Damping. Code_Aster, Salome-Meca course material GNU FDL licence (

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1 Dampin Code_Aster, Salome-Meca course material GNU FDL licence (

2 Outline Dampin in dynamic analysis Types of modelin Gettin started with Code_Aster for dynamic analysis Modal calculation with dampin Solvin the quadratic problem : principles Methods and alorithms implemented in Code_Aster 2 - Code_Aster and Salome-Meca course material GNU FDL Licence

3 Dampin : theoretical issues Dampin : enery dissipation within the different materials of the structure, in the connections between structural elements, and with the surroundin medium. Two simple models in Code_Aster viscous dampin : the dissipated enery is proportional to the speed of movement ; hysteretic dampin : the dissipated enery is proportional to the displacement & such that the dampin force is of opposite sin to the speed. Variables to characterize dampin Loss rate : η= E d par cycle 2πEp max Reduced dampin : = Code_Aster and Salome-Meca course material GNU FDL Licence

4 Viscous dampin : modellin principles Viscous dampin : time or frequency definition m u + c u + k u = f Rayleih dampin C = αk + βm Defined in the material properties DEFI_MATERIAU Or easy to implement with the command COMB_MATR_ASSE Useful for validatin alorithms Successfully introduced for transient analysis by modal recombination Does not represent heteroeneity of the structure relative to the dampin Depends heavily on the identification of the coefficients, Non physical model for dissipation, but required for some reulatory studies 4 - Code_Aster and Salome-Meca course material GNU FDL Licence

5 Viscous dampin : identification of coefficients Dampin proportional inertia characteristics Widely used for direct transient resolution can be identified with the experimental reduced dampin i of the normal mode which contributes most to the response Hiher modes are very little damped and low frequency modes are very damped Dampin proportional to stiffness characteristics Widely used for direct transient resolution can be identified with the experimental reduced dampin j of the normal mode which contributes most to the response Hiher modes are very damped and low frequency modes are very little damped i, i, are identified from two independent modes of frequencies i, j In the interval i, j, the variation of the reduced dampin remains low and outside modes will be over-damped Full proportional dampin 5 - Code_Aster and Salome-Meca course material GNU FDL Licence 0, i i, 0

6 Viscous dampin Global dampin Coefficients are defined with the material properties in DEFI_MATERIAU AMOR_ALPHA and AMOR_BETA Use in harmonic and transient analysis, physical or reduced bases DYNA_VIBRA with the keyword MATR_AMOR Modal reduced dampin DYNA_VIBRA(BASE_CALCUL= GENE with the keyword AMOR_REDUIT COMB_SISM_MODAL with the keyword AMOR_REDUIT Localized or non-proportional dampin Discrete dampers are introduced with AFFE_CARA_ELEM c = a elem i elem i Material dampin coefficients defined in DEFI_MATERIAU with AMOR_ALPHA and AMOR_BETA affected material by material in the model c elem i = α k i elem i + β i m elem i 6 - Code_Aster and Salome-Meca course material GNU FDL Licence

7 Hysteretic (or structural) dampin : principles Valid only in frequential approach noncausal model 2 mu k 1 ju f is a constant Representation of the dissipation suitable for metallic materials Used to deal with harmonic responses of structures with viscoelastic materials The coefficient is determined from a test with cyclic harmonic loadin K(1+j) m f u 7 - Code_Aster and Salome-Meca course material GNU FDL Licence 0 = j t * E cos jsin 0 e * 0 j E e 0 = jt 0 0 e * E1 E = E1 1 j with = tan E 2

8 Hysteretic dampin Used for harmonic analysis only DYNA_VIBRA(TYPE_CALCUL= HARM ) keyword MATR_RIGI Global or homoeneous dampin, identical on the whole model K c = 1+ jηk The dampin coefficient is defined with DEFI_MATERIAU, keyword AMOR_HYST The complex lobal matrix is built within ASSEMBLAGE Localized or non-proportional dampin Discrete stiffness elements are defined with AFFE_CARA_ELEM, option AMOR_HYST k = 1 c discret i + jηdiscret i kdiscret i Material dampin is defined with DEFI_MATERIAU, keyword AMOR_HYST k = 1 c elem i + jηelem i kelem i Transient analysis calculation of eienmodes and dampin coefficient ( quadratic CALC_MODES) and resolution on a real modal basis with identified modal dampin coefficients (DYNA_VIBRA(TYPE_CALCUL= TRANS,BASE_CALCUL= GENE ) 8 - Code_Aster and Salome-Meca course material GNU FDL Licence

9 Solvin the quadratic problem Initial quadratic problem : findin l & F such as K l 2 M lc F 0 Linear form : M 0 l 0 K M Usual alorithm for normal modes quad C K K 0 K quad lf F 0 Meanin of eienvalues l 2 i ii ji 1i * T i C i * T i M i = 2 Re l i * T i K i * T i M i = l i 2 i is the pulsation i is the dampin of mode i 9 - Code_Aster and Salome-Meca course material GNU FDL Licence

10 Some theory Spectral transform => different strateies Shift & Invert with complex shift 1 M u u K u lm u K M quad quad quad quad A 1 l For hih dampin : real arithmetic l 1 l Re A For low dampin : imainary arithmetic Im j l 1 l A More enerally : complex approach A 1 l 10 - Code_Aster and Salome-Meca course material GNU FDL Licence

11 In Code_Aster Power iteration methods (inverse methods) in CALC_MODES Only for a couple of frequencies Subspace iteration methods in CALC_MODES METHODE= TRI_DIAG Real approach shift = 0 => OPTION = PLUS_PETITE or CENTRE Non-zero shift => OPTION= CENTRE Imainary approach Non-zero shift => OPTION= CENTRE METHODE= SORENSEN Real approach shift = 0 => OPTION = PLUS_PETITE or CENTRE Non-zero shift => OPTION= CENTRE Imainary approach Non-zero shift => OPTION= CENTRE Complex approach Non-zero shift => OPTION= CENTRE Decomposition method in the complex space METHOD= QZ Very robust Only for small problems : computes all the modes! Nota Bene : No OPTION= BANDE and no verifications on the number of eienvalues 11 - Code_Aster and Salome-Meca course material GNU FDL Licence

12 In three steps with modal reduction Problem to solve : l c,f c such as K First step : normal modes without dampin l,f such as K l 2 M F 0 l 2 M C Second step : modal reduction by projection t t t K F KF M F MF C F CF Third step : quadratic problem on a reduced basis M l c 0 0 K C K F 0 c l c 0 Only for low dampin Far more robust & effective method use METHOD= QZ for the third step Always use PROFIL= PLEIN for numberin (and assembly) Default option in NUME_DDL and ASSEMBLAGE commands K 0 lcf F c 12 - Code_Aster and Salome-Meca course material GNU FDL Licence

13 End of presentation Is somethin missin or unclear in this document? Or feelin happy to have read such a clear tutorial? Please, we welcome any feedbacks about Code_Aster trainin materials. Do not hesitate to share with us your comments on the Code_Aster forum dedicated thread Code_Aster and Salome-Meca course material GNU FDL Licence

Continuation methods for non-linear analysis

Continuation methods for non-linear analysis FR : Méthodes de pilotage du chargement Code_Aster, Salome-Meca course material GNU FDL licence (http://www.gnu.org/copyleft/fdl.html) Outline Definition of