presentation_masarati.pdf

MBDyn - Multibody System Dynamics
LaMSID (EDF R&D - French CNRS Joint Laboratory)
July 24 - August 7, 2008
Pierangelo Masarati <masarati@aero.polimi.it>
Politecnico di Milano
Dipartimento di Ingegneria Aerospaziale

2
LaMSID – July 24 – August 7, 2008
Preamble
This presentation is split in two parts:
This presentation is split in two parts:
• Multibody System Dynamics
Multibody System Dynamics,
,
with specific reference to MBDyn,
with specific reference to MBDyn,
the free general-purpose multibody solver
the free general-purpose multibody solver
developed at “Dipartimento di Ingegneria Aerospaziale”
developed at “Dipartimento di Ingegneria Aerospaziale”
of “Politecnico di Milano”, Italy
of “Politecnico di Milano”, Italy
• FEM interaction requirements of MBDyn
FEM interaction requirements of MBDyn,
,
with the outline of possible solutions for the interaction with Aster
with the outline of possible solutions for the interaction with Aster

3
Outline
• Overview
Overview
• Multibody dynamics
Multibody dynamics
• Software architectures
Software architectures
• Problems
Problems
• Arbitrary motion description
Arbitrary motion description
• Deformable components
Deformable components
• Solving the problem
Solving the problem
• Extracting useful information
Extracting useful information
• Examples of multibody modeling with MBDyn
Examples of multibody modeling with MBDyn
• Future development
Future development
• Documentation and support
Documentation and support
• Acknowledgments
Acknowledgments

4
Multibody dynamics: overview
• Multibody
Multibody: a “buzz” word?
: a “buzz” word?
• Initial idea:
Initial idea:
 automatically write equations of motion of arbitrary mechanisms
automatically write equations of motion of arbitrary mechanisms
• Current status:
Current status:
 efficiently and accurately integrate in time
efficiently and accurately integrate in time
 exact rigid-body kinematics, plus
exact rigid-body kinematics, plus
 nonlinear finite elements, plus
nonlinear finite elements, plus
 natural inclination towards multi-physics & system integration
natural inclination towards multi-physics & system integration
• Future:
Future:
 scale to larger and larger problems, and
scale to larger and larger problems, and
 higher performances when solving more complex problems
higher performances when solving more complex problems

5
Multibody
Multibody methods:
methods:
• Usually are general-purpose: can model a wide variety
Usually are general-purpose: can model a wide variety
of mechanical systems
of mechanical systems
• Must allow to automatically write motion equations
Must allow to automatically write motion equations
• Should support an arbitrary number of a variety of parts, forces,
Should support an arbitrary number of a variety of parts, forces,
geometries, constraints, etc.
geometries, constraints, etc.
• Most often use numerical methods to compute solutions.
Most often use numerical methods to compute solutions.
• Often integrated in CAD tools, with Graphical User Interfaces (GUI)
Often integrated in CAD tools, with Graphical User Interfaces (GUI)

6
Types of analysis:
Types of analysis:
• Kinematics: direct, inverse
Kinematics: direct, inverse
• Statics (pseudo-time)
Statics (pseudo-time)
• Dynamics: direct IVP, BVP, inverse dynamics
Dynamics: direct IVP, BVP, inverse dynamics
• Miscellaneous:
Miscellaneous:
 modal analysis
modal analysis
 sensitivity analysis
sensitivity analysis
 parameter optimization
parameter optimization
 topology optimization and mechanism synthesis
topology optimization and mechanism synthesis

7
Applications:
Applications:
• Robotics
Robotics
• Automotive
Automotive
• Crashworthiness
Crashworthiness
• Aerospace engineering
Aerospace engineering
• Industrial automation
Industrial automation
• Applied mechanics
Applied mechanics
• Videogames: revenue higher than engineering software!
Videogames: revenue higher than engineering software!
• Anywhere realistic motion (or at least that looks realistic)
Anywhere realistic motion (or at least that looks realistic)
is required, multibody simulation is or can (and will) be used.
is required, multibody simulation is or can (and will) be used.

