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Reduction of Complexity through the Use of
Geometric Functional Periodicity
Reference: Chapter 7 of
Nam P. Suh, Complexity: Theory and Applications,Oxford University Press, 2005
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Definition of Complexity
Complexity is defined as a measure of uncertainty in satisfying the FRs.
According to this definition,
complexity is a relative quantity.
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Four Different Kinds of Complexity
• Time-Independent Real Complexity• Time-Independent Imaginary Complexity
• Time-Dependent Combinatorial Complexity• Time-Dependent Periodic Complexity
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“Complexity” can be reduced by taking the following actions:
• Reduce Time-Independent Real Complexity
• Eliminate Time-Independent Imaginary Complexity
• Transform Time-Dependent Combinatorial Complexity into Time-Dependent Periodic Complexity
Important concept: Functional Periodicity
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Functional Periodicity• Temporal periodicity• Geometric periodicity• Biological periodicity• Manufacturing process periodicity• Chemical periodicity• Thermal periodicity• Information process periodicity• Electrical periodicity• Circadian periodicity• Material periodicity
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Time-Dependent Combinatorial Complexity
Time-dependent combinatorial complexity arises because the future events occur in unpredictable ways and thus cannot be predicted.
For example, it occurs when the system range moves away from the design range as a function of time.
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Time-Dependent Combinatorial Complexity
Prob. Density
FR
Des ignRan ge
T h e System Range changescontinuously as a function of time.
T im e
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“Complexity” can be reduced by taking the following actions:
• Reduce Time-Independent Real Complexity
• Eliminate Time-Independent Imaginary Complexity
• Transform Time-Dependent Combinatorial Complexity into Time-Dependent Periodic Complexity
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Reduction of Combinatorial Complexity
How?
Through “Re-initialization” of the system
by defining a “Functional Period”.
Note: Functional period is defined by a repeating set of functions, not by time period, unless “time” is a set of
functions.
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Transformation of Time-Dependent Combinatorial Complexity
• Basic Idea1. Make sure that the design satisfies the
Independence Axiom.
2. Identify a set of FRs that undergoes a cyclic change and has a functional period.
3 Identify the functional requirement that may undergo a combinatorial process
4.
6. "Reinitialization"5. Set the beginning of the cycle as t=0
T Ccom ( FRi ) ⇒ Cper (FRi )
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Functional Periodicity• The functional periodicity are the following types:
(1) Temporal periodicity
(2) Geometric periodicity(3) Biological periodicity(4) Manufacturing process periodicity(5) Chemical periodicity(6) Thermal periodicity(7) Information process periodicity(8) Electrical periodicity(9) Circadian periodicity(10) Material periodicity
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Example: Design of Low Friction Sliding Surfaces
Consider the following task:
Reduce friction between two sliding surfaces under load
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Coefficient of friction versus sliding distance
µs
µi
µ
Distance slid(b)
∆µρ
µs
µi
µ
Distance slid(a)
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Effect of removing wear particles for an Armco iron slider sliding against an Armco iron specimen
µs
µi
µ
Distance slid
µI’
Wear particles removed
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Friction Space
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Friction at Dry Sliding InterfacePlowing Mechanism
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Friction at Dry Sliding InterfaceParticle Agglomeration
Why do particles agglomerate?
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Friction at Dry Sliding InterfaceHeight of Agglomerated Particles
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Friction at Dry Sliding InterfaceFriction Coefficient and the Number of
Agglomerated Particles
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Friction at Dry Sliding InterfaceReduction of Friction by Elimination of Particles
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Design of Low Friction Sliding Surfaces without Lubricants
• What are the FRs?
• What are the constraints?
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Design of Low Friction Sliding Surfaces without Lubricants
FR1 = Support the normal loadFR2 = Prevent particle generationFR3 = Prevent particle agglomerationFR4 = Remove wear particles from the
interface
• Constraint: No lubricant
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Design of Low Friction Sliding Surfaces without Lubricants
Figures removed for copyright reasons.See Figure 7.11 and 7.13 in [Complexity]: Suh, N. P. Complexity: Theory and Applications. New York, NY: Oxford University Press, 2005. ISBN: 0195178769.
