Christopher A. Brown. Surfaces on architectural and civil engineering structures should often fulfill several functions, e.g., aesthetics, cleanability, heat transfer, and friction. A road or runway surface, for example, must satisfy two functional requirements (FRs): FR1) provide a comfortable ride without excessive vertical motion, and FR2) facilitate stopping and turning with sufficient friction. The FR1 requires smooth surface and FR2 favors rough. Suh’s axioms requires a DP for each FR. The DPs are physical attributes that fulfill the FRs. The complete design solution includes an integrated CEME (collectively exhaustive, mutually exclusive) decomposition hierarchy, from abstract to detailed, of the design problem. Pairs of FRs and their corresponding DPs, are related by design equations. Axiomatic design and multiscale analysis in surface metrology address these kinds of specifications.

1. Axiomatic Design and the Specification of Surface
Topographies
Christopher A. Brown
Professor of Mechanical Engineering
Worcester Polytechnic Institute
Worcester, Massachusetts, USA

2. Brown – AD & the Specification of Surface Topographies
Objectives and rationale
• Propose evidence-based specification of surface topographies
• Surfaces cover everything
• Topography influences surface behavior, feel, and appearance
• Integrate multiple topographical behaviors
• While maintaining independent control
• Multiscale analysis and characterization (narrow bandwidth)
2

3. Brown – AD & the Specification of Surface Topographies
Sciences of design and of surface metrology
Scientific disciplines have a few underlying laws, axioms or principles
• Axiomatic design
• Domains (stakeholder, functional, physical, process)
• Axioms (independence and information)
• Multiscale analysis and characterization of topographies
• Principles to find functional correlations and the basis for discrimination
• Area-scale analysis
• Evidence-based specifications
3

4. Brown – AD & the Specification of Surface Topographies
Design
Formulations exist in four basic domains or spaces
1. Stakeholder (“customer”, customer needs, CNs)
2. Functional (functional requirements, FRs)
3. Physical (“solution”, design parameters, DPs)
4. Process (“manufacturing & construction”, process variables, PVs)
4
CNs
Customer
Needs
What adds
value
FRs
Functional
Requirements
What it does
DPs
Design
Parameters
What it looks
like
PVs
Process
Variables
How it is
made
stakeholder functional physical process
Increasing practicality for implementation
Value chain →

5. Brown – AD & the Specification of Surface Topographies
Principles of Axiomatic Design
All good designs comply with two axioms
1. Maintain the independence of the functional elements
Adjustable, controllable, avoid unintended consequences
2. Minimize the information content
Maximize the probability of success (p) , robust, avoid complexity
5
DP1 DP2 DP3
FR1 X O O
FR2 O X O
FR3 O O X
I= 𝒍𝒏
𝟏
𝒑
DP1 DP2 DP3
FR1 X O O
FR2 X X O
FR3 X X X
DP1 DP2 DP3
FR1 X X O
FR2 X X O
FR3 O O X
DP1 DP2 DP3
FR1 X X X
FR2 X X X
FR3 X X X
uncoupled decoupled partly coupled fully coupled
ideal special order of adjustment find new DP1 or DP2
examine FR1&FR2
X indicates influence of DP on FR

6. DPs
Irregular,
complex
geometriesat
sufficientlyfine
scales
Processing PVs Performance FRs
Measurement, analysis and characterizationof surface topographies
For understanding the relations between PVs, DPs and FRs
Surface Metrology
Surface
topography
(texture)
Brown – AD & the Specification of Surface Topographies
6

7. Brown – AD & the Specification of Surface Topographies
Smooth
wheel - road
RoughRough
tread-road
Road Surface
Area-scale plot of the road surface texture
1
Relative area
Scale
Smooth-rough crossover
Functional
Independence
using scale

8. Brown – AD & the Specification of Surface Topographies
8
Roughness
characterization parameter
Performance
characterizationparameter
Performance
tolerance
Roughness
tolerance
Performance
– texture
FR-DP
Relations
supports
product design

9. Brown – AD & the Specification of Surface Topographies
9
Roughness
characterization parameter
Performance
characterizationparameter
Performance
tolerance
Roughness
tolerance
How smooth can you make it?
Performance
– texture
FR-DP
Relations
supports
product design

10. Brown – AD & the Specification of Surface Topographies
10
Performance
characterizationparameter
Performance
tolerance
Roughness
tolerance

11. Brown – AD & the Specification of Surface Topographies
11
Roughness
characterizationparameter
Manufacturing
process parameter
Process
tolerance
Roughness
tolerance
Abrasive processes
Rz = b + K (F/H)½
(Brown and Savary 1991
Wear 141, 211)
Performance
– texture
DP-PV
Relations
supports
product design

12. Brown – AD & the Specification of Surface Topographies
12
PV-DP (process-texture)
0
5
10
15
20
25
0 50 100 150 200 250 300
theoretical
measured
Roughness Rzμm
Feed μm/rev
Rz = f²/8r
Rz = f²/8r + 1.4μm
Texture-process
relations for
machining by
turning
Haas SL10, with a
0.4mm tool nose
Kennametal
insert.
Unpublished 2013

13. Brown – AD & the Specification of Surface Topographies
13
62mm
Photo: medium skidabrader Surface NASA Wallops

14. Brown – AD & the Specification of Surface Topographies
WPI’s SLM
measuring
experimental
runway pavement
NASA Wallops
Space Flight
Facility
14
Wm. Johnsen, PhD, friction and wear on pavementsWm. Johnsen, PhD, friction and wear on pavements

