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2nd-order coupled nonlinear DEs

  1. Mar 27, 2008 #1
    Hi, folks. I'm new to this forum.

    I joined because i am so desperate in solving a 2nd-order coupled nonlinear differential equation for the motion of a system similar to an inverted pendulum model. The equation, which i will call here EXACT governing eqn, is this

    A(s) s" + B s' + C(s, s') = Dsin(t)

    where s(t) = {u(t) v(t)}^T [2x1 vector].

    The 2x2 matrix A(s) matrix given as

    A(s) = [ h h^2 ]
    [ h^2 h^2 + u^2]

    where h is a constant and u=u(t).

    My problem is that direct integration can not proceed because A(s) does not have an inverse when u becomes zero.

    The equation can be simplified when s(t) are small and the governing equation reduces to third-order ODE in terms of v, i will call an APPROXIMATE governing eqn. u(t) can be solved as u = f(v, v').

    In more than 6 months, i have tried several methods:

    1. Method switching.
    Switch from solving the EXACT to APPROXIMATE equation, when u(t)~0 or becomes
    less than some numerical value say e, then switch bact to EXACT soln when u(t)
    becomes large. This method, however, caused significant discontinuities at e; and
    sometimes solution diverges when amplitude D is large.

    2. Least squares.
    I added a new equation to minimize the acceleration s"(t), i.e., added

    W(u) * s'(t+dt) = W(u) * s'(t), W(u)=1 for simplicity here

    and solve a system of 6-first order ODEs in terms of s(t+dt) and s'(t+dt) using
    principle of least squares.

    This EXACT solution is identical to the APPROXIMATE solution when amplitude D is
    small, which is has to be in this cases. But when amplitude D is large, the additional
    equation (minimizing acceleration s") may be too strong to cause significant changes
    in the governing equation, thereby altering the expected motion.


  2. jcsd
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