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Analysis of 2D and 3D circular braiding processes: Modeling the interaction between the process parameters and the pre-form architecture
Mechanism and Machine Theory 69 (2013) 90–104
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Analysis of 2D and 3D circular braiding processes: Modeling the interaction between the process parameters and the pre-form architecture G. Guyader, A. Gabor ⁎, P. Hamelin Civil and Environmental Engineering Laboratory (LGCIE) — Site Bohr, University Lyon 1, 82, boulevard Niels Bohr, Campus de la Doua, 69622 Villeurbanne Cedex, France
a r t i c l e
i n f o
Article history: Received 28 August 2012 Received in revised form 28 February 2013 Accepted 29 April 2013 Available online 13 June 2013 Keywords: Braiding Process control Mathematical description Braiding angle Braiding front Transitory process
a b s t r a c t This paper focuses on the modeling of a 2D and 3D circular braiding process to establish the adapted relationships between the architecture of the complex-shaped braids and the process parameters by considering the transitory and steady-state process stages. The modeling approach is based on differential geometry. We describe the trajectory of a strand of yarn as a parameterized curve on an elementary surface of the mandrel, which is described geometrically by two radii of curvature of the reference system. The developed analytical relationships allow us to take into account such transitory or steady-state process-run phenomena as the motion of the braiding front, the slippage on the mandrel and the relaxation of the yarn. The model is successfully compared to other analytical models found in the literature. Additionally, we validate the model for the case of 2D and 3D circular braiding processes run in the transitory stage. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction The development of composites in the ground and air transportation industry is becoming more important because it addresses different structural elements such as energy absorbers, tanks and other parts of the vehicle framework [1]. Creating a sound structural element requires sufficient knowledge of the mechanical properties that are essential to the optimization of the structure's geometry. In the case of textile-reinforced composites, the architecture of the reinforcement significantly influences the mechanical properties of the composite [2,3]. Therefore, it is essential to fully understand the structuring of the reinforcement during the manufacturing process, especially in the case of complex processes such as 2D or 3D interlock braiding. The use of composites obtained by 2D or 3D interlock braiding offers the ability to produce high-performance structural elements [2], and braiding technology is widely used to manufacture pre-forms. This process has the ability to produce hollow and revolution shapes in a single pass to meet the requirements of several industrial applications. Major limitations in the design of machine-made braids include restricted width, diameter, thickness and shape selection. In the case of special or batch braids, the main limitations are productivity and product length [3]. The geometry of the obtained reinforcement structure can be quite complex, leading to marked orthotropic behaviors of the composite. One of the most important geometrical parameters is the braiding angle, which describes the relative position of the yarns on the mandrel. The braiding angle depends on several process parameters, such as the translational speed of the mandrel, the mandrel shape, or the rotational velocity of the carriers. Therefore, it is important to identify the relationships between the structural properties of the reinforcement and the main parameters of the manufacturing process. The relationship between the process parameters and the braid structure has already been covered in several studies. Two types of models have been developed that describe the structuring of the braid on the mandrel: an analytical model, based on a ⁎ Corresponding author. Tel.: +33 4 72 69 21 30; fax: +33 4 78 94 69 06. E-mail address: [email protected] (A. Gabor). 0094-114X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mechmachtheory.2013.04.015
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geometric modeling of the braiding process [4–6], and a vector model, which defines the braid as an intersection between the surface of the mandrel and the vectors representing the yarns [7]. The vector approach can be adapted to complex-shaped mandrels. These works introduced three main hypotheses: (i) a rectilinear path of the yarns in the zone situated between the carrier platform and the mandrel, neglecting the friction of the yarns during the interlacing; (ii) a perfect grip of the yarn on the mandrel, considering a lack of relative movement of yarns on the mandrel; and (iii) a circular trajectory of the carriers on the braiding platform, neglecting their sinusoidal movement around an average circle. To obtain a better approach to the structure of the braid, Kessels and Akkerman [7] proposed to model the sliding of the yarns on the mandrel, Zhang et al. [5,8] addressed the friction between the yarns in the convergence zone, considering a modified yarn trajectory, and Du and Popper [6] introduced the transitory movement of the braiding front in the case where the braiding parameters are modified during the process. In recent studies [9,10], finite element or analytical modeling is combined with experimental approaches to get a realistic image of the braiding, capable of detecting braiding imperfections, yarn path and yarn interactions. In [11] mechanical properties of the braid are obtained by micromechanical modeling based on the microstructure of the yarns and a repetitive unit cell geometry. Given the above results, in this paper, we propose to describe the analytical relationships between the specific parameters of the manufacturing process and the structural characteristics of the braid in the case of complex-shaped mandrels. The modeling approach is based on differential geometry. We describe the trajectory of a strand of yarn as a parameterized curve on an elementary surface of the mandrel, which is described geometrically by two radii of curvature of the reference system. Consequently, the paper is structured as follows: – First, we describe the braiding process and examine the associated phenomena and the resulting mechanisms. – Second, we develop adapted analytical relationships based on the existing hypotheses found in the literature. – Third, to validate the developed analytical relationships, we compare the models to the results found in the literature and to our own experimental results for the case of 2D and 3D transitory braiding processes. – The paper concludes with an analysis and discussion of the theoretical results. 2. Parameters of the braiding process 2.1. The braiding process The main component of a conventional braiding machine is a circular platform on which carriers are placed to hold the spools of yarn (Fig. 1a). The number of spools/carriers depends on the desired density of the braid. The warp carriers move in a counterclockwise direction, and the weft carriers move in a clockwise direction. Usually, the mandrel is placed in the center of the braiding platform normal to its plane (Fig. 1b). The translational movement of the mandrel allows the braid to be constructed on the entire surface of the mandrel. The circular motion of the carriers around the braiding axis rolls up the yarns on the mandrel, while the sinusoidal movement of the carriers leads to the interlacing of the yarns (Fig. 1a). The yarns leaving the spools interlace progressively, forming a convergence zone between the plane where the yarns leave the carriers and the braiding front situated at the point of contact with the mandrel (Fig. 1b). In the case of 3D braiding, the carrier platform contains several rows of carriers (Fig. 2a). The carriers travel from row to row, creating an interlocking of the two-by-two layers (Fig. 2b.)
Fig. 1. (a) Overall view of a 2D braiding machine. (b) Description of the main elements of a braiding process.
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Fig. 2. (a) Outline of the carriers on a 3D braiding machine; (b) detailed view of the motion of a horn-gear during a 3D braiding operation.
2.2. Geometric parameters of the braid At a mesoscopic level, the geometric characteristics of the braid can be given by the orientation of the constitutive yarns. In the α case of a 2D braid, three parameters are involved. The first parameter, known as the braiding angle and noted by , gives the 2 → orientation of the yarns in the plane of the fabric. We define it as the angle between the yarn and the axis ez′ , which is the → projection of the braiding axis ez on the surface of the mandrel (Fig. 3). The second parameter is the crimp, denoted by e, which characterizes the weaving of the yarn over the height of the fabric. It L−l 100: is expressed as a percentage of the length of the fabric l and the initial length L of the yarn: e ¼ L The third parameter is the coverage factor, denoted as C, describing the ratio between the area given by the yarns and the total area of the surface of the mandrel. 2.3. Main phenomena occurring during the braiding process The braiding process involves different mechanisms that impact the positioning of the yarns on the mandrel [6–8,12]. First, after leaving the carriers, the yarns are subjected to friction in the convergence zone during the interlacing (Fig. 4). The friction between the yarns causes a reduction in the rotational speed of the yarns around the braiding axis and consequently influences the final parameters of the braid (e.g., the braiding angle). Next, a variation of the process parameters (e.g., rotational speed of the platform, translation speed of the mandrel) changes the position of the braiding front, introducing a transitory stage before the establishment of the steady-state run. The variation of
Fig. 3. Definition of the braiding angle.
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Braiding front variation
93
Friction
Sliding of yarn
Fig. 4. Main phenomena occurring during the braiding process.
the position of the braiding front creates a relative movement of the yarns on the mandrel that may modify the braiding angle. Finally, during the braiding process, the yarns are stretched, and the corresponding tensile force induces the sliding of yarns on the mandrel. The orientation of the yarns will change accordingly.
2.4. Process parameters First, assuming a rectilinear trajectory of the yarns in the convergence zone, the yarns will follow the motion of the carriers on the braiding platform. It is assumed that the carriers have an overall rotational motion around the braiding axis describing the circumference of the braiding platform. The overall rotational speed Ω is linked to the spinning speed ω_ of the horn gears and ω_ . The radius of this overall rotational movement is denoted by Rg. When a braiding ring their number N by the expression: Ω ¼ 2N is used, we can specify that the yarn motion is defined similarly, and the dimensions of the braiding platform are only substituted by those of the braiding ring. The mandrel motion directly affects the formation of the braid. In this paper, we consider only a translational mandrel motion along the braiding axis with a speed denoted by V. We assume a complex mandrel shape defined in → → → a cylindrical coordinate system (Fig. 5) associated with the braiding platform er ; eϑ ; ez . An elementary surface of the mandrel is described by the distance R between the elementary surface and the braiding axis and the projections of the radii of curvature
Fig. 5. Process parameters and mandrel description.
