Generalized Coordinates new video and code 👀

[cooperrc]

Using generalized coordinates and Lagrange Equations for a system of pulleys and blocks.

In this video and notebook, we can go through the analysis of degrees of freedom and choosing generalized coordinates for the Lagrange equations.

One big lesson is that # of DOFs == # of eoms

You will have one equation of motion for every degree of freedom. Many times, like the video, they are coupled to each other.

For the system of pulleys, try changing the masses of blocks 1, 2, and 3. Is there a way to have block one stationary, but blocks 2 and 3 move?

What other ways do degrees of freedom affect dynamics?

[cjdipietrantonio]

Assuming that pulley 2 is massless and everything starts at rest, we know that block 1 remains stationary if $\ddot{y}_1 = 0$. To figure out when this is possible, we can rewrite our system of equations to solve for an expression for $\ddot{y}_1$:

$$ \ddot{y}_1 = 0 = \frac{\left[m_1 \left(m_2 + m_3\right) - 4 m_2 m_3\right]g}{m_1 \left(m_2 + m_3\right) + 4 m_2 m_3} $$

For physical masses $m_{1}, m_{2}, m_{3} > 0$, the denominator is positive. Also, we know that $g \neq 0$, which leaves us with:

$$ m_1 \left(m_2 + m_3\right) - 4 m_2 m_3 = 0 $$

Therefore, for block 1 to remain stationary, we must have:

$$ m_1 = \frac{4 m_2 m_3}{m_2 + m_3} $$

Now, if we impose $\ddot{y}_1 = 0$ on our second Lagrange equation,

$$ (-m_2 + m_3) \ddot{y}_1 + (m_2 + m_3)\ddot{y}_2 = (m_2 - m_3) g $$

we obtain the following expression for $\ddot{y}_2$:

$$ \ddot{y}_2 = \frac{(m_2 - m_3) g}{m_2 + m_3} $$

Which tells us that if $m_2 \neq m_3$, then $\ddot{y}_2 \neq 0$, and blocks 2 and 3 move! However, if $m_2 = m_3$, then $\ddot{y}_2 = 0$, and the entire system remains static. For the first example, I made the following changes to the Pluto notebook:

begin
    m2 = 0.2
    m3 = 0.8
    m1 = (4*m2*m3)/(m2+m3)

    pulley_accel(m1, m2, m3)
end

begin
	y1, p2, y2, y3 = calc_motion(m1, m2, m3)
	@gif for i in 1:length(y1)
		block_positions = Dict(
		    :block1 => (1.0, y1[i]),
		    :block2 => (1.8, y2[i]),
		    :block3 => (2.2, y3[i]),
		)
		
		pulley_positions = Dict(
		    :circle1 => (1.5, 2.0),  # big pulley
		    :circle2 => (2.0, p2[i]),  # small pulley
		)
		
		diagram_plot = plot_pulley_diagram(block_positions, pulley_positions)
		ylims!(-2, 3)
		diagram_plot
	end
end

From which we can confirm that blocks 2 and 3 move in opposite directions while block 1 remains stationary!

For the second case $m_2 = m_3$:

As expected, everything remains static. From this example, we can see that the dynamics of a system are significantly influenced by its degrees of freedom. Not only do they determine the number of equations of motion required to describe the system, but also how the constraints shape the system's motion and how forces are coupled between components.

[gwgezas123]

woah this shows this problem really well. This is pretty cool, great work!

[med17018]

In the lecture video, we observed a pulley system with multiple degrees of freedom (2 DOF), that were coupled to each other. In a system like the one we observed, with coupled DOFs and multiple masses, we know the acceleration of each of the masses is linked. Realistically, because all three accelerations are linked, m1 is not going to be stationary so long as m2 and m3 are accelerating.

However, as @cjdipietrantonio showed, if we alter the system by changing constraints (essentially adding a ground constraint to m1 and therefore changing DOF of the system), we are able to calculate mass values that will allow m2 and m3 to move while m1 is stationary.

[snowman383]

Degrees of freedom determine the number of independent equations of motion, which can be coupled to describe complex system behavior. By adjusting the masses in the pulley system, it's possible for block one to stay stationary while blocks two and three move, this demonstrates that the distribution of mass can selectively constrain specific degrees of freedom. This also shows that degrees of freedom not only dictate the minimum coordinates needed to describe a system but also influence how constraints and forces shape the resulting motion.

