ANALYSIS OF LOAD PATTERNS IN RUBBER COMPONENTS FOR … · ANALYSIS OF LOAD PATTERNS IN RUBBER...

ANALYSIS OF LOAD PATTERNS IN RUBBERCOMPONENTS FOR VEHICLES

Jerome Merelformerly at Hutchinson Corporate Research Center

Israel WanderApex Technologies

Pierangelo Masarati, Marco MorandiniDipartimento di Ingegneria Aerospaziale,

Politecnico di Milano

Multibody Dynamics 2007Milano, June 25–28 2007

Outline

Motivation

Software Description

Connectivity

Constitutive Laws

Application: Car Suspension Model

Conclusions & Acknowledgements

Outline

Motivation


Connectivity

Constitutive Laws



Motivation

I Multibody Dynamics started as formalism for mechanicsof rigid-body mechanisms

I Today: fully developed industrial-grade computational tool

I Blend of exact, arbitrary kinematics& nonlinear finite elements

I Ideal playground for multidisciplinary problems

Motivation





Motivation





Motivation





Motivation

I Hutchinson, as a worldwide leader in manufacturing of rubbercomponents, has a long tradition in accurate and detailedmodeling of their products

→ nonlinear finite elements

I The market requires to extend analysis capabilities to theentire mechanical system, in order to design and validatesingle components

→ multibody dynamics

I Hutchinson traditionally developed specialized finite elementanalysis capabilities in-house

I Industrial requirement: preserve in-house analysis capabilitiesfor multibody analysis as well; solution:

→ use free software

as an alternative to “reinventing the wheel”

Motivation









Motivation









Motivation









Motivation









Motivation









Motivation









Outline

Motivation


Connectivity

Constitutive Laws




I MBDyn: general-purpose MultiBody Dynamics softwaredeveloped at Politecnico di Milano since the early ’90s

I mainly applied to rotorcraft dynamics and aeroservoelasticity,but it is currently exploited (at Politecnico di Milano and by3rd parties) in projects involving

I Rotorcraft dynamics (helicopters, tiltrotors)I Aircraft landing gear analysisI Robotics and mechatronics, including real-time simulationI AutomotiveI Wind turbinesI Biomechanics








I Rotorcraft dynamics (helicopters, tiltrotors)

I Aircraft landing gear analysisI Robotics and mechatronics, including real-time simulationI AutomotiveI Wind turbinesI Biomechanics




I Rotorcraft dynamics (helicopters, tiltrotors)I Aircraft landing gear analysis

I Robotics and mechatronics, including real-time simulationI AutomotiveI Wind turbinesI Biomechanics




I Rotorcraft dynamics (helicopters, tiltrotors)I Aircraft landing gear analysisI Robotics and mechatronics, including real-time simulation

I AutomotiveI Wind turbinesI Biomechanics




I Rotorcraft dynamics (helicopters, tiltrotors)I Aircraft landing gear analysisI Robotics and mechatronics, including real-time simulationI Automotive

I Wind turbinesI Biomechanics




I Rotorcraft dynamics (helicopters, tiltrotors)I Aircraft landing gear analysisI Robotics and mechatronics, including real-time simulationI AutomotiveI Wind turbines

I Biomechanics






MBDyn is free software (GPL). The user is granted the rights to:

1. access the source code

2. distribute the software, for free or for money, provided thesource code is distributed as well, and

the rights are not restricted

3. modify the source code

4. distribute modified sources, for free or for money, provided themodified source code is distributed as well, and the originalrights are not restricted, but rather

extend to modifications

Further information at

http://www.aero.polimi.it/~mbdyn/














2. distribute the software, for free or for money,

provided thesource code is distributed as well, and










2. distribute the software, for free or for money, provided thesource code is distributed as well,

and

































4. distribute modified sources, for free or for money,

provided themodified source code is distributed as well, and the originalrights are not restricted, but rather










4. distribute modified sources, for free or for money, provided themodified source code is distributed as well, and the originalrights are not restricted,

but rather
























Outline

Motivation


Connectivity

Constitutive Laws



Connectivity

Mechanical systems are modeled as nodes, that provide sharedequations and degrees of freedom, connected by elements, thatcontribute to shared equations and optionally provide private ones.

