Curvature-controlled trajectory planning for variable stiffness composite laminates

Composite Structures 238 (2020) 111986

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Curvature-controlled trajectory planning for variable stiffness composite laminates

T

⁎

Xuejuan Niua,b, , Yaxin Liua, Jinchao Wua, Tao Yanga a b

School of Mechanical Engineering, Tiangong University, Tianjin 300387, China Advanced Mechatronics Equipment Technology Laboratory, Tianjin 300387, China

A R T I C LE I N FO

A B S T R A C T

Keywords: Variable stiffness composite laminates Fiber tow placement Cubic B-spline Curvature correction Trajectory planning

Variable-stiffness laminates can redistribute the applied load and increase critical buckling loads compared to traditional straight fiber laminates. To take full advantage of fiber reinforced composite materials, a practical trajectory planning method is generated based on the maximum principle stress vector field. A reference path is represented as a blend curve of a sequence of uniform cubic B-spline segments passing through some given maxistress points. Based on the local-support property of each B-spline segment, subsequent paths within single lamina can be easily obtained by shifting the reference path along a specific direction. A fast localized curvaturecorrection algorithm is proposed to control the curvatures of the reference path and strictly constrain the void gap or overlap in a variable stiffness lamina. This trajectory planning method takes the requirement of automated fiber placement machines into account, and improves the mechanical properties of the variable stiffness composite laminates by decreasing the occurrence of gap-errors, such as buckling and wrinkling between adjacent paths. A practical case of variable stiffness trajectory planning is provided to demonstrate the feasibility and efficiency of the proposed method. In this practical case, the gap-error rate has decreased from 45.8% to 4.2%.

1. Introduction Fiber reinforced polymer composites (FRPCs) exhibit superior mechanical properties, as well as versatility and the potential for lightweight structures [1]. FRPCs have been considered as a substitute material for steel in automotive, transportation, aerospace, and marine applications. Automated fiber placement (AFP) has developed into an important technology to manufacture large-scale FRPCs, due to low cost and no need of tooling or mould [2]. In current industrial applications, AFP can place 2–32 individual fiber prepreg tows. This method can be used with both gantry style automatic machines and universal industrial robots equipped with an AFP head [3]. Robotic fiber placement technology also offers the ability of placing each individual tow along a custom-designed trajectory, which has the potential to fabricate structures with complex geometries or with variable stiffness (VS). Comparing with traditional constant stiffness (CS) angle-ply placement method, VS placement could place the fiber tow in continuously changing angle, which leads to different stiffness properties on different location. Curvilinear fiber path can redistribute load paths and adjust the in-plane stiffness distribution to increase the failure resistance [4]. Many researchers have studied the advantages of VS placement through

⁎

optimizing tow-paths for structures under different load conditions. Rasool et al. [5,6] investigated dynamic stability characteristics of VS composite panels subjected to periodic in-plane loads, compressive loads or shear loads. They’ve got a conclusion that the buckling capacity of VS laminates may be significantly higher than constant stiffness ones. By evaluating the sensitivity of the buckling load with respect to local change of stiffness of each element on the plate, Setoodeh [7] redistributed the global loads to maximize buckling performance of the plate. However, the manufacturing cost of automated fiber placement laminates has a dramatic increase due to the complexity of the trajectory planning and machine automatic control, compared with the traditional constant stiffness lay-up method [8]. To quickly fabricate a high quality and well–consolidated structure by curvilinear fiber placement, automated fiber placement technology faces the challenge how to reduce the time cost comply with composite structure’s quality requirements. Gürdal et al. [9] firstly proposed the concept of variable stiffness, and defined the fiber angles as linearly varying function along x and y axes. The linear variation method has been widely adopted in the analysis, design and manufacture processes of fiber reinforced composite structures [10]. Besides, direct parametrization of the tow paths

Corresponding author. E-mail address: [email protected] (X. Niu).

https://doi.org/10.1016/j.compstruct.2020.111986 Received 17 October 2019; Received in revised form 20 December 2019; Accepted 23 January 2020 Available online 28 January 2020 0263-8223/ © 2020 Elsevier Ltd. All rights reserved.

