# Optimal Roving Winding on Toroidal Parts of Composite Frames

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## Abstract

**:**

## 1. Introduction

**Note 1.**

## 2. Materials and Methods

#### 2.1. Torus-Shaped Part of the Frame

#### 2.2. Level of Difficulty of Roving Winding

#### 2.3. Mathematical Description of Roving Winding on the Torus

#### 2.4. Determination of Winding Angle on Torus

**Note 2.**

**Proof.**

#### 2.5. Determination of Torodial Helix Parameter $\omega $

#### 2.6. Optimal Number of Rovings Used during Winding

**Note 3.**

**Note 4.**

## 3. Results and Discussion

#### 3.1. Determining the Difficulty of Torus Winding

#### 3.2. Relations between Winding Parameters

#### 3.3. Winding Angle of Rovings on the Torus

#### 3.4. Determination of Optimal Number of Rovings

**Note 5.**

#### 3.5. Recommended Procedure before Starting Winding

- Determine the suitability of winding the rovings on the non-load-bearing frame (Relations (3), (7), and (8) can be used, see Table 1). If the winding conditions are unfavorable, consider whether, for example, to use a differently shaped frame or to choose a different composite manufacturing technology.
- Calculate parameter ω using Relation (22). Based on the knowledge of this parameter, an estimate of the number of roving revolutions on the whole toroidal helix can be obtained.
- Determine the maximum winding angle of the roving ${\stackrel{~}{\alpha}}_{ext}$ on the torus at the outer circumference ${p}_{1}$ (see Figure 6b) using Relations (17) and (20). At the same time, determine the minimum winding angle ${\stackrel{~}{\alpha}}_{int}$ on the inner circumference of ${p}_{2}$ by applying Relations (18) and (21). For the required winding angle $\alpha $ for a given layer, the following relation holds: ${\stackrel{~}{\alpha}}_{int}<\alpha <{\stackrel{~}{\alpha}}_{ext}$. During the winding procedure, the winding angle $\stackrel{~}{\alpha}$on the torus changes continuously and ${\stackrel{~}{\alpha}}_{int}\le \stackrel{~}{\alpha}\le {\stackrel{~}{\alpha}}_{ext}$. Due to the continuously changing winding angle $\stackrel{~}{\alpha}$, it is useful to determine whether the changing winding angle satisfies the winding requirements with respect to the planned loading of the polymer composite frame using a suitable modelling software tool (e.g., ABAQUS, ANSYS).
- Determine the optimized number of rovings $n$ for the winding of the layers at their specified width $d$. To the selected value of $n$, calculate the overlap ${\stackrel{~}{\epsilon}}_{02}$ of two adjacent rovings on the outer circumference ${p}_{1}$ and the overlap ${\stackrel{~}{\epsilon}}_{13}$ on the inner circumference ${p}_{2}$. Following this, select the winding of the roving with the most suitable width $d$ provided by the supplier of rovings.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) The frame attached to the robot end effector passes through the winding head with a single rotating ring. One layer of winding is formed. (

**b**) An example of a 3D frame with a complicated shape.

**Figure 2.**(

**a**) Simultaneous winding of three layers of glass rovings on the open frame. (

**b**) Fixing the closed frame to the robot end effector. (

**c**) Rotating ring of winding head with coils with wound rovings.

**Figure 3.**The first rotating ring of the winding head winds one roving at an angle of 45° and the following second rotating ring winds the roving at an angle of −45°.

**Figure 4.**(

**a**) One turn of a right-hand straight helix ${h}_{R}$. (

**b**) One turn of a left-hand straight helix ${h}_{L}$. (

**c**) Characteristics triangle of a straight helix.

**Figure 5.**(

**a**) Model of the torus. (

**b**) Non-bearing polyurethane frame for winding rovings with a middle section forming part of the torus.

**Figure 6.**(

**a**) An example of a torus. (

**b**) Torus with outer circumferential circle ${p}_{1}$ and inner circumferential circle ${p}_{2}$.

**Figure 8.**(

**a**) Torus centered at the origin $S$, xy-plane cut, rotation of circle $k$ ≡ (M, r) around the x-axis. (

**b**) Roving of width d with the central axis l.

**Figure 9.**Graphical representation of the ratio of the size of partial surfaces ${s}_{1}$ and ${s}_{2}$, and ${s}_{total}={s}_{1}+{s}_{2}$.