Relevant elements can be

I rigid bodies→ contribute inertia properties to nodes

I joints→ add private algebraic equations that constrain nodes→ contribute constraint reactions to shared equations

Connectivity





Connectivity





Connectivity





Connectivity

Selected test application: car suspensions model

a wide variety of deformable components model rubber parts

Connectivity

Deformable components:

I nodes exchange configuration-dependent forces and moments

I separation between constitutive model. . .

F = F (u, u, . . .)

. . . and connectivity

u = u (u1,u2)

F1 = T1 (u1,u2)F

F2 = T2 (u1,u2)F

I Connectivity and constitutive models development isdecoupled; provided an adequately expressive API is designed,they mutually benefit from each other.

Connectivity




F = F (u, u, . . .)


u = u (u1,u2)

F1 = T1 (u1,u2)F

F2 = T2 (u1,u2)F


Connectivity




F = F (u, u, . . .)


u = u (u1,u2)

F1 = T1 (u1,u2)F

F2 = T2 (u1,u2)F


Connectivity




F = F (u, u, . . .)


u = u (u1,u2)

F1 = T1 (u1,u2)F

F2 = T2 (u1,u2)F


Connectivity




F = F (u, u, . . .)


u = u (u1,u2)

F1 = T1 (u1,u2)F

F2 = T2 (u1,u2)F


Connectivity

Deformable components implemented in MBDyn:

I 1D component: rod

I 3D component: angular spring

I 3D component: linear spring

I 6D component: linear & angular spring

I 6D component: geometrically “exact”, composite ready beam

I Component Mode Synthesis (CMS) element

Connectivity


I 1D component: rod






Connectivity


I 1D component: rod






Connectivity


I 1D component: rod






1D Component: Rod

R0

R1x1

f1

p1

R2

x2

f2p2

l

I connects two points p1, p2

I optionally offset from therespective nodes:

p1 = x1 + f1

p2 = x2 + f2

I offsets are rigidly connectedto nodes

f1 = R1f1

f2 = R2f2

I straining related to distance between points p1 and p2:

l = p2 − p1 ε =

√lT l

l0− 1

3D Component: Angular Spring

R0

R1x1

R2

x2

θ

I connects two nodes x1, x2

I straining related toperturbation θ of relativeorientation R = RT

1 R2

I the joint has no location inspace (it is typically pairedto a spherical hinge or otherrelative position constraint)


I relative orientation matrix R = RT1 R2

I → implies that the component constitutive properties areintrinsically referred to node 1

I relative orientation vector θ = ax(exp−1 (R)

)I relative angular velocity ω = RT

1 (ω2 − ω1)

I internal moment M = M (θ,ω), referred to node 1

I contribution to virtual work δL = −θTδ M

I relative orientation virtual perturbation θδ = RT1 (θ2δ − θ1δ)

I contributions to node equilibrium

M1 = M

M2 = −M

with M = R1M






1 (ω2 − ω1)





M1 = M

M2 = −M

with M = R1M





)

I relative angular velocity ω = RT1 (ω2 − ω1)





M1 = M

M2 = −M

with M = R1M






1 (ω2 − ω1)





M1 = M

M2 = −M

with M = R1M






1 (ω2 − ω1)





M1 = M

M2 = −M

with M = R1M






1 (ω2 − ω1)





M1 = M

M2 = −M

with M = R1M






1 (ω2 − ω1)





M1 = M

M2 = −M

with M = R1M






1 (ω2 − ω1)





M1 = M

M2 = −M

with M = R1M


Pros:

I the formulation is straightforward

Cons:

I the model is biased towards one node

I as a consequence, formulating any constitutive law butisotropic may not be straightforward

With linear anisotropic constitutive law, if connectivity is reversed:


Pros:


Cons:





Pros:


Cons:





Pros:


Cons:





Pros:


Cons:





Pros:


Cons:




3D Component: “Invariant” Angular Spring

From now on, the previous angular spring is termed “attached”

I the “attached” formulation issue quickly showed up whenusing orthotropic bushings in the car suspension model

I it took a while to find out that unstable oscillations werecaused by geometrical nonlinearity in the bushings connectivity

I in all cases, linear (visco)elastic properties were used

I reversing the order of the nodes always solved the problem

Rationale:

I the behavior of the component is intrinsically independentfrom the ordering of the connectivity

I if the connectivity formulation depends on its ordering, theconstitutive properties need to take care of invariance

I otherwise, connectivity must take care of invariance itself







Rationale:










Rationale:










Rationale:










Rationale:










Rationale:










Rationale:










Rationale:





