![Page 1: Tolerance Analysis of Assemblies Using Kinematically-Derived Sensitivities Paul Faerber Motorola Corporation – Lawrenceville, GA Presented by: Jeff Dabling](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d385503460f94a10dd1/html5/thumbnails/1.jpg)
Tolerance Analysis of Assemblies Using
Kinematically-Derived Sensitivities
Paul FaerberMotorola Corporation – Lawrenceville, GA
Presented by:Jeff Dabling
BYU – Graduate Student
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• Differences (Kinematic)• Dimensions not allowed to vary• Requires sensitivity to velocity
inputs• Multiple Position Analysis
Similarities• Both use vector loops• Both use kinematic joints• Both use sensitivities
• Differences (Tolerance)• Dimensions allowed to vary
• Requires sensitivity to dimensional variation
• Single position analysis
Assembly
NetworkGraph
Skeleton Diagram
VectorLoops
VariationAnalysis
IncrementPosition (ifmechanism)
DimensionalVariation
Inputs
Mechanism
NetworkGraph
Skeleton Diagram
VectorLoops
Velocity andAcceleration
Analysis
IncrementMechanism
PositionKinematicInputs
Tolerance Analysis Kinematic Analysis
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Research Objectives
• Model static assemblies with a kinematic modeler• Extract tolerance sensitivities from a kinematic
solver• Perform tolerance analysis on a mechanism in
multiple positions• Assess difficulty of adding tolerance analysis to
commercial kinematic software
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Tolerance Analysis of Mechanisms
Tolerance models for eachposition of interest
1
2 4
3
Current Method:
2
1
4
3
Solid ModelNew Position
Solid ModelNew Position
Solid ModelNew Position
OriginalSolid Model
1
2 4
3
1
2 4
3
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Merging Kinematic and Tolerance Analyses
MultiplePositions
Tolerance models for each position
SensitivitiesKinematic/ Dynamic Model
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Kinematic Analysis
2
1
4
3
r2
r3
r4
r1
3
2
1
4
4-bar mechanism Vector loop equation
Resulting velocity equations
Matrix formulation
r e r e r e r ei i i i1 2 3 4
1 2 3 4 0( ) ( ) ( ) ( )
2 2 2 3 3 3 4 4 4
2 2 2 3 3 3 4 4 4
0
0
r r r
r r r
sin sin sin
cos cos cos
3
4
12
2 2 4 4 2
3 4 3 3 4
2 2 3 3 2
4 4 3 3 4
2
B A
r
rr
r
(cos sin cos sin )
(cos sin cos sin )(cos sin cos sin )
(cos sin cos sin )
A B2
3
40
Br r
r r
3 3 4 4
3 3 4 4
sin sin
co s co s
A
r
r
2 2
2 2
sin
cos
Kinematic Solution
Outputs Input
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Tolerance Analysis
2
1
4
3
r2
r3
r4
r1
3
2 1
4
3
2
4
4-bar mechanism Vector loop equations
Linearized equations
r e r e r e r ei i i i1 2 3 4
1 1 2 1 2 3 1 2 3 4 0( ) ( ) ( ) ( )
co s co s co s co s ( sin sin sin )
( s in sin ) sin
sin sin sin sin ( co s co s co s )
( co s co s ) co s
1 1 2 2 3 3 4 4 2 2 3 3 4 4 2
3 3 4 4 3 4 4 4
1 1 2 2 3 3 4 4 2 2 3 3 4 4 2
3 3 4 4 3 4 4 4
0
0
dr dr dr dr r r r d
r r d r d
dr dr dr dr r r r d
r r d r d
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Tolerance Analysis, continued
4-bar mechanism Matrix Formulation
Tolerance Analysis Solution (non-statistical)
A
d
dr
dr
dr
dr
Bd
d
2
1
2
3
4
3
4
0
A r r r
r r r
2 2 3 3 4 4 1 2 3 4
2 2 3 3 4 4 1 2 3 4
sin sin sin cos cos cos cos
cos cos cos sin sin sin sin
B
r r r
r r r
3 3 4 4 4 4
3 3 4 4 4 4
sin sin sin
cos cos cos
dr4 dr4
d3
d4
B A
d2
dr1
dr2
dr3
Si,j
d2
dr1
dr2
dr3
1
Outputs
Inputs
2
1
4
3
r2
r3
r4
r1
3
2 1
4
3
2
4
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Estimated Tolerance Accumulation4-bar mechanism
r2
r3
r4
r1
3
2 1
4
3
2
4
d dr
drr
drr
drr
dr
dr
drr
drr
drr
dr
33
22
2 3
11
2 3
22
2 3
33
2 3
44
2
44
22
2 4
11
2 4
22
2 4
33
2 4
44
2
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
RSS
RSS
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Observations
