Geometric Analysis: Pressure Angle / Overturning Moment

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Geometric Analysis: Pressure Angle / Overturning Moment

Cam Pressure Angle, (μ)

Pressure-Angle represents the efficiency with which the cam transfers its motion and force to the follower, and vice versa.

Pressure Angle is similar to Transmission-Angle.

Imagine pushing a door to open it...

If you push the door-handle with a straight arm, with your arm at 90º to the door (perpendicular(⊥), or normal to the door), you will open the door easily. You only need to overcome the inertia force of the door and any friction in its hinges. You will not need to hold on to the door-handle.

If you push the door-handle through a straight-arm, with your arm at an angle of 45º to the door, then you will need to:

oOvercome the inertia of the door - this is the same as before if the door moves with the same speed.

oPlus, you will need to hold on to the door-handle so that your hand does not slide across the door.

The Total Force against your hand will increase.

There is a force component from the door-handle that reacts against your hand to stop you hand sliding across the door. This force component must be resisted by the door's hinges, which also pull/push on the door-frame.

The transmission angle with which your arm pushes against the door-handle is similar to the pressure-angle between a cam-profile and the cam-follower lever.

GA-Cam-PressureAngle1

The three vectors of the 'force-triangle' (usually taken at the center of the cam-follower) that result from the contact between the cam-profile and the cam-follower are the:

Contact Force. (Hypotenuse of the force-triangle). Its direction is normal to the cam at the cam-contact point, and through the ceter of the cam-follower.

Useful-Force. ( Adjacent of the force-triangle). Its instantaneous direction is in which the cam-follower must move.

Useless-Force. ( Opposite of the force-triangle). Its direction will 'stretch' a Swinging-Arm Follower, or 'bend' a Translating Follower.

From the three vectors, we can define Pressure-Angle

The Pressure Angle, μ, is the angle from the direction in which the center-of the cam-follower must move to the direction of the normal force at the cam-contact, which also acts through the center of the cam-roller.


These force vectors are 'kinetostatic' and ignore friction.

Pressure Angle = Cos-1

(

 

Useful Force

)

Contact Force


Pressure Angle Limits: Rules-of-Thumb

Translating Cam-Follower:  Pressure Angle <  ±30º

Swinging Cam-Follower:  Pressure Angle < ±35º


To reduce the Pressure Angle

Improve the motion design. Can you increase the duration of a segment? Can you change the cam-law to one with a lower peak velocity?

Move the position of the cam-follower's pivot axis. Change the pivot-point of the cam-follower. Or, if the follower is a 'translating follower', move the axis of the follower so that it is not through the center of the cam (an 'offset translating follower').

Increase the size of the cam. The pressure-angle will usually decrease as you increase the size of the cam.


Force Vectors with different Pressure Angles

We can compare Contact, Useful, and useless Forces

If Useful Force of 100N, then when:

Pressure Angle is 0º:   Contact Force = 100.0N;     Useless Force = 0N

Pressure Angle is 10º: Contact Force = 101.5N;   Useless Force = 16.7N

Pressure Angle is 60º: Contact Force = 200.0N;   Useless Force =  173.2N

Pressure Angle - 30 degrees

^^ Pressure Angle 30 degrees ^^

 

Pressure Angle = 30º

Useful Force      = 100N

Contact Force   = 115N

Useless Force    = 57.7N

The force that tries to 'stretch' the cam-follower arm is 57.7% of the useless force.

The Contact Force is 15% more than the useful force.

Pressure Angle 45 degrees

^^ Pressure Angle 45 degrees ^^

 

Pressure Angle = 45º

Useful Force       = 100N

Contact Force    = 141.4N

Useless Force     = 100N

The force that tries to 'stretch' the cam-follower arm is the same as the useless force.

The Contact Force is 41.4% more than the useful force.

Overturning Moment

The Overturning Moment applies to Flat-Faced Translating Followers.

The Overturning Moment is the product of the distance from the contact-point to the sliding axis of the translating-follower.

We define the Re-Turning-Moment as the product of the reaction-forces that act on the linear-bearing and the distance between the linear-bearings (or the length of a linear-bearing)

OverturningMoment-1a

Understanding Overturning-Moment

Imagine you want to slide a plank of wood along the floor, but between two walls. The plank is 2m long, 1m wide, and the 2m length is along the walls. There is a small gap.

The Plank is uniform. However, it is cut so that it contacts the wall only at its ends, 2m apart.

Ff (against the floor) = Friction-Coefficient (μ) × Mass × Gravity (mg). (Note: do not confuse Friction-Coefficient, μ, and Pressure-Angle, μ).

If the friction force is 'uniform', the Friction-Force vector will be through the center-of-Mass and parallel to the wall - nominally 0.5m away from the wall.

No Over-turning Motion

If you slide the plank (slowly) with a hand that is in line with the center-of-mass, the plank will not tend to rotate.

In the ideal case, the wall does not need to react against the sides of the plank. The wall does not, theoretically, need to be there. Although it would be difficult to do without the walls!


OverturningMoment-2a

IMPORTANT: ignoring Friction against the Wall.

Overturning Moment

Now, push the plank at a point nearer to one wall (so that the contact is offset from center-of-mass 'axis' and friction-force vector).

Overturning Moment = Frictional force(Ff) × perpendicular distance to your hand from plank's center-of-mass 'axis'.

Assuming the Friction-Force does not change, the further away from the plank's center-of-mass you push, the greater the over-turning moment.

The overturning-moment will tend to rotate the plank. The wall prevents it rotating, of course.

Re-turning Moment (to coin a term)

The wall resists the plank at opposite corners of the plank (nominally)

Re-turning moment = Reaction Forces from the walls against the plank × length of plank

If you use a longer plank, the reaction-forces from the wall become less, but the returning-moment remains the same.


Over-turning moment = Re-turning moment

Over-turning moment = Contact-Force × distance to the 'sliding-axis' from the contact-point.

Re-turning moment = Reaction-Force at one end of the Sliding-Joint × distance between reaction forces in Sliding-Joint.

GA-OverturningMoment

In the image: we can see the contact-force at the top of a 'cam' between the cam and the flat-faced-follower.

The contact force is 20N, and it is 50mm from the vertical sliding axis of the cam-follower.

Thus:

Overturning Moment (1000N.mm) = Contact Force (20N) × Distance to Contact Point from Sliding-Axis (50mm)

Overturning-Moment = Re-turning-Moment

Re-turning Moment (1000N.mm) = Distance between Sliding-Bearings (30mm) × Reaction Force (33.33N)


Notes:

During a typical cam-follower motion, the contact point continually moves across the flat-face from one side to the other.

This analysis ignores any friction-force in the sliding-bearings and friction-force between the cam and the cam-follower.

How to reduce the Over-turning Moment:

Improve the motion design.

Can you increase the duration of a segment? Can you change the cam-law to one with a lower peak velocity?

Move the cam-follower's sliding-axis relative to the cam's center.

Change the the 'translating follower' to an 'offset translating follower' (Move the Sliding-Axis towards the Contact-Point.

This only applies when the contact point is different for each direction of the followers motion.

Increase the size of the cam

This does not make any difference to the contact-force or overturning moment. However, the cam's radius of curvature increases, which will reduce its Hertzian Contact-stress.

Increase the length of the 'Sliding-Joint' - move bearing apart as far as possible.

The contact force at each of the sliding-joint is inversely proportional to the length of the sliding-joint.