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SPAÇAR: A Finite Element Approach in
Flexible Multibody Dynamics
UIC Seminar
September 27, 2004
University of Illinois at Chicago
Laboratory for Engineering Mechanics
Faculty of Mechanical Engineering
Arend L. Schwab
Google: Arend Schwab [I’m Feeling Lucky]
September 27, 2004
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Acknowledgement
TUdelft:
Hans Besseling
Klaas Van der Werff
Helmut Rankers
Ton Klein Breteler
Jaap Meijaard
… MSc students
UTwente:
Ben Jonker
Ronald Aarts
… MSc students
September 27, 2004
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Contents
•
Roots
•
Modelling
•
Some Finite Elements
•
Eqn’s of Motion
•
Examples
•
Discussion
September 27, 2004
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Engineering Mechanics at Delft
From Analytical Mechanics in 50’s:
Warner T. Koiter
On the Stability of Elastic Equilibrium
, 1945
To Numercial Methods in Applied Mechanics in 70’s:
Hans Besseling
The complete analogy between the matrix equations and the continuous
field equations of structural analysis
, 1963
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Mechanism and Machine Theory
Application of Numerical
Methods to:
•
Kinematic Analysis
•
Type and Dimension
Synthesis
•
Dynamic Analysis
CADOM
project:
C
omputer
A
ided
D
esign of
M
echanisms, 1972
Rankers, Van der Werff, Klein Breteler, Schwab,
et al.
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Mechanism and Machine Theory,
Kinematics
Denavit & Hartenberg, 1955
•
Rigid Bodies
•
Relative Coordinates
(few)
•
Kinematic Constraints
(few)
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Mechanism and Machine Theory,
Kinematics
Klaas Van der Werff, 1975
Finite Element Approach
•
Flexible Bodies
•
Absolute Coordinates and
Large Rotations
(many)
•
Kinematic Constraints =
Rigidity of Bodies
(many)
Note: Decoupling of the positional nodes and the orientational nodes.
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Multibody System Dynamics
Finite Element Approach
Key Idea:
Specification of Independent Deformation Modes of the Finite Elements
Coordinates:
(
x
p
,
p
,
x
q
,
q
) total 6
Deformation Modes:
total 6

3=3
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Multibody System Dynamics
Finite Element Approach
Pro’s:
•
Easy FEM assembly of the system equations
•
Easy mix of partly Rigid and/or Flexible elements
•
Small set of elements for Large class of Multibody Systems
•
Absolute Coordinates and Large Rotations
•
Gen. Deformation can act as Relative Coordinates
Con’s:
•
Many coordinates, many constraints
•
Non

Constant Mass Matrix
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Multibody System Dynamics
Finite Element Approach
Generalized Deformation can act as Relative
Coordinates
Ex. Hydraulic Cylinder
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Multibody System Dynamics
Compare to Rigid Bodies with Constraints
Milton Chace & Nicky Orlandea, DRAM, ADAMS, 1970
Constraints are at the Joints
Constraints are in the Bodies
FEM approach
Rigid Bodies with Constraints
September 27, 2004
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3D Beam Element
Coordinates:
(
x
p
,
p
,
x
q
,
q
) total 14
Deformation Modes:
total 14
–
6 = 8
Cartesian Coordinates
x
p
= (x, y, z)
p
and Euler Parameters
p
=(
0
,
1
,
2
,
3
)
p
= 0
= 0
–
2 = 6
September 27, 2004
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3D Hinge Element
Coordinates:
(
p
,
q
) total 8
Deformation Modes:
total 8
–
3 = 5
= 0
= 0
–
2 = 3
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3D Truss Element
Coordinates:
(
x
p
,
x
q
) total 6
Deformation Modes:
total 6
–
5 = 1
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3D Wheel Element
Coordinates:
(
x
w
,
p,
x
c
) total 10
Some Counting:
Pure rolling Rigid body has
3 degrees of freedom (velocities).
We need 10

3=7 Constraints on the
Velocities.
Pure rolling is 2 Velocity Constraints,
Lateral and Longitudinal.
Leaves 7

2=5 Deformation Modes
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3D Wheel Element
Coordinates:
(
x
w
,
p,
x
c
) total 10
Deformation Modes:
Generalized Slips:
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Ex. Universal or Cardan Joint
Physical Model
Two FEModels: (a) with 4 Rigid Hinges, and (b) with 2 Flexible Hinges
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Ex. Universal or Cardan Joint
Two FEModels: (a) with 4 Rigid Hinges, and (b) with 2 Flexible Hinges
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Dynamic Analysis
In the spirit of
d’Alembert
and
Lagrange
we transform the DAE
in terms of generalized independent coordinates q
j
with x
i
=F
i
(q
j
) resulting in
From which we solve
and Numerically Integrate as an ODE.
Note: the Elastic Forces are according to
and
September 27, 2004
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Ex. ILTIS Road Vehicle Benchmark
The ILTIS Vehicle
Suspension
FEM Model
85 Elements
239 Gen. Def.
70 Nodes
226 Gen. Coord.
10 DOF’s

