symbolic solve coupled second order differential equations

[ptc-5782298]

Dear all,

I have posted a similar question a while ago, and the general consensus was that it is not possible, at least with Mathcad, to symbolic solve a system of coupled second order different equation.

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 same result as in Eq. 5. So, I was wondering if someone could help me with this. Although it would be perfect if it is somehow possible with Mathcad, but any solution to this problem would be of interest.

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

Newin

[HarveyHensley]

There is an analytical solution to your problem, but Mathcad cannot fully obtain the symbolic expressions needed for the eigenvectors and eigenvalues. However, even if it could find them, the matrix form would not be very useful, I suspect.

You said that you didn't get a numerical solution that agreed with eq 5. What were the values of all of the parameters and the initial values used. Please attach a worksheet with your attempt. I wonder if eq 5 is not correct?

[HarveyHensley]

Update:

Mathcad was able to determine the symbolic eigenvector matrix after all. I must have made an error in defining the problem the first time. However, the result was too large to display, so it can't be used for your purpose of seeing an entire symbolic solution.

As an alternative to a symbolic solution, you could obtain a parametric solution to the ODE system which would allow you to show how the plots change with different values of the parameters.

[Werner_E]

It was often written that Mathcad symbolics has quite a lot of vweak points and often needs (a lot of ) help.

At least we can verify equation (3), not sure about equation (5) .

Look if the attached files help.

BTW, what are you trying to do? Why do you need a symbolic solution?

[jroth]

So, this may or may not help with your case.

I wasn't able to get mathcad to solve a system symbolically (I did try for a couple hours in my homework), but I was however able to (in matrix form) graph the system using modal superposition. It worked out quite nicely. All you have to do, is write symbolically the ending equation and apply the initial conditions.

Attached is my worksheet along with a PDF of the worksheet incase you are unable to open it. Hope it helps =).

[ptc-5782298]

Thank you all,

Harvey, numeric solution works well and I can obtain the results. Symbolic solution is, however, not possible.

Werner, many thanks for your worksheet. The reason I would like to have symbolic result is for my presentation, which it would be better to describe with an equation and not just showing graphs, and also that I saw people writting out equations like Eq. 5 in the attached copy above, so I was curious to obtain this myself. Anyway, given the time constraint, I might just stick with the numeric results for the time being.

Jake, many thanks for the worksheet. I shall have a look at it soon!