Salisbury Differential Ramp Force

[spoles]

I have been modeling lockup torque vs ramp angle for a constant input torque from the motor for our Salisbury diff and I am questioning the results. I am getting a function that gives a parabola that concaves down. Peak lockup torque is at 45 degrees according to the attached graph but intuitively, this should function should decrease as ramp angle increases. I have attached the graph.

This curve originates from my calculation of the force the thrusting ramp induces on the clutch packs, F_a = F_t*sin(alpha)*cos(alpha).

I've also attached my force diagrams of the ramps. In them, F_t (force generated by input torque) is kept constant. It can be seen the magnitude of F_a is equal at alpha=30 and =60, and greatest at alpha=45.

I've also read from a source that F_a = F_t/tan(alpha). I opted to disagree with this as I was only able to derive this from creating the normal force on the ramp greater than F_t. From my understanding, normal force is never greater than input force on an incline plane. Any exceptions?

Has anyone been able to derive a function for the thrust force?

[BillCobb]

What is the relative (lateral) displacement of the drive tabs ? You KNOW what that should be at a 90 degree ramp angle.

[spoles]

What is the relative (lateral) displacement of the drive tabs ? You KNOW what that should be at a 90 degree ramp angle.

I'm confused how displacement of the ramps affects the applied force to the clutch packs? Yes the move outward inversely proportional to the ramp angle, but I'm concerned with contact forces.

[spoles]

I suppose better way of saying that is yes, as the ramps move outward they create more compression on the clutch packs and, by consequence, more "lock". But how is F_a itself dependent upon x?

Obviously, F_a = 0 when alpha = 90, but it my mind that is because there is no horizontal component of the resultant force on the incline.

[Pete Marsh]

"I opted to disagree with this as I was only able to derive this from creating the normal force on the ramp greater than F_t. From my understanding, normal force is never greater than input force on an incline plane. Any exceptions?"

Yeah, this is a wedge, not a block on an inclined plane. This would have a different free body diagram.

[spoles]

Yeah, this is a wedge, not a block on an inclined plane. This would have a different free body diagram.

Ah, this seems to be the core of my problem. The equation from my source is correct by this diagram. I did not realize the wedge diagram would be different. Thank you Pete.

[BillCobb]

What does your clutch friction model say about engagement vs. lateral component of the Force due to input torque ? Maybe time to run into the lab and measure its characteristics... Then model what you measure. Try to observe displacements, too