8
Classification of approaches:
Classification of approaches:
• Based on coordinate set
Based on coordinate set
 Minimal coordinate set
Minimal coordinate set
 Redundant coordinate set
Redundant coordinate set
• Based on local solution methods
Based on local solution methods
 Direct linear solvers
Direct linear solvers
 Iterative solvers
Iterative solvers
• Based on problem formulation
Based on problem formulation
 Explicit
Explicit
 Implicit
Implicit
• Based on integration schemes
Based on integration schemes
 Ordinary Differential Equations (ODE) w/ constraint stabilization
Ordinary Differential Equations (ODE) w/ constraint stabilization
 Differential-Algebraic Equations (DAE)
Differential-Algebraic Equations (DAE)
classical approach
classical approach
modern approach
modern approach
specialistic
specialistic
general-purpose
general-purpose
classical approach
classical approach
modern approach
modern approach

9
Multibody dynamics
Basic equations:
Basic equations:
 mechanics of unconstrained system of bodies
mechanics of unconstrained system of bodies
 subjected to configuration-dependent loads
subjected to configuration-dependent loads
Can be obtained from many (equivalent!) approaches:
Can be obtained from many (equivalent!) approaches:
 Newton-Euler
Newton-Euler: linear/angular equilibrium of each body
: linear/angular equilibrium of each body
 d'Alembert-Lagrange
d'Alembert-Lagrange: virtual work of active forces/moments
: virtual work of active forces/moments
 Gauss
Gauss,
, Hertz
Hertz,
, Hamilton
Hamilton, ...: variational principles
, ...: variational principles
Note: non-smooth mechanics tend to resort to discrete equations
Note: non-smooth mechanics tend to resort to discrete equations
Mxẍ=f x, ẋ, t
Mxẋk−1, k
=∫k−1
k
f x, ẋ , td t

10
Multibody dynamics
Constrained system: kinematic constraints
Constrained system: kinematic constraints
 holonomic (algebraic)
holonomic (algebraic)
 Non-holonomic
Non-holonomic
(differential, not integrable to holonomic)
(differential, not integrable to holonomic)
usually
usually
• algebraic relationship between kinematic variables
algebraic relationship between kinematic variables
• explicitly dependent on time: rheonomic
explicitly dependent on time: rheonomic
• scleronomous otherwise
scleronomous otherwise
x, t=0
x , ẋ, t=0
Ax, tẋ=bx, t
a=M−1
f
ẍ=aM−1
fc
M−1
fc

11
Multibody dynamics
Minimal set:
Minimal set:
usually, this relationship:
usually, this relationship:
 is not known in advance, or
is not known in advance, or
 cannot be easily made explicit with respect
cannot be easily made explicit with respect
to Lagrangian coordinates q
to Lagrangian coordinates q
Cordinate partitioning is required, e.g.:
Cordinate partitioning is required, e.g.:
 direct elimination from derivative of constraint equation
direct elimination from derivative of constraint equation
 QR, SVD or similar decomposition
QR, SVD or similar decomposition
Results in Maggi-Kane equations and similar approaches
Results in Maggi-Kane equations and similar approaches
Small system is obtained by expensive numerical reduction
Small system is obtained by expensive numerical reduction
(unless topology knowledge can be exploited)
(unless topology knowledge can be exploited)
x=xq , t

12
Multibody dynamics
Redundant set:
Redundant set:
By Lagrange multipliers:
By Lagrange multipliers:
 dynamics of constrained system using physical coordinates
dynamics of constrained system using physical coordinates
 constraint reactions applied to equations of motion
constraint reactions applied to equations of motion
 algebraic constraints explicitly added to the system
algebraic constraints explicitly added to the system
• mechanism made of multiple bodies with multiple constraints
mechanism made of multiple bodies with multiple constraints
and few actual DoFs:
and few actual DoFs:
 system size nearly doubles
system size nearly doubles
• mechanism made of deformable bodies with few constraints (e.g. FEM):
mechanism made of deformable bodies with few constraints (e.g. FEM):
 system size not significantly altered
system size not significantly altered
• sparsity is almost preserved
sparsity is almost preserved
⋅= ⋅ x⋅/x
T