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2.2. Buckling and wrinkling phenomena of fiber tows

using parabolas [11], cubic polynomials [12], Lagrangian polynomials [13], splines [14,15] and flow field function [16,17] were also used to describe curvilinear fiber paths. However, the manufacturing of variable stiffness laminates does impose the constraint on the curvature of the fiber. Tow overlap or tow shrink of adjacent tows is inevitable if the curvature of curvilinear path is bigger than a threshold. Overlap or shrink of tows will cause the changes of the fiber volume fraction, thickness and dynamic response of the structure [18,19], which may even affect the mechanical balance properties of the structure. Wu [20] proposed a blended layup configuration method to produce “bucklefree” variable stiffness layers. In this paper, the structure and the requirements of fiber-tow placement process is introduced firstly. An innovative method is proposed to construct a reference path in Section 2.3, which take full advantage of fiber tow by aligning the fiber direction with the maximum principle stress field. Taking the curvature constrains of AFP machines and the requirements of manufacturing qualities in to consideration, a fast localized curvature-correcting algorithm is designed and applied on the reference path to smooth and blend the curvature distribution. In Section 3.1 and 3.2, the reference path is shifted with the same shifted method until all the paths covering the whole laminate are generated. With the curvature correction method, the gap-error rate of the variable stiffness laminate can be well controlled and the uniformity of the thickness of the laminate is well ensured. Finally, a practical case is investigated to demonstrate the effectiveness of the proposed trajectory planning method.

The fiber tows are made of unidirectional (UD) composite prepreg, and its width is much greater than its thickness. According to the beam mechanics theory of material mechanics, the outer side of the fiber tow is pulled while the inner side is compressed during the placing process. As a result, deformation of the fiber tow may cause defects, such as wrinkling and fiber buckling. Under certain heating or external loading conditions, the laminate may give rise to the stress concentrations near the contour of such defects [21]. The general placement methods of tows include tow-shifting method [5–7] and parallel-path method [8,9]. In both methods, a reference fiber path passing through the center of the panel should be created firstly. For tow-shifting method, subsequent fiber paths were obtained by shifting the reference path in a given direction, which means the fiber paths on the same lamina are not truly parallel. For parallel-path method, the adjacent fiber paths are parallel to the reference path, which means an adjacent path is defined as a set of points lying a constant distance from the reference curve [7]. This actually makes the fiber orientation change from path to path, and change in the vertical and the horizontal directions. The primary difference between the two methods is that each fiber path generated with tow-shifting method has an analytical expression while parallel fiber laminates don’t have. As shown in Eq. (1), the k th path, r k , can be obtained by simply shifting the former path, r k − 1, by a given distance along the direction perpendicular to the fiber.

r k = r k − 1 + k∙Δ

2.1. Requirement AFP process AFP technologies have been widely used in the aerospace structural components, such as aircraft wings, fuselage, control surfaces, etc. In our lab, the AFP procedure is completed by a 6-DOF industrial robot equipped with a tow-placement head (TPH) on the end-effector, as shown in Fig. 1. The TPH is equipped with a heat-pressure device for fixing and curing a fiber tow on to a mold. The heat-pressure device including a heating member, a pressure member, a fixing frame and a roller. The roller is surfaced with silicone rubber with excellent wear resistance and heat resistance. The individual tows, are fed through a tension system to the tow-placement head. The path width is determined by the number of individual tows, which is two in our machine. In practical applications, mainly three widths of tows: 3.17 mm, 6.35 mm and 12.7 mm. With the help of the flexibility of the industrial robot, fiber tows could be accurately pressured by appropriate compaction force acting by the roller along a proper direction. To ensure the placement quality, the axis of the TPH (z7) should be perpendicular to the plane of the plate, and the approaching direction of roller (x7) should agree with the tangent of the fiber tow path.