**Figure 10.**(

**a**) Graph of right-handed toroidal helix for specified parameters $R=100$, $r=33$, $\omega =5$ (number of winds per helix). (

**b**) δ angle clamped by vectors $\mathbf{u}\left(0\right)$ and $\mathbf{w}(0$) at point ${T}_{0}$ of toroidal helix δ. (Figure 10a and Figure 12 are generated by “Toroidal Helices—Wolfram Demonstrations Project” graphics application freely available from https://www.google.com/search?q=toroidal-helix&oq=toroidal-helix&aqs=chrome..69i57j0i13i30.10920j0j15&sourceid=chrome&ie=UTF-8#imgrc=HAw5MhPvHq4pfM, accessed on 11 June 2023).

**Figure 11.**(

**a**) Arc length part ${l}_{02}$ of circle ${p}_{1}$ and arc length part ${l}_{13}$ of circle ${p}_{2}$. (

**b**) Laying a roving of width $d$ at an angle ${\stackrel{~}{\alpha}}_{ext}$ (relation 18a/). Value ${c}_{0}$ indicates the length of the wound roving on connecting points ${T}_{0}$ and ${T}_{2}$.

**Figure 12.**Examples of regular right-handed toroidal (

**a**) 3-helix: major radius R = 10, minor radius r = 2.5, ω= 2 (number of winds per helix), (

**b**) 5—helix: major radius R = 10, minor radius r = 4.5; ω = 3.

**Figure 13.**Floor plans of tori with parameters: $R=500\left[\mathrm{m}\mathrm{m}\right]$; (

**a**) $r=50\left[\mathrm{m}\mathrm{m}\right],a=0.1$; (

**b**) $r=100\left[\mathrm{m}\mathrm{m}\right],a=0.2$; (

**c**) $r=400\left[\mathrm{m}\mathrm{m}\right],a=0.8$.

**Figure 14.**Graphical representation of $a=r/R$, ${s}_{2}/{s}_{1}$, and $o({p}_{2})/o({p}_{1})$ values for constant major radius $R=500\left[\mathrm{m}\mathrm{m}\right]$ and gradually increasing minor radius $r$.

**Figure 15.**Example of the curved part of polymer composite frame with the following parameters: major radius R = 102.5 [mm], minor radius r = 17.5 [mm], width of roving d = 5 [mm]. The non-load-bearing polyurethane frame is visible in the vertical section (light colour of the cross-section).

Major Radius (R) [mm] | Minor Radius (r) [mm] | Aspect Ratio (a) | Ratio $\frac{{\mathit{s}}_{2}}{{\mathit{s}}_{1}}$ | Ratio $\frac{\mathit{o}\left({\mathit{p}}_{2}\right)}{\mathit{o}\left({\mathit{p}}_{1}\right)}$ |
---|---|---|---|---|

1000 | 20 | 0.02 | 0.9748 | 0.9607 |

500 | 0.5 | 0.5171 | 0.3333 | |

800 | 0.8 | 0.2407 | 0.1111 | |

500 | 50 | 0.1 | 0.8802 | 0.8181 |

100 | 0.2 | 0.7741 | 0.6666 | |

400 | 0.8 | 0.3251 | 0.1111 | |

100 | 20 | 0.2 | 0.7741 | 0.6666 |

50 | 0.5 | 0.5171 | 0.3333 | |

90 | 0.9 | 0.2715 | 0.0526 | |

50 | 10 | 0.2 | 0.7741 | 0.6666 |

20 | 0.4 | 0.5941 | 0.4285 | |

30 | 0.6 | 0.4472 | 0.2500 |

**Table 2.**Interrelation of parameters when winding the straight part of the frame and the curved part of the frame in the shape of the torus part.

Major Radius (R) [mm] | Minor Radius (r) [mm] | Aspect Ratio (a) | Winding Angle (α) [°] [rad] | tg α | Parameter ω | Toroidal Pitch $\left(\mathit{H}\right)$ [mm] | |
---|---|---|---|---|---|---|---|

