1. FORCED VIBRATION & DAMPING WK 2

2. Damping
a process whereby energy is taken from the vibrating system and is being absorbed by the surroundings.
Examples of damping forces:
internal forces of a spring,
viscous force in a fluid,
electromagnetic damping in galvanometers,
shock absorber in a car.

3. Free Vibration Vibrate in the absence of damping and external force
Characteristics:
the system oscillates with constant frequency and amplitude
the system oscillates with its natural frequency
the total energy of the oscillator remains constant

4. Damped Vibration (1)
The oscillating system is opposed by dissipative forces.
The system does positive work on the surroundings.
Examples:
a mass oscillates under water
oscillation of a metal plate in the magnetic field

5. Damped Vibration (2)
Total energy of the oscillator decreases with time
The rate of loss of energy depends on the instantaneous velocity
Resistive force  instantaneous velocity
i.e. F = -bv where b = damping coefficient
Frequency of damped vibration < Frequency of undamped vibration

6. Damping Simulation
Click the Picture

7. Damping and Resonance
1 In an oscillating system such as the oscillation of a simple pendulum, the oscillation does not continue with the same amplitudes indefinitely.

8. Damping and Resonance
2 The amplitude of oscillation of the simple pendulum will gradually decrease and become zero when the oscillation stops. The decrease in the amplitude of an oscillating system is called damping.

9. Damping and Resonance
3 An oscillating system experiences damping when its energy is drained out as heat energy.
(a) External damping of the system is the loss of energy to overcome frictional forces or air resistance.

10. Damping and Resonance
(b) Internal damping is the loss of energy due to the extension and compression of the molecules in the system.

11. Damping and Resonance 4 Damping in an oscillating system causes
(a) the amplitude, and
(b) the energy of the system to decrease.

12. Types of Damped Oscillations (2)
Critical damping
No real oscillation
Time taken for the displacement to become effective zero is a minimum
Figure

13. Slight Damping

14. Critical Damping

15. Heavy Damping

16. Figure 15.12  (a) If the atoms in a molecule do not move too far from their equilibrium positions, a graph of potential energy versus separation distance between atoms is similar to the graph of potential energy versus position for a simple harmonic oscillator. (b) The forces between atoms in a solid can be modeled by imagining springs between neighboring atoms.
Fig , p.464

17. Fig. P15.26, p.478

18. Figure 15.24   (b) One type of automotive suspension system, in which a shock absorber is placed inside a coil spring at each wheel.
Fig b, p.472

19. Figure 15.24  (a) A shock absorber consists of a piston oscillating in a chamber filled with oil. As the piston oscillates, the oil is squeezed through holes between the piston and the chamber, causing a damping of the piston’s oscillations.
Fig a, p.472

20. Taken from http://bama.ua.edu/~rschad/teaching/LABs/CH15%20osc/
Figure One example of a damped oscillator is an object attached to a spring and submersed in a viscous liquid.
Fig , p.471

21. Figure 15. 22 Graph of position versus time for a damped oscillator
Figure Graph of position versus time for a damped oscillator. Note the decrease in amplitude with time.
At the Active Figures link at you can adjust the spring constant, the mass of the object, and the damping constant and see the resulting damped oscillation of the object.
Fig , p.471

22. Figure 15.23  Graphs of position versus time for (a) an underdamped oscillator, (b) a critically damped oscillator, and (c) an overdamped oscillator.
Fig , p.471

23. Figure 15.25  Graph of amplitude versus frequency for a damped oscillator when a periodic driving force is present. When the frequency  of the driving force equals the natural frequency 0 of the oscillator, resonance occurs. Note that the shape of the resonance curve depends on the size of the damping coefficient b.
Fig , p.473

24. Damped Oscillations
The previous image shows a system that is underdamped – it goes through multiple oscillations before coming to rest. A critically damped system is one that relaxes back to the equilibrium position without oscillating and in minimum time; an overdamped system will also not oscillate but is damped so heavily that it takes longer to reach equilibrium.

25. Amplitude

26. Energy
Amplitude of vibration is fixed for a specific driving frequency
Driving force does work on the system at the same rate as the system loses energy by doing work against dissipative forces
Power of the driver is controlled by damping

27. Amplitude Amplitude of vibration depends on
the relative values of the natural frequency of free oscillation
the frequency of the driving force
the extent to which the system is damped

28. Effects of Damping
Driving frequency for maximum amplitude becomes slightly less than the natural frequency
Reduces the response of the forced system

29. Damping and Resonance 4 Damping in an oscillating system causes
(a) the amplitude, and
(b) the energy of the system to decrease
(c) the frequency, f does not change.

