- #1
graffy76
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Hello,
I have a little project I'm playing with that involves calculating a series of forces and summing them to define the motion of an object (in this case, a walking pedestrian).
The force equation is modeled on time-dependent vectors and scalars and is solved using Gear's predictor-corrector numerical method.
The Gear predictor portion of the method is described thus:
x[n+1] = x + hv + ((h^2/2)a + ((h^3)/6)A + ((h^4)/24)B
with v[n+1], a[n+1], A[n+1], and B[n+1] defined as the first through fourth derivatives of the above.
The force equation (= dv/dt) has time-dependent displacement and velocity vectors and scalars, but is not explicitly defined in terms of time. So, I obviously need to express the force equation in terms of time (or in this case, the time step, h).
My only thought would be to re-write the displacement and velocity components explicitly in terms of Newton's equations for curvlinear motion.
Thus:
if F(t) = v * x(t),
then F(t) = v * (x[n] + v[n]h + 1/2*a[n](t^2) ).
Such a rewriting would allow two derivatives of F(t) for the Gear model. By doing this, I think I'm just assuming that the motion of the pedestrians in time step h ( = 0.1 seconds in this case) is curvlinear - the same assumption that is taken to establish the Gear equations in the first place.
In any case, I haven't dealt with this level of math or physics since college (going on ten years now), and I'm a bit rusty.
Am I on track, or is this the wrong way to go about getting the two extra derivatives I need?
If you want more information about the specific problem, see below:
http://public.rz.fh-wolfenbuettel.de/~apel/files/thesis.pdf"
see pages 21 - 29. The force equation is summarized on page 26 and the Gear predictor portion is summarized on page 28.
Any help is much appreciated.
Thanks.
Joel
I have a little project I'm playing with that involves calculating a series of forces and summing them to define the motion of an object (in this case, a walking pedestrian).
The force equation is modeled on time-dependent vectors and scalars and is solved using Gear's predictor-corrector numerical method.
The Gear predictor portion of the method is described thus:
x[n+1] = x + hv + ((h^2/2)a + ((h^3)/6)A + ((h^4)/24)B
with v[n+1], a[n+1], A[n+1], and B[n+1] defined as the first through fourth derivatives of the above.
The force equation (= dv/dt) has time-dependent displacement and velocity vectors and scalars, but is not explicitly defined in terms of time. So, I obviously need to express the force equation in terms of time (or in this case, the time step, h).
My only thought would be to re-write the displacement and velocity components explicitly in terms of Newton's equations for curvlinear motion.
Thus:
if F(t) = v * x(t),
then F(t) = v * (x[n] + v[n]h + 1/2*a[n](t^2) ).
Such a rewriting would allow two derivatives of F(t) for the Gear model. By doing this, I think I'm just assuming that the motion of the pedestrians in time step h ( = 0.1 seconds in this case) is curvlinear - the same assumption that is taken to establish the Gear equations in the first place.
In any case, I haven't dealt with this level of math or physics since college (going on ten years now), and I'm a bit rusty.
Am I on track, or is this the wrong way to go about getting the two extra derivatives I need?
If you want more information about the specific problem, see below:
http://public.rz.fh-wolfenbuettel.de/~apel/files/thesis.pdf"
see pages 21 - 29. The force equation is summarized on page 26 and the Gear predictor portion is summarized on page 28.
Any help is much appreciated.
Thanks.
Joel
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