7. Harmonic response calculation#

The following equation is solved in the frequency domain:

(7.1)#\[ (-j\ omega\ mathrm {³}\ mathrm {.} \ mathrm {Q} -\ omega\ mathrm {²}\ mathrm {.} \ mathrm {M} +j\ omega\ mathrm {.} \ mathrm {C} +\ mathrm {K})\ cdot\ mathrm {x} =\ left\ {\ sum _ {i=1} ^ {k} {h} _ {h} _ {i} (f)\ mathrm {.} {\ omega} ^ {{n} _ {i}}\ mathrm {.} {\ mathrm {e}} ^ {j\ pi\ frac {{\ phi} _ {j}} {180}}\ mathrm {.} {\ mathrm {g}} _ {i} (P)\ right\}\]

Where:

\(\mathrm{K}\) represents a real or complex stiffness matrix (case of hysteretic damping)

\(\mathrm{M}\) represents a mass matrix

\(\mathrm{C}\) represents a viscous damping matrix

\(\mathrm{Q}\) represents an acoustic impedance matrix derived from a displacement-pressure-potential formulation \((\mathrm{u},p,\varphi )\), cf. [R4.02.02].

\(P\) is a common point in the structure.

\(\omega \mathrm{=}2\pi f\): pulsation of excitement

\({\mathrm{g}}_{i}(P)\): spatial description of arousal

\({h}_{i}(f)\): frequency description of arousal

\(\mathrm{x}\): complex answer

The damping of the structure can be viscous or hysteretic [U2.06.03] [R5.05.04]. In the case of hysteretic damping, a complex stiffness matrix is available and the following equation is solved:

(7.2)#\[ (\ mathrm {K} -\ omega\ mathrm {²}\ mathrm {.} \ mathrm {M})\ cdot\ mathrm {x} =\ left\ {\ sum _ {i=1} ^ {k} {h} _ {i} (f)\ mathrm {.} {\ omega} ^ {{n} _ {i}}\ mathrm {.} {\ mathrm {e}} ^ {j\ pi\ frac {{\ phi} _ {j}} {180}}\ mathrm {.} {\ mathrm {g}} _ {i} (P)\ right\}\]

with \(\mathrm{K}\): complex stiffness matrix.

This operator can be used in imposed force or in imposed displacement (resolution in absolute frame of reference) or in force of inertia of imposed drive (resolution in relative frame of reference).

7.1. Operand RESULTAT#

◊ RESULTAT = harm

Name of the result data structure to be enriched. This keyword is mandatory if you are in re-entering concept mode (reuse).

7.2. Operand MODELE#

◊ MODELE = mo

Name of the concept defining the model whose elements are subject to harmonic calculation.

7.3. Operand CHAM_MATER#

◊ CHAM_MATER = chmat

Name of the concept that defines the material field assigned on the mo model.

7.4. Operand CARA_ELEM#

◊ CARA_ELEM = character

Name of the concept defining the characteristics of the elements of beams, plates, shells, etc…

7.5. Operand MATR_MASS#

♦ MATR_MASS = m

Name of the concept assembled matrix \(\mathrm{M}\) corresponding to the mass matrix of the system.

7.6. Operand MATR_RIGI#

♦ MATR_RIGI = k

Name of the concept assembled matrix \(\mathrm{K}\) corresponding to the stiffness matrix of the system. Hysterical damping is achieved with a complex stiffness matrix.

7.7. Operand MATR_AMOR#

◊ MATR_AMOR = c

Name of the concept assembled matrix \(\mathrm{C}\) corresponding to the viscous damping matrix of the system.

7.8. Keyword AMOR_MODAL#

Keyword factor to specify viscous damping in the form of lists of reduced modal damping (percentage of critical damping) with the following operands, only possible in the case of a harmonic analysis on a modal basis.

7.8.1. Operands AMOR_REDUIT/LIST_AMOR#

/AMOR_REDUIT = the

List of all reduced modal viscous amortizations: \(({\eta }_{1},{\eta }_{2},\mathrm{...},{\eta }_{n})\).

**Note: If the number of reduced depreciations given is less than the number of base vectors used in the modal base, the depreciations of the additional vectors are taken equal to the last damping in the list.*

/LIST_AMOR = l_love

Name of the listr8 type concept containing the list of reduced viscous modal damping.

7.9. Operand MATR_IMPE_PHI#

◊ MATR_IMPE_PHI = imp

Name of the concept assembled matrix \(\mathrm{Q}\) corresponding to the impedance matrix for a fluid-structure system whose formulation is in displacement-pressure-potential \((\mathrm{u},p,\varphi )\) [R4.02.02].

7.10. Operands FREQ/LIST_FREQ#

♦/FREQ = lf

List of all calculation frequencies \(f\): (f1, f2,…, fn).

/LIST_FREQ = cf

Name of the listr8 type concept containing the list of calculation frequencies \(f\).

7.11. Operands TOUT_CHAM/NOM_CHAM#

◊/TOUT_CHAM = 'OUI'

/CHAM_NAME = | “DEPL” | “FAST” | “ACCESS”

Choice of fields to calculate to represent the answer: displacement, speed, acceleration, or all three.