⋅=⋅ x⋅/ẋ
T

[M / x
T
/x
0 ]{ẍ
}=
{f
b'}

13
Multibody dynamics
Constraint equations written “
Constraint equations written “as is
as is”:
”:
problem becomes differential algebraic (DAE); issues:
problem becomes differential algebraic (DAE); issues:
• needs specific care to be solved: (nearly) L-stable integration, i.e.
needs specific care to be solved: (nearly) L-stable integration, i.e.
 unconditionally stable (A-stable), and
unconditionally stable (A-stable), and
 for
for
• the constraint equation implies the additional constraints
the constraint equation implies the additional constraints
but they are not explicitly enforced:
but they are not explicitly enforced:
may need constraint stabilization
may need constraint stabilization (Gear et al.)
(Gear et al.)
x, t=0
xk1
0  t∞
̇x , t=0
̈x , t=0

14
Multibody dynamics
Constraint equations:
Constraint equations:
• x is correct
x is correct
• derivatives may be inaccurate
derivatives may be inaccurate
• multipliers may be inaccurate
multipliers may be inaccurate
k
k
k+1
k+1
k+2
k+2
k+2
k+2
x j
, t j
=0
̇x j
, t j
=0
numerical solution
numerical solution
with constraint
with constraint
stabilization
stabilization
x j
, t j
=0
numerical solution
numerical solution
x, t=0 exact solution
exact solution

15
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 algorithms
can be solved by conditionally stable algorithms
• the constraint equation does not imply the original constraints
the constraint equation does not imply the original constraints
suffers from drift! definitely needs constraint stabilization!
suffers from drift! definitely needs constraint stabilization!
common technique: Baumgarte
common technique: Baumgarte
(violation governed by asymptotically stable linear differential eq.)
(violation governed by asymptotically stable linear differential eq.)
/ x
ẍ=b'
x, t=0
̇x , t=0
−2 ̇−2

/ x
ẍ=b'

16
• Monolithic:
Monolithic:
 user prepares specific model using built-in library elements
user prepares specific model using built-in library elements
 general-purpose solver swallows model and spits results
general-purpose solver swallows model and spits results
• Library:
Library:
 user writes specific solver using library elements
user writes specific solver using library elements
• usually needs programming skills; the solver must be compiled
usually needs programming skills; the solver must be compiled
 specific solver solves the problem and spits results
specific solver solves the problem and spits results
• Symbolic manipulators:
Symbolic manipulators:
 user writes equations
user writes equations
 symbolic manipulation engine solves equations and spits results
symbolic manipulation engine solves equations and spits results
• Modelica (and Modelica-like):
Modelica (and Modelica-like):
 user prepares model using a modeling language and libs
user prepares model using a modeling language and libs
 general-purpose interpreter generates specific solver
general-purpose interpreter generates specific solver
 specific solver solves the problem and spits results
specific solver solves the problem and spits results

17
Free software examples (surely there are more):
Free software examples (surely there are more):
• Monolithic:
Monolithic:
 MBDyn
MBDyn
• 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_MEC
3D_MEC
 EasyDyn (MuPad)
EasyDyn (MuPad)
 RoboTran (requires Matlab)
RoboTran (requires Matlab)
• Modelica:
Modelica:
 OpenModelica?
OpenModelica?
non-free counterparts omitted
non-free counterparts omitted
frequently architectures overlap
frequently architectures overlap

18
• MBDyn is monolithic
MBDyn is monolithic
• Input consists in a text file
Input consists in a text file
• The input syntax allows some flexibility, e.g.:
The input syntax allows some flexibility, e.g.:
 math expressions evaluation
math expressions evaluation
 variables definition
variables 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 modules
writing 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 facility
There is no built-in pre-post processing facility
• Help in this area would be warmly appreciated!
Help in this area would be warmly appreciated!
 MBDyn output can be translated into EasyAnim
MBDyn output can be translated into EasyAnim
 there is an independent, partial customization based on Blender
there is an independent, partial customization based on Blender
 what about a custom Salome toolbox for multibody?
what about a custom Salome toolbox for multibody?

