Oscillations and Waves Flashcards
Differential equation for mass on a spring
d^2x/dt^2=-k/m*x
Differential equation for pendulum
d^2(theta)/dt^2=-g/l*theta
Differential equation for LC circuit
d^2i/dt^2=-1/LC*i
What is the general solution for a harmonic oscillator
Acos(w0*t+phi)
for any displacement eg. x, theta, etc.
Comes from the superposition from the two general solutions.
How are circular motion and harmonic motion linked
harmonic motion is circular motion projected onto the real (x) axis
What is the total energy of a naturally oscillating system
1/2mw0^2*A^2
what is w0 for a spring
sqrt(k/m)
what is w0 for a pendulum
sqrt(g/l)
what is the damping coefficient
gamma=b/m where b is the constant of proportionality of the linear damping term
what is the term added to the equation for a damped harmonic oscillator
-b*velocity [x dot]
what is the quadratic solution for w for a damped oscillating system
w=i(gamma)/2 ± sqrt((wo)^2-(gamma/2)^2)
what is the case for light damping
gamma/2 < w0
what is the light damping solution
Ae^(-gammat/2) cos(wdt+phi)
what does an increased damping coefficient mean for a lightly damped system
a decreased time to get to equilibrium
what is the case for heavy damping
gamma/2 > w0
what is the heavy damping solution
Be^-t(gamma+)/2 + Ce^-t(gamma-)/2
what does an increased damping co-efficient mean for a heavy damped system
an increased time to get to equilibrium
what is the case for critical damping
gamma/2=w0
what is the critical damping solution
(B+C)e^-t(gamma)/2
what is particular about critical damping
there is no oscillation, there is just a decay
what is the form of the external driver force
Focos(wt)
what is the solution, and parameters, doe the driven oscillator
what type of solution is this
x=Acos(wt+phi)
STEADY STATE SOLUTION
where:
A=(Fo/m)/sqrt((w0^2-w^2)^2+w^2(gamma)^2)
phi=tan^-1(-w(gamma)/w0^2-w^2)
when does resonance occur
w=w0
What happens to phi over the resonance ‘cycle’
below resonance, phi=0 [driving force too slow]
at resonance, phi=-pi/2
above resonance, phi=-pi [driving force too fast]
what determines the width of the resonance amplitude peak
gamma (damping co-efficient)
what is a special case for the driven harmonic oscillator and what are the solutions
at resonance, w=w0:
A=(Fo/2mw0)/sqrt((w-w0)^2+(gamma/2)^2)
phi=tan^-1(gamma/w-w0)
what is the equation for Q
Q=w0/gamma
