2. 1© Manhattan Press (H.K.) Ltd. 7.9 Examples of forced vibration

3. 2 © Manhattan Press (H.K.) Ltd. Examples of forced vibration 7.9 Examples of forced vibration (SB p. 291) 1. Bridges Crosswind in valley caused the bridge to resonate and collapse Go to More to Know 3 More to Know 3

4. 3 © Manhattan Press (H.K.) Ltd. Examples of forced vibration 7.9 Examples of forced vibration (SB p. 291) 2. Vehicles Body of bus or motorcycle vibrates when it is travelling at a particular speed. 3. Glass A glass is broken by sound wave vibrating at resonant frequency of the glass.

5. 4 © Manhattan Press (H.K.) Ltd. Examples of forced vibration 7.9 Examples of forced vibration (SB p. 291) 4. Electrical circuits Radio receiver works on principle of resonance - tune to particular station, radio waves received have same frequency as that of electrical oscillations in circuits - electrical oscillations of high amplitude and same frequency as the radio waves is induced in receiver

6. 5 © Manhattan Press (H.K.) Ltd. Examples of forced vibration 7.9 Examples of forced vibration (SB p. 292) 5. Microwave ovens - Frequency of microwaves = Natural frequency of vibration of water molecules - Water molecules in food resonate and food get heated up

7. 6 © Manhattan Press (H.K.) Ltd. Examples of forced vibration 7.9 Examples of forced vibration (SB p. 292) 6. Magnetic resonance imaging Magnetic resonance imaging (MRI) - used for medical diagnoses - EM fields of varying frequencies cause nuclei of atoms to oscillate - Energy is absorbed by atoms in resonance - Pattern of energy absorption produces computer enhanced photograph Go to Example 5 Example 5

8. 7 © Manhattan Press (H.K.) Ltd. 7.1 Periodic motion and isochronous oscillation 1. In periodic motion, an object always repeats its path at equal time intervals. 2. Isochronous oscillation is an oscillation in which the period is independent of the amplitude of oscillation. If the displacement of the object in an oscillation that can be described by harmonic functions (such as sine or cosine), it can be called harmonic motion. 7.9 Examples of forced vibration (SB p. 295)

9. 8 © Manhattan Press (H.K.) Ltd. 7.1 Periodic motion and isochronous oscillation 3. If there is no friction in harmonic motions, then no energy will be lost and the oscillation will continue indefinitely. This kind of motion is known as simple harmonic motion (SHM). 7.9 Examples of forced vibration (SB p. 295)

10. 9 © Manhattan Press (H.K.) Ltd. 7.2 Kinematics of simple harmonic motion 7.9 Examples of forced vibration (SB p. 295) 4.For SHM, 5. For an object undergoes simple harmonic motion, (a) its acceleration is always directed towards the equilibrium position, and (b) its acceleration is directly proportional to its displacement.

11. 10 © Manhattan Press (H.K.) Ltd. 7.3 Relationship between SHM and circular motion 7.9 Examples of forced vibration (SB p. 295) 6. The simple harmonic motion of an object can be analyzed as the projection of the circular motion. 7. In rotating vector model, Variation with xVariation with t Displacement (x)- x = A cos  t Velocity (v) v = -  A sin  t Acceleration (a) a = -  2 xa = -  2 A cos  t

12. 11 © Manhattan Press (H.K.) Ltd. 7.4 Dynamics of simple harmonic motion 7.9 Examples of forced vibration (SB p. 295) 8. For simple pendulums: (a) The period T depends on the length of string and not the amplitude. (b) It can be used to measure the acceleration due to gravity (g). By measuring the period for different values of length of the string, a graph of  against T 2 is plotted. From the slope of the graph, g (= slope ×4π 2 ) can then be found.

13. 12 © Manhattan Press (H.K.) Ltd. 7.4 Dynamics of simple harmonic motion 7.9 Examples of forced vibration (SB p. 296) 9. For a loaded spring displaced to the right by a displacement x, it obeys Hooke’s Law. The restoring force F is directly proportional to x and the direction of F is opposite to that of x. F = –kx where k is called the force constant. 10. For horizontal mass-spring system:

14. 13 © Manhattan Press (H.K.) Ltd. 7.4 Dynamics of simple harmonic motion 7.9 Examples of forced vibration (SB p. 296) 11. For vertical mass-spring system: The period of vertical mass-spring system is the same as that of the horizontal one. 12. For combination of springs in series: The equivalent force constant for the springs (k’) in series is: If the springs are released, its period is:

15. 14 © Manhattan Press (H.K.) Ltd. 7.4 Dynamics of simple harmonic motion 7.9 Examples of forced vibration (SB p. 296) 13. For combination of springs in parallel: The equivalent force constant for the springs (k ’) in parallel is: k’ = k 1 + k 2 If the springs are released, its period is:

16. 15 © Manhattan Press (H.K.) Ltd. 7.5 Energy exchange in SHM 7.9 Examples of forced vibration (SB p. 296) 14. The total energy of the system is the sum of the kinetic and potential energies, which is a constant.

