ME451
Kinematics and Dynamics
of Machine Systems
Relative Kinematic Constraints, Composite Joints – 3.3
October 6, 2011
© Dan Negrut, 2011
ME451, UW-Madison
“I want to put a ding in the universe.”
Steve Jobs
Before we get started…
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Last Time
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Discussed relative constraints
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x, y,  relative constraints
distance constraint
For each kinematic constraint, recall the procedure that provides what it
takes to carry out Kinematics Analysis
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

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
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Today
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Covering relative constraints:
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Identify and analyze the physical joint
Derive the constraint equations associated with the joint
Compute constraint Jacobian q
Get  (RHS of velocity equation)
Get  (RHS of acceleration equation, this is challenging in some cases)
Revolute, translational, composite joints, cam-follower type constraints
Skipping gears
Assignment, due in one week

3.3.2, 3.3.4, 3.3.5 + MATLAB + ADAMS
2
Revolute Joint
Step 1: Physically imposes the condition that point P on body i and a
point P on body j are coincident at all times
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Step 2: Constraint Equations (q,t) = ?
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Step 3: Constraint Jacobian q = ?
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Step 4:  = ?
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Step 5:  = ?
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Translational Joint
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Step 1: Physically, it allows relative translation between two bodies
along a common axis. No relative rotation is allowed.
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Step 2: Constraint Equations (q,t) = ?
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Step 3: Constraint Jacobian q = ?
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Step 4:  = ?
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Step 5:  = ?
NOTE: recall notation
4
Attributes of a Constraint
[it’ll be on the exam]
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What do you need to specify to completely define a certain type of constraint?
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In other words, what are the attributes of a constraint; i.e., the parameters that define it?
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For absolute-x constraint: you need to specify the body “i”, the point P that
enters the discussion, and the value that xP should assume
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For absolute-y constraint: you need to specify the body “i”, the point P that
enters the discussion, and the value that yP should assume
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For a distance constraint, you need to specify the “distance”, but also the
location of point P in the LRF, the body “I” on which the LRF is attached to, as
well as the point of coordinates c1 and c2
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How about an absolute angle constraint? Think about it…
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The Attributes of a Constraint
[Cntd.]
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Attributes of a Constraint: That information that you are supposed to know
by inspecting the mechanism
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It represents the parameters associated with the specific constraint that
you are considering
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When you are dealing with a constraint, make sure you understand
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What the input is
What the defining attributes of the constraint are
What constitutes the output (the algebraic equation[s], Jacobian, , , etc.)
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The Attributes of a Constraint
[Cntd.]
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Examples of constraint attributes:
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For a revolute joint:
 You know where the joint is located, so therefore you know
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For a translational join:
 You know what the direction of relative translation is, so therefore
you know
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For a distance constraint:
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You know the distance C4
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Example 3.3.4
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Consider the slider-crank below. Come up with the set of kinematic
constraint equations to kinematically model this mechanism
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Use the Cartesian (absolute) generalized coordinates shown in the picture
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Example 3.3.2 – Different ways of modeling
the same mechanism for Kinematic Analysis
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Approach 1: bodies 1, 2, and 3
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Approach 3: bodies 1 and 2
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Approach 2: bodies 1 and 3
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Approach 4: body 2
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Page 68 (unbalanced
parentheses, and text)
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Page 73 (transpose and sign)
Errata:
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Page 67 (sign)
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Page 73 (perpendicular sign, both equations)
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Composite Joints (CJ)
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Just a means to eliminate one intermediate body whose
kinematics you are not interested in
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Revolute-Revolute CJ
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Also called a coupler
Practically eliminates need of
connecting rod
Given to you (joint attributes):
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Revolute-Translational CJ
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Given to you (joint attributes):
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Distance c
Point Pj (location of revolute joint)
Axis of translation vi’
Location of points Pi and Pj
Distance dij of the massless rod
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Composite Joints
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One follows exactly the same steps as for any joint:
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Step 1: Physically, what type of motion does the joint allow?
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Step 2: Constraint Equations (q,t) = ?
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Step 3: Constraint Jacobian q = ?
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Step 4:  = ?
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Step 5:  = ?
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We will skip Gears
(section 3.4)
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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
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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 before
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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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Example: 3.4.1
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
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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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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
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Rack and Pinion Pair
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Step 1: Understand the physical element
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Step 2: Constraint Equations (q,t) = ?
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Step 3: Constraint Jacobian q = ?
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Step 4:  = ?
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Step 5:  = ?
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End Gear Kinematics
Begin Cam-Follower Kinematics
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Preamble: Boundary of a Convex Body
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Assumption: the bodies we are dealing with are convex
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To any point on the boundary corresponds one value of
the angle  (this is like the yaw angle, see figure below)
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The distance from the reference
point Q to any point P on the
convex boundary is a function of :
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It all boils down to expressing two
quantities as functions of 
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The position of P, denoted by rP
The tangent at point P, denoted by g
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Cam-Follower Pair
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Assumption: no chattering takes place
The basic idea: two bodies are in contact, and at the contact point the
two bodies share:
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The contact point
The tangent to the boundaries
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Recall that a point is located by the
angle i on body i, and j on body j.
Therefore, when dealing with a camfollower, in addition to the x,y,
coordinates for each body one needs
to rely on one additional generalized
coordinate, namely the contact point
angle :
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
Body i: xi, yi, i, i
Body j: xj, yj, j, j
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Cam-Follower Constraint
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Step 1: Understand the physical element
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Step 2: Constraint Equations (q,t) = ?

Step 3: Constraint Jacobian q = ?

Step 4:  = ?

Step 5:  = ?
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