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→ → → → ρz and ρϑ onto the ez ; er and eϑ ; er planes, respectively. These radii define the centers of curvature of the elementary area Pz and Pϑ, respectively.
→
If we project the normal axis en of the elementary area in the plane
→ → → ez ; er , we obtain a longitudinal orientation axis en;z .
Therefore, the orientation of the elementary area in the longitudinal direction can be defined by the angle γz formed by the axes →
→
→
en;z and er . In a similar way, we can define the orientation angle γϑ by projecting the axis en on the radial plane formed by the axis → → eϑ; er . 3. Analytical relationships between the process parameters and the geometry of the braid We propose to directly describe the influence of the process parameters on the braiding angle and therefore to set up the process parameters according to the desired distribution of the braiding angle on the surface of the mandrel. The analytic approach is based on a geometric description of the yarn trajectory by considering an elementary area of the mandrel. On this area, we divide the yarn trajectory into two components, the longitudinal component lz(t) and the radial one lϑ(t). We obtain the analytical expression of the braiding angle as a function of the longitudinal and radial components of the yarn trajectory: α −1 lϑ ðt Þ ¼ tan : ð1Þ 2 lz ðt Þ The radial component can be associated with the roll-up of the yarn on the mandrel, while the longitudinal component corresponds to the translation of the yarns on the mandrel due to its motion and that of the braiding front. To obtain a comprehensive view of the process parameters, we establish the relationship of the radial component lϑ(t) of the yarn trajectory with that of the _ braiding front translation (speed) in the longitudinal direction h. With the assumption of non-slippage of yarns on the mandrel, the path of the yarn is given by the movement of the intersection point I(t) between the yarn trajectory and the surface of the mandrel. To determine the movement of the intersection point, we define → → → the path of the yarn as a curve parameter associated with a local Frenet reference system en ; et ; ek . Another reference system → → → eR ; eΩ ; ez is used to describe the motion of the carriers on the braiding platform (Fig. 5). Generalizing this model to the entire surface of the mandrel, we obtain an overall braiding angle re-partition on the mandrel. Fig. 6 presents the general scheme of the braiding process with these different elements. We define the points O(t) and O′(t) as the intersection of the braiding axis with the braiding front and with the braiding platform, respectively. The point I′(t) is the projection of the intersection point I(t) on the braiding platform, and the point M(t) represents a carrier. Fig. 7 presents a cross-section of the braiding operation along a yarn. Using Euler's theorem, we define the radius of curvature ρs of the parameterized curve at the point I(t) as a function of the longitudinal and radial radii of curvature of the mandrel: 2 α 2 α cos sin 1 ρρ 2 2 ¼ α z ϑ : ¼ þ 2 α ρs ρz ρϑ þ ρϑ cos ρz sin2 2 2
Fig. 6. Scheme of the braiding process and its parameters.
ð2Þ
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Fig. 7. Scheme of cross-section along the tangential and normal axes of the Frenet reference system.
The radius of curvature of the parameterized curve defines the center of curvature P around which the yarn is rolled up. The distance l(t) represents the length of the yarn between the intersection point I(t) and the carrier M(t). We can now express the → position of the carrier on the braiding platform O′ M ðt Þ via the intersection point I(t). → → → O′ M ðt Þ ¼ O′ O ðt Þ þ OM ðt Þ
ð3Þ
→ → → → with: OM ðt Þ ¼ OP þ PI ðt Þ þ IM ðt Þ → → → OM ðt Þ ¼ OP −ρs en þ lðt Þ→ et : As the yarns roll up on the mandrel, we can assume that the center of curvature P is an instantaneous center of rotation of the → _ yarn. We consequently assume a rotation of the Frenet reference system around its axis ek with a rotational speed denoted as φ. → We consider the variation of thebraiding angle as a rotation of the Frenet reference system around its normal axis en with a rotational speed denoted by α_ . With these assumptions, and expression (3), we obtain a new relationship connecting the 2
process parameters and the roll-up speed of yarns on the surface of the mandrel. α_ → ΩRg → eΩ ¼ V → ez þ l_ðt Þ−ρs φ_ → et þ lðt Þφ_ → en þ lðt Þ e 2 k
ð4Þ
The above relationship involves terms expressed in the Frenet reference system and also terms expressed in the reference system of the carrier. To overcome this inconvenience, we project the cross-section of the braiding operation along a strand of yarn on a plane coinciding with the braiding platform, as shown in Fig. 8.
M (t)
e
en
e
l (t )
et
er
I (t )
D
s
Rg
p R
o Fig. 8. Projection of the cross-section along the tangential and normal axes of the Frenet reference system on the braiding platform.
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G. Guyader et al. / Mechanism and Machine Theory 69 (2013) 90–104 →
In this figure, we define the point P′, the distance l′(t) and the axis et′ as the projections on the braiding platform of the point P, → → the distance l(t) and the tangential axis et , respectively, following the axis ez′ . → → The angle δ is the angle formed between the et′ and eR axes. The expression of the angle δ and the distance l′(t) are easily obtained from the triangle O′MD and can be written as:
cosðδÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2g −ðR þ h tan γz Þ2 cos2 γ ϑ
ð5Þ
Rg
l′ ðt Þ ¼ Rg cosðδÞ−ðR þ h tanγ z Þ sinðγϑ Þ: →
ð6Þ
→
We can now write the axes ez and eΩ of the carrier reference system in the Frenet reference system as follows: → → → ez ¼ cosγz ez′ þ sinγ z en
ð7Þ
α α → → → et þ sin e with: ez′ ¼ cos 2 2 k → → → eΩ ¼ cosδ en þ sinδ et′
ð8Þ
α α → → → ek þ sin e: with: et′ ¼ cos 2 2 t → → By substituting the relationships of the axes ez and eΩ into the general expression (4) and considering their projections on the → axes of the Frenet reference system, we obtain three independent relationships. The first, projected on the normal axis en , gives the expression of the roll-up speed φ_ of the yarn on the surface of the mandrel: φ_ ¼
ΩRg cosδ−V sinγ z lðt Þ
ð9Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 R2g −ðR þ h tan γ z Þ cos2 γϑ
Rg cosðδÞ−ðR þ h tan γz Þ sinðγ ϑ Þ cosðδÞ ¼ : α Rg sin 2 The roll-up speed is the ratio between the rate of the deposited length of yarn and the radius of the mandrel. → The second relationship, projected on the tangential axis et , connects the variation of the length of the yarn l_ ðt Þ between the carrier and the intersection point to the process parameters and the braiding angle: with: lðt Þ ¼
α α _ l_ ðt Þ ¼ ΩRg sinδ sin −V cosγ z cos þ ρs φ: 2 2
ð10Þ →
The third expression, projected on the transversal axis ek , gives the braiding angle variation as a function of the process parameters and the actual braiding angle: ΩR sinδ cos α −V cosγ sin α g z α_ 2 2 : ¼ 2 lðt Þ
ð11Þ
These three equations are able to describe several phenomena, such as the slipping of the yarn on the mandrel or the relaxation of the yarn during the braiding operation. We will focus on the first equation leading to the radial component of the yarn trajectory and to the braiding front translation speed h_ in the longitudinal direction. Assuming the center of curvature P as an instantaneous center of rotation, the motion of the intersection point I(t) can be considered as a rotation around the center of curvature of the tangential axis with a rotational speed φ_ ðt Þ. The tangential component of the deposit of yarn on the mandrel lt(t) is consequently expressed as: lt ðt Þ ¼ ρs φ_ ðt Þ:
ð12Þ
This expression can be divided into radial and longitudinal components that can be easily expressed by projection on the radial and longitudinal axes, respectively. α lϑ ðt Þ ¼ lt ðt Þ sin 2
ð13Þ
α le ðt Þ ¼ lt ðt Þ cos 2
ð14Þ
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Therefore, we obtain the radial component of the yarn trajectory lϑ(t) and the braiding front translation speed h_ in the longitudinal direction. lϑ ðt Þ ¼ ρs Ω
α Rg cos δ−V sinγ z sin 2 lðt Þ
α Rg cosδ−V sinγ z h_ ¼ V−ρs Ω cos 2 lðt Þ
ð15Þ
ð16Þ
Assuming that the radial component of the yarn trajectory on the mandrel corresponds to the roll-up speed of the yarn and the longitudinal component corresponds to the translation of the yarn due to the motion of the mandrel and the braiding front, we can construct a general relationship of the braiding angle adapted to complex mandrel shapes and transitory process run. _ ðt Þ α −1 ρs φ ¼ tan V þ dh 2
ð17Þ
4. Comparison to existing models As a first validation, we compare the results obtained with the model developed above with the results found in the literature. The most common braiding operations use a cylindrical mandrel in a stationary process run. Ko [4] studied this case, giving the analytical relationship of the braiding angle. We also cite the works of Zhang [5,8] for this type of braiding. The study developed by Du and Popper [6] considers an axisymmetric mandrel with longitudinal variation and introduces a quantification of the transitory stage, while the work of W. Michaeli [12] studied the influence of the eccentricity of a cylindrical mandrel on the structure of a braid. We can consider the eccentricity as a geometrical offset involving radial variation of the geometry of the mandrel. Therefore, we first focus on cylindrical mandrels before separately considering mandrels with longitudinal geometry variations and mandrels with radial geometry variations. 4.1. Application to cylindrical mandrel In the case of a cylindrical mandrel with a radius R, on every point of the surface, we have: γϑ = 0, γz = 0, cosðδÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2g −R2 cos2 γ ϑ R . Therefore, the relationship between the radial component of the yarn trajectory on the mandrel , ρs ¼ 2 α Rg sin 2 and the speed of the braiding front translation in the longitudinal direction becomes: lϑ ðt Þ ¼ RΩ
ð18Þ
RΩhðt Þ h_ ¼ V− qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2g −R2
ð19Þ
leading to the braiding angle expression previously given by Zhang [5]. 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 R2g −R2 α −1 @ A ¼ tan 2 hðt Þ
ð20Þ
We observe that the expression of the motion of the braiding front is similar to the expression previously given by Du and Popper [6] when applied to a cylindrical mandrel. h_ ðt Þ ¼ V−
RΩhðt Þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u u1 R hðt Þ Rg t − þ tanγz Rg Rg Rg
ð21Þ
In the case of a steady-state run with the assumption of a tangential deposit of yarns on the mandrel, Zhang et al. [8] propose the following expression of the position of the braiding front h(t).