I relative orientation R = RT1 R2 remains the same

I constitutive properties referred to mid-rotation θ = 12θ, such

that intermediate relative orientation matrix R = exp(θ ×

)yields RR = R

2= R

I perturbation of intermediate orientation θδ =(I + R

)−1θδ

I absolute mid-rotation orientation R = R1R = R2RT

I relative angular velocity ω = RT

(ω2 − ω1)

I internal moment is now M = M (θ,ω)

I internal moment in the absolute frame M = RM



I constitutive properties referred to mid-rotation θ = 12θ,

such


)yields RR = R

2= R


)−1θδ



(ω2 − ω1)







)

yields RR = R2

= R


)−1θδ



(ω2 − ω1)







)yields RR = R

2= R


)−1θδ



(ω2 − ω1)







)yields RR = R

2= R


)−1θδ



(ω2 − ω1)







)yields RR = R

2= R


)−1θδ



(ω2 − ω1)







)yields RR = R

2= R


)−1θδ



(ω2 − ω1)







)yields RR = R

2= R


)−1θδ



(ω2 − ω1)







)yields RR = R

2= R


)−1θδ



(ω2 − ω1)




Pros:

I the model is no longer biased towards one node

I simpler constitutive laws may be formulated

Cons:

I the formulation is less straightforward

I little bit more computationally expensive


Pros:

I the model is no longer biased towards one node

I simpler constitutive laws may be formulated

Cons:

I the formulation is less straightforward

I little bit more computationally expensive

3D Component: Linear Spring

R0

R1x1

f1p1

R2

x2

f2

p2

I Same as rod, but...

I straining related to distancebetween points ε = p2 − p1

I the joint does not react purerelative rotation (pin;usually paired to relativeorientation constraint)


I same issue about “attached” vs. “invariant” formulation

I formulation of invariant case even less straightforward(not presented here, but fully developed and implemented inthe software)

Same linear anisotropic constitutive law, transverse shear case

-0.5

-0.4

-0.3

-0.2

-0.1

0

0 10 20 30 40 50

Dis

plac

emen

t, m

Load, N

(a) ground(b) floating

(c) invariant

-0.15

-0.1

-0.05

0

0.05

0 10 20 30 40 50

Dis

plac

emen

t, m

Load, N


(c) invariant


I same issue about “attached” vs. “invariant” formulationI formulation of invariant case even less straightforward

(not presented here, but fully developed and implemented inthe software)


-0.5

-0.4

-0.3

-0.2

-0.1

0

0 10 20 30 40 50

Dis

plac

emen

t, m

Load, N


(c) invariant

-0.15

-0.1

-0.05

0

0.05

0 10 20 30 40 50

Dis

plac

emen

t, m

Load, N


(c) invariant


I same issue about “attached” vs. “invariant” formulationI formulation of invariant case even less straightforward

(not presented here, but fully developed and implemented inthe software)


-0.5

-0.4

-0.3

-0.2

-0.1

0

0 10 20 30 40 50

Dis

plac

emen

t, m

Load, N


(c) invariant

-0.15

-0.1

-0.05

0

0.05

0 10 20 30 40 50

Dis

plac

emen

t, m

Load, N


(c) invariant

6D Components

I linear & angular spring

I models fully coupled bushingsI formulation fully developed, but. . .I . . . only partially implemented, essentially because of limited

usefulness so far

I kinematically “exact”, composite ready beamI detailed description outside the scope of this workI see Ghiringhelli et al., AIAA Journal, 2000

6D Components

I linear & angular springI models fully coupled bushings

I formulation fully developed, but. . .I . . . only partially implemented, essentially because of limited

usefulness so far


6D Components

I linear & angular springI models fully coupled bushingsI formulation fully developed, but. . .

I . . . only partially implemented, essentially because of limitedusefulness so far


6D Components

I linear & angular springI models fully coupled bushingsI formulation fully developed, but. . .I . . . only partially implemented, essentially because of limited

usefulness so far


6D Components


usefulness so far

I kinematically “exact”, composite ready beam

I detailed description outside the scope of this workI see Ghiringhelli et al., AIAA Journal, 2000

6D Components


usefulness so far

I kinematically “exact”, composite ready beamI detailed description outside the scope of this work

I see Ghiringhelli et al., AIAA Journal, 2000

6D Components


usefulness so far


Outline

Motivation


Connectivity

Constitutive Laws



Constitutive Laws

I define the input/output relationship required by deformablecomponents

F = F (u, u)

