Geometrical modeling of near-net shape braided preforms

Abstract

Circular braiding has been successfully adapted for producing near-net shape structures for advanced fiber-reinforced composites. Net-shape manufacturing is significantly important for fabricating complex three-dimensional (3D) preforms. Geometrical modeling of braid patterns on 3D preforms plays a key role in determining their mechanical behavior. In this research work, the geometrical models of strand trajectory on the surface of cylindrical and conical mandrels with diamond, regular and triaxial braid patterns have been developed. These geometrical models of strand profiles were then simulated using Virtual Reality Modeling Language. Subsequently, the strands on complex-shaped mandrels, including ‘bottle’ and ‘funnel’, were simulated and accordingly, the braid angles have been predicted and compared with the experimental results. A virtual experiment was also conducted to compare the trajectory of the strands having constant and varying braid angles on the surface of conical mandrels.

Keywords

strand braiding modeling preform

Circular braiding is a traditional technique normally employed for producing rope-like structures, but this process has expanded its horizons to develop three-dimensional (3D) complex preforms that can be used in advanced fiber-reinforced composites. These 3D complex preforms can be easily fabricated by braiding the 3D mandrels using a conventional maypole braiding process.^1,2 Braiding over a mandrel can potentially have number of advantages, including the ability to precisely align the constituent strands and to build the desired number of layers to obtain the required level of thickness along with the possibility of forming near-net shapes.² The path of constituent strands in braids needs to be predicted as it is a first step towards engineering analysis of understanding their micromechanics and other characteristics including resin infusion, aerodynamics and crash performance.

Du and his colleagues^3,4 have successfully developed a process model of circular braiding and formulated a relationship between process parameters with the braid angles formed on the surface of axisymmetric-shaped mandrels. There was a discrepancy between the theoretical and experimental results of braid angles and it was attributed to the curved path of strands in the convergence zone near the deposit plane.⁴ Subsequently, Zhang et al.⁵ analyzed the actual strand path in the circular braiding process based on the particle velocity approach and their methodology has considered the curved strand path in the convergent zone for obtaining the realistic braid angles on the surface of 3D-shaped mandrels. On the other hand, the trajectory of strands was modeled in 3D net-shaped braided structures using the Frenet frame approach.⁶ Similarly, Akkerman and his colleagues^7,8 reported a series of publications formulating a relationship between strand paths and braid angles on 3D non-axisymmetric mandrels with the possibility of eccentric cross-sections that varied in shape and size. More recently, the braid pattern was predicted on any arbitrary mandrel shape based on the minimum path condition followed by the strand.⁹ Initially, the braid point was determined on the surface of the mandrel where the length of the constituent strand running from the previous braid point to the guide ring was minimized, resulting in the ‘minimum path’ condition. In these publications, the equations have been solved numerically in order to obtain the 3D coordinates of strand profiles present on the surface of the mandrel.^7,9 Alternatively, we have developed the geometrical models of strand paths on various mandrel shapes, including cylinder, cone with circular and elliptical cross-sections, square prism and square pyramids.^10,11 However, these investigations did not consider the local sinusoidal path of the strand, also known as ‘crimp’, in order to simplify the geometrical models. More recently, Alpyildiz¹² formulated the geometrical models of strand paths in biaxial and triaxial tubular braids by including the local sinusoidal path of the strands. Therefore, one of the main aims of the present research work is to develop the geometrical models of strand profiles present on the surface of cylindrical and conical mandrels having diamond, regular and triaxial braided structures. Furthermore, these simple mandrels have been combined together to form ‘bottle’ and ‘funnel’ shapes on which the strand profiles have been simulated. Accordingly, the braid angles have been predicted and, subsequently, compared with the experimental results.

Theoretical details

The following assumptions were made in order to develop the geometrical models of braided structures on the surface of 3D mandrels:

a tubular braided structure consists of sets of clockwise and counter-clockwise strands such that their centerline follows a local sinusoidal path corresponding to their nominal helical path;

the braid strands do not slip after forming the desired pattern on the surface of the mandrel;

the braid pattern formed on the surface of the mandrel is continuous in nature;

in triaxial braids, the axial strands are assumed to be uniform, straight and they are placed at the cross-over regions such that they locally increase the amplitude of the sinusoidal path of the crossing strand;

it is invariably assumed that the strand cross-section is lenticular in shape as the path of the constituent strands follows local sinusoidal variations whilst moving in a helical manner.

