springs & simple harmonic motion

Springs & Simple Harmonic Motion

http://www.inpoc.no/static/animation/trolli-03.gif

• A bird on a perch can put itself into a

periodic oscillation.

• An oscillation is a back and forth

motion over the same path.

• It is said to be periodic if each cycle of the motion takes place in equal periods of time.

• Simple Harmonic Motion (SHM) can be described as a projection of circular motion on one axis.

SHM and Circular Motion• An object moving with constant speed in a

circular path observed from a distant point will appear to be oscillating with simple harmonic motion.

• The shadow of a pendulum bob moves with s.h.m. when the pendulum itself is either oscillating or moving in a circle with constant speed.

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SHM and Circular MotionFor any SHM there is a corresponding circular

motion.

• the radius of the circle is equal to the amplitude of the SHM

• the time period of the circular motion is equal to the time period of the SHM

• The relationship of circular

motion and SHM is

a

rT

22 4

Cutnell & Johnson, Wiley Publishing,

Physics 5th Ed.

SHM and Circular Motion• Rearranging,

• Recall, therefore• The constant of proportionality between acceleration

and displacement for an object moving with s.h.m. is equal to the square of the angular velocity of the corresponding circular motion.

• So, to find the value of the constant for a given oscillation, we simply measure the time period and then use the relation

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Spring ReviewHooke’s Law• Fs=restoring force of a spring• k = spring constant• x = displacement of the spring

• The friction free motion shown above is known as Simple Harmonic Motion (SHM)

Fs=-kx

Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.

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Spring ReviewSHM graph

• Sinusoidal motion

• max. stretching distance (x) from the equilibrium position is equal to the amplitude of the graph.



SHM has displacement, velocity and acceleration.

Displacement:


SHM and Circular Motion

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Note: r = amplitude (A)

SHM and Circular MotionPeriod (T) – time required to complete one cycle.

• Recall, for 1 cycle &

• Therefore,

Frequency (f) – number of cycles

per second

• Units – Hertz (Hz)

t

2 Tt

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SHM Terms

Copywrited by Holt, Rinehart, & Winston

SHM and Circular MotionVelocity

• Velocity of the shadow is the vx of vT.

Recall

At x =0 m v vmax

Therefore, θ = 0°

or

t rvT

cosTvv

trv cos

rv max

rv max

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SHM and Circular MotionAcceleration

• Acceleration of the shadow is the ax of ac.

Recall

At x = r, a amax

Therefore, θ = 90°

or

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sincaa 2rac

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2max ra

2max ra

Note: r = amplitude (A)

Graphs Describing S.H.M.• Displacement against time x = rsin(ωt)

• Velocity against time v = rωcos(ωt)

• Acceleration against time a = -rω²sin(ωt)

Note: All these graphs assume that, at t = 0, the body is at the equilibrium position.

SHM EnergyIn SHM the total energy possessed by the

oscillating body does not change with time.

Recall, Total Mechanical Energy = Kinetic Energy + Potential Energy

• An oscillation in which the total energy decreases with time is described as a damped oscillation.– Due to air resistance or other similar causes

SHM K.E. & P.E.• Kinetic energy against time ½m[rωcos(ωt)]²

• Since, P.E. = T.M.E. - K.E.

• P.E. graph has the same form as the K.E. graph but is inverted.

SHM K.E. & P.E.

When,

• P.E. is a maximum, K.E. is a minimum (K.E.=0)

• K.E. is a maximum, P.E. is a minimum (P.E.=0)

T.M.E = P.E. + K.E.

http://ecommons.uwinnipeg.ca/archive/00000030/02/shmani2.gif

http://www.maths-physics.nuigalway.ie/Maple_animations/images%5CSHM13.gif


SHM of SpringsDerivation of Frequency of SHM

• Hooke’s Law: F=-kx

• F = ma = -kx

• Frequency,

• Period,

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Elastic Potential Energy (PEelastic)

• Energy that a spring contains by being stretched or compressed.

221 kxPEelastic

http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html



Total Mechanical EnergyWhat forms of Mechanical Energy have we

discussed?

