## 5 Numerical Examples
### High-Speed Flexible Slider-Crank  
This example is about a flexible slider-crank suffered from a constant moment M, which aims at studying the influence of proposed method on the rotational nonlinearity. As shown in Figure 2, Rod I is connected to the earth and Rod II through a spherical joint at the same time; the rigid slider was also hinged with Rod II by a spherical joint and is restricted to moving along the X-axis. The mass of the slider was 0.1 kg and the moment was M = 500 Nm. Initially, Rod I and Rod II were both horizontally stationary in the X-Y plane, that is, \(\alpha\) = 0. Other geometrical and material parameters are shown in Table 1. In the simulation, both rods are discretized into six elements using the 3-order NURBS basic function; the total simulation time was 0.2 s with two different time steps \(h\) = 0.0001 s and \(h\) = 0.0005 s. The elastic deformation of rod was measured by the distance between the midpoint of rod and the midpoint of the line connecting the two ends of rod. The deformation rate was the ratio between the elastic deformation of the rod and the original length of the rod. Obviously, the deformation of Rod II during the simulation increased gradually with the increase of rotation speed, as displayed in Figure 3. The maximum deformation of Rod II and the maximum rotational speed of Rod I exceeded 5% of the rod length and 320 rad/s, respectively.  
In general, if the Jacobian matrix corresponding to the external force is ignored, then the system iteration matrix consists of a generalized mass matrix, a tangent stiffness matrix, and a Jacobian matrix corresponding to the constraint equation. It is assumed that \(\mathbf{J}_{\theta}\) represents the iterative matrix composed of the generalized mass matrix and the stiffness matrix corresponding to rotation parameters, and the convergence criterion of Newton iteration is set to the incremental value of the generalized coordinate, \(||\mathbf{D}||^{2}\leq 10^{-6}\), in which \(\mathbb{D}=[\Delta\mathbf{q}^{\mathrm{T}},h^{2}\beta\Delta\mathbf{x}^{ \mathrm{T}}]^{\mathrm{T}}\). The number of times that the iteration matrix \(\mathbf{J}_{\theta}\) is updated (update times) and the total iterations in the simulation for different time discretizations and update  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Parameters** & **First Case** \\ \hline Length of Rod I & 0.3 m \\ Length of Rod II & 0.6 m \\ Cross-section & (0.03 \(\times\) 0.03) m\({}^{2}\) \\ Young’s modulus & 2.1e11 N/m\({}^{2}\) \\ Poisson coefficient & 0.3 \\ Mass density & 7850 kg/m\({}^{3}\) \\ Mass of slider & 0.1 kg \\ Gravity & 9.81 m/s\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters for the flexible slider-crank.  
Figure 2: The configuration of the slider-crank.  
conditions of the truss structure unit are given in Table 2. The \"Update\" means \(\mathbf{J}_{\Theta}\) is updated all the time, and the \"Frozen\" denotes \(\mathbf{J}_{\Theta}\) in the reference configuration is maintained for the whole simulation. Obviously, if \(\mathbf{J}_{\Theta}\) is updated in each Newton iteration step, the whole simulation needs a total of 1435 Newton iteration steps, and \(\mathbf{J}_{\Theta}\) needs to be updated 1434 times. While the iteration matrix \(\mathbf{J}_{\Theta}\) is kept for the whole simulation, the total iterations merely needs 1473 times. It is demonstrated that the rigid rotational nonlinearity is reduced dramatically based on the local frame approach, such that the iteration matrix \(\mathbf{J}_{\Theta}\) can remain unchanged all the time, especially for the case of moderate deformation rate in this example. Moreover, the rigid rotational nonlinearity can be also reduced with a smaller time step, since the total iterations remains unchanged in the condition of \"Frozen\", as shown in Table 2. From a computational point of view, however, the smaller the time step, the more iteration steps are needed, which increases the computation cost.