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Chapter 10

Chapter 10. Sinusoidally Driven Oscillations. Question of Chapter 10. How do the characteristic frequencies generated in one object (say a piano string) excite vibrations in another object (say a sounding board)?. A Simple Driving System. Natural Frequency ( w o ).

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Chapter 10

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  1. Chapter 10 Sinusoidally Driven Oscillations

  2. Question of Chapter 10 • How do the characteristic frequencies generated in one object (say a piano string) excite vibrations in another object (say a sounding board)?

  3. A Simple Driving System

  4. Natural Frequency (wo) • If the board is a door, then the natural frequency is around 0.4 Hz. • If the system is driven at 0.4 Hz, large amplitudes result. • Smaller amplitudes result for driver frequency different from 0.4 Hz.

  5. Actual motions • The door starts with complex motions (transient) that settle down to sinusoidal, no matter the motor rate. • The final frequency is always the driving frequency of the motor (w). • The amplitude of the oscillations depends on how far from the natural frequency the motor is.

  6. Driving at Various Rates

  7. Natural Frequency Amplitude vs. Frequency

  8. w << wo • Motor frequency is far below the natural frequency (w << wo) door moves almost in step with motor. • Door moves toward motor when bands are stretched most.

  9. w < wo • Door lags behind the motor.

  10. w = wo • Door lags by one quarter cycle.

  11. w >> wo • Door lags by one-half cycle.

  12. w/wo Door Lags << 1 0 < 1 Small = 1 ¼-cycle >> 1 ½-cycle Summarizing

  13. Computer Model Click on the link and experiment

  14. Nature of the Transient • Transients are reproducible • If crank starts in the same position, we get the same transient • Damped Harmonic Oscillations • Shown by changing the damping • Imagine the bottom of the door immersed in an oil bath • The amount of immersion gives the damping

  15. Small Damping

  16. Heavier Damping

  17. Two Part Motion • Damped harmonic oscillation (transient) is at the natural frequency • Driven (steady state) oscillation is at the driver frequency

  18. Driver Frequency = Natural Frequency

  19. Damping and the Steady State • As long as we are far from natural frequency, damping doesn’t affect the steady state. • Near the natural frequency, damping does have an effect.

  20. Small damping W½ Amplitude Large damping Frequency Damping and the Steady State As damping is increased the height of the peak decreases

  21. Trends with Damping • As damping increases we expect the halving time to decrease ( )Oscillations die out quicker for larger damping. • As damping increases the maximum amplitude decreases ( ) • Also notice W½ D.Larger damping means a broader curve.

  22. Percentage Bandwidth (PBW) • Range of frequencies for which the response is it least half the maximum amplitude. • Let N be the number of oscillations that the pendulum makes in T½. • Direct measurement yields PBW = 38.2/N measured in %

  23. Example of PBW • Imagine tuning an instrument by using a tuning fork (A 440) while playing A. • If you are not matching pitch, the tuning fork is not being driven at its natural frequency and the amplitude will be small. • Only at a frequency of 440 Hz will the amplitude of the tuning fork be large

  24. Example of PBW - continued • T½ = 5 sec (it takes about 5 seconds for the tuning fork to decay to half amplitude) • N = (440 Hz) (5 sec) = 2200 cycles • So when you get a good response from the tuning fork, you have found pitch to better than PBW = 38.2/2200 = 0.017% or 0.076 Hz!

  25. Caution! • You must play long, sustained tones • Short “toots” will stimulate the transient which recall is at the natural frequency of the tuning fork (440 Hz) • Without the sustained driving force of the instrument, we will never get to the steady state and the tuning fork will ring due to the transient. • You will think the instrument is in pitch when it is not.

  26. Systems with Two Natural Modes • Each mode has its own frequency, decay time, and shape. • The modes are always damped sinusoidal. • Superposition applies.

  27. Simple Two Mass Model

  28. Normal Modes of Two Mass Model (Chapter 6) Let Mode 1 have a natural frequency of 10 Hz and Mode 2 a natural frequency of 17.32 Hz.

  29. 17.32 Hz 10 Hz Amplitude Frequency Driving Point Response Function or Resonance Curve

  30. Frequencies Between Peaks • Mass one has a mode one component and should lag a half-cycle behind the driver(w > wo1) • Mass one also has a mode two component to its motion, and here the driving frequency is less than the natural frequency (w << wo2) • Mass one keeps in step with the driver • These conflicting tendencies account for the small amplitude here

  31. New Terms • Driving Point Response Curve – measure the response at the mass being driven • Transfer Point Response Curve – measure the response at another mass in the system (not a driven mass)

  32. Properties of a Sinusoidally Driven System • At startup there is a transient that is made up of the damped sinusoids of all of the natural frequencies. • Once the transient is gone the steady state is at the driving frequency. When the driving frequency is close to one of the natural frequencies, the amplitude is a maximum and resembles that natural mode.

  33. A Tin Tray The tray is clamped at three places. Sensors ( )and drivers ( )are used as pairs in the locations indicated.

  34. Response Curves

  35. General Principles • Sensor cannot pickup any mode whose nodal line runs through it. • Notice that Sensor 2 is on the centerline • It cannot pick up modes with nodal lines through the center, such as…

  36. Sensor 2 is blind to…

  37. Sensor 2 is very sensitive to…

  38. Drivers Ability to Excite Modes • If a driver falls on the nodal line of a mode, that mode will not be excited • If a driver falls between nodal lines of a mode, that mode will be excited

  39. Steady State Response • Superposition of all the modes excited and their amplitudes at the detector positions. • Some modes may reinforce or cancel other modes. • Example – consider the modes on the next screen • Colored sections are deflected up at this time and the uncolored sections are deflected down • The vertical lines show where in the pattern of each we are for a particular position on the plate

  40. Superposing Two Modes

  41. Summary • Altering the location of either the driver or the detector will greatly alter what the transfer response curve will be. • Altering the driver frequency will also change the response.

  42. Three Cases Presented • Deflections of the same sign (giving a larger deflection)Add • Deflections of opposite sign (canceling each other out)Subtract • Deflection of one mode lined up with the node of the other (deflection due to one mode only)Single

  43. The G4 Phantom at G3 196, 392, 588, 784, 980, 1176, … Depress G3 slowly Press & release G4 392, 784, 1176, 1568, 1960, 2352, …

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