[jordanbollacke]

The degrees of freedom are important to tell us the kind of constraints that are on a system, as observed through the Greubler count, which I have actually never heard of before. I was curious if there are any strategies to identifying the Greubler count, because it went pretty fast on the video. The examples identified make sense, but I was wondering if there are any constraints that we might see that are less common/might take more thinking to identify.

[jhngms]

Is there a way to have block one stationary, but blocks 2 and 3 move?

To have $m_1$ remain stationary, we must look at the governing equations provided in the example:

1. $(m_1 + m_2 + m_3)\ddot{y}_1 + (m_3 - m_2)\ddot{y}_2 = (m_1 - m_2 - m_3)g$
2. $(m_3 - m_2)\ddot{y}_1 + (m_2 + m_3)\ddot{y}_2 = (m_2 - m_3)g$

We can rewrite where block 1's acceleration equals zero, $\ddot{y}_1=0$. This removes the first term from both equations. Now, we can rewrite equation 2 to solve for $\ddot{y}_2$.

3. $\ddot{y}_2 = (m_2 - m_3)g/(m_2 + m_3)$

Plugging this into equation 1 gives us:

4. $(m_3 - m_2) (m_2 - m_3)g/(m_2 + m_3)= (m_1 - m_2 - m_3)g$

Isolating $m_1$:

5. $m_2 + m_3-\frac{(m_3 - m_2) ^2}{(m_2 + m_3)}= m_1$

I added this just above the animation block provided by Professor Cooper:

begin
    m2 = 0.5
    m3 = 1
    m1 = m2 + m3 - ((m3 - m2)^2 / (m2 + m3))
end

And edited the following line in the gif generation to be a function of these masses, rather than defined explicitly:

y1, p2, y2, y3 = calc_motion(m1,m2,m3)

Here we see $m_1$ is entirely stationary while $m_2$ and $m_3$ move.

What other ways do degrees of freedom affect dynamics?

As some folks have mentioned, the Gruebler count is used for determining mobility of a system. In another course, I used the Gruebler-Kutzbach formula to determine mobility. Interestingly, under certain geometric conditions, this formula fails. Here walks through a simple example explaining this phenomenon.

For simpler systems, it is always best for me to visualize the motion of the system, then use the equations, much like Professor Cooper did in the video.

Thanks,
John

[joshuajimenez1]

In this pulley system, although increasing the masses produces the expected behavior of heavier masses accelerating downward, the blocks cannot move independently because each pulley has its own constraint and equation of motion. Since both pulleys can move, the system has two degrees of freedom rather than three.

If block 1’s degree of freedom is removed either by fixing the top pulley or perfectly balancing the forces on m1 and m2 with m3, the small pulley cannot move, and the system reduces to just one degree of freedom motion between m2 and m3​. In the complete system, any imbalance between m2 and m3​ creates a net force on the small pulley, which in turn forces m1 to move. This means m1​ can remain stationary while m2 and m3 move only if force balance is maintained at the small pulley but in any other case all three masses are dynamically involved with each other. The number of degrees of freedom determines how motion is distributed and how forces move through the system, and whether special configurations like some parts remaining stationary while others move are dynamically possible.

[ericd3030]

Working through this pulley system really helped me understand what DOFs = equations of motion actually means in practice. At first, it looks like there should be three degrees of freedom because three masses are moving. But once I accounted for the two rope constraints, I realized the system only has one true degree of freedom. The key constraint is: 2*y1 + y2 + y3 = constant. The same goes fr acceleration.

I tested the system under two mass conditions.
Case 1: m1 = 2, m2​ = 1.2, m3 = 0.8. Here, m1 = m2 + m3. It resulted in all accelerations and velocities equaling zero.
Case 2: m1 = 10, m2​ = 5, m3 = 1. Here, m1 does not equal m2 + m3. The resulting aceration for a1 was -3.41, and a2 and a3 were the same value but positive. The same when for velocity. v1 = -6.82 and the corespoing v2 and v3 were the samve value but positive. This is exactly what I expected. Block 1 accelerates downward, and blocks 2 and 3 move upward. The signs make sense, and the magnitudes match constant acceleration motion starting from rest.

