2. Benchmark solution#

2.1. Calculation method used for the reference solution#

The spectral response is calculated by modal superposition of a 2 mass—4 spring system subjected to three distinct excitations. We determine the movement of the masses at nodes \(\mathrm{NO2}\) and \(\mathrm{NO4}\) along the \(x\) axis.

Analytically, we calculate:

natural frequencies \({f}_{i}\),
the associated eigenvectors \({\phi }_{{N}_{i}}\) normalized with respect to the modal mass,
the system’s \({\Psi }_{i}\) static support modes,
the \({P}_{\mathrm{ij}}\) modal participation factors relating to support,
\({\mathrm{Rm}}_{\mathrm{ij}}\) the maximum response of each mode based on the excitation spectra,
\({R}_{\mathrm{ij}}\) the contribution of the driving movement of each support from the differential movements,

2.2. Benchmark results#

2.2.1 Characteristic matrices and vectors

stiffness matrix \(K\)

\(K=\left[\begin{array}{ccccc}{k}_{1}& -{k}_{1}& 0& 0& 0\\ -{k}_{1}& {k}_{1}+{k}_{2}& -{k}_{2}& 0& 0\\ 0& -{k}_{2}& {k}_{2}+{k}_{3}& -{k}_{3}& 0\\ 0& 0& -{k}_{3}& {k}_{3}+{k}_{4}& -{k}_{4}\\ 0& 0& 0& -{k}_{4}& {k}_{4}\end{array}\right]\) matrix relating to degrees of freedom 1, 2, 2, 3, 4, 5

\({K}^{p}=\left[\begin{array}{ccccc}{k}_{1}+{k}_{2}& 0& -{k}_{1}& -{k}_{2}& 0\\ 0& {k}_{3}+{k}_{4}& 0& -{k}_{3}& -{k}_{4}\\ -{k}_{1}& 0& {k}_{1}& 0& 0\\ -{k}_{2}& -{k}_{3}& 0& {k}_{2}+{k}_{3}& 0\\ 0& -{k}_{4}& 0& 0& {k}_{4}\end{array}\right]\)

partitioned matrix degrees of freedom of structure 2, 4, degrees of freedom of support 1, 3, 5

\({K}^{p}=\left[\begin{array}{cc}k& {k}_{\mathrm{xs}}\\ {k}_{\mathrm{sx}}& {k}_{\mathrm{ss}}\end{array}\right]\) \(k=\left[\begin{array}{cc}{k}_{1}+{k}_{2}& 0\\ 0& {k}_{3}+{k}_{4}\end{array}\right]\) \({k}_{\mathrm{xs}}=\left[\begin{array}{cccc}-{k}_{1}& 0& -{k}_{2}& 0\\ 0& -{k}_{3}& 0& -{k}_{4}\end{array}\right]\)

mass matrix \(M\)

\(M=\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& {m}_{1}& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& {m}_{2}& 0\\ 0& 0& 0& 0& 0\end{array}\right]\)

matrix relating to degrees of freedom 1, 2, 3, 4, 5

\({M}^{p}=\left[\begin{array}{ccccc}{m}_{1}& 0& 0& 0& 0\\ 0& {m}_{2}& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right]\)

partitioned matrix degrees of freedom of structure 2, 4, degrees of freedom of support 1, 3, 5

modal calculation in embedded basis

\((K-{\lambda }_{i}M){\phi }_{i}=0\) with \({\lambda }_{i}={\omega }_{i}^{2}\)

\(\mathrm{dét}(K-{\lambda }_{i}M)=0\iff {\lambda }_{i}^{2}-(\frac{{k}_{1}+{k}_{2}}{{m}_{1}}+\frac{{k}_{3}+{k}_{4}}{{m}_{2}}){\lambda }_{i}+\frac{({k}_{1}+{k}_{2})({k}_{3}+{k}_{4})}{{m}_{1}{m}_{2}}=0\)

\({\lambda }_{1}=\frac{2k}{m}\) \({\lambda }_{2}=\frac{4k}{m}\)

natural frequencies:

