puchm@cutmcvax.cutmcvax.cs.curtin.edu.au (RichardPuchmayer) (04/23/91)
Dear netters,
Having received advice as to which books to look in, I have
looked in those books. I now know a bit about Lagrangian
dynamics:
L = T - U
where : T = kinetic energy of the system
U = potential -- " --
let d = partial derivative
D = total derivative
x = a generalized coordinate
x. = time derivative of a generalized coordinate
then:
dL D dL
-- - --(--) = 0
dx Dt dx.
For ALL the examples I have looked at the derivation of the
kinetic energy is not too difficult, but the formulation of
the potential energy seems to be very problem dependent.
Is there a general method by which to derive the potential
energy for a system?
Also:
For a double pendulum :
mass m1 is connected to a pivot by a massless rod of
length l1, mass m2 is connected to mass m1 (via a
pivot) by a massless rod of length l2. The angle
between the vertical and rod(l1) is th1 and the angle
between the vertical and rod(l2) is th2. The only forces
acting on the masses are gravity and any constraint
forces through the rods. Any variable with a dot '.'
after it denotes the first time derivative of that
variable. Two dots '..' denote the second time
derivative.
T1 = 1/2 * m1 * l1 * th1.
T2 = 1/2 * m2 * (l1 * th1. + l2 * th2.)
U1 = -m1 * g * l1 * cos(th1)
U2 = -m2 * g * l2 * cos(th2) * l1 * cos(th1)
Are the potentials correct ?
If YES, why ?
If NO, why and what are the correct ones?
The above questions should indicate a total lack of
understanding as to the formulation of the potentials.
PS: I've posted this to comp.simulation about a week ago but
there has been no traffic in that group. So I'll try this
one.
Thanks,
Richard.
Thanks,
Richard.
--
| Some of us are poets, some of us are not. | puchm@cutmcvax.cs.curtin.edu.au |
| Richard Puchmayer, Masters Student at | Sorry but I |
| Curtin University of Western Australia | don't know any other addresses |
| I know nothing, so can hold no opinions for myself or others.....:-) |weverka@boulder.colorado.edu (Robert T. Weverka) (04/24/91)
mail bounced so this gets posted... In article <puchm.672390379@cutmcvax> puchm@cutmcvax.cutmcvax.cs.curtin.edu.au (RichardPuchmayer) writes: > > T1 = 1/2 * m1 * l1 * th1. > T2 = 1/2 * m2 * (l1 * th1. + l2 * th2.) > > U1 = -m1 * g * l1 * cos(th1) > U2 = -m2 * g * l2 * cos(th2) * l1 * cos(th1) > > Are the potentials correct ? > If YES, why ? > If NO, why and what are the correct ones? > The above questions should indicate a total lack of > understanding as to the formulation of the potentials. > U = m * g * h so... you should have U2 = -m2 * g * ( l2 * cos(th2) + l1 * cos(th1) ) since the quantity in parenthesis is the height. For the kinetic T= 1/2 m v^2 so ... v1 = th1. * l v2 = l * ~th1. + l * ~th2. where ~ denotes vector quantity. Add these vectorally and compute the magnitude squared. Note: check your equations with dimensional analysis. this would have shown you the error you have in T1 and T2. have fun. -Ted