• Combining of all these

elasticalgravitaionrotationalnaltranslatio PEPEKEKETME

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21 kxmghImvE

SHM of Springs


Energy ProblemAn object of mass m = 0.200 kg is vibrating on a

horizontal frictionless table as shown. The spring has a spring constant k = 545 N/m. It is stretched initially to xo =4.50 cm and released from rest. Determine the final translational speed vf of the object when the final displacement of the spring is (a) xf = 2.25 cm and (b) xf=0 cm.


The Simple Pendulum • When displaced from its

equilibrium position by an angle θ and released it swings back and forth.

• Plotting the motion reveals a pattern similar to the sinusoidal motion of SHM



http://www.enm.bris.ac.uk/teaching/pendulum/animations/timeseries_simple.gif

The Simple Pendulum • The gravitational force (Fgx)

provides the torque.

• This restoring force (Fgx):

• Since the displacement and restoring force act in opposite directions.

• The torque of the pendulum is

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The Simple Pendulum For small angles (10°or smaller)

• θ = sinθ

• Where k‘ is a constant independent of θ

• This form is similar to F=-kx (Hooke’s Law)

So,


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The Simple Pendulum For small angles ONLY

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Energy of a PendulumTME = Constant

TME = PE + KE

PEmax KE = 0

KEmax PE = 0

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Swinging ProblemDetermine the length of a simple pendulum that

will swing back and forth in SHM with a period of 1.00 s.

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Simple Harmonic Motion


Elastic Deformation• When a spring is stretched and released it

returns to its original shape.

• Likewise, some materials when stretched or compressed and released return to their original shape

Elastic Materials

• Explained, by modeling the chemical bonds between as atoms as springs. When force causing deformation is removed, material returns to original shape


Elasticity• A property of a body that causes it to deform

when a force is exerted and return to its original shape when the deforming force is removed, within certain limits


http://www.bioeng.auckland.ac.nz/cmiss/examples/8/84/844/web_data/animation.gif

Stress ( )• The force exerted on an area divided by the

area

• Units Newton per square meter (N/m2 )

or pascal (Pa)

A

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Strain ( )• The resulting fractional change in length (ΔL/Lo)

due to a stretch/compression deformation.

• unitless

lengthoriginal

lengthinchangestrain

oL

L


Young’s Modulus (Y)• Measure of the elasticity of a material.

• Ratio of stress to strain of a material.

• A material property

• SI units Newton per square meter (N/m2 )

Ystrain

stressModulussYoung '


Shear Deformation• Change in shape of an object due to the

application of equal parallel forces in opposite directions.

Shear Stress ( )

Shear Strain ( )

oL

x

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http://scign.jpl.nasa.gov/learn/plate5.htm

Shear Modulus (G)• Ratio of shear stress to shear strain of a

material.

• A material property

• SI units Newton per square meter (N/m2 )

strainshear

stressshearModulusShear

G

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Hooke’s Law• Stress is directly proportional to strain

– Slope equal to Young’s modulus or Shear Modulus depending on measurement

• Elastic region


Stress Strain CurveProportionality limit – point on a stress strain

curve where stress and strain are no longer directly proportional.

Elastic limit – point above which the material will not return to its original shape– Above inelastic region


Elastic Properties of Selected Engineering Materials

Material Density(kg/m3)

Young's Modulus109 N/m2

Ultimate Strength Su

106 N/m2

Yield Strength Sy

106 N/m2

Steela 7860 200 400 250

Aluminum 2710 70 110 95

Glass 2190 65 50b ...

Concretec 2320 30 40b ...

Woodd 525 13 50b ...

Bone 1900 9b 170b ...

Polystyrene 1050 3 48 ...

a Structural steel (ASTM-A36), b In compression, c High strength, d Douglas firData from Table 13-1, Halliday, Resnick, Walker, 5th Ed. Extended.

http://hyperphysics.phy-astr.gsu.edu/hbase/mecref.html#c1




Applications:NONLINEAR CONSTITUTIVE BEHAVIOR OF PZT

-160

-140

-120

-100

-80

-60

-40

-20

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MP

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Strain (%)

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-20

0

-0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0

Str

ess

(M

Pa

)

Strain (%)