500 | 25 | 0.05 | 5 | 0.0815 | 0.0874 | 228.8329 | 13.7287 |

30 | 0.5235 | 0.5773 | 34.6440 | 90.68215 | |||

45 | 0.7853 | 1.0000 | 20.0000 | 157.0796 | |||

50 | 0.1 | 5 | 0.0815 | 0.0874 | 114.3118 | 27.4826 | |

30 | 0.5235 | 0.5773 | 17.32051 | 181.3799 | |||

45 | 0.7853 | 1.0000 | 10.0000 | 314.1592 | |||

100 | 0.2 | 5 | 0.0815 | 0.0874 | 61.3496 | 51.2080 | |

30 | 0.5235 | 0.5773 | 8.6610 | 362.7286 | |||

45 | 0.7853 | 1.0000 | 5.0000 | 628.3185 | |||

450 | 0.9 | 5 | 0.0815 | 0.0874 | 12.7128 | 247.1204 | |

30 | 0.5235 | 0.5773 | 2.1224 | 1480.2076 | |||

45 | 0.7853 | 1.0000 | 1.1111 | 2827.4616 |

**Table 3.**Determination of the winding angle ${\stackrel{~}{\alpha}}_{int}$ on the inner circumference of the torus (circle ${p}_{2}$ ) and the winding angle ${\stackrel{~}{\alpha}}_{ext}$ on the outer circumference of the torus (circle ${p}_{1}$ ) depending on the major radius $R$ of the torus, the minor radius $r$ of the torus and the desired winding angle $\alpha $.

Major Radius (R) [mm] | Minor Radius (r) [mm] | Aspect Ratio (a) | Winding Angle (α) [°] [rad] | tg α | Parameter $\mathit{\omega}$ | Angle ${\stackrel{~}{\mathit{\alpha}}}_{\mathit{i}\mathit{n}\mathit{t}}$ [°] | Angle ${\stackrel{~}{\mathit{\alpha}}}_{\mathit{e}\mathit{x}\mathit{t}}$ [°] | |
---|---|---|---|---|---|---|---|---|

100 | 20 | 0.2 | 5 | 0.0815 | 0.0874 | 57.2082 | 3.9968 | 5.9872 |

50 | 0.5 | 30 | 0.5235 | 0.5773 | 3.4644 | 16.1007 | 40.8909 | |

90 | 0.9 | 45 | 0.7853 | 1.0000 | 1.4148 | 4.4904 | 56.1712 | |

50 | 10 | 0.2 | 5 | 0.0815 | 0.0874 | 61.3496 | 3.73040 | 5.5857 |

20 | 0.4 | 30 | 0.5235 | 0.5773 | 4.7755 | 17.4376 | 36.2379 | |

30 | 0.6 | 45 | 0.7853 | 1.0000 | 1.6666 | 21.8021 | 57.9956 |

**Table 4.**Optimized number of rovings $n$ used in winding and the size of overlaps ${\stackrel{~}{\epsilon}}_{02}$ on the outer and ${\stackrel{~}{\epsilon}}_{13}$ on the inner circumference of the torus for given values of $R$, $r$, $d$ and $\alpha $.

Outer Radius (R) [mm] | Inner Radius (r) [mm] | Param. a | Angle Winding (α) [°] | Param. ω | Roving Width (d) [mm] | Optimized Number of Rovings (n) | Outer Overlap ( ${\stackrel{~}{\mathit{\epsilon}}}_{02}$) [mm] | Inner Overlap (${\stackrel{~}{\mathit{\epsilon}}}_{13}$) [mm] |
---|---|---|---|---|---|---|---|---|

100 | 20 | 0.2 | 10 | 28.3607 | 9 | 3 | 0.3378 | 3.1811 |

25 | 0.25 | 30 | 6.9282 | 11 | 0.7944 | 3.6238 | ||

30 | 0.3 | 45 | 3.3333 | 17 | 0.3467 | 3.2236 | ||

200 | 10 | 0.05 | 10 | 113.4429 | 5 | 3 | 1.2083 | 1.5621 |

30 | 34.6410 | 7 | 0.4055 | 0.7795 | ||||

45 | 20.0000 | 10 | 0.6526 | 0.9274 |

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## Share and Cite

**MDPI and ACS Style**

Mlýnek, J.; Rahimian Koloor, S.S.; Knobloch, R.
Optimal Roving Winding on Toroidal Parts of Composite Frames. *Polymers* **2023**, *15*, 3227.
https://doi.org/10.3390/polym15153227

**AMA Style**

Mlýnek J, Rahimian Koloor SS, Knobloch R.
Optimal Roving Winding on Toroidal Parts of Composite Frames. *Polymers*. 2023; 15(15):3227.
https://doi.org/10.3390/polym15153227

**Chicago/Turabian Style**

Mlýnek, Jaroslav, Seyed Saeid Rahimian Koloor, and Roman Knobloch.
2023. "Optimal Roving Winding on Toroidal Parts of Composite Frames" *Polymers* 15, no. 15: 3227.
https://doi.org/10.3390/polym15153227