How to solve general engineering equation (Duhammel Integral)

[Ronankeating]

Dear All,

I'll need your guidance on how to solve the equation generally encountered in engineering equations, which is mainly in that form M*X''(t)+C*X'(t)+K*X(t) = F(t), where M, C, K and F(t) are mass, damp, stiffness and external force matrices respectively. Actually those equation may come in many flavours ranging from the absence of any terms on left or right side. Additionally the solution that we are seeking may not be a function of time (transient) as conversely I've formed above. Generally it's been called Duhamel integral and may have or not inclusion of the eigenvalues, so what I'm asking is introductory info and any suggestion on the good book related to that topic.

Also I'd like to take the suggestion from you for the good programming library e.g. Lapack, Arpack, Slepc, Petsc etc.. which solves the fore mentioned equation with different type of variation of LHS and RHS.

Your comment will be appreciated.

Regards,

 

[chiro]

Hey RonanKeating.

What kind of platform were you considering using? MATLAB? Native FORTRAN?

 

[Ronankeating]

In general I'm comfortable with Fortran.

 

[Ronankeating]

Your help will be appreciated, Dear All

 

[chiro]

I've heard of people talk about LAPACK libraries which you might want to check out:

http://www.netlib.org/lapack/

 

[Ronankeating]

Sure I did, I already have the lapack library, and also the ArPack as well. But I couldn't find any subroutine/function which directly addresses fore mentioned equation solution. Probably there are some preliminary process that I need to do before calling any of the subroutines but I don't have hint of what possibly are those( factorization, decompositioning etc.. ) so I really appreciate If any summarizes curtly how its been solved.

Regards,

 

[AlephZero]

Duhamel's integral is the analytical solution to a linear single-degree-of-freedom dynamics equation. As such it is closely related to Laplace transforms, etc.

If you want to use it numerically you have to do two things:

1. Convert your equations of motion into of uncoupled SDOF equations. One way to do that is to use modal coordinates.
2. Evaluate the integrals in Duhamel's solution numerically. The practical details of that depend very much on what are the external forces on the system.

I wouldn't recommend this as a practical numerical method for general dynamics problems. One issue is that you may need to retain all the modal components in the model to get the correct results (i.e. you can't ignore the high frequency modes) and this can cause problems evaluating the Duhamel integrals.

One feature of the method is that you can find the response of the system at any arbitrary time - but unless you know beforehand when the "interesting" times will occur, that isn't of much practical use.

 

[Ronankeating]

Thank you very much indeed AlephZero,

Regarding the book that I'm reading it mentions that if the eigen values and eigenvectors are known then using mode superposition method on can reduce and decouple the forementioned simulataneous motion equation into a set of ordinary differential equation in single unknowns. I think that was what you are suggesting at and conforms with the books suggestion. But how am I supposed actually to proceed further to implement that, basically in an outline?

Any good books that you can suggest for that ??

Regards,