Multibody System Dynamics: MBDyn Overview · Multibody System Dynamics: MBDyn Overview ......

Multibody System Dynamics:MBDyn Overview

Pierangelo Masarati <[email protected]>Politecnico di MilanoDipartimento di Ingegneria Aerospaziale

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Pierangelo Masarati – MBDyn Overview

Outline

• Multibody dynamicsMultibody dynamics• Software architecturesSoftware architectures• ProblemsProblems• Arbitrary motion descriptionArbitrary motion description• Deformable componentsDeformable components• Solving the problemSolving the problem• Extracting useful informationExtracting useful information• Examples of multibody modeling with MBDynExamples of multibody modeling with MBDyn• Future developmentFuture development• Documentation and supportDocumentation and support

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Multibody dynamics

Basic equations:Basic equations:

mechanics of unconstrained system of bodiesmechanics of unconstrained system of bodies subjected to configuration-dependent loadssubjected to configuration-dependent loads

Can be obtained from many (equivalent!) approaches:Can be obtained from many (equivalent!) approaches: Newton-EulerNewton-Euler: linear/angular equilibrium of each body: linear/angular equilibrium of each body d'Alembert-Lagranged'Alembert-Lagrange: virtual work of active : virtual work of active

forces/momentsforces/moments GaussGauss, , HertzHertz, , HamiltonHamilton, ...: variational principles, ...: variational principles

M x x=f x , x , t

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Multibody dynamics

Constrained system: kinematic constraintsConstrained system: kinematic constraints• holonomicholonomic

• non-holonomic (not integrable to holonomic)non-holonomic (not integrable to holonomic)

usuallyusually

• algebraic relationship between kinematic variablesalgebraic relationship between kinematic variables• explicitly dependent on time: rheonomicexplicitly dependent on time: rheonomic• scleronomous otherwisescleronomous otherwise

x , t =0

x , x , t =0

A x , t x=b x , t a=M−1 f

x=aM−1 fc

M−1 fc

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Multibody dynamics

Minimal set:Minimal set:

usually, this relationship:usually, this relationship: is not known in advance, oris not known in advance, or cannot be easily made explicit with respect to qcannot be easily made explicit with respect to q

Coordinate partitioning is required, e.g.:Coordinate partitioning is required, e.g.: direct elimination from derivative of constraint direct elimination from derivative of constraint

equationequation QR or similar decompositionQR or similar decomposition

Results in Maggi-Kane equations and similar approachesResults in Maggi-Kane equations and similar approaches

Small system is obtained by expensive numerical reductionSmall system is obtained by expensive numerical reduction

(unless topology knowledge can be exploited)(unless topology knowledge can be exploited)

x=x q , t

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Multibody dynamics

Redundant set:Redundant set:

By Lagrange multipliers:By Lagrange multipliers: dynamics of constrained system using physical dynamics of constrained system using physical

coordinatescoordinates constraint reactions applied to equations of motionconstraint reactions applied to equations of motion algebraic constraints explicitly added to the systemalgebraic constraints explicitly added to the system

• Multiple bodies with few actual dofs:Multiple bodies with few actual dofs: system size nearly doublessystem size nearly doubles

• Multiple bodies with few constraints:Multiple bodies with few constraints: system size not significantly alteredsystem size not significantly altered

• Sparsity is almost preservedSparsity is almost preserved

⋅= ⋅ x⋅/xT

⋅= ⋅ x⋅/ xT

[ M / xT

/x

0 ]{x}={ f

b '}

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Multibody dynamics

Constraint equations written “Constraint equations written “as isas is”:”:

problem becomes differential algebraic (DAE); issues:problem becomes differential algebraic (DAE); issues:• needs specific care to be solved: (nearly) L-stable needs specific care to be solved: (nearly) L-stable

integration, i.e.integration, i.e. unconditionally stable, andunconditionally stable, and for for

• the constraint equation implies the additional constraintsthe constraint equation implies the additional constraints

but they are not explicitly enforced:but they are not explicitly enforced:may need constraint stabilizationmay need constraint stabilization (Gear et al.) (Gear et al.)

