Basic Lagrangian dynamics question (setup of problem)

[engineer23]

I am currently a grad student. Part of my PhD work will be to formulate a mathematical model of a manufacturing process using Lagrangian dynamics. I am just beginning to delve into the world of variational mechanics, having never had a formal course in the subject. The process involves a stationary tool (rigid) which rotates about a fixed axis and contacts a metal plate that is moving in the x-direction at some feed rate, fr (similar to a milling operation). The plate material is reinforced with abrasive "spheres" which come into contact with the tool at a rate that is dependent on tool rotation speed, material feed rate, percent reinforcement, etc. and result in wear of the tool.
I'm in the formative stages, but I just wanted to be clear on how many degrees of freedom (n) this problem entails. N is the number of "particles" and m indicates the number of kinematical constraints. My mechanics book relates n to N and m by the expression n = 3N -m. Does N = 2 (the rotating tool + the abrasive particle moving at fr which the tool comes in contact with)? Or is there another N I am neglecting (such as the other components of the material? What about m?

Visualize the abrasive particle carving a straight portion of the tool into a curved shape as it makes contact. My ultimate goal is to determine the shape of this curve...

Thanks for any help you can offer me! I will probably be posting a lot more in the coming year as I progress. The experimental work is somewhat complete...I am trying to obtain some sort of mathematical model which validates the shapes I have observed. This is my first of what will be many attempts...

 

[LightHero]

In this problem, N would be equal to 2, the rotating tool and the abrasive particle moving at fr which the tool comes in contact with. m is the number of kinematical constraints, which would depend on the equations of motion you are using to describe the system. For example, if you are using Newtonian mechanics, then m would be equal to 3 (one equation for each of the three spatial dimensions). If you are using Lagrangian dynamics, then m would depend on the type of constraint you are using.

 

[astrokreng]

it is great to see that you are already thinking about the mathematical model for your manufacturing process using Lagrangian dynamics. This is a complex problem and it is important to properly define the degrees of freedom (n) and kinematical constraints (m) in order to accurately model the system.

Based on the information provided, it seems like N would indeed be 2 in this case, as you have correctly identified the rotating tool and the abrasive particle as the two main components of the system. However, it is important to also consider any other components that may affect the motion of these two main components, such as the material the tool and particle are interacting with or any external forces acting on the system.

As for m, it would depend on the specific constraints in your system. It could include constraints such as the fixed axis of rotation for the tool, the feed rate of the metal plate, and any other restrictions on the motion of the tool and particle.

In terms of your ultimate goal to determine the shape of the curve created by the abrasive particle, this would involve solving the equations of motion for the system and analyzing the resulting trajectory of the particle as it interacts with the tool. This could also involve considering the wear and deformation of the tool as it is being carved by the particle.

I wish you the best of luck with your research and I am sure you will gain a deeper understanding of variational mechanics as you progress in your studies. Feel free to reach out with any further questions or updates on your work.