Flexible multibody systems - Relative coordinates approach

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1 Computer-aided analysis of multibody dynamics (part 2) Flexible multibody systems - Relative coordinates approach Paul Fisette (paul.fisette@uclouvain.be) Introduction In terms of modeling, multibody scientists must develop a «critical mind» statics «by hand», Matlab From a «customer problem» flexible bodies advanced dynamics basic dynamics advanced dynamics Matlab rigid MBS code flexible MBS code => What is the «real» problem to solve. and thus the optimal model to build

Introduction It is particularily true when dealing with «flexibility» Examples : Car suspension deformation : MBS + static FEM (in post-process) Antenna deployment : MBS + FEM (super-elements )

2 Introduction It is particularily true when dealing with «flexibility» Examples : Car suspension deformation : MBS + static FEM (in post-process) Antenna deployment : MBS + FEM (super-elements ) Flexible 2D mechanism : Chassis torsion : MBS + Finite segment / 2D beam model Full FEM or Lumped torsion (MBS-rigid)? => What is the «real» problem to solve and thus the «minimal» model to build Introduction FEM community MBS community > 1990 : «handshake» MBS community «credos» : Simulation time efficiency towards real time Lumped and/or «macro» models are still very promising High dynamics problems Flexibility : small deformation few modes High interest in system control and optimization

3 Contents Relative coordinates approach : review MBS with flexible beams : finite segment approach MBS with flexible beams : assumed mode approach =>chap. 9 of Beam Model Symbolic implementation MBS with telescopic beams Kluwer Academic Publishers, 2003 Contents Relative coordinates approach : review MBS with flexible beams : finite segment approach MBS with flexible beams : assumed mode approach =>chap. 9 of Beam Model Symbolic implementation MBS with telescopic beams Kluwer Academic Publishers, 2003

Concepts Multibody structure Leaf body 5 Rigid bodies (+flexible beams) 3 2 body joint base 1 0 4

structure) : the «inbody» vector : inbody = [ 0 1 2 2 4 4 1 7] Joints Concepts - Definitions

4 Concepts Multibody structure Leaf body 5 Rigid bodies (+flexible beams) 3 2 body joint base loop Relative joint coordinates Tree-like structures Closed structures Topology (tree-like structure) : the «inbody» vector : inbody = [ ] Joints Concepts - Definitions Classical 6 dof! More «Exotic» «Knee» joint (1 6 dof) «Wheel on rail» joint (5 dof) «Cam/Follower» joint

5 Tree-like MBS: Recursive Newton-Euler Origin : Robotics inverse dynamics (O(N body ) operations) Forward kinematics : kin. ω j = ω i + Ω ij Backward dynamics :.. F i = F j + F k + m i x i. L i = L j + L k + I i ω i +... Joint projection :... F i F j body i ω i L j body j ω j body k F k L i L k dyn. Tree-like MBS: Recursive Newton-Euler Origin : Robotics inverse dynamics (Luh, Walker, Paul, 1980) (O(N body ) operations) Objective : Recursive formulation for direct dynamics (O(N body2 ) operations) to keep the advantage of the recursive formulation

6 Closed-loop MBS Kinematic loops for unconstrained system 11 Closed-loop MBS Coordinate partitioning : u v v Exact resolution of the constraints Position : Velocity : Acceleration : 12

7 Contents Relative coordinates approach : review MBS with flexible beams : finite segment approach MBS with flexible beams : assumed mode approach =>chap. 9 of Beam Model Symbolic implementation MBS with telescopic beams Kluwer Academic Publishers, 2003 Finite segment formulation A very simple idea «Computer methods in flexible multibody dynamics», Huston R.L., IJNME 1991 (also Amirouche, 1986) Flexible rod Ressort hélicoïdal Ω = 150 rad/sec θ Glissière Flexibility effects are modeled by spring (and possible dampers) between bodies => Lumped flexibility formulation ex. bending :... i k ij {... l i l j j

8 Finite segment formulation Motivation i... k ij { l i l j... j to avoid the «difficult» mariage between rigid body dynamics and structural dynamics intuitive and direct method perfect for a first «pre-study» can be implemented in a rigid multibody code (easy to implement!) incorporate the flexibilty effects into the global dynamic equations not intrinsically limited to elastic systems (=> viscoelastic, nonlinear elastic ) Limitations restricted to slender bodies (beams, tapered bodies, rods, ) no prove to satisfy the «Rayleigh» vibration criteria (in terms of eigenvalues approx.) deformation coupling : sequence dependent (but ok for small deformation) could be used «erroneously» : requires skills, insight and intuition computer efficiency : OK in 2D, heavy in 3D Finite segment formulation Computation of the equivalent stiffness coefficients Equivalent stiffness coefficient are computed from basic principle of structual mechanics applied to bending, torsion and extension Extension (example) : Continuous : µ dx Lumped : m l Resulting Force Torque in the MBS equations => Joint force (for extension) or torque (for torsion/bending)