DP1 = Total contact area of the pad, ADP2 = Roughness of the planar surface of pads, RDP3 = Length of the pad in the sliding direction, λDP4 = Volume and depth of the pocket for wear particles, V
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Design of Low Friction Sliding Surfaces without Lubricants
The design equation:
FR1
FR2
FR3
FR4
⎧
⎨
⎪ ⎪
⎩
⎪ ⎪
⎫
⎬
⎪ ⎪
⎭
⎪ ⎪
=
X 0 0 00 X x 00 0 X 00 0 0 X
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥
DP1
DP2
DP3
DP4
⎧
⎨
⎪ ⎪
⎩
⎪ ⎪
⎫
⎬
⎪ ⎪
⎭
⎪ ⎪
=
X 0 0 00 X x 00 0 X 00 0 0 X
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥
ARλV
⎧
⎨ ⎪ ⎪
⎩ ⎪ ⎪
⎫
⎬ ⎪ ⎪
⎭ ⎪ ⎪
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Design of Low Friction Sliding Surfaces without Lubricants
Test Results:
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Friction at Dry Sliding InterfaceUndulated Surface for Elimination of Particles
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Friction at Lubricated Interface
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Effect of Boundary Lubrication
∼ µ ~ 0.1
• Cause?– Plowing
• What is the role of a lubricant?– Lower shear stress– Transport particles– Prevent particle agglomeration– Prevent adhesion
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Geometric Functional Periodicity to Decrease Face Seal Wear
Figures removed for copyright reasons.See Figure 7.16 through 7.21 in [Complexity].
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FRs of Face Seal
• FRs
FR1 = Support the normal loadFR2 = Prevent particle migration across the sealFR3 = Prevent agglomeration of particles in the seal/metal interfaceFR4 = Provide lubricant to the interfaceFR5 = Prevent the flow of the lubricant out of the sealed areaFR6 = Delay the initiation of the wear process
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DPs of Face Seal
• DPs
DP1 = Total Area of the pad, ADP2 = Seal lipDP3 = Diameter of the pad, λDP4 = LubricantDP5 = Length of the sealed tip, LDP6 = Seal material
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Design Matrix for Face Seal Wear
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Reduction of Wear of Pin-Joints By the Introduction of Geometric Functional
Periodicity
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Drive sprockets, idlers, rollers, Grouser shoes
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FRs and DPs of Conventional Pin-Joints
• FRsFR1 = Carry the loadFR2 = Allow rotational motion about the axis of the
bushingFR3 = Minimize friction forceFR4 = Minimize wear
• DPsDP1 = Shaft/bushing dimensionsDP2 = Clearance between the shaft and the bushingDP3 = Clearance between the shaft and the bushingDP4 = Material properties
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Friction in Geometrically Confined Space
Several slides removed for copyright reasons.See Mosleh, M., and N. P. Suh. “High Vacuum Undulated Sliding Bearings.”SLTE Transactions 38 (1995): 277-284.
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FRs and DPs of Pin-Joints with Geometric Functional Periodicity
• FRsFR1 = Carry the loadFR2 = Allow rotational motion about the axis of the
bushingFR3 = Minimize friction forceFR4 = Minimize wear
• DPsDP1 = Dimensions of the pin and the bushing structureDP2 = Cylindrical pin/bushing jointDP3 = Low-friction systemDP4 = Wear-minimization system
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FRs and DPs of Pin-Joints with Geometric Functional Periodicity
FR 1FR 2FR 3FR 4
⎧
⎨ ⎪ ⎪
⎩ ⎪ ⎪
⎫
⎬ ⎪ ⎪
⎭ ⎪ ⎪
=
X 0000 Xxx00 Xx000 X
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
DP 1DP 2DP 3DP 4
⎧
⎨ ⎪ ⎪
⎩ ⎪ ⎪
⎫
⎬ ⎪ ⎪
⎭ ⎪ ⎪
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Decomposition of FR3 and DP3 of Pin-Joints with Geometric Functional Periodicity
• FR3sFR31 = Remove Wear Particles from the interfaceFR32 = Apply lubricant
• DP3sDP31 = Particle removal mechanismDP32 = Externally supplied grease
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FRs and DPs of Pin-Joints with Geometric Functional Periodicity
FR 31FR 32
⎧ ⎨ ⎩
⎫ ⎬ ⎭
=X 0xX
⎡
⎣ ⎢
⎤
⎦ ⎥
DP 31DP 32
⎧ ⎨ ⎩
⎫ ⎬ ⎭
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Decomposition of FR3 and DP3 of Pin-Joints with Geometric Functional Periodicity
• FR31s (Remove wear particles from the interface)FR311 = Prevent particle agglomeration FR312 = Do not allow the increase in the normal force due
to the presence of particles at the interfaceFR313 = Trap wear particles
• DP31s (Particle removal mechanism)DP311 = The length of the pad on the liner that actually
comes in contact with the pinDP312 = Flexible ringDP313 = The indented part of the undulated surface
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Decomposition of FR3 and DP3 of Pin-Joints with Geometric Functional Periodicity
Figure removed for copyright reasons.See Figure 7.22 in [Complexity].
