Complex mathematic problem applied to engineering. How to solve it?

Question

I have a mathematical problem. The matrix describe in a well proposed math form a pratical but very complex engineering problem.

As is not possible describe here, in words, the details of the problem are in this image .GIF file:

http://img223.imageshack.us/img223/3382/engineeringproblemoi9.gif


Can you suggest how to solve it?

Answer 1

You have 5 equations in 8 variables, and 1 constraint equation. This is a simple exercise in determining a row-ranked solution. You should attempt to get rank-6 solutions.



1. Get the RHS variables to the LHS and incorporate it into your problem: i,e, define X = [x,y,z,n1,n2,n3,n4,n5]'.



2. Add the constraint equation to the set of equations, taking

the number of equations to 6: n1+n2+..+n5 = 1.



3. Form a matrix equation, AX = b, where X is defined in step 1. You will have 6 equations and 8 unknowns. Vector b will have zeros in all entries except one, where it is 1.



4. Choose any 2 variables (at random) and assign some values to them. Move the resulting computed columns to the RHS and add (subtract) to b.



5. Solve the resulting matrix by either LU or a matrix inverse.



6. If unsolvable, then choose some other pair of variables and/or assign different values till a solution is obtained.

Answer 2

This is a set of 5 equations in 8 unknowns. As such, it has an infinite number of solutions. However, given the constraints that are stated, and an assumption that all Ni are non-negative, one can find a bounded solution set.



If you start with a guess that all Ni are the same (Ni = 0.2), then you can reduce the problem to 5 equations in 3 unknowns, and find a solution based on any 3 equations. Once you have done that, you need to start varying the parameters Ni to see if you can find a set of Ni that satisfies all 5 equations, within the limits specified for x, y, and z.



There's probably a better way, but that's the best I can come up with.

Answer 3

Try apply a loan from the local bank so that you can open a small company with two motor powed boats. You can sale the ticket for traffic and hire an assistant. You can be a small boss earning some money. When you have stored up enough money you can build a pedestrian bridge for the town people.

Answer 4

You can solve the problem using the singular value decomposition method. The infinite set of solutions can be expressed in terms of 2 arbitrary constants a1 and a2 as follows:



{x, y, z, N1, N2, N3, N4, N5 } = { -0.001381, -0.0005271 0.005574, 0.1788, 0.2012, 0.2201, 0.1861, 0.2138} +

a1 {-0.01208, -0.03183, 0.03777, -0.06324, -0.1049 0.6163, 0.277, -0.7252} +

a2 { 0, 0.04739, -0.01251, -0.206, -0.711,

-0.0004926, 0.5788, 0.3387}



You can easily verify the above solution by substituting them into the original equations.

Answer 5

Don't know; but it's not obvious that the simultaneous equations are soluble - I mean if you add three of them together you are left with 3 equations and 4 unknowns.