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to be continued...
to be continued...
Created by Greg Peterson
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Revision as of 10:44, 26 October 2009

Using the Laplace Transform to solve a spring mass system that is critically damped

Problem Statement

An 98 Newton weight is attached to a spring with a spring constant k of 40 N/m. The spring is stretched 4 m and rests at its equilibrium position. It is then released from rest with an initial upward velocity of 2 m/s. The system contains a damping force of 40 times the initial velocity.

Solution

Given

m=989.81

Spring Constant k=40

Damping Constant C=40

x(0)=0

x˙(0)=4

Standard equation: 

md2xdt2+Cdxdt+khx=0

Solving the problem

Therefore the equation representing this system is.

989.8d2xdt2=40x40dxdt

Now we put the equation in standard form

d2xdt2+4010dxdt+4010x=0


Now that we have the equation written in standard form we need to send it through the Laplace Transform.

[d2xdt2+4010dxdt+205x]

And we get the equation (after some substitution and simplification).

s2X(s)+4sX(s)+4X(s)=4

X(s)(s2+4s+4)=4


X(s)=4(s+2)2

Now that we have completed the Laplace Transform and solved for X(s) we must so an inverse Laplace Transform. 

1[4(s+2)2]

and we get

x(t)=4te2t

So there you have it the equation of a Critically Damped spring mass system.

Apply the Initial and Final Value Theorems to find the initial and final values

Initial Value Theorem
limssF(s)=f(0)
Final Value Theorem
lims0sF(s)=f()


Applying this to our problem

The Initial Value Theorem

limssX(s)=4(s+2)2

limssX(s)=4(+2)2=0


So as you can see the value for the initial position will be 0. Because the infinity in the denominator always makes the function tend toward zero.

Which makes sense because the system is initially in equilibrium. 

The Final Value Theorem

lims0sX(s)=4(s+2)2


lims0sX(s)=4(0+2)2=44

This shows the final value to be 44ft

Which appears to mean the system will be right below equilibrium after a long time. 

Bode Plot of the transfer function

Transfer Function

X(s)=4(s+2)2

to be continued...

Created by Greg Peterson

Checked by