15. Iteration 368/500 Patch Area: 3.55mm2 Rel. Area: 1.1102
Iteration 452/500 Patch Area: 0.485mm2
Rel. Area: 1.5109
Iteration 500/500 Patch Area: 314µm2
Rel. Area: 2.895
Area-scale
tiling
Multiscale
analysis
Brown – AD & the Specification of Surface Topographies
15

16. Area-scale Plot - Multiscale characterization
0.001 0.01 0.1 1 10
RelativeArea
2
1
Smooth-Rough
Crossover
Scale (mm2)
Brown – AD & the Specification of Surface Topographies
16

17. Brown – AD & the Specification of Surface Topographies
17
Actual friction coefficient
Predictedfrictioncoefficient
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
R² = 0.966
slope = 0.994
Neural network
analysis, based on 4
texture parameters:
Asfc med scale,
SRC small scale,
ARp small scale,
Mean Texture
Depth
Predicted versus Actual Friction

18. y
x
pavement tire
dynamometer
lathe
Aluminum
disk (to lathe
chuck)
bracket
(to lathe
carriage)
(to A/D/A
computer)
impact
direction
Standard Pavement Specimen
0
0.5
1
1.5
2
2.5
0 20 40 60 80 100
Time (msec.)
COF
Brown – AD & the Specification of Surface Topographies
18
Wear and Mean lab test

19. Brown – AD & the Specification of Surface Topographies
19
0.000
0.200
0.400
0.600
0.800
1.000
1.200
1.400
1.600
1.800
2.000
0.7 0.8 0.9 1 1.1 1.2 1.3
Mean Friction Coefficient (µ)
Wearrate(mg/sec.)
Wear rate vs Mean Friction Coefficient
Shows independence of Wear rate and Friction

20. South
North
Brown – AD & the Specification of Surface Topographies
20
Corey Randall
Christopher A. Brown
WPI’s Surface Metrology Lab
Multiscale statistical analysis for Discrimination
Northern v. Southern Appalachians

21. Area-scale Analysis
south
north
Note - Area-scale
complexity is -1000 times
the slope, determined over
a window one order of
magnitude in width
Brown – AD & the Specification of Surface Topographies
21

22. F-test South v North
using relative area
99% confidence
Brown – AD & the Specification of Surface Topographies
22

23. F-test South v North
using area-scale complexity
99% confidence
Brown – AD & the Specification of Surface Topographies
23

24. Brown – AD & the Specification of Surface Topographies
24
FR-DP fatigue tests - four-point bending
about 80 specimens
for fatigue limit
testing

25. Brown – AD & the Specification of Surface Topographies 25
-3º stress relieved
transverse loading
45º stress relieved
longitudinal loading
Rendering of measurements

26. Brown – AD & the Specification of Surface Topographies
26

27. Finding the right scale, or spatial frequency, for
specific functional correlations and
discriminations is like tuning in a radio station
In order to understand anything the bandwidth
needs to be appropriately limited and tune the
central frequency needs to be appropriately placed
Brown – AD & the Specification of Surface Topographies
27

28. Surface Metrology and Scales
SURFACE
BEHAVIOR
SURFACE
CREATION TEXTURE
DISCRIMINATION andFUNCTIONAL CORRELATION
b-Scaleb-Scalec-Scalec-Scale
BEHAVIOR
CHARACTERIZATION
CREATION
CHARACTERIZATION
MEASURED
TEXTURE
Measurement ScaleMeasurement Scale
TEXTURE
CHARACTERIZATION
Analysis ScaleAnalysis Scale
Irregular components,
not differentiable
(finite approximations)
Brown – AD & the Specification of Surface Topographies
28

29. Brown – AD & the Specification of Surface Topographies
Principles of Surface Metrology
1. Characterization parameters, describing geometric features whose
nature is appropriate for the applications and phenomena of
interest, e.g., area for adhesion and curvature for fatigue;
2. Scales fundamentally appropriate to the application, indicated by
narrow regions in scale where there are highly confident
discriminations and strong correlations;
3. Statistics, appropriate for the application and phenomena, e.g.,
extreme values for fatigue, mean values for friction;
4. Adequate resolutions of a measurement system, a range of scales
with sufficient resolution in the measurement, which includes the
scales fundamental to the phenomena of interest.
29

30. Brown – AD & the Specification of Surface Topographies
Customer Needs (CNs) – surface topographies
• Users (runways, roads, floors)
• Traction (wet and dry friction)
• Aesthetics (gloss)
• Maintenance (buildings, roads, runways, floors)
• cleanability
• Architect and engineers
• Specification materials
• Functional correlations with performance
• Metrology
• Discrimination
• Process
• Functional correlation with process (finishing)
30

31. Brown – AD & the Specification of Surface Topographies
Concluding remarks
31
• Conventional, mono-scale height parameters, are inadequate to find
FR-DP and DP-PV relations
• Research is required to discover these relations
• Multiscale analysis and characterization can facilitate these
discoveries
• Independence of competing functional requirements can be
maintained by exploiting scale specificity in interactions with
topographies

32. Brown – AD & the Specification of Surface Topographies
Thank you
for your kind
attetion
32