\(\Rightarrow\) \({f}_{1}=\frac{{\omega }_{1}}{2\pi }\) \({f}_{2}=\frac{{\omega }_{2}}{2\pi }\)

non-standard natural modes:

\({\phi }_{1}=\left[\begin{array}{}0\\ 1\\ 0\\ 0\\ 0\end{array}\right]\) \({\phi }_{2}=\left[\begin{array}{}0\\ 0\\ 0\\ 1\\ 0\end{array}\right]\)

generalized modal masses \({\mu }_{i}={\phi }_{i}^{T}M{\phi }_{i}\):

\({\mu }_{1}={\mu }_{2}=m\)

eigenmodes normalized to the unit generalized modal mass \({\phi }_{\mathrm{Nk}}\):

\(\Rightarrow\) \({\phi }_{\mathrm{N1}}=\frac{{\phi }_{1}}{\sqrt{{\mu }_{1}}}\) \({\phi }_{\mathrm{N2}}=\frac{{\phi }_{2}}{\sqrt{{\mu }_{2}}}\)

static support modes \({\Psi }_{\mathrm{Sj}}\)

Static mode matrix reduced to structure ddls \({\varphi }_{S}=-{k}^{-1}{k}_{\mathrm{xs}}\)

\({\varphi }_{S}=\frac{-1}{4k}\left[\begin{array}{cc}2& 0\\ 0& 1\end{array}\right]\left[\begin{array}{ccc}-k& -k& 0\\ 0& -2k& -2k\end{array}\right]=\left[\begin{array}{ccc}& & 0\\ 0& & \end{array}\right]\)

static solution to a unit movement of node \(\mathrm{NO1}\):

trips: \({\Psi }_{\mathrm{S1}}=\left[\begin{array}{}1\\ \\ 0\\ 0\\ 0\end{array}\right]\)

static solution to a unit movement of node \(\mathrm{NO3}\):

trips: \({\Psi }_{\mathrm{S2}}=\left[\begin{array}{}0\\ \\ 1\\ \\ 0\end{array}\right]\)

static solution to a unit movement of node \(\mathrm{NO5}\):

trips: \({\Psi }_{\mathrm{S3}}=\left[\begin{array}{}0\\ 0\\ 0\\ \\ 1\end{array}\right]\)

rigid body fashion \({\psi }_{\mathrm{R1}}\)

Rigid mode matrix reduced to structure ddls: \({\varphi }_{R}={\varphi }_{S}{S}_{R}\)

Rigid body fashion \({\psi }_{\mathrm{R1}}=\left[\begin{array}{}1\\ 1\\ 1\\ 1\\ 1\end{array}\right]\)

We check that: \({\psi }_{\mathrm{R1}}={\Psi }_{\mathrm{S1}}+{\Psi }_{\mathrm{S2}}+{\Psi }_{\mathrm{S3}}\)

2.2.2 Modeling A: Loading 1 multi-support

modal participation factors \({P}_{\mathrm{kj}}={\phi }_{\mathrm{Nk}}^{T}M{\Psi }_{j}\):
contribution of dynamic mode 1 to the movement imposed on node \(\mathrm{NO1}\):

\(\Rightarrow\) \({P}_{11}={\phi }_{\mathrm{N1}}^{T}M{\Psi }_{1}=\frac{\sqrt{m}}{2}\)

contribution of dynamic mode 1 to the movement imposed on node \(\mathrm{NO3}\):

\(\Rightarrow\) \({P}_{12}={\phi }_{\mathrm{N1}}^{T}M{\Psi }_{2}=\frac{\sqrt{m}}{2}\)

contribution of dynamic mode 1 to the movement imposed on node \(\mathrm{NO5}\):

\(\Rightarrow\) \({P}_{13}={\phi }_{\mathrm{N1}}^{T}M{\Psi }_{3}=0\)

contribution of dynamic mode 2 to the movement imposed on node \(\mathrm{NO1}\):

