Including GD&T Tolerance Variation in a Commercial Kinematics Application Jeff Dabling Surety...
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Including GD&T Tolerance Variation in a Commercial
Kinematics Application
Jeff DablingSurety Mechanisms & IntegrationSandia National Laboratories
Research supported by:
Summary Variation Propagation Obtaining Sensitivities Variation/Velocity Relationship Equivalent Variational
Mechanisms in 2D EVMs in 3D Example in ADAMS
Dimensional and Kinematic
Geometric
3 Sources of Variation in Assemblies
U
A
A + A
U + U
R R + R
U
A
U + U
R R
ReelArm
Plunger
Baseb
re i
g
a
u
h RL
RLGap
RT
Open Loop
Closed Loop
CL
Pad
DLM Vector Assembly Model
The effect of feature variations in 3D depends upon the joint type and which joint axis you are looking down.
Rotational Variation
3D cylindrical slider joint
Nominal Circle
Cylindricity Tolerance Zone Translatio
nalVariation
Flatness Tolerance Zone
View looking down the cylinder axis
View normal to the cylinder axis
X
Y
Z
How Geometric Variation Propagates
Flatness Tolerance Zone
x
z
K
K
F
F
K Kinematic MotionF Geometric Feature Variation
x
y
z
K
KK
F
F
K K
F
Cylindrical Slider Joint Planar Joint
y
3D Propagation of Surface Variation
Variations Associated with Geometric Feature – Joint Combinations
(Gao 1993)
JointsGeom
Tol
Prismatic
Rx Rz Rx Rz Rx Rz Rx Rz RxRz Rx Rz RxRzTy
Rx RzRx Rz Rx Rz Rx Rz RxRz Rx Rz Rx Rz Tx Tz Tx Tz
Rx RzRx Rz Rx Rz Rx Rz RxRz Rx Rz Rx Rz Tx Tz Tx Tz
RxRyRz
Rx Rz
Rx Rz
RxRyRz RxRyRz RxRyRz RxRyRz RxRyRz
TxTyTz TxTyTz TxTyTz TxTyTz Tx TyTz
Ty Ty Ty Ty Ty Ty
TyTy Rx Ty Rx Ty Rx Ty Rx Rx Ty RxRxRx
EP
P
P
P
C
Pt
S
Ty Rx
Ty Rx Ty Rx
Ty RxTy Rx Ty Rx Ty Rx Ty Rx Ty Rx
Ty Rx Ty Rx Ty Rx Ty
Ty Rx
Ty Rx Ty Rx
Ty RxTy Rx Ty Rx Ty Rx Ty Rx Ty Rx
Ty Rx Ty Rx Ty Rx TyTy Rx Ty Rx Ty Rx
Ty
Ty Ty Ty Ty Ty Ty Ty
Ty
Ty Ty Ty Ty Ty Ty Ty
Ty Ty Ty
Cylindrical
Revolute
Planar
Spherical
CrsCyl
ParCyl
EdgSli
CylSli
PntSli
SphSli
Variables used have nominal values of zero
Variation corresponds to the specified tolerance value
Including Geometric Variation
LengthsticCharacteri
ZoneToleranceFlatness1tan
Rotational Variation
Flatness Tolerance = Zone
= ±
Characteristic Length
Rotational variation due to flatness variation between two planar surfaces:
Translational variation due to flatness variation:
TranslationalVariation
Flatness Tolerance = Zone
=±/2
2
VariationnalTranslatio
Geometric Variation Example
Translational: additional vector with nominal value of zero. (3, 4)
Rotational: angular variation in the joint of origin and propagated throughout the remainder of the loop. (1, 2)
A
.01
U1
R
U2
H
.02
.01
.01
U1
A
H
U2
R3
R2
R1
(3, 4)
(1, 2)
0)270cos()180cos(
)90cos()90cos()90cos(
)90cos()90cos()0cos()0cos(
2
3
211
AU
R
RHRUH x
1 1
11 1
1 1
2
22
2
2
2
2
3 4
Sensitivities from Traditional 3D Kinematics
Sandor,Erdman 1984: 3D Kinematics using 4x4 transformation
matrices [Sij] in a loop equation
Uses Derivative Operator Matrices ([Qlm], [Dlm]) to eliminate need to numerically evaluate partial derivatives
Equivalent to a small perturbation method; intensive calculations required for each sensitivity
][]][[]][[][ 0)1(230100 ISSSSS nnn
])][([)]([
lmmijm
mij QqSq
qS
Sensitivities from Global Coordinate Method
Uses 2D, 3D vector equations Derives sensitivities by evaluating
effects of small perturbations on loop closure equations
(Gao 1993)
0
0
0
cos
cos
cos
i
z
i
y
i
x
i
z
i
y
i
x
L
H
L
H
L
H
L
H
L
H
L
H
3
2
1
32
31
23
i
x
i
y
i
x
i
z
i
y
i
x
H
H
H
YXH
XZH
ZYH
Length Variation Rotational Variation
(taken from Gao, et. al 1998)
Variation – Velocity Relationship
Tolerance sensitivity solution
Velocity analysis of the equivalent mechanism
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
When are the sensitivities the same?
