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Including GD&T Tolerance Variation in a Commercial Kinematics Application. Jeff Dabling Surety Mechanisms & Integration Sandia National Laboratories. Research supported by:. Summary. Variation Propagation Obtaining Sensitivities Variation/Velocity Relationship

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Including GD&T Tolerance Variation in a Commercial Kinematics Application


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including gd t tolerance variation in a commercial kinematics application

Including GD&T Tolerance Variation in a Commercial Kinematics Application

Jeff Dabling

Surety Mechanisms & Integration

Sandia National Laboratories

Research supported by:

summary
Summary
  • Variation Propagation
  • Obtaining Sensitivities
  • Variation/Velocity Relationship
  • Equivalent Variational Mechanisms in 2D
  • EVMs in 3D
  • Example in ADAMS
3 sources of variation in assemblies
3 Sources of Variation in Assemblies

q

R

q

D

D

A

R +

R

R

A

R

q + Dq

A

+

A

U

U

D

U

U +

D

U +

U

Geometric

Dimensional and

Kinematic

dlm vector assembly model

C

L

R

L

R

T

Arm

R

L

DLM Vector Assembly Model

Gap

Open Loop

i

e

r

Plunger

Pad

u

g

Reel

a

b

Base

Closed Loop

h

how geometric variation propagates
How Geometric Variation Propagates

Y

3D cylindrical slider joint

RotationalVariation

X

FlatnessToleranceZone

Z

View normal to the cylinder axis

CylindricityToleranceZone

Nominal

Circle

The effect of feature variations in 3D depends upon the joint type and which joint axis you are looking down.

FlatnessToleranceZone

Translational

Variation

View looking down the cylinder axis

3d propagation of surface variation
3D Propagation of Surface Variation

K

Kinematic Motion

F

Geometric Feature Variation

F

F

K

K

y

y

K

K

x

x

F

z

F

z

K

F

K

K

Cylindrical Slider Joint

Planar Joint

variations associated with geometric feature joint combinations

G

e

o

m

T

o

l

J

o

i

n

t

s

R

R

R

R

R

R

R

R

y

z

x

y

z

x

y

z

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

x

y

z

x

y

z

x

y

z

x

y

z

x

y

z

T

T

T

T

T

T

y

y

y

y

y

y

T

R

T

R

T

R

T

R

R

R

R

T

R

T

y

x

y

x

y

x

y

x

x

x

x

y

x

y

T

R

T

R

T

R

T

R

T

R

T

E

y

x

y

x

y

x

y

x

y

x

y

P

T

R

T

R

T

R

T

R

T

R

T

R

T

R

y

x

y

x

y

x

y

x

y

x

y

x

y

x

T

R

T

C

T

R

T

R

T

R

T

R

T

R

T

y

x

y

x

y

x

y

x

y

x

y

x

y

P

T

R

T

R

T

R

T

R

T

R

T

R

T

R

y

x

y

x

y

x

y

x

y

x

y

x

y

x

P

t

P

S

P

Variations Associated with Geometric Feature – Joint Combinations

(Gao 1993)

P

l

a

n

a

r

R

R

R

R

R

R

R

R

R

R

R

R

R

R

T

x

z

x

z

x

z

x

z

x

z

x

z

x

z

y

R

e

v

o

l

u

t

e

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

T

T

T

T

x

z

x

z

x

z

x

z

x

z

x

z

x

z

x

z

x

z

x

z

C

y

l

i

n

d

r

i

c

a

l

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

T

T

T

T

x

z

x

z

x

z

x

z

x

z

x

z

x

z

x

z

x

z

x

z

P

r

i

s

m

a

t

i

c

R

R

R

R

R

R

R

R

R

R

x

y

z

x

y

z

x

y

z

x

S

p

h

e

r

i

c

a

l

C

r

s

C

y

l

P

a

r

C

y

l

E

d

g

S

l

i

R

T

R

y

x

y

x

C

y

l

S

l

i

T

y

P

n

t

S

l

i

T

T

T

T

T

T

T

y

y

y

y

y

y

y

T

T

T

T

y

y

y

y

S

p

h

S

l

i

T

T

T

T

T

T

T

y

y

y

y

y

y

y

including geometric variation

Rotational variation due to flatness variation between two planar surfaces:

Rotational Variation

= ±Db

Flatness Tolerance =

Zone

a

Characteristic Length

  • Translational variation due to flatness variation:

=±a/2

Translational

Variation

Flatness Tolerance = Zone

a

Including Geometric Variation
  • Variables used have nominal values of zero
  • Variation corresponds to the specified tolerance value
geometric variation example

f

.01

U

2

q

R

H

.02

A

.01

b1

b2

b1

b2

f

U

.01

1

(a3, a4)

b1

b2

b1

b2

b1

b2

R2

U2

b1

b2

b1

b2

R3

q

H

A

R1

U1

(b1, b2)

Geometric Variation Example
  • Translational: additional vector with nominal value of zero. (a3, a4)
  • Rotational: angular variation in the joint of origin and propagated throughout the remainder of the loop. (b1, b2)

a3

a4

sensitivities from traditional 3d kinematics
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
sensitivities from global coordinate method
Sensitivities from Global Coordinate Method

(Gao 1993)

  • Uses 2D, 3D vector equations
  • Derives sensitivities by evaluating effects of small perturbations on loop closure equations

Length Variation

Rotational Variation

(taken from Gao, et. al 1998)

variation velocity relationship

2

4

r

3

2

3

r

4

r

2

2

2

2

1

r

1

Variation – Velocity Relationship

(Faerber 1999)

Tolerance sensitivity solution

Velocity analysis of the equivalent mechanism

When are the sensitivities the same?

2d equivalent variational mechanisms

2D Kinematic Joints:

Kinematic Assembly

Equivalent Variational Joint:

Parallel Cylinders

Edge Slider

Planar

Cylinder Slider

Static Assembly

2D Equivalent Variational Mechanisms

(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)
3d equivalent variational mechanisms
3D EquivalentVariational Mechanisms

3D Kinematic Joints:

Equivalent Variational Joints:

Rigid (no motion)

Prismatic

Revolute

Parallel Cylinders

Cylindrical

Spherical

Planar

Edge Slider

Cylindrical Slider

Point Slider

Spherical Slider

Crossed Cylinders

geometric equivalent variational mechanisms

R

Geometric EquivalentVariational Mechanisms

f

Y

d

f

f

f

R1

f

f

f

f

f

X

f

f

d

f

f

R2

Z

f

f

Parallel Cylinders

Revolute

Prismatic

Rigid

f

Y

f

f

f

f

X

f

f

f

f

Z

f

f

f

Spherical

Planar

Edge Slider

Cylindrical

d

d

R1

d

f

f

f

R

R2

f

f

Crossed Cylinders

Spherical Slider

Point Slider

Cylindrical Slider

example model print head

A

f3

h

f

a2

2

a3

Inset A

q1

j

f

g

i

3

Inset B

k

B

f

e

f

d

e

c

a1

Z

b

d

a

c

f1

X

Example Model: Print Head

Geometric EVM

Pro/E model

print head results

A B D E G I J K L

A B D E G I J K L

C

C

f1

f1

F

F

f3

f3

Print Head Results

Results from Global Coordinate Method:

Results from ADAMS velocity analysis:

3D GEVM in ADAMS

research benefits
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
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