FR2662827A1 - METHOD AND APPARATUS FOR ANALYZING A MECHANICAL SYSTEM WITH SEVERAL BODIES - Google Patents

Abstract

The invention relates to the analysis of complex mechanical systems. In a modeling mode, the apparatus provides a model with several bodies corresponding to the studied system, which allows the analysis of complex real machines. A pre-processor (13) facilitates description of the model, and a symbolic generator (15) then generates a specific computer program. In a digital simulation mode, data from the real system is combined with the model program, allowing operation of the device by a user who is not necessarily familiar with modeling techniques, but who is familiar with the modeled system. Application to the design of machines.

Description

The present invention relates to a method of

  to effectively analyze a system with several

  any body, holonomic type, non-holonomic, or not

  piece-wise holonomist, which requires defining the topo-

  The method further includes the introduction of numerical values defining a multi-body system of a specific type. as part of the general type, simulating the operation

  multi-body system, and obtaining

  results of the simulation The process involves

  control and the indication of the acceptability

  physical limitations imposed by the system,

  to the numerical values that were introduced

  your. The invention describes a method for the analysis

  a mechanical system containing mechanical elements

  mutually related, which includes the operations of defining the basic topology of the mechanical system, selecting an inertial fixed reference for the mechanical system, specifying the constraints of the mechanical system and generating symbolic equations of motion for the constrained system, for any system

  holonomic, non-holonomic and non-holo-

  nome in pieces, from the specified constraints

for the system.

  Another aspect of the invention provides a

  apparatus for the analysis of a mechanical system

  mutually linked mechanical elements, which includes a preprocessor, data input means for the introduction of the topology and

  constraints of the mechanical system in the preprocess

  and means associated with the preprocessor to formulate symbolic motion equations for any constrained holonomic, nonholonomic and nonholonomic piecewise system, based on the topology and constraints that have

have been introduced.

  Other features and advantages of the invention will be better understood when reading the

  following description of embodiments,

  given by way of non-limiting examples.

  the description refers to the drawings appended hereto

  which: Figure l A is a representation of a multi-body mechanical system that is analyzed

according to the invention.

  Figure 1B is a representation of the system

  multi-body mechanical system of Figure l A in

a disjoint state.

  Figure 2 is a block diagram of the system

of the invention.

  Figure 3 is a block diagram of the

  pre-processor function in the system described.

  Figure 4 is a diagram of a visualization

  an example of a tree topology that produces

the pre-processor.

  Fig. 5 is a schematic block diagram of the functions performed by the system data module

of the invention.

  Figure 6 is a block diagram of

  functions performed by the postprocessor of the invention.

  tion. Figure 7 is a representation of a

  another multi-body mechanical system which is

lysed by the invention.

  The system described here makes it possible to perform an analysis of movements and forces in systems

  mechanical components consisting of rigid bodies bound together

  such as machines, robots, vehicles and spacecraft The system described here is useful for Kane's method of formulating the equations governing the motion of mechanical systems and the driving forces, as described in the book " Dynamics: Theory and Applications, "Thomas R. Kane and David A. Levinson, McGraw-Hill Book Company Copyright 1985 (which will be referred to below as

the name "Kane").

  Kane's process is associated with

  symbolic algebraic issues that are used in

  computer to generate a description of simula-

  For a given multi-body system, reference may be made to the document: "Symbol Manipulation Computer Program Generating Equations Of Motion Of Mechanical System", Yi-Chung Chang, Ph.D.

  Dissertation, Stanford University Library, 1989.

  We can always separate a mechanical system

  many bodies by breaking links between

  two of the bodies in the system, so that this last

  deny has an open topology or "tree" Figure SA represents a mechanical system with several

  body in which elements or bodies belonging to

  The mechanical system is joined to pivot points 11 and 12. FIG. 1B shows the same system disjoint at pivot points 11 and 12. It is possible to have more than one "shaft" in any system. degrees of freedom between any two adjacent elements or bodies

  in the system, by specifying general coordinates

  and generalized speeds, with regard to

  the relative position and motion between these two bodies The total number of generalized coordinates is equal to that of the generalized velocities The system is therefore holonomic The following will be described the formulations required for the simulation of a multi-body system of holonomic type If the system has been disjoined as shown in Figure 1, for the purpose of obtaining

  to define a "tree" topology, it is possible to impose

  stresses at relative positions or motions between any two bodies in the system that have been separated during the disjunction process, for

  recreate the link When doing this, the vi-

  Generalized tesses are no longer independent and therefore the system becomes non-holonomic. The formulations for simulation will also be described below.

  of a multi-body system of non-holonomic type.

  In addition, by imposing different sets of constraints on a system described by a "tree" topology, a multi-piece system can be described, and the formulations applying to such a system are also contemplated hereinafter. is

  describe here allows you to perform three types of simula-

  depending on the degrees of freedom, holo-

  nome and / or non-holonomic, of the multi-body system.

  These simulations include a kinematic simulation

  that, an inverse kinematic simulation and simula-

  This is also discussed in more detail below. An energy control function is incorporated to provide a high level of confidence in the numerical results obtained by these simulations. This control function will also be described below. next in more detail A multi-body system having a "tree" topology contains p rigid bodies Bi, (i = 1 ,, p), which will be defined as being connected to a previously defined body or to an inertial reference. N As seen in Figure 1B, one

  can use B to represent the inertial reference

  Therefore, N or B is always the

  root of the tree topology We can define the

  by a table j (i) of dimension p, and j (i) is the index of the body to which the body B1 is connected. For example, table j (i) of the multi-body system of FIG. next, when the elements of the multi-body system are designated as shown in Figure 1 B. i 1 2 3 4 5 6 7 8 j O 1 2 3 2 5 6 6 Bl has a mass mi and a sum of dyads of central inertia equal to Ii We can specify the configuration and the movement of Bl with respect to B. by an equal number of generalized coordinates qr and generalized velocities ur After having specified qr and ur for all B, with i ranging from 1 to p, we have a total number of qr 'and therefore ur, which is equal to n We say that the mechanical system has n degrees of freedom We denote by Bt and BJ the centers of mass of B1 and B We denote by n 1, N 2, N 3) and (vil, v 32, v 33) two sets of orthogonal unit vectors fixed in B1 and B, and these two sets coincide with (n1, N2, N3), q ui is a game of

  Orthogonal unit vectors fixed in N (or C).

  The existence of these vectors will be allowed when all the qr will be equal to zero We note that when j is

  equal to zero, B * is a fixed point in B 0, and the vectors

  N unit units fixed in B1 and Bj are equal to the unit vectors fixed in N.

  besides by Bl and B two points which coincide

  instantaneously and who are respectively

  instantaneously fixed on Bl and B, the configura-

  tion and movement of B1 with respect to B are given

  born by the following magnitudes, as a function of q r and u r PJJ is a position vector ranging from B * to B in bases B.

J J J

  pll is a position vector from B to B * in bases B. J 1 i J Ci is a directional cosine vector from (vi, v 2, v 3) to (nl, n I, N 3) i 'Vrel is the relative velocity between Bl and Bj, which is equal to Vi Vj in B bases

  jol is the angular velocity of Bl with respect to

port to B in bases B J i

  If i is zero, then B is not required

  o is instantaneously fixed in N, and its speed is given by: V, which is the speed of B in bases o

N or in the repository.

  Moreover, the relation between ur and qr (that is, the derivative of qr) is given by the following expression, involving the matrices Yr and Z: rs rn ur t Yrs + Z (r = 1 , ns = 1 l There are Fik forces (k = 1 ,, ii) which act on the body B1 at the point P and couples T acting on the body Bi These quantities are defined as follows: Fik is a force expressed in bases B. or bases N pik is a vector of position ranging from B * pik Ti is a pair expressed in bases B. 1- or N. The kinematics involves the position,

  orientation, partial speed, angular speed

  the partial acceleration, acceleration and acceleration of

  gular of each element or body, as well as the expressions of the derivatives with respect to the time of qri in the terms ur and qr (r = 1,, n) These quantities bound to a body B are obtained by summing i the quantities for B and "relative" quantities between B1 and Bj, following the order of the "tree" topology from the root to the branches The following relationships are applicable: Directional cosine matrix of (n 1, N 2, where n 3) to (n 1, N 2, N 3): ci = Oc J here (1) Position of B *: ip = pi + oci pjj + oci pii (2) Partial angular velocity of the body B: o ci C = J + 1 (r = 1, n) (3) rr Wr in bases B; Ut = i W t + t <(4) in Bi bases; wi = i Ci cj + jwi (5) in bases B Partial velocity of Bl: If J 0, we have: Vi = (Vr) N + i (Vj) 3 x Plrr Nr + co x pii + vrel ( 6) rr (V) N in bases N or B Vr) N 1 T = (V) + Ci E (i + Wl 'c P 3 Vt = (Vt) N + CjA l (Vt) L t PJJ i + i rel + CJ x Pii + Vt (7) tt Vr = (Vr) + (Vr) L respectively in rbN and L 'bases N and B in Vi = Vi + oci (Wi x PJJ) + Ci ( Then, if j = 0, we have: V = V + Ci x pii + vrel rrrr Vr in N bases or B r 1 bases in in tttti = VO oi ii rel Vt V + C to P + V Vi = V + ci To (O Coi x pii + vrel) bases N. Angular acceleration of Bl: di = icj jj + i + jc i + id W

Bi bases, without contribution of 6 r.

