The document presents a volumetric contact dynamics model. It begins with an introduction to point contact models and their limitations. It then describes the key aspects of the volumetric model, including the volumetric model framework, equations for normal forces, friction forces, and a stick-slip friction state model. It also discusses experiments used to validate the model and how the model was implemented in MapleSim.

1. Introduction
Volumetric Model
Experiments
Modelling
A Volumetric Contact Dynamics Model
Mike Boos
SYDE 652
March 27, 2012
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2. Introduction
Volumetric Model
Experiments
Modelling
Outline
1 Introduction
2 Volumetric Model
Volumetric model framework
Normal forces
Friction forces
3 Experiments
Normal force experiments
Friction experiments
4 Modelling
Graph theoretic contact model
MapleSim model
Demos
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3. Introduction
Volumetric Model
Experiments
Modelling
Outline
1 Introduction
2 Volumetric Model
Volumetric model framework
Normal forces
Friction forces
3 Experiments
Normal force experiments
Friction experiments
4 Modelling
Graph theoretic contact model
MapleSim model
Demos
Mike Boos A Volumetric Contact Dynamics Model 3/ 34

4. Introduction
Volumetric Model
Experiments
Modelling
Motivation
Dextre at the tip of Canadarm2 (Gonthier, 2007)
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5. Introduction
Volumetric Model
Experiments
Modelling
Point contact models
fn
Hertz theory
B1
f n = kδ p n
k (p = 3/2 for
δ
sphere-on-sphere)
B2
Figure: Point contact model.
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6. Introduction
Volumetric Model
Experiments
Modelling
Point contact models
fn
Hertz theory
B1
f n = kδ p n
k (p = 3/2 for
δ
sphere-on-sphere)
B2 Hunt-Crossley
f n = kδ p (1 + aδ) n
Figure: Point contact model.
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7. Introduction
Volumetric model framework
Volumetric Model
Normal forces
Experiments
Friction forces
Modelling
Outline
1 Introduction
2 Volumetric Model
Volumetric model framework
Normal forces
Friction forces
3 Experiments
Normal force experiments
Friction experiments
4 Modelling
Graph theoretic contact model
MapleSim model
Demos
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8. Introduction
Volumetric model framework
Volumetric Model
Normal forces
Experiments
Friction forces
Modelling
Volumetric model
fn
B1
kv
Force element
df n = kv δ(s)n
B2
Figure: Modified Winkler elastic foundation model.
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9. Introduction
Volumetric model framework
Volumetric Model
Normal forces
Experiments
Friction forces
Modelling
Volumetric model
S
S
B1 B1
ρs ρv
δ(s)
s n
s n
sc p
Contact plate
pc
B2 B2
V = S δ(s)dS
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10. Introduction
Volumetric model framework
Volumetric Model
Normal forces
Experiments
Friction forces
Modelling
Volumetric model
S
S
B1 B1
ρs ρv
δ(s)
s n
s n
sc p
Contact plate
pc
B2 B2
V = S δ(s)dS
pdV
pc = V
V
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11. Introduction
Volumetric model framework
Volumetric Model
Normal forces
Experiments
Friction forces
Modelling
Volumetric model
S
S
B1 B1
ρs ρv
δ(s)
s n
s n
sc p
Contact plate
pc
B2 B2
V = S δ(s)dS Js = S ((ρs ·ρs )I−ρs ρs )δ(s)dS
pdV
pc = V
V
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12. Introduction
Volumetric model framework
Volumetric Model
Normal forces
Experiments
Friction forces
Modelling
Volumetric model
S
S
B1 B1
ρs ρv
δ(s)
s n
s n
sc p
Contact plate
pc
B2 B2
V = S δ(s)dS Js = S ((ρs ·ρs )I−ρs ρs )δ(s)dS
pdV
pc = V
V Jv = V ((ρv · ρv )I − ρv ρv )dV
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13. Introduction
