$$$$ HBM NOTICE MB234859 17/10/02 21:15:14 9577 DATE 17/10/02 Procedure HBM Voir aussi : CONTINU ______________ HBM_POST Section : Mecanique Dynamique HBM TAB1; FRAN========================================================== Objet : ______ HBM (Harmonic Balance Method ou equilibrage harmonique) transforme le probleme dynamique non-lineaire etabli dans le domaine temporel sous la forme du systeme d'equations differentielles (1) de taille N en un systeme d'equations algebriques (2) de taille N*(2H+1) via la decomposition en serie de Fourier (3) des inconnues du probleme en vue d'une resolution par la methode de CONTINUation. .. . . M u(t) + C u(t) + K u(t) = fnl(u,u) + fext(wt) (1) avec : M : matrice de masse C : matrice d'amortissement K : matrice de raideur fnl : forces non-lineaires dans le domaine temporel fext : vecteur des forces exterieures dans le domaine temporel de frequence w u : vecteur inconnue dans le domaine temporel Z(w) U - Fnl(U) - Fext = 0 (2) avec : Z(w) = diag(K, Z_1, Z_2, ... Z_H ) Z_k = [ K - k²w² M w C ] [ -wC K - k²w² M ] U = ( U_0 U_1 V_1 ... U_H _VH ) tel que : u(t) = U_0 + \sum_{k=1..H} cos kwt U_k + sin kwt V_k (3) Entree : _______ TABHBM = TABLE . 'RIGIDITE_CONSTANTE' = K . 'AMORTISSEMENT_CONSTANT' = C . 'MASSE_CONSTANTE' = M . 'BLOCAGES_MECANIQUES' = Kblocages . 'RIGIDITE_CENTRIFUGE' . 'CORIOLIS_CONSTANT' . 'N_HARMONIQUE' = nombre d'harmoniques H . 'RESULTATS' = table des resultats attendus . i . 'POINT_MESURE' exprimes sur ddl temporel . 'COMPOSANTES' . 'COULEUR' . 'TITRE' Sortie : _______ TABHBM . 'RIGIDITE_HBM' = partie de Z relative a K . 'AMORTISSEMENT_HBM' = partie de Z relative a C (pour w=1) . 'MASSE_HBM' = partie de Z relative a M (pour w=1) . 'BLOCAGES_HBM' = partie de Z relative a Kblocages . 'CENTRIFUGE_HBM' ... . 'CORIOLIS_HBM' . 'RESULTATS_HBM' = table des resultats attendus . j . 'POINT_MESURE' exprimes sur ddl frequentiels . 'COMPOSANTES' . 'COULEUR' . 'TITRE' . 'RESULTATS' . i . 'INDICES_HBM' = liste des indices j associe au i^eme resultat . 'COMPOSANTES' . 'DEPLACEMENT' = composantes temporelles de u (max.6) . 'FORCE' = composantes temporelles de f (max.6) . 'DEPLACEMENT_HBM' = composantes frequentielles de U . 'FORCE_HBM' = composantes frequentielles de F . 'HARM_DEPLACEMENT' = table des composantes de U (par harmonique) . 'HARM_FORCE' = table des composantes de F (par harmonique) Correspondance entre inconnues temporelles et frequentielles : +-----------+--------------------------------------+ | domaine | domaine frequentiel | | temporel | k=0 k=1(cos) k=1(sin) ... | +-----------+--------------------------------------+ | UX | U1 U4 V4 ... | | UY | U2 U5 V5 ... | | UZ | U3 U6 V6 ... | +-----------+--------------------------------------+ ANGL========================================================== Description : ____________ The HBM (Harmonic Balance Method) procedur converts the non-linear dynamical problem described in the temporal domain by a set of N differential equations (1) into a set of N*(2H+1) algebraic equations (2) via a Fourier serie decomposition (3) of the unknowns of the problem with the goal to solve it by a continuation method. .. . . M u(t) + C u(t) + K u(t) = fnl(u,u) + fext(wt) (1) with : M : mass matrix C : damping matrix K : stiffness matrix fnl : non-linear forces expressed in the temporal domain fext : external forces expressed in the temporal domain of frequency w u : vector of unknowns expressed in the temporal domain Z(w) U - Fnl(U) - Fext = 0 (2) with : Z(w) = diag(K, Z_1, Z_2, ... Z_H ) Z_k = [ K - k²w² M w C ] [ -wC K - k²w² M ] U = ( U_0 U_1 V_1 ... U_H _VH ) so that : u(t) = U0 + \sum_{k=1..H} cos kwt U_k + sin kwt V_k (3) Entree : _______ TABHBM = TABLE . 'RIGIDITE_CONSTANTE' = K . 'AMORTISSEMENT_CONSTANT' = C . 'MASSE_CONSTANTE' = M . 'BLOCAGES_MECANIQUES' = Kconstraints . 'RIGIDITE_CENTRIFUGE' . 'CORIOLIS_CONSTANT' . 'N_HARMONIQUE' = number of harmonics H . 'RESULTATS' = table of expected results . i . 'POINT_MESURE' expressed with temporal dof . 'COMPOSANTES' . 'COULEUR' . 'TITRE' Sortie : _______ TABHBM . 'RIGIDITE_HBM' = part of Z relative to K . 'AMORTISSEMENT_HBM' = part of Z relative to C (pour w=1) . 'MASSE_HBM' = part of Z relative to M (pour w=1) . 'BLOCAGES_HBM' = part of Z relative to Kblocages . 'CENTRIFUGE_HBM' ... . 'CORIOLIS_HBM' . 'RESULTATS_HBM' = table of expected results . j . 'POINT_MESURE' expressed with frequential dof . 'COMPOSANTES' . 'COULEUR' . 'TITRE' . 'RESULTATS' . i . 'INDICES_HBM' = list of indices j associated to the i^th result . 'COMPOSANTES' . 'DEPLACEMENT' = temporal components for u (max.6) . 'FORCE' = temporal components for f (max.6) . 'DEPLACEMENT_HBM' = frequential components for U . 'FORCE_HBM' = frequential components for F . 'HARM_DEPLACEMENT' = table of the frequential components (by harmonic) . 'HARM_FORCE' = table of the frequential components (by harmonic) Correspondence between temporal and frequential dof : +-----------+--------------------------------------+ | temporal | frequential domain | | domain | k=0 k=1(cos) k=1(sin) ... | +-----------+--------------------------------------+ | UX | U1 U4 V4 ... | | UY | U2 U5 V5 ... | | UZ | U3 U6 V6 ... | +-----------+--------------------------------------+