hðt Þ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V R2g −R2 ΩR
ð22Þ
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Substituting the braiding front position expression into the previous expression of the braiding front motion, we have: h_ ¼ 0:
ð23Þ
Finally, we obtain the expression of the braiding angle in the case of a cylindrical mandrel and a steady-state run, which is similar to the expression previously given by Ko [4] for the same braiding conditions. α −1 RΩ ¼ tan 2 V
ð24Þ
4.2. Application to mandrels with longitudinal geometry variations The longitudinal geometry variation of a mandrel can be treated as linear for an elementary length. The entire surface of the mandrel is thus described as a succession of small cone trunks. With this condition, we obtain for every point of the surface: γϑ = 0, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 R2g −ðR þ h tanγ z Þ R and cosðδÞ ¼ , ρs ¼ . Rg sin2 α2 The expressions of the radial component of the yarn trajectory l and of the translation speed of the braiding front h_ become: ϑ
0
1
V sinγ z B C lϑ ðt Þ ¼ R@Ω− qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A R2g −ðR þ h tanγ z Þ2
ð25Þ
h_ ðt Þ ¼ V−T1−T2
ð26Þ
with: RΩhðt Þ T1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Rg −ðR þ h tanγz Þ2
T2 ¼
Vhðt Þ sinγz : R2g −ðR þ hðt Þ tan γz Þ2
We note that for a small variation of the mandrel geometry in the longitudinal direction or for a large radius Rg of the braiding platform, the second and third terms of Eqs. (25) and (26), respectively, can be neglected. With these assumptions, we obtain a similar relationship of the radial component lϑ and of the braiding front translation speed h_ to the expressions given by Du and Popper [6] for the same braiding conditions (Eqs. (27) and (28), respectively). lϑ ðt Þ ¼ RΩ
ð27Þ
RΩhðt Þ dh ¼ V− qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2g −ðR þ hðt Þ tanγ z Þ2
ð28Þ
The relationship of the braiding angle becomes: 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 R2g −ðR þ hðt Þ tanγz Þ2 α −1 A: ¼ tan @ 2 hðt Þ
ð29Þ
To obtain an overview of the effect of neglecting the terms in Eqs. (25) and (26), we take as an example an existing braiding case with 64 carriers, a radius Rg of the braiding platform of 750 mm, a translation speed of the mandrel of 1 mm/s, and a rotational speed of the braiding platform of 0.27 rad/s. Fig. 9 presents the effect of the ratio T1/T2 on the motion speed of the braiding as a function of the longitudinal orientation angle γz. We note that the effect of the third term, T2, on the motion speed of the braiding front is quite small, and thus the present model approaches the motion of the braiding front in the same manner as the model developed by Du and Popper [6].
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Fig. 9. Effect of the T1/T2 ratio on the motion speed of the braiding front.
4.3. Application to mandrels with radial geometry variations In the case of mandrels with radial geometry variations, we have at every point of the surface: γz = 0, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2g −R2 cos2 γ ϑ R α . The relationships of lϑ and h_ become: , ρs ¼ Rg sin2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2g −R2 cos2 γ ϑ lϑ ðt Þ ¼ RΩ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2g −R2 cos2 γϑ −R sinγ ϑ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2g −R2 cos2 γ ϑ _h ðt Þ ¼ V−RΩhðt Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 R2g −R2 cos2 γ ϑ −R sinγϑ
cosðδÞ ¼
ð30Þ
ð31Þ
leading to the following braiding angle expression: 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 R2g −R2 cos2 γϑ −R sin γϑ α −1 @ A: ¼ tan 2 hðt Þ
ð32Þ
To estimate the validity of these expressions, we apply Eqs. (31), (32), and (33) to the results obtained by Michaeli et al. [12], who analyzed the influence of the mandrel eccentricity on the distribution of the braiding angle. The eccentricity can be implemented in the present model as a variation of the angle γϑ. Fig. 10 shows the deviation from the expected braiding angle due to the eccentricity of the mandrel. The diameter of the mandrel is 100 mm, and the effective radius of the braiding platform is 350 mm. As can be observed in the present model, the developed model approaches the experimental results of Michaeli et al. with good accuracy for a wide range of braiding angles.
Fig. 10. Deviation from the expected braiding angle due to the eccentricity of the mandrel: comparison of the analytical model and the results of Michaeli et al. [12].
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Fig. 11. Spacing of meshes during a transitory process run.
5. From a 2D to a 3D interlock braiding process The previously cited results were developed for 2D circular braiding operations assuming a circular trajectory of the carriers on the braiding platform and neglecting their sinusoidal movement around an average circle. Considering the orientation of the yarn and the thickness of the fabric, we can construct an analogy between a 3D circular braid and a stack of 2D circular braids (neglecting the interlocking between the superposed layers). In reality, in the case of a 3D circular braiding machine, the transverse motion of the carriers around an average circle may be important, and the assumption of a circular trajectory of the carriers on the braiding platform is not immediate. We note that the average rotational speed of the carriers is similar to that of the 2D braiding, and the motion of the carriers from one level to another is quick enough to be averaged by the overall circular motion of the braiding front. Consequently, we assume that the average longitudinal component of the yarn trajectory will not be affected. 5.1. Preliminary assumptions To establish a transitory process run during the braiding operations, the translational speed of the mandrel was varied and the rotational speed of the braiding platform was held constant. Thus, the position of the braiding front can be varied from an initial value to a set-point value. Both values can be calculated using formula (22), corresponding to the position of the braiding front in a steady-state process run. Furthermore, it can be stated that in the case of a constant rotational speed of the platform, a mesh develops in the braid during a constant time interval, independent from the other process parameters. For example, in the case of a braid assimilated into a taffeta, a complete revolution is needed to create a mesh. Therefore, for a given rotational speed Ω of the platform, a mesh is constructed during a time Δt: Δt ¼
4π : Ω
ð33Þ
Consequently, by comparing the distance between a defined number of meshes (Fig. 11) and the distance traveled by the mandrel, it is possible to establish the motion speed of the braiding front: d h_ ¼ V− n nΔt
ð34Þ
with h_ as the motion speed of the braiding front, V as the speed of the mandrel and n as the number of successive meshes. Before the start of the measurement, the braiding front is positioned manually on the mandrel at its theoretical value using a laser level; the braiding is started at a constant velocity to obtain stabilization of the braiding front. Next, the motion speed of the mandrel is shifted directly to the set-up value, resulting in the variation of the position of the braiding front (Fig. 12). As it can be seen, the contact point between yarns and the mandrel (the braiding front) gets closer to the crown of the braiding machine and consequently the braiding angle changes. The same type of fiber was used for all braiding operations: E-glass Hybon 2001 Roving, PPG 2400 tex. Filaments have a diameter of 25 μm. The choice of this type of fiber is related to production requirements. Several type of glass fibers were tested before to limit friction and knot in the convergence zone to avoid interruption of the braiding process. 5.2. 2D circular braiding To assess the variation of the braiding front during a transitory process, a 2D braiding operation was set up characterized by 64 carriers, an effective radius of the braiding platform of 750 mm, a rotational speed of the platform of 7.36 rad/s, and a PVC
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Fig. 12. Motion of the braiding front during the transitory process.
mandrel with a length of 1000 mm and a diameter of 100 mm. The mandrel is moved by a six-DOF pulling unit driven by a PC unit. The mandrel is translated normal to the braiding front only (Fig. 13). The first braiding was carried out by accelerating the translation speed of the mandrel from an initial value of 5 mm/s to 15 mm/s. Fig. 14 presents a comparison of the experimental and theoretical results, confirming the ability of the analytical model to assess the motion of the braiding front. However, at the end of the braiding operation, the steady state is not yet attained, and the two curves begin to diverge. This difference can be attributed to the post-braiding operation (cutting of yarns, local tightening or loosening of yarns, etc.), given the relatively low density of the braid.
