Differential equation: Spring/Mass system of driven motion with damping

  • #1

Homework Statement


A 32 pound weight stretches a spring 2 feet. The mass is then released from an initial position of 1 foot below the equilibrium position. The surrounding medium offers a damping force of 8 times the instantaneous velocity. Find the equation of motion if the mass is driven by an external force of 2cos(5t).


Homework Equations


F=kx
m=W/g

m[itex]\frac{d^{2}x}{dt^{2}}[/itex]+[itex]\beta[/itex][itex]\frac{dx}{dt}[/itex]+kx=f(t)


The Attempt at a Solution



I found that k=16[itex]\frac{lb}{ft}[/itex] and m=1 slug. This gets me the following equation:

[itex]\frac{d^{2}x}{dt^{2}}[/itex]+[itex]\beta[/itex][itex]\frac{dx}{dt}[/itex]+16x=2cos(5t)

I'm at a loss for how to determine [itex]\beta[/itex], which is the damping force of 8 times the instantaneous velocity. I don't know how to determine instantaneous velocity. I know that once I have [itex]\beta[/itex], I can just use a LaPlace transform to find x(t). But [itex]\beta[/itex] is my stumbling block right now.

As I was writing this, it occured to me that [itex]\frac{dx}{dt}[/itex]=instantaneous velocity and that would make [itex]\beta[/itex]=8. That in turn makes the problem very easy to solve. Am I correct in this thinking?

We kind of rushed through this application in class the other day. Thanks in advance for any help you're able to provide.
 

Answers and Replies

  • #2
rock.freak667
Homework Helper
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As I was writing this, it occured to me that [itex]\frac{dx}{dt}[/itex]=instantaneous velocity and that would make [itex]\beta[/itex]=8. That in turn makes the problem very easy to solve. Am I correct in this thinking?.

Yeah I would agree that β=8 here. Just make sure your initial conditions are correct with the proper signs.
 

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