Degrees of Freedom & Constraints of Pulley & Spring System

In summary, the system has two degrees of freedom, the angular rotation of the upper pulley and the angular rotation of the lower pulley. The angular rotation of the upper pulley determines the amount of stretch of the spring, and it also determines how far down the center of the lower pulley moves. The angular rotation of the lower pulley determines how far mass 1 moves downward and how far mass 2 moves upward relative to the center of the lower pulley.
  • #1
ShayanJ
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Consider the system shown in the picture.
http://updata.ir/images/m9wcnahubdfanhq9edok.jpg
What are its degrees of freedom?
And is the Lagrangian below,the correct one for this system?
[itex]
\mathfrak{L}=\frac{1}{2} M_2 \dot{x}^2+\frac{1}{2}m_1\dot{x}_1^2+\frac{1}{2}m_2\dot{x}_2^2+\frac{1}{2}I_1\omega_1^2+\frac{1}{2}I_2\omega_2^2-\frac{1}{2}kx^2+M_2gx+m_1g(x+x_1)+m_2g(x+x_2)
[/itex]
Where x is the distance from the center of the lower pulley to the static platform that the spring is connected to directly and [itex] x_1 [/itex] and [itex] x_2 [/itex] are the distances from masses to the center of the lower pulley and the Is are the moments of inertia of the pulleys and I have taken the lower platform to be the zero point for gravitational potential energy.
What are the constraints?
I can think of the constancy of the length of ropes and the relation between [itex] \omega[/itex]s and the velocity of masses.But I have problem finding the relation between the change of length in the spring and the distance from the lower platform to the center of the lower pulley and I just assumed that they're equal.
Any idea is welcome.
Thanks
 
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  • #2
Hi Shyan,

I see only two degrees of freedom in the figure (adequate for establishing all the required features of the kinematics): The angular rotation of the upper pulley (as a function of time) and the angular rotation of the lower pulley (as a function of time).

The angular rotation of the upper pulley determines the amount of stretch of the spring, and it also determines how far down the center of the lower pulley moves.

The angular rotation of the lower pulley determines how far mass 1 moves downward and how far mass 2 moves upward relative to the center of the lower pulley.

I hope this helps.

Chet
 
  • #3
Chestermiller said:
Hi Shyan,

I see only two degrees of freedom in the figure (adequate for establishing all the required features of the kinematics): The angular rotation of the upper pulley (as a function of time) and the angular rotation of the lower pulley (as a function of time).

The angular rotation of the upper pulley determines the amount of stretch of the spring, and it also determines how far down the center of the lower pulley moves.

The angular rotation of the lower pulley determines how far mass 1 moves downward and how far mass 2 moves upward relative to the center of the lower pulley.

I hope this helps.

Chet


Hi chet
Yeah,it was helpful...and after reading it,I was like: " Oohhh...Of course! "...I should have noticed it.But I think I was a little confused.
Thanks
 

FAQ: Degrees of Freedom & Constraints of Pulley & Spring System

What is the definition of degrees of freedom?

Degrees of freedom are the number of independent variables or parameters that are required to fully describe the state of a system. In the context of a pulley and spring system, degrees of freedom refer to the number of independent motions that the system can make.

How do you calculate the degrees of freedom in a pulley and spring system?

In a pulley and spring system, the degrees of freedom can be calculated by counting the number of movable elements in the system. For example, in a single pulley system with a mass attached to one end of a spring, there are two movable elements (the pulley and the mass), so the system has two degrees of freedom.

What is a constraint in a pulley and spring system?

A constraint in a pulley and spring system is a restriction or limitation on the possible motions of the system. Constraints can be physical, such as the fixed position of a pulley, or mathematical, such as the requirement that the length of the spring must remain constant.

How do constraints affect the degrees of freedom in a pulley and spring system?

Constraints reduce the number of degrees of freedom in a pulley and spring system. For example, a fixed pulley would reduce the degrees of freedom from two to one, as it restricts the motion of the pulley in one direction. Similarly, a constraint on the length of the spring would reduce the degrees of freedom by one, as the length of the spring cannot vary independently.

How do degrees of freedom and constraints impact the behavior of a pulley and spring system?

The degrees of freedom and constraints determine the possible motions and equilibrium positions of a pulley and spring system. More degrees of freedom allow for a greater range of possible motions, while constraints restrict these motions to a smaller set of possibilities. Understanding these factors is crucial in analyzing the behavior and stability of a pulley and spring system.

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