Volumetric model framework
Volumetric Model
Normal forces
Experiments
Friction forces
Modelling
Volumetric model
S
S
B1 B1
ρs ρv
δ(s)
s n
s n
sc p
Contact plate
pc
B2 B2
V = S δ(s)dS Js = S ((ρs ·ρs )I−ρs ρs )δ(s)dS
2
J{s,v} n = rgyr V n
pdV
pc = V
V Jv = V ((ρv · ρv )I − ρv ρv )dV
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14. Introduction
Volumetric model framework
Volumetric Model
Normal forces
Experiments
Friction forces
Modelling
Normal Forces
fn
df n = kv δ(s)(1 + a vn )n
B1 τs
ft
τr
B2
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15. Introduction
Volumetric model framework
Volumetric Model
Normal forces
Experiments
Friction forces
Modelling
Normal Forces
fn
df n = kv δ(s)(1 + a vn )n
τs
Normal force
B1
ft f n = kv V (1 + a vcn )n
τr
B2
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16. Introduction
Volumetric model framework
Volumetric Model
Normal forces
Experiments
Friction forces
Modelling
Normal Forces
fn
df n = kv δ(s)(1 + a vn )n
τs
Normal force
B1
ft f n = kv V (1 + a vcn )n
τr
Rolling resistance torque
τ r = kv a Js · ω t
B2
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17. Introduction
Volumetric model framework
Volumetric Model
Normal forces
Experiments
Friction forces
Modelling
Basic friction model
fn
df t = −µ dfn vt
ˆ
B1 τs
ft
τr
B2
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18. Introduction
Volumetric model framework
Volumetric Model
Normal forces
Experiments
Friction forces
Modelling
Basic friction model
fn
df t = −µ dfn vt
ˆ
Friction force
B1 τs
ft f t = −µ fn vsct
ˆ
τr
B2
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19. Introduction
Volumetric model framework
Volumetric Model
Normal forces
Experiments
Friction forces
Modelling
Basic friction model
fn
df t = −µ dfn vt
ˆ
Friction force
B1 τs
ft f t = −µ fn vsct
ˆ
τr
Spinning friction torque
2
τ s = −µ rgyr fn ω n
ˆ
B2
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20. Introduction
Volumetric model framework
Volumetric Model
Normal forces
Experiments
Friction forces
Modelling
Stick-slip state
Average surface velocity
vavg = vsct · vsct + (rgyr |ω n |)2
2
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21. Introduction
Volumetric model framework
Volumetric Model
Normal forces
Experiments
Friction forces
Modelling
Stick-slip state
Average surface velocity
vavg = vsct · vsct + (rgyr |ω n |)2
2
Stick-slip state
2
vavg
−
v2
s=e s
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22. Introduction
Volumetric model framework
Volumetric Model
Normal forces
Experiments
Friction forces
Modelling
Stick-slip state
Average surface velocity
vavg = vsct · vsct + (rgyr |ω n |)2
2
Stick-slip state
2
vavg
−
v2
s=e s
Maximum friction coefficient
µmax = µC + (µS − µC ) s
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23. Introduction
Volumetric model framework
Volumetric Model
Normal forces
Experiments
Friction forces
Modelling
Stick-slip state
Average surface velocity
vavg = vsct · vsct + (rgyr |ω n |)2
2
Stick-slip state
2
vavg
−
v2
s=e s
Maximum friction coefficient
µmax = µC + (µS − µC ) s
Can add lag to s for dwell time dependency.
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24. Introduction
Volumetric model framework
Volumetric Model
Normal forces
Experiments
Friction forces
Modelling
Bristle model
fN
Bristle properties
Deformation: zsc
Rotation: θn
Contact sites
Surface asperities (‘bristles’) in
contact (Gonthier, 2007).
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25. Introduction
Volumetric model framework
Volumetric Model
Normal forces
Experiments
Friction forces
Modelling
Bristle model
fN
Bristle properties
Deformation: zsc
Rotation: θn
Parameters
Stiffness: σo
Contact sites
Damping: σ1
Surface asperities (‘bristles’) in
contact (Gonthier, 2007).