In the past, the braid angle has been computed based upon the rotational speed of the horn gear (

ω_{h}

), radius of the mandrel (R_m), number of horn gears (N_h) and take-up speed (v), as shown below:²

α = \tan - 1 (\frac{2 ω_{h} R_{m}}{N_{h} v})

(1)

In general, the locus of the strand axis follows a helical path around a cylindrical mandrel, as shown in the following equations:

X = R \cos θ Y = R \sin θ Z = R θ \cot α

(2)

where R is the effective radius of the helical strand, r is the strand radius, θ is the wrapping angle (

0 < θ < 2 π

) and α is the braid angle defined as the angle between the strand and braid axes.

Now consider a model of a typical cross-over strand, as illustrated in Figure 1; it is invariably assumed that the centerline of the crossing strand follows a sinusoidal path.

Figure 1.

A model of the cross-over strand in a diamond braid. Here $X' Y'$ represents the local reference frame of the strand following the helical path.

The equation of the strand path can be easily projected along the centerline of the strand crimp in the $X' Y'$ plane, as shown in the following equation:¹³

y' = h \sin (\frac{π x'}{p})

(3)

where h is the undulation height (≈ radius of crossing strand) and p is the crossing strand spacing.

Here, the strands placed under the cross-over regions experience transverse forces that causes a change in their cross-section such that the undulation height approaches the radius of the cross-over strand, as shown in Figure 1. Moreover, p can be defined as a length of diamond trellis unit formed on a typical mandrel, as shown in Figure 2. Accordingly, it can be expressed in terms of braid angle (α), effective radius of the helical strand (R) and total number of constituent strands (N), as shown below:¹⁴

p = \frac{2 π R}{N \sin α}

(4)

Figure 2.

A parametric model of tubular braid depicting the diamond trellis obtained on slitting along the braid axis and a cutaway view showing strand crimp.

Consider a small element of length ‘dS’ on a sinusoidal helix, and let ‘ $R d θ$ ’ be the projection of this element in the XY plane, as shown in Figure 3. Accordingly, $R d θ = \sin α d S$ , which can be integrated to compute the simple relationship between the length of the sinusoidal helix along the direction of strand axis ( $x'$ ) and braid angle (α), that is:

x' = \frac{R θ}{\sin α}

(5)

Figure 3.

Helical path of a strand having sinusoidal undulations on a typical cylindrical mandrel.

Thus, the expression of the strand profile following the sinusoidal path can be computed by combining Equations (3)–(5). Hence,

y' = R (θ) = h \sin (\frac{N θ}{2})

(6)

Thus, the above equation needs to be included in the helical path of the strand, as shown in Equation (2), in order to include the local sinusoidal variation. It should be noted that $R (θ)$ is considered as $R_{1} (θ)$ and $R_{2} (θ)$ for clockwise and counter-clockwise directions, respectively, which are defined for different cases. These sinusoidal variations require two sets of strands to form a crest and trough and, accordingly, a phase difference of π can be introduced for clockwise and counter-clockwise strands. Hence, combining Equations (2) and (6) would yield the following expression of the strand paths forming the tubular braid in the clockwise direction:

X_{i} = (R + R_{1} (θ)) \cos (θ + (i - 1) γ), i = 1, 2 \dots n Y_{i} = (R + R_{1} (θ)) \sin (θ + (i - 1) γ), i = 1, 2 \dots n Z_{i} = R θ \cot α

(7)

where γ is the shift angle between the two consecutive strands and it is equal to

2 π / n

, where n is the number of strands moving in the same direction.¹² For a typical braid having the same type of constituent strands in the clockwise and counter-clockwise directions, ‘n’ can be considered as ‘N/2’.

