hasan@emx.utexas.edu (David A. Hasan) (04/26/91)
I have failed from within my newsreader and my mailer in trying to send mail directly to richard. So here is my response to his questions about potential energy of the double pendulum problem. Sorry to the rest of you for its length... In article <puchm.672390379@cutmcvax> you write: > > > T1 = 1/2 * m1 * l1 * th1. > T2 = 1/2 * m2 * (l1 * th1. + l2 * th2.) > I think that you meant to *square* the velocity terms in these expressions, right? > U1 = -m1 * g * l1 * cos(th1) > U2 = -m2 * g * l2 * cos(th2) * l1 * cos(th1) > > Are the potentials correct ? Except for the typo in U2 (the two l-cosine terms should be added together instead of multiplied), this looks ok. > 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. > "Potential energy" as a fundamental principal on which to base your analysis has some difficulties if the form of the potential is not obvious to start with. In fact, potential energy is another way of representing the "work" done by so-called conservative forces. In a uniform gravity field (situations where the effect of gravity is usefully modeled as a constant acceleration due to gravity -- g), the potential has the form U = mgh where h is the distance *above* some arbitrarily selected reference. ("above" = "opposed to gravity") The reasons why the reference can be selected arbitrarily are not really important (it is because the FORCE due to gravity is calculated by differentiating the potential, and in the process of differentiating all constants drop out), but is is *crucial* that you select ONE reference (sometimes called a "datum") and use it for all your derivations. Based on the expressions you have given, the reference seems to be the "root" hinge of the system. The form U=mgh is derived from basic princples as follows: The potential energy is defined as the negative of the work done by gravity on the mass in moving it from the datum to its location. Work done is a dot-product of the gravity force and the displacement. But the gravity force is downward and the displacement is upward, so the dotproduct in the definition of work gives you work done by gravity = (force vector) . (displ. vector) = - (mg) (h) = -mgh But the potential energy is the NEGATIVE of this: U = -( work done by gravity ) = -( - mgh ) = mgh In your case, the masses are BELOW the reference, so h (which is defined as the height ABOVE the reference) is a negative quantity. Of course, it is useful in this problem to work with (positive) distance quantities such as l_1 * cos(theta_1) l_1 * cos(theta_1) + l_2 * cos(theta_2) which are distances of the masses BELOW the reference. This is where the negative sign comes in. Now, the question you ask about "how to derive the potential energy" is actually more involved than this in general. I don't know exactly what directions your work will take you, but if your simulations are going to handle more complex systems, you might be required to go beyond the discussion above. Ultimately, it all boils down to understanding what forces are acting on the system and representing as many of them as possible by a potential energy. (By the way, this is not always possible, for example if the pendulum is suspended in a fluid, you'll have to deal with the fluid forces using work principles directly.) In my work, I'm dealing with flexible vehicles in orbit. The flexibility intoduces internal stresses which do work and can be modelled by a potential energy. And the gravitational forces can also be modelled as a potential energy. However, these potential energies have a different form than the (simple) U=mgh discussed above. The differences are due primarily to the fact that the forces themselves act in a significantly more complicated manner than the force due to gravity in a uniform gravity field. If all you need to do in model point masses and rigid bodies near the surface of the earth, then the U=mgh stuff will get you quite far. Just be warned, however, that the fundamental principles are somewhate "hidden" by the notion of potential energy. Finally, beware that there are some people out there who discuss "potential functions". The field of celestial mechanics is full of these. It is an unfortunate and often confusing fact that potential functions and energies differ in their definitions by a minus sign. So if you go beyond U=mgh in your efforts, pay attention to the small print. -- | David A. Hasan | hasan@emx.utexas.edu