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Chapter 14 - Oscillations

Chapter 14 - Oscillations. Harmonic Motion Circular Motion Simple Harmonic Oscillators Linear - Horizontal/Vertical Mass-Spring Systems Angular - Simple Pendulum Energy of Simple Harmonic Motion Damped Oscillators Driven Oscillators - Resonance. Harmonic. Horizontal mass-spring.

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Chapter 14 - Oscillations

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  1. Chapter 14 - Oscillations • Harmonic Motion Circular Motion • Simple Harmonic Oscillators • Linear - Horizontal/Vertical Mass-Spring Systems • Angular - Simple Pendulum • Energy of Simple Harmonic Motion • Damped Oscillators • Driven Oscillators - Resonance

  2. Harmonic

  3. Horizontal mass-spring Hooke’s Law:

  4. Solutions to differential equations • Guess a solution • Plug the guess into the differential equation • You will have to take a derivative or two • Check to see if your solution works. • Determine if there are any restrictions (required conditions). • If the guess works, your guess is a solution, but it might not be the only one. • Look at your constants and evaluate them using initial conditions or boundary conditions.

  5. Our guess

  6. The restriction on the solution

  7. Vertical Springs

  8. The constant – phase angle

  9. Definitions • Amplitude - (A, qm) Maximum value of the displacement (radius of circular motion). Determined by initial displacement and velocity. • Period - (T) Time for a particle/system to complete one cycle. • Frequency - (f) The number of cycles or oscillations completed in a period of time • Phase - (wt + f)Time varying argument of the trigonometric function. • Phase Constant - (f)Initial value of the phase. Determined by initial displacement and velocity. • Angular Frequency (Velocity) - (w)Time rate of change of the phase.

  10. Relation to circular motion

  11. Energy in the SHO

  12. Simple pendulum

  13. The restriction on the solution

  14. Damped Oscillations “Dashpot” Equation of Motion Solution

  15. Damped Oscillations “Dashpot” Equation of Motion Solution

  16. Damped frequency oscillation B - Critical damping (=) C - Over damped (>)

  17. Forced vibrations

  18. Resonance Natural frequency

  19. Quality (Q) value • Q describes the sharpness of the resonance peak • Low damping give a large Q • High damping gives a small Q • Q is inversely related to the fraction width of the resonance peak at the half max amplitude point.

  20. Tacoma Narrows Bridge

  21. Tacoma Narrows Bridge (short clip)

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