19
MBDyn interacts with other software:
MBDyn interacts with other software:
• Generic streamed input/output
Generic streamed input/output
 Input: “drive” (scalar); can be fed to several entities:
Input: “drive” (scalar); can be fed to several entities:
• imposed load
imposed load
• imposed joint motion
imposed joint motion
• imposed deformable component prestrain
imposed deformable component prestrain
• Inputs can be aggregated and manipulated
Inputs can be aggregated and manipulated
 Output: “measurement” (scalar); can be any exportable entity:
Output: “measurement” (scalar); can be any exportable entity:
• any model state
any model state
• internal parameters of elements and nodes
internal parameters of elements and nodes
• outputs can be aggregated and manipulated
outputs can be aggregated and manipulated
 Used to interact with Simulink, Scicos, ... also in hard real-time
Used to interact with Simulink, Scicos, ... also in hard real-time
• Generic motion output/force input
Generic motion output/force input
 Used for interaction with CFD solvers
Used for interaction with CFD solvers
 Could be (easily?) adapted for interaction with Aster?
Could be (easily?) adapted for interaction with Aster?
• Other software needed to pre/post-process data (e.g. FEM)
Other software needed to pre/post-process data (e.g. FEM)

20
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 are used
Momentum and momenta moment are used
instead of pseudo-velocities
instead of pseudo-velocities
• allows multiple contributions to inertia of a single node
allows multiple contributions to inertia of a single node
(but constraint stabilization is no longer straightforward...)
(but constraint stabilization is no longer straightforward...)
Constrained equations are directly written
Constrained equations are directly written
in differential-algebraic form:
in differential-algebraic form:
M ẋ= p
ṗ= f x , ẋ, t
M ẋ= p
ṗ/ x
T
= f x , ẋ, t
x , t= 0
(for dynamic nodes only)
(for dynamic nodes only)
(=0 for static nodes)
(=0 for static nodes)

21
Problems
• Fundamental problem:
Fundamental problem:
 integration of Initial Value Problem (IVP) in time
integration of Initial Value Problem (IVP) in time
 also in Hard Real-Time (by way of RTAI, RT-Net & Scicos)
also in Hard Real-Time (by way of RTAI, RT-Net & Scicos)
for Hardware-In-the-Loop (HIL) simulations
for Hardware-In-the-Loop (HIL) simulations
• Static analysis as degeneration of IVP dynamic analysis:
Static analysis as degeneration of IVP dynamic analysis:
 momentum and momenta moment definitions omitted
momentum and momenta moment definitions omitted
 only gravity is considered
only gravity is considered
 system determination must be provided by kinematic constraints
system determination must be provided by kinematic constraints
and deformable components
and deformable components
• Kinematic analysis as degeneration of IVP dynamic analysis:
Kinematic analysis as degeneration of IVP dynamic analysis:
 inertia elements omitted
inertia elements omitted
 system determination must be provided by kinematic constraints
system determination must be provided by kinematic constraints
 deformable components can act as “regularization”
deformable components can act as “regularization”

22
Problems
[K' / x
T
/x
0 ]{x
0
}=
{ 0
−x, t}
Experimental inverse dynamics problem
Experimental inverse dynamics problem
• inverse kinematics
inverse kinematics
(allows under-
(allows under-
constrained
constrained
systems):
systems):
• the RHS contains the desired motion and its derivatives
the RHS contains the desired motion and its derivatives
• the (regularized) static analysis provides the kinematic inversion
the (regularized) static analysis provides the kinematic inversion
[K' / x
T
/x
0 ]{ẋ
1
}=
{ 0
bx , t}
[K' / x
T
/x
0 ]{ẍ
2
}=
{ 0
b'x , ẋ, t}
/ x
T
=f 'x, ẋ , ẍ, t
same matrix!
same matrix!
nonlinear
nonlinear
linear
linear
linear
linear
linear
linear
K'=I : Moore-Penrose pseudo-inverse
: Moore-Penrose pseudo-inverse

23
Problems
• Direct eigenanalysis (eXperimental)
Direct eigenanalysis (eXperimental)
 issues with constraints formulation
issues with constraints formulation
(mainly with rotations)
(mainly with rotations)
 issues with equations implementation
issues with equations implementation
(matrices not available for some elements; e.g. aerodynamics)
(matrices not available for some elements; e.g. aerodynamics)
• Future development: relative coordinate frame dynamics
Future development: relative coordinate frame dynamics
 imposed frame motion: modifications only to RHS inertia elems
imposed frame motion: modifications only to RHS inertia elems
 instrumental for many helicopter rotor/wind turbine
instrumental for many helicopter rotor/wind turbine
dynamics problems
dynamics problems

