Lagrangian problem of a cylinder on inclined plane and two springs

[elano Blumer]

Homework Statement

A cylinder on a inclined plane is rolling without slipping. Inclined plane is connected to wall with a spring and cylinder is connected to wall with a spring too. All frictions will be neglected, and all the given data has shown on the image below.

As seen above, k2 spring and cylinder is only moving at vertical axis which can be thought as y.

First of all, i have doubts on generalized coordinates. I think only one generalized coordinate should be used and it should be y. I need to solve this problem with D'Alembert Principle and Lagrangian Dynamics.

Thank you in advance for your help.

2. The attempt at a solution

in order to apply Lagrangian dynamics,

Does it look right or am i missing something about rotational kinetic energy?

 

[elano Blumer]

Kinetic energy;

or
for x=x1 and y=x2

is second one right?

 

[OldEngr63]

You are correct in saying that this system has only one degree of freedom.

It would be helpful if you would very carefully define your coordinates on the figure. Where is x = 0? Where is y = 0? You may also find it useful to define an "up-slope coordinate, s" as a secondary variable.

The biggest part of your difficulty right now is that you have not adequately dealt with the kinematics of this problem. This is very common, because everyone says, "oh, kinematics ... that's trivial," but it is not. It is what is causing most of your difficulty right now.

 

[elano Blumer]

I can define our generalized coordinates like this. This graphic is designed for D!alembert principle.

In this situation our Lagrange eqn is;

and

we can write

then we can solve this problem,isnt it?

 

[elano Blumer]

I think it is true.

 

[OldEngr63]

Your equation says that x1 = 0 implies x2 = 0 also. Why is this true?

Exactly what does x1 measure? You show the left end of the arrow at the wall, but the right end is to what?

x2 reference is how far above the bottom of the ramp?

What is the relation between rotation of the disk and x1?

I think you still have not really dealt with the kinematics of the problem in a complete way. It will make your life much, much simpler if you do so.

 

[elano Blumer]

Ohh.Sorry.. T is not equal the zero. You re right.

and

-x2 reference is how far above the bottom of the ramp?

I think it is not important. It defines cylinder's motion on vertival axis. If I say x2 ,It starts from O.

-What is the relation between rotation of the disk and x1?

and

and my answer is;

 

[elano Blumer]

I think Lagrange equation is more simle method than others(D'alembert, Hamilton...etc) to solve this problem.

 

[OldEngr63]

Actually, since there is only a single degree of freedom in this system, the Eksergian's equation is available and it is usually more simple than Lagrange, although they give the same result.

 

[elano Blumer]

I think my result is exactly true.
Now I try to solve this problem with d'Alembert principle.
I hope that I will find same result.

Then I want to find natural frequency of the system. Finally when alpha=0, what is meaning?

 

[elano Blumer]

Ohhh. Nooo. My result is a bit wrong. True result is;

NOT

İS WRONG.

 

[OldEngr63]

I think you have an algebraic error in your original work; I have done a very similar analysis but got you last equation (sort of!).

This problem is not well posed in this respect: Nothing is given about the free length of the springs. For what values of x and y are the springs relaxed? You assumed (in your form for V) that both displacements were measured from the relaxed position, but his may violate the kinematic relations.

To be properly posed, the problem should specify the free length conditions (ie, values of x and y that make the springs relaxed).

 

[elano Blumer]

Thanks for your helping.

But I don't understand your saying. What is your last solution? Can you write it here?

 

[OldEngr63]

Oh, that would not be fair! You will learn a lot more if you puzzle this out yourself.

I suggest that you accept my observation that there is not enough information given about the relaxed state of the springs, and simply assume that there are values xo and yo that apply at the same system configuration for which both springs are relaxed. Based on that, re-formulate your potential energy expression.

In order to do this, you will have to do a more complete kinematic analysis than you have done thus far (as it appears to me). I suggest that you assume that the length of the base of the wedge is B and that the horizontal distance from the wall at left to the centerline of the vertical spring is C.

To do the kinematics, I strongly recommend that you employ vector loop equations, or better, their scalar equivalents.

 

[elano Blumer]

Ok.