Approach based on Cartesian coordinates

Transcription

1 GraSMech course Computer-aided analysis of rigid and flexible multibody systems Approach based on Cartesian coordinates Prof. O. Verlinden Faculté polytechnique de Mons GraSMech Multibody 1 Modelling steps Choose the configuration parameters (q) Set up the kinematics: express position, velocity and acceleration (rotational and translational) in terms of q and its first and second time derivatives Express the forces in terms of q, its time derivatives and time t Build the differential equations of motion Numerical treatment of the equations (lesson 5) GraSMech Multibody 2 General principle The motion of each body is expressed independently 3 configuration parameters per body in planar case 6 (at least) configuration parameters per body in spatial case All joints are treated through constraint equations -> numerous but simple (sparse) equations of motion -> differential-algebraic equations of motion -> systematic and easy to program GraSMech Multibody 3 1

2 Some references Computer-aided Kinematics and Dynamics of Mechanical Systems Edward. J. HAUG (University of Iowa, DADS) Allyn and Bacon, 1989 Volume I: Basic methods Volume II: Advanced and interdisciplinary methods Kinematic and Dynamic Simulation of Multibody Systems. The Real-Time Challenge Javier G. de Jalon (University of San Sebastian) Eduardo Bayo (University of Santa Barbara) Springer-Verlag, 1993 Adams theory in a nutshell Dan Negrut, Brett, Harris (University of Michigan) available from GraSMech Multibody 4 Structure of the presentation Formulation in the case of planar systems Kinematics Equations of motion without constraints Constraints associated with typical joints Equations of motion with constraints Formulation in the case of spatial systems Kinematics (rotation parameters!) Constraints Equations of motion GraSMech Multibody 5 Kinematics of a body (planar case) 3 configuration parameters for each body x i, y i and θ i GraSMech Multibody 6 2

3 Configuration parameters The vector of configuration parameters is of dimension 3xn B GraSMech Multibody 7 System without constraints Only applied efforts are exerted (no joint reaction forces), characterized by the resultant force on body i the resultant moment on body i (wrt center of mass) GraSMech Multibody 8 Examples Spring Damper of damping coefficient c GraSMech Multibody 9 3

4 Equations of motion (Newton-Euler) Translation equilibrium (Newton) Rotation equilibrium 3 equations of motion per body GraSMech Multibody 10 Complete set of equations of motion The n cp equations of motion will be Or in general with a constant and diagonal mass matrix (only for 2D) GraSMech Multibody 11 Constraints Revolute joint: defined between point P of body i and point Q of body j Corresponding constraints GraSMech Multibody 12 4

5 Revolute joint P and Q are defined by their local coordinates Generic form of the equations of constraints => 2 natural routines in the program GraSMech Multibody 13 Example: simple pendulum Revolute joint between O (fixed body 0) O of body 1 Constraint equations GraSMech Multibody 14 Example: double pendulum Joint in O Constraints GraSMech Multibody 15 5

6 Example: double pendulum Joint in A Constraints GraSMech Multibody 16 Prismatic (translational) joint Defined by the position of the axis of motion on each body Point P (ξ P/i, η P/i ) and angle α i Point Q (ξ Q/i, η Q/i ) and angle α j 2 constraints Axis on i parallel to axis on j PQ is parallel to the axis (on i or j) GraSMech Multibody 17 Prismatic (translational) joint Parallelism of the axes PQ parallel to axis with n i axis on i GraSMech Multibody 18 6

7 Example: slider-crank mechanism Prismatic joint between slider (body 3) and ground (body 0) Location of the axis on 0 Location of the axis on 3 GraSMech Multibody 19 Derivation of constraints Complete system: n C constraint equations Jacobian matrix of constraints B (n C xn cp ) Time derivative of constraints (if the constraints are scleronoumous) Second time derivative of constraints GraSMech Multibody 20 Initial constraints Revolute joint Time derivative of constraints Or in matrix form => another expected routine in the program GraSMech Multibody 21 7

