# Why are the angular velocities of concentrated masses in Lagrangian independent?

There is an ordinary linear three-mass system. If we write its Lagrangian, we get the following equation. where $$W_k$$ and $$W_n$$ - kinetic and potential energy.

To find the moment of rotation of the first mass, we must differentiate lagrangian first by the angle of rotation of the first mass $$\phi_1$$, then find the rate of change in time of the Lagrangian derivative with respect to speed $$\omega_1$$.

Lagrange equation

Suppose we want to find the moment of the first mass $$J_1$$ (we assume that on the shaft of the drive motor).

$$\frac{\partial L}{\partial q} = \frac{\partial L}{\partial \phi_1} = -c_{12} (\phi_1 - \phi_2)$$

$$\frac{d}{dt}\frac{\partial L}{\partial \dot{q}} = \frac{d}{dt}\frac{\partial L}{\partial \omega_1} = J_1 \frac{\partial \omega_1}{\partial t}$$

Which ultimately gives us a dynamic equation:

$$M - c_{12} (\phi_1 - \phi_2) = J_1 \frac{\partial \omega_1}{\partial t}$$

My questions are as follows:

1. Where did the masses of $$J_2$$ and $$J_3$$ go? Because we are looking for a derivative and speeds of this masses $$\omega_2$$ and $$\omega_3$$ do not explicitly depend on $$\omega_1$$, no matter how massive they are, they do not include into the equation of motion and do not affect the torque M.

2. Do I understand correctly that taking into account the influence of the moments of inertia of the remaining masses $$J_2$$ and $$J_3$$ is possible only by bringing the moments of inertia to the motor shaft $$J_1$$ (by correcting the coefficients of the square of the gear ration).

3. Is it possible to do without this operation and include the gear ratio in the model?

## 1 Answer

Welcome to Robotics, Andrew Sol. I think I'm a little confused with your question as it appears to be about independent rotational masses connected by rotational springs, but then later you're asking about gear ratios.

This is a succinct as I can think to put it, and again I may have misunderstood the question so please feel free to comment on this answer or edit your question to clarify:

The equations of motion govern the dynamics of the system. If you have several bodies that are rigidly coupled, as in a gearbox, then you would (I would, at least) modify the moments of inertia to account for the gearbox ratio and then treat the entire gear train as one system.

Consider this: If $$\omega_2$$ could be calculated by $$\omega_1$$ and a gear ratio, then what dynamics do you need to solve?

• Lagrangian formula contains components of kinetic energy. We differentiate the Lagrangian by velocity $\omega_1$. In this case, the components in $\omega_2$ and $\omega_3$ will "leave", because they do not depend on $\omega_1$, which means that when finding the derivative they can be removed., i.e. no matter how massive they are (i.e. $J_2$ and $J_3$), they have no effect on the moment of the first mass, in this case ... – dtn Jan 13 '20 at 4:55
• Do I understand correctly that they can only be taken into account when correcting the moment of inertia of the first mass? Can you do without this adjustment? I need to get a multi-mass system taking into account stiffness, and then convert it into a state space. I can not consider the system as absolutely rigid. – dtn Jan 13 '20 at 4:59
• Pay attention to how the structural diagram of a two-mass system looks like, taking into account the gear ratio. Figure 5. Servo controller structure for two-mass model Design and servo control of a leak tightness machine working based on hydrostatic pressure aging method. – dtn Jan 13 '20 at 5:25
• @AndrewSol - The thing that's confusing me about your question is that there's no gearbox in your diagram and no description anywhere of where a gearbox would be or what it's connecting, but then your questions all seem to be about gear ratios. If the motion of masses 2 and 3 don't depend on the motion of mass 1 then they're not coupled and are thus two separate systems. I think it would help if you could please edit your question to post a diagram that better reflects the system you're asking about. – Chuck Jan 13 '20 at 14:05
• I specified the problem and tried to make the questions consistent and not confusing. – dtn Jan 13 '20 at 16:15