Another thing this problem made clear is that degrees of freedom affect more than just how many equations you write down. They also control how complex the motion can be. With only one DOF, everything in this system is locked together. If one mass moves, they all have to move in a very specific way. There’s no flexibility or “independent behavior.” In systems with more DOFs, you can get coupled motion, internal oscillations, energy transferring between parts, and multiple natural modes of motion. So increasing the number of DOFs usually makes the dynamics more complex.

[danny-kruzick]

Can block 1 remain stationary while blocks 2 and 3 move?

Yes, but only for a very specific mass ratio.
From the equations of motion, block 1 stays stationary when:
y1_ddot = 0
Solving the system gives the condition:
m1 = (4 m2 m3) / (m2 + m3)
So there is a special combination of masses that makes block 1’s acceleration zero while blocks 2 and 3 still move.
What’s interesting is that block 1 is not constrained. The system still has two degrees of freedom. The equations are still coupled. It’s just that the internal force balance through the pulleys causes the net force on block 1 to cancel out.
If m2 = m3, then everything stays static. So the motion really depends on how the mass is distributed.
To me, this is a good example to show that zero acceleration does not mean a DOF disappeared it just means the dynamics happened to suppress motion in that coordinate.

What else do DOFs affect?

Degrees of freedom do more than just tell us how many equations we need.
They affect:
How energy moves between parts of the system
How strongly the coordinates are coupled
How many natural frequencies or vibration modes exist
How sensitive the system is to adding or removing constraints
More DOFs usually mean more ways for energy to redistribute internally.

[JulianPojano]

To get blocks 2 and 3 to move while block 1 is stationary I needed to account for the acceleration of blocks 2 and 3 due to their difference in weight. Combining the forces, I equated the weight of block 1 on the left side of the pulley to the total forces on the right side of the pulley:
$m_1 g = g(m_2 + m_3) + g \frac{(m_3 - m_2)(m_2-m_3)}{m_3 + m_2} $

Assuming blocks 1 and 3 have a weight of 1 kg, I solved for $m_2$ and found that it should have a weight of 1/3 kg to ensure block 1 remains stationary. This ended up working as seen in the gif.

Understanding how components of a system contribute to the total degrees of freedom is essential to creating and simplifying a model. At first glance for this system I would assume 3 degrees of freedom and then take a overcomplicated approach when trying to solve for the equations of motion. Knowing the minimum number of variables needed to a system is very helpful when trying to determine the equations of motion and construct a model, especially in complex systems.

[juliamachaj]

Using this code setup was a useful way to visualize the degrees of freedom alongside the equations of acceleration. The degrees of freedom within a system obviously limit how all parts within a system moved, and sometimes it is hard to differentiate degrees of freedom when things are interconnected. (E.g., the mistake Prof Cooper pointed out that can happen when you double count for the cords in a pulley system.) This is why FBDs are extremely useful to simplifying a dynamics problem, because then the equations of motion that result can be simplified (as many others have already shown in this discussion post).

I messed around a little but with the masses of the system but was ultimately getting the same gifs as everyone else. What would happen if we added more degrees of freedom? If we added some pendulum motion to the masses hanging from the pulleys? I'll poke around and see if Copilot can help me model something like this but I wonder if that's where project 2 will take us.

[matlabkid]

For the system of pulleys, changing the masses of blocks 1, 2, and 3 leads interesting results. The way to have block one stationary, but blocks 2 and 3 move, after static equating I fonud that I need to make mass 1 or two a little heavier. while this only got me so far i believed that the acceleration had to be negative while mass 1 had to be almost near 0 for this to work. When solving for y1_ddot = 0 we can simplify the 2nd order equations to equal the y2_ddot and leave the masses, m1*(m2+m3) = 4m2m3; m1 = 3, m2 and m3 either 3 or 1 each.

What other ways do degrees of freedom affect dynamics? Other than how do degrees of freedom affect dynamics in pulleys and how many coordinates do we need, I guess we can expand on what motions are possible, how forces are coupled, what contrains the behavior and what emerges such as damping or special oscailltions from these motions.

[bgoncal22]

Yes, it is possible to have block 1 remain stationary while blocks 2 and 3 still move, as long as the masses are chosen so the forces balance in a way that keeps block 1 from accelerating. In that case, blocks 2 and 3 can still move relative to each other because the system has more than one degree of freedom.

Degrees of freedom affect dynamics by determining how many independent ways a system can move and how many equations of motion are needed to describe it. As the number of degrees of freedom increases, the motion usually becomes more coupled and more complex.