\(\Rightarrow\) \({P}_{21}={\phi }_{\mathrm{N2}}^{T}M{\Psi }_{1}=0\)

contribution of dynamic mode 2 to the movement imposed on node \(\mathrm{NO3}\):

\(\Rightarrow\) \({P}_{22}={\phi }_{\mathrm{N2}}^{T}M{\Psi }_{2}=\frac{\sqrt{m}}{2}\)

contribution of dynamic mode 2 to the movement imposed on node \(\mathrm{NO5}\):

\(\Rightarrow\) \({P}_{23}={\phi }_{\mathrm{N2}}^{T}M{\Psi }_{3}=\frac{\sqrt{m}}{2}\)

mode response \(i\) to the support movement \(j\)

\({\mathrm{Rm}}_{\mathrm{kj}}={\phi }_{\mathrm{Nk}}{P}_{\mathrm{kj}}\frac{{A}_{\mathrm{kj}}}{{\omega }_{i}^{2}}\)

Combined modal oscillator responses

Mode 1 response to support movement1: \({\mathrm{Rm}}_{11}={\phi }_{\mathrm{N1}}{P}_{11}\frac{{A}_{11}}{{\omega }_{1}^{2}}=\frac{{A}_{11}}{2{\omega }_{1}^{2}}(\begin{array}{c}1\\ 0\end{array})\)

Mode 1 response to support movement2: \({\mathrm{Rm}}_{12}={\phi }_{\mathrm{N1}}{P}_{12}\frac{{A}_{12}}{{\omega }_{1}^{2}}=\frac{{A}_{12}}{2{\omega }_{1}^{2}}(\begin{array}{c}1\\ 0\end{array})\)

Mode 1 response to support movement3: \({\mathrm{Rm}}_{13}={\phi }_{\mathrm{N1}}{P}_{13}\frac{{A}_{13}}{{\omega }_{1}^{2}}=(\begin{array}{c}0\\ 0\end{array})\)

Mode 2 response to support movement1: \({\mathrm{Rm}}_{21}={\phi }_{\mathrm{N2}}{P}_{21}\frac{{A}_{21}}{{\omega }_{2}^{2}}=(\begin{array}{c}0\\ 0\end{array})\)

Mode 2 response to the support2 movement: \({\mathrm{Rm}}_{22}={\phi }_{\mathrm{N2}}{P}_{22}\frac{{A}_{22}}{{\omega }_{2}^{2}}=\frac{{A}_{22}}{2{\omega }_{2}^{2}}(\begin{array}{c}0\\ 1\end{array})\)

Mode 2 response to support movement3: \({\mathrm{Rm}}_{23}={\phi }_{\mathrm{N2}}{P}_{23}\frac{{A}_{23}}{{\omega }_{2}^{2}}=\frac{{A}_{23}}{2{\omega }_{2}^{2}}(\begin{array}{c}0\\ 1\end{array})\)

intra-group accumulation (algebraic sum)

Fashion 1: \({\mathrm{Rm}}_{\mathrm{1groupe1}}={\mathrm{Rm}}_{11}+{\mathrm{Rm}}_{12}=\frac{{A}_{11}+{A}_{12}}{2{\omega }_{1}^{2}}(\begin{array}{c}1\\ 0\end{array})\)

mode 2: \({\mathrm{Rm}}_{\mathrm{2groupe1}}={\mathrm{Rm}}_{21}+{\mathrm{Rm}}_{22}=\frac{{A}_{22}}{2{\omega }_{2}^{2}}(\begin{array}{c}0\\ 1\end{array})\)

contribution of support \(j\) to the training movement

\({R}_{j}={\Psi }_{j}{D}_{j}\)

contributions of groups 1 and 2 to the training movement

\({R}_{\mathrm{groupe1}}={R}_{1}+{R}_{2}=\frac{1}{2}(\begin{array}{c}{D}_{1}+{D}_{2}\\ {D}_{2}\end{array})\) \({R}_{\mathrm{groupe2}}={R}_{3}=\frac{1}{2}(\begin{array}{c}0\\ {D}_{3}\end{array})\)

accumulation of modes (quadratic)

