2. Benchmark solution#

2.1. Calculation method used for the reference solution#

The spectral response is calculated by modal superposition of a spring mass system subjected to two distinct excitations. The movement of the masses and the support reactions are determined at nodes \(\mathit{NO1}\) and \(\mathit{NO4}\) along the \(x\) axis.

Analytically, we calculate:

  • natural frequencies \({f}_{i}\),

  • the associated eigenvectors \({\varphi }_{\text{Ni}}\) normalized with respect to the modal mass,

  • the system’s \({\psi }_{j}\) static support modes,

  • the \({P}_{\mathit{ij}}\) modal participation factors relating to support,

  • \({\mathrm{Rm}}_{\mathrm{ij}}\) the maximum response of each mode based on the excitation spectra,

  • \({\text{Re}}_{j}\) the contribution of the driving movement of each support from the differential movements,

  • \({\mathrm{Rc}}_{j}\) the term static correction,

  • the primary and secondary components of the response according to the accumulation rules adopted.

2.2. Benchmark results#

  • stiffness matrix \(K\)

\(K=\left[\begin{array}{cccc}k& -k& 0& 0\\ -k& \mathrm{2k}& -k& 0\\ 0& -k& \mathrm{11k}& -\mathrm{10k}\\ 0& 0& -\mathrm{10k}& \mathrm{10k}\end{array}\right]\)

\({K}^{p}=\left[\begin{array}{cccc}\mathrm{2k}& -k& -k& 0\\ -k& \mathrm{11k}& 0& -\mathrm{10k}\\ -k& 0& k& 0\\ 0& -\mathrm{10k}& 0& \mathrm{10k}\end{array}\right]\)

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

\({K}^{p}\mathrm{=}\left[\begin{array}{cc}{k}_{\mathit{xx}}& {k}_{\mathit{xs}}\\ {k}_{\mathit{sx}}& {k}_{\mathit{ss}}\end{array}\right]\) \({k}_{\mathit{xx}}\mathrm{=}\left[\begin{array}{cc}\mathrm{2k}& \mathrm{-}k\\ \mathrm{-}k& \mathrm{11k}\end{array}\right]\) \({k}_{\mathit{xs}}\mathrm{=}\left[\begin{array}{cc}\mathrm{-}k& 0\\ 0& \mathrm{-}\mathrm{10k}\end{array}\right]\)

\({k}_{\mathit{sx}}\mathrm{=}\left[\begin{array}{cc}\mathrm{-}k& 0\\ 0& \mathrm{-}\mathrm{10k}\end{array}\right]\) \({k}_{\mathit{ss}}\mathrm{=}\left[\begin{array}{cc}k& 0\\ 0& \mathrm{10k}\end{array}\right]\)

  • mass matrix \(M\)

\(M=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& m& 0& 0\\ 0& 0& m& 0\\ 0& 0& 0& 0\end{array}\right]\)

  • modal calculation in embedded basis

\({k}_{\mathrm{xx}}=\left[\begin{array}{cc}\mathrm{2k}& -k\\ -k& \mathrm{11k}\end{array}\right]\)

\(({k}_{\mathrm{xx}}-{\lambda }_{i}{m}_{\mathrm{xx}}{\varphi }_{i})=0\) \({\lambda }_{i}={\omega }_{i}^{2}\)

Let’s :math:`` \({\lambda }_{2}=\frac{k}{\mathrm{2m}}(13+\sqrt{85})\)

  1. natural frequencies:

Let’s :math:`` \({f}_{2}=\frac{{\omega }_{2}}{2\pi }\)

  1. non-standard modes:

Let’s \({\varphi }_{1}=(\begin{array}{c}0\\ 1\\ (\text{-}9+\sqrt{85})/2\\ 0\end{array})\) \({\varphi }_{2}=(\begin{array}{c}0\\ \text{-}1\\ (9+\sqrt{85})/2\\ 0\end{array})\)

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

Let’s :math:`` \({\mu }_{2}=\frac{m}{4}(170+18\sqrt{85})\)

  1. eigenmodes normalized to the unit generalized modal mass \({\varphi }_{\text{Ni}}\):

Let’s \({\varphi }_{\mathrm{N1}}=\frac{{\varphi }_{1}}{\sqrt{{\mu }_{1}}}\) \({\varphi }_{\mathrm{N2}}=\frac{{\varphi }_{2}}{\sqrt{{\mu }_{2}}}\)

  • modal reactions \({\mathrm{Fm}}_{i}\):

\({r}_{i}={k}_{\mathrm{sx}}{\varphi }_{\mathrm{Nis}}\) \({\varphi }_{\text{Ni}}^{p}=(\begin{array}{c}{\varphi }_{\text{Nix}}\\ {\varphi }_{\text{Nis}}\end{array})\) \({\mathrm{Fm}}_{i}^{p}=(\begin{array}{c}0\\ {r}_{i}\end{array})\)

