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 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 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( )dV

F x xVdx

Oscillators are ubiquitous

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

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 f

T

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 motion

First, 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