  Total angular acceleration: n T + "i Jr * Ur r = 1 in bases B: 1 (8) (9) (10) (11) (12) (13) Acceleration of B *: If j O o ai = (a ') N + here (a J) L + Wl x (x P 33 JJ) + Ojjx PP J +) ix (ix pii) + xp id + 2 ix Vrel + d (Vrel) (14) dt (ai ) N in bases N, or bases Bi; N d () rel is the derivative with respect to the time of Vrel dt (in bases Bi, without contribution of 6 r, which is the

  derived from ur with respect to time.

  Total acceleration: i = airna T a + L Vir * r (15) r = 1 in bases N. If j = O o: ai Nd o '; ii) dt (V) + x (1 x P) iel d (rel) + ix pii + 22 i rx V tel (l (16) qr is expressed as a function of ur and qr using: lW rsl = Yrsl (17) {Xr} = lWrsl {Z (18) and n qr 7 rs + s Xr (19) s = i We can write the dynamic equations in the following form: lMrsl {Ur} = R (20) In the above, lMrsl is a matrix of dimensions N xn, {r} is a matrix of dimensions N x 1 containing 6 r, which is the derivative of ur by

  compared to time, and {Rr J is a dimensional matrix

  sions N x 1 We use to build M and R the kinematic quantities we have just done

reference.

  M is the mass matrix of the system.

  vii i i i i M = m g Vl (& <21) Mrs = mi r + Vs + r s) l (21) with the following notations: m is the mass of the body B; Vr, Vi are given by equations (6) or (9); Ii is the sum of the inertial dyads of body B, that is to say: ## EQU1 ## + I 31 (n 3 N 1 + N 1 N 3) + I 12 (nN 2 + NN (22) ir O Ci are given by equation (3) R is the right-hand side of the dynamic equations

  First, the resulting force is determined

  te F and the resulting moment T with respect to the center of mass of all the forces acting on each Bi body, using: ui ik Fi = Fik (23) k = 1 in N bases or Bi bases, Ti = T + Pik x F (24) k = 1 in N bases or B bases One builds then the moment of inertia, without contribution from 6 r, of each body B. T = Ii Xi + Cui (i wi) (25) I in bases B. Ii is the sum of the inertial dyads of B. (equation (22)); i is given by equation (12), wi is given by equation (12), i west given by equation (5) We obtain Rr by the relation Vli R = V r (mi a) + F i i = l + r * '(T + T) 1 (27) where Vr is given by equation (6) or (9) r M is the mass of the body B. i 1 ia is given by the equation (4) or (16) F is given by equation (23) iwr is given by equation (3) r Ti is given by equation (24) T * is given is by equation (25) ) The resultant force and the resulting moment with respect to a point, fixed in B or B (the body to which B is connected according to the topology in "tree"), that B must be the force b on the body B. J exercise on Bi, in addition to strengths and couples

  given assets, to produce the solutions for movement

  The problem is posed in the following way: If the system motions, all forces and active couples, a body Bij and a point P1 set to either B or Bj, and expressed by a ci J position vector P ranging from B * or B * to the c 1 l point of interest in the fixed base of the body, are

  all given, it remains to find Fci, that is to say the

  this constraint that B exerts on B in these bases or references, and Tci that is to say the couple of contrait

  that B exerts on B in these bases or references.

  To solve this problem, we perform the following procedures: 1 We determine all the bodies, including Bi, which are on the branches of the tree of Bi There is a total number of NTR bodies whose numbers are recorded in the table ITR (k), with

k = 1,, NTR.

  NTR 2 Fi = Z (ma T-F) ITR (k) (30) c k = 1 in N bases, with the following notations: the index ITR (k) applies to m, a T and F; m ITR (k) is the mass of the body BL TR (k) a Test obtained by equation (15); FITR (k) is obtained by equation (23), solved in N bases.

i o J-

  3 If P is fixed on Bi, the quantity Pc, that is to say the position of pi is: Coiioi Op = p + C (28) cc in bases N, P given by the equation (2), while Ci is given by equation (1), or if P is fixed on Bj, we have: cj Opi = pi + CP (29) cc (29) cc

  4 T * is determined for all the spe-

  cified in the table ITR (k): (T *) i = IZT + Xi x (II J) (31) in bases Bi, with the following notations: I is the sum of the diads of inertia, given by the equation (24), i T is given by equation (13); i

wi is given by equation (5).

  NTR T = T-T + (P-i P) x (ma T-F) lTR (k) C k = 1 c T ITR (k)

(32)

  in N bases, with the following notations: the index ITR (k) applies to T *, T, P, m,

T ''

  a T and F (T) ITR (k) is given by equation (31) TITR (k) is given by equation (24)

  PITR (k) is given by equation (2).

  Once a system has been defined

  If a body with a topology in "tree", one can impose constraints on the system, to produce a system with several bodies having constraints of configuration or movement One defines in the way

  following configuration and movement constraints

  It should be noted that these two types of constraints

  relate to constraints applying to the configuration of

  relative ration or relative motion between two

  body We will consider later a type of

  more general training that can be used to

  to draw the movements of a system We define a

  configuration constraint by the following quantities

  your. Two bodies B1 and B. Two position vectors appearing as vectors G H vectors P i Pk respectively in bases B

  and B P is the position vector of B jus-

  k H at a point G, and Pk is the position vector of B * up to a point H These two vectors are

  functions of qr and t, or constants.

  A unit vector b fixed in B1 or Bk 1 expressed

in bases Bl or Bk.

Constants.

  A function of the time f V defined by the following relation: f A (NPG _ NPH b (33) in which f T is or can be a constant, P and P are respective positions of G and H in N. We can define a constraint of movement by the following quantities: Two bodies B1 and Bk ik GHG Two vectors of position P, PP goes from Bt to a point G, and it can be decomposed into two components

  P 'Get Pi, who are respectively

  in B bases and bi bases Bi is a

  body in the system other than Bi.

  p G = p + G (34) p ii Similarly:

H H H

  = k kk (35) A unit vector b in bases of a body

  selected, for example Bi bases.

Constant components.

  A function of the time f T 'defined by the relation: f T (V V) b (36) in which VG and VH are speeds of the respective points G and H, assuming that they are instantaneously fixed respectively in B.

and Bk-

  A general constraint can be specified by defining coefficients A (r = 1,, n) and B, r as functions of time and qr of the following non-holonomic constraint equation: n- z Ar Ur + B = O (37) r = 1

  When imposing all the constraints

  cedentes, which amount to m equations of stress, the system has p degrees of freedom, with: p = N m (38) The purpose of the previous kinematic calculation is

  to generate a constraint equation of the form

  generated by equation (37) for each of the configuration or motion constraints and, by differentiations, to generate the associated constraint equation for r, in order to generate quantities D such that: n rl Ar 6 r + D = O (39) r = the quantity D which can be expressed as follows: n D (40 u) D = r 1 rr U rr (40) By combining these equations for all the constraints, we can put under a matrix format the global constraint equations: lAerl {Ur + {Be} = O (41) lAerl {6 r} + {De = O (42) In equations (41) and (42), lAerl is a matrix of dimensions mxn, {ur} and Pur} are matrices of dimensions N x 1, containing ur and r, and lBel and

  lDel are matrices of dimensions m x 1

  configuration trainings are explicitly expressed as a function of qr and t, to facilitate the determination of a set of qr and t satisfying

  these constraint equations of configuration.

  ## EQU1 ##

  Ci o Ck given by equation (1).

