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I'm attempting to model a crank-rocker mechanism (see picture). A motor is connected to the crank side and is driving the mechanism with a torque $\tau_{motor}$. A load is attached to the rocker side which applies a torque $\tau_{load}$. The moment of inertia on the motor side is $J_\theta$ and the moment of inertia on the load side is $J_\varphi$. I want to find the equivalent moment of inertia $J_{eq}$ that the motor experiences from $J_\varphi$ and $J_\theta$ combined. system overview

I have used Freudenstein's equation to find the relationships $\varphi(\theta)$ and $\frac{\partial \varphi}{\partial \theta}(\theta)$.

For a gearbox (with constant gear ratio) the solution would be $J_{eq} = (\frac{N1}{N2})^2 J_\varphi + J_{\theta}$ (see https://electronics.stackexchange.com/questions/325607/dc-motor-differential-equation?noredirect=1&lq=1).

What is $J_{eq}(\theta)$ in my case? My intuition is that the relationship should become something like $J_{eq}(\theta) = ( \frac{\partial \varphi}{\partial \theta}(\theta))^2 J_\varphi + J_{\theta}$. However, I have not been able to prove this.

Filip Karlsson
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  • If the load is attached to a bearing, there really isn't a torque at the support. There would be a torque at the long bar where it attaches to the weight. You will need to create the movement equations for the 3-bar linkage. The resulting torque could only be graphed; there will not be a constant value. The motor torque would not be constant either unless you set it to constant, in which case the motor speed will vary with the load's acceleration. – Tiger Guy Mar 12 '20 at 20:11
  • Thank you @ScottDunnington, the load is actually attached directly to the rocker and not to the bearing. I have updated the picture to show this.
    The motor speed is controlled by a PI controller, so the torque will indeed vary. What I find difficult to understand is what will happen to the equivalent moment of inertia experienced at the motor side when the rocker changes direction.     – Filip Karlsson Mar 12 '20 at 23:57
  • By bearing I mean the bearing black around which you are placing the torque load. That location cannot experience a torque because it is free moving. You need to create an equation of motion for the load, convert that to an equation of angular inertia, and then translate that inertia back to the motor. You might have negative torque (produced) at the motor if the load is swinging down at the right time. Also don't forget gravity. I feel like this is akin to deriving the pendulum equation. – Tiger Guy Mar 13 '20 at 00:14

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