[C-vonk]

For my code, I set the right-hand side to have a total mass of 1kg; however i made it so that the m3 was much larger than m2 with this i was able to lower the left hand side mass down to aproximately .4 kg before i hit equalibrium.

[GoateT33]

When experimenting with the pulley system, it seems possible for block 1 to remain stationary while blocks 2 and 3 move if the masses are balanced so that the net force acting on block 1 is essentially zero. For example, if the forces transmitted through the ropes from blocks 2 and 3 cancel each other out, block 1 would experience no acceleration while the other two blocks could still move relative to each other. This would require a specific combination of the three masses so that the tensions in the system balance at block 1 (m1 = m2 + m3).

This example also highlights how degrees of freedom directly influence the dynamics of a system. Each DOF introduces another equation of motion and another variable that can change with time, which can lead to coupled motion between components. In systems like this pulley setup, the motion of one block affects the others because the coordinates are linked through the constraints of the ropes. The number of DOFs therefore determines not only how many equations are needed to describe the system, but also how complex the motion becomes. Fewer DOFs usually mean simpler dynamics, while more DOFs allow for more independent movement and more complicated interactions between components.

[jadongs]

Degrees of freedom determine how many independent motions a system can have, and they also determine how many equations of motion are required when applying the Lagrange equations. In the pulley system, the blocks are connected by constraints from the ropes and pulleys, so the motion of one block influences the others. Because of this coupling, changing the masses can affect how forces and tensions are distributed throughout the system.

It seems possible for block 1 to remain stationary while blocks 2 and 3 move if the masses are chosen such that the net force acting on block 1 balances out. In that case, the tensions in the ropes could cancel at block 1, preventing it from accelerating while the other two blocks still move relative to each other. This highlights how mass distribution can influence which degrees of freedom are active in motion. @cjdipietrantonio and @jhngms, example work helped me work out my own problem and test my own mass changes.

More generally, degrees of freedom affect system dynamics by determining how many variables can change independently and how strongly different parts of the system are coupled. Systems with fewer degrees of freedom tend to have simpler motion, while systems with more degrees of freedom allow for more complex interactions between components.

[ojmoney00]

This idea that the DoF equaling to the equations of motion is what I'm learning in my vibrations class currently. We are able to use this idea to create a system of equations using matrices for a 2 spring, 2 mass system or a mass with two perpendicular spring system and figure out the motion graph for it, just to name a few systems! Usually as well, the number of DoF means the number of modes of a vibration, where each mode has its own specific frequency and shape. This can get out of hand however, as if you are analyzing a model in something like ANSYS, then there is a lot of DoF that is literally impossible to solve by hand without a computer.

Also having too many DoF in the right way can cause what's called "chaos," which can cause a system to diverge and cause rapid, uncontrollable motion.

In terms of the pulley animation, cjdipietrantonio's answer for this solution is spot on. At first, you would think that it's simply just the m1 = m2+ m3 where m2 doesn't equal m3, but this isn't the case because of the motion of the pulleys. I wasn't able to find an answer for m1 m2 and m3 in terms of just numbers of just guessing and checking, so maybe the only way to solve this is based on the equation from cjdipietrantonio.

[LRand7]

In the lecture video, we looked at a pulley system with two degrees of freedom, and the motion of the masses was clearly coupled. Because of this, the accelerations of each mass are linked through the constraints of the system, so if two blocks are moving, it’s not realistic for one mass (like m1) to remain completely stationary while the others accelerate. Degrees of freedom determine how many independent equations of motion are needed, and in constrained systems these equations are often coupled. Changing the masses can shift how the system behaves, and in some cases it may be possible to arrange conditions where one block stays still while the others move, depending on how forces and constraints balance out.

[willhennessey428]

There is a way for m1 to stay stationary while m2 and m3 move. This can only happen if the forces from m2 and m3 balance the weight from m1. The total of m2 and m3 can not simply equal m1. The acceleration, velocity, and weight of m2 and m3 must be taken into account. This can be solved for using equations of motion and visualized using the attached code. I have seen multiple iterations of it above which further prove that it is possible.

Knowing that degrees of freedom equals the number of equations of motion we can get a better idea of what else degrees of freedom mean to a dynamic system. These equations of motions define how a system moved. Depending on the plane in which the system is moving, different equations of motion can mean the same thing. Think cartesian and polar coordinates. They also can tell us the behavior if the system.

