Benchmark solution
=====================


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 :math:`\mathit{NO1}` and :math:`\mathit{NO4}` along the :math:`x` axis.

Analytically, we calculate:


* natural frequencies :math:`{f}_{i}`,


* the associated eigenvectors :math:`{\varphi }_{\text{Ni}}` normalized with respect to the modal mass,


* the system's :math:`{\psi }_{j}` static support modes,


* the :math:`{P}_{\mathit{ij}}` modal participation factors relating to support,


* :math:`{\mathrm{Rm}}_{\mathrm{ij}}` the maximum response of each mode based on the excitation spectra,


* :math:`{\text{Re}}_{j}` the contribution of the driving movement of each support from the differential movements,


* :math:`{\mathrm{Rc}}_{j}` the term static correction,


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


Benchmark results
----------------------


* **stiffness matrix** :math:`K`

:math:`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]`

:math:`{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

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

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


* **modal calculation in embedded basis**

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

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

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


1. natural frequencies:

Let's :math:`` :math:`{f}_{2}=\frac{{\omega }_{2}}{2\pi }`


1. non-standard modes:

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


* generalized modal masses :math:`{\mu }_{i}={}^{T}\varphi _{i}M{\varphi }_{i}`:

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


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

Let's :math:`{\varphi }_{\mathrm{N1}}=\frac{{\varphi }_{1}}{\sqrt{{\mu }_{1}}}` :math:`{\varphi }_{\mathrm{N2}}=\frac{{\varphi }_{2}}{\sqrt{{\mu }_{2}}}`


* modal reactions :math:`{\mathrm{Fm}}_{i}`:

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

Let's :math:`{\mathrm{Fm}}_{1}=\frac{k}{\sqrt{{\mu }_{1}}}(\begin{array}{c}-1\\ 0\\ 0\\ 5(9-\sqrt{85})\end{array})` :math:`{\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 :math:`{P}_{\mathrm{ij}}={}^{T}\varphi _{i}M{\psi }_{j}`:


* contribution of dynamic mode 1 to the movement imposed on node :math:`\mathit{NO1}`:

:math:`{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 :math:`\mathit{NO4}`:

:math:`{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 :math:`\mathit{NO1}`:

:math:`{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 :math:`\mathit{NO4}`:

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


* dynamic mode participation factor 1 in direction :math:`X`:

:math:`{P}_{\mathrm{1X}}={P}_{11}+{P}_{12}`


* dynamic mode 2 participation factor in direction :math:`X`:

:math:`{P}_{\mathrm{2X}}={P}_{21}+{P}_{22}`


* **static support modes** :math:`{\psi }_{j}`


* static solution to a unit movement of node :math:`\mathit{NO1}`:

displacements: :math:`{\psi }_{1}=\frac{1}{21}(\begin{array}{c}21\\ 11\\ 1\\ 0\end{array})` nodal reactions: :math:`{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 :math:`\mathit{NO4}`:

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


* **mode response** :math:`i` **to the support movement** :math:`j`

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


* **static correction**


* static modes :math:`{u}_{j}` solution from :math:`K{u}_{j}=M{\psi }_{j}`:

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

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


* static correction relating to the movement of the :math:`j` press if mode 2 is not selected:

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


* **contribution of support** :math:`j` **to the training movement**

:math:`R{e}_{j}={r}_{j}{D}_{j}` with :math:`{r}_{j}={\psi }_{j}\mathrm{ou}{\mathrm{Fs}}_{j}`

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


Uncertainty about the solution
---------------------------

None (exact analytical solution).