damped harmonicoscillator

8/7/2019 Damped harmonicoscillator

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Examine the Damped Oscillation of spring mass system using webcam & Matlab.

OBJECT:

Examine the amplitude of freely oscillating and under

damped harmonic oscillator of a spring mass system by

using Webcam & Mat lab.


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APPARATUS:

The apparatus used in this experiment was:

y Stand

y Spring mass system

y Set of masses

y Webcam

y Beaker

y PC

y Matlab

Stand Spring Mass

Set of masses Beaker


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PC Webcam

Matlab


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APPARATUS SETUP:

The schematic diagram of the apparatus setup is shown in this figure,


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THEORY:

Vibratory Motion:

To and fro motion of a body about its mean position is known as vibratory motion.

Restoring Force:

The force that brings or tends to bring the oscillatory body toward its mean positionis known as restoring force.

Fr is the Restoring force in this figure. It bring body back to it mean position orequilibrium position.


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Characteristics ofharmonic oscillator:

y Vibration:

One Complete round trip is known as vibration.

This figure is showing one vibration. As the body move from it position, completesone complete trip mean its one vibration.

y Time period:

Time requires to complete one vibration is known as Time period. And is given by

formula.

T=

T= 2

These equations demonstrate that period and frequency are independent of amplitude & the

initial phase of motion.


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y Frequency:

Frequency is the number of occurrences of a repeating event per unit time.

= 2f

f =

(5)

Inserting values of in equation (5)

f =

It will give us the frequency of body having vibratory motion.

Simpleharmonic motion:

It is defined as:

The acceleration causes the motion of a particle or object is directly proportional and inopposition to its displacement from its equilibrium position

Mathematically

a(t) x(t)

a(t) = kx(t)

Where,

a= acceleration; x= displacement (amplitude); k= constant of proportionality

Now consider the x-component of motion with time:


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X (t)=x0 cos (t + ) (1)

Where,

t= time; =phase

The velocity and acceleration as a function of time can be found as:

Differentiating equation (1) w.r.t t(Time)

v(t) =

V(t)= -x0 sin (t + )

-ive sign shows the direction of velocity, it says that direction of Velocity will beopposite that of displacement.

Now,

a(t) = (

) =

= - 2x0 cos (t + )

x(t)= x0 cos (t + )

Therefore,

a (t) = - 2x(t)

The above equation describes SHM.

Conditions for a body to show SHM are:

y System should have restoring force.y System should possess inertia.


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PROCEDURE:

Pull the mass down by about by few centimeters in water and then

release it so that it start moving in w ater. W e will watch this and catch this

proceeding with help of webcam. And then study the behavior using

MA TLAB . W e will use the following Com man ds.

(In the script, Mat lab commands are preceded with the symbol >>.)

y STEP 1 :

>> W ebcam

The command above act ivates a l ive video preview window and we can

see the ma ss osci llating on our comp uter screen. Re adjust the app aratus

to obtain a vertically oscillating m ass o n the co mp uter screen.

y STEP 2:

In order to acquire frame s, we need to type

>> start(vid);

This wil l acquire 300 frames. Observe the parameters updating

themselves in the bottom of the video preview window.

y STEP 3:

In order to store these frame s in the me mory, we need to type

>> savedata

This wi l l create two arrays named frames and t ime and store the acquired


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frame s and their t ime stamp ed values.

y STEP 4:

>> size(frames)

y STEP 5:

The f irst frame can be viewed by typing the comm and

>> firstframe= frame s(: , : , : , 1)

>> Im view(firstframe );

y STEP 6:

Next we need to crop the acquired frames so that the irrelevant

information is f i l tered out. Mat lab has a built- in command known as

imcrop which crops an image to a specified rectangle. W e can crop the

first fram e b y typing

>> rect=[xm in y min (xmax- xm in) (ym ax-y min) ]

y STEP 7:

In order to crop the defined rect we will use the following com ma nd

>> cropregion= im crop(firstfram e,rect);

y STEP 8:

In order to crop all the 30 0 frame s in one g o we wil l use a for loop

for i= 1:300


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region s(:,:,:,i)= imcrop (frame s(:,:,:,i), rect);

end

Ma t lab saves this cropped array with the na me regions.

y STEP 9:

>> imageprocess

Will process all the cropped images in regions and wil l calculate the

center of ma ss of the oscillating m ass.

y STEP 10:

In order to ca lculate the displaceme nt coordinates for the center of ma ss , use the

command

>> masscenter

y STEP 11

The x and y coordinates of the center of mass through the 300 frames

can be viewed by typing

>> x= center(:,1)

>> y= center(:,2)

y STEP 12

Now the time can be defined by

>> t ime= t ime (1:300);


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y STEP 13

For plotting of graph of x coordinates against t ime we wil l use the

following comma nd

>> figure; plot(time,x);

Similarly for y coordinates

>> Figure; plot(time,y);

Ou r graph plot against t ime and Am plitude x is ;

4 6 8 10 12 14 1688

88.5

89

89.5

90

90.5

91

91.5

92

92.5

T ime " t "

Am

plitude

"x"

Damp ing


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damped harmonicoscillator

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