1. 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

2. 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 v 2 = p 2 /(2m), Potential energy V = -  F dx displacement Example: Morse potential

3. Morse potential for H 2 Sketch the potential: Consider asymptotic behavior at x=0 and x= , Find x 0 for minimum V 0 (at dV/dx=0) Think about how to find x(t) near the bottom of potential well. Preview: Near x 0, motion can be described by

4. Oscillators are ubiquitous

5. 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

6. 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 = x 1 + x 2. 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 v max and a max.

7. 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 v 0 (t) and x 0 (t): E = ____________ Do Ch.3 # 5: given x 1, v 1, x 2, v 2, find  and A.

8. Springs in series and parallel Do Ch.3 # 7: Find effective frequency of each case.

9. 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 

10. Damped harmonic motion First, watch simulation and predict behavior for various b. Then, model damping force proportional to velocity, F d = - 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: e i  = cos  + i sin  Superpose two linearly independent solutions: x = x 1 + x 2. Apply BC to find unknown coefficients.

11. Solutions to Damped HO: x = e  t (A 1 e qt +A 2 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 +A 2 sin  t) is a solution. Do Examples 3.4 #1-4 p.91. Setup Problem 9. p.129

12. 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