SPAAR: A Finite Element Approach in Flexible Multibody Dynamics

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SPAÇAR A Finite Element Approach inFlexible
Multibody Dynamics

• UIC Seminar

Arend L. Schwab Google Arend Schwab Im
Feeling Lucky
September 27, 2004 University of Illinois at
Chicago
Laboratory for Engineering MechanicsFaculty of
Mechanical Engineering
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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
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Contents

• Roots
• Modelling
• Some Finite Elements
• Eqns of Motion
• Examples
• Discussion

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Engineering Mechanics at Delft

• From Analytical Mechanics in 50s
• Warner T. Koiter
• On the Stability of Elastic Equilibrium, 1945
• To Numercial Methods in Applied Mechanics in
  70s
• 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 Computer Aided Design of
Mechanisms, 1972Rankers, 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 DynamicsFinite Element Approach
Key IdeaSpecification of Independent
Deformation Modes of the Finite Elements
Coordinates (xp, ?p , xq , ?q) total
6 Deformation Modes total 6-33
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Multibody System DynamicsFinite Element Approach

• Pros
• 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
• Cons
• Many coordinates, many constraints
• Non-Constant Mass Matrix

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Multibody System DynamicsFinite 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
FEM approach
Rigid Bodies with Constraints
Constraints are at the Joints
Constraints are in the Bodies
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3D Beam Element
Coordinates (xp, ?p , xq , ?q) total
14 Deformation Modes total 14 6 8
2 6
0 0
Cartesian Coordinates xp (x, y, z)p and Euler
Parameters ?p (?0, ?1, ?2, ?3)p
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3D Hinge Element
Coordinates (?p, ?q) total 8 Deformation
Modes total 8 3 5
2 3
0 0
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3D Truss Element
Coordinates (xp, xq ) total 6 Deformation
Modes total 6 5 1
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3D Wheel Element
Coordinates (xw, ?p, xc ) total 10 Some
Counting Pure rolling Rigid body has3 degrees
of freedom (velocities). We need 10-37
Constraints on the Velocities. Pure rolling is 2
Velocity Constraints, Lateral and
Longitudinal. Leaves 7-25 Deformation Modes
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3D Wheel Element
Coordinates (xw, ?p, xc ) 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 dAlembert and Lagrange we
transform the DAE
in terms of generalized independent coordinates
qj with xiFi(qj) resulting in
From which we solve and Numerically Integrate as
an ODE. Note the Elastic Forces are according
to and
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Ex. ILTIS Road Vehicle Benchmark
- Rigid Cabin - 4 Independently Suspended
Wheels - CALSAP Tire Model
The ILTIS Vehicle
85 Elements 239 Gen. Def. 70 Nodes 226 Gen.
Coord. 10 DOFs
Suspension
FEM 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
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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
?qk Kinematic Coordinates variations A
Non-Holonomic Constraints B Tangent Reonomic
Constraints Matrix
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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
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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 T2?/? Transform these Matrices into
Fourier Series
and assume a periodic solution of the form
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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 (2k1)dof linear equations from which
we can solve the2k1 harmonics Which form the
solution of the small Vibration problem
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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
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Ex. Slider-Crank Mechanism
Maximal Midpoint Deflection/l for a range of ?s
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
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Ex. Slider-Crank Mechanism
Individual Harmonics of the Midpoint Deflection/l
for a range of ?s
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
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Ex. Dynamics of an Uncontrolled Bicycle
Cornell University, Ithaca, NY, 1987 Yellow Bike
in the Car Park
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Ex. Dynamics of an Uncontrolled Bicycle
Cornell University, Ithaca, NY, 1987 Yellow Bike
in the Car Park
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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
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Ex. Dynamics of an Uncontrolled Bicycle
FEModel 2 Wheels, 2 Beams, 6 Hinges
3 Degrees of Freedom
4 Kinematic Coordinates
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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
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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 lt v
lt 5.7 m/s
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Ex. Dynamics of an Uncontrolled Bicycle
What happens for vgt5.7 m/s?
Forward Speed v 6.3 m/s
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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.