24
• Mechanical degrees of freedom:
Mechanical degrees of freedom:
 structural node positions in the absolute reference frame
structural node positions in the absolute reference frame
 structural node orientations with respect to the absolute frame
structural node orientations with respect to the absolute frame
(but in updated form...)
(but in updated form...)
• Kinematics is always written with respect to the absolute frame
Kinematics is always written with respect to the absolute frame
• Newton-Euler equations are written in the absolute frame
Newton-Euler equations are written in the absolute frame
 moment equilibrium (Euler) equations are written
moment equilibrium (Euler) equations are written
with respect to the (moving) nodes
with respect to the (moving) nodes
• Special elements may introduce further approximations
Special elements may introduce further approximations
 e.g. Component Mode Synthesis (CMS) element
e.g. Component Mode Synthesis (CMS) element

25
Orientation handling:
• orientation variables: Cayley-Gibbs-Rodrigues parameters
orientation variables: Cayley-Gibbs-Rodrigues parameters
• orientation matrix:
orientation matrix:
• orthonormality:
orthonormality:
• derivative:
derivative:
• incremental approach from step
incremental approach from step k
k to
to k
k+1 to eliminate the
+1 to eliminate the
singularity issue that arises from orientation parameters
singularity issue that arises from orientation parameters
(increments must be small anyway for accuracy):
(increments must be small anyway for accuracy):
R=Rg
Ṙ RT
=×=Ggġ×
RT
=R−1
Rk
=Rgk
Rk−1

26
• Orientation handling:
 the actual orientation variables are the Cayley-Gibbs-Rodrigues
the actual orientation variables are the Cayley-Gibbs-Rodrigues
parameters relative to the correction phase of each time step
parameters relative to the correction phase of each time step
 k
k: time step counter
: time step counter
 i
i: correction iteration counter (0: predicted value)
: correction iteration counter (0: predicted value)
• Orientation matrix:
Orientation matrix:
• Derivative:
Derivative:
Rk
i
=Rg
i
Rk
0
Ṙk
i
Rk
i
T
=k
i
×=Rg
i
k
0
×Gg
i
ġ
i
×
k-1 k(i)
k(0)
“updated-updated”
lagrangian
updated lagrangian

27
• Incremental orientation from previous step:
Incremental orientation from previous step:
 Orientation parameters order of magnitude:
Orientation parameters order of magnitude:
• Incremental orientation from prediction:
Incremental orientation from prediction:
 Orientation parameters order of magnitude:
Orientation parameters order of magnitude:
where
where n
n is the min between the order of the predictor and that
is the min between the order of the predictor and that
of the integration method (MBDyn: 3 and 2, respectively)
of the integration method (MBDyn: 3 and 2, respectively)
 As a consequence:
As a consequence:
(only in Jacobian
(only in Jacobian
matrix, of course!)
matrix, of course!)
g~O∥∥ t
g
~O tn1

Rg
≈I
Gg
≈I
Ġg
≈0

28
• Lumped deformable components
Lumped 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 element
Intrinsic, composite-ready Finite-Volume beam element
 arbitrary constitutive law
arbitrary constitutive law
 piezoelectric constitutive law
piezoelectric constitutive law
 aerodynamic (strip-theory with inflow model) beam element
aerodynamic (strip-theory with inflow model) beam element
• Component Mode Synthesis (CMS)
Component Mode Synthesis (CMS)
 attached to a floating frame (a multibody node) for rigid body motion
attached to a floating frame (a multibody node) for rigid body motion
 linear state-space representation of unsteady aerodynamics
linear state-space representation of unsteady aerodynamics
ẋ= A xBqs
fa
= qC xD0
qs
b/V D1
q̇s
b/V 2
D2
q̈s


29
Lumped deformable components (3D, 6D):
Lumped deformable components (3D, 6D):
• Attached form:
Attached form:
 constitutive properties referred to either of the connected nodes
constitutive properties referred to either of the connected nodes
• Intrinsic form (invariant: ):
Intrinsic form (invariant: ):
 constitutive properties referred to a floating reference frame
constitutive properties referred to a floating reference frame
 intrinsically handles geometrical nonlinearity related to rotations
intrinsically handles geometrical nonlinearity related to rotations
 correctly captures bending-torsion buckling behavior
correctly captures bending-torsion buckling behavior
 essential for anisotropic deformable components
essential for anisotropic deformable components
The intrinsic/invariant form is unique to MBDyn.
The intrinsic/invariant form is unique to MBDyn.
It resulted from a work for Hutchinson CdR
It resulted from a work for Hutchinson CdR
= axexp
−1
Ra
T
Rb

