A geometric approach for filament winding pattern generation and study of the influence of the slippage coefficient

Abstract

A special feature of the Filament Winding (FW) process is known as pattern: diamond-shaped mosaic that results from the sequence of movements of the mandrel and tow delivery eye. One of the main factors to generate different patterns is the return path of the tow and, for a non-geodesic trajectory, the path depends on the friction between tow and mandrel. Aiming at a practical description of the FW process, a novel geometric approach on pattern construction is presented. Pattern generation, skip configurations and definitions of geodesic and non-geodesic trajectories in regular winding and return regions are described based on developed surfaces, residue classes and modular arithmetic. The influence of mandrel’s length, mandrel’s rotation angle and variation of the winding angle in the return region are presented, for they are important parameters of the process. Examples of winding angle, mandrel rotation and non-geodesic path in cylindrical and non-cylindrical surfaces of revolution are shown and discussed.

Appendix

1.1 Determination of the relation between the pattern and dwell

Let insert Eq. (9) into Eq. (11). It results in

$$\begin{aligned} 2 p_{\mathrm{tr}} \left( 360 \left\lfloor \dfrac{L}{L_r} \right\rfloor + {\hat{d}}_w \right)&\equiv 0 \; \left( \mathrm {mod} \; 360 \right) \\ 2 p_{\mathrm{tr}} 360 \left\lfloor \dfrac{L}{L_r} \right\rfloor + 2 p_{\mathrm{tr}} {\hat{d}}_w&\equiv 0 \; \left( \mathrm {mod} \; 360 \right) \end{aligned}$$

By the same rule that infers the reduction of the pattern given a skip, one obtains

$$\begin{aligned} p_{\mathrm{tr}} 360 \left\lfloor \dfrac{L}{L_r} \right\rfloor + p_{\mathrm{tr}} {\hat{d}}_w \equiv 0 \; \left( \mathrm {mod} \; 180 \right) \end{aligned}$$

This simplification is possible given the symmetry of the strokes inside a circuit. And due to the properties of sum in modular arithmetic: the value $p_{\mathrm{tr}} 360 \left\lfloor \frac{L}{L_r} \right\rfloor$ belongs to the residue class $\left[ 0 \right] _{180}$ as $p_{\mathrm{tr}} \in {\mathbb {N}} \setminus 0$ and $\left\lfloor \frac{L}{L_r}\right\rfloor \in {\mathbb {N}}$ and, therefore, $p_{\mathrm{tr}} \left\lfloor \frac{L}{L_r}\right\rfloor \in {\mathbb {N}}$. Any non-negative integer multiplied by 360 belongs to the residue class $\left[ 0 \right] _{180}$. This proves that the region of regular winding with geodesic trajectory have influence of any kind in the pattern generation. Then

$$\begin{aligned} \left[ 0 \right] + p_{\mathrm{tr}} {\hat{d}}_w \equiv p_{\mathrm{tr}} {\hat{d}}_w \equiv 0 \; \left( \mathrm {mod} \; 180 \right) \end{aligned}$$

in which is Eq. (12).