Download Introduction to Electromagnetism

Oscillators
fall CM lecture, week 3, 17.Oct.2002, Zita, TESC
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Review forces and energies
Oscillators are everywhere
Restoring force
Simple harmonic motion
Examples and energy
Damped harmonic motion
Phase space
Resonance
Nonlinear oscillations
Nonsinusoidal drivers
Review: Force, motion, and energy
Acceleration a = dv/dt, velocity v = dx/dt, displacement x = v dt
For time-dependent forces:
v(t) = 1/m F(t) dt
For space-dependent forces: v dv = 1/m F(x) dx.
Total mechanical energy E = T + V is conserved in the absence of
dissipative forces:
Kinetic T = (1/2) m v2 = p2 /(2m), Potential energy V = - F dx
displacement x   v dt   
Example: Morse potential
2
m
( E  V ( x) dt

V ( x)  V0 1  e
x

 V
2
0
Morse potential for H2

V ( x)  V0 1  e
x

 V
2
0
Sketch the potential: Consider asymptotic behavior at x=0 and x=,
Find x0 for minimum V0 (at dV/dx=0)
Think about how to find x(t) near the bottom of potential well.
Preview: Near x0, motion can be described by
F ( x)  
dV
  xV0
dx
Oscillators are ubiquitous
Restoring forces
Restoring force is in OPPOSITE direction to displacement.
Which are restoring forces for mass on spring? For _________
Spring force
Gravity
Friction
Air resistance
Electric force
Magnetic force
other
Simple harmonic motion:
Ex: mass on spring
First, watch simulation and predict behavior for various m,k. Then:
S F = ma
- k x = m x”
Guess a solution: x = A cost wt? x = B sin wt? x = C e wt?
Second-order diffeq needs two linearly independent solutions:
x = x1 + x2. Unknown coefficients to be determined by BC.
Sub in your solution and solve for angular frequency w  2 f 
(1): Apply BC: What are A and B if x(0) = 0? What if v(0) = 0?
(2): Do Ch3 # 1 p.128: Given w and A, find vmax and amax.
2
T
Energies of SHO (simple harmonic oscillator)
Find kinetic energy in terms of v(t): T(t) = _________
Find potential energy in terms of x(t): V(t) = _________
Find total energy in terms of initial values v0(t) and x0(t):
E = ____________
Do Ch.3 # 5: given x1, v1, x2, v2, find w and A.
Springs in series and parallel
Do Ch.3 # 7: Find effective frequency of each case.
Simple pendulum
S F = ma
- mg sin q = m s”
Small oscillations: sin q ~ q
Sub in:
Guess solution of form q = A cos wt.
Differentiate and sub in:
Solve for w
arclength: s = L q
Damped harmonic motion
First, watch simulation and predict behavior for various b. Then,
model damping force proportional to velocity, Fd = - c v:
S F = ma
- k x - cx’ = m x”
Simplify equation: multiply by m, insert w=k/m and g = c/(2m):
Guess a solution: x = C 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  w02
Use (3.4.7)
where w0=k/m
Write q = i wd. Then wd =______
Show that x = e gt (A cos wt +A2 sin wt) is a solution.
Do Examples 3.4 #1-4 p.91. Setup Problem 9. p.129
More oscillators next week:
Damped HO:
energy and “quality factor”
Phase space (see DiffEq CD)
Driven HO and resonance
Damped, driven HO
Electrical - mechanical analogs
Nonlinear oscillator
Nonsinusoidal driver: Fourier series