2.3. Generating the reference path represented as cubic B-spline 2.3.1. Initialize the reference path based on paths of stress trajectories The maximum-principal-stress criterion postulates that the growth of the crack will occur in a direction perpendicular to the maximum principal stress[21]. To make a full use of the unidirectional fiber composite, the fiber orientations in the VS laminate need to follow the paths of internal forces and stresses. The trajectory of the maximum principal stress, here we name it as “stress trajectories”, are obtained from the results of the FE model in Abaqus, and used to initialize the reference path. Here, stress trajectories are generated from a sequence of points whose maximum principal stress are the local peaks on the variable stiffness laminates. Without loss of generality, we evenly divide the long side of the composite laminate into n regions. Here, we denote the long side of the laminate as x - axis . Then the local peak of maximum principle stress on each region should be searched, and the corresponding point Vi (i = 0, ...,n) will be used as the trajectory points of the cubic curve. The polyline through all the trajectory points is defined as the stress trajectory of the laminate. For a given laminate, the stress trajectory is different for different load condition. Our target here is to interpolate n + 1 trajectory points

Fixing Frame

Tow #1

Compaction Roller

Plate Tow #2

y7

Placement Direction

(1)

where the superscript n denotes the number of the total paths on the lamina. Δ is the shifting parameter, which usually equal to the width of fiber tow. Due to the mathematical simplicity, tow-shifting method has been a widely adopted method for the trajectory planning of variable stiffness laminate. For the application of tow-shifting method in the curvilinear path generation of a fiber tow, there are many disadvantages, such as specific inherent defects induced by gaps, overlaps, buckling and wrinkling of tows. The effects of these defects on the structural performance of composite laminates have been elaborate [19]. To avoid the occurrence of gaps between tows, the shifting distance between two adjacent paths should be less than the tow width, while the defects, such as the overlapping, bubbles and wrinkling, have been introduced in the tow placements process as shown in Fig. 2. When the curvature of the curve path is smaller, the phenomena of overlapping and wrinkling can be more serious, which will cause the unbalanced stiffness of the laminate.

2. The requirements and constrains of fiber tow placement

z7

for k = 1, 2, ⋯, n

x7

Fig. 1. Structural diagram of tow-placement head (TPH) attached on an industrial robot. 2

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Fig. 2. Disadvantages of tow shifting: a) buckling phenomena, b) wrinkling phenomena.

Bubbles Overlap

Wrinkling

a)

b)

Trajectory Point

P3

P1

V2 V0

P0 0

r3 V3

Pi(0) = Vi ; ⎧ ⎨ Pi(k + 1) = Pi(k ) + δi = 1 (6Vi − Pi(−k )1 − Pi(+k )1). 4 ⎩

P4 Vn

2

3

4

…

(t ) denotes the ith uniwhere εp is the user-defined tolerance, and form cubic B-spline curve segment at the kth iteration. Constraining tangential direction on top of trajectorial points requires increasing the degree of freedom of the curve. Here, a new knot is inserted in the middle of each knot span, shown in Fig. 4. For each uniform cubic B-spline segment, the range of parameter u is [0,1]. So, the values of all the new added knots equal to 1 . Fig. 4 illustrates the 2 uniform B-spline curve interpolating the data points [V ] with control points [P ] as well as the refined control points [Q]. The refined control points [Q]j (j = 2i) are the control points of curve segment in term of u . 2 Then the curve segment is expressed in term of new knot parameter u as follow

Fig. 3. Interpolate stress trajectories by a uniform cubic B-spline curve.

