How Do You Choose Generalized Coordinates for a Timber Beams and Springs System?

[Trenthan]

Homework Statement

We two beams of timber, of identical length joined together at the middle, perpendicular forming a "X" in a sense. Underneath the end of each beam we have a spring attached, thus 4 in total. 3 have identical spring constants and the forth is greater than the other 3. We are told each springs natural length, and to assume the COM is at the join between the beams

I need to model this using lagrangian mechanics. Now where i am stuck is picking the generalized coordinates. Checking the answer it says there are 3 but I am un-sure as to what they are

Homework Equations

NOT really relevant
"lagrangian" L = T(kinetic energy) - V(potential energy).

The Attempt at a Solution

Thinking in cartesion intially i thought sphereical corrdinates, thus (theta, thi, and r). This would allow me to find the spring height wether in compression or extension. This didnt really help me form a "lagrangian" L = T(kinetic energy) - V(potential energy).

So i think i made a mistake after some reading and the generalized coordinates should be related to describing the Centre of mass of the system similar to post by "phagist", titled "Generalized coordinates of a couple harmonic oscillator" looks very similar.

1. So in my case would i still need 2 angles to describe tilt between x-z, y-z, and one for the centre of mass height or is this the height of each individual spring?
2. I thought generalized coordinates can only describe a distance? how can a angle a generalied coordinate?,

Thanks in advance TRENT

 

[anathema]

Dear Trent,

Thank you for your post. It seems that you are on the right track with your thinking about using spherical coordinates. However, in this case, the generalized coordinates that are needed are not related to the orientation of the beams, but rather to the displacement of the springs.

To fully describe the system, you will need three generalized coordinates, which can be chosen as the displacements of the three identical springs from their natural lengths. These can be denoted as x1, x2, and x3, where x1 and x2 represent the displacements of the two springs on the longer beams, and x3 represents the displacement of the spring on the shorter beam.

To form the Lagrangian, you will need to calculate the kinetic energy and potential energy of the system. The kinetic energy can be expressed as the sum of the kinetic energies of the individual springs, which can be calculated using the generalized coordinates. The potential energy can be expressed as the sum of the potential energies of the individual springs, which will depend on their spring constants and the displacements given by the generalized coordinates.

I hope this helps guide you in the right direction. Best of luck with your work!

 

[nlightner]

I would first commend the individual for their efforts in trying to understand the problem and using their knowledge of spherical coordinates. However, I would also suggest that they reconsider their approach to the problem.

Generalized coordinates are a powerful tool in Lagrangian mechanics because they allow us to simplify the equations of motion by reducing the number of variables we need to consider. In this case, the system is already fairly simple, with only four springs and two beams. I would recommend using Cartesian coordinates, as they are the most intuitive and easiest to work with in this case.

To answer your questions:
1. The generalized coordinates in this case should be the positions of the springs, as they are the variables that will be changing as the system moves. The tilt of the beams can be described using these coordinates, as the springs will change in length as the beams tilt.
2. Generalized coordinates can describe any variable that is changing in the system, not just distance. In this case, we are interested in the positions of the springs, so these are the appropriate generalized coordinates to use.

I would also suggest breaking down the problem into smaller parts and then combining them to form the overall Lagrangian. For example, you can first consider the motion of one beam and its associated spring, and then combine it with the other beam and spring. This will help make the problem more manageable.

In summary, I would suggest using Cartesian coordinates and breaking down the problem into smaller parts to solve it using Lagrangian mechanics. Good luck!