Solving Rolling Disc Torque: Newton-Euler & Lagrange

In summary, the conversation discussed solving for the torque T by using Newton-Euler equations and taking moments about C. It was determined that there is no torque acting on the disc about point C, as the disc is rolling without slipping and there is no external torque acting on it. The parallel axis theorem was used to calculate the moment of inertia of the disc about point C.
  • #1
t_dawolf
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Homework Statement



The center of mass (C) of the circular rolling disc is offset from its centroid (O) by an eccentric distance [tex]OC = \epsilon[/tex]. The radius of the wheel is R. The disc is rolling without slipping at a constant rotational speed [tex]\omega[/tex] by a variable torque T. Solve for this torque T by:
(a) Newton-Euler equations, taking moments about C
(b) N-E eqns, taking moments about P
(c) Lagrange's equations

(please note in the diagram Th = [tex]\theta[/tex] and ep = [tex]\epsilon[/tex])

Homework Equations



[tex] \phi = (\theta + \frac{\pi}{2}) [/tex]

[tex]r_{C} = -\epsilon cos(\phi) i + (R - \epsilon sin(\phi))j[/tex]

[tex]v_{C} = \epsilon \omega sin(\phi)i - \epsilon \omega cos(\phi))[/tex]

[tex]a_{C} = \epsilon \omega^{2} cos(\phi)i + \epsilon \omega^{2} sin(\phi)[/tex]


[tex][F_x = 0 ; F_y = mg ; T] = Diag([m m I_c])[\ddot{x} ; \ddot{y} ; \alpha = 0] + [ 1 0 ; 0 1 ; -\epsilon cos(\phi) -\epsilon sin(\phi)][R_x ; R_y][/tex]

[tex]T = \epsilon m (w^2(cos^2(\phi)i + sin^2(\phi)j) - g sin(\phi)j)[/tex]



The Attempt at a Solution



First I'm just trying to solve for (a), and I'm not sure that I've set up the Reactions coefficients matrix correctly for the Torque equation. I can imagine that what I've derived for the torque is correct, but I'm still wrapping my head around the concepts and processes and would really appreciate some guidance regarding my equation setup and approach.

Thanks!
 

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  • #2




Thank you for your question. I will try my best to guide you in solving for the torque T using Newton-Euler equations and taking moments about C.

First, let's review the Newton-Euler equations for a rigid body:

Sum of forces = mass x acceleration

Sum of moments = moment of inertia x angular acceleration

We will use these equations to solve for the torque T.

To begin, let's take moments about C. This means we will sum the moments of all the forces acting on the disc about point C. This will result in a single equation with the unknown torque T as the only variable.

To do this, we will need to calculate the moment of inertia of the disc about point C. This can be done using the parallel axis theorem, which states that the moment of inertia about any point is equal to the moment of inertia about the center of mass plus the mass of the object times the square of the distance from the new point to the center of mass. In this case, the distance from C to the center of mass is \epsilon, so we can write:

I_C = I_O + m\epsilon^2

where I_O is the moment of inertia of the disc about its center of mass, which can be calculated using the formula for a solid cylinder:

I_O = \frac{1}{2}mR^2

Now, we can set up our equation for the sum of moments about C:

T = I_C\alpha

Recalling that \alpha = 0 for a rolling disc, we can simplify this to:

T = I_C\alpha = I_C(0) = 0

This means that there is no torque acting on the disc about point C. This makes sense, as the disc is rolling without slipping and there is no external torque acting on it.

I hope this helps guide you in solving for the torque T using Newton-Euler equations and moments about C. Please let me know if you have any further questions or need clarification on any of the steps.



Scientist
 

What is the difference between Newton-Euler and Lagrange methods for solving rolling disc torque?

The Newton-Euler method uses the laws of motion and the principles of angular momentum to analyze the motion of a rolling disc. On the other hand, the Lagrange method uses the concept of energy and the Lagrange equations to solve for the motion of the disc.

How do the initial conditions affect the solution for rolling disc torque using Newton-Euler or Lagrange?

The initial conditions, such as the initial position, velocity, and angular velocity, play a crucial role in determining the solution for rolling disc torque using both Newton-Euler and Lagrange methods. They are used to derive the equations of motion and ultimately, the solution.

What are the key assumptions made in the Newton-Euler and Lagrange methods for solving rolling disc torque?

The key assumptions made in both methods include: the disc is rigid and homogeneous, there is no air resistance or rolling friction, and the disc is rolling without slipping. These assumptions allow for simplified equations and solutions to be derived.

Can the Newton-Euler and Lagrange methods be applied to other rolling objects besides a disc?

Yes, the Newton-Euler and Lagrange methods can be applied to any homogeneous rolling object, as long as the key assumptions mentioned above hold true. However, the equations and solutions may vary depending on the shape and properties of the object.

How can the solutions obtained from Newton-Euler and Lagrange methods be verified?

The solutions obtained from both methods can be verified by comparing them to experimental data or by using numerical simulations. Additionally, the solutions can also be checked for physical consistency and accuracy by analyzing the behavior of the object in different scenarios.

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