Computer-aided analysis of rigid and flexible multibody...

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GraSMech – Multibody 1

Computer-aided analysis of rigid and

flexible multibody systems (Part II)

Component mode synthesis

of flexible bodies

Prof. O. Verlinden

GraSMech course 2005-2006


Kinematics of a flexible body

(floating reference frame)

Superimposition of

a gross rigid-body motion defined by the floating (moving)

reference frame

a small elastic displacement measured with respect to the

undeformed configuration

The elastic displacement is

expressed as a weighted

summation of predefined

deformation (component) modes

with r0 the position in undeformed

configuration


Principle of substructuring

Used initially in structural analysis to get lowerorder systems for computingeigenproperties

Idea:

Each structure isanalysed separately=> typical deformationmodes

The model of the complete structure ismodelled from the selected deformationmodes of each subpart

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Substructuring in multibody systems

Applied naturally to

multibody systems

Substructure=body

Interface nodes =

connection points

(joints or applied

forces)


Classical substructuring approaches


Classical modal bases

Qualities of a mode set

Efficiency (convergence): accurate system response with a

minimal number of modes

Independence: the modes of a component do not depend on the

other ones (universality)

Generality: applicable in any circumstances

Classical solutions mix static and vibration modes: Craig-Bampton,

Mac Neal-Rubin, Benfield-Hruda

Major reference: « A review of time-domain and frequency-domain

component mode synthesis methods, R.R. Craig, Jnal of

Analytical and experimental modal analysis, 2 (2), 1987

Other reference: M. Géradin, A. Cardona, Flexible Multibody

Dynamics, John Wiley & Sons, Chichester, 2001

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Driving example: a beam

Case of a beam with 2 boundary nodes


Substructure equations

Equations of motion of the substructure (n d.o.f., small

deflections, linear material, ...)

Partition between internal (I) and boundary (B) degrees of

freedom (n=nI+nB)


Substructure equations

Boundary degrees of freedom can be splitted once more into R

(statically determinate=rigid-body prevention) and E (excess or

redundant boundary conditions) (n=nI+nE+nR, nR=6)

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Case of the beam

2 boundary nodes

planar case (3 dof per node)

nB=6

nR=3, nE=3


Rigid-body modes

Rigid-body modes ΨΨΨΨR can be determined in several ways

By geometric considerations: r=d+Rr0 (r=new position, d=global

displacement, R=rotation matrix, r0=initial position)

Null frequency free boundaries vibration modes obtained from

Static solution for a unit displacement of one the R boundary

coordinates

Number of rigid-body modes=6

1 mode per

column

6x6 unit

matrix


Rigid-body modes of the beam

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Constraint modes

Constraint mode: deformation modes ΨΨΨΨC obtained whenimposing a unit displacement of one of the boundary degrees of

freedom while all other boundary degrees of freedom are fixed

No distinction between E and R boundary d.o.f.

Obtained by solving

Number of static constraint modes: nB


Constraint modes of the beam


Redundant constraint modes

Redundant constraint mode: same as constraint modes but only

on the redundant boundary degrees of freedom (E), with

statically determinate degrees of freedom fixed (R)

Obtained by solving

Number of static redundant constraint modes: nE = nB-6

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Redundant constraint modes of the beam


Attachment modes

Attachment modes: deformation modes ΨΨΨΨA obtained when

applying a unit force on one of of the redundant boundary

degrees of freedom (E), with statically determinate degrees of

freedom fixed (R)

Obtained by solving

Number of attachment modes : nE = nB-6


Attachment modes of the beam

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Summary

Substructure: n d.o;F., n=nI+nB, nB=nR+nE, nR=6 (in 3D)

Defined modes

Rigid-body modes (6 modes)

Constraint modes: one unit displacement on B (nB modes)

Redundant constraint modes: one unit displacement on E,

with R fixed (nE=nB-6 modes)

Attachment modes: unit effort on E with R fixed (nB-6 modes)

These 3 mode sets are equivalent (linear combinations of each

other)

Rigid body modes ΨΨΨΨR and redundant constraint modes ΨΨΨΨRC

Rigid body modes ΨΨΨΨR and attachment modes ΨΨΨΨA

Constraint modes ΨΨΨΨB

=> Constraint modes span the rigid-body modes !