Similarly, the equation of strand paths in the counter-clockwise direction can be obtained by changing their orientation in the XY plane, as shown below:

X_{i} = (R + R_{2} (θ)) \cos (θ + (i - 1) γ), i = 1, 2 \dots n Y_{i} = - (R + R_{2} (θ)) \sin (θ + (i - 1) γ), i = 1, 2 \dots n Z_{i} = R θ \cot α

(8)

In Equations (7) and (8), the positive and negative values of y-coordinate describe the positions of the strands in the clockwise and counter-clockwise directions, respectively.

Case I: Cylindrical mandrel with diamond braids

Diamond braids (1/1 weave type) can be easily represented with the centerline of the crossing strands that follow sinusoidal paths on the surface of the cylindrical mandrel. Thus, the local strand profile in both clockwise and counter-clockwise directions can be easily computed, as shown below:

R_{1} (θ) = r \sin (\frac{N θ}{2} + \frac{π}{2}) forclockwisestrands

(9)

R_{2} (θ) = r \sin (\frac{N θ}{2} + \frac{3 π}{2}) forcounter-clockwisestrands

(10)

Hence, the strand paths forming the diamond braided structure on the surface of the cylindrical mandrel can be modeled by combining Equations (7)–(10). It should be noted that the effective radius (R) is the sum of the mandrel radius (R_m) and strand diameter (2r), as shown in Equation (11):

R = R_{m} + 2 r

(11)

Case II: Cylindrical mandrel with regular braids

The equations of diamond braid can be extended to regular braid by considering the strand path to follow 2/2 twill weave, as shown in Figure 4. Here, the centerline of the crossing strand passes over and under two sets of strands in the local $X' Y'$ plane. Accordingly, the strand profile can be divided in terms of straight (KL) and undulated (LM) segments, as illustrated in Figure 4. Thus, the local strand profile moving in the clockwise direction is shown below:

R_{1} (θ) = {- r; 0 < θ < \frac{γ}{2} r \sin (\frac{N θ}{2} + \frac{π}{2}); \frac{γ}{2} < θ < γ r; γ < θ < \frac{3 γ}{2} r \sin (\frac{N θ}{2} + \frac{π}{2}); \frac{3 γ}{2} < θ < 2 γ}

(12)

Figure 4.

Path of cross-over strand in a typical regular braid. Here $X' Y'$ represents the local reference frame of the strand following the helical path.

Similarly, the strand profile following the counter-clockwise direction is shown below:

R_{2} (θ) = {r; 0 < θ < \frac{γ}{2} r \sin (\frac{N θ}{2} + \frac{3 π}{2}); \frac{γ}{2} < θ < γ - r; γ < θ < \frac{3 γ}{2} r \sin (\frac{N θ}{2} + \frac{3 π}{2}); \frac{3 γ}{2} < θ < 2 γ}

(13)

Hence, the strand paths forming the regular braided structure on the surface of the cylindrical mandrel can be modeled by combining Equations (7) and (8) and Equations (11)–(13).

Case III: Cylindrical mandrel with triaxial braids

The triaxial braided structure can be developed by placing the axial strands at the cross-over regions of the braider strands in the direction of the braid axis. These axial strands locally increase the amplitude of the sinusoidal path of the crossing strands, as shown in Figure 5. It is assumed that these axial strands have the same minor diameter as that of braider strands. Thus, the local braider strand profiles following clockwise and counter-clockwise directions can be obtained, as shown below:

R_{1} (θ) = {2 r \sin (\frac{N θ}{2} + \frac{π}{2}); 0 < θ < γ r \sin (\frac{N θ}{2} + \frac{π}{2}); γ < θ < 2 γ}

(14)

R_{2} (θ) = {2 r \sin (\frac{N θ}{2} + \frac{3 π}{2}); 0 < θ < γ r \sin (\frac{N θ}{2} + \frac{3 π}{2}); γ < θ < 2 γ}

(15)

Figure 5.

Path of cross-over strand in a typical triaxial braid. Here $X' Y'$ represents the local reference frame of the strand following the helical path.