m= R 
m
=0,=1
=1/2

30
Intrinsic, composite-ready beam
Intrinsic, composite-ready beam
• Topology:
Topology:
 1D reference line p, 1D reference structure R
1D reference line p, 1D reference structure R
 2D section characterization
2D section characterization
reference line
reference line
reference
reference
orientation
orientation
reference motion:
reference motion:
warping
warping
x= pR t
x/
= p/

×R t
p R
t

p, R

31
Intrinsic, composite-ready beams
Intrinsic, composite-ready beams
• strain measure:
strain measure:
• equilibrium (from VWP):
equilibrium (from VWP):
• constitutive properties:
constitutive properties:
= R
T
p/
−R0
T
p0/
= RT

−R0
T
 0
f /
= 
m/
p/
×f= 
f = f , 
m= m,

32
Intrinsic, composite-ready beams: 3-node discretization
Intrinsic, composite-ready beams: 3-node discretization
• Finite Volume approach: equilibrium of finite portions of beam
Finite Volume approach: equilibrium of finite portions of beam
• internal forces function of node kinematics thru constitutive laws
internal forces function of node kinematics thru constitutive laws
• warping goes into constitutive properties computation
warping goes into constitutive properties computation
node 1
node 1
node 2
node 2
node 3
node 3
point I
point I
point II
point II
fI
, mI
fII
, mII
f1
, m1
f2
, m2
f3
, m3
constitutive properties from detailed 2D FEM
constitutive properties from detailed 2D FEM

33
Solving the problem
Numerical integration
Numerical integration
• implicit, (quasi-)L stable 2 step algorithm
implicit, (quasi-)L stable 2 step algorithm
• tunable algorithmic dissipation: asymptotic spectral radius 1
tunable algorithmic dissipation: asymptotic spectral radius 1→
→0
0
 asymptotic spectral radius = 0: 2
asymptotic spectral radius = 0: 2nd
nd
order BDF
order BDF
 “
“optimal” dissipation: spectral radius ~ 0.6
optimal” dissipation: spectral radius ~ 0.6
• second-order accurate, with third-order accurate predictor
second-order accurate, with third-order accurate predictor
• variable time step
variable 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 implemented
different integrators can be used; new ones can be implemented
yk
=a1
yk−1
a2
yk−2
 tb0
ẏk
b1
ẏk−1
b2
ẏk−2


34
Solving the problem
• Prediction:
Prediction:
• Correction iteration:
Correction iteration:
but
but
the problem becomes algebraic
the problem becomes algebraic
ẏk
0
=m1
yk−1
m2
yk−2
/ tn1
ẏk−1
n2
ẏk−2
yk
0
=a1
yk−1
a2
yk−2
 tb0
ẏk
0
b1
ẏk−1
b2
ẏk−2

f /ẏ
ẏi
f /y
yi
=−f ẏk
i−1
, yk
i−1
, tk

yi
= t b0
ẏi
f /ẏ
 t b0
f / y
ẏi
=−f ẏk
i−1
, yk
i−1
, tk

ẏk
i
=ẏk
i−1
 ẏi
yk
i
=yk
i−1
 t b0
ẏ
i

35
Solving the problem
Model assembly
Model assembly
• model could be input incorrectly
model could be input incorrectly
• initial values of the state (position, velocity, reactions) are needed
initial values of the state (position, velocity, reactions) are needed
• this task might not be trivial
this task might not be trivial
• 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
/x
T
'= f '
C'ẋ−ẋ0
/x
T
'= ḟ '
x, t0
= 0
̇x, t0
= 0

36
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 duplication
convergence 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)
ẏ=f y , t
0=f y, ẏ , t
f /ẏ
c f /y
ẏi
=−f ẏ0
i−1
, y0
, t0