Inertia-relief modes

Inertia-relief mode (Hintz/Herting form) ΨΨΨΨH deformation of the body subjected to a rigid-body acceleration field, with boundarydegrees of freedom fixed (interpretation: the motion is imposedthrough the boundaries)

Obtained by solving

Number of inertia-relief modes: 6 (one per rigid-body mode)

rigid-body

modes


Inertia-relief modes of the beam

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Attachment modes of an unrestrained body

Attachment modes of an unrestrained body: ΨΨΨΨU other kind of

inertia- relief mode obtained by applying unit forces on the

boundary degrees of freedom of the substructure

Principle: a force on the free substructure induces

A rigid-body acceleration field

An elastic deflection

The mode is obtained from the elastic deflection, made mass-

orthogonal to the rigid-body modes

Number of unrestrained attachment modes: nB


Equivalence between static modal bases

It can be shown that the following mode sets are equivalent (linear

combinations of each other)

Rigid body modes ΨΨΨΨR, redundant constraint modes ΨΨΨΨRC and

inertia-relief modes ΨΨΨΨH (6+nE+6=nB+6)

Rigid body modes ΨΨΨΨR and attachment modes ΨΨΨΨA and inertia-

relief modes ΨΨΨΨH (6+nE+6=nB+6)

Constraint modes ΨΨΨΨB and inertia-relief modes ΨΨΨΨH (nB+6)

Unrestrained attachment modes ΨΨΨΨU and rigid-body modes ΨΨΨΨR

(nB+6)

They are all statically complete mode sets ->able to represent

exactly any response to a static load on interfaces, including

global accelerations (important for driven bodies, like in

earthquake)


Vibration modes

Free boundary vibration modes (n modes among which rigid-

body modes)

Fixed boundary vibration modes (nI=n-nB modes)

Loaded boundary vibration modes (n modes)

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Free boundary vibration modes


Fixed boundary vibration modes


Classical reductions

Craig-Bampton: constraint modes and fixed interface vibration modes (=Guyan-Irons condensation if no vibration modes)Advantage: good kinematic conditioning

vibration modes are internal (no kinematic coupling withboundary motion)

each boundary mode corresponds to a unit displacement of one and only one boundary degree of freedom -> looks likea traditional element (easy for assembly)

Mac Neal-Rubin: residual attachment modes (equivalent to attachment modes) and free interface vibration modesAdvantage: good conditioning of global mass and stiffnessmatrices!!! Redundancy is possible (more modes than d.o.f.)

Note: if inertia-relief modes are added, redundancy can appear in CB

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Efficiency of modal bases

Example: free-free modes recomputed from Craig-Bampton


Situation in MBS

The floating reference frame brings the rigid-body modes -> any

mode set can be used as far as rigid-body modes are

suppressed

Example: in CB, ΨΨΨΨB replaced by ΨΨΨΨRC (ΨΨΨΨB=ΨΨΨΨR+ΨΨΨΨRC)

Inertia-relief modes are important due to gross motion

accelerations of the bodies

Craig-Bampton is often used although it lacks inertia-relief

modes (automatic generation from ANSYS, ABAQUS,

NASTRAN to ADAMS)

The choice of the reference frame is not so important (any mode

can be expressed wrt any reference frame)


Quality test of a mode set in MBS

Quality test of a mode set for a given simulation

Simulation of the body with a ``finite element'' mode set (Guyan

condensation on privileged nodes)

=> reference deformation ΨΨΨΨ0 (t)

Least-squares => best modal approximation of reference

deformation by modal base X

Error: ε(t)

