Symbolic solve coupled second order differential equations

[newin]

Dear all,

I have posted a similar question in another forum and the general consensus seems to suggest that it is not possible to symbolic solve a system of coupled second order different equation with damping (dissipation) and driving forces.

However, I have found in many papers and books writing out analytical formula of the solutions to the coupled equations. Attached is the system I am trying to solve. I have seen analytical formula even for one more mass. Although solving this sort of equation with two masses, no damping (dissipation) and with only one driving force is simple enough, even by hand, it is impossible to do the same for a system with two different damping constants and two driving forces. I have been trying for two weeks now, but could not figure out the solution.

Of course, I could get the numeric solution, but I could not get the result shown in Eq. 5. So, I was wondering if someone could help me with solve this analytically. Any solution to this problem would be of interest.

Any help would be gratefully appreciated! Many thanks in advance.



Newin

 

[the_wolfman]

I don't see why you can't. The algebra might get a little messy but the solution method is pretty straight forward.

Start by Laplace transforming each equation. This will give you a algebraic system of equations of the form Ax=b.
Solve this equation using standard linear algebra techniques to invert A.
Finally using an inverse Laplace transform on x_i to get x_i(t).