r2
r3
r4
r1
3
2
1
4 3
2
4
3
42
2 2 4 4 2
3 4 3 3 4
2 2 3 3 2
4 4 3 3 4
2
J
r
rr
r
ij
(co s sin co s sin )
(co s sin co s sin )(co s sin co s sin )
(co s sin co s sin )
d
dB A
d
dr
dr
dr
dr
S
d
dr
dr
dr
dr
i j
3
4
1
2
1
2
3
4
2
1
2
3
4
,
Kinematic Analysis Solution
Tolerance analysis solution (non-statistical)
2 x 1
2 x 5
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Kinematic Analysis of an Equivalent Variational Mechanism
r2
r3
r4
r1
3
2
1
4
r e r e r e r ei i i i1 2 3 4
1 2 3 4 0( ) ( ) ( ) ( )
co s co s co s co s sin sin sin
sin sin sin sin co s co s co s
r r r r r r r
r r r r r r r1 1 2 2 3 3 4 4 2 2 2 3 3 3 4 4 4
1 1 2 2 3 3 4 4 2 2 2 3 3 3 4 4 4
0
0
A
r
r
r
r
B
2
1
2
3
4
3
40
Br r
r r
3 3 4 4
3 3 4 4
sin sin
cos cos
3
4
1
2
1
2
3
4
2
1
2
3
4
B A
r
r
r
r
J
r
r
r
r
i j
,
Ar
r
2 2 1 2 3 4
2 2 1 2 3 4
sin co s co s co s co s
co s sin sin sin sin
Kinematic analysis solutionMatrix formulation
Resulting velocity equations
Vector loop equation
(Dimension ri not constant)
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Comparisons
Tolerance analysis solution (non-statistical)
Kinematic analysis of equivalent mechanism solution
d S d S dr S dr S dr S dr
d S d S dr S dr S dr S dr
3 1 1 2 1 2 1 1 3 2 1 4 3 1 5 4
4 2 1 2 2 2 1 2 3 2 2 4 3 2 5 4
, , , , ,
, , , , ,
4
3
2
1
2
,
4
3
2
1
2
1
4
3
r
r
r
r
J
r
r
r
r
AB ji
4
3
2
1
2
,
4
3
2
1
2
1
4
3
dr
dr
dr
dr
d
S
dr
dr
dr
dr
d
ABd
dji
3 1 1 2 1 2 1 1 3 2 1 4 3 1 5 4
4 2 1 2 2 2 1 2 3 2 2 4 3 2 5 4
J J r J r J r J r
J J r J r J r J r, , , , ,
, , , , ,
Are the sensitivities the same?
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Transformation to Relative AnglesRelative rotations Differential rotations Psuedo-velocities
Kinematic analysis solution
Tolerance analysis solution (after substitution)
dt
drJ
dt
drJ
dt
drJ
dt
drJ
dt
dJ
dt
d 45,1
34,1
23,1
12,1
21,1
3
dt
drJ
dt
drJ
dt
drJ
dt
drJ
dt
dJ
dt
d 45,2
34,2
23,2
12,2
21,2
4
d
dtJ
d
dtJ
dr
dtJ
dr
dtJ
dr
dtJ
dr
dt
31 1
21 2
11 3
21 4
31 5
41 ( ), , , , ,
d
dtJ J
d
dtJ J
dr
dtJ J
dr
dtJ J
dr
dtJ J
dr
dt4
2 1 1 12
2 2 1 21
2 3 1 32
2 4 1 43
2 5 1 54 ( ) ( ) ( ) ( ) ( ), , , , , , , , , ,
3
dt
d
dt
d
dt
d 233
dt
d
dt
d
dt
d 344 344 ddd
233 ddd 233
344
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• Uses kinematic elements to represent dimensional variations in a kinematic model of the assembly
• ri are kinematic inputs
• ri are proportional to dimensional tolerances
Equivalent Variational Mechanisms
r2
r3
r4
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• Vector loops identical for both types of analyses
• Use tolerance analysis techniques to develop vector loops for assemblies
• Use these vector loops as a starting point in developing EVM
Equivalent Variational Mechanisms
Mechanism
NetworkGraph
Skeleton Diagram
VectorLoops
Velocity andAcceleration
Analysis
IncrementMechanism
Position
Kinematic Analysis
EquivalentVariational
Inputs
KinematicInputs
Tolerance Analysis
Assembly
NetworkGraph
Skeleton Diagram
VectorLoops
VariationAnalysis
IncrementPosition (ifmechanism)
DimensionalVariation
Inputs
EquivalentVariationalMechanism
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The Stack Blocks Assembly
Frame
BlockCylinder
Gap
p
b
c
a
r
dn
e
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Creating Vector Loop Assembly Models
Gap
Frame
Cylinder
Block
Loop 2
Loop 1
Loop 3
Kinematic JointsNetwork Graph
Cylindrical DRF
FeatureDatum Cylinder
Block
Frame
Rectangular DRF
Part and Feature Reference Frames
Closed Loop #1 Closed Loop #2 Open Loop #1
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Dimensional Variations
± d ddt
Skeleton modelEquivalent jointAngular dimension
ddt
dl
Skeleton modelEquivalent JointIndependent variation
l ± dl
dldt
Slider Joint
Each vector represents a link in the EVM
Angular Variations
Linear Variations
