10.1.1. Equations of Motion using Modal Approach

Kinematics

../_images/image00321.png

Figure 10.2 Configuration of a flexible body

Figure 10.2 shows the configuration of a flexible body \(f\) in the undeformed and deformed state. In the figure,

  • \(XYZ\) : Inertia reference frame

  • \(O\) : Origin of XYZ

  • \({{x}_{f}}{{y}_{f}}{{z}_{f}}\) : Flexible body reference frame

  • \({{O}_{f}}\) : Origin of \({{x}_{f}}{{y}_{f}}{{z}_{f}}\)

  • \({{P}_{i0}}\) : Node \(i\) in the undeformed state

  • \({{P}_{i}}\) : Node \(i\) in the deformed state

  • \({{r}_{f}}\) : Position vector from the point \(O\) to the point \({{O}_{f}}\)

  • \({\mathbf{u}_{{f}_{i0}}}\) : Position vector from the point \({{O}_{f}}\) to the \({{P}_{i0}}\)

  • \(\mathbf{u}_{{f}_{i}}^{'}\) : Nodal elastic deformation position vector from the point \({{P}_{i0}}\) to the \({{P}_{i}}\) measured with respect to \({{x}_{f}}{{y}_{f}}{{z}_{f}}\). \(\mathbf{u}_{{{f}_{i}}}^{'}\) is nodal elastic deformation

  • \(\mathbf{d}_{{{f}_{i}}}^{'}\) : \(\mathbf{u}_{{{f}_{i0}}}+\mathbf{u}_{{{f}_{i}}}^{'}\), position vector from the point \({{O}_{f}}\) to the point \({{P}_{i}}\)

For a generic node \(i\) of a flexible body \(f\), position vector of the node can be written as

(10.1)\[{\mathbf{r}_{{{f}_{i}}}}={\mathbf{r}_{f}}+{\mathbf{A}_{f}}\mathbf{d}_{{{f}_{i}}}^{'}\]

where,

(10.2)\[\mathbf{d}_{{{f}_{i}}}^{'}\cong \mathbf{u}_{{{f}_{i0}}}^{'}+\boldsymbol\Phi_{T}^{f_i}\,{\mathbf{a}_{f}}\]

In above equation, \(\boldsymbol\Phi_{T}^{{{f}_{1}}}\) is the translational modal matrix of a node \(i\).

By taking time differentiation of the position vector, velocity of the node \(i\) is obtained as follows;

(10.3)\[{\dot{\mathbf{r}}}_{{f}_{i}}={\mathbf{A}_{f}}\dot{\mathbf{r}}_{f_i}^{'}-{\mathbf{A}_{f}}{\tilde{\mathbf{d}}}_{{f_i}}^{'}{\boldsymbol{\omega}}_f^{'}+\mathbf{A}_{f}\boldsymbol{\Phi}_{T}^{f_i}\dot{\mathbf{a}}_{f}\]

By taking time differentiation of the velocity vector, acceleration of the node \(i\) is obtained as follows;

(10.4)\[\ddot{\mathbf{r}}_{f_i}={\mathbf{A}_{f}}\left( \ddot{\mathbf{r}}_{f}^{'}-{\tilde{\mathbf{d}}}_{{f_i}}^{'}\dot{\boldsymbol{\omega}}_f^{'}+\ddot{\mathbf{d}}_{f_i}^{'}+\tilde{\boldsymbol{\omega}}_{f}^{'}\tilde{\boldsymbol{\omega}}_{f}^{'}\mathbf{d}_{{f_i}}^{'}+2\tilde{\boldsymbol{\omega}}_{f}^{'}\dot{\mathbf{d}}_{{f_i}}^{'}\right)\]

where,

(10.5)\[\dot{\mathbf{d}}_{f_i}^{'}=\dot{\mathbf{u}}_{f_i}^{'}=\boldsymbol{\Phi}_{T}^{f_i}\dot{\mathbf{a}}_{f}\]
(10.6)\[\ddot{\mathbf{d}}_{f_i}^{'}=\ddot{\mathbf{u}}_{f_i}^{'}=\boldsymbol{\Phi}_{T}^{f_i}\ddot{\mathbf{a}}_{f}\]

EOM

By the virtual work principle, equations of motion can be obtained as

\(\mathbf{B}^{T}\left( \mathbf{M}\begin{Bmatrix} \dot{\mathbf{Y}} \\ \ddot{\mathbf{a}} \end{Bmatrix} + \mathbf{C}_{\mathbf{z}_c}^{T} \boldsymbol{\lambda}_1 -\mathbf{Q}^C -\mathbf{Q}^E-\mathbf{Q}^G -\mathbf{Q}^D -\mathbf{Q}^F \right) + \mathbf{C}_{\mathbf{q}_c}^{T}\boldsymbol{\lambda}_2 -\mathbf{Q}^J = 0\)

where,
\(\mathbf{z}:\text{ Cartesian coordinates}\text{.}\)
\(\mathbf{a}:\text{ Modal coordinates}\text{.}\)
\(\mathbf{q}:\text{ Relative coordinates}\text{.}\)
\(\mathbf{M}\text{=}\left[ \begin{matrix} {{\mathbf{M}}_{zz}} & {{\mathbf{M}}_{za}} \\ {{\mathbf{M}}_{az}} & {{\mathbf{M}}_{aa}} \\ \end{matrix} \right]:\text{ Mass matrix}\text{.}\)
\(\dot{\mathbf{Y}}:\text{ Local acceleration of cartesian coordinates}\text{.}\)
\(\mathbf{C}_{\mathbf{z}_{c}}^{{}}:\text{ Partial derivative terms of constraint equations with respect to }{{\mathbf{z}}_{c}}\text{.}\)
\(\mathbf{C}_{\mathbf{q}_{c}}^{{}}:\text{ Partial derivative terms of constraint equations with respect to }{{\mathbf{q}}_{c}}\text{.}\)
\(\boldsymbol{\lambda}_{1}:\text{ Lagrangian multipliers}\text{.}\)
\(\boldsymbol{\lambda}_{2}:\text{ Lagrangian multipliers}\text{. (Driving constraint of a joint)}\)
\(\mathbf{Q}_{{}}^{C}:\text{ Quadratic velocity term}\text{.}\)
\(\mathbf{Q}_{{}}^{E}:\text{ Elastic force term}\text{.}\)
\(\mathbf{Q}_{{}}^{G}:\text{ Gravitational force term}\text{.}\)
\(\mathbf{Q}_{{}}^{D}:\text{ Damping force term}\text{.}\)
\(\mathbf{Q}_{{}}^{F}:\text{ Etc}\text{. force terms}\text{. (Modal force, Generalized force, Contact force ...)}\)
\(\mathbf{Q}_{{}}^{J}:\text{ Joint friction force term}\text{.}\)