1 oscillations_03 Damped Oscillations Forced Oscillations and Resonance SHM

2 CP 445 Review: SHM mass/spring system

3 CP445

5 CP 455

6 oscillations_03: MINDMAP SUMMARY Reference frame, restoring force, damping force, driving force (harmonic driving force), net (resultant) force, Newton’s Second Law, natural frequency of vibration, free oscillations, damping (underdamped, critical damping, overdamped), exponential decay, graphical interpretation of damping, forced oscillations, resonance, driving frequency, resonance curve, self-excited oscillations, examples of resonant phenomena, strategy for answering examination questions, ISEE

7 Mathematical modelling for harmonic motion Newton’s Second Law can be applied to the oscillating system  F = restoring force + damping force + driving force  F(t) = - k x(t) - b v(t) + F d (t) For a harmonic driving force at a single frequency F d (t) = F max cos(  t +  ). This differential equation can be solved to give x(t), v(t) and a(t). CP 463

8 Damped Oscillations Oscillations in real systems die away (the amplitude steadily decreases) over time - the oscillations are said to be damped For example: The amplitude of a pendulum will decrease over time due to air resistance If the oscillating object was in water, the greater resistance would mean the oscillations damp much quicker. CP 463

9 Damped Oscillations

10 CP 463

11 The graphs show the vertical motion of an object attached to the end of a spring. Describe the motion of the object. Up is the positive direction.

12 Forced oscillations Driven Oscillations & Resonance If we displace a mass suspended by a spring from equilibrium and let it go it oscillates at its natural frequency If a periodic force at another frequency is applied, the oscillation will be forced to occur at the applied frequency - forced oscillations CP 465

13 Resonance Forced oscillations are small unless the driving frequency is close to the natural frequency When the driving frequency is equal to the natural frequency the oscillations can be large - this is called resonance Away from resonance, energy transfer to the oscillations is inefficient. At resonance there is efficient energy transfer which can cause the oscillating system to fail - see wine glass experiment. Famous example of resonance: soldiers marching on bridge CP 465

14 Resonance phenomena occur widely in natural and in technological applications: Emission & absorption of light Lasers Tuning of radio and television sets Mobile phones Microwave communications Machine, building and bridge design Musical instruments Medicine nuclear magnetic resonance (magnetic resonance imaging) x-rays hearing Nuclear magnetic resonance scan CP 465

15 CP 465 Response Curve f d / f o b is the damping factor

16 Sinusoidal driving force f d / f o = 0.1 Sinusoidal driving force f d / f o = 1 CP 465

17 Sinusoidal driving force f d / f o = 2 Impulsive force – constant force applied for a short time interval. CP 465

18 Forced vibrations - resonance f D = 0.4 f o f D = 1.1 f o f D = 1.6 f o Animation courtesy of Dr. Dan Russell, Kettering University

An optical technique called interferometry reveals the oscillations of a wine glass Great Links to visit

20 Eardrum Auditory canal Cochlea Basilar membrane vibrations of small bones of the middle ear vibration of eardrum due to sound waves Inner air – basilar membrane – as the distance increases from the staples, membrane becomes wider and less stiff – resonance frequency of sensitive hair cells on membrane decreases staples 3000 Hz 30 Hz Resonance and Hearing CP467 staples

21 Self excited oscillations Sometimes apparently steady forces can cause large oscillations at the natural frequency Examples singing wine glasses (stick-slip friction) Tacoma Narrows bridge (wind eddies) CP 466

22 CP 466

23 What is a good strategy for answering examination questions ???

24 1 Read and answer the question 2 Type of problem Identify the physics – what model can be used? Use exam formula sheet 3 Answer in point form Break the question into small parts, do step by step showing all working and calculations (if can’t get a number in early part of a question, use a “dummy” number. Explicit physics principles (justification, explanation) Annotated diagrams (collect and information & data – implicit + explicit Equations Identify  Setup  Execute  Evaluate

25 Problem 1 Why do some tall building collapse during an earthquake ?

26 I S E E Vibration motion can be resolved into vertical and horizontal motions

27 Vertical motion m k natural frequency of vibration driving frequency f d Resonance f d  f o large amplitude oscillations – building collapses

28 Driver frequency f d nodes antinodes 2 nd floor disappeared – driving frequency matches natural frequency (3 rd harmonic) Horizontal Motion Resonance Standing Waves setup in building 2 nd floor

29 ground vibrates horizontally setting up a standing wave antinode at top of building node antinode  this floor will collapse node at ground level

30 Problem 2 Consider a tractor driving across a field that has undulations at regular intervals. The distance between the bumps is about 4.2 m. Because of safety reasons, the tractor does not have a suspension system but the driver’s seat is attached to a spring to absorb some of the shock as the tractor moves over rough ground. Assume the spring constant to be 2.0  10 4 N.m -1 and the mass of the seat to be 50 kg and the mass of the driver, 70 kg. The tractor is driven at 30 km.h -1 over the undulations. Will an accident occur?

31  x= 4.2 m k= 2x10 4 N.m -1 v= 30 km.h -1 m= ( ) kg = 120 kg Solution 2 I S E E

32 Tractor speed v =  x /  t = 30 km.h -1 = (30)(1000) / (3600) m.s -1 = 8.3 m.s -1 The time interval between hitting the bumps (  x = 4.2 m)  t =  x / v = (4.2 / 8.3) s = 0.51 s Therefore, the frequency at which the tractor hits the bumps and energy is supplied to the oscillating system of spring-seat-person f = 1 /  t = 1 / 0.51 = 2.0 Hz. The natural frequency of vibration of the spring-seat-person is This is an example of forced harmonic motion. Since the driving frequency (due to hitting the bumps) is very close to the natural frequency of the spring-seat- person the result will be large amplitude oscillations of the person and which may lead to an unfortunate accident. If the speed of the tractor is reduced, the driving frequency will not match the natural frequency and the amplitude of the vibration will be much reduced.