Transcript shm - Atlanta International School Moodle

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.
• 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.
http://www.ul.ie/~nolk/s_h_m_%20train.gif
http://www.physics.uoguelph.ca/tutorials/shm/Animation1.gif
SHM and Circular Motion
SHM and Circular Motion
For 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
4 r
T 
a
2
2
Cutnell & Johnson, Wiley Publishing,
SHM and Circular Motion
2


4

4

r
• Rearranging,
 a
2
T 
 T 2 r
a


2
2
2
• Recall,   T therefore a   x
• 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
2

T
Spring Review
Hooke’s Law
Fs=-kx
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
• Fs=restoring force of a spring
• k = spring constant
• x = displacement of the spring
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
• The friction free motion shown above is known as
Simple Harmonic Motion (SHM)
Spring Review
SHM graph
• Sinusoidal motion
• max. stretching distance (x) from the
equilibrium position is equal to the amplitude of
the graph.
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
SHM and Circular Motion
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
SHM has displacement, velocity and acceleration.
Displacement:
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
x  r sin 
  t
x  r sin t
Note: r = amplitude (A)
SHM and Circular Motion
Period (T) – time required to complete one cycle.



t  T



2

• Recall,
for
1
cycle
&
t
• Therefore,
2
T
Frequency (f) – number of cycles
per second
cycles
Hz 
• Units – Hertz (Hz)
s
1
f 
T

f 
2
  2f
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.

SHM Terms
Copywrited by Holt, Rinehart, & Winston
SHM and Circular Motion
Velocity
• Velocity of the shadow is the vx of vT.
  t
v  vT cos Recall
vT  r
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
v  r cos t
At x =0 m v  vmax
Therefore, θ = 0°
vmax  r or
v max  r
Note: r = amplitude (A)
SHM and Circular Motion
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
Acceleration
• Acceleration of the shadow is the ax of ac.
  t
Recall
a  ac sin 
a c  r 2
a  r 2 sin t
At x = r, a  amax
Therefore, θ = 90°
amax  r 2
or
amax  r 2
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 Energy
In 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.
http://ecommons.uwinnipeg.ca/archive/00000030/02/shmani2.gif
Copywrited by Holt, Rinehart, & Winston
http://www.maths-physics.nuigalway.ie/Maple_animations/images%5CSHM13.gif
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.
SHM of Springs
Derivation of Frequency of SHM
• Hooke’s Law: F=-kx
• F = ma = -kx 
k
 r sin t   r sin t
m

2
k
m
• Frequency,

1
f 

2 2
• Period,
1
m
T   2
f
k
k
m
Elastic Potential Energy (PEelastic)
• Energy that a spring contains by being
stretched or compressed.
PEelastic  kx
1
2
2
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
Total Mechanical Energy
What forms of Mechanical Energy have we
discussed?
• Combining of all these
TME  KEtranslational  KErotational  PE gravitaional  PEelastic
E  mv  I  mgh  kx
1
2
2
1
2
2
1
2
2
SHM of Springs
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
Energy Problem
An 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.
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
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
Copywrited by Holt, Rinehart, & Winston
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
The Simple Pendulum
Fgx  Fg sin 
• Since the displacement and
restoring force act in opposite
directions. F  mg sin 
• The torque of the pendulum is
   mg sin  L
a   g sin 
Copywrited by Holt, Rinehart, & Winston
Copywrited by Holt, Rinehart, & Winston
• The gravitational force (Fgx)
provides the torque.
• This restoring force (Fgx):
The Simple Pendulum
I
m
I  mL
2
  2f 
g
L
I
a   g sin 
Copywrited by Holt, Rinehart, & Winston
Copywrited by Holt, Rinehart, & Winston
For small angles (10°or smaller)
• θ = sinθ   mgL