9 Finite segment formulation Proposed combinations (not exhaustive) Finite segment formulation Proposed combinations (not exhaustive) COMBINED TAPERED SEGMENTS

10 Finite segment formulation Example : a flexible slider-crank Ressort hélicoïdal θ Ω = 150 rad/sec A y A Glissière Rod modal analysis around equilibrium (horizontal configuration) Finite segment formulation Example : a flexible slider-crank {Y} Ressort hélicoïdal θ Ω = 150 rad/sec A y A Glissière Lateral deflection of the mid-point A in frame {Y} y A ya FEM FSM

11 Finite segment formulation Example : a flexible slider-crank {Y} Ressort hélicoïdal θ Ω = 150 rad/sec B Glissière Shortening of the rod (point B) in frame {Y} x B x B xb FEM FSM Finite segment formulation Kane s benchmark : 2D rotating beam : FEM (+). : monomials shape function (+) : FSM (+) -----:modal shape function (-)

12 Contents Relative coordinates approach : review MBS with flexible beams : finite segment approach MBS with flexible beams : assumed mode approach =>chap. 9 of Beam Model Symbolic implementation MBS with telescopic beams Kluwer Academic Publishers, 2003 MBS with flexible beams Flexible beam model Shape functions Beam Kinematics Multibody kinematics (forward) Joint dynamic equations( backward) Deformation equations (beam per beam) Symbolic computation Applications

13 MBS with flexible beams Flexible beam model Shape functions Beam Kinematics Multibody kinematics (forward) Joint dynamic equations( backward) Deformation equations (beam per beam) Symbolic computation Applications Flexible beam model : hypotheses Geometry : prismatic beams - rectilinear centroidal axis Material : homogeneous and isotropic - conforms to linear elasticity Deformation model : Timoshenko 3D, conservation of plane cross sections, shear deformation and rotary inertia included Kinematics : angular and curvature of the beam must remain small (rotation matrix linearized) - but still compatible to «capture» geometic stiffening effect Topology : a beam has the same status as a rigid body in the MBS : Rigid body Flexible beam Joint

14 Flexible beam model : notations Beam local rotation Linearized rotation matrix Flexible beam model : notations C Centroidal axis deformation Vector position (section S) : Current = undeformed + deformed : Displacement field v :

15 Flexible beam : shape functions ^ E z θ ^ E x v For the x, y, z, linear deformation of the centroidal axis C with : generalized coordinates (=amplitude of shape functions) For the x, y, z, angular deformation of the cross sections S with : generalized coordinates (=amplitude of shape functions) Flexible beam : shape functions ^ E z ^ E x Which kind of shape functions? θ v «global» shape functions From a previous (FEM) modal analysis => assumed modes From a purely mathematical set of functions: cubic splines Legendre polynomials Monomials

16 Flexible beam : monomials Why monomials? They can «theoretically» approximate any plausible deformation (think at a Taylor series which combines them) Being an invariable set of functions, there is no need for a prior modal analysis* They are perfectly suitable for symbolic computation of integrals, ex.: =! But they do not form a set of orthogonal functions (=> numerical unstabilities) *Monomials being not eigenmodes, the beam configuration (q,qd,qdd) may «move away» from the equilibrium state of a prior modal analysis MBS with flexible beams Flexible beam model Shape functions Beam Kinematics Multibody kinematics (forward) Joint dynamic equations( backward) Deformation equations (beam per beam) Symbolic computation Applications

17 Flexible beam : kinematics «Relative» kinematics Angular velocity and acceleration Linear velocity and acceleration MBS with flexible beams Flexible beam model Shape functions Beam Kinematics Multibody kinematics (forward) Joint dynamic equations( backward) Deformation equations (beam per beam) Symbolic computation Applications

18 MBS with flexible beam : kinematics Forward kinematic recursion etc. for accelerations MBS with flexible beam : joint dynamics MBS Virtual power principle: Virtual velocity field :