  Equation (3) is then applied to form

  sea of constraint equations of configuration, expressed explicitly according to qr and t: g (qrt) = O (45)

  Conquest equations are then determined.

  associated motion train, in the form of equations (37) and (39): VG = V + Xi ip G + ld (p G) J (46) in bases N or bases B i Vr is given by equation (6) or (9), where is given by equation (3), id G ld (Pi) lr is the third partial speed of id GG dti P) which is the derivative with respect to time of Pl in the reference Bl in bases Bi (equation (19), etc). VG = Vt + Gt id (47) in bases N or bases B i V is given by equation (7), ti ctest given by equation (4), id G ld (PG) lt is the rest velocity of dt (Pi) H kk H lkd H v = V + & r x Pk + Ldt (P) Ir (48) r = rrk xt <4 H kktk + VG V i xp G Gd P) (50) VtVt + at x P + -d (P) t <50) V = v + C Ai X dt i + Vi in N bases or Bi, is given by equation (8) or (11), given by the equation (5), and Hk k H kd Hp) (1 VH = Vk x + dk x P k + d (51) G = ai + ix G + i lwix PG + 2d (PG) lj aa + XX i 2 dt i id 2 + d (p G) (52) dt 2 ia is expressed in N bases or bases B i a is given by equation (14) or (16), i is given by equation (12), given by equation (12), it is given by equation (5), and we also have: H kk H kxrwkx kd HP 1) a = a + tx Pk + Hk k P HkdH) l + (k Ix l Pk + 2 (Pk) kd (Pl Hdtk (53) + dd (Pk) (3 We take b = i if b is fixed in Bi, and b = k if b is fixed in Bk, we have then: Lr = b lVG VH -bx ( Np G _Np H 54) Vr is given by equation 46), v Gest given by equation (46), r VH is given by equation (47), rwr is given by equation (3) , r Np G, p are given by equations (43)

and (44).

  GV H b Np G Np HB = b Vt Vt bx (Np Np H) dt tt dt (T) (55) D = b the G a H 2 W bx (VG V) bx (PG _ p H) wb x ( wbx (Np G-Np H)) ld 2 dt 2 (f T) (56)

  These equations are the equations of

  in the form of equations (37) and (39).

  We will describe below how one obtains a set of values of qr and t which satisfies all the constraints of configuration If one denotes by L the total number of constraints of configuration, one can combine all the constraints of configuration to obtain: ge (qrt ) = O (e = 1,, L) (57) lAerl {ur + Be = O (r = 1,, n) (58)

  To show the use of Newton's method

  Raphson to solve the system of non-linear equations

  corresponding to equations (57), it is assumed that the first L qr are unknowns whose values must be determined, while the remaining qrs

  are fixed known quantities.

  The following is a first matrix (59) in which lYrsl is a matrix of dimensions n x N which is defined in the relation relative to the first column (59).

  solution for ur given previously With the equa-

  tions (57), (59) and an initial assumption concerning

  the value of qr (r = 1,, n), we can find the

  lution relative to qr (r = 1,, L) by the method of Newton-Raphson. The computation of the motion stress equations is as follows: G ii GV = V + oi Jx p G (60) rrr in bases N or bases B. ii Vi is given by equation (6) or (9) , Cr is given by equation (3), G p G is given by equation (2) in bases Bi, and v G ji o Gi p G (61) Vt = Vt + tx (61) H Vk + (> kp H (62) Vr = Vr + Cor xp H (62) rrr VH = Vk + cuk xp H (63) Vt = V ttx VG = i + xp G (64) with Vi in N bases or of Bi, Vi is given by equation (8), Xi is given by equation (5), and H = k + kxp HV = V + W x P (65) G Nd G a = (V) = ai + Cg + PG + d (p G + ii) G i + d (PG) 1 + "xp G + dt x PG ii (66) with ai in N bases or Bi bases, Nd G -d (VG) is the derivative with respect to the time dt of V, which is the velocity of G, assuming G fixed instantaneously, in B1 or in N, ii a is given by equation (14) or (16) ui W ii are given by equation (5), GGP Pii are given by equation (34), d (p Gi) are derivatives with respect to the time lapse of P in bases B. i I times of ii dt (p G dt ii) are derivatives with respect to P in bases Bii, i is given by equation (12 ), and z (is given by equation (12), and H Nd H a = dt (V) kkkx H kd a + W xlx + H (P) kk Hkkd p Hkp H + C x Pkk + dt (Pkk) lxzx P (67) A = b lVG VH (68) b is given by equation (36), Vr is given by equation (60), r VH is given by equation (62), r B = bl VVH f T (69) D = b the G a H bx VG V v H) -d (70) dt (f T) a, a are given by equations (66) and

(67),

  b 6 b is given by equation (5), G v H V, V are given by equations (64) and

(65).

  Since in this case A and B are given in the form of functions of qr and t, in the case of the general constraint it suffices to determine D by the relation: nd D = N d (Ar) u ur dt ( ) (71) r = lt r where d ND ArAr -t (A r) = 51 <a qsqs t (72) dt (r) qs qsa + ts = ld NB _ B (73) dt (B) = S ts = l Dynamic equations are the combination of holonomic dynamical equations, that is, those for the tree-topology system before constraints are imposed, with the addition of the multiplier contribution. of Lagrange X e which is associated with each constraint, and of the totality of the constraint equations We can rewrite these

  equations (20), with the contribution of

  Ae and equations (41) and (44), in the form IMI {6} = JR} + Al (74) 1Al {u J + {B i = O (75) lAl l + {D = O (76) In equations (74), (75) and (76), M is a matrix of dimensions N xn, R is a matrix of dimensions N x 1, X is a matrix of dimensions m x1, A is a matrix of dimensions mxn, and B and D are matrices of dimensions mx 1 It should be noted at

  again that N represents holonic degrees of freedom

  mes of the mechanical system, and that m is the total number of constraints imposed on the system The number of non-holonomic degrees of freedom p is therefore: p = n-m,

as shown in equation (38).

  To simplify the explanations, we suppose that the first P ur, designated by ui, are chosen

  in order to be independent generalized speeds

  are, and that the remaining ones, are denoted by u D and are dependent generalized velocities. We can thus decompose the matrix A in two parts, AI and AD, and we can write the equations (75) and (76) under the following form: lA 1 AD 1 Or} + {BJ = O (77) fu lAI AD 1u + Dj = O (78) Similarly, we can decompose M and R, and we can rewrite the equation (74) of the following way:

T M T + {") (79)

  The problem can be posed as follows

  If M, R, A, B, D and u are given, we can

  determine u D, 1 I (to be integrated), 6 D (to be used alone)

  to calculate accelerations), and A The following explanation of the resolution of this mathematical problem will be divided into two categories:

  The first category concerns a dynamic case, ie a case in which p is not equal to

  The second category concerns a cinematic case

  that in which p is zero In the case of dyna-

  in which p does not equal zero,

  truce in the following way the matrices for Ac, Bc and D c

A = 1 A (80)

c D I -1 Bc = -AD B (81)

D = -A D (82)

  dA is a matrix of dimensions m xp and B and D are c CC matrices of dimensions mx 1 We then obtain u D by the relation: {u D 3 = Al 1 {u I + Bi} (83) We construct following the matrices for M (pxp) and R (px 1):

C C

M + M + M A + (MDA) AT M A

  c I ID c (MID A c D Ac (84) c I + Ac RD - (MID + Ac MD) D (85)

CC C D I

  Then 6 i is determined by the relation: I ûA} = 1Mel {ROR (86) One determines D by the relation: UDJ = lA% lt J} + {From (87) One determines by the relation: j AJ = A- T L-RD + (MID + MDA) + MD) (88)

D.} ID + M-D + (MZD

  We note that equation (86) represents the equations

  dynamics of the non-holonomic system.

  In the kinematic category, where p is zero, A is a dimension matrix

  N x n, and all urs are dependent variables.

  We determine u and by the relations: 1-1 {us = -A -B (89)

 3 = -A D (90)

  A is determined by the relation: A = AT 1 -R + M 1 (91)

  Not all r will be integrated, but will be used

  only to obtain the accelerating information

  tion. The forces of stress for a non-holonomic system correspond to two types of forces or couples. The first type comprises the force that is necessary to maintain each constraint imposed on the

  topology system in "tree" described above.

  These forces are related to Lagrange multipliers.

  The second type of stress force is the force or the couple of stress that exists between two bodies

  adjacent to the "tree" topology We must, however,

  to modify these forces and these couples to include

  the contribution of the first type of forces or

  In the case of a force of con-

  stresses resulting from stress equations, no additional computation is necessary It is only the way in which the Lagrange multiplier, obtained by the equations (88) or (91), is interpreted in the form of forces of interaction between the two bodies whose relative motion is subject to constraints via stress equations This physical interpretation is possible only for configuration and motion constraints, but is not possible for a general constraint. For a configuration constraint, we can interpret the Lagrange multiplier A in the following way: if b is fixed in Bi, A is the number corresponding to the measurement of the force that Bk exerts on B1 at the point H, in the direction b If at

  where b is set in Bk, A is the number corresponding to

  weighing to the extent of the force that Bk exerts on Bl at the point G, in the direction b For a constraint

  of movement, the Lagrange multiplier is

  preted as being the number corresponding to the measure of the force that Bk exerts on Bl at point G in the

* direction b, and À is interpreted as the name

  corresponding to the measure of the force that Bl exerts

  this on Bk at the point G in the direction b.

