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Oscillators fall 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. - PowerPoint PPT Presentation
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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
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