In the case of triaxial braids, the expression of effective radius (R) has been modified as illustrated below:

R = R_{m} + 3 r

(16)

The equations of braider strand paths forming the triaxial braids in the clockwise and counter-clockwise directions can be obtained by combining Equations (7), (8), (14)–(16). Furthermore, the equations of the axial strands can be obtained by placing these strands at the cross-over positions in the XY plane. Thus,

X_{i} = R \cos ((i - 1) γ), i = 1, 2 \dots n Y_{i} = R \sin ((i - 1) γ), i = 1, 2 \dots n Z_{i} = R θ \cot α

(17)

Case IV: Conical mandrel with diamond braids

In the past, the equation of a typical strand path following the surface of a conical mandrel has been computed without considering the local sinusoidal variations, as shown below:¹⁰

x = R_{0} e θ \cot α \tan τ \cos θ y = R_{0} e θ \cot α \tan τ \sin θ z = R_{0} e θ \cot α \tan τ \cot τ

(18)

where

R_{0} = R_{b} + r - (R θ \cot α) \tan τ

(19)

where R₀ is the effective radius of the helical strand at

θ = θ_{0}

, R_b is the base radius of the cone and τ is the taper angle of the cone.

In general, a cone consists of a series of cylindrical mandrels with varying radii. Accordingly, the above equation can be generalized for strands moving in the clockwise and counter-clockwise directions similar to the strand profiles obtained for the cylindrical mandrel as discussed in Case I. Thus, the strand profile moving in the clockwise direction is given by the following expression:

X_{i} = (R' + R_{1} (θ)) \cos (θ + (i - 1) γ), i = 1, 2 \dots n Y_{i} = (R' + R_{1} (θ)) \sin (θ + (i - 1) γ), i = 1, 2 \dots n Z_{i} = R' \cot τ

(20)

where

R' = R_{0} e θ \cot α \tan τ and R_{1} (θ) = r \sin (\frac{N θ}{2} + \frac{π}{2})

(21)

Similarly, the equation of the strand profile in the counter-clockwise direction is given by

X_{i} = (R' + R_{2} (θ)) \cos (θ + (i - 1) γ), i = 1, 2 \dots n Y_{i} = - (R' + R_{2} (θ)) \sin (θ + (i - 1) γ), i = 1, 2 \dots n Z_{i} = R' \cot τ

(22)

where

R_{2} (θ) = r \sin (\frac{N θ}{2} + \frac{3 π}{2})

(23)

Case V: Conical mandrel with regular and triaxial braids

As mentioned in Case II, the equations of the diamond braid can be extended to regular braid by dividing the path of the crossing strand in terms of straight and undulated segments. Thus, the strand profiles, that is, $R_{1} (θ)$ and $R_{2} (θ)$ as defined in Equations (12) and (13), can be used in combination with Equations (20)–(23) for obtaining the expressions of strand profiles in the clockwise and counter-clockwise directions on the surface of the conical mandrel. Similarly, the expressions of constituent strand profiles following the braider strand direction in triaxial braids can be obtained by combining Equations (14), (15) and (20)–(23). Furthermore, the equations of axial strand profiles can be computed by replacing R in Equation (17) by $R'$ , as defined in Equation (21).

Experimental details

Regular braids consisting of 16 strands of multifilament hybrid filaments were braided on the surface of ‘bottle’ and ‘funnel’-shaped mandrels, as illustrated in Figure 6. These multifilament hybrid strands (linear density of 128 tex) consist of Kevlar, modacrylic, high-tenacity nylon filament, fine steel wires, textured nylon and polyester filaments in predefined weight proportions. The circular braiding machine consists of eight horn gears and the rotational speed of the horn gear was kept at 16.49 rad/s. Similarly, the take-up speed was also kept constant at 2.4 m/min. The screenshot images of these braided structures were captured for determining braid angles using the IMAGEJ®, a public domain image processing software.

Figure 6.

Models of mandrels with (a) bottle (b) funnel shapes. All dimensions are in mm.

