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Harmonic Oscillator

Harmonic Oscillator. Hooke’s Law. The Newtonian form of the spring force is Hooke’s Law. Restoring force Linear with displacement. The Lagrangian form uses the potential energy. L. L+ x. L - x. The spring force has a potential energy V = ½ kx 2 . Minimum energy at equilibrium.

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Harmonic Oscillator

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  1. Harmonic Oscillator

  2. Hooke’s Law • The Newtonian form of the spring force is Hooke’s Law. • Restoring force • Linear with displacement. • The Lagrangian form uses the potential energy. L L+ x L - x

  3. The spring force has a potential energy V = ½ kx2. Minimum energy at equilibrium. No velocity, K = ½ mv2 = 0 A higher energy has two turning points. Corresponds to K = 0 In between K > 0 Motion forbidden outside range Energy Curve V E E0 x1 x0 x2 x

  4. Potential Well • An arbitrary potential near equilibrium can be approximated with a spring potential. • Second order series expansion • First derivative is zero V E0 x x0

  5. Stability • For positive k, the motion is like a spring. • Stable oscillations about a point • For negative k, the motion is unstable. V E0 x xS xU unstable

  6. The differential equation at stable equilibrium has a complex solution. Euler’s formula Real part is physical Complex Solutions Im r ir sin q q Re r cos q Complex conjugate for real solution

  7. Small damping forces are velocity dependent. Not from a potential Generalized force on right side The differential equation can be solved with an exponential. Possibly complex Quadratic expression must vanish Damping Force

  8. The quadratic equation in l has three forms depending on the constants. If g > w0, W is real. Overdamped solution If g = w0, W is zero. Critically damped solution If g > w0, W is imaginary. Underdamped solution Three Cases

  9. The energy in a damped oscillator is dissipated. Work done by friction Lightly damped systems have periods close to undamped. Damping g<< w0 Quality factor Q measures energy loss per radian. Quality Factor next

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