Rigid Cabin

4 Independently
Suspended Wheels

CALSAP Tire Model
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Ex. ILTIS Road Vehicle Benchmark
Static Equilibrium Results
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Ex. ILTIS Road Vehicle Benchmark
Handling Performance Test:
Ramp

to

Step Steer Manoeuvre at v = 30 m/s.
CALSPAN tire model
Zero Lateral Slip
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Ex. Slider

Crank Mechanism
Slider

Crank Mechanism from Song & Haug, 1980
Rigid Crank,Flexible Connecting Rod
=150 rad/s, 2% damping
Transient Solution
Periodic Solution
First Eigenfrequency of pinned joint
connecting rod
0
= 832 rad/s
September 27, 2004
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Linearized Equations of Motion
Equations of Motion can be Analytically Linearized at a Nominal Motion
Even for Systems having Non

Holonomic Constraints!
with:
M: reduced Mass Matrix
C: Tangent Velocity dependant Matrix
K: Tangent Stiffness Matrix
q
k
: Kinematic Coordinates variations
A: Non

Holonomic Constraints
B: Tangent Reonomic Constraints Matrix
September 27, 2004
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Ex. Slider

Crank Mechanism
Nominal Periodic Motion and small Vibrations described by the
Linearized Equations of Motion
Slider

Crank Mechanism from Song & Haug, 1980
Rigid Crank,Flexible Connecting Rod
=150 rad/s, 2% damping
Transient Solution
Periodic Solution
September 27, 2004
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Linearized Equations of Motion at
Nominal Periodic Motion
Periodic Solutions for small Vibrations superimposed on a
Nominal Periodic Motion
Linearized Equations of Motion at Nominal Motion:
The Coefficients in the Matrices are Periodic with Period T=2
/
Transform these Matrices into Fourier Series:
and assume a periodic solution of the form:
September 27, 2004
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Linearized Equations of Motion at
Nominal Periodic Motion
Periodic Solutions for small Vibrations superimposed on a
Nominal Periodic Motion
Substitution into the Linearized Equations of motion and balance of every
individual Harmonic leads to:
These are (2k+1)*dof linear equations from which we can solve the
2k+1 harmonics:
Which form the solution of the small Vibration problem:
September 27, 2004
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Ex. Slider

Crank Mechanism
Slider

Crank Mechanism from Song & Haug, 1980
FEModel: 2 Beam Elements for
the Flexible Connecting Rod
Rigid Crank,Flexible Connecting Rod
=150 rad/s, 2% damping
Transient Solution
Periodic Solution
September 27, 2004
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Ex. Slider

Crank Mechanism
Damping 1% and 2%
First Eigenfrequency of pinned joint
connecting rod
0
= 832 rad/s
Resonace at 1/5, 1/4, and 1/3 of
0
Linearized Results
Full Non

Linear Results
Maximal Midpoint Deflection/l for a range of
’s
September 27, 2004
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Ex. Slider

Crank Mechanism
First Eigenfrequency
of pinned joint
connecting rod
0
= 832 rad/s
Resonace at 1/5, 1/4,
and 1/3 of
0
Quasi Static Solution
Individual Harmonics of the Midpoint Deflection/l for a range of
’s
September 27, 2004
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Ex. Dynamics of an Uncontrolled Bicycle
Cornell University, Ithaca, NY, 1987:
Yellow Bike in the Car Park
September 27, 2004
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Ex. Dynamics of an Uncontrolled Bicycle
Cornell University, Ithaca, NY, 1987:
Yellow Bike in the Car Park
September 27, 2004
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Ex. Dynamics of an Uncontrolled Bicycle
Modelling Assumptions:

rigid bodies

fixed rigid rider

hands

free

symmetric about vertical plane

point contact, no side slip

flat level road

no friction or propulsion
Note: This model is energy conservative
September 27, 2004
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Ex. Dynamics of an Uncontrolled Bicycle
3 Degrees of Freedom:
4 Kinematic Coordinates:
FEModel: 2 Wheels, 2 Beams, 6 Hinges
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Ex. Dynamics of an Uncontrolled Bicycle
Forward Full Non

Linear Dynamic Analysis with an initial small side

kick
Forward Speed:
v = 3.5 m/s
v = 4.5 m/s
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Ex. Dynamics of an Uncontrolled Bicycle
Investigate the Stability of the Steady Forward Upright Motion by
means of the Linearized Equations of Motion at this Steady Motion
Linearized Equations of Motion for Systems having Non

Holonomic Constraints
in State

Space form:
Assume an exponential motion for the small variations:
September 27, 2004
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Ex. Dynamics of an Uncontrolled Bicycle
Rootloci of
from the Linearized Equations of Motion with as a Parameter the
Forward Speed v
Asymptotically Stable in the
Speed Range:
4.1 < v < 5.7 m/s
September 27, 2004
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Ex. Dynamics of an Uncontrolled Bicycle
What happens for v>5.7 m/s?
Forward Speed:
v = 6.3 m/s
September 27, 2004
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Conclusions

SPAÇAR is a versatile FEM based Dynamic Modeling System for Flexible
and/or Rigid Multibody Systems .

The System is capable of modeling idealized Rolling Contact (Non

Holonomic Constraints).

The System uses a set of minimal independent state variables, which
avoid the use of differential

algebraic equations.

The Equations of Motion can be Linearized Analytically at any given
state.
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