  The resolution procedures indicated

  previously for restraint forces are valid, except that it is necessary to modify

  F and T before applying equations (30) and (32).

  i The purpose of this amendment is to take into account the contribution of the Lagrange multiplier to Comme

  previously indicated, the assessment of the strength of con-

  between B and B is not possible if the system

  the multi-body test that is analyzed contains con-

  general stresses Each configuration constraint must therefore be modified as follows: If b is fixed in B: F 4 F + A b (92) 1 i Fk Fk b (93) TT + (P H_-P) x (Ab ) (94) i Tk Tk + Pk x (Xb) (95) If best fixed in Bk: FF + A b (96)

i 1.

  Fk Fk b (97) Ti T + P? x (b) (98) ik i N) x (-A (i Tk Tk + (NP G_pk) x (-b) (99) kk (9 For each motion constraint, it is necessary to make the following modifications Fi P Fi + X b (100) Fk Fk At b (101) TAT + P x (At b) (102) i Tk T + p H x (-b) (103)

  Up to this point three

  to impose constraints on a multi-body mechanical system having a "tree" topology, to obtain a non-holonomic multi-body system,

  of a simple type A system can be defined

  body by piece, imposing in turn a part or a "subset" of total constraints

  to the system having the topology in "tree", and in

  thus blending different non-holonomic systems.

  Because these non-holonomic systems are established

  through the application of different subsets of equa-

  In the case of the same "tree" topology system, the term "multi-piece system" is used to denote all of these different systems. This set of systems can be grouped together in a single simulation program for analysis If we have already determined kinematic and dynamic characteristics for the multi-body system with "tree" topology, and if we have built Ar,

  B or D for each constraint (respectively

  figuration, movement or general), it is then sufficient to gather a subset of matrices Ar, B and D in accordance with which constraints must be imposed, and to form matrices lAerl, Be} and {De}, defined in the equations (41) and (42), for

  the interval or "piece" of non-holonomic time con-

  We can then apply the resolution equations (74) to (91) to any non-holonomic piece. Two additional pieces of information are needed to perform the simulation of a multi-piece mechanical system.

  first element is the way in which the system

  The second piece of information needed is the action the system can take to determine the piece to be simulated at the present moment. This involves the determination of the initial piece to be used, and the criteria for moving from one piece to another. of analysis must accomplish at this transition, whether the configuration or the movement of the system just before

  transition does not meet the constraints of

  figuration or movement after the passage to the new

  piece for simulation In the simulation system

  these two pieces of information are provided by

  the person who models the mechanical system.

  The energy control function is implemented in the following way: Gr 1 Qr S, Z, Kz and Ko are defined as follows: Gr = X lvr Fi + 03 Ii (r = l ,, n) ( 104) r 1 = 1 r Vr is given by equations (6) where (9) Fi is given by equation (23), Cr is given by equation (3), Ti is given by the equation ( 24) 0 if m = 0 3 m ('y 1Ar * if m OO (105) A is given by equation (41) sr X is given by equation (88) or (91) P i N dii Nd i S = J lmi V t (Vt) + (t) li = 1

(106)

  m is the mass of the body B. V is given by equations (8) or (11), Nd i d (Vt) is the derivative with respect to time, in N, of Vi given by equations (7) or

(10),

  i W i is given by equation (5), Ii is the sum of the inertia dyads of Bi, given by equation (22), Nd i dt (t) is the derivative with respect to time, i in N, of Co t, given by equation (4), n Z = _ (Gr + Qr) U r + S (107) r = 1 K 2 = {UT EMl {ur (108) ur is the generalized velocity ,

  M is the holonomic mass matrix, giving

  born by the equation (21), 1 Ai i +, i I i Ko mi Vt-Vt + "Jt-Ittl (109) i = l mi is the mass of the body Bi, Vt is given by the equations (7 ) or (10), Ii is the sum of the inertia dyads of Bi,

  Wt is given by equation (14).

  The energy control function E is defined as follows: E = Z + K 2 Ko (110) In equation (110), Z is the integral of Z which is given by equation (107) The integration of Z must be accompanied by the integration of Or (r = l, r

  , n-m) and qr (r = l ,, n) If E does not remain constant

  so, the results of the simulation are incorrect.

  If E remains constant, this does not guarantee the validity of the simulation, but it is considered to be a

presumption of validity.

  The kinematic simulation indicates that no generalized coordinates qr or generalized velocities ur have been introduced into the definition of the multi-body mechanical system with "tree" topology. In other words, N = zero In this case, only Z in

  equation (107) will be integrated in the function of

  computer energy control, E It should be noted that the quantities in equation (108) and the summation n which is defined by (Gr + Qr) ur in the equation r = 1 (107) are equal to zero more, because V = i

  Vt, the energy control function E which is defined

  denoted by equation (110) becomes: i Nd i i i Nd i E = | Lmi v (Vi) + OI d (d) l dt L = 1 1 i P ii _ 2 lmi Vi v + i To Oil (112) L = 1 The inverse kinematic simulation indicates that the total number of qr that are introduced into the topology mechanical system in "tree" is equal to

  total number of constraints In other words, N = m.

  The system therefore has zero degrees of freedom, that is, p is zero In such a case, only r (r = 1,, n), which is provided by equation (16) and

  Z that is provided by equation (107) will be integrated.

  ur and (r = 1,, n) are obtained using res-

  equations (89) and (90), and \ r is obtained by the equation (91) The movements of the bodies are obtained by the equations (1), (2), (5), (8), (11) ), (13), and (15) at each time step, and the quantities Fi and T (i = 1 v, p) will be calculated using equations (23) and (24), and equations (92) to (103), to facilitate the assessment of the

  constraint, as previously described.

  Dynamic simulation of a mechanical system

  that to multiple bodies means that after imposing constraints, if any, the multi-body system has a number of degrees of freedom different from

  zero, i.e., p = N m is greater than zero.

  We can divide the dynamic simulation into two categories

  gories: (1) a tree topology system (m = 0);

  and (2) a multi-body system with constraints

are (m different from zero).

  In both cases, the quantity qr indicated in equation (19) and the quantity Z indicated in equation (107) will be integrated. The movements of bodies are calculated using equations (1),

  (2), (5), (8), (11), (13) and (18) at each time step

  In addition to these calculations, the following quantities will also be calculated: In the case of a system with

  topology in "tree" (m = zero), we will solve the equation

  (20) vis-à-vis, and we will integrate to get ur (r = 1,, n) We calculate F and Ti (i = 1,

  p) using equations (23) and (24) with

  than temporary printing, to facilitate the evaluation

  Evaluation of Stress Forces Considered Previously In the case of a system with stresses (m is not equal to zero), the quantity I given by equation (86) is integrated to obtain u I * The quantity u D which is given by the equation (83), the quantity D which is given by the equation (87) and the quantity To which is given by the equation (88),

  will also be evaluated at each time step of

  In addition, the quantities Fi and Ti (i = 1, P) will be calculated by using equations (23), (24) and equations (92) to (103), to facilitate

  the assessment of the forces of restraint that we have

previously.

  As can be seen in Figure 2, the

  implementation of this system begins with an operation

  information model that is carried out by a modeller, who is defined as a person with a relatively high level of competence in the field of characteristics and analysis of

  mechanical systems Information entry operations

  Mation are performed by means of a keyboard or other input device 14, and the information that is introduced is applied to a preprocessor 13 The output of the preprocessor is connected to a generator

  symbolic 15 which acts as a transformation

  to generate new information The general

  symbolic driver generates some files that appear

  In the form of source codes and documentation for the simulation, the output signal of the module 16 of FIG. 2 is transmitted by media such as disks, tapes and printed documents to a text editor and compiler. , 17 Other input information can be introduced by the same or another keyboard 14, and by the modeler or a user who is familiar with the specific mechanical systems that are being analyzed, but who does not necessarily have the high level knowledge of the mechanical systems possessed by the modeler It should be noted that the output information, on a medium, that the simulation source code module 16 provides, is autonomous and portable, and can be used on site or elsewhere by a user or a modeller, as desired and convenient The output information of the text editor and compiler 17 is transmitted to a system data module 18, along with the i user input information that comes from the keyboard 14 The system data module accepts

  symbols, initial conditions and parameters

  very time-sensitive for simulation Files that are generated by the system module are input to the simulator 19 Files are provided to the

  executable code 20 for the simulation, so as to

  to visualize the real simulation of a multi-body mechanical system Simulation result data

  are transmitted to a post-processor 21, which is

  keyed to the user information input keyboard

  14 and the system data module 18 The post-

  processor gives a user the ability to manipulate simulation result data, for

  generate files containing selective quantities

  22 in a user-specified format, xy data for a plot operation, at 23, and animated simulation results at 24.

  files, plots and animation, respectively bearing

  references 22, 23 and 24, may be

  in accordance with the requirements of the user.

