1. Chapter 10 Elasticity & Oscillations

2. Elasticity and Oscillations
Elastic Deformations
Hooke’s Law
Stress and Strain
Shear Deformations
Volume Deformations
Simple Harmonic Motion
The Pendulum
Damped Oscillations, Forced Oscillations, and Resonance
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Chap 10c - Elas & Vibrations - Revised

3. Elastic Deformation of Solids
A deformation is the change in size or shape of an object.
An elastic object is one that returns to its original size and shape after contact forces have been removed. If the forces acting on the object are too large, the object can be permanently distorted.
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Chap 10c - Elas & Vibrations - Revised

4. Chap 10c - Elas & Vibrations - Revised 4-3-10
Hooke’s Law F F
Apply a force to both ends of a long wire. These forces will stretch the wire from length L to L+L.
Could incorporate personal response system questions from the College Physics by G/R/R 2E ARIS site ( Instructor Resources: CPS by eInstruction, Chapter 10, Questions 1, 2, 3, 4, 5, and 6.
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Chap 10c - Elas & Vibrations - Revised

5. Chap 10c - Elas & Vibrations - Revised 4-3-10
Stress and Strain
Define:
The fractional change in length
Force per unit cross-sectional area
Stretching ==> Tensile Stress
Squeezing ==> Compressive Stress
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Chap 10c - Elas & Vibrations - Revised

6. Chap 10c - Elas & Vibrations - Revised 4-3-10
Hooke’s Law
Hooke’s Law (Fx) can be written in terms of stress and strain (stress  strain).
The spring constant k is now
Y is called Young’s modulus and is a measure of an object’s stiffness. Hooke’s Law holds for an object to a point called the proportional limit.
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Chap 10c - Elas & Vibrations - Revised

7. Chap 10c - Elas & Vibrations - Revised 4-3-10
A steel beam is placed vertically in the basement of a building to keep the floor above from sagging. The load on the beam is 5.8104 N and the length of the beam is 2.5 m, and the cross-sectional area of the beam is 7.5103 m2. Find the vertical compression of the beam.
Force of ceiling on beam
Force of floor on beam
For steel Y = 200109 Pa.
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Chap 10c - Elas & Vibrations - Revised

8. Chap 10c - Elas & Vibrations - Revised 4-3-10
Example (text problem 10.7): A 0.50 m long guitar string, of cross-sectional area 1.0106 m2, has a Young’s modulus of 2.0109 Pa.
By how much must you stretch a guitar string to obtain a tension of 20 N?
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Chap 10c - Elas & Vibrations - Revised

9. Chap 10c - Elas & Vibrations - Revised 4-3-10
Beyond Hooke’s Law
Elastic Limit
If the stress on an object exceeds the elastic limit, then the object will not return to its original length.
Breaking Point
An object will fracture if the stress exceeds the breaking point. The ratio of maximum load to the original cross-sectional area is called tensile strength.
Could incorporate personal response system questions from the College Physics by G/R/R 2E ARIS site ( Instructor Resources: CPS by eInstruction, Chapter 10, Question 7.
MFMcGraw
Chap 10c - Elas & Vibrations - Revised

10. Chap 10c - Elas & Vibrations - Revised 4-3-10
Ultimate Strength
The ultimate strength of a material is the maximum stress that it can withstand before breaking.
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Chap 10c - Elas & Vibrations - Revised

11. Chap 10c - Elas & Vibrations - Revised 4-3-10
An acrobat of mass 55 kg is going to hang by her teeth from a steel wire and she does not want the wire to stretch beyond its elastic limit. The elastic limit for the wire is 2.5108 Pa.
What is the minimum diameter the wire should have to support her?
Want
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Chap 10c - Elas & Vibrations - Revised

12. Chap 10c - Elas & Vibrations - Revised 4-3-10
Shear Deformations
A shear deformation occurs when two forces are applied on opposite surfaces of an object.
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Chap 10c - Elas & Vibrations - Revised

13. Chap 10c - Elas & Vibrations - Revised 4-3-10
Stress and Strain
Define:
Hooke’s law (stressstrain) for shear deformations is
where S is the shear modulus
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Chap 10c - Elas & Vibrations - Revised

14. Chap 10c - Elas & Vibrations - Revised 4-3-10
Example (text problem 10.25): The upper surface of a cube of gelatin, 5.0 cm on a side, is displaced by 0.64 cm by a tangential force. The shear modulus of the gelatin is 940 Pa.
What is the magnitude of the tangential force? F
From Hooke’s Law:
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Chap 10c - Elas & Vibrations - Revised

