = inductance (henrys),
\u
$R$ = resistance (ohms)
\u
$C$ = capacitance (farads)
\u
$Q(t)$ = charge at time $t$ (coulombs)
\u
$I(t)$ = current at time $t$ (amperes)
\u
$E(t)$ = impressed voltage (volts).

1 volt = 1 ohm $\cdot$ 1 ampere = 1 coulomb / 1 farad = 1 henry $\cdot$ 1 amperes/ 1 second


\eject
\centerline{$mu''(t) + \gamma u'(t) + ku(t) = F_{external}, ~~m, \gamma, k \geq 0$}

$r_1, r_2 = {-\gamma \pm \sqrt{\gamma^2 - 4km} \over 2m}$

$\gamma^2 - 4km > 0$:  $u(t) = Ae^{r_1t} + Be^{r_2t} + \psi(t)  $
\w
$\gamma^2 - 4km = 0$:  $u(t) = (A + Bt)e^{r_1t}+ \psi(t)  $
\w
$\gamma^2 - 4km < 0$:  $u(t) = e^{-{\gamma t \over 2m}}(A cos \mu t + B sin \mu t)+ \psi(t)   $

\rightline{$= e^{-{\gamma t \over 2m}}R cos( \mu t - \delta)+ \psi(t)   $~~~~~~~}

\rightline{where $A = Rcos(\delta)$,  $B = Rsin(\delta)$}


$\mu$ =  quasi frequency, ${2 \pi \over \mu}$ = quasi period
\w\w\vfill

Note if $\gamma = 0$, then

Critical damping: $\gamma = 2\sqrt{km}$

Overdamped: $\gamma > 2\sqrt{km}$
\eject

Suppose a mass weighs 64 lbs stretches a spring 4 ft.  If there is no damping and the spring is 
stretched an additional 
foot and set in motion with an upward velocity of $\sqrt{8}$ ft/sec, find the equation of 
motion of the mass.
\v

$Weight = mg$:  $m = {weight \over g} = {64 \over 32} = 2$
\v

$mg - kL = 0$ implies $k = {mg \over L} = {64 \over 4} = 16$
\v

$mu''(t) + \gamma u'(t) + ku(t) = F_{external}$
\v
[$\gamma^2 - 4km < 0$:  $u(t) = e^{-{\gamma t \over 2m}}(A cos \mu t + B sin \mu t)$  \break 
Hence $u(t)= 
A cos \mu t + B sin \mu t $ since $\gamma = 0$].
\v


$2u''(t) + 16u(t) = 0$
\v

$u''(t) + 8u(t) = 0$
\v

$u(0) = 1$, $u'(0) = -\sqrt{8}$
\v

$r^2 + 8 = 0 \rightarrow$
$r^2  = -8 \rightarrow$
$r  = 
\sqrt{-8}
=i\sqrt{8}
=0 + i\sqrt{8}$
\end
\v

$u(t) = e^{-{\gamma t \over 2m}}(A cos \mu t + B sin \mu t) $
\v

$u(t) = A cos \sqrt{8} t + B sin \sqrt{8} t $
\v

$u(0) = 1$:  $1 = Acos(0) + B sin(0) = A$
\v
$u'(t) = -\sqrt{8}A sin \sqrt{8} t + \sqrt{8}B cos \sqrt{8} t $

\v
$u'(0) =  -\sqrt{8}:$  
$-\sqrt{8} =-\sqrt{8}A sin (0) + \sqrt{8}B cos (0) $
\v
$B = -1 $

\v

$u(t) =  cos \sqrt{8} t - sin \sqrt{8} t $


\end