x , t =0

xk1

0 t∞

x , t =0

x , t =0

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Multibody dynamics

Constraint equations:Constraint equations:• x is correctx is correct• derivatives may be inaccuratederivatives may be inaccurate• multipliers may be inaccuratemultipliers may be inaccurate

kk

k+1k+1

k+2k+2

k+2k+2x

j, t

j=0

x j , t j=0

xj, t

j=0

x , t =0

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Multibody dynamics

Alternative: constraint equations differentiated to second order:Alternative: constraint equations differentiated to second order:

problem remains ordinary differential (ODE);problem remains ordinary differential (ODE);• can be solved by conditionally stable algorithmscan be solved by conditionally stable algorithms• the constraint equation does not imply the original constraintsthe constraint equation does not imply the original constraints

definitely needs constraint stabilization!definitely needs constraint stabilization!

common technique: Baumgartecommon technique: Baumgarte

(violation governed by asymptotically stable linear differential eq.)(violation governed by asymptotically stable linear differential eq.)

/ x

x=b'

x , t =0

x , t =0

/ x

x=b'−2 −2

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Software architectures

• Monolithic:Monolithic: user prepares specific model using built-in library elementsuser prepares specific model using built-in library elements general-purpose solver swallows model and spits resultsgeneral-purpose solver swallows model and spits results

• Library:Library: user writes specific solver using library elementsuser writes specific solver using library elements

• usually needs programming skills;usually needs programming skills;the solver must be compiledthe solver must be compiled

specific solver solves the problem and spits resultsspecific solver solves the problem and spits results• Symbolic manipulators:Symbolic manipulators:

user writes equationsuser writes equations symbolic manipulation engine solves equationssymbolic manipulation engine solves equations

and spits resultsand spits results• Modelica (and Modelica-like):Modelica (and Modelica-like):

user prepares model using a modeling language and libsuser prepares model using a modeling language and libs general-purpose interpreter generates specific solvergeneral-purpose interpreter generates specific solver specific solver solves the problem and spits resultsspecific solver solves the problem and spits results

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Free software examples (surely there are more):Free software examples (surely there are more):• Monolithic:Monolithic:

MBDynMBDyn• Library:Library:

DynaMechs (C++)DynaMechs (C++) Mbs3d (requires Matlab)Mbs3d (requires Matlab) Open Dynamics Engine (ODE) (C++)Open Dynamics Engine (ODE) (C++)

• Symbolic manipulators:Symbolic manipulators: 3D_MEC3D_MEC EasyDyn (MuPad)EasyDyn (MuPad) RoboTran (requires Matlab)RoboTran (requires Matlab)

• Modelica:Modelica: OpenModelica?OpenModelica?

non-free counterparts omittednon-free counterparts omitted

frequently architectures overlapfrequently architectures overlap

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• MBDyn is monolithicMBDyn is monolithic• Input consists in a text fileInput consists in a text file• The input syntax allows some flexibility, e.g.:The input syntax allows some flexibility, e.g.:

math expressions evaluationmath expressions evaluation variables definitionvariables definition ““rigorous” syntax checking, but free style, indentation, ...rigorous” syntax checking, but free style, indentation, ...

• Relevant portions of the code are modular and can be extended by:Relevant portions of the code are modular and can be extended by: writing run-time loadable moduleswriting run-time loadable modules hacking the code (it's free, all in all!)hacking the code (it's free, all in all!)

• There is no built-in pre-post processing facilityThere is no built-in pre-post processing facility• Help in this area is warmly appreciated!Help in this area is warmly appreciated!

