Kinematics Review

1
Kinematics Review
Kinematics Study of relative motion Mechanism
Mechanical device that transfers or transforms
motion or energy from an input source to an
output
2
Four-bar Linkage
Four-bar Linkage - simplest close-loop mechanism
has four links and four joints.
Tracer or Coupler Point
P
Coupler link
Input link
3
Output link
4
2
1
1
Fixed link (ground)
3
Linkage Analysis Synthesis

• Analysis - analysis is determining motion of a
  mechanism with known geometry
• Synthesis - determining linkage geometry to
  perform a specified task

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Link Types
Binary Link
Ternary Link
Quaternary Link
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Joint Types
Slider and link
Pin (revolute) joint and ground
Slider (prismatic joint) and ground
Cam and follower (may roll or slide)
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Kinematic Pairs

• Lower Pairs - joints that allow one degree of
  freedom of relative motion. Pin (revolute)
  joints, sliders (prismatic joints), and screws
  are examples.
• Higher Pairs - allow two degrees of freedom of
  relative motion between mating links. Cams are
  examples

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Degrees of Freedom

• Degrees of Freedom (DOF) - the number of
  independent inputs required to define the
  position of all links of a mechanism

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Grueblers Equation

• Grueblers Equation
• Where
• n number of links
• f1 number of lower pairs
• f2 number of higher pairs

F 3(n-1) - 2f1 - f2
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Mechanism Inversions
Mechanism inversions are produced by making
another link the fixed link.
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Grashofs Law

• Grashofs Law - for at least one link to have
  full rotation
• Where
• s - the length of the shortest link
• l - the length of the longest link
• p and q - the lengths of the other links

s l lt p q
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Position Displacement
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Displacement Vector Method
R2 R3 R1 R4
R3
or r2eiq2 r3eiq3 r1eiq1 r4eiq4
r3
q3
R4
r2
the real part r2cosq2 r3cosq3 r1cosq1
r4cosq4
r4
R2
q4
q2
r1
The imaginary part ir2sinq2 ir3sinq3
ir1sinq1 ir4sinq4
R1
q1
Two equations and two unknowns (q3 and q4)
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Case 1B Motion
Velocity of points on a pinned link
Position vector of point A RA RAeiqA
Velocity of point A
dRA
dqA
VA i RAeiqA iw2RA
dt
dt
w - counter clockwise is positive
Note The velocity vector is perpendicular to the
position vector
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Instant Centers

• An instant center is a point at which there is no
  relative velocity between two links of a
  mechanism at that instant
• Does not work for acceleration

V
A
A
V


B
B
A
0
(1
,
2
)
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Kennedys Theorem

• The three instantaneous centers of three bodies
  moving relative to one another must lie along a
  straight line

3
4
(2
,
3
)
w
2
w
(1,4 -
2,4)
2

w
(1,2 -
2,4)
4
2
(1
,
4
)
(1
,
2
)
Relates an angular velocity ratio to a ratio of
distances between instant centers 1. If the
relative IC lies between the absolute ICs the
angular velocity ratio is negative 2. If the
relative IC lies outside the other two, the
angular velocity ratio is positive
1
Where is (2,4)?
(1,3)
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Instant Centers for Mechanisms
N of instant centers n of links
6
1
5
3
2
4
1
1
Sliders one IC is at the link and the slider and
the other IC has infinite locations in the
direction perpendicular to the sliding direction
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Complex Velocity Analysis

• Differentiate vector loop equations from
  displacement analysis
• Solve for unknown velocity terms (these are
  linear equations)
• Solve for velocities of other points (such as
  coupler points)

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Complex Number Method 4-bar
r2eiq2 r3eiq3 r1eiq1 r4eiq4 differentiate
displacement equation ir2w2eiq2 ir3w3eiq3
ir4w4eiq4
R3
q3
r3
R4
r2
r4
R2
the real part -r2w2sinq2 - r3w3sinq3 -
r4w4sinq4 the imaginary part ir2w2cosq2
ir3w3cosq3 ir4w4cosq4
q4
q2
w2
r1
R1
q1
Two equations and two unknowns, solve for (w3 and
w4)
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Acceleration of a Point Pinned to Ground
Position vector of point A RA
RAeiqA Velocity of point A
dRA
dqA
iw2RA
VA
iRA
eiqA
dt
dt
Acceleration of point A
d2RA
dVA
AA

iRAa eiqA - w2RA eiqA
dt2
dt
AA RA (ia - w2 )
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Relative Acceleration
AB AA ABA Analytically
dVBA
d(RBAiw)
RBAia eiqA - RBAw2 eiqA
ABA

dt
dt
RBA(ia -w2 )
Every point on a rigid body has the same angular
accleration, a.
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Coriolis Acceleration
Find the acceleration of point A when a2 0
where RA r2eiq2
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Acceleration of Center of Mass

• Accelerations

A2 rg2a2 ieiq2 - rg2w22eiq2 rg2eiq2 (a2 i -
w22)
a
3
A
A3
3
a
4
A
a
4
2
w
2
A
2
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Kinetostatic Analysis

• Inertia Forces

Fo2 -m2A2 -m2 rg2eiq2 (a2 i - w22) To2
-Ig2a2
F
o3y
T
o3
F
o3x
F
Fo3 To3
o4y
F
o4x
T
F
o4
o2y
T
o2
F
o2x
T
in
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Kinetostatic Analysis
Note The sum of the vector ground forces and
torques is called the shaking force and shaking
torque, respectively.
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Linkage Synthesis and Analysis

• Analysis is used to determine a mechanisms
  performance when its desired dimensions are known
• Synthesis is used to design a mechanism when the
  desired performance is known
• Synthesis usually involves multiple analyses

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Precision Points

• In general, a mechanism cannot be designed to
  follow an arbitrarily specified path. Therefore,
  a mechanism is designed to match the desired path
  exactly in some positions and to minimize the
  overall error.
• Precision points - points where the mechanism
  motion exactly meets the prescribed motion
• Structural error - difference between mechanism
  path and the prescribed path

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Graphical Synthesis - Two Positions
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Graphical Synthesis - Three Positions
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Complex Number Method
d
j
r
P
S
P
6
j
6
q
6
B
Z
r
5
5
g
a
j
j
V
r
3
link
4
4
r
3
q
5
q
U
link
3
3
4
A
b
r
j
2
posit
ion
j
pos
it
ion
1
link
2
W
2
q
q
r
4
2
1
A
o
B
G
o
1
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Dyad or Standard Form - First Dyad
d
j
P
P
j
From the vector loop shown W2 Z5 dj - W2eibj
- Z5eiaj 0 W2(eibj - 1) Z5(eiaj - 1) dj
Z
5
a
j
b
j
W
2
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Dyad or Standard Form - Second Dyad
From the vector loop shown U4 S6 dj - U4eigj
- S6eiaj 0 U4(eigj - 1) S6(eiaj - 1) dj
d
j
P
P
j
S
6
a
j
g
j
U
4
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Three Position Synthesis Cont.
Use Cramers rule to solve for the unknowns