15. Chap 10c - Elas & Vibrations - Revised 4-3-10
Volume Deformations
An object completely submerged in a fluid will be squeezed on all sides.
The result is a volume strain;
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Chap 10c - Elas & Vibrations - Revised

16. Chap 10c - Elas & Vibrations - Revised 4-3-10
For a volume deformation, Hooke’s Law is (stressstrain):
where B is called the bulk modulus. The bulk modulus is a measure of how easy a material is to compress.
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Chap 10c - Elas & Vibrations - Revised

17. Chap 10c - Elas & Vibrations - Revised 4-3-10
An anchor, made of cast iron of bulk modulus 60.0109 Pa and a volume of m3, is lowered over the side of a ship to the bottom of the harbor where the pressure is greater than sea level pressure by 1.75106 Pa.
Find the change in the volume of the anchor.
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Chap 10c - Elas & Vibrations - Revised

18. Deformations Summary Table
Bulk Modulus (B)
Shear modulus (S)
Young’s modulus (Y)
Constant of proportionality
Fractional change in volume
Ratio of the relative displacement to the separation of the two parallel surfaces
Fractional change in length
Strain
Pressure
Shear force divided by the area of the surface on which it acts
Force per unit cross-sectional area
Stress
Volume
Shear
Tensile or compressive
This table appears in this chapter’s Master the Concepts.
MFMcGraw
Chap 10c - Elas & Vibrations - Revised

19. Simple Harmonic Motion
Simple harmonic motion (SHM) occurs when the restoring force (the force directed toward a stable equilibrium point) is proportional to the displacement from equilibrium.
Could incorporate personal response system questions from the College Physics by G/R/R 2E ARIS site ( Instructor Resources: CPS by eInstruction, Chapter 10, Questions 8, 9, 11, and 12.
MFMcGraw
Chap 10c - Elas & Vibrations - Revised

20. Chap 10c - Elas & Vibrations - Revised 4-3-10
The Spring
The motion of a mass on a spring is an example of SHM. x
Equilibrium position y
The restoring force is F = kx.
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Chap 10c - Elas & Vibrations - Revised

21. Equation of Motion & Energy
Assuming the table is frictionless:
Classic form for SHM
Also,
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Chap 10c - Elas & Vibrations - Revised

22. Spring Potential Energy
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Chap 10c - Elas & Vibrations - Revised

23. Simple Harmonic Motion
At the equilibrium point x = 0 so a = 0 too.
When the stretch is a maximum, a will be a maximum too.
The velocity at the end points will be zero, and
it is a maximum at the equilibrium point.
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Chap 10c - Elas & Vibrations - Revised

24. Chap 10c - Elas & Vibrations - Revised 4-3-10
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Chap 10c - Elas & Vibrations - Revised

25. What About Gravity?
When a mass-spring system is oriented vertically, it will exhibit SHM with the same period and frequency as a horizontally placed system.
The effect of gravity is cancelled out.
Could incorporate personal response system questions from the College Physics by G/R/R 2E ARIS site ( Instructor Resources: CPS by eInstruction, Chapter 10, Questions 10, 13, 14, and 15.
MFMcGraw
Chap 10c - Elas & Vibrations - Revised

26. Spring Compensates for Gravity
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Chap 10c - Elas & Vibrations - Revised

27. Representing Simple Harmonic Motion
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Chap 10c - Elas & Vibrations - Revised

28. Chap 10c - Elas & Vibrations - Revised 4-3-10
A simple harmonic oscillator can be described mathematically by:
where A is the amplitude of the motion, the maximum displacement from equilibrium, A = vmax, and A2 = amax.
Or by:
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Chap 10c - Elas & Vibrations - Revised

29. Linear Motion - Circular Functions
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Chap 10c - Elas & Vibrations - Revised

30. Projection of Circular Motion
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Chap 10c - Elas & Vibrations - Revised

31. The Period and the Angular Frequency
The period of oscillation is
where  is the angular frequency of the oscillations, k is the spring constant and m is the mass of the block.
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Chap 10c - Elas & Vibrations - Revised

32. Chap 10c - Elas & Vibrations - Revised 4-3-10
The period of oscillation of an object in an ideal mass-spring system is 0.50 sec and the amplitude is 5.0 cm.
What is the speed at the equilibrium point?
At equilibrium x = 0:
Since E = constant, at equilibrium (x = 0) the KE must be a maximum. Here v = vmax = A.
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Chap 10c - Elas & Vibrations - Revised

33. Chap 10c - Elas & Vibrations - Revised 4-3-10
Example continued:
The amplitude A is given, but  is not.
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Chap 10c - Elas & Vibrations - Revised