MBDyn output can be translated into EasyAnimMBDyn output can be translated into EasyAnim there is an independent, partial customization based on Blenderthere is an independent, partial customization based on Blender

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Problems

Equations of motion: for each node (purely geometrical entity),Equations of motion: for each node (purely geometrical entity),• Newton-Euler, written as first-order system of equations:Newton-Euler, written as first-order system of equations:

• Momentum and momenta moment insteadMomentum and momenta moment insteadof pseudo-velocitiesof pseudo-velocities

• allows multiple contributions to inertia of a single nodeallows multiple contributions to inertia of a single node

Constrained equations in differential-algebraic form:Constrained equations in differential-algebraic form:

M x= pp= f x , x , t

M x= p

p/ xT

= f x , x , t

x , t = 0

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Problems

• Fundamental problem:Fundamental problem: Integration of Initial Value Problem (IVP) in timeIntegration of Initial Value Problem (IVP) in time

• Static analysis as degeneration of IVP dynamic analysis:Static analysis as degeneration of IVP dynamic analysis: momentum and momenta moment definitions omittedmomentum and momenta moment definitions omitted only gravity is consideredonly gravity is considered system determination only provided by kinematic constraints system determination only provided by kinematic constraints

and deformable componentsand deformable components

• Kinematic analysis as degeneration of IVP dynamic analysis:Kinematic analysis as degeneration of IVP dynamic analysis: inertia elements omittedinertia elements omitted system determination only provided by kinematic constraintssystem determination only provided by kinematic constraints deformable components can act as “regularization”deformable components can act as “regularization”

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Problems

Experimental inverse dynamics problemExperimental inverse dynamics problem• inverse kinematics:inverse kinematics:

• the RHS contains the desired motion and its derivativesthe RHS contains the desired motion and its derivatives• the (regularized) static analysis provides the kinematic the (regularized) static analysis provides the kinematic

inversioninversion

[ K' / xT

/x

0 ]{ x 0

}={ 0

−x , t}

[ K' / xT

/x

0 ]{ x 1

}={ 0

b x , t }

[ K' / xT

/x

0 ]{ x 2

}={ 0

b'x , x , t }

/ xT =f ' x , x , x , t

same matrix!same matrix!

nonlinearnonlinear

linearlinear

linearlinear

linearlinear

K'=I : Moore-Penrose pseudo-inverse: Moore-Penrose pseudo-inverse

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Problems

• Experimental direct eigenanalysisExperimental direct eigenanalysis issues with constraints formulationissues with constraints formulation

(mainly with rotations)(mainly with rotations) issues with equations implementationissues with equations implementation

(matrices not available)(matrices not available)

• Relative coordinate frame dynamics Relative coordinate frame dynamics imposed frame motion: modifications only to RHS imposed frame motion: modifications only to RHS

inertia elemsinertia elems instrumental for many helicopter rotor dynamics instrumental for many helicopter rotor dynamics

problemsproblems

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Arbitrary motion description

• Mechanical degrees of freedom:Mechanical degrees of freedom: structural node positions in the absolute reference framestructural node positions in the absolute reference frame structural node orientation with respect to the absolute structural node orientation with respect to the absolute

frameframe

• Kinematics is always written with respect to the absolute frameKinematics is always written with respect to the absolute frame

• Newton-Euler equations are written in the absolute frameNewton-Euler equations are written in the absolute frame moment equilibrium (Euler) equations are writtenmoment equilibrium (Euler) equations are written

with respect to the respective (moving) nodewith respect to the respective (moving) node

• Special elements may introduce further approximationsSpecial elements may introduce further approximations e.g. Component Mode Synthesis (CMS) elemente.g. Component Mode Synthesis (CMS) element

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Orientation handling:Orientation handling:• orientation variables: Cayley-Gibbs-Rodrigues parametersorientation variables: Cayley-Gibbs-Rodrigues parameters• orientation matrix:orientation matrix:

• orthonormality:orthonormality:

• derivative:derivative:

• incremental approach from step incremental approach from step kk to to kk+1 to eliminate the +1 to eliminate the orientation parameters singularity issue (increments are orientation parameters singularity issue (increments are necessarily small for accuracy):necessarily small for accuracy):

R=R g

R RT=×=G g g×

RT=R−1

Rk=R g

kR

k−1

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• Orientation handling:Orientation handling: the actual orientation variables are the Cayley-Gibbs-the actual orientation variables are the Cayley-Gibbs-

Rodrigues parameters relative to the correction Rodrigues parameters relative to the correction phase of each stepphase of each step

kk: time step counter: time step counter ii: correction iteration counter (0: predicted value): correction iteration counter (0: predicted value)

• Orientation matrix:Orientation matrix:

