a volumetric contact dynamics model

IntroductionVolumetric Model

ExperimentsModelling

A Volumetric Contact Dynamics Model

Mike Boos

SYDE 652

March 27, 2012

Mike Boos A Volumetric Contact Dynamics Model 1/ 34



Outline

1 Introduction

2 Volumetric ModelVolumetric model frameworkNormal forcesFriction forces

3 ExperimentsNormal force experimentsFriction experiments

4 ModellingGraph theoretic contact modelMapleSim modelDemos




Outline

1 Introduction







Motivation

Dextre at the tip of Canadarm2 (Gonthier, 2007)




Point contact models

B1

B2

fn

kδ

Figure: Point contact model.

Hertz theory

fn = kδp n(p = 3/2 forsphere-on-sphere)

Hunt-Crossley

fn = kδp(1 + aδ)n




Volumetric model frameworkNormal forcesFriction forces

Outline

1 Introduction








Volumetric model

B1

B2

fn

kv

Figure: Modified Winkler elastic foundation model.

Force element

dfn = kvδ(s)n





Volumetric model

B1

B2

n

S

δ(s)

s

Contact plate

B1

B2

n

S

pc

sc

ρs ρv

sp

V =∫S δ(s)dS

pc =∫V pdV

V

Js =∫S((ρs ·ρs)I−ρsρs)δ(s)dS

Jv =∫V ((ρv · ρv)I− ρvρv)dV

J{s,v}n = r2gyrV n





Normal Forces

B1

B2

τ s

τ r

fn

f t

dfn = kvδ(s)(1 + a vn)n

Normal force

fn = kvV (1 + a vcn)n

Rolling resistance torque

τ r = kv aJs · ωt





Basic friction model

B1

B2

τ s

τ r

fn

f t

df t = −µdfnvt

Friction force

f t = −µ fnvsct

Spinning friction torque

τ s = −µ r2gyrfn ωn





Stick-slip state

Average surface velocity

v2avg = vsct · vsct + (rgyr|ωn|)2

Stick-slip state

s = e−

v2avgv2s

Maximum friction coefficient

µmax = µC + (µS − µC) sCan add lag to s for dwell time dependency.





Stick-slip state



Stick-slip state

s = e−

v2avgv2s


µmax = µC + (µS − µC) s

Can add lag to s for dwell time dependency.





Stick-slip state



Stick-slip state

s = e−

v2avgv2s


µmax = µC + (µS − µC) sCan add lag to s for dwell time dependency.





Bristle model

fN

Contact sites

Surface asperities (‘bristles’) incontact (Gonthier, 2007).

Bristle properties

Deformation: zsc

Rotation: θn

Parameters

Stiffness: σo

Damping: σ1





Tangential friction forces

Friction force

f t = −fn (sat(σo zsc + σ1 zsc, µmax) + σ2 vsct)

Bristle deformation rate

zsc = svsct + (1− s)(1σ1 µC dirε(vsct, vε)− σo

σ1 zsc

)







τ s = −r2gyr fn(sat(σo θn + σ1 θn,

µmaxrgyr

)+ σ2 ωn

)n

Bristle deformation rate

θn = s ωn + (1− s)(

µCσ1 rgyr sgn(ωn)− σo

σ1 θn

)





The Contensou effect

Translational friction forcestend to ‘cancel out’ as angularvelocity increases.

Contensou factors

Cv =|vsct|vavg

Cω =rgyr|ωn|vavg

We now need to update theslipping coefficient in ourbristle dyanmics equations toinclude these factors.

C

A Bv

ωvA

vB

vC

vDD

ω r

ω r

ω r

ω r

v << ωr (Gonthier, 2007)




Normal force experimentsFriction experiments

Outline

1 Introduction








Contact properties

Focus on simple geometric pairs:

Cylinder-on-plane

Sphere-on-plane

Payload material: Stainless steelContact plane materials: Al, Mg





Apparatus in normal configuration

Force sensorPayload/specimen (stainless steel)

Encoder reference Contact surface (Al or Mg)

Linear

encoder





Selected results: Normal forces

Quasi-static loading of SScylinder on Mg plane

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−5

0

5

10

15

20

25

Displacement (µm)

Conta

ct fo

rce (

N)

Measured data

Perpendicular fit

Perpendicular contact point

Misaligned fit

Misaligned contact point

kv = 5.17× 1012N/m3

Damping factors (a) measured

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8

9

10x 10

4

Impact velocity (mm/s)D

am

pin

g facto

r (s

/m)

Estimated factors for AlFit of a ∝ 1/v

i for Al

Estimated factors for MgFit of a ∝ 1/v

i for Mg

a ≈ 1−e2eff

eeff vin





Experimental procedure

Identify parameters

Verify parameters Contensou effect






Identify parameters Verify parameters

Contensou effect






Identify parameters Verify parameters Contensou effect





Friction apparatus

Cylindrical

payload

Rotational

motor

Linear

motor

Encoder

reference

Linear

encoder

3DOF force sensors

Contact

surface

z

y

x

z

y

x





Selected results: Contensou effect

0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Time (s)

Co

eff

icie

nt

of

Frictio

n

Measured coefficients

Model coefficients

0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Time (s)C

oe

ffic

ien

t o

f F

rictio

n

Measured coefficients

Model coefficients




Graph theoretic contact modelMapleSim modelDemos

Outline

1 Introduction








Model

c1

c2

f

m1

m2

d

g

Bodies: m1, m2

Measured displacement: d

Volume centroid (relative toeach body): c1, c2

Contact forces: f





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, relativevelocity) calculated for frame ‘a’

Js ≈ JV - volume inertia tensor is easier to calculate thansurface inertia

Volume centroid and surface centroid are very close (normaland friction forces at same location)





Model

c1

c2

f

m1

m2

d

g





Parameters





Forces Block





Geometry Calculation Block





Geometry: Sphere-on-plane





Geometry: Cylinder-on-plane





Demo: ‘wobbly’ clutch





Demo: tippe top





Questions


a volumetric contact dynamics model

Education

v pdvvjs

s sdspc

r2gyrv nmike

ss sisssdsjv

force elementdfn

hertz theoryfn

kp1 anmike

tip of canadarm2 gonthier