Analytic Solution For Equilibrium Point

[DLinkage]

I recently made an interesting problem involving a uniform rod. I would like to find an analytic solution to this problem however I do not know if this is possible. The equation yielding the equilibrium points is given by

omega^2*R*cos(phi) + omega^2*(L/2)*cos(phi)*sin(phi) + g*sin(phi) = 0

omega is a constant being a measure of angular velocity
R is also a predetermined radius
L is a constant but we are to consider the variations L < R, L = R, L > R

phi is the coordinate we are trying tofind equilibrium points for.

The problem is a uniform rod rotating freely in a vertical plane whos basepoint is a radial distance R away from the primary axis of rotation. The length of the rod is L, and its vertical orientation is given by phi measured clockwise from the vertical.

Note that there are 4 angles that yield a solution if L is long enough, (this is not to say that all 4 are stable). Only two will exist however (between 0 and 180 degrees if L is less than this k*R (k > 1)). We can easily find stability by either looking at the numerical solution and checking concavity (When L = k*R the concavity becomes zero at the critical points and it is tangent to the line y = 0). So this is all great for insight and such but is there an analytic solution? Pretty much this is becoming a math problem with independent coefficients. So if we normalize all the terms by omega^2*r the nwe have something like

cos(phi) + A*cos(phi)*sin(phi) + B*sin(phi) = 0

Anyone have any ideas? I'm completely stumped (tried s(2a) = 2cos(a)sin(a))
also sin(a) = sqrt(1-cos(a)^2) hasnt gotten me anywhere either. Thanks

 

[Aaron Bex]

It sounds like you've done a lot of work on this problem already and have made some progress. Have you tried using the double angle formula to simplify the equation? It might be helpful to break it down into two parts and solve it that way. Another approach might be to use trigonometric identities such as the law of cosines or the law of sines. I'm not sure if any of these will help you find an analytic solution, but it's worth a shot. Good luck!

 

[charng]

I would first commend you on your interesting problem involving a uniform rod. The equation you have provided is certainly complex, but it is not impossible to find an analytic solution. It may require some mathematical manipulation and possibly the use of numerical methods, but it is certainly within the realm of possibility.

One approach you could take is to try to simplify the equation by reducing the number of variables. For example, you could try to eliminate the constant omega by substituting it with a variable, say x, and then try to solve for x in terms of phi. This may lead to a more manageable equation that can be solved analytically.

Another approach would be to use numerical methods to approximate the solution. This involves using a computer program to iteratively solve the equation and find the equilibrium points. This may not give you an exact solution, but it can provide valuable insights and allow you to visualize the problem in different scenarios.

I would also recommend consulting with other experts in the field, such as mathematicians or physicists, who may have more experience in solving similar problems. Collaborating with others can often lead to new insights and solutions.

In conclusion, while finding an analytic solution to your problem may be challenging, it is certainly possible with the right approach and resources. Keep exploring and don't be afraid to seek help from others in the scientific community.