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Oct 31, 2013 (3 years and 7 months ago)

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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