Inverse Laplace of mass spring damper system.

[jabwemet]

hello All, I have a problem finding the inverse laplace of mass spring damper system.

Here are my equations. Please help me in finding the solution of x(t)

Actually, I am trying to find the value of x(t) for an underdamped condition.


(1/M)/(S^2+2ζω_n+〖ω_n〗^2 ) = (c_1 (S+a)+c_2 (ω))/((〖s+a)〗^2+ω^2 )

Now, If I apply the inverse laplace transform, I know that I need to get the value of x(t) as
e^(-at) (c_1 cosωt+c_2 sinωt)
So, If I can get to know the values of a and ω in (c_1 (S+a)+c_2 (ω))/((〖s+a)〗^2+ω^2 ) I can apply inverse laplace and find x(t).
Can someone help me in finding the values of a and ω

 

[Jhero]

?

Thank you for reaching out for help with your problem. As a scientist specialized in dynamics and control systems, I may be able to provide some guidance in finding the solution to your mass spring damper system.

First, let's take a closer look at the equations you have provided. The equation on the left side is the transfer function for your system, and the one on the right side is the partial fraction expansion of the transfer function. The terms a and ω are known as the natural frequency and the damping ratio, respectively.

To find the values of a and ω, we need to first determine the values of M and ζω_n. The value of M can be obtained from the physical properties of your mass spring damper system, while ζω_n can be calculated using the damping ratio and natural frequency values.

Once we have these values, we can substitute them into the equation on the right side and solve for a and ω. This can be done using algebraic manipulation or by using a numerical method such as the method of least squares.

Once you have the values for a and ω, you can then apply the inverse Laplace transform to find the solution for x(t), as you have correctly noted in your post. It is important to keep in mind that the solution will depend on the values of a and ω, so make sure to double-check your calculations before using the solution.

I hope this helps in finding the solution to your problem. Let me know if you need any further assistance. Good luck!