8 Revolute joint Second time derivative of constraints GraSMech Multibody 22 Prismatic joint Initial constraint equations Time derivative of constraints (!!! only if we assume θ i +α i =θ j +α j ) GraSMech Multibody 23 Assembly of the jacobian matrix Only 2 bodies are concerned by a joint -> sparsity of the matrix GraSMech Multibody 24 8

9 Example: simple pendulum n B =1 n cp =3 n c =2 Dimension of B=2x3 GraSMech Multibody 25 Example: double pendulum n B =2 n cp =6 n c =4 Dimension of B=4x6 Body 1 (x 1,y 1,θ 1 ) Body 2 (x 2,y 2,θ 2 ) Joint (O) Joint (A) GraSMech Multibody 26 Example: slider-crank mechanism n B =3 n cp =9 n c =8 Dimension of B=8x9 GraSMech Multibody 27 9

10 Constraints - summary All the joints lead to a set of n c constraint equations of the form The time derivatives of the constraint equations is of the form with B the jacobian matrix of constraints GraSMech Multibody 28 Equations of motion with constraints The equations of equilibrium become Joint reaction forces? Link with the constraints? GraSMech Multibody 29 Joint reaction forces The virtual power developed by the joint reaction forces Can be written in matrix form GraSMech Multibody 30 10

11 Joint reaction forces The virtual power developed by the joint reaction forces must be null for any admissible motion The virtual velocities are admissible if they verify the constraints The joint reaction forces must be of the form The Lagrange multipliers λ are the amplitudes of the joint reaction forces while B gives the direction (along the constrained motion) GraSMech Multibody 31 Equations of motion with constraints The (n cp +n C ) equations of motion are in the (n cp +n C ) unknowns q (differential ) and λ (algebraic) Set of differential-algebraic equations (DAE) which can only be integrated by particular methods (stability problems) => lesson 5 GraSMech Multibody 32 Alternative forms With constraints at velocity level better conditioned but drift of constraints With constraints at acceleration level usual form but drift still worse GraSMech Multibody 33 11

12 Examples: simple pendulum n cp =3 n c =2 5 eq. of motion Equations of motion with constraints at position level GraSMech Multibody 34 Examples: simple pendulum n cp =3 n c =2 5 eq. of motion Equations of motion with constraints at acceleration level GraSMech Multibody 35 Examples: simple pendulum Equilibrium of the arm So that the joint reaction force in O corresponds to GraSMech Multibody 36 12

13 Examples: slider-crank mechanism n cp =9 n c =8 17 eq. of motion GraSMech Multibody 37 Theorem of virtual power Alternative formalisms for any virtual motion, which leads to with Q the generalized force, obtained by identification from Example: force F on point P of body i GraSMech Multibody 38 Theorem of virtual power In the planar case, we have The equations have exactly the same form GraSMech Multibody 39 13

14 Lagrange s equations Lagrange s equations are written or with L the Lagrangian=T-V, T the kinetic energy and V the potential energy Q(nc) the generalized forces developed by non conservative forces GraSMech Multibody 40 Lagrange s equations If we replace L by T-V, we get GraSMech Multibody 41 Lagrange s equations The kinetic energy is written from which gives The Lagrange s equations become or GraSMech Multibody 42 14

15 Lagrange s equations in planar case The mass matrix is constant The Lagrange s equations become GraSMech Multibody 43 Canonical or Hamilton s form The canonical momenta are defined by The Lagrange s equations can the be written in an equivalent first-order form (2n cp +n c equations) (formulation used in ADAMS) GraSMech Multibody 44 Spatial case The situation of each body i is expressed by with u i the translation parameters of body i And θ i the rotation parameters of body i, for example (but not necessarily) 3 angles φ ι, θ ι and ψ i GraSMech Multibody 45 15

16 Configuration parameters The vector of configuration parameters of the complete system is of dimension 6xn B GraSMech Multibody 46 Translational kinematics The translational kinematics is trivial GraSMech Multibody 47 Choice of rotation parameters Several solutions exist for the rotation Succession of rotations (Euler angles, Bryant angles, ) Euler parameters (4 parameters), quaternions, conformal rotation vector, Rodrigues parameters Problems of rotation Cannot be added Not integrable GraSMech Multibody 48 16