\({\mathit{Rm}}_{\mathit{groupe}1}=\sqrt{{({\mathit{Rm}}_{11}+{\mathit{Rm}}_{12})}^{2}+{({\mathit{Rm}}_{21}+{\mathit{Rm}}_{22})}^{2}}=\left(\begin{array}{c}\frac{{A}_{11}+{A}_{12}}{2{\omega }_{1}^{2}}\\ \frac{{A}_{22}}{2{\omega }_{2}^{2}}\end{array}\right)\)

\({\mathrm{Rm}}_{\mathrm{groupe2}}=\sqrt{{\mathrm{Rm}}_{13}^{2}+{\mathrm{Rm}}_{23}^{2}}=\frac{{A}_{23}}{2{\omega }_{2}^{2}}(\begin{array}{c}0\\ 1\end{array})\)

response from support groups 1 and 2

\({R}_{1}=\sqrt{{\mathrm{Rm}}_{\mathrm{groupe1}}^{2}+{R}_{\mathrm{groupe1}}^{2}}\) \({R}_{1}^{2}=\frac{1}{4}\left[\begin{array}{c}\frac{{({A}_{11}+{A}_{12})}^{2}}{{\omega }_{1}^{2}}+{({D}_{1}+{D}_{2})}^{2}\\ \frac{{A}_{22}^{2}}{{\omega }_{2}^{2}}+{D}_{2}^{2}\end{array}\right]\)

\({R}_{2}=\sqrt{{\mathrm{Rm}}_{\mathrm{groupe2}}^{2}+{R}_{\mathrm{groupe2}}^{2}}\) \({R}_{2}^{2}=\frac{1}{4}\left[\begin{array}{c}0\\ \frac{{A}_{23}^{2}}{{\omega }_{2}^{2}}+{D}_{3}^{2}\end{array}\right]\)

inter-group accumulation (quadratic)

\(R=\sqrt{{R}_{1}^{2}+{R}_{2}^{2}}\) \({R}^{2}=\frac{1}{4}\left[\begin{array}{c}\frac{{({A}_{11}+{A}_{12})}^{2}}{{\omega }_{1}^{2}}+{({D}_{1}+{D}_{2})}^{2}\\ \frac{{A}_{22}^{2}+{A}_{23}^{2}}{{\omega }_{2}^{2}}+{D}_{2}^{2}+{D}_{3}^{2}\end{array}\right]\)

2.2. 3 M B modeling

It is noted here that, only the answer to DDS is considered to validate the combination rules to be used in this B modeling.

contribution of support \(j\) to the training movement

\({R}_{j}={\Psi }_{j}{D}_{j}\)

\({R}_{e1}={\mathrm{\Psi }}_{S1}{D}_{1}=\left[\begin{array}{c}{D}_{1}\\ \raisebox{1ex}{{D}_{1}}\!\left/ \!\raisebox{-1ex}{2}\right.\\ 0\\ 0\\ 0\end{array}\right]\)

\({R}_{e2}={\mathrm{\Psi }}_{S2}{D}_{1}=\left[\begin{array}{c}0\\ \raisebox{1ex}{{D}_{1}}\!\left/ \!\raisebox{-1ex}{2}\right.\\ {D}_{1}\\ \raisebox{1ex}{{D}_{1}}\!\left/ \!\raisebox{-1ex}{2}\right.\\ 0\end{array}\right]\)

\({R}_{e3}={\mathrm{\Psi }}_{S3}{D}_{2}=\left[\begin{array}{c}0\\ 0\\ 0\\ \raisebox{1ex}{{D}_{2}}\!\left/ \!\raisebox{-1ex}{2}\right.\\ {D}_{2}\end{array}\right]\)

and to the support reactions:

\({F}_{1}=k{D}_{1}\left[\begin{array}{c}\raisebox{1ex}{1}\!\left/ \!\raisebox{-1ex}{2}\right.\\ 0\\ \raisebox{1ex}{-1}\!\left/ \!\raisebox{-1ex}{2}\right.\\ 0\\ 0\end{array}\right]\)

\({F}_{2}=k{D}_{1}\left[\begin{array}{c}\raisebox{1ex}{-1}\!\left/ \!\raisebox{-1ex}{2}\right.\\ 0\\ \raisebox{1ex}{3}\!\left/ \!\raisebox{-1ex}{2}\right.\\ 0\\ -1\end{array}\right]\)

\({F}_{3}=k{D}_{2}\left[\begin{array}{c}0\\ 0\\ -1\\ 0\\ 1\end{array}\right]\)

In this modeling B, the accumulation of uncorrelated support groups with DDS equal is presented in calculation 1 where the intra-group accumulation is in absolute value and the inter-group accumulation is quadratic. The absolute value combination for intra-group accumulation is used here because the differential movements at the supports are unsigned. As a counterexample, calculations 2 and 3 are also established where the intra-group and inter-group accumulations are respectively linear and quadratic.

Calculation 1: Combining uncorrelated support groups with DDS equal intra-group:

math:: `{R} _ {e} =sqrt {{(left| {R} | {R} _ {e1}right|+left| {R} _ {e2}right|)} ^ {2} |)} ^ {2}} +} ^ {2}} =\ left [\ begin {array} {c}\ right|)} ^ {2}} =left [begin {array} {c}left| {D} {2}} + {D} {2}} + {D} {2}} + {D} _ {2}right| {D} {2}} + {D} {2}} + {D} {2}right| {D} {2}} + {D} {2}} + {D} {2} |\\\ left| {D} _ {1}right|\ left|\ left| {D} _ {1}right|\ sqrt {{(raisebox {1ex} {1ex}} {{ex}} {{D} _ {1}}! left/! raisebox {-1ex} {2}right.)} ^ {2} + {(raisebox {1ex} {{D} _ {2}}}! left/! raisebox {-1ex} {2}right.)} ^ {2}}}\left| {D} _ {2}right|end {array}right] `
math:: `F=sqrt {{(left| {left| {F} | {F} _ {1} _ {2}right|)} ^ {2} + {({F} _ {3})}}} ^ {3})}} ^ {2}}} =kleft [begin {array} {c}left| {D} _ {1}} + {({F} _ {3})}} {3})} ^ {3})}} ^ {3})}} ^ {2}}} =kleft [begin {array} {c}left| {D} _ {1}right|\ 0\sqrt {{(2 {D} _ {1})} ^ {1})} ^ {1})} ^ {2})} ^ {2}}\ 0\ sqrt {{({D} _ {1})} ^ {1})} ^ {2})} ^ {2}}end {array} _ {1})} ^ {2}}end {array}right] `

Calculation 2: Intra-group and inter-group linear**combination*

\({R}_{e}={R}_{e1}+{R}_{e2}+{R}_{e3}=\left[\begin{array}{c}{D}_{g1}\\ {D}_{1}\\ {D}_{1}\\ \raisebox{1ex}{({D}_{1}+{D}_{2})}\!\left/ \!\raisebox{-1ex}{2}\right.\\ {D}_{2}\end{array}\right]\)

\(F={F}_{1}+{F}_{2}+{F}_{3}=k\left[\begin{array}{c}0\\ 0\\ {D}_{1}\\ 0\\ -({D}_{1}+{D}_{2})\end{array}\right]\)