• Derivative:Derivative:

Rk i =R g

i Rk0

Rk i R

k i T=

k i ×=R g

i k0×G g

i g

i ×

k-1 k(i)

k(0)

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• Incremental orientation from previous step:Incremental orientation from previous step: Orientation parameters order of magnitude: Orientation parameters order of magnitude:

• Incremental orientation from prediction (as in MBDyn):Incremental orientation from prediction (as in MBDyn): Orientation parameters order of magnitude:Orientation parameters order of magnitude:

where where nn is the min between the order of the predictor is the min between the order of the predictorand of the integration methodand of the integration method(MBDyn: 3 and 2, respectively, so (MBDyn: 3 and 2, respectively, so nn = 2) = 2)

As a consequence:As a consequence:

(only in Jacobian)(only in Jacobian)

g~O∥∥ t

g~O tn1

R g≈I

Gg≈I

G g≈0

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Deformable components

• Lumped deformable componentsLumped deformable components rod (1D)rod (1D) linear, angular components (3D)linear, angular components (3D) linear & angular component (6D)linear & angular component (6D)

• Intrinsic, composite-ready Finite-Volume beam elementIntrinsic, composite-ready Finite-Volume beam element arbitrary constitutive lawarbitrary constitutive law piezoelectric constitutive lawpiezoelectric constitutive law aerodynamic beam elementaerodynamic beam element

• Intrinsic, composite-ready shell and membrane elementsIntrinsic, composite-ready shell and membrane elements

• Component Mode Synthesis (CMS)Component Mode Synthesis (CMS) attached to a floating frame (a node)attached to a floating frame (a node) linear state-space representation of unsteady aerodynamicslinear state-space representation of unsteady aerodynamics

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Lumped deformable components (3D, 6D):Lumped deformable components (3D, 6D):

• Attached form:Attached form: Constitutive properties referred to either of the connected Constitutive properties referred to either of the connected

nodesnodes

• Intrinsic form (invariant: ):Intrinsic form (invariant: ): Constitutive properties referred to a floating reference frameConstitutive properties referred to a floating reference frame Intrinsically handles geometrical nonlinearity related to Intrinsically handles geometrical nonlinearity related to

rotationsrotations Correctly captures bending-torsion buckling behaviorCorrectly captures bending-torsion buckling behavior Essential for anisotropic deformable componentsEssential for anisotropic deformable components

= ax exp−1Ra

T Rb

m= R m

=0,=1

=1/2

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Intrinsic, composite-ready beamIntrinsic, composite-ready beam• Topology: Topology:

1D reference line p, 1D reference structure R1D reference line p, 1D reference structure R 2D section characterization2D section characterization


reference linereference line

reference reference orientationorientation

reference motion:reference motion:

warpingwarping

x= pR tx

/= p

/

×R t

p R

t

p, R

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Intrinsic, composite-ready beamsIntrinsic, composite-ready beams• strain measure: strain measure:

• equilibrium (from VWP):equilibrium (from VWP):

• constitutive properties:constitutive properties:


= RT p/−R0

T p0 /

= RT −R

0T

0

f/=

m/p

/×f=

f = f ,

m= m ,

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Intrinsic, composite-ready beams: 3-node discretizationIntrinsic, composite-ready beams: 3-node discretization• Finite Volume approach: equilibrium of finite portions of beamFinite Volume approach: equilibrium of finite portions of beam• internal forces function of node kinematics thru constitutive lawsinternal forces function of node kinematics thru constitutive laws• warping goes into constitutive properties computationwarping goes into constitutive properties computation


node 1node 1

node 2node 2

node 3node 3

point Ipoint Ipoint IIpoint II

fI, m

I

fII

, mII

f1, m

1

f2, m

2

f3, m

3

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Solving the problem

Numerical integrationNumerical integration• implicit, (quasi-)L stable 2 step algorithmimplicit, (quasi-)L stable 2 step algorithm

• tunable algorithmic dissipation: asymptotic spectral radius 1tunable algorithmic dissipation: asymptotic spectral radius 1→→00 asymptotic spectral radius = 0: 2asymptotic spectral radius = 0: 2ndnd order BDF order BDF ““optimal” dissipation: spectral radius ~ 0.6optimal” dissipation: spectral radius ~ 0.6