17 Rotation kinematics We need the expression of And for the velocity or GraSMech Multibody 49 Euler angles Sucession of rotations about local Z (precession φ), X (nutation θ) and Z (spin angle ψ) Singularity when θ=0 or θ=π (axis of φ=axis of ψ) Rotation velocity Parameters used in ADAMS! GraSMech Multibody 50 Bryant angles Sucession of rotations about local axes Z (yaw), Y (pitch) and X (roll) Singularity when θ=+/-π/2 (axis of φ=axis of ψ) Rotation velocity GraSMech Multibody 51 17

18 Euler parameters The rotation of an angle φ about an axis specified by the unit vector n canbewritten with e 0, e 1, e 2 and e 3 the Euler parameters linked by the constraint GraSMech Multibody 52 Euler parameters Let us define the 3x4 matrices E and G The kinematics is given by and Parameters used in LMS/DADS! GraSMech Multibody 53 Equilibrium of a body The equilibrium of body i is given by which becomes after projection and after replacing the accelerations or if the rotation term is projected in i (Φ cstt) GraSMech Multibody 54 18

19 Equations of motion without constraints General form with The mass matrix is no longer diagonal nor constant GraSMech Multibody 55 Principle of virtual power Virtual power developed by inertia forces of body i with GraSMech Multibody 56 Principle of virtual power Virtual power developed by applied forces with GraSMech Multibody 57 19

20 Equations of motion By applying We get with GraSMech Multibody 58 Lagrange s equations Lagrange s equations are the same as the ones obtained by TVP From which we deduce GraSMech Multibody 59 Spatial constraints Constraints defined between frames i.a and j.b Generic constraint equations GraSMech Multibody 60 20

21 Example: spherical joint 3 constraint equations (same x,y and z coordinates) GraSMech Multibody 61 Example: prismatic joint along X 5 constraint equations (same X axis and same orientation) ξ=x GraSMech Multibody 62 Example: revolute joint about Z 5 constraint equations (same Z axis and same origin) ξ=z GraSMech Multibody 63 21

22 Other examples Cylindrical joint (4 constraints) Universal joint (4 constraints) GraSMech Multibody 64 Distance constraint Fixed distance between i.a and j.b Composite joint: realized by a bar (with negligible mass) connected to i.a and j.b by spherical joints GraSMech Multibody 65 Other kinds of joints Point on curve (eventually straight) Point on surface (eventually plane) Surface on surface with or without slip (non holonomic) Gears Rack and pinion GraSMech Multibody 66 22

23 Derivation of constraints The derivative of a constraint between frames i.a and j.b can be written Example which leads to GraSMech Multibody 67 From Derivation of constraints as the derivative of the constraint can also be expressed with GraSMech Multibody 68 Link with the jacobian matrix By projecting we get from which we can deduce B by identication with => the matrix B can be built from vectors associated with constraints GraSMech Multibody 69 23

24 Reaction force related to a constraint Reaction efforts related to constraint k The associated virtual power must be null if the constraint is verified => Constraint forces can be treated as external forces GraSMech Multibody 70 Example With constraint we have constraint along X and the corresponding effort is Force along X GraSMech Multibody 71 Equations of motion with constraints We know that The (6xn B +n C ) equations become With Euler angles, we have (8xn B +n C ) equations 7xn B differential equations n B +n C constraint equations GraSMech Multibody 72 24

25 Case of time-dependent constraints Time dependent (rheonomous) constraints are of the form Equations of motion with constraints at position level Equations of motion with constraints at velocity level GraSMech Multibody 73 Case of non holonomic constraints Non holonomic constraints are generally of the form and are not integrable Equations of motion become GraSMech Multibody 74 Conclusions Highly systematic approach => easy to implement (used in ADAMS and DADS) Numerous but sparse equations Differential-algebraic equations with potential integration problems GraSMech Multibody 75 25