2. Benchmark solution#

2.1. Calculation method used for the reference solution#

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

Analytically, we calculate:

  • natural frequencies \({f}_{i}\),

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

  • the system’s \({\psi }_{j}\) 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,

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

These analytical calculations are described in the Matlab file sdld301.55.

2.2. Reference quantity#

  • stiffness matrix \(K\)

  • \(K=\left[\begin{array}{cccc}k& -k& 0& 0\\ -k& \mathrm{3k}& -\mathrm{2k}& 0\\ 0& -\mathrm{2k}& \mathrm{3k}& -k\\ 0& 0& -k& k\end{array}\right]\)

    \({K}^{p}=\left[\begin{array}{cccc}\mathrm{3k}& -\mathrm{2k}& -k& 0\\ -\mathrm{2k}& \mathrm{3k}& 0& -k\\ -k& 0& k& 0\\ 0& -k& 0& k\end{array}\right]\)

partitioned matrix structural degrees of freedom \(\mathrm{2,}3\), support degrees of freedom \(\mathrm{1,}4\)

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

  • mass matrix \(M\)

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

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

partitioned matrix structural degrees of freedom \(\mathrm{2,}3\), support degrees of freedom \(\mathrm{1,}4\)

  • modal calculation in embedded basis

\({K}_{\mathrm{xx}}=\left[\begin{array}{cc}\mathrm{3k}& -\mathrm{2k}\\ -\mathrm{2k}& \mathrm{3k}\end{array}\right]\) \({m}_{\mathrm{xx}}=\left[\begin{array}{cc}m& 0\\ 0& m\end{array}\right]\)

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

\({\lambda }_{1}=\frac{k}{m}\) \({\lambda }_{2}=\frac{\mathrm{5k}}{m}\)

  • natural frequencies:

\(\Rightarrow \text{}{\mathrm{freq}}_{1}=\frac{{\omega }_{\mathrm{p1}}}{2\pi };{\mathrm{freq}}_{2}=\frac{{\omega }_{\mathrm{p2}}}{2\pi }\)

  • non-standard modes:

  • \({\phi }_{1}=(\begin{array}{c}0\\ 1\\ 1\\ 0\end{array})\) \({\phi }_{2}=(\begin{array}{c}0\\ 1\\ -1\\ 0\end{array})\)

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

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

  • eigenmodes normalized to the unitary generalized modal mass \({\phi }_{\text{Ni}}\):

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

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

    • \(\Rightarrow \text{}{\mathrm{Fm}}_{1}=K{\phi }_{\mathrm{N1}}=\frac{k}{\sqrt{\mathrm{2m}}}(\begin{array}{c}1\\ 5\\ -5\\ 1\end{array})\) \({\mathrm{Fm}}_{2}=K{\phi }_{\mathrm{N2}}=\frac{k}{\sqrt{\mathrm{2m}}}(\begin{array}{c}1\\ 1\\ 1\\ -1\end{array})\)

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

Static mode matrix reduced to structural degrees of freedom \({\varphi }_{s}=-{k}_{\mathrm{xx}}^{\text{-1}}{k}_{\mathrm{xs}}\)

\({\varphi }_{s}=-\frac{1}{\mathrm{5k}}\left[\begin{array}{cc}3& 2\\ 2& 3\end{array}\right]\left[\begin{array}{cc}-k& 0\\ 0& -k\end{array}\right]=\frac{1}{5}\left[\begin{array}{cc}3& 2\\ 2& 3\end{array}\right]\)

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

displacements: \({\psi }_{1}=\frac{1}{5}(\begin{array}{c}5\\ 3\\ 2\\ 0\end{array})\) nodal reactions: \({F}_{1}=K{\psi }_{1}=\frac{k}{5}(\begin{array}{c}-8\\ 0\\ 0\\ -2\end{array})\)

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

displacements: \({\psi }_{2}=\frac{1}{5}(\begin{array}{c}0\\ 2\\ 3\\ 5\end{array})\) nodal reactions: \({F}_{2}=K{\psi }_{2}=\frac{k}{5}(\begin{array}{c}-2\\ 0\\ 0\\ -3\end{array})\)

  • factors of modal participation in multi-support: \({P}_{\mathrm{ij}}{=}^{T}{\phi }_{i}M{\psi }_{j}\)

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

\({P}_{11}{=}^{T}{\phi }_{\mathrm{N1}}M{\psi }_{1}=\frac{1}{5}\sqrt{\frac{m}{2}}\)

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

\({P}_{12}{=}^{T}{\phi }_{\mathrm{N1}}M{\psi }_{2}=\frac{-1}{5}\sqrt{\frac{m}{2}}\)

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

\({P}_{21}{=}^{T}{\phi }_{\mathrm{N2}}M{\psi }_{1}=\sqrt{\frac{m}{2}}\)