Calculation 3: C**combination* quadratic within-group and inter-group

math:: `{R} _ {e} =sqrt {{({R} _ {e1})} ^ {2} + {({R} _ {e2})} ^ {2} + {({R} _ {e3}) =sqrt {{({R} _ {e3}) =sqrt {({R} _ {e}) ={2}}} =left [begin {array} {c}left| {D} _ {1}\\{1}right|\ raise| {D} _ {1}right|\ raise| box {1ex} {left| {D} _ {1}right|}! left/! raisebox {-1ex} {2}right.\left| {D} _ {1}right|\ sqrt {{(raisebox {1ex} {{ex} {{D}} _ {1}}! left/! raisebox {-1ex} {2}right.)} ^ {2} + {(raisebox {1ex} {{D} _ {2}}}! left/! raisebox {-1ex} {2}right.)} ^ {2}}}\left| {D} _ {2}right|end {array}right] `
math:: `F=sqrt {{({F} _ {1})} ^ {1})} ^ {2})} ^ {2})} ^ {2} + {({F} _ {3})}}} ^ {2})}} ^ {2}}} =kleft [left [begin {array} {c} {c}raisebox {1ex} {left| {D} _ {3})}} ^ {3})}} ^ {3})} ^ {2})} ^ {3})} ^ {2})} ^ {2})} ^ {2})} ^ {2})} ^ {2})} ^ {2})} ^ {2})} ^ {2})} ^ {2})}! left/! raisebox {-1ex} {sqrt {2}}}right.\ 0\sqrt {raisebox {1ex} {{(sqrt {10} {D} {D} {D} _ {1}})}{1})}! left/! raisebox {-1ex} {4}right. + {({D} _ {2})} ^ {2}}}\ 0\sqrt {{({D} _ {1})} ^ {2})} ^ {2})} ^ {2})} ^ {2}}}end {array}right] `

2.2. 4 M C modeling

C modeling consists in testing the output options for a spectral modal calculation in correlated multi-support, that is to say, all the supports correlated with each other. Thus, a single support group is formed. All support contributions are combined in LINE by default. Modal accumulation is performed by rule SRSS.

The seismic load consists of three spectra in VITE, which are converted from the spectra in ACCE in models A and B by dividing them by the pulsations associated with the two modes \({f}_{1}=2.25079\) Hz and \({f}_{1}=3.1831\) Hz. The numerical values are as follows:

at node \(\mathit{NO}1\):

\({\mathit{SRO}}_{\mathit{NO}1}^{\mathit{VITE}}({f}_{1})={A}_{11}/(2\mathrm{\pi }{f}_{1})=7/(2\mathrm{\pi }2.25079)m/s\)

\({\mathit{SRO}}_{\mathit{NO}1}^{\mathit{VITE}}({f}_{2})={A}_{21}/(2\mathrm{\pi }{f}_{2})=5/(2\mathrm{\pi }3.1831)m/s\)

at node \(\mathit{NO}3\):

\({\mathit{SRO}}_{\mathit{NO}3}^{\mathit{VITE}}({f}_{1})={A}_{12}/(2\mathrm{\pi }{f}_{1})=7.7/(2\mathrm{\pi }2.25079)m/s\)

\({\mathit{SRO}}_{\mathit{NO}3}^{\mathit{VITE}}({f}_{2})={A}_{22}/(2\mathrm{\pi }{f}_{2})=5.5/(2\mathrm{\pi }3.1831)m/s\)

at node \(\mathit{NO}5\):

\({\mathit{SRO}}_{\mathit{NO}5}^{\mathit{VITE}}({f}_{1})={A}_{13}/(2\mathrm{\pi }{f}_{1})=12/(2\mathrm{\pi }2.25079)m/s\)

\({\mathit{SRO}}_{\mathit{NO}5}^{\mathit{VITE}}({f}_{2})={A}_{23}/(2\mathrm{\pi }{f}_{2})=6/(2\mathrm{\pi }3.1831)m/s\)

Analytical solution for responses on the go ( DEPL ) :

The mode 1 response:

\({\mathit{Rm}}_{1}={\mathit{Rm}}_{11}+{\mathit{Rm}}_{12}+{\mathit{Rm}}_{13}=\frac{1}{2{\mathrm{\omega }}_{1}^{2}}\left[\begin{array}{c}0\\ {A}_{11}+{A}_{12}\\ 0\\ 0\\ 0\end{array}\right]\)