  For convenience, the invention has been implemented

  than now using C-language input information for the symbolic generator 15, and

  information in Fortran language at the end of this gen-

  Other languages may be used, although the best mode known at present is that which is described.

  It is necessary to define several ter-

  to facilitate the understanding of this presentation

  symbol is defined as a string of characters

  res containing not more than 20 characters, in this mode of

  realization, and starting with a character of the alpha-

  bet or by a numeric value Any string starts

  T is a reserved string for a symbol.

  A time function is a function of time only

  the name of a temporal function begins

  by the characters TT, followed by not more than 15 characters

  res A function Q is a function of time, generalized coordinates qi and position or

  body orientation The name of a Q function

  Mence by TQ, followed by not more than 15 characters A sensor function is a function of time, generalized coordinates qi, generalized velocity ui, position or orientation of body,

  speed or angular velocities of the body, or of

  very large quantities of sensors which are defined

  reads a sensor function to specify quantities

  Kinematic Tenses Required for Determining Command Forces The name of a sensor function starts with the characters TS, followed by at most 15 characters. A force function is a function of time, generalized coordinates qi and generalized velocities ui On uses a force function to calculate forces or torques that are applied to the multi-body system during simulation The name of a force function starts with TF, followed by up to 15 characters A user-defined function is a any function among a time function, a function Q, a function of

  sensor or a force function, which may be

  set as a user-defined function if it is not explicitly defined in the preprocessor 13D or DD are added at the end of the chain of a user-defined function, for designate the first or second derivatives of the

function with respect to time.

  The system receives input data in a symbolic or digital format, from a keyboard or terminal 14, or from data files that

  previously created The input data is divided into

  four categories: (1) definitional data;

  multi-body system; (2) simulation data; (3) body shape data; and (4) Expressions for post-treatment A multi-body system includes bodies, joints, stresses, drives, forces, sensors, and piece information.

  this information is not applicable to each system

  Figure 1, previously described, is an example of a multi-body system that is analyzed by the system and method of analysis described herein. As previously described, a modeler may first sever the system. to multiple bodies, to define bodies such that the system has a topology in "tree" The modeler

  subsequently introduced the "loop" or the constraints

  of movement When the system is in the topo-

  In the case of a disjointed "tree", it must be brought into a reference configuration (all qi equal to zero), and the three orthogonal unit vectors fixed to the body, on each body, must be defined.

  so that in the reference configuration, always

  the bases coincide with each other At the moment of definition, a body must be connected to another body by a joint The system provides a library of joints, including the following types of joints: square; free (attached to 6 of freedom);

  cylinder; plan; prismatic; revolution; screw; spherical

  than; universal and defined by the user.

  create a body, a user must provide the information

  following: body name; mass; inertia; body

  located on the inside (the name of the body or

  multi-body definition defined previously); seal; internal joint relation (position vector from center of mass of inner body to joint in bases of inner body, three symbols); joint-body relation (vector of position going from the joint to the center of mass of the body in bases of the body, three symbols) A modeller can introduce the constraint to define a topology in "loop",

  and to impose movement constraints on the system

  There are five types of constraints: position;

  speed; gear; cam; and defined by the user.

  To define a constraint other than the user-defined constraint, the user must provide the following information: name; type; first name

  body; name of the second body and parameters as men-

  previously, the modeler must have sufficed

  knowledgeable about how

  holonomized generalized tesses must be defined

  for the analysis system that is described.

  must provide the name of the constraint, the type of

  the constraint and functions of qt.

  A drive is used to

  describe the time history of a general speed

  Therefore, a modeler must be familiar with the way in which generalized speeds are defined and controlled for

  a joint A modeler will provide the following information

  te for a drive member: name; name of the joint; number of degrees of freedom; and function The system allows to have a library of types of force Six types of force are currently included

  In the library For example, a compelling force

  requires the name of the first body, the name of the second body, the vector of position from the center of mass of the first body to an intermediate point, in bases of the first body, and therefore the point

  in the bases of the second corps (six sym-

  boles), the position vector from the center of mass of the second body to an intermediate point in the bases of the second body, and therefore the point of application in bases of the first body (six

  symbols), the name of the force function and the direc-

  force on the second body, in bases

  of the first body or the second body (three symbols).

  To define a force, a user will specify the name of the force, the type of force, and the parameters

  such as those listed here for the control force.

  The purpose of a sensor is to provide kinematic quantities, with a defined scale, which are necessary for the evaluation of the torque and the

  These quantities include the posi-

  and the speed between two points fixed on two

  different bodies, and the orientation and speed

  between the two bodies Parameters provided by the modeler for the calculation of these quantities are necessary for each type of sensor.

  a translation sensor and a rotation sensor.

  To specify a sensor, a modeler must provide

  the name of the sensor, the type of the sensor, the name of the

  first body, second body name and parameters

of the sensor.

  As previously discussed, to specify a piecewise system, a modeler must provide a total number of "pieces" in the system. For each piece, the system asks the modeler to provide a subset of constraints that are

  defined in the multi-body system The criteria

  for the transition between these pieces and those

  around how to make the transition, will be provided by the symbolic processor, to generate

  expressions for variables and parameters

necessary for transitions.

  The simulation data includes

  digital resolution settings, sub-programs

  my and initial conditions that are defined by

  the user, and time parameters

  numeric resolution dice are needed to solve linear equations, equations

  ordinary differentials and algebraic equations

  non-linear models, from geometric constraints

  The necessary parameters that a modeler must

  specify for ordinary differential equations

  res include the size of the minimum integration step

  male, a tolerance, a fixed step size or a

  variable pitch size Parameters for equa-

  non-linear algebraic

  maximum of iterations and tolerance on that number.

  In the data definition for a system with several

  In the case of a body, a modeler can specify function names, instead of explicit functions, to describe constraints, drives, forces, and sensors. These functions are then calculated during simulation by calling a

  subroutine defined by the modeler This sub-pro-

  gram is generated by the symbolic generator 15, and all the functions are set to zero A modeller must modify this subroutine to implement the functions Finally, a user or a modeller will have to specify initial conditions for q and u.

  The user will have at his disposal a system of

  solution of equations, to verify that the initial q

  that have been specified satisfy the geometric constraints

  which are defined in the data

  multi-body system The initial time, the final time, and the print step size for

  simulation are also specified by a user.

  In order to pre-visualize a system with several

  body before simulation, and animate the simulated movement, after simulation, a user

  must define a geometric shape for each body.

  The user will provide the shape in a data file The origin of each body is at the center of mass, and the repository databases follow the one used in the reference configuration, at the time of the definition of the multi-body system For the convenience of the user, the system

  provides a library of simple geometric shapes

  ples, comprising elements of the parallelepiped type, sphere, cylinder, cone, etc. For post-processing simulation data results, a user can perform mathematical operations on the results before

  As an example, instructions for

  Post-processor 21 provides position, velocity, acceleration values, vector quantities and stress-related quantities, etc. The information that generates the system that is described includes system definition data to

  Several bodies were provided with sufficient explanations

  all the data The pre-trial

  13 can also read this data on demand

  of the modeler An additional explanation will be presented

  of generalized coordinates q and generalized velocities u It should be noted that if all q are

  equal to zero, the multi-body system pre-

  the input configuration is also specified to the system data module 18 This input information contains a list of symbols whose values are to be assigned, and a list of function names which are defined by the user Moreover, the input information that is

  applied to the symbolic generator 15 is defined.

  Certain subprogrammes which form part of the

  mes of the system are incorporated in order to find the position and the orientation of each body for a given moment and a given parameter q, to calculate the sum of the squares of the residues of each geometrical constraint for a given instant and a parameter q

  given, to calculate the gradient of the sum of the car-

  res residues of each geometric constraint, by

  compared to parameters q selected by the user

  for a given moment and a given parameter q,

  and to calculate functions defined by the user

  related to the position and orientation of the body as well as geometric constraints The input data files 19 of FIG.

  numerical values for all symbols, para-

  meters of digital resolution, the initial conditions

  for the simulation and the temporal parameters, a table of correspondence between defined symbols

  programs and symbols specified by the user.

  a topology of the multi-body system before

  the application of constraints, and an identification number

  provided by the user for the purpose of identifying

  Proud simulation cases Simulation results

  which are obtained by the executable code for the

  simulation, 20, include general coordinates

  q, generalized velocities u, derivatives of u with respect to time, Lagrange multipliers, a center of mass position, a speed and an acceleration of each body, an orientation, an angular velocity and an angular acceleration of each body, the sum of all active forces

  on each body, including the contribution of the

  Lagrange tiplicators, and defined functions

  by the user The data files 22 include

  the time history of user specified quantities A data file for the animation

  is incorporated, together with a graphic representation

  of the multi-body system, to enable

* a user to view the system before the simulation

  lation, and to visualize the way in which the system configuration changes when the values of q are changed In addition, this helps the user to select values of q, hypothetically

  initial, for solving the equations of con-

  geometric type trainte Graphic means incorporated in the postprocessor 21 are used for visualization.

