Transcript Introduction to Electromagnetism

Oscillators
fall CM lecture, week 4, 24.Oct.2002, Zita, TESC
• Review simple harmonic
oscillators
• Examples and energy
• Damped harmonic motion
• Phase space
• Resonance
• Nonlinear oscillations
• Nonsinusoidal drivers
Review Simple harmonic motion
Mass on spring:
w2 
S F = ma
- k x = m x”
- k x = - m w2 x
k
m
Simple pendulum:
w2 
g
L
S F = ma
- mg sin q = m s”
- g q = L q” = -L w2 q
Solutions: x = A cost wt + B sin wt or x = C+ e iwt + C- e -i wt
vmax = w A, amax = w2 A
Potential energy: V = (1/2) k x2.
Ch.11: for any conservative force, F = -kx where k = V”(x0)
Energies in SHO
(Simple Harmonic Oscillator)
LC circuit as a SHO
Instead of S F = ma, use Kirchhoff’s loop law S V = 0. Find the
voltage across a capacitor from C = Q/Vc. The voltage across an
inductor is VL = L dI/dt. Use I= - dQ/dt to write a diffeq for Q(t)
(current flows as capacitor discharges):
Show that Q(t) = Q0 e -iwt is a solution. Find frequency w and I(t)
Energy in capacitor = UE = (1/2) q V= (1/2) q2 /C
Energy in inductor = UB =(1/2) L I2
Oscillations in LC circuit
Damped harmonic motion
(3.4 p.84)
First, watch simulation and predict behavior for various drag
coefficients c. Model damping force proportional to velocity, Fd= -cv:
S F = ma
- k x - cx’ = m x”
Simplify equation: divide by m, insert w=k/m and g = c/(2m):
Guess a solution: x = A e lt
Sub in guessed x and solve resultant “characteristic equation” for l.
Use Euler’s identity: eiq = cos q + i sin q
Superpose two linearly independent solutions: x = x1 + x2.
Apply BC to find unknown coefficients.
Solutions to Damped HO: x = e -gt (A1 e qt +A2 e -qt )
Two simply decay: critically damped (q=0) and overdamped (real q)
One oscillates: UNDERDAMPED (q = imaginary).
Predict and view: does frequency of oscillation change? Amplitude?
q  g 2 - w 02
Use (3.4.7)
where w0=k/m
Write q = i wd. Then wd =______
Show that x = e -gt (A cos wd t +A2 sin wd t) is a solution.
Do Examples 3.4.2, 3.4.4 p.91. Setup Problem 9. p.129
Examples of Damped HO
G.14.55 ( 385): A block of mass m oscillates on the end of spring of force
constant k. The black moves in a fluid which offers a resistive force
F= - bv. (a) Find the period of the motion. (b) What is the fractional
decrease in amplitude per cycle? © Write x(t) if x=0 at t=0, and if
x=0.1 m at t=1 s.
Do this first in general, then for m = 0.75 kg, k = 0.5 N/m, b = 0.2 N.s/m.
RLC circuit as a DHO
Capacitor: Vc.=Q/C
Inductor: VL = L dI/dt. Resistor: VR = IR
Use I= - dQ/dt to write a diffeq for Q(t):
Note the analogy to the diffeq for a mass on a spring!
Inertia: Inductance || mass; Restoring: Cap || spring; Dissipation: Resistance || friction
d 2Q
dQ Q
d 2x
dx
L 2 R
 0
m 2  c  kx  0
dt
dt C
dt
dt
Don’t solve the diffeq all over again - just use the form of solution you
found for mass on spring with damping! Solve for Q(t):
RLC circuit
Ex: (G.30.8.p.766) At t=0, an inductor (L = 40 mH = milliHenry) is
placed in series with a resistance R = 3 W (ohms) and charged
capacitor C = 5 mF (microFarad). (a) Show that this series will
oscillate.
(b) Determine its frequency with and without the resistor.
© What is the time for the charge amplitude to drop to half its
starting value?
(d) What is the amplitude of the current?
(e) What value of R will make the circuit non-oscillating?
Driven HO and Resonance
As in your DiffEq Appendix A, the solution to a nonhomogeneous
differential equation m x” + c x’ + kx = F0eiwt has two parts:
y(t) = yh(t) + yp(t)
The solution yh(t) to the homogeneous equation (driver = F = 0)
gives transient behavior (see phase diagrams).
For the steady-state solution to the nonhomogeneous equation,
guess yp(t) = A F0ei(wt-f). Plug it into the diffeq and apply initial
conditions to find A and f.
Show that the amplitude A (3.6.9) peaks at resonance (wr2 = w02 - 2g2
= wd2 - g2) and levels out to the steady-state value in (3.16.13a) p.103.
Set up Problem 3.10 p.129 if time.
Resonance
wd w0 m w0
Q


2g 2g
c