French Chap2

8/3/2019 French Chap2

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Questions?

Lecture 25Superposition of periodic motions


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Superposed vibrations in 1d

In linear systems resultant of more than oneharmonic displacements is the sum of individualvibrations

x1=A

1cos( t+

1) x

2=A

2cos( t+

2)

Same frequency

x=x1+x

2=Acos( t+)


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Complex exponential formalism

x1=A

1cos( t+

1)=(A

1e

i ( t+1))

x2=A2cos

( t+2)=(A2 ei ( t+

2)

)

x=x1+x

2=(A

1e

i (1

t+1)+A

2e

i (2

t+2))

z=z1+z

2=e i (1 t+1)(A

1+A

2e

i (2

1))


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z

=e

i (1t+

1)

(A

1

+A

2e

i (2

1)

)

z1

z2

Phase differencebetween the

vibrations

A


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Superposition of vibrations different frequencies

x1=A

1cos(

1t) x2=A2 cos(2 t)


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Beats

x=x1+x2=A(cos

(1 t)+cos

(2 t))

Equal amplitudes, different frequencies

= 2A cos

1

2

2t cos

1+

2

2t


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Beats

x=x1+x

2=A(cos(

1t)+cos(

2t))

Equal amplitudes, different frequencies

=2A cos

1

2

2t cos

1+

2

2t


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Questions?

Lecture 26Superposition of periodic motions


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Questions?

Lecture 27Vibrations at right angles


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Vibrations at right angles

x=A1cos(

1t+

1) y=A2cos(2 t+2)

1=2

Possible cases

12


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Perpendicular motion samefrequencies

x=A1cos( t) y=A2 cos( t+)

Initial phase difference


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A1=A

2=1

=0 =2

=4

The motion starts atthe dark red spot andgoes towards the lightred spot, alwaysfollowing the blue line


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x=cos( t)

y=cos( t+4)


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x=cos( t)

y=cos( t+4 )


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Questions?

Lecture 28Vibrations at right angles

( different frequencies )


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Perpendicular motion differentfrequencies Lissajous figures

x=cos( t)

y=cos(2 t+4)


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x=cos( t)

y=cos(2 t+4) Step 1

Identify the initial points

on the circles.


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x=cos( t)

y=cos(2 t+4) Step 2

Draw a horizontal line

through the point onthe circle for y-axismotion

Draw a vertical linethrough the point onthe circle for x-axismotion

The intersection of thelines will give a pointon the Lissajous figure.


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x=cos( t)

y=cos(2 t+4) Step 3

Find the positions on

the circles after time dt

Repeat Step 2

Join the points foundfor the Lissajous figure


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x=cos( t)

y=cos(2 t+4) Step 4

Repeat Step 3, i.e.,

Find the positions onthe circles after time2 dt

Repeat Step 2

Join the points foundfor the Lissajous figure

Note: Angle covered bythe point on the circleswill be proportional to

the correspondingan ular s eeds.


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x=cos( t)

y=cos(2 t+4)

Repeat the last stepand you will constructthe Lissajous figure.

Any checkpoint?


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x=cos( t)

y=cos(2 t+4)

Checks:

After completing the

Lissajous figure :

Count how many timesthe curve reaches anextreme value of x (ory) to complete thecurve.

Here Max[x] is reached

once

Max[y] is reached twice

Notice the ratio of thefrequencies.


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Homework

Use this check on all the figures given in the textbook.


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Questions?

Lecture 29Coupled Oscillators


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Coupled oscillators

Two independent pendula are connected by aspring

Set one of them in motion

Observe the exchange of energy between thetwo


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Coupled oscillators

Restoring forces on each massF

res A=m02xA+k(xAxB)

Fres B=m0

2xBk(xAxB)

Due to motion ofthe pendulum

Due to the spring

Note the sign


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Equations of motion

m xA+m02xA+k(xAxB)=0

m

xB+m

0

2x

Bk(x

Ax

B)=0

q1+

0

2q

1=0 q1=xA+xB

q2+ '2 q

2=0 q2=xAxB '

2=02+2

k

m

Add

Subtract

Solution ??


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q1=xA+xB=Ccos(0 t+1)

q2=xAxB=D cos( ' t+2)

xA=[Ccos(0 t+1)+D cos( ' t+2)]/2

xB=[Ccos(0 t+1)D cos( ' t+2)]/2

C ,1, D ,2 : from the initial conditions on the masses


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Special cases

Initially both masses are given the same displacementfrom equilibrium (in the same direction) and releasedfrom rest

Initially both masses are given the same displacementfrom equilibrium (in opposite directions) and releasedfrom rest

One of the masses (say A) is displaced and releasedfrom rest (Exercise : Find Cand D).

D=0q2=xAxB=0 at t=0

C=0q1=xA+xB=0 at t=0

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French Chap2