2. Dynamic system modeling#

2.1. Average finite element model#

2.1.1. Transitional resolution in absolute coordinates#

In the absolute coordinate system, the mechanical system is modelled using the finite element method. This basic model (generally the one that would have been used in the deterministic study) is referred to as the « mean finite element model ». All sizes relating to the average models are underlined.

Let \(t\mapsto \underline{\mathrm{y}}(t)\) be the transitory response in the absolute coordinate system of the « mean finite element model » defined on the study interval \(\mathrm{[}\mathrm{0,}T\mathrm{]}\) and set to value in \({ℝ}^{k}\) where \(k\) is the number of degrees of freedom. The mass, damping, and stiffness matrices are respectively noted \(\left[\underline{M}\right]\), \(\left[\underline{D}\right]\), and \(\left[\underline{K}\right]\).

The transient response \(\underline{y}(t)\) of the « mean finite element model » verifies the following discretized nonlinear differential equation:

\(\left[\underline{M}\right]\underline{\ddot{y}}(t)+\left[\underline{D}\right]\underline{\dot{y}}+{f}_{c}(t,\underline{\dot{y}(t)},\underline{y(t)},\underline{w})=f(t)\), t, eq 2.1.1-1

with the initial conditions,

\(\underline{y}(0)=\dot{\underline{y}}(0)=0\), eq 2.1.1-2

  • \(f(t)\in {ℝ}^{m}\) represents the discretization of external forces by finite elements.

  • \({f}_{c}(t,\underline{y}(t),\underline{\dot{y}}(t),\underline{w})\in {ℝ}^{m}\) corresponds to localized nonlinearities (for example due to elastic shock stops). The elements \({\underline{w}}_{\mathrm{1,}}\mathrm{...},{\underline{w}}_{v}\) of the \(\underline{w}\in {ℝ}^{\nu }\) vector represent a set of parameters defining these nonlinearities (for example game, shock stiffness, shock damping, etc.).

2.1.2. Transitional resolution in relative coordinates (earthquake)#

As in the transient case in absolute coordinates, the mechanical system is modelled by a basic model, the « mean finite element model ».

We denote \(t\to \underline{z}(t)\) the transient response in absolute coordinates of this model over the study interval \([\mathrm{0,}T]\) with value in \({ℝ}^{k}\) (attention: notice the change in notation compared to the previous paragraph.).

The transient response of the « mean finite element model » verifies the following discretized nonlinear differential equation:

\(\left[\begin{array}{cc}\left[\underline{M}\right]& \left[{\underline{M}}_{\mathrm{ls}}\right]\\ {\left[{\underline{M}}_{\mathrm{ls}}\right]}^{T}& \left[{\underline{M}}_{s}\right]\end{array}\right]\left[\begin{array}{c}\underline{\ddot{z}}(t)\\ {\underline{\ddot{z}}}_{s}(t)\end{array}\right]+\left[\begin{array}{cc}\left[\underline{D}\right]& \left[{\underline{D}}_{\mathrm{ls}}\right]\\ {\left[{\underline{D}}_{\mathrm{ls}}\right]}^{T}& \left[{\underline{D}}_{s}\right]\end{array}\right]\left[\begin{array}{c}\underline{\dot{z}}(t)\\ {\underline{\dot{z}}}_{s}(t)\end{array}\right]+\left[\begin{array}{cc}\left[\underline{K}\right]& \left[{\underline{K}}_{\mathrm{ls}}\right]\\ {\left[{\underline{K}}_{\mathrm{ls}}\right]}^{T}& \left[{\underline{K}}_{s}\right]\end{array}\right]\left[\begin{array}{c}\underline{z}(t)\\ {\underline{z}}_{s}(t)\end{array}\right]+\left[\begin{array}{c}{F}_{c}(t,\underline{z}(t),\underline{\dot{z}}(t);\underline{w})\\ {0}_{d}\end{array}\right]=\left[\begin{array}{c}\underline{g}(t)\\ {\underline{g}}_{s}(t)\end{array}\right]\)

\(t\in [\mathrm{0,}T]\). Eq 2.1.2-1

with the initial conditions,

\(\underline{z}(0)=\underline{\dot{z}}(0),{\underline{z}}_{s}(0)={\underline{\dot{z}}}_{s}(0)\) eq 2.1.2-2

  • \(\underline{g}(t)\in {ℝ}^{m}\) represents the discretization by finite elements of external forces and \(\underline{{g}_{s}}(t)\in {ℝ}^{d}\) corresponds to the discretization of reaction forces due to the \(d\) Dirichlet conditions.

