predicting vibration frequencies of the diamond wafer
DESCRIPTION
Predicting Vibration Frequencies of the Diamond Wafer. Presented by: Austin Antoniou Site: Nuclear Physics Mentor: Dr. Richard Jones. Modes of oscillation. Oscillation on the Z axis Rotation About X Axis Rotation About Y Axis Rotation About Z Axis. y. x. z. Model of the diamond. - PowerPoint PPT PresentationTRANSCRIPT
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Predicting Vibration Frequencies of the Diamond Wafer
Presented by: Austin AntoniouSite: Nuclear PhysicsMentor: Dr. Richard Jones
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Modes of oscillation
x
y
z
Oscillation on the Z axis
Rotation About X Axis
Rotation About Y Axis
Rotation About Z Axis
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Model of the diamond
Wire length=w
Diamond length=aDistance between wire joints=s Mass of
diamond=mVelocity of a wave traveling on wire=v
Tension Forces
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Oscillation on the Z axis
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Diagram of Z Axis Oscillation
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Equation Used for Z axis Oscillation(continued)
z(y,t)=A+sin(ky-ωt)+A-sin(ky-ωt)
z(0,t)=A+sin(-ωt)+A-sin(ωt)
z(0,t)=(-A++A-)sin(ωt)z(y,t)=A[sin(ky)cos(ωt)+cos(ky)sin(ωt)+
sin(ky)cos(ωt)-cos(ky)sin]z(y,t)=2Asin(ky)cos(ωt)Where 2Asin(kw)=z0, A=z0/(2sin(kw))
z’(w,t)=z0kcot(kw)cos(ωt)z’(w,t)=kcot(kw)
z’(w,t)=kcot(kw)
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Equation Used for Z Axis Oscillation
(Continued)
z’(w,t)=kcot(kw)
This equation can be related to the restoring force:
FR= -kFTcot(kw)z
FR=(d2z/dt2)m=-kFTcot(kw)z(d2z/dbt2)+[-(kFTcot(kw)/m]z=0
The generic equation used for ω is ω2=(K/m)
In this case, K is equal to
Therefore,
cot(kw)z-kFmF Tdtzd
R 2
2
0zmcot(kw)kF
dtzd T2
2
mK2
w)cot(FT vv
m
w)cot(F vT2
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Graph of ω2= ωcot(ω)
ω (rad/s)
f(ω
) (
Rad
2/s
2)
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Rotation about the X axis
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Diagram of X Axis Rotation
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Rotation About the Y Axis
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Diagram of Y Axis Rotation
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Rotation about the Z axis
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Diagram of Z Axis Rotation
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Predicted frequencies of the diamond wafer
Oscillation on Z Axis
Rotation About X Axis
Rotation About Y Axis
Rotation About Z Axis
45.1 78 55.5 2560
5060 5060 506010100 10100 10100
Frequencies(Hz) (Hz)(Hz)(Hz)
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ω2=ωcot(ω)
ω (rad/s)
f(ω
) (
Rad
2/s
2)
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What should we make of these calculations?
The higher-frequency oscillations are nearly identicalMathematically, this is because the slopes of the graphs of the graphs of ω only change slightly from mode to modePhysically, this is because the system has natural frequencies due to properties such as the length of the wires
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Questions?