Discretization of inertia terms (principal axis coordinate frame)

A Galerkin discretization of the (virtual) displacements and rotations $\boldsymbol{U}$

$$\boldsymbol{U} \approx \boldsymbol{U}^h = \left[ \begin{array}{c}... ...\ \delta \psi_3(\xi) \end{array} \right] = \boldsymbol{N} \delta \boldsymbol{d}$$ (30)

leads along with the nodal (virtual) displacements and rotations

$$\boldsymbol{d} = \left[ \begin{array}{c} (U_1)_1 \\ (U_2)_1 \\ : ... ...delta (U_1)_1 \\ \delta (U_2)_1 \\ : \\ \delta (\psi_{3})_2 \end{array} \right]$$ (31)

and the with isoparametric shape functions $\boldsymbol{N}(\xi)$

$$\boldsymbol{N} = \left[ \begin{array}{cccccccccccc} \varphi_1 & 0... ... & 0 & 0 & 0 & 0 & \varphi_1& 0 & 0 & 0 & 0 & 0 & \varphi_2 \end{array} \right]$$ (32)

where

$$\varphi_1(\xi) = \frac{1}{2}(1-\xi) \ , \varphi_2(\xi) = \frac{1}{2}(1+\xi)$$ (33)

to the following weak formulation of the inertia terms:

$$\int_x \delta \boldsymbol{U}^T \cdot \left[\begin{array}{c} \text... ...t] \textrm{d}X_1 = \delta \boldsymbol{d}^T \boldsymbol{M} \ddot{\boldsymbol{d}}$$ (34)

Requiring a constant cross-section along the entire element thus yields constant material parameters, which can be pulled out of the above integration of $\boldsymbol{M}$ over $X_1$. Thus the integration of the product of the shape functions can be performed analytically (under consideration of the transformation of the length differential $\textrm{d}X_1$ to $\textrm{d}\xi$ according to

$$\textrm{d}X_1 = \textrm{det}\boldsymbol{J} \textrm{d}\xi = \frac{1}{2}(X_2 - X_1)\textrm{d}\xi$$ (35)

$$\int_L \varphi_i \varphi_j \textrm{d}X_1 = \frac{1}{2}(X_2 - X_1)... ... \ , \quad i = j\\ \frac{1}{6}(X_2 - X_1) \ , \quad i \ne j \end{array} \right.$$ (36)