• second-order accurate, with third-order accurate predictorsecond-order accurate, with third-order accurate predictor• variable time stepvariable time step• not ideal for non-smooth problems (multi-step)not ideal for non-smooth problems (multi-step)• different integrators can be used; new ones can be implementeddifferent integrators can be used; new ones can be implemented

yk=a

1y

k−1a

2y

k−2 t b

0y

kb

1y

k−1b

2y

k−2

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Solving the problem

• Prediction:Prediction:

• Correction iteration:Correction iteration:

butbut

the problem becomes algebraicthe problem becomes algebraic

yk0 =m

1y

k−1m

2y

k−2/ tn

1y

k−1n

2y

k−2

yk0 =a

1y

k−1a

2y

k−2 tb

0y

k0b

1y

k−1b

2y

k−2

f/ y

y i f/y

yi =−f yk i−1 , y

k i−1 , t

k

y i = t b0 y i

f/ y t b

0f

/ y y i =−f y

ki−1 , y

ki−1 , t

k

yk i =y

k i−1 yi

y k i

=yk i−1

t b0 y i

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Solving the problem

Model assemblyModel assembly• model could be input incorrectlymodel could be input incorrectly• initial values of the state (position, velocity, reactions) are neededinitial values of the state (position, velocity, reactions) are needed• this might not be a trivial taskthis might not be a trivial task• initial state values must comply with constraints:initial state values must comply with constraints:

• a dummy static nonlinear problem is solved (regularization):a dummy static nonlinear problem is solved (regularization):

x0,

t0=0

x0, t0=0

K' x−x0/xT

'= f '

C' x−x0

/xT '= f '

x , t0= 0

x , t0= 0

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Solving the problem

Solution initialization (so-called “derivatives”)Solution initialization (so-called “derivatives”)• explicit problem:explicit problem:

• implicit problem:implicit problem:

• modified correction phase to initialize solution:modified correction phase to initialize solution:

• convergence no longer quadratic, but saves lots of code duplicationconvergence no longer quadratic, but saves lots of code duplication• Setting might not work (problem can be structurally singular) Setting might not work (problem can be structurally singular)

y=f y , t

0=f y , y , t

f/ yc f

/y y i =−f y

0 i−1 , y

0, t

0

y0 i =y

0 i−1 yi

y0=y

0

c=0

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Extracting useful information

• Detailed analysis requires detailed models, but...Detailed analysis requires detailed models, but...• excessive details endanger the chance to extract useful informationexcessive details endanger the chance to extract useful information• Proper Orthogonal Decomposition allows to extract information from Proper Orthogonal Decomposition allows to extract information from

redundant measuresredundant measures• Consider a set of Consider a set of NN measurements X for measurements X for nn time steps; their SVD: time steps; their SVD:

• The singular values allow to determine the The singular values allow to determine the mm most relevant signals most relevant signals

• Note thatNote that

• This allows to efficiently compute the singular values and the POMsThis allows to efficiently compute the singular values and the POMs

XT∈ℝ

n×N=U VT

X1 : m , nT =U

n ,1 : m

1: m ,1 : mV

N ,1 : mT

XT X= U 2 UT

UT XT= VT

X XT= V

2 VT

XT V= U

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Extracting useful information

• The POMs can be used to identify a transition matrixThe POMs can be used to identify a transition matrix

• If X contains the free response of the system, the transition If X contains the free response of the system, the transition matrix allows to estimate the relevant eigenvalues (AR model)matrix allows to estimate the relevant eigenvalues (AR model)

• More sophisticated system identification techniques can be usedMore sophisticated system identification techniques can be used(model order reduction is an open research field)(model order reduction is an open research field)

• A technique based on covariance estimates from time histories A technique based on covariance estimates from time histories has been recently proposed; works for:has been recently proposed; works for:

free responsefree response forced responseforced response unmeasured forced responseunmeasured forced response

Xk1= Xk

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Examples of multibody modeling with MBDyn