ẏ0
i
=ẏ0
i−1
 ẏi
y0
=y0
c=0

37
• Detailed analysis requires detailed models, but...
Detailed analysis requires detailed models, but...
• excessive details endanger the chance to extract useful information
excessive details endanger the chance to extract useful information
• Proper Orthogonal Decomposition allows to extract information
Proper Orthogonal Decomposition allows to extract information
from redundant measures
from redundant measures
• Consider a set of
Consider a set of N
N measurements X for
measurements X for n
n time steps; their SVD:
time steps; their SVD:
• The singular values allow to determine the
The singular values allow to determine the m
m most relevant signals
most relevant signals
• Note that
Note that
• This allows to efficiently compute the singular values and the POMs
This allows to efficiently compute the singular values and the POMs
XT
∈ℝn×N
=U  VT
X1: m, n
T
=Un,1: m
1: m,1: m
VN ,1: m
T
X
T
X= U 
2
U
T
UT
XT
=  VT
X X
T
= V 
2
V
T
XT
V= U 

38
• The POMs can be used to identify a transition matrix
The POMs can be used to identify a transition matrix
• If X contains the free response of the system, the transition matrix
If X contains the free response of the system, the transition matrix
allows to estimate the relevant eigenvalues (AR model)
allows to estimate the relevant eigenvalues (AR model)
• More sophisticated system identification techniques can be used
More sophisticated system identification techniques can be used
• A technique based on covariance estimates from time histories has
A technique based on covariance estimates from time histories has
been recently proposed; works for:
been recently proposed; works for:
 free response
free response
 forced response
forced response
 unmeasured forced response
unmeasured forced response
[1] G. Quaranta, P. Masarati, and P. Mantegazza: “Continuous-Time Covariance Approaches for
Modal Analysis”, Journal of Sound and Vibration, Vol.310/1-2 pp. 287-312, 5 February 2008.
[2] G. Quaranta, P. Masarati, and P. Mantegazza: “Assessing the Local Stability of Periodic
Motions for Large Multibody Nonlinear Systems Using POD”, Journal of Sound and
Vibration, Vol 271/3-5, pp. 1015-1038, 2004.
Xk1
= Xk

39
Robotics:
Robotics:
delta robot
delta robot
inverse dynamics
inverse dynamics
for computed
for computed
torque control
torque control

40
Robotics: PA-10
Robotics: PA-10
inverse kinematics
inverse kinematics
with path optimization
with path optimization
of cooperating robots
of cooperating robots

41
Robotics:
Robotics:
biomimetic robot
biomimetic robot
real-time motion planning
real-time motion planning
by inverse kinematics
by inverse kinematics
with fault detection
with fault detection

42
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 assembly
check stresses introduced during assembly
 check loads on supports introduced during assembly
check loads on supports introduced during assembly
 check interference with other parts during assembly
check interference with other parts during assembly
 check interference with other parts during operation
check interference with other parts during operation
• the model has been developed by a rubber manufacturer
the model has been developed by a rubber manufacturer
(Hutchinson CdR)
(Hutchinson CdR)
• it is used for product design and certification
it 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 the standard MBDyn (the “hints”)
partitioning, which are now part of the standard MBDyn (the “hints”)

43
Automotive: mechanical modeling of suspensions
Automotive: mechanical modeling of suspensions
purpose: determine loads in rubber bushings and other components
purpose: determine loads in rubber bushings and other components

44
Rotorcraft dynamics and aeroservoelasticity:
Rotorcraft dynamics and aeroservoelasticity:
• WRATS (NASA/Army) tiltrotor aeromechanics
WRATS (NASA/Army) tiltrotor aeromechanics

45
Rotorcraft dynamics
Rotorcraft dynamics
and aeroservoelasticity:
and aeroservoelasticity:
• ERICA (AgustaWestland)
ERICA (AgustaWestland)
tiltrotor aeromechanics
tiltrotor aeromechanics
(ADYN, NICETRIP EU 6FP,
(ADYN, NICETRIP EU 6FP,
in cooperation with
in cooperation with
Eurocopter, DLR, ONERA
Eurocopter, DLR, ONERA
and more...)
and more...)