[AuZwick]

This example made it clear that even though multiple blocks are moving, the constraints reduce the system to one true degree of freedom. Because everything is tied together by the rope, the motion of each block depends on the others, so you don’t get independent movement.

For block 1 to stay stationary while blocks 2 and 3 move, the forces would have to balance so that m1 = m2 + m3. In that case, the system is in equilibrium and nothing accelerates. Otherwise, if that balance isn’t met, all the blocks have to move together based on the constraint.

This also shows that degrees of freedom really affect how the system behaves. With only one DOF, the motion is very restricted, but with more DOFs you can get more complex behavior like different parts moving independently or energy being transferred between components.

[Johnsw9]

One thing I found interesting in this notebook was how generalized coordinates simplify a complicated system into only the motions that actually matter. In the pulley example, even though there are multiple moving blocks, the constraints couple the motion together so the system still has a limited number of true degrees of freedom. That also explains why changing one mass changes the acceleration of the entire system, since the equations of motion are linked through the constraints. I think it would be cool to showcase a more complex mechanism next, maybe something like a double pendulum or slider-crank system, where choosing the wrong generalized coordinates makes the derivation much harder than it needs to be.

[michaelsbriscoe]

Working through the pulley system helped me better understand what it really means for the number of degrees of freedom to match the number of equations of motion. At first it seems like there should be three DOFs since there are three moving masses, but once the rope constraints are applied, the motion is restricted. The positions of the blocks are tied together, so they can’t move independently, which reduces the number of true DOFs and leads to coupled equations.

For the case where block 1 stays stationary while blocks 2 and 3 move, this is only possible for a specific relationship between the masses. If the masses are chosen correctly, the forces balance in a way that keeps block 1 from accelerating even though the other two are moving. Otherwise, because everything is coupled, motion in blocks 2 and 3 will cause block 1 to move as well.

More generally, degrees of freedom affect how flexible a system’s motion can be. With fewer DOFs, the motion is highly constrained and everything moves together in a fixed way. As you increase the number of DOFs, the system can exhibit more complex behavior, like multiple modes of motion and energy transfer between components.

[cbull0]

The pulley example helped me understand why the # of DOF determines how many equations of motion are needed. Even though multiple blocks are moving the rope constraints connect all of their motions together so the system has fewer independent motions. I also found it interesting that block 1 can stay stationary while blocks 2 and 3 still move. This is only when where the tensions and accelerations from m2 and m3 balance out m1 correctly. It is not just based on the weights alone because the dynamics of the system also affect the motion.

[tjs21014]

This example helped clarify how degrees of freedom (DOFs) determine both the number of equations of motion and how motion is coupled in a system. At first glance, the pulley setup seems like it should have three DOFs since three masses move but the rope constraints reduce this to fewer independent coordinates. This made it clear why carefully identifying constraints is just as important as counting moving parts. By adjusting the mass values in the notebook it is possible for block 1 to remain stationary while blocks 2 and 3 move, but only for a very specific mass ratio. In this case block 1 is not constrained; instead, the internal force balance through the pulleys causes its net force to be zero. This was a useful reminder that zero acceleration does not mean a degree of freedom has been removed, it just means the dynamics suppress motion in that coordinate. Overall, this discussion reinforced that DOFs affect more than just how many equations we write. They control how forces and energy are transmitted through the system, how strongly motions are coupled, and how complex the resulting dynamics can be. Choosing good generalized coordinates makes these relationships much easier to see and analyze.

[bliew52]

Yes, there is a way to have mass 1 stay stationary but have mass 2 and 3 move. By setting the acceleration of block 1 to be zero and substituting two equations of motion, we can solve for the relationship between (m2 and m3) to equal m1. As long as this ratio is satisfied, m1 will always be stationary in this case.
I remember using degrees of freedom in my Vibrations class. Since the number of degrees of freedom is connected with the number of equations of motion, it was always used to define the complexity of the problem. It is helpful to have constraints (such as the pulleys in this example) to better predict the outcomes of a system.

[gwgezas123]

I think with these problems I am starting to see how coding can really help visualize a mechanical problem. As there are more degrees of freedom, and linearily more equations of motion, it starts to get very difficult to visualize mentally or pen and paper the system. I could not imagine trying to understand a 4+ pronged pullley system with multiple weights in different directions and multiple tensile loads maybe not just vertically. Coding is really sweet and it is pretty cool that it can show this. Degrees of freedom affect dynamics by how much crazier the system can move or how much more independence it has. Something that has 1 degree of freedom is really restricted, something with a million degrees of freedom can in theory basically act like water. It is really interesting to think about from an engineering perspective and consider what is the minimum amount of DOF to not complicate a system but to still achieve the goal required.