Fig. 13. 2D braiding operation set up to validate the advancement of the braiding front in a transitory process in the case of a cylindrical mandrel.
Fig. 14. Comparison of the translation of the braiding front (left) and of the variation of the braiding angle (right) in the case of 2D circular braiding when the mandrel speed is increased.
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Fig. 15. Comparison of the translation of the braiding front (left) and of the variation of the braiding angle (right) in the case of 2D circular braiding when the mandrel speed is decreased.
A second braiding was produced by slowing the translation speed from 10 mm/s to 5 mm/s. The results are presented in Fig. 15, which show a good correlation between the experimental and theoretical results.
5.3. 3D circular braiding To validate the analytical relationship of the motion of the braiding front during a transitory state process (Fig. 16), we used a 3D braiding platform consisting of five carrier levels that define an effective internal and external diameter of 2000 mm and 3000 mm, respectively. Each carrier level is composed of 64 carriers, the rotational speed of the braiding platform is 7.36 rad/s, and the mandrel consists of a PVC pipe with a length of 2000 mm and a diameter of 50 mm. An automated pulling unit allows control of the variation of the speed and the movement of the mandrel normal to the plane of the braiding platform (Fig. 16). Four types of braids were produced, permitting analysis of the transitory state for braiding angles ranging from 35° to 65°. The first braiding operation was carried out by varying the speed of the mandrel from 15.95 mm/s to 7.5 mm/s. A second braiding operation was set up by increasing the speed of the mandrel from 7.5 mm/s to 15.95 mm/s. Fig. 17 shows the comparison of the experimental and the theoretical results for the chosen braiding angles. We note that the experimental and theoretical curves fit with good accuracy, and the gradient of the motion of the braiding front is the same when approaching or moving away from the braiding platform. Furthermore, we can remark that the curves diverge slightly at the start of the braiding, which could be a consequence of the compaction of yarns due to the sudden change in the process parameters. Two other situations were considered in which the mandrel motion's speed was changed from 13.12 mm/s to 18.75 mm/s and from 8.44 mm/s to 6.19 mm/s. In Fig. 18 the experimental and theoretical curves show a good accuracy of the model in describing the progress of the braiding front, validating again the pertinence of the proposed model.
Fig. 16. Setup for a 3D circular braiding process used to validate the motion of the braiding front during a transitory process run in the case of a 100-mm circular mandrel.
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Fig. 17. Comparison of the translation of the braiding front in the case of 3D circular braiding when the mandrel speed is changed. (a) Decrease of the mandrel speed; (b) increase of the mandrel speed.
6. Conclusions and outlook In this paper, we presented an analytical model of the yarn trajectory during a circular braiding operation. This mathematical model uses a parameterized curve approach and produces selected relationships that are characteristic of the interactions between the process parameters and the braiding architecture. The positioning of the braiding front during a transitory process run, the slippage of the yarn on the mandrel and the relaxation of the yarn can be described even in the case of complex-shaped mandrels. The results were compared to other analytical models found in the literature. In the case of a steady-state process and symmetric revolution shapes, the developed model is perfectly adapted to link the process parameters (e.g., braiding platform rotational speed or mandrel speed) to the characteristics of the braid (e.g., braiding angle). The results also approach the distribution of the braiding angle in the case of circular eccentric mandrels with good accuracy. To validate the ability of the model to describe the motion of the braiding front in the case of a transitory process run, we performed a prototype braiding operation in which the mandrel translational speed was suddenly increased or decreased. The model described the advancement of the braiding front for multiple speed ranges with good accuracy. Nevertheless, certain other aspects of this work must be validated by taking into account more complex mandrel shapes or the yarn relaxation behavior due to the friction in the convergence zone. The developed model considers a linear trajectory of yarns in the convergence zone. But in the case of an important friction between yarns, this hypothesis is no longer valid and it would be suitable to model the path of the yarns with a parabolic function or equivalent. The final objective is to establish the process parameters (e g., the rotational speed of the platform and the motion of the mandrel in the braiding space) to obtain the requested braiding angle distribution on a complex-shaped mandrel. Acknowledgments We gracefully acknowledge the enterprise DJP-Composites for participating in the experimental validation of the theoretical parameters by allowing us to use their 3D interlock braiding equipment. This work was supported by the 7th FUI Project “Sagane” of the Ministry of the Economy, Finance and Industry of France, to whom the authors are very grateful.
Fig. 18. Comparison of the translation of the braiding front in the case of 3D circular braiding when the mandrel speed is changed. (a) Increase of the mandrel speed from 13.12 mm/s to 18.75 mm/s. (b) Decrease of the mandrel speed from 8.44 mm/s to 6.19 mm/s.
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References [1] T.W. Chou, F.K. Ko, Textile Structural Composites, vol. 3Elsevier, 1989. [2] L. Wenning, C. Guangming, Q. Xin, Proc. of the 38th Int. SAMPE Symposium, 1993, pp. 10–13. [3] A.B. Macander, R.M. Crane, E.T. Camponeschi, Fabrication and mechanical properties of multidimensionally (X-D) braided composite materials, Composite Materials (1986) 422–443. [4] F.K. Ko, Braiding engineered materials handbook, Composites, ASM International, vol. 1, 1987. 519–528. [5] Q. Zhang, D. Beale, R.M. Broughton, Analysis of circular braiding process. Part 1: theoretical investigation of kinematics of the circular braiding process, Journal of Manufacturing Science and Engineering vol. 121 (1999) 345–350, no3. [6] G.W. Du, P. Popper, Analysis of a circular braiding process for complex shapes, Journal of the Textile Institute (1994) 316–337. [7] J.F.A. Kessels, R. Akkerman, Prediction of the yarn trajectories on complex braided preforms, Composites Part A: Applied Science and Manufacturing 33 (8) (2002) 1073–1081. [8] Q. Zhang, D. Beale, R.M. Broughton, Analysis of circular braiding process. Part 2: mechanics analysis of the circular braiding process and, Journal of Manufacturing Science and Engineering vol. 121 (1999) 351–359, (no3). [9] A. Pickett, A. Erber, T. von Reden, K. Drechsler, Comparison of analytical and finite element simulation of 2D braiding, Plastics, Rubber and Composites 38 (9–10) (2009) 387–395. [10] A. Rawal, P. Potluri, C. Steele, Prediction of yarn paths in braided structures formed on a square pyramid, Journal of Industrial Textiles 35 (2) (2005) 115–135. [11] P. Potluri, A. Manan, Mechanics of non-orthogonally interlaced textile composites, Composites Part A 34 (2003) 481–492. [12] W. Michaeli, U. Rosenbaum, M. Jehrke, Processing strategy for braiding of complex-shaped parts based on a mathematical process description, Composites Manufacturing vol. 1 (4) (December 1990) 243–251.