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Example: railway engine

Finite element model of the vehicle


Maneuvers

Two maneuvers

a rail defect (impulse) at 5 m/s (l=100 mm, w=4 mm, h=10 mm)

a lane change at 7 m/s (soft: l=50 m, w=4 m, h=1 m))


Results for the impulse maneuver

BHM: Benfield-Hruda mode set (vibration modes on suspensions)

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Results for the soft maneuver

BHM: Benfield-Hruda mode set (vibration modes on suspensions)


Geometric and natural boundary conditions

Component-mode synthesis=Rayleigh-Ritz method

Geometric boundary conditions -> displacements and slopes

Natural boundary conditions -> forces and moments: stiffness,

inertia, ...

Meirovitch: « in the case of multibody systems, ..., for the most part,

the only boundary conditions are natural

Mac Neal-Rubin: null natural boundary conditions for the vibration

modes -> not adapted when efforts are applied on the interfaces

(except for the static modes)


From the finite element model

to the flexible body

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From finite element model to flexible body

The substructure is usually defined in a finite element software

which allows to easily compute the different deformation modes

How to retrieve the rigid body characteristics (mass, inertia

tensor) ?

How to compute the invariants of the deformation modes

necessary to write the equations of motion ?

References

O. Walrapp, « Standard input data of flexible members in multibody

systems », in Advanced Multibody System Dynamics, W.

Schiehlen, Kluwer, Dordrecht, 1993


Equilibrium of a flexible body

Flexible body with a floating reference frame denoted E


Acceleration of a particle

Expression of the acceleration

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Translational equilibrium of a flexible body

with

Rigid-body contribution

Deformation

contribution

With Tisserand frame, 1Cn=0

(first mean axis condition)


Rotational equilibrium of a flexible body

with


Deformation contribution

With Tisserand frame, 2Cn=0

(first mean axis condition)


Equilibrium for mode n

with

For small deformation and a linear material

Internal equilibrium of a flexible body

Deformation contribution


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Translation and rotation coordinates

The degrees of freedom of the finite element model can be

partitioned so as to identify the displacements and rotations of

each node

The position vector of node i wrt to the reference frame is

denoted by ri

Important note: all subsequent vector relationships are assumed to

be expressed in the reference frame of the body !

translation displacements of node i (dx,dy,dz)

rotation angles of node i (θx,θy,θz)


Translation and rotation coordinates

The mass matrix and the deformation modes can then be

partitioned in this way


Forces and accelerations

If we denote

Fi, Ti : the force and torque applied on node i

a_i, ΩΩΩΩ i the translational and rotational accelerations undergoneby node i

Forces able to produce the acceleration field are given by

The resultant force Ftot is given by

projected on E !

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The resultant torque Ttot (reduced at the origin of the reference

frame) is given by

projected on E !



If we define ΨΨΨΨR1 and ΨΨΨΨR2 (dimension 3N x 3)

Matrix ΨΨΨΨR1 gathers 3 rigid-body translational modes

Matrix ΨΨΨΨR2 gathers 3 rigid-body rotational modes (about the origin

of the reference frame of the body)


Retrieving the mass

If the body undergos a uniform acceleration field a, the total force is

given by

and according to the Newton’s law, we must have

and then

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If the body undergos a uniform acceleration field a, the total torque

is given by

and according to the Newton’s law, we must have

and then

An equivalent formula can be used

Retrieving the center of mass


Retrieving the first invariant

Translational equilibrium of the flexible body

=> the invariant 1Cn is the force generated by an acceleration field

proportional to the nth deformation mode

which gives


Retrieving the second invariant

Rotational equilibrium of the flexible body

=> the invariant 2Cn is the torque (wrt the reference frame)

generated by an acceleration field proportional to the nth

deformation mode

which gives

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Retrieving the third invariant

