## 2 Skewing Effects  
### Investigated Model  
Generally, the skewing approach is applied to rotating and linear machines, but the number of papers about its impacts on a tubular generator is still modest [11; 12; 13; 14]. Unlike the unbalanced model or the addition of auxiliary teeth [15; 16; 17], the skewing method does not require any extra components, hence, the volume and mass of the machine remain the same. In [12], PMs are in triangular skewing, which introduces an angular component, and consequently alters the radial and axial component in the form of amplitude. 2.5D and 3D analysis prove that the detent force in a tubular linear machine is decreased when the skewing amplitude \(\Delta\)s is increased. In [12; 13], PMs are skewed and divided into a multi-slice shape. In the first case, small PM segments are parallel to each other, following two helical paths with opposite screw direction. The multi-slice method was used to avoid calculating the full 180\({}^{\circ}\) model, thus, reducing the calculation time. In another case, individual PM pieces were cut in one piece out of a hollow shaft with two parallel oblique planes. This is called a plane skewed method. Computational results show that force ripple is reduced from 41.6 N (on the un-skewed model) to 8.32 N (on the plane skewed model) [13]. In term of reducing force ripple, the skewing method using multi-slice PM is more effective the than plane skewed method. However, the feasibility of this method needs to be further investigated in detail. According to the back-up data from [12], a single-slot single-pole model is considered to reduce the complexity and time consumed in 3D FEM analysis. By assuming that the 8-slot 8-pole machine is symmetrical, the performance of the full model is simply multiplied by 8 times. In the other approach, when a single-slot single-pole model has a boundary condition applied (periodicity), its performance is equivalent to the full model. Figure 1 shows a cross-section of the single-slot, single-pole model, which was taken from an optimal 8-slot 8-pole machine proposed in [9]. The axial PM width is 5 mm, the radial PM width is 18 mm, and the tooth width is 6.4 mm. The iron core and spacers are still composed of carbon steel S20C, a magnetic material. With the same number of turns per slot, phase resistance is smaller by eight times (5.37 \(\Omega\)). As mentioned in [9], PM ring is wrapped by a cover comprised of stainless steel 1.1 mm thick. In addition to a 0.3 mm mechanical air gap, the total magnetic air gap is 1.4 mm. It should be noted that in this paper, the standard operating conditions for all models are 0.25 m/s of the vibration speed, 10 Hz of the vibration frequency and 11.25 mm of the peak-to-peak stroke length. To increase the machine's performance, a hybrid PM structure proposed in [8; 9; 10] was also implemented. Specifications of the model to be investigated are summarized again in Table 1.  
Because one pole pitch is \(\tau_{\text{p}}\) = 23 mm, when the PMs are skewed by 30\({}^{\circ}\), it means that the PM ring is shifted by:  
\[S_{L}=\frac{23\times 30^{\circ}}{360^{\circ}}=1.917\ (\text{mm}) \tag{1}\] \begin{table}
\begin{tabular}{l c} \hline \hline
**Item** & **Value** \\ \hline Magnetic air gap, \(g\) (mm) & 1.4 \\ Radial permanent magnet (PM) thickness, \(t_{\text{m}}\) (mm) & 7.0 \\ Axial PM width, \(w_{\text{am}}\) (mm) & 5.0 \\ Radial PM width, \(w_{\text{rm}}\) (mm) & 18.0 \\ Spacer thickness, \(t_{\text{sp}}\) (mm) & 2.0 \\ Spacer width, \(w_{\text{sp}}\) (mm) & 18.0 \\ Tooth width, \(w_{\text{t}}\) (mm) & 6.4 \\ Tooth thickness, \(t_{\text{t}}\) (mm) & 14.0 \\ Pole pitch, \(\tau_{\text{p}}\) (mm) & 23.0 \\ Slot pitch, \(\tau_{\text{s}}\) (mm) & 23.0 \\ Back iron thickness, \(t_{\text{b}}\) (mm) & 4.0 \\ Number of turns/slot & 361 \\ Phase resistance, (\(\Omega\)) & 5.37 \\ Material of the axial PM/radial PM & 50 SH, \(B_{\text{r}}\) = 1.4 T/40 SH, \(B_{\text{r}}\) = 1.26 T \\ Material of the tooth, back iron, spacer & S20C \\ \hline \hline \end{tabular}
\end{table}