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:  F = ma - k x = m x” - k x = - m  2 x Simple pendulum:  F = ma - mg sin  = m s” - g  = L  ” = -L  2  Solutions: x = A cost  t + B sin  t or x = C + e i  t + C - e -i  t v max =  A, a max =  2 A Potential energy: V = (1/2) k x 2. Ch.11: for any conservative force, F = -kx where k = V”(x 0 )

Energies in SHO (Simple Harmonic Oscillator)

LC circuit as a SHO Instead of  F = ma, use Kirchhoff’s loop law  V = 0. Find the voltage across a capacitor from C = Q/V c. The voltage across an inductor is V L = L dI/dt. Use I= - dQ/dt to write a diffeq for Q(t) (current flows as capacitor discharges): Show that Q(t) = Q 0 e -i  t is a solution. Find frequency  and I(t) Energy in capacitor = U E = (1/2) q V= (1/2) q 2 /C Energy in inductor = U B =(1/2) L I 2

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, F d = -cv:  F = ma - k x - cx’ = m x” Simplify equation: divide by m, insert  =  k/m and  = c/(2m): Guess a solution: x = A 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.

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  d  t +A 2 sin  d  t) is a solution. Do Examples 3.4.2, p.91. Setup Problem 9. p.129

Examples of Damped HO G ( 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: V c.=Q/C Inductor: V L = L dI/dt. Resistor: V R = 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 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  (ohms) and charged capacitor C = 5  F (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 = F 0 e i  t has two parts: y(t) = y h (t) + y p (t) The solution y h (t) to the homogeneous equation (driver = F = 0) gives transient behavior (see phase diagrams). For the steady-state solution to the nonhomogeneous equation, guess y p (t) = A F 0 e i(  t-  ). Plug it into the diffeq and apply initial conditions to find A and . Show that the amplitude A (3.6.9) peaks at resonance (w r 2 = w  2 = w d 2 -  2 ) and levels out to the steady-state value in ( a) p.103. Set up Problem 3.10 p.129 if time.

Resonance