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?