Rotational equilibrium of the flexible body

⇒ the invariant 3Cn (3x3 matrix) is the torque (wrt the reference

frame) generated by a Coriolis acceleration field with a relative

velocity proportional to the nth deformation mode

For each node i, the accelerations are given by



The identification leads to


Retrieving the fourth invariant

Internal equilibrium of the flexible body

⇒ the invariant 4Cn is the contribution on mode n of a Coriolis

acceleration field with a relative velocity proprotional to the lth

deformation mode

For each node i, the accelerations are given by

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The identification leads to


Mass and stiffness matrices

Internal equilibrium of the flexible body

The elements of the modal mass and stiffness matrices are

classically obtained by


Link with co-rotational formulation

Some invariants can also be retrieved by identification between

a first expression of the kinetic energy

which can also be developed in terms of some of the invariants

and an expression of the kinetic energy similar to the one used

in the corotational formulation presented by O. Brüls

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Accuracy of the evaluations

The rigid-body growths m, ΦΦΦΦ and rG are determined exactly if the

shape functions are able to reproduce exactly the rigid-body

velocity field (normally the case when 6 dof per node: beam,

plate, ...)

The invariants 1C, 2C, M* and K* are exact if the rigid-body

modes and the mode shapes are represented exactly by the

shape functions (case of a static mode for a beam)

Otherwise, the accuracy increases with the density of the mesh.

The invariants 3C and 4C, are always an approximation, getting

better with the number of elements


Exemple: simulation of a kart

Particularities of a kart

No suspensions

No differential: the rear

wheels are rigidly mounted on

the same axle

=> in cornering, the pilot twists

the chassis to raise the rear

interior wheel

Weight of the pilot = weight of

the vehicle

=> A flexible model of the

chassis is necessary !


Elements of the kart

Main elements of a competition kart

1. chassis made of steel tubes

2. rear axle

3. bearings of the rear axle

4. disc brake

5. engine (100 cm3 2 stroke)

6. steering mechanism

1 2

3

4

65 3

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FiniteFinite elementelement model of the model of the chassischassis alonealone


FiniteFinite elementelement model (model (chassischassis++rearrear axleaxle))


FiniteFinite elementelement model (model (chassischassis++engineengine))

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Validation of the finite element model

Static measurements

Experimental modal analysis

In different configurations

chassis alone

chassis with engine

chassis with rear axle


ExampleExample: torsion mode: torsion mode


ExampleExample: : bendingbending modemode

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Flexible bodyFlexible body

The engine and the rear axle are included in the model

10 interface nodes defined in

the finite element model

1: wheels attachements

2: steering shaft

3: pilot seat

The flexible body can then be

exported from ANSYS to

ADAMS (modal neutral

file) once the number of

vibration modes is defined

(Craig-Bampton)


ADAMS modelADAMS model

1 flexible body

4 tires

pilot=1 rigid body

attached to the

chassis by bushings

steering mechanism


Simulation in cornering

Simulation

Straight line for stabilization

Maneuver: step of steering

wheel angle (0.082 rad in 0.1

sec)

The velocity is controlled by a

torque on the wheels

C = K ( (ωωωωtarget - ωωωω) – λ dωωωω/dt )

Results (lateral acceleration)

trajectory radius larger with

flexible body

better transient behaviour with

flexible body

haversine

time

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Distribution of contact forces

Rear wheels: less lateral load transfer with the flexible case

Front wheels: nearly no load transfer in flexible case

Explanation: steering mechanism

Vertical forces

Straight line

Cornering, rigid

Cornering,

flexible


Understeer/oversteer behaviour

Constant steering angle and different speeds

understeer at low speed

oversteer above 6 m/s

critical speed higher

with flexible chassis

yaw rate gain


Conclusions

Flexible bodies can be easily exported from finite element

softwares to multibody softwares

rigid-body characteristics

invariants of the deformation modes

The most widespread reduction is the one of Craig-Bampton

(constraint modes and fixed boundaries vibration modes)

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