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How to solve 2nd order ODE with matrix parameters in Matlab

  1. Dec 3, 2011 #1
    1. The problem statement, all variables and given/known data

    I have a frequency equation to solve for the displacement for a spring mass damper truss system, as seen below,

    [m]u''+[c]u'+[k]u=f(t),
    where m,c,k, are all matrices (2x2), and f(t) is a graph-defined forcing function. I am to use 3 nodes, using the central difference approximation method.

    I am to solve for the displacement and the time history of the displacement.

    2. Relevant equations

    What I have learned so far involved solving matrix equations using [A]{x}={b}

    3. The attempt at a solution
    I am just confused as to the basic concepts- I should have to be able to decouple this equation, because this is essentially trying to solve a space and time dependent problem.

    I am not sure how I can solve for this, and also doing that using the CDM- I would presume that I would somehow solve for the displacement at each time "point", but do i have to use the CDM to solve for the displacements as well? If that is the case, how can I solve the equation that involves matrices and not values?

    any help to get me started will be of immense help.

    thanks!
     
  2. jcsd
  3. Dec 4, 2011 #2

    AlephZero

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    There is something wrong there. If you have 3 nodes your matrices should be 3x3 not 2x2.

    Or do you mean "there are 3 nodes in the structure, but one of them is fixed, so it can be eliminated."

    Start by understanding how to use CDM for a system with just one variable, for example a mass oscillating on a spring.

    You get an equation that says [itex]x_{t+h}[/itex] = something involving [itex]x_t[/itex], [itex]x_{t-h}[/itex], f(t), and k c and m.

    For a system with several degrees of freedom, you use exactly the same equation, except x and f are vectors and k c and m are matrices. The only difference is that instead of dividing by a scalar value of m, you multiply by the inverse matrix [itex]m^{-1}[/itex].

    CDM is often used when the mass matrix m is diagonal, and inverting a diagonal matrix is easy.
     
  4. Dec 4, 2011 #3
    Hey alephzero,

    Thanks so much for your help. It definitely helped in me understanding the approach. I have been consulting some numerical methods handbooks to get myself familiar with when mck are single variables. It seems that I will have to build a tri-diagonal matrix and make it solve for a vector of x values(it seems at the very minimum, I will need x1, x2, xj, xn-2 xn-1 values, with j being the iteration i am solving for, and n-1 being the last iteration. If I am going for more iterations, however, then the xj portion will be lengthened, right?)


    However, I am still a bit puzzled in how I can solve this for a matrix of k and c values (since m will be multiplied through by the inverse method)- for example, in row 1 column 1 of the tridiagonal matrix, I will be multiplying by 2+delT*q_1, where q_1 stands for the c matrix corresponding at a certain time. So will my tridiagonal matrix involve another matrix multiplication?

    Also, your earlier assumption is correct in that my problem requires a 3 node system. Then the way to solve that is to iterative this code 3 times for the 3 different node and force locations, is that correct?

    thanks, as always.

    iqjump123
     
  5. Dec 5, 2011 #4

    AlephZero

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    You seem be getting confused about the the number of variables at different points in space, and the values of their displacement at diffferent times.

    If you have a single variable problem like a mass on a spring, the equation of motion equation is mu'' + cu' + ku = f(t)

    You convert that into an equation connecting the values of x at three different times, using difference approximations. Call the three times t-h, t, and t+h where h is the time step.

    You can approximate u'(t) by [u(t+h) - u(t-h)] / 2h
    and u''(t) by [u(t+h) - 2u(t) +u(t-h)] / h^2

    So the finite difference version of the equation of motion is

    m [u(t+h) - 2u(t) +u(t-h)] / h^2 + c [u(t+h) - u(t-h)] / 2h + k u(t) = f(t)
    or
    m [u(t+h) - 2u(t) +u(t-h)] + (ch/2) [u(t+h) - u(t-h)] + kh^2 u(t) = h^2 f(t)

    As you work through the solution, you already know u(t) and u(t-h) so you can rearrange this to find u(t+h)

    [m + ch/2] u(t+h) = (2m - kh^2) u(t) + (-m + ch/2) u(t-h) + h^2 f(t)

    If you are starting from t = 0, the initial conditons give you the values of u(0) and u'(0).
    To get the calculation started you need the values of u(0) and u(-h). One way to get u(-h) is use a Taylor series approximation and say
    u(-h) = u(0) - h u'(0).

    So the first time step of the CDM is to calculate u(h) from
    [m + ch/2] u(h) = (2m - kh^2) u(0) + (-m + ch/2) u(-h) + h^2 f(0)
    For the next time steps, you calculate
    [m + ch/2] u(2h) = (2m - kh^2) u(h) + (-m + ch/2) u(0) + h^2 f(h)
    [m + ch/2] u(3h) = (2m - kh^2) u(2h) + (-m + ch/2) u(h) + h^2 f(2h)
    etc.
    You don't need to make one big set of equations for all the time steps.

    If you have more than 1 variable (3 in your case), at each step you have to solve a 3x3 set of equations. The equation
    [m + ch/2] u(t+h) = (2m - kh^2) u(t) + (-m + ch/2) u(t-h) + h^2 f(t)
    is a matrix equation like Au(t+h) = b, where A = m + ch/2. You have to do the matrix multiplcations on the right hand side to calculate the vector b, and then solve the simultaneous equations for u(t+h).

    Note, in my first post I said you just needed to invert m. Sorry, I forgot you had the damping matrix c in your model so that wasn't quite right.
     
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