chapter 10 elasticity & oscillations. mfmcgraw-phy1401chap 10d - elas & vibrations - revised...

Chapter 10

Elasticity & Oscillations

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 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

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 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.

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 4

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.

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 5

Define:

L

Lstrain The fractional

change in length

A

Fstress Force per unit cross-

sectional area

Stress and Strain

Stretching ==> Tensile Stress

Squeezing ==> Compressive Stress

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 6

Hooke’s Law (Fx) can be written in terms of stress and strain (stress strain).

L

LY

A

F

The spring constant k is nowL

YAk

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.

Hooke’s Law

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 7

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 5.8104 N and the length of the beam is 2.5 m, and the cross-sectional area of the beam is

7.5103 m2. Find the vertical compression of the beam.

Force of floor on beam

Force of ceiling on beam

Y

L

A

FL

L

LY

A

F

For steel Y = 200109 Pa.

m 100.1N/m 10200

m 5.2

m 105.7

N 108.5 42923

4

Y

L

A

FL

Compressive Stress

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 8

Example (text problem 10.7): A 0.50 m long guitar string, of cross-sectional area 1.0106 m2, has a Young’s modulus of 2.0109 Pa.

By how much must you stretch a guitar string to obtain a tension of 20 N?

mm 5.0m 100.5

N/m 100.2

m 5.0

m 100.1

N 0.20

3

2926

Y

L

A

FL

L

LY

A

F

Tensile Stress

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 9

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.

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 10

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 11

The ultimate strength of a material is the maximum stress that it can withstand before breaking.

Ultimate Strength

Materials support compressive stress better than tensile stress

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 12

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 2.5108 Pa.

What is the minimum diameter the wire should have to support her?

Want limit elastic stress A

F

limit elasticlimit elastic

mgFA

mm 1.7m 107.1limit elastic

4

limit elastic2

3

2

mgD

mgD

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 13

Different Representations

0

0

0

0 0

0 0

0

F ΔL= Y

A L

ΔL 1 F=

L Y A

1 FΔL = L

Y A

1 FL - L = L

Y A

1 FL = L + L

Y A

1 FL = L 1+

Y A

Original equation

Fractional change

Change in length

New length

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 14

Shear Deformations

A shear deformation occurs when two forces are applied on opposite surfaces of an object.

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 15

A

F

Area Surface

ForceShear StressShear

L

x

surfaces of separation

surfaces ofnt displaceme Strain Shear

Hooke’s law (stressstrain) for shear deformations is

L

xS

A

F

Define:

where S is the shear modulus

Stress and Strain

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 16

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

F

N 30.0cm 5.0

cm 64.0m 0025.0N/m 940 22

L

xSAF

From Hooke’s Law:

L

xS

A

F

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 17

A

Fpressurestress volume

An object completely submerged in a fluid will be squeezed on all sides.

The result is a volume strain;V

Vstrain volume

Volume Deformations

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 18

For a volume deformation, Hooke’s Law is (stressstrain):

V

VBP

where B is called the bulk modulus. The bulk modulus is a measure of how easy a material is to compress.

Volume Deformations

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 19

An anchor, made of cast iron of bulk modulus 60.0109 Pa and a volume of 0.230 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 1.75106 Pa.

Find the change in the volume of the anchor.

36

9

63

m 1071.6

Pa 100.60

Pa 1075.1m 230.0

B

PVV

V

VBP

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 20

Examples

• I-beam

• Arch - Keystone

• Flying Buttress

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 21

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

PressureShear force divided by the area of the surface on which it acts

Force per unit cross-sectional area

Stress

VolumeShearTensile or

compressive

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 22

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.

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 23

Characteristics of SHM

• Repetitive motion through a central equilibrium point.

• Symmetry of maximum displacement.

• Period of each cycle is constant.

• Force causing the motion is directed toward the equilibrium point (minus sign).

• F directly proportional to the displacement from equilibrium.

Acceleration = - ω2 x Displacement

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 24

The motion of a mass on a spring is an example of SHM.

