Hi,
I am working on a method to evaluate a distributed parameter beam system to determine its natural frequencies. To do so, I need to find the exponential matrix of a six by six symbolic matrix, and then integrate the individual entries using the symbol as the variable from 0 to 1.
So I start with this matrix:
bigPhis = {{0, 1, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0}, {0, 0, 0, 1, 0,
0}, {0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 1}, {c1, 0, c2, 0,
alpha^2, 0}};
where c1 is a complex number, and c2 and alpha are both real. I am able to get the exponential matrix output (point for Mathematica over Matlab!) using
bigPsixs = MatrixExp[bigPhis*(x - xo)]
where x is a symbol and xo is a value of 0.5. I take the first entry of this matrix, which gives me an output that starts like this:
{RootSum[(8.29521*10^6 -
55.5556 I) - (9.95425*10^7 - 666.667 I) x + (4.97712*10^8 -
3333.33 I) x^2 - (1.32723*10^9 -
8888.89 I) x^3 + (1.99085*10^9 -
13333.3 I) x^4 - (1.59268*10^9 -
10666.7 I) x^5 + (5.30893*10^8 - 3555.56 I) x^6 -
331809. #1^2 + 3.98171*10^6 x #1^2 - 1.99086*10^7 x^2 #1^2 +
5.30895*10^7 x^3 #1^2 - 7.96342*10^7 x^4 #1^2 + ... ... ...
(continues)
So I have read the articles about #1 being a placeholder for the first root solution, which I kind of sort of understand, but the next step for this problem is:
om11 = Integrate[Exp[alpha*(x - xo)]*bigPsixs[[1, 1]], {x, 0, xo}] + Integrate[Exp[alpha*(xo - x)]*bigPsixs[[1, 1]], {x, xo, 1}];
So you can see that I still need x in the bigPsixs input to be able to integrate it. Does anyone have any tips or ideas of how to make this work? I don't really know how to solve for #1 numerically and still keep x in the equation for the next step integration, but maybe I am confused on the Mathematica output... please help?
Thanks,
Kaitlin