1. Gyroscopic damping and stiffness#

1.1. Characteristics of gyroscopy#

The models currently available in Code_Aster for rotating machines involve structural elements such as beams (for rotors) and discrete elements (for disks, bearings, supports, etc.).

Calculations of the kinetic energy of the Timoshenko beam element (considering the flexural deformations) or of the discrete element reveal the terms of gyroscopic effect.

For a rotating system, the classical equation for structural dynamics depends on the speeds of rotation as follows:

\(M\ddot{\delta }+{C}_{\mathit{gyro}}(\Omega )\dot{\delta }+(K\text{+}{K}_{\mathit{gyro}}(\dot{\Omega }))\delta \mathrm{=}0\)

Where \(M\) and \(K\) are the classic stiffness and mass matrices of the rotating mechanical system. \({C}_{\mathit{gyro}}(\Omega )\) is a non-symmetric matrix, a function of the speed of rotation \(\Omega\), including gyroscopic damping characteristics. This gyroscopic damping matrix is antisymmetric, and its contribution is proportional to the angular speed.

\({K}_{\mathit{gyro}}(\dot{\Omega })\) is a non-symmetric matrix, a function of rotational acceleration \((\dot{\Omega })\), including gyroscopic stiffness characteristics. This gyroscopic stiffness matrix is full. Its contribution is proportional to the angular acceleration and is therefore zero in case of constant rotation speed. The gyroscopic stiffness matrix only makes sense when there is a speed transition.

Gyroscopic damping and stiffness are currently available in Code_Aster for straight beams (Timoshenko Elements POU_D_T and Euler POU_D_E) of constant or variable cross section and circular shape as well as for discrete elements. The hypotheses used for the beams are:

Timoshenko hypothesis: transverse shear and all inertia terms are taken into account. This hypothesis is to be used for low levels of movement (Elements POU_D_T).
Euler hypothesis: transverse shear is neglected. This hypothesis is verified for strong impulses (Elements POU_D_E).

The natural speed of rotation (along the axis of the beam) can be constant or variable (case of a transient rotation speed: increase in speed or deceleration).

Disks are cylinders of revolution whose axis coincides with the axis of the beam. These disks are assumed to be undeformable and are modelled by discrete elements. The material is homogeneous and isotropic.

Finally, for more details, the reader can refer to the reference documentation for gyroscopic damping and stiffness in Code_Aster [R5.05.07].

1.2. Calculation of gyroscopic matrices#

It is possible to build the gyroscopic damping and stiffness matrices from each element of the model as for stiffness and mass. For structural finite elements such as beams or discrete elements, it is possible to calculate the real elementary matrices corresponding to the calculation options” MECA_GYRO “and” RIGI_GYRO “. These operations are called as follows:

AMGYELEM [matr_elem_ DEPL_R] = CALC_MATR_ELEM

(/♦ OPTION: “ MECA_GYRO “

♦ MODELE: mo [model]

♦ CHAM_MATER: chmat [cham_master]

◊ CARA_ELEM: cara [cara_elem]

)

RIGYELEM [matr_elem_ DEPL_R] = CALC_MATR_ELEM

(/♦ OPTION: “ RIGI_GYRO “

♦ MODELE: mo [model]

♦ CHAM_MATER: chmat [cham_master]

◊ CARA_ELEM: cara [cara_elem]

)

The assembly of the elementary gyroscopic matrices is obtained with the usual operator ASSE_MATRICE [U4.61.22].

AMGYASS = ASSE_MATRICE (MATR_ELEM = AMGYELEM,

NUME_DDL = NUMEDDL,)

RIGYASS = ASSE_MATRICE (MATR_ELEM = RIGYELEM,

NUME_DDL = NUMEDDL,)

It should be noted that the same numberings and the same storage mode must be used as for the stiffness and mass matrices (operator NUME_DDL [U4.61.11]).

This operation can also be carried out at the same time as the construction of the other matrices (mass, stiffness, damping, etc.) using the macro command ASSEMBLAGE

ASSEMBLAGE (MODELE = MODELE,

CHAM_MATER = CHMAT,

CARA_ELEM = CARELEM,

CHARGE = BLOQUAGE,

NUME_DDL =CO (“NUMEDDL”),

MATR_ASSE =( _F (MATRICE =CO (“RIGIDITE”),

OPTION =” RIGI_MECA “,),

_F (MATRICE =CO (“MASSE”),

OPTION =” MASS_MECA “,),

_F (MATRICE =CO (“AMOR”),

OPTION =” AMOR_MECA “,),

_F (MATRICE =CO (“AMGY”),

OPTION =” MECA_GYRO “,),

_F (MATRICE =CO (“RIGY”),

OPTION =” RIGI_GYRO “,),),)

1.3. Use of gyroscopic matrices#

The calculated gyroscopic damping and stiffness matrices \({C}_{\mathit{gyro}}\) and \({K}_{\mathit{gyro}}\) are unitary and must therefore be multiplied by the angular speed and acceleration respectively to obtain the gyroscopic effect of a given speed of rotation.