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26. Introduction
Volumetric model framework
Volumetric Model
Normal forces
Experiments
Friction forces
Modelling
Tangential friction forces
Friction force
f t = −fn (sat(σo zsc + σ1 zsc , µmax ) + σ2 vsct )
˙
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27. Introduction
Volumetric model framework
Volumetric Model
Normal forces
Experiments
Friction forces
Modelling
Tangential friction forces
Friction force
f t = −fn (sat(σo zsc + σ1 zsc , µmax ) + σ2 vsct )
˙
Bristle deformation rate
1 σo
zsc = s vsct + (1 − s) σ1 µC dir (vsct , v ) − σ1 zsc
˙
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28. Introduction
Volumetric model framework
Volumetric Model
Normal forces
Experiments
Friction forces
Modelling
Spinning friction torque
Spinning friction torque
2 ˙ µ
τ s = −rgyr fn sat σo θn + σ1 θn , rmax + σ2 ωn n
gyr
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29. Introduction
Volumetric model framework
Volumetric Model
Normal forces
Experiments
Friction forces
Modelling
Spinning friction torque
Spinning friction torque
2 ˙ µ
τ s = −rgyr fn sat σo θn + σ1 θn , rmax + σ2 ωn n
gyr
Bristle deformation rate
µC
θn = s ωn + (1 − s) σ1 rgyr sgn(ωn ) − σo θn
˙
σ1
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30. Introduction
Volumetric model framework
Volumetric Model
Normal forces
Experiments
Friction forces
Modelling
The Contensou effect
Translational friction forces vC
tend to ‘cancel out’ as angular ωr
C
velocity increases.
vB
ωr
v
A B
vA
ωr ω
vD
D
ωr
v << ωr (Gonthier, 2007)
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31. Introduction
Volumetric model framework
Volumetric Model
Normal forces
Experiments
Friction forces
Modelling
The Contensou effect
Translational friction forces vC
tend to ‘cancel out’ as angular ωr
C
velocity increases.
vB
ωr
Contensou factors
v
|vsct | rgyr |ωn | A B
Cv = vavg Cω = vavg vA
ωr ω
vD
D
ωr
v << ωr (Gonthier, 2007)
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32. Introduction
Volumetric model framework
Volumetric Model
Normal forces
Experiments
Friction forces
Modelling
The Contensou effect
Translational friction forces vC
tend to ‘cancel out’ as angular ωr
C
velocity increases.
vB
ωr
Contensou factors
v
|vsct | rgyr |ωn | A B
Cv = vavg Cω = vavg vA
ωr ω
We now need to update the D
vD
slipping coefficient in our ωr
bristle dyanmics equations to
v << ωr (Gonthier, 2007)
include these factors.
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33. Introduction
Volumetric Model Normal force experiments
Experiments Friction experiments
Modelling
Outline
1 Introduction
2 Volumetric Model
Volumetric model framework
Normal forces
Friction forces
3 Experiments
Normal force experiments
Friction experiments
4 Modelling
Graph theoretic contact model
MapleSim model
Demos
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34. Introduction
Volumetric Model Normal force experiments
Experiments Friction experiments
Modelling
Contact properties
Focus on simple geometric pairs:
Cylinder-on-plane
Sphere-on-plane
Payload material: Stainless steel
Contact plane materials: Al, Mg
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35. Introduction
Volumetric Model Normal force experiments
Experiments Friction experiments
Modelling
Apparatus in normal configuration
Force sensor
Payload/specimen (stainless steel)
Linear
encoder
Encoder reference Contact surface (Al or Mg)
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36. Introduction
Volumetric Model Normal force experiments
Experiments Friction experiments
Modelling
Selected results: Normal forces
Quasi-static loading of SS Damping factors (a) measured
cylinder on Mg plane 4
x 10
10
25 Estimated factors for Al
Measured data 9 Fit of a ∝ 1/vi for Al
Perpendicular fit
Perpendicular contact point Estimated factors for Mg
20 8 Fit of a ∝ 1/vi for Mg
Misaligned fit
Misaligned contact point
7
Damping factor (s/m)
15
6
Contact force (N)
5
10
4
5 3
2
0
1
0
−5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Impact velocity (mm/s)
Displacement (µm)
1−e2eff
kv = 5.17 × 1012 N/m3 a≈ i
eeff vn
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37. Introduction
Volumetric Model Normal force experiments