Glossary α: braiding angle e: shrinkage C: coverage factor Ω: overall rotational speed ω_ : horn-gear rotational speed N: number of horn gears V: mandrel translation speed ρz: radius of curvature along the longitudinal axis ρϑ: radius of curvature along the radial axis Pz: center of curvature along the longitudinal axis Pϑ: center of curvature along the radial axis γz: angle of orientation along the longitudinal axis γϑ: angle of orientation along the radial axis lϑ: radial component of the yarn trajectory lz: longitudinal component of the yarn trajectory _ motion speed of the braiding front h: I: intersection point M: carrier point on the braiding platform O: geometrical center of the mandrel in the plane of the braiding front O′: center of the braiding platform ρs: radius of curvature of the trajectory of the yarn on the mandrel l: length between the intersection point of the yarn on the mandrel and the carrier P: center of curvature of the trajectory of a yarn on the mandrel _ yarn roll-up speed φ: α_ : variation of the braiding angle → → δ: angle formed between the axes e′t and eR R: radius of the mandrel Rg: effective radius of the braiding platform h: braiding front position lt: tangential component of the yarn deposit on the mandrel Δt: time necessary for the formation of a mesh dn: distance between n meshes n: number of considered meshes
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Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt
Analysis of 2D and 3D circular braiding processes: Modeling the interaction between the process parameters and the pre-form architecture G. Guyader, A. Gabor ⁎, P. Hamelin Civil and Environmental Engineering Laboratory (LGCIE) — Site Bohr, University Lyon 1, 82, boulevard Niels Bohr, Campus de la Doua, 69622 Villeurbanne Cedex, France
a r t i c l e
i n f o
Article history: Received 28 August 2012 Received in revised form 28 February 2013 Accepted 29 April 2013 Available online 13 June 2013 Keywords: Braiding Process control Mathematical description Braiding angle Braiding front Transitory process
a b s t r a c t This paper focuses on the modeling of a 2D and 3D circular braiding process to establish the adapted relationships between the architecture of the complex-shaped braids and the process parameters by considering the transitory and steady-state process stages. The modeling approach is based on differential geometry. We describe the trajectory of a strand of yarn as a parameterized curve on an elementary surface of the mandrel, which is described geometrically by two radii of curvature of the reference system. The developed analytical relationships allow us to take into account such transitory or steady-state process-run phenomena as the motion of the braiding front, the slippage on the mandrel and the relaxation of the yarn. The model is successfully compared to other analytical models found in the literature. Additionally, we validate the model for the case of 2D and 3D circular braiding processes run in the transitory stage. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction The development of composites in the ground and air transportation industry is becoming more important because it addresses different structural elements such as energy absorbers, tanks and other parts of the vehicle framework [1]. Creating a sound structural element requires sufficient knowledge of the mechanical properties that are essential to the optimization of the structure's geometry. In the case of textile-reinforced composites, the architecture of the reinforcement significantly influences the mechanical properties of the composite [2,3]. Therefore, it is essential to fully understand the structuring of the reinforcement during the manufacturing process, especially in the case of complex processes such as 2D or 3D interlock braiding. The use of composites obtained by 2D or 3D interlock braiding offers the ability to produce high-performance structural elements [2], and braiding technology is widely used to manufacture pre-forms. This process has the ability to produce hollow and revolution shapes in a single pass to meet the requirements of several industrial applications. Major limitations in the design of machine-made braids include restricted width, diameter, thickness and shape selection. In the case of special or batch braids, the main limitations are productivity and product length [3]. The geometry of the obtained reinforcement structure can be quite complex, leading to marked orthotropic behaviors of the composite. One of the most important geometrical parameters is the braiding angle, which describes the relative position of the yarns on the mandrel. The braiding angle depends on several process parameters, such as the translational speed of the mandrel, the mandrel shape, or the rotational velocity of the carriers. Therefore, it is important to identify the relationships between the structural properties of the reinforcement and the main parameters of the manufacturing process. The relationship between the process parameters and the braid structure has already been covered in several studies. Two types of models have been developed that describe the structuring of the braid on the mandrel: an analytical model, based on a ⁎ Corresponding author. Tel.: +33 4 72 69 21 30; fax: +33 4 78 94 69 06. E-mail address: [email protected] (A. Gabor). 0094-114X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mechmachtheory.2013.04.015
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geometric modeling of the braiding process [4–6], and a vector model, which defines the braid as an intersection between the surface of the mandrel and the vectors representing the yarns [7]. The vector approach can be adapted to complex-shaped mandrels. These works introduced three main hypotheses: (i) a rectilinear path of the yarns in the zone situated between the carrier platform and the mandrel, neglecting the friction of the yarns during the interlacing; (ii) a perfect grip of the yarn on the mandrel, considering a lack of relative movement of yarns on the mandrel; and (iii) a circular trajectory of the carriers on the braiding platform, neglecting their sinusoidal movement around an average circle. To obtain a better approach to the structure of the braid, Kessels and Akkerman [7] proposed to model the sliding of the yarns on the mandrel, Zhang et al. [5,8] addressed the friction between the yarns in the convergence zone, considering a modified yarn trajectory, and Du and Popper [6] introduced the transitory movement of the braiding front in the case where the braiding parameters are modified during the process. In recent studies [9,10], finite element or analytical modeling is combined with experimental approaches to get a realistic image of the braiding, capable of detecting braiding imperfections, yarn path and yarn interactions. In [11] mechanical properties of the braid are obtained by micromechanical modeling based on the microstructure of the yarns and a repetitive unit cell geometry. Given the above results, in this paper, we propose to describe the analytical relationships between the specific parameters of the manufacturing process and the structural characteristics of the braid in the case of complex-shaped mandrels. The modeling approach is based on differential geometry. We describe the trajectory of a strand of yarn as a parameterized curve on an elementary surface of the mandrel, which is described geometrically by two radii of curvature of the reference system. Consequently, the paper is structured as follows: – First, we describe the braiding process and examine the associated phenomena and the resulting mechanisms. – Second, we develop adapted analytical relationships based on the existing hypotheses found in the literature. – Third, to validate the developed analytical relationships, we compare the models to the results found in the literature and to our own experimental results for the case of 2D and 3D transitory braiding processes. – The paper concludes with an analysis and discussion of the theoretical results. 2. Parameters of the braiding process 2.1. The braiding process The main component of a conventional braiding machine is a circular platform on which carriers are placed to hold the spools of yarn (Fig. 1a). The number of spools/carriers depends on the desired density of the braid. The warp carriers move in a counterclockwise direction, and the weft carriers move in a clockwise direction. Usually, the mandrel is placed in the center of the braiding platform normal to its plane (Fig. 1b). The translational movement of the mandrel allows the braid to be constructed on the entire surface of the mandrel. The circular motion of the carriers around the braiding axis rolls up the yarns on the mandrel, while the sinusoidal movement of the carriers leads to the interlacing of the yarns (Fig. 1a). The yarns leaving the spools interlace progressively, forming a convergence zone between the plane where the yarns leave the carriers and the braiding front situated at the point of contact with the mandrel (Fig. 1b). In the case of 3D braiding, the carrier platform contains several rows of carriers (Fig. 2a). The carriers travel from row to row, creating an interlocking of the two-by-two layers (Fig. 2b.)
Fig. 1. (a) Overall view of a 2D braiding machine. (b) Description of the main elements of a braiding process.
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Fig. 2. (a) Outline of the carriers on a 3D braiding machine; (b) detailed view of the motion of a horn-gear during a 3D braiding operation.
2.2. Geometric parameters of the braid At a mesoscopic level, the geometric characteristics of the braid can be given by the orientation of the constitutive yarns. In the α case of a 2D braid, three parameters are involved. The first parameter, known as the braiding angle and noted by , gives the 2 → orientation of the yarns in the plane of the fabric. We define it as the angle between the yarn and the axis ez′ , which is the → projection of the braiding axis ez on the surface of the mandrel (Fig. 3). The second parameter is the crimp, denoted by e, which characterizes the weaving of the yarn over the height of the fabric. It L−l 100: is expressed as a percentage of the length of the fabric l and the initial length L of the yarn: e ¼ L The third parameter is the coverage factor, denoted as C, describing the ratio between the area given by the yarns and the total area of the surface of the mandrel. 2.3. Main phenomena occurring during the braiding process The braiding process involves different mechanisms that impact the positioning of the yarns on the mandrel [6–8,12]. First, after leaving the carriers, the yarns are subjected to friction in the convergence zone during the interlacing (Fig. 4). The friction between the yarns causes a reduction in the rotational speed of the yarns around the braiding axis and consequently influences the final parameters of the braid (e.g., the braiding angle). Next, a variation of the process parameters (e.g., rotational speed of the platform, translation speed of the mandrel) changes the position of the braiding front, introducing a transitory stage before the establishment of the steady-state run. The variation of
Fig. 3. Definition of the braiding angle.
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Braiding front variation
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Friction
Sliding of yarn
Fig. 4. Main phenomena occurring during the braiding process.
the position of the braiding front creates a relative movement of the yarns on the mandrel that may modify the braiding angle. Finally, during the braiding process, the yarns are stretched, and the corresponding tensile force induces the sliding of yarns on the mandrel. The orientation of the yarns will change accordingly.
2.4. Process parameters First, assuming a rectilinear trajectory of the yarns in the convergence zone, the yarns will follow the motion of the carriers on the braiding platform. It is assumed that the carriers have an overall rotational motion around the braiding axis describing the circumference of the braiding platform. The overall rotational speed Ω is linked to the spinning speed ω_ of the horn gears and ω_ . The radius of this overall rotational movement is denoted by Rg. When a braiding ring their number N by the expression: Ω ¼ 2N is used, we can specify that the yarn motion is defined similarly, and the dimensions of the braiding platform are only substituted by those of the braiding ring. The mandrel motion directly affects the formation of the braid. In this paper, we consider only a translational mandrel motion along the braiding axis with a speed denoted by V. We assume a complex mandrel shape defined in → → → a cylindrical coordinate system (Fig. 5) associated with the braiding platform er ; eϑ ; ez . An elementary surface of the mandrel is described by the distance R between the elementary surface and the braiding axis and the projections of the radii of curvature
Fig. 5. Process parameters and mandrel description.