Robotics:Robotics:

delta robotdelta robot

inverse dynamicsinverse dynamics

for computedfor computed

torque controltorque control

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Robotics: PA-10Robotics: PA-10

inverse kinematicsinverse kinematics

with path optimizationwith path optimization

of cooperating robotsof cooperating robots

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Robotics:Robotics:

biomimetic robotbiomimetic robot

real-time motionreal-time motion

planning by inverseplanning by inverse

kinematics withkinematics with

fault detectionfault detection

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Industrial processes:Industrial processes:• simulation of automotive components assembly (car brake pipe) to:simulation of automotive components assembly (car brake pipe) to:

check stresses introduced during assemblycheck stresses introduced during assembly check loads on supports introduced during assemblycheck loads on supports introduced during assembly check interference with other parts during assemblycheck interference with other parts during assembly check interference with other parts during operationcheck interference with other parts during operation

• the model has been developed by a rubber manufacturerthe model has been developed by a rubber manufacturer• it is used for product design and certificationit is used for product design and certification• it required the development of specific features for solution it required the development of specific features for solution

partitioning, which are now part of MBDynpartitioning, which are now part of MBDyn

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Automotive: mechanical modeling of suspensionsAutomotive: mechanical modeling of suspensions

purpose: determine loads in rubber bushings and other componentspurpose: determine loads in rubber bushings and other components

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Rotorcraft dynamics and aeroservoelasticity:Rotorcraft dynamics and aeroservoelasticity:• WRATS (NASA/Army) tiltrotor aeromechanicsWRATS (NASA/Army) tiltrotor aeromechanics

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Rotorcraft dynamicsRotorcraft dynamics

and aeroservoelasticity:and aeroservoelasticity:• ERICA (AgustaWestland)ERICA (AgustaWestland)

tiltrotor aeromechanicstiltrotor aeromechanics(ADYN, NICETRIP)(ADYN, NICETRIP)

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Future development

• Multiscale handling of submodels with different dynamicsMultiscale handling of submodels with different dynamics aircraft flight mechanics (~1 to 5 Hz: very slow)aircraft flight mechanics (~1 to 5 Hz: very slow) main rotor dynamics (~5 to 40 Hz: intermediate)main rotor dynamics (~5 to 40 Hz: intermediate) tail rotor dynamics (~25 to >100 Hz: fast)tail rotor dynamics (~25 to >100 Hz: fast)

• Interfacing with different domainsInterfacing with different domains Fluid-structure (Lagrangian/Eulerian modeling of workflows)Fluid-structure (Lagrangian/Eulerian modeling of workflows) structure-structurestructure-structure active control of large deformable systemsactive control of large deformable systems

• Better abstraction/modularization of components/solution phasesBetter abstraction/modularization of components/solution phases more freedom in model customizationmore freedom in model customization tight integration into nonlinear structural analysis (Aster?)tight integration into nonlinear structural analysis (Aster?)

• More...More...

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Documentation and support

• Theory manual:Theory manual: Incomplete; needs lots of workIncomplete; needs lots of work

• User manualUser manual available and up to dateavailable and up to date

• TutorialsTutorials available, but reportedly too simple; need workavailable, but reportedly too simple; need work

• Applications manualApplications manual available, but only few applications so faravailable, but only few applications so far

• Installation manualInstallation manual available, incomplete and outdated (not critical)available, incomplete and outdated (not critical)

• Mailing listsMailing lists available: announce, users, develavailable: announce, users, devel the “users” list also serves as issue tracking provisionthe “users” list also serves as issue tracking provision

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Documentation and support

• Another important item that is missing is an automated test suiteAnother important item that is missing is an automated test suite can be run automatically after building the softwarecan be run automatically after building the software allows to check build errorsallows to check build errors allows to check regressions in new releasesallows to check regressions in new releases serves as example of modeling functionalitiesserves as example of modeling functionalities

• The rest is underway (always a work in progress)The rest is underway (always a work in progress)

Given the nature of the project, contributions are always welcome!Given the nature of the project, contributions are always welcome!

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Questions?

Multibody System Dynamics: MBDyn Overview · Multibody System Dynamics: MBDyn Overview ......

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