I task separation allows to exploit similar constitutive laws indifferent contexts, without bothering about connectivity

δF =∂F

∂uδu +

∂F

∂uδu

I this aspect is emphasized in the implementation exploitingC++ templates for different dimensionalities (1D, 3D, 6D)

δFn =∂Fn

∂un︸︷︷︸n×n

δun +∂Fn


δun

Constitutive Laws


F = F (u, u)


δF =∂F

∂uδu +

∂F

∂uδu


δFn =∂Fn


δun +∂Fn


δun

Constitutive Laws


F = F (u, u)


δF =∂F

∂uδu +

∂F

∂uδu


δFn =∂Fn


δun +∂Fn


δun

Constitutive Laws

Use:

I call Update() at each iteration

I subsequent calls to F(), FDE(), FDEPrime() allow to accessthe force and its partial derivatives (e.g. to compute theresidual or the contribution to the Jacobian matrix)

I as soon as AfterConvergence() is called, the solutionconverged, and the final state can be consolidated, if needed

I the paper contains examples of C++ meta-code forconstitutive laws, including isotropic and orthotropic templates

I or, refer to mbdyn/base/constltp* files in the source code

Constitutive Laws

Use:






Constitutive Laws

Use:






Constitutive Laws

Use:






Constitutive Laws

Use:






Constitutive Laws

Use:






Outline

Motivation


Connectivity

Constitutive Laws




I the deformable components have been applied to the analysisof a car suspension model

I the model consists in the full front and rear suspension systemof a generic car (not representative of a specific vehicle)


I the deformable components have been applied to the analysisof a car suspension model

I the model consists in the full front and rear suspension systemof a generic car (not representative of a specific vehicle)


I the chassis is modeled as a rigid body

(the use of a CMS model is foreseen for further validation)

I the front torsion bar and the rear twist beam are modeled bybeam elements

I the model consists in about 1300 equations, related toI about 100 structural nodesI about 60 nonlinear beam elementsI more than 80 joints


I the chassis is modeled as a rigid body(the use of a CMS model is foreseen for further validation)












Typical analysis:

I consists in evaluating the loads in rubber components whenthe model is subjected to test rig excitation

I excitation pattern: acceleration imposed to front right axle

-80

-60

-40

-20

0

20

40

60

0 0.1 0.2 0.3 0.4 0.5

Acc

eler

atio

n, m

/s^2

Time, s

xyz


Typical analysis:

I consists in evaluating the loads in rubber components whenthe model is subjected to test rig excitation

I excitation pattern: acceleration imposed to front right axle

-80

-60

-40

-20

0

20

40

60

0 0.1 0.2 0.3 0.4 0.5

Acc

eler

atio

n, m

/s^2

Time, s

xyz


-10000

-8000

-6000

-4000

-2000

0

2000

4000

0 0.1 0.2 0.3 0.4 0.5

For

ce, N

Time, s

xyz

-2

0

2

4

6

8

10

12

14

0 0.1 0.2 0.3 0.4 0.5

Mom

ent,

Nm

Time, s

xyz

Force and moment in front right shock absorber top bushing


-1500

-1400

-1300

-1200

-1100

-1000

-900

-800

0 0.1 0.2 0.3 0.4 0.5

For

ce, N

Time, s

Force in front right shock absorber

-10

-8

-6

-4

-2

0

2

4

6

8

10

0 0.1 0.2 0.3 0.4 0.5

Acc

eler

atio

n, m

/s^2

Time, s

xyz

CG acceleration

Outline

Motivation


Connectivity

Constitutive Laws




I The work illustrates versatile modeling of structuralcomponents for mechanics analysis of rubber components

I Component behavior dependence on connectivity has beeneliminated by invariant deformable components, withoutformulating connectivity-dependent constitutive properties

I The features illustrated in this work will be distributed shortly,with the next release of the software (1.3.0)

I MBDyn is considered by Hutchinson as a viable tool forindustrial exploitation

I The authors acknowledge the support of Hutchinson SA R&DCentre and particularly of Dr. Daniel Benoualid in developingfree multibody software.

























ANALYSIS OF LOAD PATTERNS IN RUBBERCOMPONENTS FOR VEHICLES

Jerome Merel, Israel Wander,Pierangelo Masarati, Marco Morandini

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ANALYSIS OF LOAD PATTERNS IN RUBBER COMPONENTS FOR … · ANALYSIS OF LOAD PATTERNS IN RUBBER...

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