  The system itself controls all data

  modelers and users who are introduced

  in the system The symbols that are introduced in the preprocessor 13 are checked to verify the respect of the conventions This check concerns words which are reserved for symbols, on the dual use of names for the elements of the multi-body system, on the convention concerning function names, and all the conventions described

  here for the elements of the multi-body system.

  The numbers that are entered in the preprocessor 13 and the numerical values for symbols that a modeller or user enters into the system data module are controlled according to the following rules: 1 The body mass must be greater than or equal to

equal to zero.

  2 Moments of inertia of the body must

  be greater than or equal to zero.

  3 The sum of two moments of inertia

  conch of a body must be greater than or equal to

third.

  4 Moments of inertia crossed must

  meet the requirements of physical possibilities.

  The standard of a vector of direction must

to be equal to one.

  6 All restrictions on parameters of an element of the multi-body system

  that are described here must be respected.

  7 Spring recall constants and

  depreciation constants must be higher than

res or equal to zero.

  8 Constraints and coaching bodies

  must be mutually independent.

  9 Initial conditions must satisfy

  make a geometric constraint, a constraint of

  motion and a motion organ constraint.

  We previously defined information

  of entry, and a description has been established previously

  how the information is conveyed

  formed while it is being processed by the On

  will now present a description concerning

  orders or opportunities available in each system module, to allow a modeler or user to define a multi-body system and to monitor the execution of system software, to achieve the goals of the simulation As described

  previously, the system contains five functional modules

  the preprocessor 13; the symbol generator

  15; the system data module 18; simulation modules 19 and 20; and the post-processor 21 A typical procedure that a modeler can use to implement the functions of the system comprises the following operations

1 Call the preprocessor 13.

2 Define bodies and joints.

3 Define constraints.

  4 Define drive elements.

Define the system in pieces.

  6 Define sensors and forces.

  7 Save input information, and ask the system to generate input files

for the symbolic generator 15.

  8 Operate the symbol generator

  15 to establish symbolic formats for

  movements of the mechanical system.

  The modeler or user then selects the process that will follow the following sequence: 1 Modify defined subroutines

by the modeler.

  2 Calculation of simulation codes and

  position in the text editor and compile

17.

  3 Calling the System Data Module 18.

  4 Definition of body shapes or

elements of the mechanical system.

  5 Introduction of numerical values for

  symbols and call means of control of

  to control numeric values.

6 Preview the system.

  7 Introduction of initial values for q and control of the configuration.

  8 Solving equations giving the values

  their initials of q compatible with geometric constraints. 9 Introducing initial values for the

  independent parameter u, and control of the singularity.

  Introduction of resolution settings

  numerical and temporal simulation parameters.

  11 Safeguarding body shapes and issuing

  a request to the system for the

  generates an input data file for simula-

  dynamic and post-processor 21.

  12 Execution of simulation subroutines

tion.

13 Post-processor call 21.

  14 Entering a file name.

  introduction of expressions for quantities

  interesting parts of the mechanical system.

  16 Graphic representation of the quantities of interest (23) and data backup in the

file (22).

  17 Adding system points of interest

mechanical animation (24).

  18 Selection of mechanical system bodies for animation, and introduction of parameters

animation.

  19 Animation of simulation results and /

  or generating an animation data file.

  We will now undertake a description

  detailed description of the various elements of the system.

  processor 13 allows a modeler to define a multi-body system and then generate a FORTRAN code (in this embodiment), for the

  control of data in the system data module.

  18, as well as a data file for the gen-

  symbolic identifier 15, on the basis of the information that is introduced by the modeler The input file

  which is generated can also be read by the pre-

  processor at a later time, for further modification Figure 3 shows the control structure of the preprocessor, as a

  general representation of the possibilities of

  processor The FILE function is for six subsystems

  functions The first subfunction NEW rejects

  the current model found in the pre-trial

  if there is one, and it opens a NEW file specified by the modeler, to build a new model The INCLUSION subfunction appended to the current model, if there is one, the model that comes from a file specified by the modeler The subfunction

  SAVE saves system data at several

  bodies which have been introduced by the model

  The SAVE and GENERATION subfunctions

  instruct the modeler to manipulate system data

  multibody system defined by the modeler,

  concerning coordinates and general speeds

  symbols, which require the assignment of values, and other input information for the symbolic generator 15 The DIRECTORY subfunction

  view all the files in the working directory.

vail current of the pre-processor.

  The EDITION function allows the selection of an element of the mechanical system, before the call of one of the five sub-functions which are represented on FIG. 3. The CREATION sub-function allows the creation of a selected element. function MODIFICATION allows to introduce a change in symbols and

  parameter values of a previously defined element

  The DELETE subfunction allows you to

  primer a previously defined element

  COPY function is used to transfer one of the parameters

  elements to another sub-function OBSER-

  VATION allows visual examination of all items

  defined from the selected item.

  The ASSISTANCE function provides messages

  associated with a function or sub-function

selected function.

  The ELEMENT function offers a choice of

  all the aspects available to define a system

  Several of these aspects are represented attached to the ELEMENT function on

Figure 3.

  The ASSEMBLY function is used to perform a

  control of the definition of the mechanical system

  body, and it generates the general coordinates

  q and generalized speeds u The sub-function

  The TREE OF CONTROL enables the construction of a

  topology table in "tree" for the defined model.

  The subfunction DEFINITION OF Q U allows the assignment

  q and u at each joint following changes in

  body and joints The sub-function LIST OF Q U, allows the visualization of the definition of q

  and u in the case where no changes have been made

  killed after the last subfunction DEFINITION OF Q. U The DOUBLE SYMBOLS subfunction provides a list of all the symbols that have been used more than once in the system definition, and provides a list of the elements of the system. mechanical

  in which these symbols have been used.

  function PRESENTATION OF SYMBOLS provides a list of

  all the symbols that have been used in the definition

  mechanical system The RESUME subfunction

  visualize the total number of items that have been

  for each element of the multi-body system.

  If the definition of q and u is up to date, this sub-

  function then displays the total number of q's and

  degrees of freedom of the multi-body system.

  The VISUALIZATION function provides a choice for system background visualization The BODY TREE sub-function provides the system's "tree" topology, including the joint names. Figure 4 shows an example of the topology of the system.

  "tree" If a joint name appears in a definition

  of a body, but has not been defined as a joint-type element, an asterisk will be placed

  before the name of the seal The symbol N represents

  days an inertial frame The "chain" aspect of the system is indicated in the horizontal direction and

  the "branch" aspect is indicated in the green direction

of Figure 4.

  The purpose of the symbolic generator 15 is to

  read the data file that is generated by the pre-

  processor 13, and generate a symbol / function list file, two routines used by the system data module 18, and a program of

simulation in FORTRAN language.

  The purpose of the System Data Module 18 is to provide modellers and users

  means for introducing numerical values

  of symbols, initial conditions and time parameters for simulation A control for the detection of numeric values physically

  impossible will be performed, as previously described.

  Initial conditions will also be checked to see if the geometric constraints and

  movement are satisfied, and at the request of

  readers or users, solutions of the equa-

  constraints will be provided for the conditions

  initial coordinates and general speeds

  are selected by the modeler or the user. A kinematic singularity will also be

  detected, and the system will automatically select

  independent generalized speeds for

  solve the problem The system allows a model

  user or user to create body shapes and graphically visualize selected bodies

  position and orientation of each of these separate bodies

  selected can be changed in accordance with

  generalized coordinate values that are introduced

  by the modeler or the user It is possible

  sible to change the general coordinate values

  keyboard 14 of Figure 2, or by other

  means, if available. All calculations

  With respect to body position, body orientation and stress equations are performed in this embodiment using FORTRAN codes that are generated by the symbolic generator.

  FORTRAN codes must therefore be modified if

  kinematical determinations of time must be

  updated, and these codes must then be compiled

  The compiled object code can then be chained

  to the system data module 18 According to a varian-

  te, a modeler or a user can chain these

  object codes in FORTRAN to build an executable code

  stand-alone table which can then be implemented by the system data module 18 This requires the symbolic generator 15 to generate a main program

  in FORTRAN, instead of simple subroutines.

