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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: w2 S F = ma - k x = m x” - k x = - m w2 x k m Simple pendulum: w2 g L S F = ma - mg sin q = m s” - g q = L q” = -L w2 q Solutions: x = A cost wt + B sin wt or x = C+ e iwt + C- e -i wt vmax = w A, amax = w2 A Potential energy: V = (1/2) k x2. Ch.11: for any conservative force, F = -kx where k = V”(x0) Energies in SHO (Simple Harmonic Oscillator) LC circuit as a SHO Instead of S F = ma, use Kirchhoff’s loop law S V = 0. Find the voltage across a capacitor from C = Q/Vc. The voltage across an inductor is VL = L dI/dt. Use I= - dQ/dt to write a diffeq for Q(t) (current flows as capacitor discharges): Show that Q(t) = Q0 e -iwt is a solution. Find frequency w and I(t) Energy in capacitor = UE = (1/2) q V= (1/2) q2 /C Energy in inductor = UB =(1/2) L I2 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, Fd= -cv: S F = ma - k x - cx’ = m x” Simplify equation: divide by m, insert w=k/m and g = c/(2m): Guess a solution: x = A 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 wd t +A2 sin wd t) is a solution. Do Examples 3.4.2, 3.4.4 p.91. Setup Problem 9. p.129 Examples of Damped HO G.14.55 ( 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: Vc.=Q/C Inductor: VL = L dI/dt. Resistor: VR = 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 d 2Q dQ Q d 2x dx L 2 R 0 m 2 c kx 0 dt dt C dt dt 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 W (ohms) and charged capacitor C = 5 mF (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 = F0eiwt has two parts: y(t) = yh(t) + yp(t) The solution yh(t) to the homogeneous equation (driver = F = 0) gives transient behavior (see phase diagrams). For the steady-state solution to the nonhomogeneous equation, guess yp(t) = A F0ei(wt-f). Plug it into the diffeq and apply initial conditions to find A and f. Show that the amplitude A (3.6.9) peaks at resonance (wr2 = w02 - 2g2 = wd2 - g2) and levels out to the steady-state value in (3.16.13a) p.103. Set up Problem 3.10 p.129 if time. Resonance wd w0 m w0 Q 2g 2g c