34. Chap 10c - Elas & Vibrations - Revised 4-3-10
The diaphragm of a speaker has a mass of 50.0 g and responds to a signal of 2.0 kHz by moving back and forth with an amplitude of 1.8104 m at that frequency.
(a) What is the maximum force acting on the diaphragm?
The value is Fmax=1400 N.
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Chap 10c - Elas & Vibrations - Revised

35. Chap 10c - Elas & Vibrations - Revised 4-3-10
Example continued:
(b) What is the mechanical energy of the diaphragm?
Since mechanical energy is conserved, E = Kmax = Umax.
The value of k is unknown so use Kmax.
The value is Kmax= 0.13 J.
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Chap 10c - Elas & Vibrations - Revised

36. Chap 10c - Elas & Vibrations - Revised 4-3-10
Example (text problem 10.47): The displacement of an object in SHM is given by:
What is the frequency of the oscillations?
Comparing to y(t) = A sint gives A = 8.00 cm and  = 1.57 rads/sec. The frequency is:
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Chap 10c - Elas & Vibrations - Revised

37. Chap 10c - Elas & Vibrations - Revised 4-3-10
Example continued:
Other quantities can also be determined:
The period of the motion is
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Chap 10c - Elas & Vibrations - Revised

38. Chap 10c - Elas & Vibrations - Revised 4-3-10
The Pendulum
A simple pendulum is constructed by attaching a mass to a thin rod or a light string. We will also assume that the amplitude of the oscillations is small.
Could incorporate personal response system questions from the College Physics by G/R/R 2E ARIS site ( Instructor Resources: CPS by eInstruction, Chapter 10, Questions 16, 17, 18, and 19.
MFMcGraw
Chap 10c - Elas & Vibrations - Revised

39. Chap 10c - Elas & Vibrations - Revised 4-3-10
The pendulum is best described using polar coordinates.
The origin is at the pivot point. The coordinates are (r, φ). The r-coordinate points from the origin along the rod. The φ-coordinate is perpendicualr to the rod and is positive in the counterclock wise direction.
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Chap 10c - Elas & Vibrations - Revised

40. Chap 10c - Elas & Vibrations - Revised 4-3-10
Apply Newton’s 2nd Law to the pendulum bob.
If we assume that φ <<1 rad, then sin φ  φ and cos φ 1, the angular frequency of oscillations is then:
The period of oscillations is
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Chap 10c - Elas & Vibrations - Revised

41. Chap 10c - Elas & Vibrations - Revised 4-3-10
Example (text problem 10.60): A clock has a pendulum that performs one full swing every 1.0 sec. The object at the end of the string weighs 10.0 N.
What is the length of the pendulum?
Solving for L:
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Chap 10c - Elas & Vibrations - Revised

42. Chap 10c - Elas & Vibrations - Revised 4-3-10
The gravitational potential energy of a pendulum is
U = mgy.
Taking y = 0 at the lowest point of the swing, show that y = L(1-cos).  L
y=0
Lcos
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Chap 10c - Elas & Vibrations - Revised

43. Chap 10c - Elas & Vibrations - Revised 4-3-10
The Physical Pendulum
A physical pendulum is any rigid object that is free to oscillate about some fixed axis. The period of oscillation of a physical pendulum is not necessarily the same as that of a simple pendulum.
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Chap 10c - Elas & Vibrations - Revised

44. Chap 10c - Elas & Vibrations - Revised 4-3-10
The Physical Pendulum
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Chap 10c - Elas & Vibrations - Revised

45. Chap 10c - Elas & Vibrations - Revised 4-3-10
Damped Oscillations
When dissipative forces such as friction are not negligible, the amplitude of oscillations will decrease with time. The oscillations are damped.
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Chap 10c - Elas & Vibrations - Revised

46. Chap 10c - Elas & Vibrations - Revised 4-3-10
Graphical representations of damped oscillations:
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Chap 10c - Elas & Vibrations - Revised

47. Chap 10c - Elas & Vibrations - Revised 4-3-10
Damped Oscillations
·Overdamped: The system returns to equilibrium without oscillating. Larger values of the damping the return to equilibrium slower.
·Critically damped : The system returns to equilibrium as quickly as possible without oscillating. This is often desired for the damping of systems such as doors.
·Underdamped : The system oscillates (with a slightly different frequency than the undamped case) with the amplitude gradually decreasing to zero.
Source: Wikipedia
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Chap 10c - Elas & Vibrations - Revised