30. Damping and Resonance
5 To enable an oscillating system to go on continuously, an external force must be applied to the system.

31. Damping and Resonance
6 The external force supplies energy to the system. Such a motion is called a forced oscillation.

32. Damping and Resonance
7 The frequency of a system which oscillates freely without the action of an external force is called the natural frequency.

33. Resonance (1)
Resonance occurs when the frequency of the driving force is equal to the natural frequency of the oscillating system or body.
The amplitude reaches a maximum at resonance
The energy of the system becomes a maximum at resonance

34. Demonstration of Resonance (1)
Resonance tube
Place a vibrating tuning fork above the mouth of the measuring cylinder
Vary the length of the air column by pouring water into the cylinder until a loud sound is heard
The resonant frequency of the air column is then equal to the frequency of the tuning fork

35. Demonstration of Resonance (2)
Sonometer
Press the stem of a vibrating tuning fork against the bridge of a sonometer wire
Adjust the length of the wire until a strong vibration is set up in it
The vibration is great enough to throw off paper riders mounted along its length

36. Damping and Resonance
8 Resonance occurs when a system is made to oscillate at a frequency equivalent to its natural frequency by an external force. The resonating system oscillates at its maximum amplitude.

37. Barton’s Pendulum (1)
The paper cones vibrate with nearly the same frequency which is the same as that of the driving bob
Cones vibrate with different amplitudes

38. Barton’s Pendulum (2)
Cone 3 shows the greatest amplitude of swing because its natural frequency is the same as that of the driving bob
Cone 3 is almost 1/4 of cycle behind D. (Phase difference = /2 )
Cone 1 is nearly in phase with D. (Phase difference = 0)
Cone 6 is roughly 1/2 of a cycle behind D. (Phase difference = )
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39. Hacksaw Blade Oscillator (1)

40. Hacksaw Blade Oscillator (2)
Damped vibration
The card is positioned in such a way as to produce maximum damping
The blade is then bent to one side. The initial position of the pointer is read from the attached scale
The blade is then released and the amplitude of the successive oscillation is noted
Repeat the experiment several times
Results

41. Damping and Resonance
(a) When pendulum X oscillates, all the other pendulums are forced to oscillate. It is found that pendulum B oscillates with the largest amplitude, that is, pendulum B resonates.

42. Damping and Resonance
(b) The natural frequency of a simple pendulum depends on the length of the pendulum. Note that pendulum X and pendulum B are of the same length. Therefore, pendulum X causes pendulum B to oscillate at its natural frequency.

43. Damping and Resonance
10 Some effects of resonance observed in daily life:
(a) The tuner in a radio or television enables you to select the programmes you are interested in. The circuit in the tuner is adjusted until resonance is achieved, at the frequency transmitted by a particular station selected. Hence a strong electrical signal is produced.

44. Damping and Resonance
10 Some effects of resonance observed in daily life:
(b) The loudness of music produced by musical instruments such as the trumpet and flute is the result of resonance in the air.

45. Damping and Resonance
(c) The effects of resonance can also cause damage. For example, a bridge can collapse when the amplitude of its vibration increases as a result of resonance.
THE POWER OF RESONANCE CAN DESTROY A BRIDGE. ON NOVEMBER 7, 1940, THE ACCLAIMED TACOMA NARROWS BRIDGE COLLAPSED DUE TO OVERWHELMING RESONANCE.

46. Damping and Resonance
(d)Cracking of wine glass

47. Resonance (2) Examples Mechanics: Sound:
Oscillations of a child’s swing
Destruction of the Tacoma Bridge
Sound:
An opera singer shatters a wine glass
Resonance tube
Kundt’s tube

48. Resonance (3) Electricity Light Radio tuning
Maximum absorption of infrared waves by a NaCl crystal

49. Resonant System
There is only one value of the driving frequency for resonance, e.g. spring-mass system
There are several driving frequencies which give resonance, e.g. resonance tube

50. Resonance: undesirable
The body of an aircraft should not resonate with the propeller
The springs supporting the body of a car should not resonate with the engine

51. Resonance Curves

52. Swing

53. Tacoma Bridge
Video

54. Resonance Tube
A glass tube has a variable water level and a speaker at its upper end

55. Kundt’s Tube

56. Sonometer

57. PPT]Simple Harmonic Motion
faculty.tnstate.edu/louyang/teach/Phys2010/.../Walker3_Lecture_Ch13.p...‎Cached
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