# Fabrication of High-Quality Straight-Line Polymer Composite Frame with Different Radius Parts Using Fiber Winding Process

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Manufacturing of Polymer Composite Frame

#### 2.1. Determination of the Winding Plane

_{3}. The distance of the rotational guide-line k of the fiber-processing head from the plane of fiber winding (element of this plane on Figure 5 is the cross-section of the frame containing point T(t

_{0})) depends on the radius R of the rotational guide-line k, the radius r of wound part of the frame (each part of the frame has circular cross-section) and required angle $\alpha $ of winding of the fiber on the frame. We suppose the fiber is wound on the frame as a right-handed helix p

_{R}(the second option would be left-handed helix) with axis o ≡ z, helix radius r (radius of the considered part of the frame). We consider the angle of winding of the fiber onto the surface of the frame to be the angle slope $\alpha $ of the corresponding helix (see Figure 6 on the right and in detail [48], Chapter 2). However, technicians in the manufacture of composite frames often call δ the angle of winding of the fibers onto the frame (see Figure 6 on the right), where

_{3}(the fourth coordinate of the point is 1 and of the vector is 0), in which the detail is presented elsewhere [48]. Then the parametric equation of helix p

_{R}can be expressed in the form of ${p}_{R}\left(t\right)=\left(r\mathrm{cos}t,r\mathrm{sin}t,{v}_{0}t,1\right),t\in 0,\infty )$ (see Chapter 2 in [48]). Parameter ${v}_{0}$ determines reduced pitch of helix (length of translation during rotation of fiber by one radian, ${v}_{0}=r.\mathrm{tg}\alpha $), see also Figure 6 on the right. Then derivative ${p}_{R}^{\prime}\left(t\right)=\left(-r\mathrm{sin}t,r\mathrm{cos}t,{v}_{0},0\right)$ (see Chapter 2 in [48]). Equation of tangent $m\left(t\right)$ of helix p

_{R}(t) at point $T\left({t}_{0}\right)=\left(r\mathrm{cos}{t}_{0},r\mathrm{sin}t,{v}_{0}{t}_{0},1\right)$ for given t

_{0}is defined in form

_{R}during the winding process is represented by point C. This point C is the element of the cylindrical surface with central axis z and radius R. For each point A = [x

_{A}, y

_{A}, z

_{A,}1] of considered cylindrical surface, it is true ${x}_{A}^{2}+{y}_{A}^{2}={R}^{2}$. At the same time, point C lies on tangent m(t). Therefore, exists the real value $\tilde{t}$ the equation holds

#### 2.2. Controlling of the Speed of Fibers Winding

#### Note

#### General Procedure

- Determination of the distance h1 of the fiber winding on Part I of the frame from guide-line (use Relation (4)).
- Calculation of angular speed ω1 needed to ensure the required angle of the fibers winding on Part I (use Relation (7)).
- Determination of distance h2 of the fiber winding on Part II of the frame from guide-line (use Relation (4)).
- Calculation of angular speed ω2 needed to ensure the required angle of winding of the fibers on Part II (use Relation (7)).

- 5.
- Termination of angular speed ω1 of guide-line in distance h1 before the end of Part I when the frame goes through the guide-line.
- 6.
- Smooth transition from angular velocities ω1 to ω2 of guide-line.
- 7.
- Starting angular speed ω2 of guide-line, when the beginning of Part II is in distance before guide-line h2.

## 3. Results and Discussions

#### 3.1. Verification Example

_{1}= 40 [mm], r

_{2}= 30 [mm], r

_{3}= 20 [mm]. The lengths of the individual parts are l

_{1}= l

_{2}= l

_{3}= 500 [mm]. Radius R of the guide-line of the fiber-processing head is R = 50 [mm] (see Figure 6 on the left). In this article, we consider, generally, a fiber-processing head with three guide-lines k1, k2, and k3 (see Figure 2 on the right, Figure 4). Three layers of fibers are gradually formed during the passing of the frame through the fiber-processing head. However, for clarity, we consider only winding of one layer using guide-lines k1 (see Figure 6 on the left) in this, our verification example. Part I will be wound under angle $\frac{\pi}{4}$, Part II under angle $\frac{\pi}{6}$ and Part III under angle $\frac{\pi}{3}$. The winding angles on the transition Part II-I and Part III-II, respectively, will be smoothly changed from angle $\frac{\pi}{6}$ to angle $\frac{\pi}{4}$ and from angle $\frac{\pi}{3}$ to angle $\frac{\pi}{6}$, respectively. Lengths d1 and d2 of both transition Parts II-I and III-II (see Figure 3a (bellow)) are the same value d1= d2= 80 [mm]. We suppose a constant speed w = 50 [mm/s] passage of the frame through guide-line k1 of the fiber-processing head.

