Transcript Gears (convex-convex, concave

ME451
Kinematics and Dynamics
of Machine Systems
(Gears)
Cam-Followers and Point-Follower
3.4.1, 3.4.2
September 27, 2013
Radu Serban
University of Wisconsin-Madison
Before we get started…
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Last time:
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Today:
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Relative constraints (revolute, translational)
Composite joints (revolute-revolute, revolute-translational)
Gears
Cam – Followers
Point – Follower
Assignments:
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HW 5 – due September 30, in class (12:00pm)
Matlab 3 – due October 2, Learn@UW (11:59pm)
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3.4.1
Gears
(convex-convex, concave-convex, rack and pinion)
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Gears
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Convex-convex gears
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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
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What’s known:
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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
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
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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)
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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
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Framework:
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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:
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A rack-and-pinion type kinematic
constraint is a limit case of a pair of
convex-convex gears
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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
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Kinematic constraints that define
the relative motion:
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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
Rack and Pinion Constraints
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Errata:
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Page 73
(transpose and signs)
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Page 73
(perpendicular sign,
both equations)
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3.4.2
Cam – Followers
Cam – Follower Pair
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Setup:
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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:
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The two bodies share a common point
The tangents to their boundaries are collinear
Source: Wikipedia.org
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Interlude: Boundary of a Convex Shape (1)
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Convex shape assumption ⇒ any point on the boundary is
defined by a unique value of the angle .
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The distance from the reference point 𝑄𝑖
to any point 𝑃𝑖 on the convex boundary is
a function of :
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We need to express two quantities as
functions of :
𝑃
 The position of 𝑃𝑖 , that is 𝐫𝑖
 The tangent at 𝑃𝑖 , that is 𝐠
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[handout]
Interlude Boundary of a Convex Shape (2)
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In the LRF:
where
and therefore
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In the GRF:
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Cam – Follower Pair
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Step 1
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The two bodies share the contact point:
(2 constraints)
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The two tangents are collinear:
(1 constraint)
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Recall that points 𝑃𝑖 and 𝑃𝑗 are
located by the angles i and j,
respectively.
Therefore, in addition to the 𝑥, 𝑦, 𝜙 𝑇
coordinates for each body, one needs
to include one additional generalized
coordinate, namely the angle :
Cam – Follower Constraints
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Example 3.4.3
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Determine the expression of the tangents g1 and g2
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Cam – Flat-Faced Follower Pair
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A particular case of the general cam-follower pair
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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:
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Page 80
(subscript ‘j’ instead of ‘i’)
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Page 83
(Q instead of P)
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3.4.3
Point – Follower
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Point – Follower Pair
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Setup:
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Modeling basic idea:
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Note: this diagram is more general than the
one in the textbook (includes point 𝑄𝑗 )
Pin 𝑃, attached to body 𝑖 can move (slide
and rotate) in a slot attached to body 𝑗
Very similar to a revolute joint, except…
…point 𝑃 moves on body 𝑗
Location of point 𝑃 on body 𝑗 is
parameterized by the angle 𝛼𝑗
Therefore, in addition to the 𝑥𝑗 , 𝑦𝑗 , 𝜙𝑗
coordinates for body 𝑗, one needs to
include one additional generalized
coordinate, namely the angle 𝛼𝑗 :
𝑇
Point – Follower Constraints
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