eating
03-28-2008, 02:29 AM
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 has to be in this case. 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.
ARE THERE WAYS TO SOLVE THIS TYPE OF PROBLEM????
I WOULD APPRECIATE ANY COMMENTS. THANK YOU.
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 has to be in this case. 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.
ARE THERE WAYS TO SOLVE THIS TYPE OF PROBLEM????
I WOULD APPRECIATE ANY COMMENTS. THANK YOU.