17. 16 © Manhattan Press (H.K.) Ltd. 7.6 Phase relationship in SHM 7.9 Examples of forced vibration (SB p. 296) 15. Phasor diagram shows that a leads v by  /2 and v leads x by  /2.

18. 17 © Manhattan Press (H.K.) Ltd. 7.7 Damped oscillation 7.9 Examples of forced vibration (SB p. 296) 16. In real-life mechanical oscillatory systems performing oscillations, there are damping forces such as air resistances, which cause a loss of energy and a decay of the amplitude of oscillation.

19. 18 © Manhattan Press (H.K.) Ltd. 7.8 Forced oscillation and resonance 7.9 Examples of forced vibration (SB p. 296) 17. An oscillating system such as the simple pendulum or loaded spring can be forced to oscillate at any frequency by an external periodic force. Under the influence of this force, the system is then said to be performing forced oscillations. The frequency of such a system is the same as the frequency of the external periodic force, which is called the driving frequency.

20. 19 © Manhattan Press (H.K.) Ltd. 7.8 Forced oscillation and resonance 7.9 Examples of forced vibration (SB p. 296) 18. If the frequency of the driving force is equal to the natural frequency of the oscillating system, maximum energy will be transferred to the system and its amplitude of vibration becomes maximum. This phenomenon is called resonance and this frequency is known as resonant frequency.

21. 20 © Manhattan Press (H.K.) Ltd. 7.9 Examples of forced vibration 7.9 Examples of forced vibration (SB p. 296) 19. The collapse of the Tacoma Narrows suspension bridge in 1940, the applications of microwave ovens and the magnetic resonance imaging (MRI) scanners are the examples of forced vibrations.

22. 21 © Manhattan Press (H.K.) Ltd. 7.9 Examples of forced vibration (SB p. 296)

23. 22 © Manhattan Press (H.K.) Ltd. End

24. 23 © Manhattan Press (H.K.) Ltd. 7.9 Examples of forced vibration (SB p. 291) To avoid producing a frequency equal to the natural frequency of a bridge, soldiers should not march through the bridge. They should walk randomly across the bridge instead. Return to Text

25. 24 © Manhattan Press (H.K.) Ltd. Q: Q: (a) How would you determine whether a motion remains simple harmonic at large amplitude? (b) A motor car is driven at steady speed over a rough road on which the surface height varies sinusoidally. The shock absorber mechanism which normally damps vertical oscillation is not working. (i) At a certain critical speed, the amplitude of vertical oscillation of the car becomes very large. Explain why this happens. (ii) In terms of the quantities listed below, derive a formula for (1) the natural frequency of vertical oscillation of the car, 7.9 Examples of forced vibration (SB p. 292)

26. 25 © Manhattan Press (H.K.) Ltd. Q: Q: (2) the critical speed when the amplitude of vertical oscillations is a maximum. Calculate this speed from the data given below. Mass of car and passengers (M) = 2.0 × 10 3 kg Vertical rise of car when passengers get out (s) = 1.0 × 10 –1 m Mass of passengers (m) = 5.0 × 10 2 kg Wavelength of road surface corrugations (λ) = 20 m Acceleration of free fall (g) = 9.8 m s –2 (c) Discuss the behaviour of the car when the shock absorber mechanism is working correctly, giving an appropriate sketched graph. Solution 7.9 Examples of forced vibration (SB p. 292)

27. 26 © Manhattan Press (H.K.) Ltd. 7.9 Examples of forced vibration (SB p. 293) Solution: (a) To determine whether a SHM remains simple harmonic at large amplitude, the period for oscillations of large amplitude is compared with the period for oscillations of small amplitude. If the periods are the same, the oscillation remains simple harmonic for large amplitude and vice versa. (b) (i) When the car with faulty shock absorber moves over the corrugated road surface, it is forced into vertical oscillation. The frequency of the forced oscillation f = v/ where v = speed of car. As the speed of the car v varies, the frequency of the forced oscillations changes. At the critical speed, the frequency f is equal to the natural frequency of the car and resonance occurs. The amplitude of oscillation becomes very large.

28. 27 © Manhattan Press (H.K.) Ltd. 7.9 Examples of forced vibration (SB p. 293) Solution (cont’d): (ii) (1) If k = force constant of the spring, M = mass of car and passengers, x o = spring compression when M is in equilibrium on top of the spring. Then R = Mg = kx o................ (1) Suppose that at any instant during the vertical oscillation, the additional compression is x.

29. 28 © Manhattan Press (H.K.) Ltd. 7.9 Examples of forced vibration (SB p. 294) Solution (cont’d): (2) When resonance occurs, Frequency of forced oscillation = Natural frequency (c) If the shock absorber mechanism is working properly, the vertical motion will be critically damped and no vertical oscillation will occur when the car moves over the corrugated road surface. The graph below shows that the amplitude of the displacement is small even when there is resonance under heavy damping. Return to Text