The solid 3D element ====================== A mobile structure can be put into equation, either in the fixed Galilean coordinate system (OXYZ), or in the inertia frame (O'X'Y'Z') attached to the structure. .. image:: images/10000200000003610000032CADB5BF5026A952AE.png :width: 4.4634in :height: 4.1563in .. _RefImage_10000200000003610000032CADB5BF5026A952AE.png: For dynamic analysis, the choice of the inertia coordinate system as a reference system simplifies the formulation of the differential equations governing their dynamic behavior. This choice makes it possible to characterize the Coriolis effect (gyroscopic effect seen from the rotor) which is added to the damping. Also, two other apparent stiffness effects are added to the stiffness matrix: these are the centrifugal effect and the gyroscopic effect associated with the variation in speed. Position in motion in the rotating coordinate system --------------------------------------------- In general, it will be considered that the mobile frame of reference, whose origin is characterized by a translation :math:`\mathrm{s}(t){\mathrm{=}}^{t}({s}_{x},{s}_{y},{s}_{y})` with respect to the inertia frame, rotates at the angular speed :math:`\omega {\mathrm{=}}^{t}({\omega }_{x},{\omega }_{y},{\omega }_{z})` around any axis passing through the origin. It is important to note that the components of the speed of rotation are measured in the corotational coordinate system. Note :math:`\omega \text{'}` the projection of this speed of rotation into the fixed coordinate system, i.e. :math:`\omega \mathrm{=}{t}^{}\mathrm{R}\omega \text{'}` where where :math:`\mathrm{R}(t)` is the base change matrix, formed by the direction cosines of the base vectors of the inertia coordinate system, expressed in the fixed frame of reference. The position of a point P in the inertia frame of reference (OXYZ), noted :math:`\mathrm{y}`, then has the expression: :math:`\mathrm{y}\mathrm{=}\mathrm{s}(t)+\mathrm{R}(t)\left[\mathrm{x}+\mathrm{u}(\mathrm{x},t)\right]` where :math:`\mathrm{x}` is the initial position of the point P in the inertial system and :math:`\mathrm{u}` is the displacement vector resulting from the dynamic deformation of the structure at a given time t. The absolute speed :math:`\dot{\mathrm{y}}` of the point P is defined as being the first derivative with respect to time of the vector :math:`\mathrm{y}`. It is written: :math:`\dot{\mathrm{y}}\mathrm{=}\dot{\mathrm{s}}+\dot{\mathrm{R}}(\mathrm{x}+\mathrm{u})+\mathrm{R}\dot{\mathrm{u}}` where :math:`\dot{\mathrm{s}}` is the speed of translation of the origin O' of the rotating coordinate system, written in the fixed coordinate system. It is shown that the derivative :math:`\dot{\mathrm{R}}` of the base change matrix can also be written as being the vector product between the rotation vector in the fixed coordinate system :math:`\omega \text{'}` and the base change matrix :math:`\mathrm{R}`, or as being the product of the matrix :math:`\mathrm{R}` and the antisymmetric matrix :math:`\Omega`, which includes the three components of the rotation tensor in the inertia coordinate system. :math:`\omega`: :math:`\dot{\mathrm{R}}\mathrm{=}\omega \text{'}\mathrm{\wedge }\mathrm{R}\mathrm{=}\mathrm{R}\Omega` with :math:`\Omega \mathrm{=}\left[\begin{array}{ccc}0& \mathrm{-}{\omega }_{z}& {\omega }_{y}\\ {\omega }_{z}& 0& \mathrm{-}{\omega }_{x}\\ \mathrm{-}{\omega }_{y}& {\omega }_{x}& 0\end{array}\right]` Inserting this definition into the expression for absolute speed, the result is that: :math:`\dot{\mathrm{y}}\mathrm{=}\dot{\mathrm{s}}+\omega \text{'}\mathrm{\wedge }\mathrm{R}(\mathrm{x}+\mathrm{u})+\mathrm{R}\dot{\mathrm{u}}` The absolute acceleration :math:`\ddot{\mathrm{y}}` of point P is defined as being the second derivative with respect to time of the speed vector :math:`\mathrm{y}`. It is written: :math:`\ddot{\mathrm{y}}\mathrm{=}\ddot{\mathrm{s}}+\dot{\omega }\text{'}\mathrm{\wedge }\mathrm{R}(\mathrm{x}+\mathrm{u})+\omega \text{'}\mathrm{\wedge }\mathrm{[}\text{'}\omega \text{'}\mathrm{\wedge }\mathrm{R}(\mathrm{x}+\mathrm{u})\mathrm{]}+2\omega \text{'}\mathrm{\wedge }\mathrm{R}\dot{\mathrm{u}}+\mathrm{R}\ddot{\mathrm{u}}` with :math:`\ddot{\mathrm{s}}` the translation acceleration of the origin O' of the rotating coordinate system, written in the fixed coordinate system and :math:`\dot{\omega }\text{'}` is the instantaneous rotation acceleration defined by the relation :math:`\dot{\omega }\mathrm{=}{t}^{}\mathrm{R}\dot{\omega }\text{'}`. For the sake of clarity and without losing the generality, it will be assumed in the remainder of the document that the origin O' of the rotating coordinate system is fixed, i.e. :math:`\dot{\mathrm{s}}\mathrm{=}\ddot{\mathrm{s}}\mathrm{=}0`. After pre-multiplication by the inverse transformation :math:`{t}^{}\mathrm{R}`, we then obtain the absolute speed and acceleration expressed in the moving frame of reference: :math:`{t}^{}\mathrm{R}\dot{\mathrm{y}}\mathrm{=}\dot{\mathrm{u}}+\omega \mathrm{\wedge }\mathrm{R}(\mathrm{x}+\mathrm{u})` :math:`{t}^{}\mathrm{R}\ddot{\mathrm{y}}\mathrm{=}\ddot{\mathrm{u}}+\dot{\omega }\mathrm{\wedge }(\mathrm{x}+\mathrm{u})+\omega \mathrm{\wedge }\mathrm{[}\omega \mathrm{\wedge }(\mathrm{x}+\mathrm{u})\mathrm{]}+2\omega \mathrm{\wedge }\dot{\mathrm{u}}` In this expression, which represents the theorem of the composition of the accelerations of a material point, we recognize the terms: * of relative acceleration :math:`\ddot{\mathrm{u}}`, which contributes to the mass matrix; * of training acceleration :math:`\dot{\omega }\mathrm{\wedge }(\mathrm{x}+\mathrm{u})+\omega \mathrm{\wedge }\mathrm{[}\omega \mathrm{\wedge }(\mathrm{x}+\mathrm{u})\mathrm{]}` (sum of the Euler effect due to the acceleration of rotation and the centrifugal softening effect, which contribute to the stiffness matrix); * complementary acceleration or Coriolis :math:`2\omega \mathrm{\wedge }\dot{\mathrm{u}}`, which contributes to the damping matrix. Expression of kinetic and potential energies ------------------------------------------------ In its general form, kinetic energy is obtained from the absolute speed :math:`\mathrm{y}` as follows: :math:`\mathrm{T}\mathrm{=}\frac{1}{2}\underset{\Omega }{\mathrm{\int }}\rho {\dot{\mathrm{y}}}^{t}\dot{\mathrm{y}}\text{d}\Omega` Expanding the terms gives the following expression as a function of displacement :math:`\mathrm{u}`: :math:`\begin{array}{c}\mathrm{T}\mathrm{=}\frac{1}{2}\underset{\Omega }{\mathrm{\int }}\rho {\dot{\mathrm{u}}}^{t}\dot{\mathrm{u}}\text{d}\Omega +\underset{\Omega }{\mathrm{\int }}\rho {\dot{\mathrm{u}}}^{t}\Omega \mathrm{u}\text{d}\Omega \mathrm{-}\frac{1}{2}\underset{\Omega }{\mathrm{\int }}\rho {\mathrm{u}}^{t}{\Omega }^{2}\mathrm{u}\text{d}\Omega \mathrm{-}\underset{\Omega }{\mathrm{\int }}\rho {\mathrm{u}}^{t}{\Omega }^{2}\mathrm{x}\text{d}\Omega +\underset{\Omega }{\mathrm{\int }}\rho {\dot{\mathrm{u}}}^{t}\Omega \mathrm{x}\text{d}\Omega \\ \mathrm{-}\frac{1}{2}\underset{\Omega }{\mathrm{\int }}\rho {\mathrm{x}}^{t}{\Omega }^{2}\mathrm{x}\text{d}\Omega \end{array}` The potential energy of the system (internal deformation energy and the work of external forces) has the classical expression: :math:`\mathrm{U}\mathrm{=}\frac{1}{2}\underset{\Omega }{\mathrm{\int }}{\varepsilon }^{t}\Lambda \varepsilon \text{d}\Omega \mathrm{-}\underset{\Omega }{\mathrm{\int }}{\mathrm{u}}^{t}\mathrm{f}\text{d}\Omega \mathrm{-}\underset{\mathrm{\partial }{\Omega }_{\sigma }}{\mathrm{\int }}{\mathrm{u}}^{t}\mathrm{t}\text{d}(\mathrm{\partial }\Omega )` where :math:`\varepsilon` is the vector associated with the strain tensor, :math:`\Lambda` is the behavior matrix and where :math:`\mathrm{f}` and :math:`\mathrm{t}` are, respectively, the vectors of the external volume and surface forces. The displacement vector is approximated by the finite element method. To do this, we will use the classical form functions described in the document [:ref:`R3.01.01 `]. The displacement is then written in the form of the product of a displacement interpolation matrix, noted :math:`\mathrm{B}`, and a generalized coordinate vector :math:`\mathrm{q}`. The kinetic and potential energies of the deformable body can then be written according to the structural matrices as follows: :math:`\begin{array}{c}\mathrm{T}\mathrm{=}\frac{1}{2}{\dot{\mathrm{q}}}^{t}\left[\mathrm{M}\right]\dot{\mathrm{q}}+\frac{1}{2}{\dot{\mathrm{q}}}^{t}\left[\mathrm{G}\right]\mathrm{q}\mathrm{-}\frac{1}{2}{\mathrm{q}}^{t}\left[\mathrm{N}\right]\mathrm{q}\mathrm{-}{\mathrm{q}}^{t}\underset{\Omega }{\mathrm{\int }}\rho {\mathrm{B}}^{t}{\Omega }^{2}\mathrm{x}\text{d}\Omega +{\dot{\mathrm{q}}}^{t}\underset{\Omega }{\mathrm{\int }}\rho {\mathrm{B}}^{t}\Omega \mathrm{x}\text{d}\Omega \\ \mathrm{-}\frac{1}{2}\underset{\Omega }{\mathrm{\int }}\rho {\mathrm{x}}^{t}{\Omega }^{2}\mathrm{x}\text{d}\Omega \end{array}` :math:`\mathrm{U}\mathrm{=}\frac{1}{2}{\mathrm{q}}^{t}\left[\mathrm{K}\right]\mathrm{q}\mathrm{-}{\mathrm{q}}^{\mathrm{t}}\underset{\Omega }{\mathrm{\int }}{\mathrm{B}}^{t}\mathrm{f}\text{d}\Omega \mathrm{-}{\mathrm{q}}^{\mathrm{t}}\underset{\mathrm{\partial }{\Omega }_{\sigma }}{\mathrm{\int }}{\mathrm{B}}^{t}\mathrm{t}\text{d}(\mathrm{\partial }\Omega )` The first three terms of kinetic energy highlight matrices: * mass :math:`\left[\mathrm{M}\right]\mathrm{=}\underset{\Omega }{\mathrm{\int }}\rho {\mathrm{B}}^{t}\mathrm{B}\text{d}\Omega`; * of Coriolis :math:`\left[\mathrm{G}\right]\mathrm{=}2\underset{\Omega }{\mathrm{\int }}\rho {\mathrm{B}}^{t}\Omega \mathrm{B}\text{d}\Omega`; * centrifugal acceleration :math:`\left[\mathrm{N}\right]\mathrm{=}\underset{\Omega }{\mathrm{\int }}\rho {\mathrm{B}}^{t}{\Omega }^{2}\mathrm{B}\text{d}\Omega`. The first term for potential energy highlights the stiffness matrix :math:`\left[\mathrm{K}\right]\mathrm{=}\underset{\Omega }{\mathrm{\int }}{\mathrm{\nabla }\mathrm{B}}^{t}\Lambda \mathrm{\nabla }\mathrm{B}\text{d}\Omega`. Calculation of the equilibrium equations of the rotating system ---------------------------------------- Disregarding a possible classical dissipation function, characterized by the damping matrix, the Lagrange equations for the kinetic and potential energies of the solid are written as follows: :math:`\frac{\mathrm{d}}{\text{dt}}(\frac{\mathrm{\partial }\mathrm{T}}{\mathrm{\partial }\dot{\mathrm{q}}})\mathrm{-}\frac{\mathrm{\partial }\mathrm{T}}{\mathrm{\partial }\mathrm{q}}+\frac{\mathrm{\partial }\mathrm{U}}{\mathrm{\partial }\mathrm{q}}\mathrm{=}0` By inserting the expressions for energies into the Lagrange equations, we find: :math:`\begin{array}{c}\frac{\mathrm{d}}{\text{dt}}\left[\left[\mathrm{M}\right]\dot{\mathrm{q}}+\frac{1}{2}\left[\mathrm{G}\right]\mathrm{q}+\frac{1}{2}\left[\mathrm{G}\right]\dot{\mathrm{q}}+\left[\mathrm{N}\right]\mathrm{q}+\underset{\Omega }{\mathrm{\int }}\rho {\mathrm{B}}^{t}\Omega \mathrm{x}\text{d}\Omega +\underset{\Omega }{\mathrm{\int }}\rho {\mathrm{B}}^{t}{\Omega }^{2}\mathrm{x}\text{d}\Omega +\left[\mathrm{K}\right]\mathrm{q}\mathrm{-}\underset{\Omega }{\mathrm{\int }}{\mathrm{B}}^{t}\mathrm{f}\text{d}\Omega \right]\\ \mathrm{-}\frac{\mathrm{d}}{\text{dt}}\left[\underset{\mathrm{\partial }{\Omega }_{\sigma }}{\mathrm{\int }}{\mathrm{B}}^{\mathrm{t}}\mathrm{t}\text{d}(\mathrm{\partial }\Omega )\right]\mathrm{=}0\end{array}` By explaining the time derivative and taking into account the derivative of the transformation matrix in the Lagrange equations, after simplification and rearrangement of the terms, the following matrix form is obtained: :math:`\left[\mathrm{M}\right]\mathrm{\langle }\ddot{\mathrm{q}}\mathrm{\rangle }+\left[\mathrm{G}\right]\mathrm{\langle }\dot{\mathrm{q}}\mathrm{\rangle }+(\left[\mathrm{K}\right]+\left[\mathrm{P}\right]+\left[\mathrm{N}\right])\mathrm{\langle }\mathrm{q}\mathrm{\rangle }\mathrm{=}\mathrm{\langle }\mathrm{r}\mathrm{\rangle }` :math:`\left[\mathrm{P}\right]` is the angular acceleration matrix, defined by :math:`\left[\mathrm{P}\right]\mathrm{=}\frac{1}{2}\left[\dot{\mathrm{G}}\right]\mathrm{=}\underset{\Omega }{\mathrm{\int }}\rho {\mathrm{B}}^{t}\dot{\Omega }\mathrm{B}\text{d}\Omega`, with: :math:`\dot{\Omega }\mathrm{=}\left[\begin{array}{ccc}0& \mathrm{-}{\dot{\omega }}_{z}& {\dot{\omega }}_{y}\\ {\dot{\omega }}_{z}& 0& \mathrm{-}{\dot{\omega }}_{x}\\ \mathrm{-}{\dot{\omega }}_{y}& {\dot{\omega }}_{x}& 0\end{array}\right]` :math:`\mathrm{\langle }\mathrm{r}\mathrm{\rangle }` is the vector combining the terms on the right-hand side. It includes external excitations :math:`\underset{\Omega }{\mathrm{\int }}{\mathrm{B}}^{\mathrm{t}}\mathrm{f}\text{d}\Omega +\underset{\mathrm{\partial }{\Omega }_{\sigma }}{\mathrm{\int }}{\mathrm{B}}^{t}\mathrm{t}\text{d}(\mathrm{\partial }\Omega )` and centrifugal prestress :math:`\mathrm{-}\underset{\Omega }{\mathrm{\int }}\rho {\mathrm{B}}^{t}(\dot{\Omega }\mathrm{x}+{\Omega }^{2}\mathrm{x})\text{d}\Omega`).