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ME451 Kinematics and Dynamics
of Machine Systems
(Gears)
Cam-Followers and Point-Follower 3.4.1, 3.4.2
September 27, 2013
Radu SerbanUniversity of Wisconsin-Madison
2
Before we get started…
Last time: Relative constraints (revolute, translational) Composite joints (revolute-revolute, revolute-translational)
Today: Gears Cam – Followers Point – Follower
Assignments: HW 5 – due September 30, in class (12:00pm) Matlab 3 – due October 2, Learn@UW (11:59pm)
Gears (convex-convex, concave-convex, rack and pinion)
3.4.1
4
Gears Convex-convex gears
Gear teeth on the periphery of the gears cause the pitch circles shown to roll relative to each other, without slip
First Goal: find the angle , that is, the angle of the carrier
What’s known: Angles i and j
The radii Ri and Rj
You need to express as a function of these four quantities plus the orientation angles i and j
Kinematically: PiPj should always be perpendicular to the contact plane
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Gears - Discussion of Figure 3.4.2 (Geometry of gear set)
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Gears - Discussion of Figure 3.4.2 (Geometry of gear set)
Note: there are a couple of mistakes in the book, see Errata slide
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Gear Set Constraints
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Example: 3.4.1
Gear 1 is fixed to ground Given to you: 1 = 0 , 1 = /6, 2=7/6 , R1 = 1, R2 = 2
Find 2 as gear 2 falls to the position shown (carrier line P1P2 becomes vertical)
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Gears (Convex-Concave)
Convex-concave gears – we are not going to look into this class of gears
The approach is the same, that is, expressing the angle that allows on to find the angle of the
Next, a perpendicularity condition using u and PiPj is imposed (just like for convex-convex gears)
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Rack and Pinion Preamble
Framework: Two points Pi and Qi on body i
define the rack center line Radius of pitch circle for pinion is Rj
There is no relative sliding between pitch circle and rack center line
Qi and Qj are the points where the rack and pinion were in contact at time t=0
NOTE: A rack-and-pinion type kinematic
constraint is a limit case of a pair of convex-convex gears Take the radius Ri to infinity, and
the pitch line for gear i will become the rack center line
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Rack and Pinion Kinematics
Kinematic constraints that define the relative motion: At any time, the distance between
the point Pj and the contact point D should stay constant (this is equal to the radius of the gear Rj)
The length of the segment QiD and the length of the arc QjD should be equal (no slip condition)
Rack-and-pinion removes two DOFs of the relative motion between these two bodies
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Rack and Pinion Constraints
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Errata:
Page 73 (transpose and signs)
Page 73 (perpendicular sign, both equations)
Cam – Followers3.4.2
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Cam – Follower Pair
Setup: Two shapes (one on each body) that are always in contact (no chattering) Contact surfaces are convex shapes (or one is flat) Sliding is permitted (unlike the case of gear sets)
Modeling basic idea: The two bodies share a common point The tangents to their boundaries are collinear
Source: Wikipedia.org
16
Interlude: Boundary of a Convex Shape (1)
Convex shape assumption any point on the boundary is defined by a unique value of the angle .
The distance from the reference point to any point on the convex boundary is a function of :
We need to express two quantities as functions of : The position of , that is The tangent at , that is
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In the LRF:
where
and therefore
[handout]
Interlude Boundary of a Convex Shape (2)
In the GRF:
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Cam – Follower Pair
Step 1 The two bodies share the contact point: (2 constraints)
The two tangents are collinear: (1 constraint)
Recall that points and are located by the angles i and j, respectively.
Therefore, in addition to the coordinates for each body, one needs to include one additional generalized coordinate, namely the angle :
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Cam – Follower Constraints
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Example 3.4.3
Determine the expression of the tangents g1 and g2
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Cam – Flat-Faced Follower Pair
A particular case of the general cam-follower pair Cam stays just like before Flat follower Typical application: internal combustion engine Not covered in detail, HW touches on this case
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Errata:
Page 83(Q instead of P)
Page 80(subscript ‘j’ instead of ‘i’)
Point – Follower3.4.3
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Point – Follower Pair
Setup: Pin , attached to body can move (slide
and rotate) in a slot attached to body Modeling basic idea:
Very similar to a revolute joint, except… …point moves on body Location of point on body is
parameterized by the angle Therefore, in addition to the coordinates
for body , one needs to include one additional generalized coordinate, namely the angle :
Note: this diagram is more general than theone in the textbook (includes point )
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Point – Follower Constraints