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
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Before we get started…
Last Time
Discussed relative constraints
x, y,
relative constraints
distance constraint
For each kinematic constraint, recall the procedure that provides what it
takes to carry out Kinematics Analysis
Identify and analyze the physical joint
Derive the constraint equations associated with the joint
Compute constraint Jacobian
q
Get
(RH of velocity equation
Get
(RH of acceleration equation, this is challenging in some cases
Today
Covering relative constraints:
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
Step 2: Constraint Equations
(
q
,
t
) = ?
Step 3: Constraint Jacobian
q
= ?
Step 4:
= ?
Step 5:
= ?
3
Translational Joint
Step 1: Physically, it allows relative translation between two bodies
along a
common axis
. No relative rotation is allowed.
Step 2: Constraint Equations
(
q
,
t
) = ?
Step 3: Constraint Jacobian
q
= ?
Step 4:
= ?
Step 5:
= ?
4
NOTE:
recall
notation
Attributes of a Constraint
[it’ll be on the exam]
What do you need to specify to completely define a certain type of constraint?
In other words, what are the attributes of a constraint; i.e., the parameters that define it?
For absolute

x constraint: you need to specify the body “
i
”, the point P that
enters the discussion, and the value that
x
P
should assume
For absolute

y constraint: you need to specify the body “
i
”, the point P that
enters the discussion, and the value that
y
P
should assume
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 c
1
and c
2
How about an absolute angle constraint? Think about it…
5
The Attributes of a Constraint
[
Cntd
.]
Attributes of a Constraint: That information that you are supposed to know
by inspecting the mechanism
It represents the
parameters
associated with the
specific
constraint that
you are considering
When you are dealing with a constraint, make sure you understand
What the input is
What the defining attributes of the constraint are
What constitutes the output (the algebraic equation[s], Jacobian,
,
, etc.)
6
The Attributes of a Constraint
[
Cntd
.]
Examples of constraint attributes:
For a revolute joint:
You know where the joint is located, so therefore you know
For a translational join:
You know what the direction of relative translation is, so therefore
you know
For a distance constraint:
You know the distance C
4
7
Example 3.3.4
Consider the slider

crank below. Come up with the set of kinematic
constraint equations to kinematically model this mechanism
Use the Cartesian (absolute) generalized coordinates shown in the picture
8
Example 3.3.2
–
Different ways of modeling
the same mechanism for
Kinematic
Analysis
9
Approach 1: bodies 1, 2, and 3
Approach 2: bodies 1 and 3
Approach 3: bodies 1 and 2
Approach 4: body 2
Errata:
Page 67 (sign)
10
Page 68 (unbalanced
parentheses, and text)
Page 73 (transpose and sign)
Page 73 (perpendicular sign, both equations)
Composite Joints (CJ)
Just a means to eliminate one intermediate body whose
kinematics you are not interested in
11
Revolute

Revolute CJ
Also called a coupler
Practically eliminates need of
connecting rod
Given to you (joint attributes):
Location of points P
i
and P
j
Distance d
ij
of the massless rod
Revolute

Translational CJ
Given to you (joint attributes):
Distance c
Point P
j
(location of revolute joint)
Axis of translation
v
i
’
Composite Joints
One follows exactly the same steps as for any joint:
Step 1: Physically, what type of motion does the joint allow?
Step 2: Constraint Equations
(
q
,
t
) = ?
Step 3: Constraint Jacobian
q
= ?
Step 4:
= ?
Step 5:
= ?
12
We will skip
Gears
(section 3.4)
13
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
14
What’s known:
Angles
i
and
j
The radii R
i
and R
j
You need to express
as a
function of these four
quantities plus the
orientation angles
i
and
j
Kinematically: P
i
P
j
should
always be perpendicular to
the contact plane
Gears

Discussion of Figure 3.4.2
(Geometry of gear set)
15
Gears

Discussion of Figure 3.4.2
(Geometry of gear set)
16
Note: there are a couple of mistakes
in the book, see Errata slide before
Example: 3.4.1
Gear 1 is fixed to ground
Given to you:
1
= 0 ,
1
=
/6,
2
=7
/6 , R
1
= 1, R
2
= 2
Find
2
as gear 2 falls to the position shown (carrier line P
1
P
2
becomes vertical)
17
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 P
i
P
j
is
imposed (just like for
convex

convex gears)
18
Example: 3.4.1
Gear 1 is fixed to ground
Given to you:
1
= 0 ,
1
=
/6,
2
=7
/6 , R
1
= 1, R
2
= 2
Find
2
as gear 2 falls to the position shown (carrier line P
1
P
2
becomes vertical)
19
Rack and Pinion Preamble
Framework:
Two points P
i
and Q
i
on body i
define the rack center line
Radius of pitch circle for pinion is R
j
There is no relative sliding between
pitch circle and rack center line
Q
i
and Q
j
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 R
i
to infinity, and
the pitch line for gear i will become
the rack center line
20
Rack and Pinion Kinematics
Kinematic constraints that define
the relative motion:
At any time, the distance between
the point P
j
and the contact point
D should stay constant (this is
equal to the radius of the gear R
j
)
The length of the segment Q
i
D
and the length of the arc Q
j
D
should be equal (no slip condition)
Rack

and

pinion removes two
DOFs of the relative motion
between these two bodies
21
Rack and Pinion Pair
Step 1: Understand the physical element
Step 2: Constraint Equations
(
q
,
t
) = ?
Step 3: Constraint Jacobian
q
= ?
Step 4:
= ?
Step 5:
= ?
22
End Gear Kinematics
Begin Cam

Follower Kinematics
23
Preamble
: Boundary of a Convex Body
Assumption: the bodies we are dealing with are convex
To any point on the boundary corresponds
one
value of
the angle
(this is like the yaw angle, see figure below)
24
The distance from the reference
point Q to any point P on the
convex
boundary is a function of
:
It all boils down to expressing
two
quantities as functions of
The position of P, denoted by
r
P
The tangent at point P, denoted by
g
Cam

Follower Pair
Assumption: no chattering takes place
The basic idea: two bodies are in contact, and at the contact point the
two bodies share:
The contact point
The tangent to the boundaries
25
Recall that a point is located by the
angle
i
on body i, and
j
on body j.
Therefore, when dealing with a cam

follower, in addition to the x,y,
coordinates for
each
body one needs
to rely on one additional generalized
coordinate, namely the contact point
angle
:
Body i: x
i
, y
i
,
i
,
i
Body j: x
j
, y
j
,
j
,
j
Cam

Follower Constraint
Step 1: Understand the physical element
Step 2: Constraint Equations
(
q
,
t
) = ?
Step 3: Constraint Jacobian
q
= ?
Step 4:
= ?
Step 5:
= ?
26
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