1. Vermelding onderdeel organisatie 1 SPAÇAR: A Finite Element Approach in Flexible Multibody Dynamics UIC Seminar September 27, 2004University of Illinois at Chicago Laboratory for Engineering Mechanics Faculty of Mechanical Engineering Arend L. Schwab Google: Arend Schwab [I’m Feeling Lucky]

2. September 27, 20042 Acknowledgement TUdelft: Hans Besseling Klaas Van der Werff Helmut Rankers Ton Klein Breteler Jaap Meijaard … MSc students UTwente: Ben Jonker Ronald Aarts … MSc students

3. September 27, 20043 Contents Roots Modelling Some Finite Elements Eqn’s of Motion Examples Discussion

4. September 27, 20044 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

5. September 27, 20045 Mechanism and Machine Theory Application of Numerical Methods to: Kinematic Analysis Type and Dimension Synthesis Dynamic Analysis CADOM project: Computer Aided Design of Mechanisms, 1972 Rankers, Van der Werff, Klein Breteler, Schwab, et al.

6. September 27, 20046 Mechanism and Machine Theory, Kinematics Denavit & Hartenberg, 1955 Rigid Bodies Relative Coordinates (few) Kinematic Constraints (few)

7. September 27, 20047 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.

8. September 27, 20048 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

9. September 27, 20049 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

10. September 27, 200410 Multibody System Dynamics Finite Element Approach Generalized Deformation can act as Relative Coordinates Ex. Hydraulic Cylinder

11. September 27, 200411 Multibody System Dynamics Compare to Rigid Bodies with Constraints Milton Chace & Nicky Orlandea, DRAM, ADAMS, 1970 Constraints are at the JointsConstraints are in the Bodies FEM approachRigid Bodies with Constraints

12. September 27, 200412 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 – 2 = 6

13. September 27, 200413 3D Hinge Element Coordinates: ( p, q ) total 8 Deformation Modes: total 8 – 3 = 5 = 0 – 2 = 3

14. September 27, 200414 3D Truss Element Coordinates: ( x p, x q ) total 6 Deformation Modes: total 6 – 5 = 1

15. September 27, 200415 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

16. September 27, 200416 3D Wheel Element Coordinates: ( x w, p, x c ) total 10 Deformation Modes: Generalized Slips:

17. September 27, 200417 Ex. Universal or Cardan Joint Physical Model Two FEModels: (a) with 4 Rigid Hinges, and (b) with 2 Flexible Hinges

18. September 27, 200418 Ex. Universal or Cardan Joint Two FEModels: (a) with 4 Rigid Hinges, and (b) with 2 Flexible Hinges

19. September 27, 200419 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

20. September 27, 200420 Ex. ILTIS Road Vehicle Benchmark The ILTIS Vehicle SuspensionFEM Model 85 Elements 239 Gen. Def. 70 Nodes 226 Gen. Coord. 10 DOF’s - Rigid Cabin - 4 Independently Suspended Wheels - CALSAP Tire Model

21. September 27, 200421 Ex. ILTIS Road Vehicle Benchmark Static Equilibrium Results

22. September 27, 200422 Ex. ILTIS Road Vehicle Benchmark Handling Performance Test: Ramp-to-Step Steer Manoeuvre at v = 30 m/s. CALSPAN tire model Zero Lateral Slip

23. September 27, 200423 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

24. September 27, 200424 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

25. September 27, 200425 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

26. September 27, 200426 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:

27. September 27, 200427 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:

28. September 27, 200428 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

29. September 27, 200429 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

30. September 27, 200430 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

31. September 27, 200431 Ex. Dynamics of an Uncontrolled Bicycle Cornell University, Ithaca, NY, 1987: Yellow Bike in the Car Park

32. September 27, 200432 Ex. Dynamics of an Uncontrolled Bicycle Cornell University, Ithaca, NY, 1987: Yellow Bike in the Car Park

33. September 27, 200433 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

34. September 27, 200434 Ex. Dynamics of an Uncontrolled Bicycle 3 Degrees of Freedom: 4 Kinematic Coordinates: FEModel: 2 Wheels, 2 Beams, 6 Hinges

35. September 27, 200435 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

36. September 27, 200436 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:

37. September 27, 200437 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

38. September 27, 200438 Ex. Dynamics of an Uncontrolled Bicycle What happens for v>5.7 m/s? Forward Speed: v = 6.3 m/s

39. September 27, 200439 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.