Vi (i = 0, ...,n) together with the unit tangent vectors ti (i = 0, ...,n) by a uniform cubic B-spline curve, which is suitable for the automatic placement of AFP procedure. Generally, B-spline curves are defined by a sequence of control points. To define a cubic spline curve with n + 1 trajectory points, we need n + 3 control points named P0 to Pn + 2 , we can define a cubic curve segments ri (i = 1, ...,n) using groups of 4 consecutive points [P ]i = [ Pi − 1 Pi Pi + 1 Pi + 2 ]T , as shown in Fig. 3. Segments ri of the uniform B-spline curve are defined over a knot interval [0 , 1]. In this case, the kth segment can be written as

2

ri

u 2

[S ] =

(2)

3

⎡1 u u u ⎤ [M ][P ]i ⎣ 2 4 8⎦ = [1 u u2 u3 ][M ][S ][P ]i

(9)

⎡4 1 ⎢1 8 ⎢0 ⎢ ⎣0

4 6 4 1

0 1 4 6

0⎤ 0⎥ 0⎥ 1⎥ ⎦

(10)

() u 2

, which we called Then, the control points of the curve segment ri [Q]j the refined control points, [Q]j = [S ][P ]i can be constructed as follows:

0⎤ 0⎥ 0⎥ 1⎥ ⎦

(3)

1

⎛ 8 (4Pi − 1 + 4Pi ) ⎞ ⎡Qj − 1 ⎤ ⎜1 ⎟ (P + 6Pi + Pi + 1) ⎢ Qj ⎥ 8 i−1 ⎟ [Q]j = ⎢ ⎥ =⎜ 1 ⎜ ⎟ + (4 P 4 P ) ⎢Qj + 1 ⎥ + i i 1 8 ⎢Qj + 2 ⎥ ⎜1 ⎟ ⎜ (Pi + 6Pi + 1 + Pi + 2) ⎟ ⎦ ⎣ ⎝8 ⎠

In this study, this reference path should be constrained to pass through the given trajectory points Vi (i = 0, ...,n) , and the tangential direction of the curve at Vi should align with the direction of the maximum principal stress at the point. Mathematically, the interpolation of the reference path ri (u) at the given sequence of trajectory can be constrained as follows,

⎧ ri (0) = Vi r ̇ (0) ⎨ ∥ ri̇ (0) ∥ = ti, ⎩ i

2

( )=

where

where

4 1 ⎡ 1 1 −3 0 3 [M ] = ⎢ 6⎢ 3 −6 3 ⎢ ⎣− 1 3 − 3

(8)

ri(k )

n-2 n-1 n

ri (u) = [1 u u2 u3 ][M ][P ]i , i = 1, ...,n; u ∈ [0, 1]

(7)

∥ri(k ) (0) − Vi ∥ ⩽ εp

P2 1

i = 1, ...,n.

where the superscript (k ) denotes the k th iteration. We repeat this process until

rn V n-1 Vn-2 Pn+2 rn-1 Pn+1 Pn

r2

r1 V1

Then the sequence of the original control points Pi can be constructed iteratively using the formulas as follow:

Control Point

(11)

i = 0, ...,n. (4)

where the dot ‘·’ denotes differentiation with respect to parameter u . According to Eqs. (2)–(4), it can be derived that

Vi = ri (uk ) = [1 0 0 0][M ][P ]i =

1 (P 6 i−1

+ 4Pi + Pi + 1),

i = 0, ...,n.

(5)

Yamaguchi [22] uses Eq. (5) to derive

δi = Vi − Pi +

1 1 ⎛Vi − (Pi − 1 + Pi + 1)⎞ 2⎝ 2 ⎠

(6)

Fig. 4. Generation of refined control points constraining tangential direction. 3

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where j = 2i . Then, the trajectory point ri (0) and its first differentiation u ri̇ (0) can be expressed in term of the control points of ri 2 as follows:

()

1 ⎧ ri (0) = 6 (Qj − 1 + 4Qj + Qj + 1); ⎪ 1 ri̇ (0) = 2 (Qj + 1 − Qj − 1); ⎨

⎪ r¨i (0) = Qj + 1 − 2Qj + Qj + 1 ⎩

(12)

Substituting Eq. (12) into Eq. (4) yields 1

⎧ 6 (Qj − 1 + 4Qj + Qj + 1) = Vi , ⎨ ⎩

Qj + 1 − Qj − 1 ∥ Qj + 1 − Qj − 1 ∥

= ti ,

for i = 0, ...,n, j = 2i. (13) Fig.6. Diagram of the iterative algorithm for curvature correction.