r2
r3
r4
r2
r3
r4
r1
3
2
1
4
(Faerber 1999)
Add dimensional variations to a kinematic model using kinematic elements
Converts kinematic analysis to variation analysis
Extract tolerance sensitivities from velocity analysis
Even works for static assemblies (no moving parts)
2D Equivalent Variational Mechanisms
r2
r3
r4
r1
3
2
1
4
r2
r3
r4
Kinematic Assembly
Static Assembly
(Faerber 1999)
Cylinder SliderPlanarParallel
CylindersEdge Slider
2D Kinematic Joints:
Equivalent Variational Joint:
Edge Slider Planar Cylinder SliderParallel Cylinders
3D Kinematic Joints: Equivalent Variational Joints:
3D EquivalentVariational Mechanisms
Rigid (no motion) Prismatic Revolute Parallel Cylinders
Cylindrical Spherical Planar Edge Slider
Cylindrical Slider Point Slider Spherical Slider Crossed Cylinders
Parallel Cylinders (2)
Edge Slider (4)
Cylindrical Slider (4)
Point Slider (5)
Spherical Slider (5) Crossed Cylinders (5)
R2
R1 R
R R1
R2
Geometric EquivalentVariational Mechanisms
R2
R1f
d
Crossed CylindersSpherical Slider
f Rd
Point Slider
f
Cylindrical Slider
ff
Rd
Edge Slider
ff
Planar
f
ff
Spherical
ff
f
Cylindrical
f
f
Y
X
Zf
f
Parallel Cylinders
R2
R1 f
f
d
d
Revolute
f
f
Y
X
Zf
f
Prismatic
f
ff
Rigid
ff
ff
f f
Example Model: Print Head
Pro/E model
Z
Xa
b
cd
g
h
ij
k
2
A
3
ef
Inset A
Inset B
f
e
c d
1
1
B
3
2 31
Geometric EVM
Print Head Results
3D GEVM in ADAMS
02410.00002410.02410.02410.00
10602.00010602.00602.00602.01
02410.00002410.02410.02410.00
001100000
A B D E G I J K LC1
F3
Results from Global Coordinate Method:
02410.00002410.02410.02410.00
10602.00010602.00602.00602.01
02410.00002410.02410.02410.00
001100000
A B D E G I J K LC1
F3
Results from ADAMS velocity analysis:
Research Benefits Comprehensive system for including
geometric variation in a kinematic vector model
More efficient than homogeneous transformation matrices
Allows use of commercial kinematic software to perform tolerance analysis
Allows static assemblies to be analyzed in addition to mechanisms
Ability to perform variation analysis in more widely available kinematic solvers increases availability of tolerance analysis
Current Limitations Implementing EVMs is currently a
manual system, very laborious Manual implementation of EVMs
can be very complex when including both dimensional and geometric variation
Difficulty with analysis of joints with simultaneous rotations
Questions?