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Driven oscillators Dependence of x on time, t Power. This lecture. x. Solution. t. Transient motion. Steady-state motion. Driven oscillators. Newton equation of motion. Driving force. Method of differentiation and substitution with complex notation to find A 0 and j.

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  1. Driven oscillators • Dependence of x on time, t • Power This lecture

  2. x Solution t Transient motion Steady-state motion Driven oscillators Newton equation of motion Driving force

  3. Method of differentiation and substitution with complex notation to find A0 and j COMPLEX NOTATION • Newton equation in complex form DIFFERENTIATION AND SUBSTITUION gives:

  4. Magnitude of displacement x: A0 A0 depends onw A0 Increasing damping, g A0 has a maximum if The frequency corresponding to the maximum of A0 is w 0

  5. Magnitude of velocity x: wA0 wA0 Increasing damping, g wA0has always a maximum The frequency corresponding to the maximum of A0 is w 0

  6. Resonance frequency The frequency at which a relevant parameter becomes maximum is known as the resonance frequency. Two resonance frequencies for the displacement for the velocity The two values are equal if The sharpness of the resonance curves depends on the strength of damping (g). In the limit g =0, both displacement and velocity tend to infinity.

  7. Displacement and force are out of phase Phase of displacement, j j depends on w

  8. Phase of displacement, j The displacement always lags the force

  9. +p/2 j* 0 -p/2 w w0 Phase of velocity, j* At resonance the velocity is in phase with the force

  10. “Sloshing” of water Find the resonance frequency in the absence of damping Examples Lakes Tea-cup Bathtub…

  11. “Sloshing” of water • In the absence of dampingthe resonance frequency wres is equal to the natural frequency w0 • Natural frequency w0 2R Axis of rotation d R • w0= natural frequency of a physical pendulum d = distance of the centre of mass from axis of rotation I = momentum of inertia m = mass of water

  12. “Sloshing” of water 2R Natural frequency w0 d

  13. “Sloshing” of water Consider general geometry L h

  14. Lock Ness monster or a sloshing mode?

  15. Light, w increasing damping Atom, w0 Power The oscillator absorbs more energy when the frequency of the driving force, w, matches the natural frequency, w0 w = w0 Power, P w Driving frequency,

  16. Light, w Atom, w0 Power The oscillator absorbs more energy when the frequency of the driving force, w, matches the natural frequency, w0 w00 w02 w01 w04 w03 w05

  17. increasing damping Power The oscillator absorbs more energy when the frequency of the driving force, w, matches the natural frequency, w0 w = w0 Power, P w Driving frequency,

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