  The system data module 18 functions

  do the following way A FILE OF function

  holds five sub-functions, as shown in FIG. 5 The OPEN subfunction opens a file which is created by the symbolic generator 15 and which

  contains all symbols and user functions

  which is used to define the mechanical system

  In addition, the previously generated simulation data file can be opened if this condition is specified, and the pattern file

  of body that has been generated previously can be-

  green, if this condition is specified The sub-function

  SAVE SHAPE saving to a file the

  body shape defined in the present moment

  function SAVING SIMULATION DATA saves

  of the simulation data defined at the pre-

  These include initial symbol values, parameters, and conditions. This file is

  used as input information for the pro-

  simulation program The subfunction DIRECTORY

  view all files with a file extension

  system data input / output le to the current directory The OUTPUT subfunction allows the modeler or user to exit the system data module, with confirmation

  data that is not saved.

  The SYMBOL function allows the modeler or

  the user to introduce or modify

  their digital of all the symbols listed, in a similar way to the on-screen edition The explanation of each symbol can be presented on the screen, at

the user's request.

  The SYMBOL CONTROL function allows you to control all numerical values entered, by calling the data control subroutine

  which is generated by the pre-processor 13, as men-

previously mentioned.

  The OBJECT function allows the creation of objects

  to define the shape of a body of a mechanical system

  that we use the shape of a body for the purpose of

  visualization, and it can be used for

  subsequent simulation results.

  The PARAMETER function allows a modeler or user to enter or modify parameters related to the accuracy of numerical solutions

  and the timing conditions of simulation.

  The INITIAL CONDITIONS function allows the entry or modification of initial conditions for q and u, in a similar way to on-screen editing. The GEOMETRY subfunction allows the calculation of residuals of geometric constraint equations, and the modification of the dependent parameter values q,

  selected by the user, which satisfy the

  geometric strokes, if the user requests it.

  The MOTION subfunction allows the calculation of the residues of motion constraint equations and the modification of the values of dependent parameters u,

  selected by the user, which satisfy the

  movement strokes, at the request of the user.

  The VISUALIZATION function contains five

  sub-functions, as seen in Figure 5.

  The GROUP subfunction allows the selection

  bodies to be visualized in the mechanical system.

  The subfunction OBSERVATION allows you to change

  manage the observation direction of the group.

  Sub-function DRAWING draws the selected body on the screen, in accordance with

  initial conditions of the parameters q.

  The ZOOM subfunction allows the user or modeler to define a part of a scene that

must be enlarged.

  The DEPLACEMENT subfunction allows a

  modeler or a user to select a

  generalized data, and to move the mechanical system

  according to a given increment of the value of the selected parameter q, at the request of the user, by means of a mouse or the keyboard This sub-function also allows a user to change the

  initial conditions to replace them with the

their current.

  The simulation code generated by the genera-

  symbolic driver 15 and the simulation data that

  generates the system data module 18 are required

  models for simulation A modeler or a user

  The following procedure should be followed for the simu-

  lation. 1 Modify the subroutine that is generated by the symbolic generator 15, concerning the forces

and / or couples.

  2 Compile and chain the simulation code

* tion to generate executable code.

3 Run the executable code.

  The purpose of the post-processor 21 is to enable

  to a modeler or a user to manipulate

  simulation result data, to generate

  data files containing select quantities

  in a user-specified format, to graph x-y data, and to animate

  simulation results The functions of the post-

  processor are shown in Figure 6 of the drawings.

  sins The FILE function concerns five sub-functions, as can be seen in the drawing. The OPEN subfunction allows the opening of a simulation results file.

  previously opened file The CLOSED subfunction

  TURE closes a previously opened file The sub-file

  function DIRECTORY visualizes all files with

  extensions related to the post-processor The sub-

  EXIT function allows the process to exit the post-processor The ASSISTANCE subfunction presents

  messages of assistance that are associated with

selected parts of the process.

  The INFORMATION function concerns seven sub-

  functions The FILE sub-function allows the visualization

  names of files that have been opened, and

  it marks the file that is in use.

  The HEADER subfunction displays the header that is defined in the simulation results file

  active The TOPOLOGY subfunction displays the topology

  of the multi-body mechanical system, in a manner identical to that previously described for the pre-processor 13 The ENTER subfunction displays

  the definition of the multi-body system

  Q AND U function visualizes the definition of coordinates

  generalized speeds and generalized speeds The USER FUNCTIONS subfeat gives the list

  of all the names of the functions defined by the user.

  The SIMULATION subfunction displays the values

  numerical symbols and simulation parameters

tion.

  The DATA function concerns four subsets

  functions The EXPRESSION subfunction allows

  modelers and users to define their

  variables, vector or scalar The sub-

  EXTERNAL function allows the modeler or user to

  to define variables by reading

  their in an external file The subfunction

  PRESENTATION allows the visualization of numerical values

  variables of a variable in a range of repositories that is defined by the modeler or the user The subfunction DIRECTORY allows the visualization of all the variables defined by the modeler or

  the user, and their definitions.

  The TRACE function concerns seven sub-functions

  The CURVE sub-function allows the user

  to define curves to plot, by editing on the screen.

  The sub-function WINDOW allows the processor to select a window to cover all the data and to spread this window to be as large as possible. The user can also define a window and a number of divisions in x

  and / or y The QUADRILLAGE subfunction allows a user to

  or a user to make appear and

  to appear grid lines The sub-function

  TITLE allows a modeler or a user to specify

  proud of a title The MARKING X and Y MARKING subfunctions allow the user to specify a marking. The DESK subfunction draws the curves

  set on a viewing screen.

  The WRITE function contains two sub-

  functions The ASCII subfunction allows model

  users or users to write time histories of a selected variable, in a specified ASCII format The RESTART subfunction allows the generation of an input file for simulation, at a restart time that is specified by a

modeler or a user.

  The ANIMATION function, which is represented in FIG. 6, concerns three sub-functions. The BODY / MARKER SELECTION sub-function allows a modeler or a user to select bodies.

  or markers to include in the animation The sub-

  REFERENCE SELECT function allows a model to be

  user or user to select a range of

  standards, as well as the benchmarks to jump

  animation The ANIMATION START sub-function

  TION starts the animation on a visual screen

  lisation. The validation performed by the system described here makes it possible to determine whether the system calculations have

  have been successfully completed The system is certainly

  The following are the performance limits, which are given below. The complexity and size of the multi-body mechanical system that can be dealt with depend on

  available space In the mode of realization

  the preferred mode which is described here, and in accordance with the best known mode, the system is limited to fifty

  bodies or elements of mechanical systems, with one hundred

  generalized data (holonomic degrees of freedom), one hundred constraints and drives, one hundred user-defined functions, and six hundred

  These standards constitute a goal of com-

  plexity, and they do not necessarily represent the constraints that are due to the memory size

  of a particular computer system.

  Input data from model

  users and users through the pre-

  processor 13, the system data module 18 and

  of the post-processor 21, are checked with respect to

  identifies conflicts of conventions,

  or physically impossible properties, and a

  Multi-body mechanical system size transition that is defined The software that was developed to implement the analysis system that is described here is portable on any system

  computer system that operates with an operating system

  UNIX, a standard C compiler, and a program

  me management / library for PHIGS (or X-WINDOW).

  As mentioned earlier, the simula-

  generated is portable on-site or off-site

  for any analysis system described here

  posing of a FORTRAN-77 compiler of standard type. As an example of how the system works

  7 shows a structure called "Stanford Arm" which comprises six elements or bodies designated by B 1, B 2, B 3, B 4, B 5. and B 6 Bl can be rotated in a Newtonian reference frame N, around a vertical axis N 2 which is fixed in both N and B 1, and B 1 supports B 2, which can be rotated with respect to B 1 around a horizontal axis which is fixed both in B1 and B 2 B 3 is supported by B 2, and can perform a translational movement with respect to B 2, while supporting B 4, which can be rotated relative to B 3, about an axis which is fixed in both B 3 and B 4 B 5 is mounted on B 4 and B 6 is mounted on B 5 in such a way that each

  of them can turn relative to the body that

  door, around an axis that is fixed both in the

  body that supports the other and in the body supported.

  To run the FORTRAN program for the Stanford Arm, a modeler first needs to generate executable code, and then create a

  data file containing the information or the in-

following instructions:

  1 A description of the execution (in one

line).

  2 The numerical values of the following parameters (system dimensions of interest): Li, L 2, L 3, L 4, L 5, L 6, MA,

IA 1, IA 2, IB 1, IB 2, IB 3, MC,

  IC 1, IC 2, MD, ID 1, ID 2, ME, IE 1,

IE 2, MF, IF 1, IF 2, IF 3, G, K

3 The initial values of

Q 1, Q 2, Q 3, Q 4, Q 5, Q 6

  4 The initial values of U1, U2, U3, U4, U5, U6 The choice of the integration method:

1: Fixed time step.