Setup

Loading Fixture

Alumina Block

Electrode

Specimen

Oil Bath Fixture

Tilt Table

Adjustable Screws

Alumina Spacer

Brass Electrode

Grafoil

PZT Sample (electroded surface)

Oil Bath

Strain GagesGrafoil

Brass Electrode

Alumina Spacer

Force

Force

Setup

+- 5 V

C=10 F-

+

+ 2000

20 KV Power Amplifier

Function Generator

Electrometer

Vout

ground

D = Q/A = C(Vout)/A

+

-

Strain Gage Amplifier

Force

Load Cell

ADC

Electrodes

Macroscopic Depth-Sensing Indentation

TestsWill Stoll, Physics Teacher, Norcross High School

International Site Mentor: Dr. T. H. Zhang and Professor F. J. Ke, Institute of Mechanics Beijing, China

Georgia Tech Mentor: Professor Min Zhou

Macroscopic Depth Sensing Indentation (DSI) Tests

• A Vicker indenter is inserted into the surface of the specimen.

• The load (P) and the displacement (h) of the indenter into the specimen are measured and plotted in a P vs h curve.

• Material properties of hardness (H) and modulus of elasticity (E) can be calculated from the P vs h curve.

Research Focus• Well established test for the nano-scale and

micro-scale regions.

• Investigate the applicability of DSI to the macro-scale region.

• Determine whether DSI testing is independent of scale.

•Hardness (H) of a specimen is calculated from:

where Ac is the contact area which is a function of the

displacement (h) of the indenter into the specimen .

•The reduced modulus Er is derived from the relationship

where S is the stiffness which is dependent on the unloading curve.

Background

2r

c

SE

A

max

c

PH

A

(The reduced modulus is used to account for the elastic deformation in both the indenter and specimen)

Test Setup• Load applied with a 250 N

load cell attached to an Instron Microtester

• Vicker’s Indenter attached to piston connected to the load cell.

• Specimen glued to attached base plate of test frame.

Instron 5848 Mirotester

frame

Load cell

DWS

Base jig

Upper jig

Indenter

Specimen

Wing

Piston

• Used a displacement capacitance sensor

• Decouples displacement from the load cell eliminating frame compliance.

Sensor

Capacitance gauge

Signal process and show

Target

FIG. 4. Measurement schematic of the capacitance displacement sensor

Independent Displacement Measurement

Loading Profile• Load tests of 2.5 N, 5 N, 10 N, 25 N, 50 N, 100

N, 150 N and 200 N performed.• Slow ramp to maximum load by controlling

displacement rate– For 2.5 N, 5 N and 10 N tests - displacement rate of 0.1 m/s. – For 25 N, 50 N tests - displacement rate of 0.5 m/s.– For 100 N, 150 N and 200 N tests - displacement rate of 1.0

m/s.

• Maximum load held for 30 seconds• Unloaded to 90% of the maximum load at a

constant rate in 80 seconds

Analysis• Hardness calculated by assuming ideal

indenter tip area. – ISO 14577 standard recommends this for

depths greater than 6 m

• Modulus calculated by finding stiffness which is the slope of the unloading curve.– Power law fit used for top 50% of unloading

curve.

Test Results

0 20 40 60 80 100

0

50

100

150

200

L

oa

d ()

Depth (m)

200N 150N 100N 50N 25N 10N 5N 2.5N

Hardness and Modulus Results

Peak Load (N)

Hardness (GPa)

Modulus (GPa)200 1.09 68.4150 1.06 67.8100 1.18 73.050 1.14 64.925 1.12 65.210 1.16 69.45 1.19 70.22.5 1.18 81.7

Aluminum accepted values: H = 1.1 GPa E = 71 GPa

Testing Challenges• Non-uniform adhesive layer

– Change mounting of specimen

• Load cell drift– Complicates calculation of the zero point– After unloading include a holding period to establish

the drift rate and then correct the displacement data for this

Conclusion• Results reinforce the validity of macroscopic DSI tests• DSI tests allow the hardness and modulus to be

determined without direct measurements of the contact area at different scale lengths.

• DSI testing holds great promise yet some refinement is still needed before widespread use.

springs & simple harmonic motion

Documents

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circular motionrearranging

circular motionshm

circular motionfor

circular motionperiod