The gyroscopic damping matrix is generally combined after assembly with the conventional damping matrix by the COMB_MATR_ASSE operator as follows:

OM=124.14*pi/30. # Rotation speed in rpm transformed into rad/s

GYOM = COMB_MATR_ASSE (COMB_R =( _F (MATR_ASSE = AMGY, COEF_R =OM,),

_F (MATR_ASSE = AMOR, COEF_R =1.,),))

where AMGY and AMOR are the assembled gyroscopic and viscous damping matrices.

Gyroscopic matrices can be used for direct linear dynamic analysis with linear dynamic response operators:

linear transient on a physical basis* DYNA_LINE_TRAN ** [U4.53.02] (cf. modeling A in [V2.02.126])

linear transient on a modal basis* DYNA_TRAN_MODAL ** [U4.53.21] (cf. modeling B in [V2.02.126])

harmonic* DYNA_LINE_HARM ** [U4.53.11] (cf. C modeling in [V2,02,126])

DYNA_VIBRA [U4.53.03] (cf. C modeling in [V2,02,126])

In the case of a rotational speed transient treated on a modal basis, the two assembled gyroscopic matrices are combined with the conventional damping and rigidity matrices in a manner that is completely transparent to the user. To do this, simply enter in DYNA_TRAN_MODAL or DYNA_VIBRA the parameters defining this speed transient (laws of imposed angular speed and acceleration and gyroscopic matrices assembled, cf. [U4.53.21] and [U2.06.32]).

These matrices are essential for the complex modal analysis of a system running with the eigenvalue search operator:

CALC_MODES [U4.52.02]

For systems with a reduced number of degrees of freedom, a direct modal calculation can be carried out with the operator mentioned above and with one or the other of the resolution algorithms (here algorithm “SORENSEN” for example):

MODEG = CALC_MODES (MATR_RIGI = RIGIDITE,

MATR_MASS = MASSE,

MATR_AMOR = GYOM,

OPTION =” CENTRE “,

CALC_FREQ =_F (FREQ =1. ,

NMAX_FREQ =16),

SOLVEUR_MODAL =_F (METHODE =” SORENSEN “),

VERI_MODE =_F (SEUIL =1.E-3),

)

For a robust quadratic modal calculation (amortized mode calculation), a two-step strategy can be used.

The QZ method is a reliable method for a quadratic modal calculation. However, it is expensive. It is therefore to be restricted to systems with a small number of degrees of freedom (a few thousand degrees of freedom) or to problems reduced by modal truncation. A calculation must be carried out in two stages:

1a - calculate a reduced undepreciated modal base:

MODES = CALC_MODES (MATR_RIGI = RIGIDITE,

MATR_MASS = MASSE,

OPTION =” BANDE “,

CALC_FREQ =_F (FREQ =( 3.,3000.))

);

1b - project mass, stiffness, and total damping matrices GYOM based on this

(modal truncation):

PROJ_BASE (BASE = MODES,

PROFIL =” PLEIN “,

MATR_ASSE_GENE =( _F (MATRICE =CO (“MAGE”),

MATR_ASSE = MASSE,),

_F (MATRICE =CO (“RIGE”),

MATR_ASSE = RIGIDITE,),

_F (MATRICE =CO (“OMGE”),

MATR_ASSE = GYOM,),),

);

2 - do a modal calculation on the generalized matrices obtained by projection:

MOD2ETAG = CALC_MODES (MATR_RIGI = RIGE,

MATR_MASS = MAGE,

MATR_AMOR = OMGE,

OPTION =” TOUT “,

SOLVEUR_MODAL =_F (METHODE =”QZ”),

);

We are therefore required to project the antisymmetric and non-symmetric matrices \({C}_{\mathit{gyro}}\) and \({K}_{\mathit{gyro}}\) into the subspace defined by a set of real eigenmodes. This operation is possible with the PROJ_MATR_BASE [U4.63.12] operator. Note that in the general case, the projected matrices are not symmetric. However, they remain usable for calculating the dynamic response in force or in imposed motion in the modal space, with the linear dynamic response operator:

linear transient on a modal basis* DYNA_TRAN_MODAL ** [U4.53.21] (cf. modeling B in [V2,02,126])