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→ → → → ρz and ρϑ onto the ez ; er and eϑ ; er planes, respectively. These radii define the centers of curvature of the elementary area Pz and Pϑ, respectively.
→
If we project the normal axis en of the elementary area in the plane
→ → → ez ; er , we obtain a longitudinal orientation axis en;z .
Therefore, the orientation of the elementary area in the longitudinal direction can be defined by the angle γz formed by the axes →
→
→
en;z and er . In a similar way, we can define the orientation angle γϑ by projecting the axis en on the radial plane formed by the axis → → eϑ; er . 3. Analytical relationships between the process parameters and the geometry of the braid We propose to directly describe the influence of the process parameters on the braiding angle and therefore to set up the process parameters according to the desired distribution of the braiding angle on the surface of the mandrel. The analytic approach is based on a geometric description of the yarn trajectory by considering an elementary area of the mandrel. On this area, we divide the yarn trajectory into two components, the longitudinal component lz(t) and the radial one lϑ(t). We obtain the analytical expression of the braiding angle as a function of the longitudinal and radial components of the yarn trajectory: α −1 lϑ ðt Þ ¼ tan : ð1Þ 2 lz ðt Þ The radial component can be associated with the roll-up of the yarn on the mandrel, while the longitudinal component corresponds to the translation of the yarns on the mandrel due to its motion and that of the braiding front. To obtain a comprehensive view of the process parameters, we establish the relationship of the radial component lϑ(t) of the yarn trajectory with that of the _ braiding front translation (speed) in the longitudinal direction h. With the assumption of non-slippage of yarns on the mandrel, the path of the yarn is given by the movement of the intersection point I(t) between the yarn trajectory and the surface of the mandrel. To determine the movement of the intersection point, we define → → → the path of the yarn as a curve parameter associated with a local Frenet reference system en ; et ; ek . Another reference system → → → eR ; eΩ ; ez is used to describe the motion of the carriers on the braiding platform (Fig. 5). Generalizing this model to the entire surface of the mandrel, we obtain an overall braiding angle re-partition on the mandrel. Fig. 6 presents the general scheme of the braiding process with these different elements. We define the points O(t) and O′(t) as the intersection of the braiding axis with the braiding front and with the braiding platform, respectively. The point I′(t) is the projection of the intersection point I(t) on the braiding platform, and the point M(t) represents a carrier. Fig. 7 presents a cross-section of the braiding operation along a yarn. Using Euler's theorem, we define the radius of curvature ρs of the parameterized curve at the point I(t) as a function of the longitudinal and radial radii of curvature of the mandrel: 2 α 2 α cos sin 1 ρρ 2 2 ¼ α z ϑ : ¼ þ 2 α ρs ρz ρϑ þ ρϑ cos ρz sin2 2 2
Fig. 6. Scheme of the braiding process and its parameters.
ð2Þ
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Fig. 7. Scheme of cross-section along the tangential and normal axes of the Frenet reference system.
The radius of curvature of the parameterized curve defines the center of curvature P around which the yarn is rolled up. The distance l(t) represents the length of the yarn between the intersection point I(t) and the carrier M(t). We can now express the → position of the carrier on the braiding platform O′ M ðt Þ via the intersection point I(t). → → → O′ M ðt Þ ¼ O′ O ðt Þ þ OM ðt Þ
ð3Þ
→ → → → with: OM ðt Þ ¼ OP þ PI ðt Þ þ IM ðt Þ → → → OM ðt Þ ¼ OP −ρs en þ lðt Þ→ et : As the yarns roll up on the mandrel, we can assume that the center of curvature P is an instantaneous center of rotation of the → _ yarn. We consequently assume a rotation of the Frenet reference system around its axis ek with a rotational speed denoted as φ. → We consider the variation of thebraiding angle as a rotation of the Frenet reference system around its normal axis en with a rotational speed denoted by α_ . With these assumptions, and expression (3), we obtain a new relationship connecting the 2
process parameters and the roll-up speed of yarns on the surface of the mandrel. α_ → ΩRg → eΩ ¼ V → ez þ l_ðt Þ−ρs φ_ → et þ lðt Þφ_ → en þ lðt Þ e 2 k
ð4Þ
The above relationship involves terms expressed in the Frenet reference system and also terms expressed in the reference system of the carrier. To overcome this inconvenience, we project the cross-section of the braiding operation along a strand of yarn on a plane coinciding with the braiding platform, as shown in Fig. 8.
M (t)
e
en
e
l (t )
et
er
I (t )
D
s
Rg
p R
o Fig. 8. Projection of the cross-section along the tangential and normal axes of the Frenet reference system on the braiding platform.
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G. Guyader et al. / Mechanism and Machine Theory 69 (2013) 90–104 →
In this figure, we define the point P′, the distance l′(t) and the axis et′ as the projections on the braiding platform of the point P, → → the distance l(t) and the tangential axis et , respectively, following the axis ez′ . → → The angle δ is the angle formed between the et′ and eR axes. The expression of the angle δ and the distance l′(t) are easily obtained from the triangle O′MD and can be written as:
cosðδÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2g −ðR þ h tan γz Þ2 cos2 γ ϑ
ð5Þ
Rg
l′ ðt Þ ¼ Rg cosðδÞ−ðR þ h tanγ z Þ sinðγϑ Þ: →
ð6Þ
→
We can now write the axes ez and eΩ of the carrier reference system in the Frenet reference system as follows: → → → ez ¼ cosγz ez′ þ sinγ z en
ð7Þ
α α → → → et þ sin e with: ez′ ¼ cos 2 2 k → → → eΩ ¼ cosδ en þ sinδ et′
ð8Þ
α α → → → ek þ sin e: with: et′ ¼ cos 2 2 t → → By substituting the relationships of the axes ez and eΩ into the general expression (4) and considering their projections on the → axes of the Frenet reference system, we obtain three independent relationships. The first, projected on the normal axis en , gives the expression of the roll-up speed φ_ of the yarn on the surface of the mandrel: φ_ ¼
ΩRg cosδ−V sinγ z lðt Þ
ð9Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 R2g −ðR þ h tan γ z Þ cos2 γϑ
Rg cosðδÞ−ðR þ h tan γz Þ sinðγ ϑ Þ cosðδÞ ¼ : α Rg sin 2 The roll-up speed is the ratio between the rate of the deposited length of yarn and the radius of the mandrel. → The second relationship, projected on the tangential axis et , connects the variation of the length of the yarn l_ ðt Þ between the carrier and the intersection point to the process parameters and the braiding angle: with: lðt Þ ¼
α α _ l_ ðt Þ ¼ ΩRg sinδ sin −V cosγ z cos þ ρs φ: 2 2
ð10Þ →
The third expression, projected on the transversal axis ek , gives the braiding angle variation as a function of the process parameters and the actual braiding angle: ΩR sinδ cos α −V cosγ sin α g z α_ 2 2 : ¼ 2 lðt Þ
ð11Þ
These three equations are able to describe several phenomena, such as the slipping of the yarn on the mandrel or the relaxation of the yarn during the braiding operation. We will focus on the first equation leading to the radial component of the yarn trajectory and to the braiding front translation speed h_ in the longitudinal direction. Assuming the center of curvature P as an instantaneous center of rotation, the motion of the intersection point I(t) can be considered as a rotation around the center of curvature of the tangential axis with a rotational speed φ_ ðt Þ. The tangential component of the deposit of yarn on the mandrel lt(t) is consequently expressed as: lt ðt Þ ¼ ρs φ_ ðt Þ:
ð12Þ
This expression can be divided into radial and longitudinal components that can be easily expressed by projection on the radial and longitudinal axes, respectively. α lϑ ðt Þ ¼ lt ðt Þ sin 2
ð13Þ
α le ðt Þ ¼ lt ðt Þ cos 2
ð14Þ
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Therefore, we obtain the radial component of the yarn trajectory lϑ(t) and the braiding front translation speed h_ in the longitudinal direction. lϑ ðt Þ ¼ ρs Ω
α Rg cos δ−V sinγ z sin 2 lðt Þ
α Rg cosδ−V sinγ z h_ ¼ V−ρs Ω cos 2 lðt Þ
ð15Þ
ð16Þ
Assuming that the radial component of the yarn trajectory on the mandrel corresponds to the roll-up speed of the yarn and the longitudinal component corresponds to the translation of the yarn due to the motion of the mandrel and the braiding front, we can construct a general relationship of the braiding angle adapted to complex mandrel shapes and transitory process run. _ ðt Þ α −1 ρs φ ¼ tan V þ dh 2
ð17Þ
4. Comparison to existing models As a first validation, we compare the results obtained with the model developed above with the results found in the literature. The most common braiding operations use a cylindrical mandrel in a stationary process run. Ko [4] studied this case, giving the analytical relationship of the braiding angle. We also cite the works of Zhang [5,8] for this type of braiding. The study developed by Du and Popper [6] considers an axisymmetric mandrel with longitudinal variation and introduces a quantification of the transitory stage, while the work of W. Michaeli [12] studied the influence of the eccentricity of a cylindrical mandrel on the structure of a braid. We can consider the eccentricity as a geometrical offset involving radial variation of the geometry of the mandrel. Therefore, we first focus on cylindrical mandrels before separately considering mandrels with longitudinal geometry variations and mandrels with radial geometry variations. 4.1. Application to cylindrical mandrel In the case of a cylindrical mandrel with a radius R, on every point of the surface, we have: γϑ = 0, γz = 0, cosðδÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2g −R2 cos2 γ ϑ R . Therefore, the relationship between the radial component of the yarn trajectory on the mandrel , ρs ¼ 2 α Rg sin 2 and the speed of the braiding front translation in the longitudinal direction becomes: lϑ ðt Þ ¼ RΩ
ð18Þ
RΩhðt Þ h_ ¼ V− qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2g −R2
ð19Þ
leading to the braiding angle expression previously given by Zhang [5]. 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 R2g −R2 α −1 @ A ¼ tan 2 hðt Þ
ð20Þ
We observe that the expression of the motion of the braiding front is similar to the expression previously given by Du and Popper [6] when applied to a cylindrical mandrel. h_ ðt Þ ¼ V−
RΩhðt Þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u u1 R hðt Þ Rg t − þ tanγz Rg Rg Rg
ð21Þ
In the case of a steady-state run with the assumption of a tangential deposit of yarns on the mandrel, Zhang et al. [8] propose the following expression of the position of the braiding front h(t).