7.12. Keyword EXCIT#

♦ EXCIT

Keyword factor used to define several types of excitations. Either by indicating an assembled vector corresponding to a loading, or loads that will lead to the calculation and assembly of a second member. For each occurrence of the keyword factor, a component of arousal is defined in the form \((h(f),\mathrm{g}(P),\phi )\), which can also be multiplied by multiplier functions. This keyword must be repeated as many times as there are load vectors.

7.12.1. Operands VECT_ASSE/VECT_ASSE_GENE/CHARGE#

Allow to define \({\mathrm{g}}_{i}(P)\) spatial discretization of the load, in the form of a field at the nodes corresponding to one or more loads of force or imposed movement.

♦/VECT_ASSE = vecti

Name of the concept produced by:

  • the operator ASSE_VECTEUR in imposed force or in force of inertia driving in acceleration imposed in an absolute frame of reference. Excitation amplitudes can be defined in the corresponding load type concepts. The expected field is a field with nodes of magnitude DEPL_R, DEPL_C, or PRES_C.

/VECT_ASSE_GENE = vect_gene

Name of the concept produced by:

  • the PROJ_VECT_BASE operator which allows you to project an assembled vector (imposed force or imposed acceleration training force in an absolute frame of reference) on a modal basis or a Ritz base.

  • the ASSE_VECT_GENE operator which allows you to project a loading on a base defined on a generalized model for dynamic substructuring calculations.

/CHARGE = chi

chi name of the load concept created by calling command AFFE_CHAR_MECA [U4.44.01].

The MODELE keyword must be entered if the CHARGE keyword is used.

7.12.2. Operands FONC_MULT_C/COEF_MULT_C/FONC_MULT/COEF_MULT#

Allow to define \({h}_{i}(f)\) function, complex or real, defining the dependence of the excitation as a function of the frequency, applied to all the components of the field at the node associated with this occurrence. Several possibilities are offered:

♦/FONC_MULT_C = hci

Name of the function_C or formula_C concept defining a complex \({h}_{i}(f)\) function of the frequency \(f\),

/COEF_MULT_C = aci

Complex coefficient multiplying the load, independent of the load,

/FONC_MULT = hi

Function concept, formula or sheet defining a real \({h}_{i}(f)\) function of the frequency \(f\),

/COEF_MULT = ai

Real factor that multiplies the load, independent of the load.

7.12.3. Operand PUIS_PULS#

◊ PUIS_PULS = neither

Allows you to define the power of the pulse, see eq. (), when the load is a function of the frequency; by default \({n}_{i}=0\). For example to take into account the time derivation of a load; example: imposed displacement vs imposed speed…

7.12.4. Operand PHAS_DEG#

◊ PHAS_DEG = phi

Allows you to define the phase of each component of the excitation in degrees in relation to a single phase reference for all excitations \({h}_{i}(f)\); by default \({\varphi }_{i}\mathrm{=}0\).

7.12.5. note#

For a problem with movement imposed while moving, we define the blocked degrees of freedom (kinematic conditions prior to the construction of the fiel_no); we can then choose an excitation:

  • in imposed displacement \(n\mathrm{=}0\), \(\varphi \mathrm{=}0\) degrees,

  • in imposed speed \(n\mathrm{=}1\), \(\varphi \mathrm{=}90\) degrees,

  • in imposed acceleration \(n\mathrm{=}2\), \(\varphi \mathrm{=}180\) degrees.

7.13. Operand EXCIT_RESU#

◊ EXCIT_RESU

This keyword factor makes it possible to define several loads in the form of a dyna_harmo-type harmonic evolution of vectors assembled second members, calculated on a physical basis. This possibility is available on a physical basis and on a reduced modal or Ritz basis.

7.13.1. Operand RESULTAT#

This keyword makes it possible to define the second members to be extracted for each calculation frequency from a result already calculated including fields of nodal forces.

♦ RESULTAT = strength

Name of the concept of harmonic evolution of second members produced by the combination of the operator CALC_FORC_NONL [U4.84.21] and then the operator REST_SPEC_TEMP [U4.63.34]. The operator CALC_FORC_NONLpermet to produce a transitory evolution of second members. The operator REST_SPEC_TEMP transforms this transitory evolution into a harmonic evolution. An example of use is provided in test case SDLS119A.

7.13.2. Operand COEF_MULT_C#

♦ COEF_MULT_C = aci

Complex multiplier coefficient of the second member vector extracted from the resuforc result for each calculation frequency.

7.14. Operand INFO#

◊ INFO = under

Allows you to perform various intermediate prints in the message file to monitor the progress of the calculation.

By default, if INFO =1, the progress of the harmonic calculation is printed with a step that corresponds to a maximum of 5% of the total number of frequencies. The general rule is to always print the first and last frequencies as well as a maximum number of 20 frequencies in the middle.

If INFO =2, an impression is made for each frequency and makes it possible to monitor the progress of the calculation more precisely.