The restoring force is F = kx.

x

Equilibrium

position

x

y

The Spring

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 25

Assuming the table is frictionless:

xx

2x

F = - kx = ma

ka t = - x t = - ω x t

m

Also, 2 21 1

2 2E t K t U t mv t kx t

Equation of Motion & Energy

Classic form for SHM

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 26

Spring Potential Energy

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 27

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.

Simple Harmonic Motion

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 28

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 29

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.

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 30

Spring Compensates for Gravity

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 31

Representing Simple Harmonic Motion

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 32

A simple harmonic oscillator can be described mathematically by:

tAt

vta

tAt

xtv

tAtx

cos

sin

cos

2

Or by:

tAt

vta

tAt

xtv

tAtx

sin

cos

sin

2

where A is the amplitude of the motion, the maximum displacement from equilibrium, A = vmax, and A2 = amax.

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 33

Linear Motion - Circular Functions

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 34

Projection of Circular Motion

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 35

The period of oscillation is .2

T

where is the angular frequency of the oscillations, k is the spring constant and m is the mass of the block.

m

k

The Period and the Angular Frequency

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 36

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:

222

2

1

2

1

2

1mvkxmvUKE

Since E = constant, at equilibrium (x = 0) the KE must be a maximum. Here v = vmax = A.

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 37

cm/sec 862rads/sec 612cm 05 and

rads/sec 612s 500

22

...Aωv

..T

The amplitude A is given, but is not.

Example continued:

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 38

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 1.8104 m at that frequency.

(a) What is the maximum force acting on the diaphragm?

2222maxmax 42 mAffmAAmmaFF

The value is Fmax=1400 N.

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 39

(b) What is the mechanical energy of the diaphragm?

Since mechanical energy is conserved, E = Kmax = Umax.

2maxmax

2max

2

12

1

mvK

kAU

The value of k is unknown so use Kmax.

2222maxmax 2

2

1

2

1

2

1fmAAmmvK

The value is Kmax= 0.13 J.

Example continued:

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 40

Example (text problem 10.47): The displacement of an object in SHM is given by:

tty rads/sec 57.1sincm 00.8

What is the frequency of the oscillations?

Comparing to y(t) = A sint gives A = 8.00 cm and = 1.57 rads/sec. The frequency is:

Hz 250.02

rads/sec 57.1

2

f

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 41

222

max

max

max

cm/sec 719rads/sec 571cm 008

cm/sec 612rads/sec 571cm 008

cm008

...Aa

...Av

.Ax

Other quantities can also be determined:

The period of the motion is sec 00.4rads/sec 57.1

22

T

Example continued:

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 42

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.

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 43

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.

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 44

Apply Newton’s 2nd Law to the pendulum bob.

2

sin

cos 0r

F mg ma

vF T mg m

r

If we assume that φ <<1 rad, then sin φ φ and cos φ 1, the

angular frequency of oscillations is then:

L

g

The period of oscillations isg

LT

22

sin

sin

( / )sin

( / )

F mg ma mL

mg mL

g L

g L

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 45

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?

m 250

4

s 01m/s 89

4L

2

2

22

2

2

...gT

g

LT

Solving for L:

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 46

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

LLcos

)cos1( Ly

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 47

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.

The Physical Pendulum

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 48

http://hyperphysics.phy-astr.gsu.edu/HBASE/pendp.html

The Physical Pendulum

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 49

Damped Oscillations

When dissipative forces such as friction are not negligible, the amplitude of oscillations will decrease with time. The oscillations are damped.

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 50

Graphical representations of damped oscillations:

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 51

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: Damping @ Wikipedia

Damped Oscillations

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 52

Damped Oscillations

The larger the damping the more difficult it is to assign a frequency to the oscillation.

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 53

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.

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 54

Tacoma Narrows Bridge

Nov. 7, 1940

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 55

Tacoma Narrows Bridge

Nov. 7, 1940

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 56

The first Tacoma Narrows Bridge opened to traffic on July 1, 1940. 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 1940. - Wikipedia

http://www.youtube.com/watch?v=3mclp9QmCGs

Tacoma Narrows Bridge

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 57

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-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 58

Extra

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 59

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."

Aeroelasticity

MFMcGraw-PHY1401 Chap 10d - Elas & Vibrations - Revised 7-12-10 60

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.

Aeroelastic Flutter