#### 3.2. Composite Frame Production

#### Note

_{1}= 30 [mm], r

_{2}= 20 [mm] and prescribed winding angle α= $\frac{\pi}{6}$(as mentioned above, technologists of composite production usually call the winding angle δ = $\frac{\pi}{2}-\alpha $) is higher in the case of winding a frame part with a smaller cross-sectional radius. The ratio of the winding density on both parts of the frame is given by the ratio of pitch of helix v

_{2}for part of the frame with radius r

_{2}and pitch of helix v

_{1}for part of the frame with radius r

_{1}(see the relation (5) and Figure 6 on the right), $\frac{{v}_{2}}{{v}_{1}}=\frac{tg\alpha 2\pi {r}_{2}}{tg\alpha 2\pi {r}_{1}}=\frac{{r}_{2}}{{r}_{1}}=\frac{2}{3}$. It means that on the part of the frame with radius r

_{2}, the winding of the fiber is denser in proportion 3:2 (see Figure 9 on the right).

- realization of simultaneous winding of three layers of fibers on the frame at different angles,
- ensure the winding of individual parts (including parts with different radii of their cross-sections) of the frame at specified different angles for given winding layer,
- possibility of ensuring the same winding angle for each winding layer for all parts of the frame with different radii of their cross-sections,
- adjacent parts of the frame with different radii have a continuous transition of wound fibers for a given layer.

## 4. Conclusions

- Calculate the distance of winding the fibers (on the frame) from the corresponding guide-line. This distance depends on the size of the required winding angle, the radius of the guide-line, and the radius of the relevant wound part of the frame.
- Determine the angular speed of the guide-line so that the fibers are wound at the desired angle. Angular speed depends on the defined winding angle, the radius of the relevant part of the frame, and the constant size of passage speed of the frame through the head.
- Based on the previous two points and assuming a constant speed of passage of the frame through the head, we can determine:
- when and with what angular speed is started and ended winding by guide-line so that the fibers are wound at a desired angle on a specified part of the frame;
- the continuous transition between two individual parts of the frame with different radii is wound at a gradually varying angle from the finished winding angle to the next desired winding angle on the following part of the frame, while the gradual change of winding angle is performed based on the continuous change of angular speed of guide-line;
- the continuous change of winding angle on a straight-line frame with a constant cross-sectional radius;
- the same winding angle on the parts of the frame with different radii of the cross-section.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Example of frame structures in the form of (

**left**) pipe structure for oil and gas application, as well as (

**right**) road and street light poles.

**Figure 2.**The process of winding of fibers on a frame; fiber-processing head only with one guide-line creates one layer of fibers on a frame (on the (

**left**)), detail of fiber-processing head with three guide-lines that creates gradually three layers of glass fibers on the frame (on the (

**right**)).

**Figure 3.**(

**a**) Example of a longitudinal cross-section of a straight frame with three parts with different radii, jump transition between individual parts of the frame ((

**a**) upper), continuous transition between the individual parts of the frame ((

**a**) lower), and (

**b**) test non-bearing polyurethane frame, continuous transition between the individual parts of the frame were created on a 3D printer. The metal part adjacent to the frame (in the right part of the picture) is used to attach the frame to the robot-end-effector (REE).

**Figure 4.**Diagram of the passage frame through the fiber-processing head with three guide-lines. For the first guide-line k1, the plane of fibers winding onto the frame is marked ${\rho}_{1}$, which is perpendicular to the axis s of the winding head (and in this case also to the axis o of the frame).

**Figure 5.**Schematic representation of the winding of a fiber by means of a rotation guide-line k of the fiber-processing head on a frame with a circular cross-section.

**Figure 6.**Schematic representation of the passage of the frame through the rotating guide-line k1, the plane of the winding of the fibers ρ

_{1}and the distance h1 of the plane from the guide-line (on the left), h1 = ‖S1M‖. Characteristic triangle of the helix on the right (r

_{1}—radius of the helix, v

_{0}—reduced pitch, v—the pitch of the helix, α – angle slope that is referred to as winding angle, ǁǁp(t)ǁǁ—length of helix p(t) for t$\in \langle 0,2\pi \rangle )$ (on the right).

**Figure 7.**The resulting winding of one fiber on the composite frame of this section using only one guide-line, one carbon fiber (for clarity), and an industrial robot (on the (

**left**)), see the frame in Figure 3b. Passage of the composite frame through the fiber-processing head using one guide-line and only two carbon fibers and the industrial robot (on the (

**right**)).

**Figure 8.**Detail of guide-line of the fiber-processing head containing spools with fibers before the start of winding (on the left). The resulting winding of two carbon fibers on the composite frame using two guide-lines (first guide-line winds one fiber under angle $\frac{\pi}{4}$, second winds one fiber under angle $-\frac{\pi}{4}$ on three parts of the frame with different radii of cross-section (on the right).

**Figure 9.**The attachment of the frame to REE and the fiber-processing head before starting the winding process (on the left). Fiber winding density for constant winding angle α = $\frac{\pi}{6}$ and frame parts with different cross-sectional radii r

_{1}= 30 [mm], r

_{2}= 20 [mm] (on the right).

**Table 1.**Calculation of distance winding plane from guide-line depending on radius R of guide-line, frame radius r, and the required size of winding angle α.