48. Chap 10c - Elas & Vibrations - Revised 4-3-10
Damped Oscillations
The larger the damping the more difficult it is to assign a frequency to the oscillation.
MFMcGraw
Chap 10c - Elas & Vibrations - Revised

49. Forced Oscillations and Resonance
A force can be applied periodically to a damped oscillator (a forced oscillation).
When the force is applied at the natural frequency of the system, the amplitude of the oscillations will be a maximum. This condition is called resonance.
Could incorporate personal response system questions from the College Physics by G/R/R 2E ARIS site ( Instructor Resources: CPS by eInstruction, Chapter 10, Question 20.
MFMcGraw
Chap 10c - Elas & Vibrations - Revised

50. Chap 10c - Elas & Vibrations - Revised 4-3-10
Tacoma Narrows Bridge
Nov. 7, 1940
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Chap 10c - Elas & Vibrations - Revised

51. Chap 10c - Elas & Vibrations - Revised 4-3-10
Tacoma Narrows Bridge
Nov. 7, 1940
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Chap 10c - Elas & Vibrations - Revised

52. Chap 10c - Elas & Vibrations - Revised 4-3-10
Tacoma Narrows Bridge
The first Tacoma Narrows Bridge opened to traffic on July 1, It collapsed four months later on November 7, 1940, at 11:00 AM (Pacific time) due to a physical phenomenon known as aeroelastic flutter caused by a 67 kilometres per hour (42 mph) wind.
The bridge collapse had lasting effects on science and engineering. In many undergraduate physics texts the event is presented as an example of elementary forced resonance with the wind providing an external periodic frequency that matched the natural structural frequency (even though the real cause of the bridge's failure was aeroelastic flutter[1]).
Its failure also boosted research in the field of bridge aerodynamics/ aeroelastics which have themselves influenced the designs of all the world's great long-span bridges built since Wikipedia
MFMcGraw
Chap 10c - Elas & Vibrations - Revised

53. Chapter 10: Elasticity & Oscillations
Elastic deformations of solids
Hooke's law for tensile and compressive forces
Beyond Hooke's law
Shear and volume deformations
Simple harmonic motion
The period and frequency for SHM
Graphical analysis of SHM
The pendulum
Damped oscillations
Forced oscillations and resonance
MFMcGraw
Chap 10c - Elas & Vibrations - Revised

54. Chap 10c - Elas & Vibrations - Revised 4-3-10
Extra
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Chap 10c - Elas & Vibrations - Revised

55. Chap 10c - Elas & Vibrations - Revised 4-3-10
Aeroelasticity
Aeroelasticity is the science which studies the interactions among inertial, elastic, and aerodynamic forces. It was defined by Arthur Collar in 1947 as "the study of the mutual interaction that takes place within the triangle of the inertial, elastic, and aerodynamic forces acting on structural members exposed to an airstream, and the influence of this study on design."
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Chap 10c - Elas & Vibrations - Revised

56. Chap 10c - Elas & Vibrations - Revised 4-3-10
Aeroelastic Flutter
Flutter
Flutter is a self-feeding and potentially destructive vibration where aerodynamic forces on an object couple with a structure's natural mode of vibration to produce rapid periodic motion. Flutter can occur in any object within a strong fluid flow, under the conditions that a positive feedback occurs between the structure's natural vibration and the aerodynamic forces. That is, that the vibrational movement of the object increases an aerodynamic load which in turn drives the object to move further. If the energy during the period of aerodynamic excitation is larger than the natural damping of the system, the level of vibration will increase, resulting in self-exciting oscillation. The vibration levels can thus build up and are only limited when the aerodynamic or mechanical damping of the object match the energy input, this often results in large amplitudes and can lead to rapid failure. Because of this, structures exposed to aerodynamic forces - including wings, aerofoils, but also chimneys and bridges - are designed carefully within known parameters to avoid flutter. It is however not always a destructive force; recent progress has been made in small scale (table top) wind generators for underserved communities in developing countries, designed specifically to take advantage of this effect.
In complex structures where both the aerodynamics and the mechanical properties of the structure are not fully understood flutter can only be discounted through detailed testing. Even changing the mass distribution of an aircraft or the stiffness of one component can induce flutter in an apparently unrelated aerodynamic component. At its mildest this can appear as a "buzz" in the aircraft structure, but at its most violent it can develop uncontrollably with great speed and cause serious damage to or the destruction of the aircraft.
In some cases, automatic control systems have been demonstrated to help prevent or limit flutter related structural vibration.
Flutter can also occur on structures other than aircraft. One famous example of flutter phenomena is the collapse of the original Tacoma Narrows Bridge.
MFMcGraw
Chap 10c - Elas & Vibrations - Revised