hðt Þ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V R2g −R2 ΩR
ð22Þ
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Substituting the braiding front position expression into the previous expression of the braiding front motion, we have: h_ ¼ 0:
ð23Þ
Finally, we obtain the expression of the braiding angle in the case of a cylindrical mandrel and a steady-state run, which is similar to the expression previously given by Ko [4] for the same braiding conditions. α −1 RΩ ¼ tan 2 V
ð24Þ
4.2. Application to mandrels with longitudinal geometry variations The longitudinal geometry variation of a mandrel can be treated as linear for an elementary length. The entire surface of the mandrel is thus described as a succession of small cone trunks. With this condition, we obtain for every point of the surface: γϑ = 0, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 R2g −ðR þ h tanγ z Þ R and cosðδÞ ¼ , ρs ¼ . Rg sin2 α2 The expressions of the radial component of the yarn trajectory l and of the translation speed of the braiding front h_ become: ϑ
0
1
V sinγ z B C lϑ ðt Þ ¼ R@Ω− qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A R2g −ðR þ h tanγ z Þ2
ð25Þ
h_ ðt Þ ¼ V−T1−T2
ð26Þ
with: RΩhðt Þ T1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Rg −ðR þ h tanγz Þ2
T2 ¼
Vhðt Þ sinγz : R2g −ðR þ hðt Þ tan γz Þ2
We note that for a small variation of the mandrel geometry in the longitudinal direction or for a large radius Rg of the braiding platform, the second and third terms of Eqs. (25) and (26), respectively, can be neglected. With these assumptions, we obtain a similar relationship of the radial component lϑ and of the braiding front translation speed h_ to the expressions given by Du and Popper [6] for the same braiding conditions (Eqs. (27) and (28), respectively). lϑ ðt Þ ¼ RΩ
ð27Þ
RΩhðt Þ dh ¼ V− qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2g −ðR þ hðt Þ tanγ z Þ2
ð28Þ
The relationship of the braiding angle becomes: 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 R2g −ðR þ hðt Þ tanγz Þ2 α −1 A: ¼ tan @ 2 hðt Þ
ð29Þ
To obtain an overview of the effect of neglecting the terms in Eqs. (25) and (26), we take as an example an existing braiding case with 64 carriers, a radius Rg of the braiding platform of 750 mm, a translation speed of the mandrel of 1 mm/s, and a rotational speed of the braiding platform of 0.27 rad/s. Fig. 9 presents the effect of the ratio T1/T2 on the motion speed of the braiding as a function of the longitudinal orientation angle γz. We note that the effect of the third term, T2, on the motion speed of the braiding front is quite small, and thus the present model approaches the motion of the braiding front in the same manner as the model developed by Du and Popper [6].
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Fig. 9. Effect of the T1/T2 ratio on the motion speed of the braiding front.
4.3. Application to mandrels with radial geometry variations In the case of mandrels with radial geometry variations, we have at every point of the surface: γz = 0, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2g −R2 cos2 γ ϑ R α . The relationships of lϑ and h_ become: , ρs ¼ Rg sin2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2g −R2 cos2 γ ϑ lϑ ðt Þ ¼ RΩ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2g −R2 cos2 γϑ −R sinγ ϑ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2g −R2 cos2 γ ϑ _h ðt Þ ¼ V−RΩhðt Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 R2g −R2 cos2 γ ϑ −R sinγϑ
cosðδÞ ¼
ð30Þ
ð31Þ
leading to the following braiding angle expression: 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 R2g −R2 cos2 γϑ −R sin γϑ α −1 @ A: ¼ tan 2 hðt Þ
ð32Þ
To estimate the validity of these expressions, we apply Eqs. (31), (32), and (33) to the results obtained by Michaeli et al. [12], who analyzed the influence of the mandrel eccentricity on the distribution of the braiding angle. The eccentricity can be implemented in the present model as a variation of the angle γϑ. Fig. 10 shows the deviation from the expected braiding angle due to the eccentricity of the mandrel. The diameter of the mandrel is 100 mm, and the effective radius of the braiding platform is 350 mm. As can be observed in the present model, the developed model approaches the experimental results of Michaeli et al. with good accuracy for a wide range of braiding angles.
Fig. 10. Deviation from the expected braiding angle due to the eccentricity of the mandrel: comparison of the analytical model and the results of Michaeli et al. [12].
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Fig. 11. Spacing of meshes during a transitory process run.
5. From a 2D to a 3D interlock braiding process The previously cited results were developed for 2D circular braiding operations assuming a circular trajectory of the carriers on the braiding platform and neglecting their sinusoidal movement around an average circle. Considering the orientation of the yarn and the thickness of the fabric, we can construct an analogy between a 3D circular braid and a stack of 2D circular braids (neglecting the interlocking between the superposed layers). In reality, in the case of a 3D circular braiding machine, the transverse motion of the carriers around an average circle may be important, and the assumption of a circular trajectory of the carriers on the braiding platform is not immediate. We note that the average rotational speed of the carriers is similar to that of the 2D braiding, and the motion of the carriers from one level to another is quick enough to be averaged by the overall circular motion of the braiding front. Consequently, we assume that the average longitudinal component of the yarn trajectory will not be affected. 5.1. Preliminary assumptions To establish a transitory process run during the braiding operations, the translational speed of the mandrel was varied and the rotational speed of the braiding platform was held constant. Thus, the position of the braiding front can be varied from an initial value to a set-point value. Both values can be calculated using formula (22), corresponding to the position of the braiding front in a steady-state process run. Furthermore, it can be stated that in the case of a constant rotational speed of the platform, a mesh develops in the braid during a constant time interval, independent from the other process parameters. For example, in the case of a braid assimilated into a taffeta, a complete revolution is needed to create a mesh. Therefore, for a given rotational speed Ω of the platform, a mesh is constructed during a time Δt: Δt ¼
4π : Ω
ð33Þ
Consequently, by comparing the distance between a defined number of meshes (Fig. 11) and the distance traveled by the mandrel, it is possible to establish the motion speed of the braiding front: d h_ ¼ V− n nΔt
ð34Þ
with h_ as the motion speed of the braiding front, V as the speed of the mandrel and n as the number of successive meshes. Before the start of the measurement, the braiding front is positioned manually on the mandrel at its theoretical value using a laser level; the braiding is started at a constant velocity to obtain stabilization of the braiding front. Next, the motion speed of the mandrel is shifted directly to the set-up value, resulting in the variation of the position of the braiding front (Fig. 12). As it can be seen, the contact point between yarns and the mandrel (the braiding front) gets closer to the crown of the braiding machine and consequently the braiding angle changes. The same type of fiber was used for all braiding operations: E-glass Hybon 2001 Roving, PPG 2400 tex. Filaments have a diameter of 25 μm. The choice of this type of fiber is related to production requirements. Several type of glass fibers were tested before to limit friction and knot in the convergence zone to avoid interruption of the braiding process. 5.2. 2D circular braiding To assess the variation of the braiding front during a transitory process, a 2D braiding operation was set up characterized by 64 carriers, an effective radius of the braiding platform of 750 mm, a rotational speed of the platform of 7.36 rad/s, and a PVC
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Fig. 12. Motion of the braiding front during the transitory process.
mandrel with a length of 1000 mm and a diameter of 100 mm. The mandrel is moved by a six-DOF pulling unit driven by a PC unit. The mandrel is translated normal to the braiding front only (Fig. 13). The first braiding was carried out by accelerating the translation speed of the mandrel from an initial value of 5 mm/s to 15 mm/s. Fig. 14 presents a comparison of the experimental and theoretical results, confirming the ability of the analytical model to assess the motion of the braiding front. However, at the end of the braiding operation, the steady state is not yet attained, and the two curves begin to diverge. This difference can be attributed to the post-braiding operation (cutting of yarns, local tightening or loosening of yarns, etc.), given the relatively low density of the braid.