Now we have obtained a cubic B-spline curve interpolating n + 1 data points V0, V1, ...,Vn together with the unit tangent vectors t0, t1, ...,tn . As the iterative algorithm mentioned above, Eq. (13) can equivalently be expressed as follow,

Edge Constrained

z

1

(k + 1) = 4 (6Vi − Qj(−k )1 − Qj(+k )1); ⎧ Qj i = 0, ...,n; j = 2i. ( 1) k + ⎨Qj + 1 = Qj(−k )1 + ∥Qj(+k )1 − Qj(−k )1∥·ti; ⎩

Shaft

y

Variable Stiffness Laminate Flange

x

(14)

The iteration process starts from the determination of [Pi] as shown in Fig. 4. To determine the end refined control points P0 and Pn , the virtual end data points V−1 and Vn + 1 must be computed beforehand. According to the second formula in Eq. (12) and the second formula in Eq. (14), the two virtual end data are calculated based on the formulas as follows,

⎧ V−1 = V1 − 2((V1 − V0)·t0) t0; ⎨ ⎩Vn + 1 = Vn − 1 + 2((Vn − Vn − 1)·tn ) tn.

Fig.7. The geometry and load condition of the designed FE model.

(15)

according to the third formula in Eq. (12). Then the trajectory points with small curvature, κi < κthr , are searched one by one along the initial reference path. 3) Based on the unique local support property of Bspline curves, curvature correction is conducted on the corresponding B-spline segment ri by adjusting the position of the corresponding control point. [Q]j = [Qj − 1 Qj Qj = 1]T , (j = 2i) For a planar curve used in this application, the curvature can be calculated with following equation,

Using the point interpolation algorithm introduced above, the control points [Pi] (i = 0, ...,n) are computed from the data points [Vi ] (i = −1, ...,n + 1). The virtual end control points P−1 and Pn + 1 are simply set to P−1 = V−1 and Pn + 1 = Vn + 1 as shown in Fig. 5, which gives the diagram of the generation of the refined control points [Qj] from the control points [Pi]. Applying Eq. (11) to Pi , (i = −1, ...,n + 1) yields Qj , (i = −1, ...,2n + 1) , and all the uniform cubic B-spline segments are defined.

κi = 2.3.2. Fast localized curvature correction algorithm The reference path r generated according to the interpolation algorithm described above is a free form curve. In order to avoid wrinkling and voids of the fiber tow, it is necessary to control the curvature of the curve. To constrain the curvature of the curve, a localized curvature-correction algorithm including following steps: 1) the threshold of the curvature, κthr , should be determined according to the properties of the fiber tow and the requirements of the quality of the fiber placement. 2) The curvature κi, (i = 0, ...,n) corresponding to each refined control points Qi , (i = 0, ...,n) should be calculated

(r ̇ (0) × r¨ (0))·ez ∥r ̇ (0) ∥3

(16)

where ez = [0 0 1]T ,and r ̇ (u), r¨ (u) are the first derivative and the second derivative of a curve r (u) respectively, and they can be calculated by using Eq. (12). Fig. 6 shows the iterative process of curvature correction. Our goal now is to decrease the curvature of a trajectory point Vi . The unit tangent vector of the curve on Vi is denoted as ti . By moving Qj(−k )1 and Qj(+k )1 → along the direction of Qj(−k )1Qj(+k )1, the curvature of curve segment ri is enlarged. With reference to Fig. 6, we denote three parameters here:

a = ∥Qj(−k +1 1) − Qj(−k )1∥ = ∥Qj(+k +1 1) − Qj(+k )1∥; b = ∥Qj(+k )1 − Qj(−k )1∥; c = ∥Qj(+k +1 1) − Qj(−k +1 1)∥ = 2a + b.