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

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

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

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

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

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

  • modal participation factors in mono-support \({P}_{i}=\frac{{\phi }_{\text{Ni}}M{\psi }_{\mathrm{R1}}}{{\mu }_{i}}\)

  • dynamic mode contribution \(1\):

\({P}_{1}{=}^{T}{\phi }_{\mathrm{N1}}M{\psi }_{\mathrm{R1}}={\phi }_{\mathrm{N1}}M({\psi }_{\mathrm{s1}}+{\psi }_{\mathrm{s2}})={P}_{11}+{P}_{12}\)

  • dynamic mode contribution \(2\):

\({P}_{2}{=}^{T}{\phi }_{\mathrm{N2}}M{\psi }_{\mathrm{R1}}={\phi }_{\mathrm{N2}}M({\psi }_{\mathrm{s1}}+{\psi }_{\mathrm{s2}})={P}_{21}+{P}_{22}\)

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

\({P}_{\mathrm{1X}}={P}_{1}+{P}_{2}\)

  • mode response \(i\) to press movement \(j\) in multi-pressure

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

Modeling \(A\):

\({A}_{11}=\frac{{a}_{1}{\mathrm{freq}}_{1}^{2}}{∣{f}_{1}^{2}-{\mathrm{freq}}_{1}^{2}∣}\): fashion \(1\), knot \(1\)

\({A}_{12}=\frac{{a}_{2}{\mathrm{freq}}_{1}^{2}}{∣{f}_{2}^{2}-{\mathrm{freq}}_{1}^{2}∣}\): fashion \(1\), knot \(2\)

\({A}_{21}=\frac{{a}_{1}{\mathrm{freq}}_{2}^{2}}{∣{f}_{1}^{2}-{\mathrm{freq}}_{2}^{2}∣}\): fashion \(2\), knot \(1\)

\({A}_{22}=\frac{{a}_{2}{\mathrm{freq}}_{2}^{2}}{∣{f}_{2}^{2}-{\mathrm{freq}}_{2}^{2}∣}\): fashion \(2\), knot \(2\)

Modeling \(B\):

\({A}_{11}={A}_{12}=\frac{{a}_{1}{\mathrm{freq}}_{1}^{2}}{∣{f}_{1}^{2}-{\mathrm{freq}}_{1}^{2}∣}\): fashion \(1\) \({A}_{21}={A}_{22}=\frac{{a}_{1}{\mathrm{freq}}_{2}^{2}}{∣{f}_{1}^{2}-{\mathrm{freq}}_{2}^{2}∣}\): fashion \(2\)

  • mode response \(i\) in single press

\({\mathrm{Rm}}_{i}={r}_{i}{P}_{i}\frac{{A}_{i}}{{\omega }_{i}^{2}}\) with \({r}_{i}={\phi }_{\text{Ni}}\mathrm{ou}{\mathrm{Fm}}_{i}\)

Combined responses of modal oscillators

\(1\) mode response: \({\mathrm{Rm}}_{1}={\phi }_{\mathrm{N1}}{P}_{1}\frac{{A}_{1}}{{\omega }_{1}^{2}}={\mathrm{Rm}}_{11}+{\mathrm{Rm}}_{12}\)

\(2\) mode response: \({\mathrm{Rm}}_{2}={\phi }_{\mathrm{N2}}{P}_{2}\frac{{A}_{2}}{{\omega }_{2}^{2}}={\mathrm{Rm}}_{21}+{\mathrm{Rm}}_{22}\)

  • static correction

  • static modes \({u}_{j}\) solution from \({k}_{\mathrm{xs}}{u}_{\mathrm{sj}}={m}_{\mathrm{xs}}{\phi }_{\mathrm{sj}}\):

\(\psi\) modes reduced to structural degrees of freedom: \({\psi }_{\mathrm{S1}}=\frac{1}{5}(\begin{array}{c}3\\ 2\end{array})\) \({\psi }_{\mathrm{S2}}=\frac{1}{5}(\begin{array}{c}2\\ 3\end{array})\)

displacements: \({u}_{1}=\frac{m}{\mathrm{25k}}(\begin{array}{c}0\\ 13\\ 12\\ 0\end{array})\) nodal reactions: \({\mathrm{Fu}}_{1}=\frac{m}{25}(\begin{array}{c}-13\\ 3\\ 10\\ -1\end{array})\)

displacements: \({u}_{2}=\frac{m}{\mathrm{25k}}(\begin{array}{c}0\\ 12\\ 13\\ 0\end{array})\) nodal reactions: \({\mathrm{Fu}}_{2}=\frac{m}{25}(\begin{array}{c}-13\\ 3\\ 10\\ -12\end{array})\)

2.3. Uncertainty about the solution#

None (exact analytical solution).