2. Benchmark solution

2.1. Calculation method

Empirical modes are calculated. The modes are tested by non-regression point values on different points and different ddls.

2.2. Reference quantities and results

We are trying to solve the following system:

(2.1)\[ \ left (-i {\ mathrm {\ omega}}} ^ {3} I (\ mathrm {\ mu}) - {\ mathrm {\ omega}} ^ {2} (\ mathrm {\ mu}) M (\ mathrm {\ mu})} ^ {3}} I (\ mathrm {\ mu}}) =F () +K (\ mathrm {\ mu}) +K (\ mathrm {\ mu}) =F (\ mathrm {\ mu})\]

The coefficient \(\mathrm{\mu }\) represents the variation parameters, \(I\) the impedance matrix, the impedance matrix, \(M\) the mass matrix, \(K\) the stiffness matrix and \(F\) the second member (applying the nodal force). The variable parameters are as follows: \(\mathrm{\omega }\), \({E}_{a}\),, \({\mathrm{\rho }}_{a}\), and \({\mathrm{\rho }}_{v}\).

The system is written in the following form:

(2.2)\[ {f} _ {1} (\ mathrm {\ omega}, {E} _ {a})\ mathrm {.} {K} _ {v} + {f} _ {2} (\ mathrm {\ omega}, {\ mathrm {\ rho}} _ {a})\ mathrm {.} {K} _ {e} + {K} _ {f} + {f} _ {3} (\ mathrm {\ omega}, {\ mathrm {\ rho}}} _ {v})\ mathrm {.} {M} _ {e} - {\ mathrm {\ omega}}} ^ {2} {2} {M} _ {f} -j {\ mathrm {\ omega}} ^ {3}\ mathrm {\ omega}}\ mathrm {.} I=F\]

We have separated the different contributions: \({K}_{v}\) for the stiffness coming from the viscoelastic part (calculated by RIGI_MECA_HYST), \({K}_{e}\) for the stiffness coming from the elastic part, \({K}_{f}\) for the stiffness coming from the fluid part, \({M}_{e}\) for the mass coming from the elastic part, \({M}_{f}\) for the mass coming from the elastic part, for the mass coming from the elastic part, for the mass coming from the fluid part and \(I\) the matrix impedance.

And the functions:

\[\]

: label: eq-4

{f} _ {1} (mathrm {omega}, {E} _ {a}) =2times {E} _ {a} (1.0+ {mathrm {nu}} _ {v})frac {omega})frac {omega}})frac {frac {{G} _ {0} _ {a} (1.0+ {mathrm {nu}} _ {v})frac {frac {{omega}})frac {frac {{G} _ {g} _ {v})frac {frac {{G} _ {a}) thrm {omega}mathrm {tau})}}} ^ {mathrm {tau})}} {1+ {(mathit {jc}mathrm {omega}mathrm {tau})}}} ^ {mathrm {tau})}} ^ {mathrm {tau})}} ^ {tau})}} ^ {mathrm {alpha}}}} {{mathrm {alpha}}}} {{mathrm {rho}}} _ {e} {c} {tau})}} ^ {tau})} ^ {tau})}} ^ {mathrm {tau})}} ^ {mathrm {tau})}} ^ 2j {E} _ {a} (1.0+ {mathrm {nu}}} _ {v})frac {frac {{G} _ {0} + {G} _ {mathrm {infty}}} {mathrm {infty}}} {mathrm {infty}}} {infty}} {infty}} {mathrm {infty}}} {(mathrm {infty}}} {(mathrm {infty}}} {infty}}} {(mathrm {infty}}} {(mathrm {infty}}}} {(mathrm {infty}}}} {mathrm {infty} {1+ {(mathit {jc}mathrm {omega}mathrm {omega}mathrm {tau})}} {{mathrm {rho}}} _ {e} {rho}}} _ {e} {c} {c}} ^ {2}}timesfrac {mathrm {alpha}}}}} {{mathrm {rho}}}} {{mathrm {rho}}} _ {0} _ {0}} _ {0} + {G} _ { mathrm {infty}} {(mathit {jc}mathrm {omega}mathrm {tau})} ^ {mathrm {alpha}}} {1+ {(mathit {jc}mathrm {jc}mathrm {jc}}mathrm {omega}}right)} {omega}mathrm {tau})} ^ {mathrm {alpha}}} {alpha}}} {alpha}}} {alpha}} {alpha}}} {alpha}} {(mathit {jc}}right)} {jc}right)} {jc}right)} {omega}} mathrm {Re}left (frac {{G} _ {0} _ {0} _ {G} _ {mathrm {infty}} {(mathit {jc}mathrm {omega}mathrm {omega}}mathrm {tau})}} ^ {mathrm {tau})}} ^ {mathrm {alpha}}} {1+ {(mathit {jc}}mathrm {omega}mathrm {tau})}} ^ {mathrm {tau})}} ^ {mathrm {tau}})} {mathrm {alpha}}} {1+ {(mathit {jc}}mathrm {omega} mathrm {tau})}}} ^ {mathrm {alpha}}}right)}

\[\]

: label: eq-5

{f} _ {2} (mathrm {omega}}, {mathrm {rho}} _ {a}) =frac {{mathrm {rho}} _ {a} {E}} _ {E}} _ {e} {e} {mathrm {rho}} _ {e} {c} ^ {2}}}

(2.3)\[ {f} _ {3} (\ mathrm {\ omega}}, {\ mathrm {\ rho}} _ {v}) =\ frac {- {\ mathrm {\ omega}} ^ {2} {\ mathrm {\ omega}} {\ mathrm {\ rho}} _ {\ mathrm {\ rho}} _ {\ mathrm {\ rho}} _ {a}} {{\ mathrm {\ rho}} _ {a}} {\ mathrm {\ rho}}} _ {e}}\]

For the calculation, we vary the four parameters over ten values (we do not copy here the value of these parameters that were drawn at random, see the command file)

2.3. Uncertainties about the solution

The error on the solution depends on the degree of reduction (number of empirical modes).