[dugganjake]

I played with the masses of the pulley system to see how the motion is affected. Of course, the motion of the different blocks are not independent as they are connected by rope in the pulley system, but it was fun to see regardless.

I used the condition for block 1 to not move and set (m_1 = \frac{4m_2m_3}{m_2+m_3}). I used values like (m_2 = 0.2) and (M_3 = 0.8) and block 1 stayed stationary, and blocks 1 and 2 moved in opposite directions.

I did a simple graph of the blocks motion since my pulley demonstration is looking a bit rough.

[Coen-uc]

As @cjdipietrantonio provided an excellent explanation on getting mass 1 to remain stationary, I will not copy their work.

I think that other ways dynamics can be affected by degrees of freedom when it comes to a system-level design of a dynamic model. While we deal with providing mathematical solutions and diagrams to already defined objects and bodies, it is often the other way around that a system with a dynamic response is designed and built. Requirements are given to the engineers, and they have to use those requirements to build in the degrees of freedom in a system. In the case of a valve, it may be that the requirement only defines a few degrees of freedom; it is now the engineer's job to design in other constraints that will not interfere with the purpose of the valve.

[JimInput]

I ended up building the simulation myself, and this was the animation I got. I ended up using a callback terminator to stop the animation once the motion went out of bounds for any section.

`function simulate_blocks(sys; L1_val=1.5, L2_val=0.75, H_val=2.0, m1_val=2.0, m2_val=0.5, m3_val=1.0, g_val=9.8, y1_0=0.5, y1_dot_0=0.0, y2_0=0.5, y2_dot_0=0.0, tspan=(0.0,4.0))

u0 = Dict(a1 => y1_0, a2 => y1_dot_0, b1 => y2_0, b2 => y2_dot_0)
p = Dict(L1 => L1_val, L2 => L2_val, H => H_val, m1 => m1_val, m2 => m2_val, m3 => m3_val, g => g_val)
prob = ODEProblem(sys, merge(u0, p), tspan)


length_condition(u, t, integrator) = begin
    y1 = u[4]
    y2 = u[2]
    s1 = H_val - y1
    sp = L1_val - y1
    y3 = L2_val - y2
    s2 = H_val - sp - y2
    s3 = H_val - sp - y3
    min(y1, y2, s1, sp, y3, s2, s3)              # root when the shortest hits 0
end

terminate_cb!(integrator) = terminate!(integrator)
cb = ContinuousCallback(length_condition, terminate_cb!)

sol = solve(prob, Tsit5(); callback=cb, saveat=0.0333)

return sol, p

[brookerichard23]

For the pulley system, it is possible to keep block 1 stationary while blocks 2 and 3 move, but only for a specific relationship between the masses. Even though there are three moving masses, the rope constraints couple the motion together and reduce the number of true degrees of freedom. Because of this, the blocks cannot move independently, and the equations of motion become coupled. By choosing the masses correctly, the forces on block 1 can balance so that it does not accelerate while blocks 2 and 3 continue to move relative to each other.
Degrees of freedom also strongly affect the overall dynamics of a system. Systems with fewer DOFs are more constrained, so the motion is more simple and more predictable. As the number of DOFs increases, the system can develop more complex behaviors like coupled motion, multiple vibration modes, and energy transfer between components. Understanding the true number of DOFs is important because it determines how many equations of motion are needed and helps simplify the modeling process when using methods like Lagrange equations.

[HCastillo03]

I was changing the masses in the notebook and found that it is possible for block 1 to stay stationary and m2 and m3 move with m1​=4m2​m3 / m2​+m3 by setting y"1 = 0​​​. Setting m2 ​to not equal m3 and adjusting m1, using the equation m1 stayed still. The setup keeps block 1 still while blocks 2 and 3 move in opposite directions from each other. This doesn't take away a degree of freedom, the forces acting on the pulley balance in a way that gives block 1 zero acceleration of motion. Degrees of freedom affect dynamics by determining the amount of independent ways a system can move and how they are coupled. This pulley system has two degrees of freedom, but constraints and the way mass is distributed determines how those motions appear.