Fig. 13. 2D braiding operation set up to validate the advancement of the braiding front in a transitory process in the case of a cylindrical mandrel.
Fig. 14. Comparison of the translation of the braiding front (left) and of the variation of the braiding angle (right) in the case of 2D circular braiding when the mandrel speed is increased.
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Fig. 15. Comparison of the translation of the braiding front (left) and of the variation of the braiding angle (right) in the case of 2D circular braiding when the mandrel speed is decreased.
A second braiding was produced by slowing the translation speed from 10 mm/s to 5 mm/s. The results are presented in Fig. 15, which show a good correlation between the experimental and theoretical results.
5.3. 3D circular braiding To validate the analytical relationship of the motion of the braiding front during a transitory state process (Fig. 16), we used a 3D braiding platform consisting of five carrier levels that define an effective internal and external diameter of 2000 mm and 3000 mm, respectively. Each carrier level is composed of 64 carriers, the rotational speed of the braiding platform is 7.36 rad/s, and the mandrel consists of a PVC pipe with a length of 2000 mm and a diameter of 50 mm. An automated pulling unit allows control of the variation of the speed and the movement of the mandrel normal to the plane of the braiding platform (Fig. 16). Four types of braids were produced, permitting analysis of the transitory state for braiding angles ranging from 35° to 65°. The first braiding operation was carried out by varying the speed of the mandrel from 15.95 mm/s to 7.5 mm/s. A second braiding operation was set up by increasing the speed of the mandrel from 7.5 mm/s to 15.95 mm/s. Fig. 17 shows the comparison of the experimental and the theoretical results for the chosen braiding angles. We note that the experimental and theoretical curves fit with good accuracy, and the gradient of the motion of the braiding front is the same when approaching or moving away from the braiding platform. Furthermore, we can remark that the curves diverge slightly at the start of the braiding, which could be a consequence of the compaction of yarns due to the sudden change in the process parameters. Two other situations were considered in which the mandrel motion's speed was changed from 13.12 mm/s to 18.75 mm/s and from 8.44 mm/s to 6.19 mm/s. In Fig. 18 the experimental and theoretical curves show a good accuracy of the model in describing the progress of the braiding front, validating again the pertinence of the proposed model.
Fig. 16. Setup for a 3D circular braiding process used to validate the motion of the braiding front during a transitory process run in the case of a 100-mm circular mandrel.
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Fig. 17. Comparison of the translation of the braiding front in the case of 3D circular braiding when the mandrel speed is changed. (a) Decrease of the mandrel speed; (b) increase of the mandrel speed.
6. Conclusions and outlook In this paper, we presented an analytical model of the yarn trajectory during a circular braiding operation. This mathematical model uses a parameterized curve approach and produces selected relationships that are characteristic of the interactions between the process parameters and the braiding architecture. The positioning of the braiding front during a transitory process run, the slippage of the yarn on the mandrel and the relaxation of the yarn can be described even in the case of complex-shaped mandrels. The results were compared to other analytical models found in the literature. In the case of a steady-state process and symmetric revolution shapes, the developed model is perfectly adapted to link the process parameters (e.g., braiding platform rotational speed or mandrel speed) to the characteristics of the braid (e.g., braiding angle). The results also approach the distribution of the braiding angle in the case of circular eccentric mandrels with good accuracy. To validate the ability of the model to describe the motion of the braiding front in the case of a transitory process run, we performed a prototype braiding operation in which the mandrel translational speed was suddenly increased or decreased. The model described the advancement of the braiding front for multiple speed ranges with good accuracy. Nevertheless, certain other aspects of this work must be validated by taking into account more complex mandrel shapes or the yarn relaxation behavior due to the friction in the convergence zone. The developed model considers a linear trajectory of yarns in the convergence zone. But in the case of an important friction between yarns, this hypothesis is no longer valid and it would be suitable to model the path of the yarns with a parabolic function or equivalent. The final objective is to establish the process parameters (e g., the rotational speed of the platform and the motion of the mandrel in the braiding space) to obtain the requested braiding angle distribution on a complex-shaped mandrel. Acknowledgments We gracefully acknowledge the enterprise DJP-Composites for participating in the experimental validation of the theoretical parameters by allowing us to use their 3D interlock braiding equipment. This work was supported by the 7th FUI Project “Sagane” of the Ministry of the Economy, Finance and Industry of France, to whom the authors are very grateful.
Fig. 18. Comparison of the translation of the braiding front in the case of 3D circular braiding when the mandrel speed is changed. (a) Increase of the mandrel speed from 13.12 mm/s to 18.75 mm/s. (b) Decrease of the mandrel speed from 8.44 mm/s to 6.19 mm/s.
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References [1] T.W. Chou, F.K. Ko, Textile Structural Composites, vol. 3Elsevier, 1989. [2] L. Wenning, C. Guangming, Q. Xin, Proc. of the 38th Int. SAMPE Symposium, 1993, pp. 10–13. [3] A.B. Macander, R.M. Crane, E.T. Camponeschi, Fabrication and mechanical properties of multidimensionally (X-D) braided composite materials, Composite Materials (1986) 422–443. [4] F.K. Ko, Braiding engineered materials handbook, Composites, ASM International, vol. 1, 1987. 519–528. [5] Q. Zhang, D. Beale, R.M. Broughton, Analysis of circular braiding process. Part 1: theoretical investigation of kinematics of the circular braiding process, Journal of Manufacturing Science and Engineering vol. 121 (1999) 345–350, no3. [6] G.W. Du, P. Popper, Analysis of a circular braiding process for complex shapes, Journal of the Textile Institute (1994) 316–337. [7] J.F.A. Kessels, R. Akkerman, Prediction of the yarn trajectories on complex braided preforms, Composites Part A: Applied Science and Manufacturing 33 (8) (2002) 1073–1081. [8] Q. Zhang, D. Beale, R.M. Broughton, Analysis of circular braiding process. Part 2: mechanics analysis of the circular braiding process and, Journal of Manufacturing Science and Engineering vol. 121 (1999) 351–359, (no3). [9] A. Pickett, A. Erber, T. von Reden, K. Drechsler, Comparison of analytical and finite element simulation of 2D braiding, Plastics, Rubber and Composites 38 (9–10) (2009) 387–395. [10] A. Rawal, P. Potluri, C. Steele, Prediction of yarn paths in braided structures formed on a square pyramid, Journal of Industrial Textiles 35 (2) (2005) 115–135. [11] P. Potluri, A. Manan, Mechanics of non-orthogonally interlaced textile composites, Composites Part A 34 (2003) 481–492. [12] W. Michaeli, U. Rosenbaum, M. Jehrke, Processing strategy for braiding of complex-shaped parts based on a mathematical process description, Composites Manufacturing vol. 1 (4) (December 1990) 243–251.
Glossary α: braiding angle e: shrinkage C: coverage factor Ω: overall rotational speed ω_ : horn-gear rotational speed N: number of horn gears V: mandrel translation speed ρz: radius of curvature along the longitudinal axis ρϑ: radius of curvature along the radial axis Pz: center of curvature along the longitudinal axis Pϑ: center of curvature along the radial axis γz: angle of orientation along the longitudinal axis γϑ: angle of orientation along the radial axis lϑ: radial component of the yarn trajectory lz: longitudinal component of the yarn trajectory _ motion speed of the braiding front h: I: intersection point M: carrier point on the braiding platform O: geometrical center of the mandrel in the plane of the braiding front O′: center of the braiding platform ρs: radius of curvature of the trajectory of the yarn on the mandrel l: length between the intersection point of the yarn on the mandrel and the carrier P: center of curvature of the trajectory of a yarn on the mandrel _ yarn roll-up speed φ: α_ : variation of the braiding angle → → δ: angle formed between the axes e′t and eR R: radius of the mandrel Rg: effective radius of the braiding platform h: braiding front position lt: tangential component of the yarn deposit on the mandrel Δt: time necessary for the formation of a mesh dn: distance between n meshes n: number of considered meshes