(17)

We now make use of Qj(−k )1 and Qj(+k )1 to define ti according to the second equation in Eq. (13).

ti =

1 (k ) (Qj + 1 − Qj(−k )1) b

(18)

Now, relying on Eq. (18), the following equation can be derived from Eq. (14),

Qj(k + 1) =

Fig. 5. Generation of refined cubic B-spline curve. 4

1 (6Vi − 2Qj(−k )1 − bti ) 4

(19)

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a)

b)

Fig.8. Initial reference path: a) initial reference path, b) distribution of curvature.

0.010 0.008 0.006

Curvature

0.004 0.002 0

--0.002 -0.004 -0.006 -0.008 -0.010 0 0

100

200

a)

300 0

400

x, mm

500

600

700

b)

Fig. 9. Curvature-controlled reference path: a) modified reference path, b) distribution of curvature.

800 700

7 Reference path p Remainder path

6 5 Dist ance betw een path s, mm

y, mm

600 500 400 300

4 3

200

2

100

1

0 0

100

200

300

400

x, mm

500

600

0 0

700

a)

100

200

300

400

x, mm

500

600

700

b)

Fig. 10. New fiber path shifted from reference path: a) position relationship of two paths, b) gap distribution.

→ moving distance of the control points ∥ Δ ∥ = d . The direction of the → vector Δ is along y − axis . With this method, all the remainder path could be generated and represented as B-spline curve.

3. Generation of subsequent paths based on the same shifted method 3.1. Obtain new path by shifting adjacent path

3.2. Using the golden section method to determine the big overlapped area

Here, the first new path r 1 (u) is obtained by shifting the control points of the reference path r 0 (u) along y − axis . Then the ith segment on the (k + 1)th path, rik + 1 (u) , is obtained by shifting the ith segment of the kth path, rik (u) , as shown below,

→ rik + 1 (u) = [1 u u2 u3 ][M ]([P ]ik + Δ), i = 0, ...,n; u ∈ [0, 1]

r k (u) represents the previous path, r k + 1 (u) represents the new path. The Euclidean distance from a discrete point qi (i = 1, 2…n) on the curve r k + 1 (u) to the previous curve r k (u) represented as,

(20)

Dh (u) = min ∥qi − qi∗∥

Denote the width of the fiber tow is d ,(d = 6.35mm in our lab), the

(21)

In order to calculate the distance, the golden section method is used 5

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800 700

6.8

Reference path Remainder path

6.6

y, mm

600 6

Dis tan ce bet we en pat hs, mm

500 5 400 4 300 200 100

0 0

6.4 6.2 6 5.8 5.6 5.4 5.2

100

200

300

400

x,mm

500

600

5 0

700

100

200

a)

300 400 x, mm

500

600

700

b)

Fig. 11. Curvature correction algorithm is used to correct the gap error phenomena: a) position relationship of two paths, b) gap distribution.

to search the corresponding point on the new curve r k + 1 (u) , who has the shortest distance to the given point on the previous r k (u) . The knot vector for each segment is [0, 1]. According to the golden section method, the knot value for each parameter space is shown as,

u = ui + 0.618ui + 1

be clearly seen that the gap between the two curves vary greatly, and there are many spots less than 6.35 mm in some areas, which further proves that the same shifted method will produce overlap of fiber tows. Fig. 10b shows the distribution of the gap between the two curves, which is obtained by using the golden section method. The gap between the two fiber tows ranges from 0.2052 mm to 6.8277 mm. If the threshold of tow gap, εd is set as 1mm , there are over 45.8% of the curve could not satisfied the requirement of ∥Dh − d∥ ⩽ εd . We call these areas as gap-error area, whose fiber-tow gap is too bigger or smaller. The percent of gap-error area is called as gap-error rate. Fig. 11 shows the curvature-controlled curve modified from the original shifted curve. The fast localized curvature-controlled algorithm described in Section 2.3.2 is used to correct the tow gap by adjusting the curvature of curve segments. The gap between the reference and the modified new curve ranges from 5.078 mm to 6.76 mm. Under the same threshold of tow gap, εd = 1mm , the gap-error rate has reduced to 4.2%. It can be easily found that the gap-error phenomenon is well controlled, which further confirms the practicality and effectiveness of the method.

