Oscillatorsfall 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 it + C- e -i t

vmax = A, amax = 2 A

Potential energy: V = (1/2) k x2.

Ch.11: for any conservative force, F = -kx where k = V”(x0)

2 k

m

2 g

L

Energies in SHO (Simple Harmonic Oscillator)

LC circuit as a SHOInstead ofF = ma, use Kirchhoff’s loop law 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 -it is a solution. Find frequency 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:

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: 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 dt +A2 sin dt) is a solution.

Do Examples 3.4.2, 3.4.4 p.91. Setup Problem 9. p.129

2 20q

Examples of Damped HOG.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 DHOCapacitor: 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

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):

2

20

d Q dQ QL R

dt dt C

2

20

d x dxm c kx

dt dt

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 ResonanceAs in your DiffEq Appendix A, the solution to a nonhomogeneous differential equation m x” + c x’ + kx = F0eit 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(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 (wr2 = w0

2 - 22

= wd2 - 2) and levels out to the steady-state value in (3.16.13a) p.103.

Set up Problem 3.10 p.129 if time.

Resonance

0 0

2 2d m

Qc