Let’s \({\mathrm{Fm}}_{1}=\frac{k}{\sqrt{{\mu }_{1}}}(\begin{array}{c}-1\\ 0\\ 0\\ 5(9-\sqrt{85})\end{array})\) \({\mathit{Fm}}_{2}\mathrm{=}\frac{k}{\sqrt{{\mu }_{2}}}(\begin{array}{c}1\\ 0\\ 0\\ \mathrm{-}5(9+\sqrt{85})\end{array})\)

  • modal participation factors \({P}_{\mathrm{ij}}={}^{T}\varphi _{i}M{\psi }_{j}\):

  • contribution of dynamic mode 1 to the movement imposed on node \(\mathit{NO1}\):

\({P}_{11}={}^{T}\varphi _{1}M{\psi }_{1}=\frac{m}{42\sqrt{{\mu }_{1}}}(13+\sqrt{85})\)

  • contribution of dynamic mode 1 to the movement imposed on node \(\mathit{NO4}\):

\({P}_{12}={}^{T}\varphi _{1}M{\psi }_{2}=\frac{10m}{21\sqrt{{\mu }_{1}}}(-8+\sqrt{85})\)

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

\({P}_{21}={}^{T}\varphi _{2}M{\psi }_{1}=\frac{m}{42\sqrt{{\mu }_{2}}}(-13+\sqrt{85})\)

  • contribution of dynamic mode 2 to the movement imposed on node \(\mathit{NO4}\):

\({P}_{22}={}^{T}\varphi _{2}M{\psi }_{2}=\frac{10m}{21\sqrt{{\mu }_{2}}}(8+\sqrt{85})\)

  • dynamic mode participation factor 1 in direction \(X\):

\({P}_{\mathrm{1X}}={P}_{11}+{P}_{12}\)

  • dynamic mode 2 participation factor in direction \(X\):

\({P}_{\mathrm{2X}}={P}_{21}+{P}_{22}\)

  • static support modes \({\psi }_{j}\)

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

displacements: \({\psi }_{1}=\frac{1}{21}(\begin{array}{c}21\\ 11\\ 1\\ 0\end{array})\) nodal reactions: \({F}_{1}=K{\psi }_{1}=\frac{10}{21}k(\begin{array}{c}1\\ 0\\ 0\\ -1\end{array})\)

  • static solution to a unit movement of node \(\mathit{NO4}\):

displacements: \({\psi }_{2}=\frac{1}{21}(\begin{array}{c}0\\ 10\\ 20\\ 21\end{array})\) nodal reactions: \({\mathit{Fs}}_{2}\mathrm{=}K{\psi }_{2}\mathrm{=}\frac{10}{21}k(\begin{array}{c}\mathrm{-}1\\ 0\\ 0\\ 1\end{array})\)

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

\({\mathit{Rm}}_{\mathit{ij}}\mathrm{=}{r}_{i}{P}_{\mathit{ij}}\frac{{A}_{\mathit{ij}}}{{\omega }_{i}^{2}}\) with \({r}_{i}\mathrm{=}{\varphi }_{\text{Ni}}\) or \({\mathit{Fm}}_{i}\)

  • static correction

  • static modes \({u}_{j}\) solution from \(K{u}_{j}=M{\psi }_{j}\):

displacements: \({u}_{1}=\frac{m}{441k}(\begin{array}{c}0\\ 122\\ 13\\ 0\end{array})\) nodal reactions: \(F{u}_{1}=\frac{m}{441}(\begin{array}{c}-122\\ 231\\ 21\\ -130\end{array})\)

displacements: \({u}_{2}=\frac{m}{441k}(\begin{array}{c}0\\ 130\\ 50\\ 0\end{array})\) nodal reactions: \(F{u}_{2}=\frac{m}{441}(\begin{array}{c}-130\\ 210\\ 420\\ -500\end{array})\)

  • static correction relating to the movement of the \(j\) press if mode 2 is not selected:

\({\mathrm{Rc}}_{j}=({\mathrm{ru}}_{j}-\frac{{P}_{\mathrm{1j}}{r}_{1}}{{\omega }_{1}^{2}}){A}_{\mathrm{1j}}\) with: \({\mathrm{ru}}_{j}={u}_{j}\mathrm{ou}{\mathrm{Fu}}_{j}\) and \({r}_{1}={\varphi }_{\mathrm{N1}}\mathrm{ou}{\mathrm{Fm}}_{1}\)

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

\(R{e}_{j}={r}_{j}{D}_{j}\) with \({r}_{j}={\psi }_{j}\mathrm{ou}{\mathrm{Fs}}_{j}\)

These analytical calculations are described in the Matlab file sdld30a.55.

2.3. Uncertainty about the solution#

None (exact analytical solution).