Mass - Spring - damper in Parallel

[kjay262]

Homework Statement

The problem is to determine the transfer function where force F is input and displacement x is output in the mass-spring-damper mechanism.

Homework Equations

Spring Force = kx [k:spring constant]
Damping Force = Cx [C:damping coefficient]
Force = (Mass)(acceleration)

The Attempt at a Solution

Attempt at solution is in picture. I am interested to know if I am following the correct methodology and if I am missing anything.

 

[kjay262]

****EDIT**** not in parallel, IN SERIES

 

[Chestermiller]

The differential equations look correct. That was the hard part. I haven't look over the part about the development of the transform, but that shouldn't have been a problem.

Chet

 

[kjay262]

If the input force is a unit impulse the transfer function is equal to the output displacement in the complex domain. I'm struggling to determine the x(t) that is the output in the time domain, which is basically the inverse laplace of the transfer function. I using the identity (attached below), however the answer introduces a complex value, am I missing something.

 

[kjay262]

Inverse Laplace Identity

Using the identity attached to determine the inverse laplace of the transfer function

 

[Chestermiller]

Multiply numerator and denominator by cs+k. See if this simplifies things.

Chet

 

[kjay262]

What if you have a situation where $\xi$ is greater than one (according to the identity image attached above) this would result in a complex number, yes? How would that be represented in graph?

 

[Chestermiller]

What if you have a situation where $\xi$ is greater than one (according to the identity image attached above) this would result in a complex number, yes? How would that be represented in graph?

Factor the denominator using the quadratic formula, and then resolve the transform into partial fractions, and you will then be able to answer your own question.