Stacked Blocks Model Dimensional Variations
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Equivalent Kinematic Joints
PlanarJoint
Joint SkeletonModel
EquivalentJoint
CylindricalSlider Joint
EdgeSliderJoint
ParallelCylinders
Jointcenter of curvature
Stacked Blocks Model Dimensional Variations Kinematic Variations
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Stacked Blocks Model (completed) Dimensional Variations Kinematic Variations Fixed Joints Pin Joints
EVM Stack Blocks Assembly
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EVM Modeling Techniques
• Extracting Sensitivities from Kinematic Solver• Unit velocities are applied to each
independent joint, one at a time. Resulting dependent variables represent the row of the tolerance sensitivity matrix corresponding to that joint.
Independent variables:r2, r3, r4, 2
Dependent variables:2, 3, 4
r2 = 1
r3 = 0
r4 = 0
4
3
2
1
2
5,24,23,22,21,2
5,14,13,12,11,1
4
3
r
r
r
r
JJJJJ
JJJJJ
22
3
4
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Variation Results for Stack Blocks
a b c e n g q r theta
dgap -0.306 0.306 -1 -1.046 1 0 1.231 -3.495 -0.197
Gap Sensitivities
dgap S dxi j j ( ),2
dgapda db dr de dn
dg dq dr d theta
( . ) ( . ) ( ) ( . ) ( )
( ) ( . ) ( . ) ( . )
0 3 0 6 0 3 0 6 1 1 0 4 6 1
0 1 2 3 1 3 4 9 5 1 9 7
2 22
2 2 2
2 2 2 2
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Tolerance Analysis Using Equivalent Variational Mechanisms
• Variations in link lengths are allowed by including slider elements in each link.
• Sensitivities are used to form RSS expressions used in statistical tolerance analysis.
• Kinematic modeler moves to the next location, and the process is repeated.
Dependent variables:2, 3, 4
Independent variables:r2, r3, r4, 2
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Modeling Techniques for Mechanisms
• Apply independent velocities as a reciprocating time function• Each new time step (second) is a new analysis point at the nominal
link-lengths
• Controlling the period of the function controls the resolution of the analysis, which affects the accuracy of the analysis
• Cosine Function• Full magnitude at time = 0
• Period equal to the time step
r
a
-a
period r= acos(bt)
period =2b
amplitude = a
r
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Variation Results for Four-bar Mechanism
Tolerance Sensitivities
-4.50
-4.00
-3.50-3.00
-2.50
-2.00
-1.50
-1.00
-0.500.00
0.50
1.00
0.00E+00 1.00E+02 2.00E+02 3.00E+02 4.00E+02
Angular Position of Input Link
Se
nsi
tivi
ty
dalpha3/dtheta2
dalpha4/dtheta2
dalpha3/dr2
dalpha4/dr2
dalpha3dr3
dalpha4/dr3
dalpha3/dr4
dalpha4/dr4
r2
r3
r4
r1
3
21
4
3
2
4
Output Variation
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 60 120 180 240 300 360
Input Angle
Var
iati
on
dalpha3
dalpha4
Sensitivities vary with position(r1, 1 can also vary)
Greatest sensitivity at 270º
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Contributions
• Defined relationship between kinematic and tolerance analyses
• Developed method for creating and analyzing equivalent variational mechanisms (EVM)• Equivalent 2-D kinematic joints presented• Demonstration of method on static assemblies, as well as
kinematic
• Demonstrated using commercial kinematic software, ADAMS• Method for extracting kinematic sensitivities• Method for returning the model to its nominal dimensions at each
time step
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Recommendations for Future Research
• Investigate relationship between the higher kinematic derivatives (acceleration and jerk) and the higher statistical moments (skewness and kurtosis)
• Integrate with commercial kinematics CAD applications– Develop a user interface
– Study degree of freedom problems
• Extend into three-dimensional assemblies
• Include form tolerances in this method