  • \({F}_{c}(t,\underline{z}(t),\underline{\dot{z}}(t),\underline{w})\in {ℝ}^{m}\) corresponds to localized nonlinearities with \(\underline{w}\in {ℝ}^{\mathrm{\nu }}\) representing a set of parameters defining these nonlinearities as before.

After static raising, the matrix equations [éq 2.1.2-1] and [éq 2.1.2-2] in the absolute coordinate system are rewritten in « relative » coordinates:

\(\left[\underline{M}\right]\underline{\ddot{y}}(t)+\left[\underline{D}\right]\underline{\dot{y}}+\left[\underline{K}\right]\underline{y(t)}+{f}_{c}(t,\underline{\dot{y}(t)},\underline{y(t)},\underline{w})=f(t)\), \(t\in [\mathrm{0,}T]\). Eq 2.1.2-3

\(\underline{y}(0)=\dot{\underline{y}}(0)=0\), eq 2.1.2-4

  • \(\underline{y}(t)\in {ℝ}^{m}\) is the vector of free degrees of freedom in the « relative » coordinate system such as

\(\underline{z}(t)=\underline{y}(t)+[\underline{R}]{\underline{z}}_{s}(t)\) with \([\underline{R}]=-{[\underline{K}]}^{-1}[\underline{{K}_{\mathrm{ls}}}]\)

  • The \(t\to \underline{\mathrm{f}}(t)\) function set to \(\mathrm{[}\mathrm{0,}T\mathrm{]}\) and set to value in \({ℝ}^{m}\) and the nonlinear application \((x,y)\mapsto {f}_{c}(t,x,y;\underline{w})\) of \({ℝ}^{m}\times {ℝ}^{m}\) in \({ℝ}^{m}\) are such that:

\(f(t)=\underline{g}(t)-([\underline{M}][\underline{R}]+[\underline{{M}_{\mathrm{ls}}}])\underline{\ddot{z}}(t)-([\underline{D}][\underline{R}]+[\underline{{D}_{\mathrm{ls}}}])\underline{\dot{z}}(t)\) eq 2.1.2-5

\({f}_{c}(t,x,y;\underline{w})={F}_{c}(t,x+\left[\underline{R}\right]{\underline{z}}_{s}(t),y+\left[\underline{R}\right]{\underline{\dot{z}}}_{s}(t);\underline{w})\text{.}\) eq 2.1.2-6

Notes:

  • In the following, depending on whether or not a static bearing has been carried out, \(\underline{y}(t)\) corresponds either to the transient response in absolute coordinates defined by [§ 2.1.1], or to the transient response in « relative » coordinates defined by the [§ 2.1.2].

  • It is assumed that if the \(d\) Dirichlet conditions were homogeneous no rigid body movement could occur. Therefore, \(\left[\underline{K}\right]\) is symmetric definite positive and its inverse \({\left[\underline{K}\right]}^{\mathrm{-}1}\) is defined, which makes it possible to introduce the real matrix \([\underline{R}]=-{[\underline{K}]}^{-1}[\underline{{K}_{\mathrm{ls}}}]\) of dimension \((m\times d)\) .

  • In Code_Aster the depreciation term in [éq 2.1.2-5] is overlooked.

2.1.3. Harmonic resolution#

As in the transitory case, the mechanical system is modelled by a basic model, the « mean finite element model ». On a frequency band \(\left[{\omega }_{1},{\omega }_{2}\right]\) , the \(q(\mathrm{\omega })\) harmonic response of the linear « mean finite element model » verifies the following equation:

\((-{\omega }^{2}\left[\underline{M}\right]+i\omega \left[\underline{D}\right]+\left[\underline{K}\right])q(\omega )=F(\omega )\), \(\omega \in \left[{\omega }_{1},{\omega }_{2}\right]\) eq 2.1.3-1

with \(f(\mathrm{\omega })\) representing finite element discretization of external forces.

