Rigid Body Mechanics - 2 or more bodies

[Jason NIcholson]

When I derive the equations of motion for 2 or more bodies where one is rotating and the other is a mixture of rotation and translation, I get a term multiplying angular velocity squared. I know its right but I don't know what to call it. Can some help me with what to call it (it means C₁*ω² below)?After eliminating one of the constrained degrees of freedom, I get an ODE for the rotational degree of freedom:
sum of moments = Coefficient1*angularVelocity²+equavelentInertia*angularAcceleration
ΣM = C₁*ω² + I_eq*α

What's interesting about C₁ is that it position dependent and it can flip signs. C₁ has units of mass*length² thus the units are consistent since ω has units of radians/sec.

 

[DrClaude]

A priori, this looks like a term for rotational kinetic energy, with C₁ related to the moment of inertia. I don't see why it is position dependent,. You should probably give some details of the derivation if you want to get help with this.

 

[Jason NIcholson]

The video below shows the motion.

thickness = 2*10mm = 20mm. member is extruded 10mm each way.

thickness = 2*5mm = 10 mm. member is extruded 5mm each way.

member1 rotates around the z axis. member2 is constrained at one end to move vertically at distance of 80mm from the YZ plane. The two body are constrained to each other via a revolute joint. Solving the equations motion for one ODE in terms of the rotational displacement, velocity, and acceleration of member1 yields an equation that looks like:

$$\sum M = C_1(\beta')^2 + I_eq\beta''$$

$\beta$ is the rotational position of member1. Both $C_1$ and $I_{eq}$ are a functions of position.

I am not convinced it is rotational energy. I have compared my results against Rigid Body Dynamics solver Recurdyn and Adams. I can match their answers so I am confident I have got the equations of motion correct.