The mode 2 response:

\({\mathit{Rm}}_{2}={\mathit{Rm}}_{21}+{\mathit{Rm}}_{22}+{\mathit{Rm}}_{23}=\frac{1}{2{\mathrm{\omega }}_{2}^{2}}\left[\begin{array}{c}0\\ 0\\ 0\\ {A}_{22}+{A}_{23}\\ 0\end{array}\right]\)

The total dynamic response by quadratic accumulation of two modes:

math:: mathit {Rm} =sqrt {{mathit {Rm}}} _ {1} ^ {2} + {mathit {Rm}} _ {2} ^ {2}} =left [begin {array} {2}}} =left [begin {array} {{array} {c}} {c}} 0\frac {left| {A} _ {11}} + {A} _ {12} {2}}} =left [begin {array} {{array} {c} {c} 0\ frac {left| {A} _ {11} + {A} _ {12}\ right|} {2}right|} {2}right|} {2}right|} {mathrm {omega}} _ {1} ^ {2}}}\ 0\frac {left| {A} _ {22} + {A} _ {23}right|} {2 {mathrm {omega}}}\ 0end {array}right|}} {2}}\ 0end {array}right|}} {2}}\ 0end {array}right|}} {2}}\ 0end {array}right|}} {2}} {2}}\ 0end {array}right|} {2}right]

Responses in reaction to supports: (REAC_NODA) :

The mode 1 response:

\({F}_{1}=K({\mathit{Rm}}_{11}+{\mathit{Rm}}_{12}+{\mathit{Rm}}_{13})=\left[\begin{array}{c}-{k}_{1}\frac{{A}_{11}+{A}_{12}}{2{\mathrm{\omega }}_{1}^{2}}\\ 0\\ -{k}_{2}\frac{{A}_{11}+{A}_{12}}{2{\mathrm{\omega }}_{1}^{2}}\\ 0\\ 0\end{array}\right]\)

The mode 2 response:

\({F}_{2}=K({\mathit{Rm}}_{21}+{\mathit{Rm}}_{22}+{\mathit{Rm}}_{23})=\left[\begin{array}{c}0\\ 0\\ -{k}_{3}\frac{{A}_{22}+{A}_{23}}{2{\mathrm{\omega }}_{2}^{2}}\\ 0\\ -{k}_{4}\frac{{A}_{22}+{A}_{23}}{2{\mathrm{\omega }}_{2}^{2}}\end{array}\right]\)

The total dynamic response by quadratic accumulation of two modes:

math:: `F=sqrt {{F} _ {1} _ {1} ^ {2} + {F} _ {2}}} =left [begin {array} {c}frac {{k} _ {1} _ {1} _ {1}left| {1}left| {1}left| {1}left| {A} _ {11}} + {A} _ {11}} + {A} _ {11}} + {A} _ {12}right|} {2}right|} {2 {mathrm {omega}} _ {1} _ {1}} ^ {2}}\ 0\ sqrt {{(frac {{k}} _ {1}left| {A} _ {11} + {A} _ {12}right|}} {2 {mathrm {sqrt {sqrt {sqrt {omega}}}} _ {omega}}} _ {omega}}} _ {omega}} _ {omega}} _ {1} ^ {2}})} ^ {2}} + {(frac {{k} _ {4}left|}} {2} + {(frac {{k}} | right|}}} {2}right|}} {2}right|}} {2}right|}} {2}right|}} {| {A} _ {22} + {A} _ {23}right|} {2 {mathrm {omega}} _ {2}})} ^ {2}}\ 0\frac {{\ frac {{k} _ {23}}}frac {{23}\ frac {{23}frac {frac {{k} _ {4} _ {4}left| {4}left| {A} _ {23}}right|} {2}right|} {2}right|} {2}right|} {2}right|} {2}right|} {2}right|} {2 {mathrmomega}} _ {2} ^ {2}}}end {array}right] `

2.3. Uncertainty about the solution#

None (analytical solution)