# Forced response of a mass-spring-damper system

1. May 12, 2009

### JolileChat

1. The problem statement, all variables and given/known data

I need to find the forced response of a mass-spring-damper system by using Laplace Transform.

2. Relevant equations

Equation of motion:
$$\ddot{x} + 2 \zeta \omega_n \dot{x} + {\omega_n}^2 x = \frac{F}{m} cos \left( \omega t \right)$$

3. The attempt at a solution

At the Laplace domain, we have:

$$X \left( S \right) = \frac{F/m}{S^2+ 2 \zeta \omega_n S + {\omega_n}^2} \frac{S}{S^2+\omega^2}$$

After this point, I try to make partial fractions, but I must be missing something. It results in a page full of algebra, and it does not look like the correct solution.

Please, could you help me on figuring out what I should do?

Last edited: May 12, 2009
2. May 12, 2009

### nickjer

You could complete the squares in the denominator of the first term and then use the convolution theorem. Or use partial fractions on the first term and then use the convolution theorem.

3. May 12, 2009

### JolileChat

You mean doing the convolution between the impulse response and the cosine forcing function?

I'd like to solve the problem in the Laplace domain, then apply the inverse transform to get the time response.

4. May 12, 2009

### nickjer

You did solve it in the Laplace domain, assuming that equation you gave in the first post is correct. Now you need to take the inverse transform of that equation. I just recommended using the convolution theorem when taking the inverse.

5. May 12, 2009

### nickjer

If you want to break it all up by partial fractions you can, but it will get very messy. Another trick would be taking the partial fraction of just the first term and then using the convolution theorem.