Creating a One Shaft Model for a Geared System w/ Inertia

[simlar]

Homework Statement

Develop a one shaft model of a geared system if the moments of inertia of the gear elements are included.
The geared system consists of two discs with inertia J1 and J2 and rotation θ1 (anticlockwise) and θ4 (clockwise). The discs are connected by two shafts with stiffness K1 and K2 respectively. The shafts are connected by two gears with inertia J3 and J4, rotation θ2 (anticlockwise) and θ3 (clockwise) and radii r1 and r2.

Homework Equations

-The time derivative of the angular momentum is equal to the sum of torque acting on the body.

-Equation of motion with constant moment of inertia:

J$\ddot{θ}$+Kθ=0

No slipping in the gear:

θ3=nθ2

n=r1/r2

The Attempt at a Solution

My strategy was to reduce the system to a system with only two degrees of freedom by some manipulation of the equations of motion.

The assumption was made that:

θ1≤θ2≤θ3≤θ4

I tried to make a one shaft model as I would have done in the case with massless gears. I constructed the equations of motion, from the FBD, for the geared system like this:

J1$\ddot{θ}$1=K1(θ2-θ1)
J2$\ddot{θ}$4=-K2(θ4-θ3)

Then I assumed that the gears move without slipping:

θ3=nθ2

n=r1/r2

To deal with the last degree of freedom I made a FBD for the gear, including contact forces and moments. I tried to include the inertia forces at this stage by assuming them have opposite direction to defined positive direction of rotation. From the FBD I got:

M1=F*r1-J3$\ddot{θ}$2=K2(θ2-θ1)

M2=F*r2+J4$\ddot{θ}$3=K2(θ4-nθ2)

where M1 is the torque in the first shaft and M2 is the torque in the second shaft.

Then I eliminated F from the above equations and found an expression for θ2.

But this is where I run into trouble. At this stage in the massless gear problem there are only two degrees of freedom left and the system can be written as a single shaft model. But now there will be additional degrees of freedom from the gears, namely: $\ddot{θ}$2 and $\ddot{θ}$3.

I thought that one possibility could be to use the transmission ratio on the angular accelerations as well. But there would still be one degree of freedom to take care about after doing this.

What can be done to develop a one shaft model of the problem. Please let me know if I did any fundamental errors or if I'm on the right track.
Any help is greatly appreciated!

 

[KataKoniK]

Thank you for your question. Developing a one shaft model for a geared system with moments of inertia can be a challenging task, but I believe your approach is on the right track. However, there are a few things that can be improved in your attempt at a solution.

Firstly, in your equations of motion for the geared system, you have not included the moments of inertia of the gears themselves. This is important as the gears also have rotational inertia that needs to be taken into account. So your equations should be:

J1\ddot{θ}1=K1(θ2-θ1)+J3\ddot{θ}2
J2\ddot{θ}4=-K2(θ4-θ3)-J4\ddot{θ}3

Secondly, when you assume that the gears move without slipping, you have to consider the effect of the gear ratio on the rotational velocities as well. So your equations should be:

θ3=nθ2
\dot{θ}3=n\dot{θ}2

This will give you two additional equations to work with when eliminating the contact forces from your gear FBDs.

Lastly, to eliminate the additional degrees of freedom from the gears, you can use the constraint equations for the gear ratios and the fact that the gears are not slipping. This will give you two additional equations that can be used to eliminate the remaining degrees of freedom.

I hope this helps you in developing a one shaft model for the geared system. Please let me know if you have any further questions or if you need clarification on any of the concepts. Good luck with your research!


(Scientist)