Benchmark solution ===================== 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 :math:`\mathrm{NO1}` and :math:`\mathrm{NO4}` along the :math:`x` axis. Analytically, we calculate: * natural frequencies :math:`{f}_{i}`, * the associated eigenvectors :math:`{\phi }_{\text{Ni}}` normalized with respect to the modal mass, * the system's :math:`{\psi }_{j}` static support modes, * the :math:`{P}_{\mathrm{ij}}` modal participation factors relating to support, * :math:`{\mathrm{Rm}}_{\mathrm{ij}}` the maximum response of each mode based on the excitation spectra, * :math:`{\mathrm{Rc}}_{j}` the term static correction. These analytical calculations are described in the Matlab file sdld301.55. Reference quantity --------------------- * **stiffness matrix** :math:`K` * :math:`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]` :math:`{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 :math:`\mathrm{2,}3`, support degrees of freedom :math:`\mathrm{1,}4` :math:`{K}^{p}=\left[\begin{array}{cc}{k}_{\mathrm{xx}}& {k}_{\mathrm{xs}}\\ {k}_{\mathrm{sx}}& {k}_{\mathrm{ss}}\end{array}\right]` :math:`{K}_{\mathrm{xx}}=\left[\begin{array}{cc}\mathrm{3k}& -\mathrm{2k}\\ -\mathrm{2k}& \mathrm{3k}\end{array}\right]` :math:`{K}_{\mathrm{xs}}=\left[\begin{array}{cc}-k& 0\\ 0& -k\end{array}\right]` * **mass matrix** :math:`M` :math:`M=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& m& 0& 0\\ 0& 0& m& 0\\ 0& 0& 0& 0\end{array}\right]` :math:`{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 :math:`\mathrm{2,}3`, support degrees of freedom :math:`\mathrm{1,}4` * **modal calculation in embedded basis** :math:`{K}_{\mathrm{xx}}=\left[\begin{array}{cc}\mathrm{3k}& -\mathrm{2k}\\ -\mathrm{2k}& \mathrm{3k}\end{array}\right]` :math:`{m}_{\mathrm{xx}}=\left[\begin{array}{cc}m& 0\\ 0& m\end{array}\right]` :math:`({k}_{\mathrm{xx}}-{\lambda }_{i}{m}_{\mathrm{xx}}){\phi }_{i}=0` :math:`{\lambda }_{i}={\omega }_{\mathrm{pi}}^{2}` :math:`{\lambda }_{1}=\frac{k}{m}` :math:`{\lambda }_{2}=\frac{\mathrm{5k}}{m}` - natural frequencies: :math:`\Rightarrow \text{}{\mathrm{freq}}_{1}=\frac{{\omega }_{\mathrm{p1}}}{2\pi };{\mathrm{freq}}_{2}=\frac{{\omega }_{\mathrm{p2}}}{2\pi }` - non-standard modes: * :math:`{\phi }_{1}=(\begin{array}{c}0\\ 1\\ 1\\ 0\end{array})` :math:`{\phi }_{2}=(\begin{array}{c}0\\ 1\\ -1\\ 0\end{array})` - generalized modal masses: :math:`{\mu }_{i}{=}^{T}{\phi }_{i}M{\phi }_{i}` * :math:`{\mu }_{1}=\mathrm{2m}` :math:`{\mu }_{2}=\mathrm{2m}` - eigenmodes normalized to the unitary generalized modal mass :math:`{\phi }_{\text{Ni}}`: * :math:`\Rightarrow \text{}{\phi }_{\mathrm{N1}}=\frac{{\phi }_{1}}{\sqrt{{\mu }_{1}}}` :math:`{\phi }_{\mathrm{N2}}=\frac{{\phi }_{2}}{\sqrt{{\mu }_{2}}}` - modal reactions :math:`{\mathrm{Fm}}_{i}`: * * :math:`\Rightarrow \text{}{\mathrm{Fm}}_{1}=K{\phi }_{\mathrm{N1}}=\frac{k}{\sqrt{\mathrm{2m}}}(\begin{array}{c}1\\ 5\\ -5\\ 1\end{array})` :math:`{\mathrm{Fm}}_{2}=K{\phi }_{\mathrm{N2}}=\frac{k}{\sqrt{\mathrm{2m}}}(\begin{array}{c}1\\ 1\\ 1\\ -1\end{array})` * **static support modes** :math:`{\Psi }_{j}` Static mode matrix reduced to structural degrees of freedom :math:`{\varphi }_{s}=-{k}_{\mathrm{xx}}^{\text{-1}}{k}_{\mathrm{xs}}` :math:`{\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 :math:`\mathrm{NO1}`: displacements: :math:`{\psi }_{1}=\frac{1}{5}(\begin{array}{c}5\\ 3\\ 2\\ 0\end{array})` nodal reactions: :math:`{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 :math:`\mathrm{NO4}`: displacements: :math:`{\psi }_{2}=\frac{1}{5}(\begin{array}{c}0\\ 2\\ 3\\ 5\end{array})` nodal reactions: :math:`{F}_{2}=K{\psi }_{2}=\frac{k}{5}(\begin{array}{c}-2\\ 0\\ 0\\ -3\end{array})` * factors of modal participation in multi-support: :math:`{P}_{\mathrm{ij}}{=}^{T}{\phi }_{i}M{\psi }_{j}` * contribution of dynamic mode :math:`1` to the movement imposed on node :math:`\mathrm{NO1}`: :math:`{P}_{11}{=}^{T}{\phi }_{\mathrm{N1}}M{\psi }_{1}=\frac{1}{5}\sqrt{\frac{m}{2}}` * contribution of dynamic mode :math:`1` to the movement imposed on node :math:`\mathrm{NO4}`: :math:`{P}_{12}{=}^{T}{\phi }_{\mathrm{N1}}M{\psi }_{2}=\frac{-1}{5}\sqrt{\frac{m}{2}}` * contribution of dynamic mode :math:`2` to the movement imposed on node :math:`\mathrm{NO1}`: :math:`{P}_{21}{=}^{T}{\phi }_{\mathrm{N2}}M{\psi }_{1}=\sqrt{\frac{m}{2}}` * contribution of dynamic mode :math:`2` to the movement imposed on node :math:`\mathrm{NO4}`: :math:`{P}_{22}{=}^{T}{\phi }_{\mathrm{N2}}M{\psi }_{2}=\sqrt{\frac{m}{2}}` * dynamic mode participation factor :math:`1` in the :math:`X` direction: :math:`{P}_{\mathrm{1X}}={P}_{11}+{P}_{12}` * dynamic mode participation factor :math:`2` in the :math:`X` direction: :math:`{P}_{\mathrm{2X}}={P}_{21}+{P}_{22}` * modal participation factors in mono-support :math:`{P}_{i}=\frac{{\phi }_{\text{Ni}}M{\psi }_{\mathrm{R1}}}{{\mu }_{i}}` * dynamic mode contribution :math:`1`: :math:`{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 :math:`2`: :math:`{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 :math:`1` in the :math:`X` direction: :math:`{P}_{\mathrm{1X}}={P}_{1}+{P}_{2}` * **mode response** :math:`i` **to press movement** :math:`j` **in multi-pressure** :math:`{\mathrm{Rm}}_{\mathrm{ij}}={r}_{i}{P}_{\mathrm{ij}}\frac{{A}_{\mathrm{ij}}}{{\omega }_{i}^{2}}` with :math:`{r}_{i}={\phi }_{\text{Ni}}\mathrm{ou}{\mathrm{Fm}}_{i}` Modeling :math:`A`: :math:`{A}_{11}=\frac{{a}_{1}{\mathrm{freq}}_{1}^{2}}{∣{f}_{1}^{2}-{\mathrm{freq}}_{1}^{2}∣}`: fashion :math:`1`, knot :math:`1` :math:`{A}_{12}=\frac{{a}_{2}{\mathrm{freq}}_{1}^{2}}{∣{f}_{2}^{2}-{\mathrm{freq}}_{1}^{2}∣}`: fashion :math:`1`, knot :math:`2` :math:`{A}_{21}=\frac{{a}_{1}{\mathrm{freq}}_{2}^{2}}{∣{f}_{1}^{2}-{\mathrm{freq}}_{2}^{2}∣}`: fashion :math:`2`, knot :math:`1` :math:`{A}_{22}=\frac{{a}_{2}{\mathrm{freq}}_{2}^{2}}{∣{f}_{2}^{2}-{\mathrm{freq}}_{2}^{2}∣}`: fashion :math:`2`, knot :math:`2` Modeling :math:`B`: :math:`{A}_{11}={A}_{12}=\frac{{a}_{1}{\mathrm{freq}}_{1}^{2}}{∣{f}_{1}^{2}-{\mathrm{freq}}_{1}^{2}∣}`: fashion :math:`1` :math:`{A}_{21}={A}_{22}=\frac{{a}_{1}{\mathrm{freq}}_{2}^{2}}{∣{f}_{1}^{2}-{\mathrm{freq}}_{2}^{2}∣}`: fashion :math:`2` * **mode response** :math:`i` **in single press** :math:`{\mathrm{Rm}}_{i}={r}_{i}{P}_{i}\frac{{A}_{i}}{{\omega }_{i}^{2}}` with :math:`{r}_{i}={\phi }_{\text{Ni}}\mathrm{ou}{\mathrm{Fm}}_{i}` Combined responses of modal oscillators :math:`1` mode response: :math:`{\mathrm{Rm}}_{1}={\phi }_{\mathrm{N1}}{P}_{1}\frac{{A}_{1}}{{\omega }_{1}^{2}}={\mathrm{Rm}}_{11}+{\mathrm{Rm}}_{12}` :math:`2` mode response: :math:`{\mathrm{Rm}}_{2}={\phi }_{\mathrm{N2}}{P}_{2}\frac{{A}_{2}}{{\omega }_{2}^{2}}={\mathrm{Rm}}_{21}+{\mathrm{Rm}}_{22}` * **static correction** * static modes :math:`{u}_{j}` solution from :math:`{k}_{\mathrm{xs}}{u}_{\mathrm{sj}}={m}_{\mathrm{xs}}{\phi }_{\mathrm{sj}}`: :math:`\psi` modes reduced to structural degrees of freedom: :math:`{\psi }_{\mathrm{S1}}=\frac{1}{5}(\begin{array}{c}3\\ 2\end{array})` :math:`{\psi }_{\mathrm{S2}}=\frac{1}{5}(\begin{array}{c}2\\ 3\end{array})` displacements: :math:`{u}_{1}=\frac{m}{\mathrm{25k}}(\begin{array}{c}0\\ 13\\ 12\\ 0\end{array})` nodal reactions: :math:`{\mathrm{Fu}}_{1}=\frac{m}{25}(\begin{array}{c}-13\\ 3\\ 10\\ -1\end{array})` displacements: :math:`{u}_{2}=\frac{m}{\mathrm{25k}}(\begin{array}{c}0\\ 12\\ 13\\ 0\end{array})` nodal reactions: :math:`{\mathrm{Fu}}_{2}=\frac{m}{25}(\begin{array}{c}-13\\ 3\\ 10\\ -12\end{array})` Uncertainty about the solution --------------------------- None (exact analytical solution).