19 MBS with flexible beam : joint dynamics Backward dynamic recursion F i L i MBS with flexible beams Flexible beam model Shape functions Beam Kinematics Multibody kinematics (forward) Joint dynamic equations( backward) Deformation equations (beam per beam) Symbolic computation Applications

20 Flexible beam : deformation dynamics Beam deformation : Displacement gradient : Strain vector of the centroïdal axis : Beam curvature vector : with where Flexible beam : deformation dynamics MBS Virtual Power Principle: (see next slides) requires the relative derivatives of Γ and K : Real : Virtual :

21 Flexible beam : deformation dynamics MBS Virtual power principle: Virtual velocity field : Flexible beam : deformation dynamics Constitutive equations (linear elasticity) : Local equations of motion (of the beam portion ds) :

22 Flexible beam : deformation dynamics By integrating by part the terms : Final form : for «each» beam i Flexible beam : deformation dynamics Example of computation : recall : for «each» beam i

23 Flexible beam : deformation dynamics Example of computation : Using monomials Analytical integrals Flexible beam : deformation dynamics Example of computation : Interpretation :

24 Flexible beam : deformation dynamics Example of computation : where :!!! Second order terms required in Γ to be consistent with first order kinematics in the VPP)!!! Example : pure bending (=> Γ 1 = 0 (by definition)) First order : Second order : pure bending => = 0!! pure bending => MBS + flexible beams : eq. of motion Joint equations of body i : Rigid : Flexible (beam) : Deformation equation of beam i : Implicit equations of motion of the MBS

25 MBS with flexible beams Flexible beam model Shape functions Beam Kinematics Multibody kinematics (forward) Joint dynamic equations( backward) Deformation equations (beam per beam) Symbolic computation Applications MBS + flexible beams : eq. of motion with Symbolic computation of the tangent matrices M, G, K with ROBOTRAN => recursive (efficient) derivation!!!!

26 MBS + flexible beams : eq. of motion Global computation scheme (Robotran) : MBS + flexible beams : symbolic generation Input file (for flexible beams) Input file (for rigid bodies) (standard file) ROBOTRAN Symbolic model

27 MBS + flexible beams : symbolic generation Symbolic model (example - Matlab) Implicit form : MBS with flexible beams Flexible beam model Shape functions Beam Kinematics Multibody kinematics (forward) Joint dynamic equations( backward) Deformation equations (beam per beam) Symbolic computation Applications

28 MBS + flexible beams : examples A cantilevered L-shaped structure : modal analysis For each beam : FEM : 10 beam elements FSM : 10 interconnected bodies Monomials : 5 shape functions in x, y, θ MBS + flexible beams : examples Kane s benchmark : 2D rotating beam CPU time reduction factor :14

29 MBS + flexible beams : examples Fisette et al. benchmark : 3D rotating beam CPU time reduction factor :38 MBS + flexible beams : examples Jahnke, Popp s benchmark : flexible slider-crank FEM

30 Contents Relative coordinates approach : review MBS with flexible beams : finite segment approach MBS with flexible beams : assumed mode approach Beam Model Symbolic implementation MBS with telescopic beams MBS + telescopic beam : principle Sliding section S, frame {t}

31 MBS + telescopic beam : principle Position constraints : Orientation constraints : Loop closure : pseudo-rotation constraints recall (part 1) 62

32 Loop closure : pseudo-rotation constraints Let s choose a subset of 3 independent constraints (general 3D rotation): 63 Loop closure : pseudo-rotation constraints if all the constraints are satisfied : because : = E => Coordinate partitioning method can be used to reduce the system => ODE 64

33 Loop closure : pseudo-rotation constraints Conclusion 1 : Pseudo-rotation constraints Pseudo-gradient with : 65 MBS + telescopic beam : example A telescopic flexible slider-crank

34 Conclusions Flexible multibody systems in relative coordinates Finite segment method : a «pragmatic» technique MBS with flexible beams Floating frame approach set of «problem-independent» shape functions Beam Model : Timoshenko (Euler-Bernouilli would certainly be a bit more efficient) Symbolic implementation : OK and «fully» in case of monomial shape functions CPU time : very powerfull (but limitation in terms of flexibility modeling) MBS with telescopic beams : closed loop approach symbolic implementation

COPYRIGHTED MATERIAL INTRODUCTION CHAPTER 1

COPYRIGHTED MATERIAL INTRODUCTION CHAPTER 1 CHAPTER 1 INTRODUCTION Modern mechanical and aerospace systems are often very complex and consist of many components interconnected by joints and force elements such as springs, dampers, and actuators.