46
Wind-turbine dynamics simulation,
Wind-turbine dynamics simulation,
also in Real-Time
also in Real-Time
gearbox
yaw teeter
pitch
tower
blades
nacelle

47
Future development
• Multiscale handling of submodels with different dynamics;
Multiscale handling of submodels with different dynamics;
for example, in the flying helicopter case:
for example, in the flying helicopter case:
 aircraft flight mechanics (~0 to 5 Hz: very slow)
aircraft flight mechanics (~0 to 5 Hz: very slow)
 airframe & main rotor dynamics (~5 to 40 Hz: intermediate)
airframe & 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 domains
Interfacing with different domains
 fluid-structure (Lagrangian/Eulerian modeling of work flows)
fluid-structure (Lagrangian/Eulerian modeling of work flows)
 structure-structure (dynamic loads to detailed static analysis)
structure-structure (dynamic loads to detailed static analysis)
 active control of large deformable systems
active control of large deformable systems
• Better abstraction/modularization of components/solution phases
Better abstraction/modularization of components/solution phases
 more freedom in model customization
more freedom in model customization
 tight integration into nonlinear structural analysis
tight integration into nonlinear structural analysis
(structural damage growth under dynamic loads?)
(structural damage growth under dynamic loads?)
• More...
More...

48
Future development
• Development has always been problem (and customer!) driven
Development has always been problem (and customer!) driven
• Development model:
Development model:
 research (possibly new) solutions in (possibly new) areas
research (possibly new) solutions in (possibly new) areas
 procure research grants and contracts in those fields
procure research grants and contracts in those fields
 never ask funds just for software development
never ask funds just for software development
 sell expertise, not just software; as a consequence...
sell expertise, not just software; as a consequence...
 ...software is not the result of a contract: it's a pre-existing tool
...software is not the result of a contract: it's a pre-existing tool
 software remains free even when customers are narrow-minded
software remains free even when customers are narrow-minded
 when possible, work with open-minded customers :-)
when possible, work with open-minded customers :-)
• Drawback: cannot stick with tight development plans :-(
Drawback: cannot stick with tight development plans :-(
We are a University: we should not sacrifice our research freedom
We are a University: we should not sacrifice our research freedom
to industry schedules
to industry schedules

49
• Freedom and features are not enough: software needs to be usable
Freedom and features are not enough: software needs to be usable
• Users need:
Users need:
 documentation
documentation
 forums to discuss issues
forums to discuss issues
 issue tracking provisions
issue tracking provisions
 reasonable guarantee of maintenance
reasonable guarantee of maintenance
 reasonable guarantee of stability across releases
reasonable guarantee of stability across releases
• The main guarantee is the freedom of the software:
The main guarantee is the freedom of the software:
 in the worst case, one can always fork and go on its own; but...
in the worst case, one can always fork and go on its own; but...
 ... first, better talk to the developers: cooperation saves efforts
... first, better talk to the developers: cooperation saves efforts

50
• Theory manual:
Theory manual:
 Incomplete; needs lots of work
Incomplete; needs lots of work
• User manual
User manual
 available and complete, but probably hard to read; needs improvements
available and complete, but probably hard to read; needs improvements
• Tutorials
Tutorials
 available, but reportedly too simple; need work
available, but reportedly too simple; need work
• Applications manual
Applications manual
 just started
just started
• Installation manual
Installation manual
 available, but incomplete and outdated
available, but incomplete and outdated
• Mailing lists
Mailing lists
 available: announce, users, devel
available: announce, users, devel
 right now, the “users” list also serves as issue tracking provision
right now, the “users” list also serves as issue tracking provision

51
• Another important item that is missing is an automated test suite, that
Another important item that is missing is an automated test suite, that
 can be run automatically after building the software
can be run automatically after building the software
 allows to check build errors
allows to check build errors
 allows to check regressions in new releases
allows to check regressions in new releases
 serves as example of modeling functionalities
serves 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!

52
Acknowledgments
• Paolo Mantegazza (“grandfather” of MBDyn)
Paolo Mantegazza (“grandfather” of MBDyn)
• Marco Morandini (lots of optimizations, friction, ...)
Marco Morandini (lots of optimizations, friction, ...)
• Giuseppe Quaranta (schur decomposition, FSI, ...)
Giuseppe Quaranta (schur decomposition, FSI, ...)
• Alessandro Fumagalli (inverse dynamics, ...)
Alessandro Fumagalli (inverse dynamics, ...)
• many Graduate and PhD students
many Graduate and PhD students
• few contributions from outside as well (mostly bug fixes)
few contributions from outside as well (mostly bug fixes)

53
Questions?

presentation_masarati.pdf

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