  2: Variable time step. 6 The initial moment and the final moment of

the simulation.

7 The temporal step of integration.

  8k: the results will be printed all

the k temporal integration steps.

  Example 1 (in the system data module):

1 2 3 4 5 6 7

8 9 10 11 12 13 14

15 16 17 18 19 20 21

22 23 24 25 26 27 28

1 1 1 1 1 1 1

0 5 O 1 1

  SYSTEM PARAMETERS (simulation codes /

  echo of the input information, for the purpose of

mentation)

L 1 O 100000 D + 00

L 2 O 100000 D + 00

L 3 0 700000 D + 00

L 4 -0 100000 D + 00

L 5 O OOOOOOD-01

L 6 0 140000 D + 00

MA O 900000 D + 01

IA 1 O 100000 D-01

IA 2 0 200000 D-01

MB 0 600000 D + 01

IB 1 0 600000 D-01

IB 2 O 100000 D-01

IB 3 0 500000 D-01

MC 0 500000 D + 01

I Ci 0 400000 D + 00

IC 2 0 100000 D-01

MD 0 100000 D + 01

ID 1 0 500000 D-03

ID 2 0 100000 D-02

ME 0 600000 D + 00

I El 0 500000 D-03

IE 2 0 200000 D-03

MF O 500000 D + 00

IF 1 0 100000 D-02

IF 2 O 200000 D-02

IF 3 0 300000 D-02

G O 981000 D + 01

K O 000000 D + 00

INITIAL CONDITION:

Q 1 = 0.

Q 2 = 0.

Q 3 = 0.

Q 4 = 0.

Q 5 = 0.

Q 6 = 0.

U 1 = 0.

U 2 = 0.

U 3 = 0.

U 4 = 0.

U 5 = 0.

U 6 = 0.

  INTEGRATION WITH NOT TEMPOREL VARIABLE (echo of instructions):

INITIAL INSTANT = 0.

FINAL INSTANT = 5.00

  NOT TIME OF INTEGRATION = 0 100000 SECOND

  RESULT PRINT ALL K NOT; k = 1

(ALL 0 1 SECOND HERE)

OUTPUT SELECTIONS:

DESCRIPTION OF THE SYSTEM

  POSITION OF THE MASS CENTER IN COORDINATES

GLOBAL

  ORIENTATION OF EACH BODY IN COORDINATES

GLOBAL

  SPEED OF THE CENTER OF MASS IN COORDINATES

GLOBAL

  ANGULAR SPEED OF EACH BODY IN COOR-

LOCAL DATA

  ACCELERATION IN GLOBAL COORDINATES

  ANGULAR ACCELERATION IN LOCAL COORDINATES

SIMULATION CONTROL

  ACTIVE FORCE IN GLOBAL COORDINATES AND

COUPLE IN LOCAL COORDINATES

TIME HISTORY OF Q

TEMPORAL HISTORY OF U

TEMPORAL HISTORY OF UD

  ANIMATION FILE, TOPOLOGICAL INFORMATION

  DIGITAL USER ENTRY FILE

ORIGINAL

  It goes without saying that many modifications can be made to the device and the process

  described and represented, without departing from the

vention.

Claims (28)

  1 Method for efficiently analyzing a   multi-body system of holonomic type, non-holo-   nome or non-piece holonomic, characterized in that it comprises the following steps: the topology of a general type of multi-system is defined   body; we generate symbolic equations representing   of the system's motion equations   Sieurs corps of general type; numerical values defining a multi-body system are introduced as part of the general type; we simulate the function of the multi-body system; the results of the simulation are determined; and we control physical restrictions and imposed by the system, concerning the numerical values introduced, and we   indicates whether these restrictions are acceptable.   The process according to claim 1, wherein   in that the step of defining the topology comprises the step of defining the topology in basic open loop.   The process according to claim 1, wherein   in that the step of defining the topology comprises the step of defining a topology with tree structure.   4. Process according to claim 1, characterized   in that the step of defining the topology   includes the step of defining constraints   which define a closed-loop topology and non-holonomic restrictions.   The method of claim 4, wherein   in that the step of defining constraints   includes the step of defining combinations of   sounds of constraint equations that apply at different times, to define a dynamic system not piecewise holonomic.   The method of claim 1, wherein   in that it includes the determination step   initial conditions appropriate for the system.   The method of claim 1, wherein   it includes the step of manipulating the simulation results to obtain   various amounts of system of interest.   The method of claim 1, wherein   characterized in that it includes the step of recording the time history of the simulation results.   The method of claim 7, wherein   in that it includes the step of recording the temporal history of the quantities of   system of interest.   Process according to claim 1,   characterized in that it comprises the step of drawing in a two-dimensional format the simulation results that are obtained, to thus visualize   overall a large amount of information.   The method of claim 1, wherein   characterized by including the step of animating the movements of the multi-body system,   to thus provide intuitive visual information.   The method of claim 1, wherein   in that the step of simulating the   functioning of the system, includes the step that   to solve ordinary differential equations   and to study the results of integration   responding to the solutions of these equations, to   undermine the accuracy and correctness of the inte-   gration of the formulation of symbolic equations.   13 A method of analyzing a mechanical system containing mechanical elements connected to each other, characterized in that it comprises the following steps: the basic topology of the mechanical system is defined; we select for the mechanical system a referential fixed inertial manner, we specify the constraints of the mechanical system, and we generate symbolic equations for the movement of the system with constraints, for any holonomic system, non-holonomic and dynamic non-holonomic by pieces, from the constraints that have been specified for the system.   The method of claim 13, wherein   characterized in that it comprises the following step: the time history of a generalized speed associated with a joint between elements in the mechanical system is fixed.   The process of claim 13, wherein   characterized in that it comprises the step of   define the types of forces acting on the elements mechanical elements.   The method of claim 13, wherein   characterized in that it comprises the step of defining scalar kinematic quantities for   the evaluation of the forces and couples of the system.   The method of claim 13, wherein   characterized in that it comprises the step of controlling and indicating the acceptable and correct character of the generated symbolic equations, taking into consideration the conventions and restrictions of the system.   The method of claim 13, wherein   characterized in that it comprises the step of   create an autonomous executable code, ensuring the   for deferred execution, on-site and Offsite.   The method of claim 13, wherein   characterized in that it comprises the following steps: numerical values are introduced for symbols and initial conditions, so as to obtain specific motion equations for the system; and the system-specific motion equations are solved to determine simulation results regarding the position and orientation of the   body, as well as constraint equations.   Method according to claim 19,   characterized in that it the step of manipulating the simulation results to obtain quantities   of predetermined systems that are of interest.   The method of claim 19, wherein   characterized by including the step of recording the time history of the results of simulation.   The method of claim 20, wherein   characterized by including the step of recording the time history of the quantities of   system of interest.   The method of claim 13, wherein   characterized by including the step of graphing the simulation results into a two-dimensional format, thereby providing a   visualization of large quantities of training.   The method of claim 13, wherein   characterized in that it comprises the step of   animate the movements of the mechanical system, to obtain   thus create intuitive visual information.   Apparatus for analyzing a mechanical system containing mechanical elements (B 1 -B 8) connected to each other, characterized in that it comprises: a pre-processor (13); data input means (14) for input into the preprocessor   (13) the topology and constraints of the mechanical system   lunch; and means (15) connected to the preprocessor   (13) to formulate symbolic motion equations   for any mechanical constraint system   holonomically, non-holonomically and non-holonomically, on the basis of topology and have been introduced.   Apparatus according to claim 25,   characterized in that it comprises: means (14)   for the introduction of numerical values of sym-   boles and initial conditions in motion equations; and means that are intended to provide solutions of the equations of motion, with the   numerical values and initial conditions intm -   in order to obtain simulation results   simulating the position and direction of   mechanical elements, as well as constraints.   Apparatus according to claim 26,   characterized by comprising means (21) for manipulating the simulation results, thereby obtaining system quantities   predetermined values that are of interest.   Apparatus according to claim 26,   characterized in that it comprises means (23) which   are intended to represent graphically the results   simulation states, in two-dimensional format.   Apparatus according to claim 26,   characterized in that it comprises means (24) which are intended to animate the movements of the elements mechanical.   Apparatus according to claim 25,   characterized in that it comprises means which are intended to validate the topology and the constraints of the   mechanical system, on the basis of the conventions of the   tem and physical possibilities.   Apparatus according to claim 26,   characterized in that it includes means for   validate numerical values of symbols and   initial readings, taking into consideration system conventions and physical possibilities.