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FRs and DPs of Pin-Joints with Geometric Functional Periodicity
FR 311FR 312FR 313
⎧
⎨ ⎪
⎩ ⎪
⎫
⎬ ⎪
⎭ ⎪ =
X 00xX 000 X
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
DP 311DP 312DP 313
⎧
⎨ ⎪
⎩ ⎪
⎫
⎬ ⎪
⎭ ⎪
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Decomposition of FR3 and DP3 of Pin-Joints with Geometric Functional Periodicity
• FR4s (Minimize wear)FR41 = Prevent particle penetration into both surfacesFR42 = Minimize the shear deformation of the pin and the
flexible liner
• DP4s (Wear-minimization system)DP41 = Hardness ration of the two materialsDP42 = Coating of the pin with a thin layer of low shear
strength material
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FRs and DPs of Pin-Joints with Geometric Functional Periodicity
FR 1FR 2FR 311FR 312FR 313FR 32FR 41FR 42
⎧
⎨
⎪ ⎪ ⎪ ⎪ ⎪
⎩
⎪ ⎪ ⎪ ⎪ ⎪
⎫
⎬
⎪ ⎪ ⎪ ⎪ ⎪
⎭
⎪ ⎪ ⎪ ⎪ ⎪
=
X 00000000 XxxxX 0000 X 0000000 xX 00000000 X 0000000 xX 00000000 X 000000 X 0 X
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
DP 1DP 2DP 311DP 312DP 313DP 32DP 41DP 42
⎧
⎨
⎪ ⎪ ⎪ ⎪ ⎪
⎩
⎪ ⎪ ⎪ ⎪ ⎪
⎫
⎬
⎪ ⎪ ⎪ ⎪ ⎪
⎭
⎪ ⎪ ⎪ ⎪ ⎪
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Pin-Joint Design
Figures removed for copyright reasons.See Figure 7.25 and 7.26 in [Complexity].
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Geometric Functional Periodicity for Electrical Connectors
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Electrical Connectors
Figure removed for copyright reasons.See Figure 7.29 in [Complexity].
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Conventional Electrical Connectors
Male connector
Female connector
Plastic overmolding
Plastic overmolding
Compliant pin (for permanent connection)
Multiple layers will be stacked together to obtain an entire connector.
Figure by MIT OCW.
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FRs of a Data Electrical Connector
FR1 = Mechanically connect and disconnect electrical terminals
FR2 = Control contact resistance (should be less than 20mΩ)FR3 = Prevent the cross-talk (i.e., interference)
between the connections
Subject to the following constraints (Cs):C1 = Low costC2 = Ease of useC3 = Long life (> million cycles)C4 = Maximum temperature rise of 30 oCC5 = Low insertion force
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FRs of a Power Electrical Connector
FR1 = Mechanically connect and disconnect electrical terminals
FR2 = Control contact resistance (should be less than 20mΩ)FR3 = Maximize power density
Subject to the following constraints (Cs):C1 = Low costC2 = Ease of useC3 = Long life (> million cycles)C4 = Maximum temperature rise of 30 oCC5 = Low insertion force
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DPs of a Data Electrical Connector
DP1 = Cylindrical assembly of the woven tube and the pin
DP2 = Locally compliant electric contact DP3 = Number of conducting wires
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Decomposition of FR1 (Mechanically connect and disconnect electrical terminals) and DP1 (Cylindrical
assembly of the woven tube and the pin)
FR11 = Align the rod axially inside the tubeFR12 = Locate the axial position of the rod in the tubeFR13 = Guide the pin
DP11 = Long aspect ratio of the rod and the tubeDP12 = Snap fitDP13 = Tapered tip of the pin
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Decomposition of FR2 (Control contact resistance to be less than 20mΩ) and DP2 (Locally compliant electric
contact)
FR21 = Prevent oxidation of the conductorFR22 = Remove wear particlesFR23 = Control line tension/deflection of the non-
conducting fiber
DP21 = Gold plated metal surfaceDP22 = Space created in the crevices between fibersDP23 = Spring
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Design Matrix
FR1FR2FR3
⎧
⎨ ⎪
⎩ ⎪
⎫
⎬ ⎪
⎭ ⎪ =
X 00XX 00XX
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
DP1DP2DP3
⎧
⎨ ⎪
⎩ ⎪
⎫
⎬ ⎪
⎭ ⎪
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Tribotek Electrical Connectors(Courtesy of Tribotek, Inc. Used with permission.)
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Tribotek Electrical Connectors(Courtesy of Tribotek, Inc. Used with permission.)
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Performance of “Woven” Power Connectors
• Power density => 200% of conventional connectors
• Insertion force => less than 5% of conventional connectors
• Electric contact resistance = 5 m ohms• Manufacturing cost -- Flexible manufacturing• Capital Investment
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Tribotek, Inc.Busbar ApplicationBusbar Application Embedded Embedded
ConfigurationConfigurationFeatures…
LowR 125 socketPress-Fit into busbar130 A
Benefits…Highest current densityLow insertion forceLow contact resistance, min voltage dropAvailable in various contact sizes
Figures removed for copyright reasons.
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Tribotek, Inc.Busbar Application Busbar Application -- Mounted ConfigurationMounted Configuration
Figures removed for copyright reasons.
Benefits…Highest current densityLow insertion forceLow contact resistance, min voltage dropAvailable in various contact sizes
Features…LowR 125 socketPress-Fit onto busbar130 A