k
• Where k‘ is a constant
independent of θ
• This form   k  is similar to
F=-kx (Hooke’s Law)
So,   k    k   mgL
The Simple Pendulum
For small angles ONLY
(10°or smaller)
a   g sin 
1
f 
2
g
L
L
T  2
g
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
  2f 
g
L
Copywrited by Holt, Rinehart, & Winston
Energy of a Pendulum
http://image-ination.com/hints/pendulum.GIF
TME = Constant
TME = PE + KE
PEmax  KE = 0
KEmax  PE = 0
Determine the length of a simple pendulum that
will swing back and forth in SHM with a period
of 1.00 s.
T  2
L
g
http://physics.mtsu.edu/~wmr/pend.gif
Swinging Problem
Simple Harmonic Motion
Copywrited by Holt, Rinehart, & Winston
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
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
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
Cutnell & Johnson, Wiley Publishing, Physics 5th
Ed.
http://www.bioeng.auckland.ac.nz/cmiss/examples/8/84/844/web_data/animation.gif
• The force exerted on an area divided by the
area
• Units Newton per square meter (N/m2 )
or pascal (Pa)
Force
stress 
Area
F

A
http://www.esm.psu.edu/courses/emch13d/design/design-fund/design-notes/stress-animation.gif
Stress ( )

Strain ( )
• The resulting fractional change in length (ΔL/Lo)
due to a stretch/compression deformation.
• unitless
change  in  length
strain 
original  length
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
L

Lo
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 )
stress
Young ' s  Modulus 
strain
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

Y

Shear Deformation
• Change in shape of an object due to the
application of equal parallel forces in opposite
directions.
Shear Stress ( )
Shear Strain (  )
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
x
 
Lo
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 )
shear  stress
Shear  Modulus 
shear  strain
G


http://octavia.ce.washington.edu/DrLayer/Exercises/L-MModel_files/elastic.jpg
http://www.geoforum.com/knowledge/texts/broms/images/47.gif
Hooke’s Law
• Stress is directly proportional to strain
– Slope equal to Young’s modulus or Shear Modulus
depending on measurement
• Elastic region
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
Stress Strain Curve
Proportionality 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
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
Elastic Properties of Selected Engineering Materials
Material
Density
(kg/m3)
Young's
Modulus
109 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 fir
Data from Table 13-1, Halliday, Resnick, Walker, 5th Ed. Extended.
Applications:
0
0
-20
-20
-40
-40
Stress (MPa)
Stress (MPa)
NONLINEAR CONSTITUTIVE BEHAVIOR OF PZT
-60
-80
-100
-60
-80
-100
-120
-120
-140
-140
-160
-0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1
-160
-0.8
Strain (%)
0
-0.6
-0.4
-0.2
Strain (%)
0
Setup
Force
Alumina Spacer
Brass Electrode
Grafoil
PZT Sample
(electroded surface)
Oil Bath
Strain Gages
Grafoil
Brass Electrode
Loading Fixture
Alumina Block
Electrode
Specimen
Oil Bath Fixture
Tilt Table
Alumina Spacer
Force
Adjustable Screws
Setup
20 KV Power
Amplifier
Load
Cell
Function
Generator
Force
Electrodes
+ 2000
+
-
+ 5V
-
Electrometer
C=10 
F
+
Vout
-
D = Q/A = C(Vout)/A
ground
Strain Gage Amplifier
ADC
Macroscopic
Depth-Sensing
Indentation Tests
Will 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 macroscale region.
• Determine whether DSI testing is independent
of scale.
Background
•Hardness (H) of a specimen is calculated from:
Pmax
H
Ac
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
Er 

2
S
Ac
where S is the stiffness which is dependent on the
unloading curve.
(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
Piston
Wing
Upper jig
DWS
Indenter
Base jig
Specimen
Independent Displacement Measurement
• Used a displacement
capacitance sensor
• Decouples displacement
from the load cell
eliminating frame
compliance.
Capacitance
gauge
Sensor
Target
Signal process
and show
FIG. 4. Measurement schematic of
the capacitance displacement sensor
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
Load ()
200
200N
150N
100N
50N
25N
10N
5N
2.5N
150
100
50
0
0
20
40
60
Depth (m)
80
100
Hardness and Modulus Results
Peak Load (N)
200
150
100
50
25
10
5
2.5
Hardness (GPa)
1.09
1.06
1.18
1.14
1.12
1.16
1.19
1.18
Modulus (GPa)
68.4
67.8
73.0
64.9
65.2
69.4
70.2
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.