(22)

After the corresponding knot values u is searched, corresponding point qi∗ on the curve r k + 1 (u) could be calculated according to Eq. (20). Then the distance Dh is calculated with the discrete method. To control the occurrence of the overlapping and void gap between two adjacent paths, the distance Dh should be constrained. The threshold of tow gap is denoted as εd . If ∥Dh − d∥ > εd , it means the overlap phenomena or the big void gap appeared between the two paths, and the curvature correcting process similar as the curvature correction of the reference path is needed to correct the distance between adjacent paths. By controlling the occurrence of the gap-error phenomena, the evenness of the thickness of the variable stiffness lamina over the whole laminate is improved greatly. This can greatly benefits the mechanical balance of the laminate structure and the load redistribution of variable stiffness laminate. This process of the generation and correction of a new subsequent fiber path continues until the shifted paths cover the whole area needed to be placed by the AFP machine.

4. Conclusion The manuscript presents an innovative trajectories planning method for variable stiffness laminate. By aligning the fiber with the direction of the maximum principal stress, this path planning takes full advantage of fiber reinforce composite material to enhance the mechanical property of laminate with no need for increasing the thickness. Based on the theory of uniform cubic B-spline, the original fiber path is interpolated from the given the maximum principal stress vectors, the trajectory points with given tangential vector. According to the local support property of cubic B-spline, a fast local curvature correction method is proposed, and applied in to curvature correction of the reference path to avoid the occurrence of buckling and wrinkling phenomena. By shifting the curvature-controlled reference path along y - axis , new fiber paths are generated one by one. By applying the fast local curvature correction method to the new paths, the gap-error phenomena between two adjacent paths are constrained. The wrinkling index of the modified reference path dropped by 38.2% than the original one by correcting the curvature of the reference path. The gap-error rate has decreased from 45.8% to 4.2% when the gap threshold εd is set to 1mm . All these guarantee the machinability of the designed variable stiffness lamina for APF machines, and improve the uniformity of the thickness and the stability mechanical properties of the laminate.

3.3. Case analysis This is a trajectory planning of a laminate with a central hole connecting a shaft through a flange, as shown in Fig. 7. The size of the laminate is 700mm × 700mm . The diameter of the central hole on the variable stiffness laminate is 28mm , and the outer diameter of the flange is 150mm . The diameter of position circle of the flange is 120mm . As shown in Fig. 7, two edge paralleling with x − axis on the laminate is constrained. The shaft is rotated by a torque of 1Nm . Based on the generation algorithm of reference path descripted in Section 2.3.1, the initial reference path is shown in Fig. 8a. The distribution of curvature along the initial reference path is shown in Fig. 8b, and the range of the initial path’s curvature is [-1.66e-2,1.03e-2]. The threshold of the curvature, κthr in Section 2.3.2, is set to 0.01 according to the requirements AFP machine in our lab. With the fast local curvature correction algorithm proposed in Section 2.3.2, the initial reference fiber path is modified and the curvature alone the modified reference path is shown in Fig. 9. The range of curvature alone the modified path is [-8.3e-3, 8.0e-3]. The wrinkling index of the modified reference path decreased by 38.2% than the original one. According to the curve shifting method descripted in Section 3.1, a new fiber path is generated by shifting the curvature-controlled reference path off 6.35 mm along y - axis , shown in Fig. 10a. The blue curve is the original path, and the red one is the new fiber path. It can

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to 6

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influence the work reported in this paper. [9]

Acknowledgment

[10]

The author(s) received financial support from the Natural Science Foundation of Tianjin (No: 18JCYBJC89000). The author(s) received financial support of China Scholarship Council.

[11] [12]

Appendix A. Supplementary data

[13]

Supplementary data to this article can be found online at https:// doi.org/10.1016/j.compstruct.2020.111986.

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