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In the above mechanism it is required to find the generalized coordinates to write equation of motion of the MDOF system. I just want to ask how do we actually approach such a problem? How to identify which kinematic degrees of freedom are relevant and independent?

The answer given in this particular case is 4. Two rotation of the pulleys, and two vertical motions of the masses B and C. Now how to visualize this? How do I methodically come to this conclusion? Again, is there any other combination possible?

All pulleys are massless and frictionless.


2 Answers 2


It's a question of how many independent variables you need in order to write out the equations of motion.

Let's walk through it step by step.

Start by just considering Pulley 1 and Mass A. If they're joined by a massless inextensible cable (as shown in your figure) then the displacement of Mass A can be determined as a function of the rotation and radius of Pulley 1. So we only need 1 degree of freedom - the rotation of Pulley 1.

If we added a spring in between Pulley 1 and Mass A then we would no longer be able to directly relate the position of Mass A with the rotation of Pulley 1. Now we need 2 degrees of freedom - the rotation of Pulley 1 and the displacement of Mass A.

Now add Pulley 2 into the mix. If Pulley 1 were connected to Pulley 2 by a massless inextensible cable, then we could define the rotation of Pulley 2 as a function of the rotation of Pulley 1. However, we've got a spring in there so we need to know the rotation of both pulleys.

Continuing with this logic and looking at Mass B and Mass C -- if either one were connected to Pulley 2 using a massless inextensible cable, we could define its position as a function of the rotation of Pulley 2. However, we've got springs attached to both masses, so we need independent variables to describe their positions.

Then Mass A and Mass C need to be connected to the ground via springs to keep the whole thing from unraveling and dropping all the weights on the floor. (I think)

Thus, we need four degrees of freedom -- rotation of Pulley 1, rotation of Pulley 2, displacement of Mass B, and displacement of Mass C. The displacement of Mass A is a function of the rotation of Pulley 1.


I often approach this type of problem by identifying constraints. Each independent constraint limits the number of degrees of freedom, required to completely define the system.

$$DOF = DOF_{system} - \sum \text{Contraints}$$

In 3D systems, each body has 6 degrees of freedom:

  • Translations in $(x, y, z) \rightarrow 3 \;\text{DOF}$
  • Rotations $(\theta_{x}, \theta_{y}, \theta_{z}) \rightarrow 3 \;\text{DOF}$ $$\therefore DOF_{system} = 6(N_{bodies})$$

In 2D systems, each body has 3 degrees of freedom:

  • Translation in $(x, y) \rightarrow 2 \;\text{DOF}$
  • Rotation $(\theta) \rightarrow 1 \;\text{DOF}$ $$\therefore DOF_{system} = 3(N_{bodies})$$

For the 2D system shown (with 5 bodies), calculate the degrees of freedom for the unconstrained system. $$\therefore DOF_{system} = 15$$

Next, determine the imposed constraints on each body:

  • The pulleys are constrained by $(x, y)$- each pulley removes $2 \;\text{DOF}$ from the system.
  • The masses are constrained (though not explicitly) by $(x, \theta)$- each mass removes $2 \;\text{DOF}$ from the system.

Lastly, recognize that body A and its adjacent pulley are not independent- they are kinematically linked by $y=r \theta$, where both move (according to the systems differential equations). This additional constraint to the system requires that:

$$DOF = 15 - \sum [2(2 \;\text{DOF})_{pulleys} + 3(2 \;\text{DOF})_{masses} + (1 \;\text{DOF})]$$ $$= 4 \;\text{DOF}$$


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