Results and discussion

The equations of the strand paths both in the clockwise and counter-clockwise directions for Cases I–V (as described in Theoretical details section) have formed the basis of simulated braided structures. A Matlab® program was written to generate the 3D coordinates of strand paths and it was subsequently interfaced with Virtual Reality Modeling Language (VRML) to render braided structures on various mandrel shapes. In the past, VRML has been successfully used as a ‘simulation’ tool for generating models of tubular braided structures.^2,10,11 The simulated diamond, regular and triaxial braided structures formed on the surface of cylindrical mandrels are shown in Figure 7. In these simulations, the braid angle is assumed to be 45° and there are 16 constituent strands forming tubular braided structures. For visualization purposes, the ratio of mandrel diameter to strand minor diameter is assumed to be 20. Similarly, the braided structures were formed on the surface of the conical mandrel having a taper angle of 5°, as illustrated in Figure 8. In the case of conical mandrels, it is well-known that highly irregular patterns of braid angle are developed when braiding is carried out from small to large diameter shapes.⁴ In the past, Du and Popper⁴ have varied the take-up speed along the height of the conical mandrel in order to uniformize the braid angle. Therefore, a virtual experiment was carried out to visually compare the strands having constant and varying braid angles along the height of conical mandrels. A constant braid angle was simulated on a conical mandrel based on the assumption that a cone is a series of cylindrical mandrels with varying radii such that the ratio of their corresponding heights with varying radius and wrapping angle, that is, $Z / R θ$ , has attained a constant value. Figure 9 shows a comparison between the path of strands with constant and varying braid angles following the surface of conical mandrels. For strands with varying braid angle on the surface of the conical mandrel, their value ranges between 16° and 45° at the top and base diameters, respectively. In another case, a braid angle of 45° was kept constant for strands following the sinusoidal variations along the surface of conical mandrels. This has given the possibility to increase the strand coverage and its volume fraction on the surface of the conical mandrel.

Figure 7.

Simulated models of cylindrical mandrels having (a) diamond (b) regular (c) triaxial braided structures.

Figure 8.

Simulated models of conical mandrels having (a) diamond (b) regular (c) triaxial braided structures.

Figure 9.

Simulated models of conical mandrels having strands of (a) varying (b) constant braid angles.

Comparison between theoretical and experimental results

The ‘bottle’ and ‘funnel’ mandrels were simulated by combining the cylindrical and conical shapes. For instance, the ‘bottle’-shaped mandrel consists of three constituents, that is, a smaller cylinder (top), a cone (middle) and a larger cylinder (base), and it is presumed that the braiding of strands take place from the surface of the smaller cylinder (top) followed by the cone (middle) and the larger cylinder (base). Accordingly, the strand paths were predicted by computing the braid angle on the surface of a smaller cylinder using Equation (1). The strand paths on the top surface of the conical mandrel were then simulated using the same braid angle as that of the smaller cylinder. Subsequently, the braid angle increased from the top to the base surface of the cone with varying radii. Finally, the braid angle of the larger cylinder follows the braid angle of the base of the conical mandrel. Similarly, the strand paths were theoretically predicted for the ‘funnel’-shaped mandrel that consists of a cylinder and a cone. A visual comparison can be observed between the simulated models and ‘real’ images of the bottle and funnel-shaped mandrels, as shown in Figures 10 and 11. A comparison between the theoretical and experimental braid angles along with the normalized height of bottle and funnel-shaped mandrels is depicted in Figure 12. In general, an excellent agreement has been observed between the theoretical and experimental values of braid angles.

Figure 10.

A visual comparison between (a) real (b) simulated models of a bottle-shaped mandrel.

Figure 11.

A visual comparison between (a) real (b) simulated models of a funnel-shaped mandrel.

Figure 12.

Comparison between theoretical and experimental braid angles in (a) bottle (b) funnel-shaped mandrels.

Conclusions

In the present work, the geometrical models of strand paths formed on the surface of cylindrical and conical mandrels having diamond, regular and triaxial braid patterns have been successfully developed and simulated with the aid of VRML. Subsequently, the ‘bottle’ and ‘funnel’-shaped mandrels were also simulated by combining the cylindrical and conical shapes in a systematic manner. In general, an excellent agreement has been observed between the theoretical and experimental values of braid angles determined along the height of bottle- and funnel-shaped mandrels. Furthermore, a virtual comparison between the trajectory of the strands simulated on the surface of conical mandrels having constant and varying braid angles has been clearly elucidated. This study has clearly formed the basis of simulating the complex-shaped mandrels using simple geometrical shapes (cylinder and cone). In the future, further complex-shaped preforms can be developed using various combinations of these simple geometrical shapes.

Footnotes

Funding

This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors.

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