Experiments Friction experiments
Modelling
Experimental procedure
Identify parameters
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38. Introduction
Volumetric Model Normal force experiments
Experiments Friction experiments
Modelling
Experimental procedure
Identify parameters Verify parameters
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39. Introduction
Volumetric Model Normal force experiments
Experiments Friction experiments
Modelling
Experimental procedure
Identify parameters Verify parameters Contensou effect
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40. Introduction
Volumetric Model Normal force experiments
Experiments Friction experiments
Modelling
Friction apparatus
Rotational
motor
Linear
motor
Linear
Cylindrical encoder
payload
Contact Encoder
surface reference
x x
y y
z 3DOF force sensors z
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41. Introduction
Volumetric Model Normal force experiments
Experiments Friction experiments
Modelling
Selected results: Contensou effect
0.45 0.45
Measured coefficients Measured coefficients
0.4 Model coefficients 0.4 Model coefficients
0.35 0.35
Coefficient of Friction
Coefficient of Friction
0.3 0.3
0.25 0.25
0.2 0.2
0.15 0.15
0.1 0.1
0.05 0.05
0 0
0 1 2 3 4 5 0 1 2 3 4 5
Time (s) Time (s)
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42. Introduction
Graph theoretic contact model
Volumetric Model
MapleSim model
Experiments
Demos
Modelling
Outline
1 Introduction
2 Volumetric Model
Volumetric model framework
Normal forces
Friction forces
3 Experiments
Normal force experiments
Friction experiments
4 Modelling
Graph theoretic contact model
MapleSim model
Demos
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43. Introduction
Graph theoretic contact model
Volumetric Model
MapleSim model
Experiments
Demos
Modelling
Model
c1 d
Bodies: m1 , m2
Measured displacement: d
f
m1
Volume centroid (relative to
each body): c1 , c2
c2
Contact forces: f
g m2
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44. Introduction
Graph theoretic contact model
Volumetric Model
MapleSim model
Experiments
Demos
Modelling
MapleSim Model: Assumptions
One deformable body model - body ‘a’ rigid, ‘b’ deformable
Normal axis is frame ‘a’ z-axis (body a is flat)
All vectors and tensors (i.e. inertia tensor, centroid, relative
velocity) calculated for frame ‘a’
Js ≈ JV - volume inertia tensor is easier to calculate than
surface inertia
Volume centroid and surface centroid are very close (normal
and friction forces at same location)
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45. Introduction
Graph theoretic contact model
Volumetric Model
MapleSim model
Experiments
Demos
Modelling
Model
c1 d
f
m1
c2
g m2
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46. Introduction
Graph theoretic contact model
Volumetric Model
MapleSim model
Experiments
Demos
Modelling
Parameters
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47. Introduction
Graph theoretic contact model
Volumetric Model
MapleSim model
Experiments
Demos
Modelling
Forces Block
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48. Introduction
Graph theoretic contact model
Volumetric Model
MapleSim model
Experiments
Demos
Modelling
Geometry Calculation Block
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49. Introduction
Graph theoretic contact model
Volumetric Model
MapleSim model
Experiments
Demos
Modelling
Geometry: Sphere-on-plane
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50. Introduction
Graph theoretic contact model
Volumetric Model
MapleSim model
Experiments
Demos
Modelling
Geometry: Cylinder-on-plane
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51. Introduction
Graph theoretic contact model
Volumetric Model
MapleSim model
Experiments
Demos
Modelling
Demo: ‘wobbly’ clutch
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52. Introduction
Graph theoretic contact model
Volumetric Model
MapleSim model
Experiments
Demos
Modelling
Demo: tippe top
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53. Introduction
Graph theoretic contact model
Volumetric Model
MapleSim model
Experiments
Demos
Modelling
Questions
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