2.2. Average reduced matrix model#

It is assumed that the vibration energy of the dynamic response is mainly localized in the low frequency domain. We can therefore build the average reduced matrix model by projecting \(\underline{y}(t)\) or \(\underline{y}(\mathrm{\omega })\) onto the subspace generated by the first n modes of the associated linear dynamic system (infinite games) homogeneous conservative (blocked supports) which is written,

\(\left[\underline{K}\right]\underline{\varphi }=\lambda \left[\underline{M}\right]\varphi\). Eq 2.2-1

Since the matrices \(\left[\underline{M}\right]\) and \(\left[\underline{K}\right]\) are positive (for \(\left[\underline{K}\right]\) cf. note 2 [§2.1.2]), the eigenvalues are real and positive,

\(0\le \underline{{\lambda }_{1}}\le \underline{{\lambda }_{2}}\le \mathrm{...}\le {\lambda }_{n}\) eq 2.2-2

The associated natural vibration modes \(\mathrm{\{}{\phi }_{\mathrm{1,}}{\phi }_{\mathrm{2,}}\dots \mathrm{\}}\) verify the orthogonality properties,

\(\text{<}\left[\underline{M}\right]{\varphi }_{\alpha }\text{,}{\varphi }_{\beta }\text{>}=\underline{{\mu }_{\alpha }}{\delta }_{\alpha \beta }\) eq 2.2-3

\(\text{<}\left[\underline{K}\right]{\varphi }_{\alpha }\text{,}{\varphi }_{\beta }\text{>}=\underline{{\mu }_{\alpha }}\underline{{\omega }_{\alpha }^{2}}{\delta }_{\alpha \beta }\) eq 2.2-4

with

\({\omega }_{\alpha }=\sqrt{({\lambda }_{\alpha })}\) eq 2.2-5

The generalized mass matrix, the generalized stiffness matrix and the generalized damping matrix respectively are denoted by:

\(\left[\underline{{M}_{n}}\right]={\left[\underline{{\Phi }_{n}}\right]}^{T}\left[\underline{M}\right]\left[\underline{{\Phi }_{n}}\right]\) eq 2.2-6

\(\left[{\underline{K}}_{n}\right]={\left[{\underline{\Phi }}_{n}\right]}^{T}\left[\underline{K}\right]\left[{\underline{\Phi }}_{n}\right]\) eq 2.2-7

\(\left[{\underline{D}}_{n}\right]={\left[{\underline{\Phi }}_{n}\right]}^{T}\left[\underline{D}\right]\left[{\underline{\Phi }}_{n}\right]\) eq 2.2-8

2.2.1. Transitional resolution#

The \(\underline{{y}^{n}}(t)\) projection of \(\underline{y}(t)\) onto the subspace generated by the first n modes of the associated homogeneous conservative linear dynamic system is written:

\(\underline{{y}^{n}}(t)=\left[\underline{{\Phi }_{n}}\right]\underline{{q}^{n}}(t)=\sum _{\alpha =1}^{n}{\underline{q}}_{\alpha }^{n}(t){\underline{\varphi }}_{\alpha }\), eq 2.2.1-1

Generalized displacements are solutions of the average reduced matrix model (nonlinear dynamic system),

\(\left[{M}_{n}\right]{\ddot{q}}^{n}(t)+\left[{D}_{n}\right]{\dot{q}}^{n}(t)+\left[{K}_{n}\right]{q}^{n}(t)+{F}_{c}^{n}(t,{q}^{n}(t),{\dot{q}}^{n}(t);W)={F}^{n}(t),\) eq 2.2.1-2

\(\dot{\underline{{q}^{n}}}(0)=\underline{{q}^{n}}(0)=0\), eq 2.2.1-3

with

\({F}^{n}(t)={\left[\underline{{\Phi }_{n}}\right]}^{T}f(t)\), eq 2.2.1-4

\({F}_{c}^{n}(t,q,p;\underline{w})={[{\Phi }_{n}]}^{T}{f}_{c}(t,[{\Phi }_{n}]q,[{\Phi }_{n}]p,\underline{w})\) eq 2.2.1-5

2.2.2. Harmonic resolution#

The projection \({\underline{y}}^{n}(\mathrm{\omega })\) of \(\underline{y}(\mathrm{\omega })\) onto the subspace generated by the first n modes of the associated homogeneous conservative linear dynamic system is written

\({q}^{n}(\mathrm{\omega })\) = \({[{\Phi }_{n}]}^{T}f(\mathrm{\omega }),\) eq 2.2.2-1

Generalized displacements \({q}^{n}(\mathrm{\omega })\) are solutions of the average reduced matrix model

\((-{\omega }^{2}\left[{\underline{M}}_{n}\right]+i\omega \left[{\underline{D}}_{n}\right]+\left[{\underline{K}}_{n}\right]){q}^{n}(\omega )={F}^{n}(\omega )\) eq 2.2.2-2

with

\({F}^{n}(\omega )={\left[\underline{{\Phi }_{n}}\right]}^{T}f(\omega )\), eq 2.2.2-3