FORCED VIBRATION & DAMPING

1. FORCED VIBRATION & DAMPING

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. Types of Damped Oscillations (1) • Slight damping (underdamping) • Characteristics: • - oscillations with reducing amplitudes • - amplitude decays exponentially with time • - period is slightly longer • - Figure • -

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

8. Types of Damped Oscillations (3) • Heavy damping (Overdamping) • Resistive forces exceed those of critical damping • The system returns very slowly to the equilibrium position • Figure • Computer simulation

9. Example: moving coil galvanometer (1) • the deflection of the pointer is critically damped

10. Example: moving coil galvanometer (2) • Damping is due to induced currents flowing in the metal frame • The opposing couple setting up causes the coil to come to rest quickly

11. Forced Oscillation • The system is made to oscillate by periodic impulses from an external driving agent • Experimental setup:

12. Characteristics of Forced Oscillation (1) • Same frequency as the driver system • Constant amplitude • Transient oscillations at the beginning which eventually settle down to vibrate with a constant amplitude (steady state)

13. Characteristics of Forced Oscillation (2) • In steady state, the system vibrates at the frequency of the driving force

14. 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

15. 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 • Figure

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

17. Phase (1) • The forced vibration takes on the frequency of the driving force with its phase lagging behind • If F = F0 cos t, then • x = A cos (t - ) • where  is the phase lag of x behind F

18. Phase (2) • Figure • 1. As f  0,  0 • 2. As f ,  • 3. As f  f0, /2 • Explanation • When x = 0, it has no tendency to move. maximum force should be applied to the oscillator

19. Phase (3) • When oscillator moves away from the centre, the driving force should be reduced gradually so that the oscillator can decelerate under its own restoring force • At the maximum displacement, the driving force becomes zero so that the oscillator is not pushed any further • Thereafter, F reverses in direction so that the oscillator is pushed back to the centre

20. Phase (4) • On reaching the centre, F is a maximum in the opposite direction • Hence, if F is applied 1/4 cycle earlier than x, energy is supplied to the oscillator at the ‘correct’ moment. The oscillator then responds with maximum amplitude.

21. 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

22. 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 = ) Previous page

23. Hacksaw Blade Oscillator (1)

24. 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

25. Forced Vibration (1) • Adjust the position of the load on the driving pendulum so that it oscillates exactly at a frequency of 1 Hz • Couple the oscillator to the driving pendulum by the given elastic cord • Set the driving pendulum going and note the response of the blade

26. Forced Vibration (2) • In steady state, measure the amplitude of forced vibration • Measure the time taken for the blade to perform 10 free oscillations • Adjust the position of the tuning mass to change the natural frequency of free vibration and repeat the experiment

27. Forced Vibration (3) • Plot a graph of the amplitude of vibration at different natural frequencies of the oscillator • Change the magnitude of damping by rotating the card through different angles • Plot a series of resonance curves

28. Resonance (1) • Resonance occurs when an oscillator is acted upon by a second driving oscillator whose frequency equals the natural frequency of the system • The amplitude of reaches a maximum • The energy of the system becomes a maximum • The phase of the displacement of the driver leads that of the oscillator by 90

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

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

31. 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

32. 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

33. 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

34. 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

35. EXAMPLES

36. Oscillation of a metal plate in the magnetic field

37. Slight Damping

38. Critical Damping

39. Heavy Damping

40. Amplitude

41. Phase

42. Barton’s Pendulum

43. Damped Vibration

44. Resonance Curves

45. Swing

46. Tacoma Bridge Video

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

48. Kundt’s Tube

49. Sonometer