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Oscillatorsfall CM lecture, week 3, 17.Oct.2002, Zita, TESC
• 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 energyAcceleration 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
Example: Morse potential
2 ( ( )mx v dt E V x dt
2
0 0( ) 1x
V x V e V
Morse potential for H2 2
0 0( ) 1x
V x V e V
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 0( ) dVF x xVdx
Oscillators are ubiquitous
Restoring forcesRestoring 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: F = ma
- k x = m x”
Guess a solution: x = A cost t? x = B sin t? x = C e t?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
(1): Apply BC: What are A and B if x(0) = 0? What if v(0) = 0?
(2): Do Ch3 # 1 p.128: Given and A, find vmax and amax.
22 fT
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 and A.
Springs in series and parallel
Do Ch.3 # 7: Find effective frequency of each case.
Simple pendulum
F = ma- mg sin = m s”
Small oscillations: sin ~ arclength: s = L Sub in:
Guess solution of form = A cos t. Differentiate and sub in:
Solve for
Damped harmonic motionFirst, watch simulation and predict behavior for various b. Then, model damping force proportional to velocity, Fd = - c v:
F = ma- k x - cx’ = m x”
Simplify equation: multiply by m, insert =k/m and = c/(2m):
Guess a solution: x = C e t
Sub in guessed x and solve resultant “characteristic equation” for .
Use Euler’s identity: ei = cos + i sin Superpose two linearly independent solutions: x = x1 + x2. Apply BC to find unknown coefficients.
Solutions to Damped HO: x = e t (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?
Use (3.4.7) where =k/m
Write q = i d. Then d =______
Show that x = e t (A cos t +A2 sin t) is a solution.
Do Examples 3.4 #1-4 p.91. Setup Problem 9. p.129
2 20q
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