Re: Spreadsheet Analysis of Rotating Stick

[brian whatcott <inet@INTELLISYS.NET>]

At 12:08 11/28/99 -0500, you wrote:
> Brian Whatcott writes:
>
> > But in system simulation one is advised to start with the highest
> > difference and accumulate a sum over successive time slice values...

> ...calculate these
> values using a quadratic approximation as such:
>
> ao = Acos(q0)
> q1 = qo+wo(dt)+0.5ao(dt)^2
> w1 = wo+ao(dt)
> a1 = Acos(q1)
...
> q(i) = 4t^2Acos(q(i-2))+2q(i-2)-q(i-4)
(q(i) depends on q(i-2) and q(i-4))
...
> Bob Carlson
Although this is evidently a step in the right direction I was still
a little uncomfortable.
If you model a meter stick pivoted at one end, starting from the upper
 vertical, I would have thought you could set the initial conditions
as angular velocity = a small starting rate
angular position = a small displacement to avoid the quasi stable upright.

! initial conditions
ao = Acos(q0)
w0 = 1E-6
q0 = 1E-6

!real time model, iterate from here.
a1 = Acos(q0)
w1 = wo+a1(dt)
q1 = qo+w1(dt)

I seem to recall that with dt set small enough the approximations
built into this very simple approach are not onerous. But you have
a spread sheet ready to go. So I expect you can soon tell if this
has sacrificed accuracy.   It is true that in particularly crucial
numerical integrations, some form of predictor/corrector or other
elaboration like Newton-Raphson is preferred for the
summastion/integration.

This expression...
> q(i) = 4t^2Acos(q(i-2))+2q(i-2)-q(i-4)
(q(i) depends on q(i-2) and q(i-4))

...was evocative of a sample digital signal processing scheme
for filtering in the frequency domain. If it discards i-1 and i-3
there is a curve fitting loss, it seems to me. I wish I were better
equiped to discuss this in more detail.





brian whatcott <inet@intellisys.net>
Altus OK