R [mm] | r [mm] | α [º] | δ [º] | h [mm] |
---|---|---|---|---|

50 | 20 | 30 | 60 | 26.4575 |

45 | 45 | 45.8258 | ||

60 | 30 | 79.3735 | ||

30 | 30 | 60 | 23.0940 | |

45 | 45 | 40.0000 | ||

60 | 30 | 69.2820 | ||

40 | 30 | 60 | 17.3205 | |

45 | 45 | 30.0000 | ||

60 | 30 | 51.9615 | ||

80 | 20 | 30 | 60 | 44.7213 |

45 | 45 | 77.4597 | ||

60 | 30 | 134.1641 | ||

30 | 30 | 60 | 42.8174 | |

45 | 45 | 74.1620 | ||

60 | 30 | 128.4524 | ||

40 | 30 | 60 | 39.1000 | |

45 | 45 | 69.2820 | ||

60 | 30 | 119.1000 |

w [mm/s] | r [mm] | α [º] | tgα | ω [rad/s] |
---|---|---|---|---|

25 | 20 | 30 | 0.5774 | 2.1649 |

45 | 1 | 1.2500 | ||

60 | 1.7321 | 0.7217 | ||

30 | 30 | 0.5774 | 1.4433 | |

45 | 1 | 0.8333 | ||

60 | 1.7321 | 0.4811 | ||

40 | 30 | 0.5774 | 1.0824 | |

45 | 1 | 0.6250 | ||

60 | 1.7321 | 0.3608 | ||

100 | 20 | 30 | 0.5774 | 8.6600 |

45 | 1 | 5.0000 | ||

60 | 1.7321 | 2.8867 | ||

30 | 30 | 0.5774 | 5.7730 | |

45 | 1 | 3.3333 | ||

60 | 1.7321 | 1.9244 | ||

40 | 30 | 0.5774 | 4.3467 | |

45 | 1 | 2.5000 | ||

60 | 1.7321 | 1.4433 |

**Table 3.**An overview of the calculated values of the distance h of the winding plane from the guide-line k1 and the rotational angular speed ω of the guide-line for individual parts of the frame. Constant speed of passage of the frame through guide-line k1 is equal of value w = 50 [mm/s].

R [mm] | w [mm/s] | Part I, α1 = $\frac{\mathit{\pi}}{4}$ | ||
---|---|---|---|---|

50 | 50 | r_{1} [mm] | h1 [mm] | ω1 [rad/s] |

40 | 30,000 | 1250 | ||

Part II, α2 =$\frac{\pi}{6}$ | ||||

r_{2} [mm] | h2 [mm] | ω2 [rad/s] | ||

30 | 23,094 | 2887 | ||

Part III, α3 =$\frac{\pi}{3}$ | ||||

r_{3} [mm] | h3 [mm] | ω3 [rad/s] | ||

20 | 79,371 | 1443 |

**Table 4.**An overview of the locations of the starts and ends of fiber winding on the frame by guide-line k1 at the required angles.

Value of the Point Lying on Axis o of the Frame [mm] | Description Position of the Point | Angular Speed ω [rad/s] | Action When Point Goes Through Guide-Line k1 |
---|---|---|---|

A = 0 | corresponding point of begin of Part III is B = A+h3 | ω3 = 1.443 | start of fiber winding on Part III at angle of α3 = $\frac{\pi}{3}$ |

C= 500 | C = A + l3 | end of fiber winding on Part III at angle of α3 = $\frac{\pi}{3}$ | |

D = 636,287 | D = h3 + l3+ d2 − h2 | ω2=2.887 | start of fiber winding on Part II at angle of α2 = $\frac{\pi}{6}$ |

E = 1136,277 | E = h3 + l3 + d2 + l2 − h2 | end of fiber winding on Part II at angle of α2 = $\frac{\pi}{6}$ | |

F = 1209,371 | F = h3 + l3 + d2 + l2 + d1 − h1 | ω1 = 1.250 | start of fiber winding on Part I at angle of α1 = $\frac{\pi}{4}$ |

G= 1709,371 | G = h3 + l3 + d2 + l2 +d1 + l1 − h1 | end of fiber winding on Part I at angle of α1 = $\frac{\pi}{4}$ |

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**MDPI and ACS Style**

Mlýnek, J.; Rahimian Koloor, S.S.; Martinec, T.; Petrů, M. Fabrication of High-Quality Straight-Line Polymer Composite Frame with Different Radius Parts Using Fiber Winding Process. *Polymers* **2021**, *13*, 497.
https://doi.org/10.3390/polym13040497

**AMA Style**

Mlýnek J, Rahimian Koloor SS, Martinec T, Petrů M. Fabrication of High-Quality Straight-Line Polymer Composite Frame with Different Radius Parts Using Fiber Winding Process. *Polymers*. 2021; 13(4):497.
https://doi.org/10.3390/polym13040497

**Chicago/Turabian Style**

Mlýnek, Jaroslav, Seyed Saeid Rahimian Koloor, Tomáš Martinec, and Michal Petrů. 2021. "Fabrication of High-Quality Straight-Line Polymer Composite Frame with Different Radius Parts Using Fiber Winding Process" *Polymers* 13, no. 4: 497.
https://doi.org/10.3390/polym13040497