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Design and production of filament-wound composite square tubes
Accepted Manuscript Design and production of filament-wound composite square tubes Lei Zu, Hui Xu, Bing Zhang, Debao Li, Bin Zi, Bingzhan Zhang PII: DOI: Reference:
S0263-8223(18)30469-0 https://doi.org/10.1016/j.compstruct.2018.02.069 COST 9419
To appear in:
Composite Structures
Received Date: Revised Date: Accepted Date:
31 January 2018 13 February 2018 22 February 2018
Please cite this article as: Zu, L., Xu, H., Zhang, B., Li, D., Zi, B., Zhang, B., Design and production of filamentwound composite square tubes, Composite Structures (2018), doi: https://doi.org/10.1016/j.compstruct.2018.02.069
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Design and production of filament-wound composite square tubes Lei Zu a, Hui Xu a, Bing Zhang a,*, Debao Li a, Bin Zi a, Bingzhan Zhang b a
School of Mechanical Engineering, Hefei University of Technology, Hefei 230009, China b
School of Automotive and Transportation Engineering, Hefei University of Technology, Hefei 230009, China
Abstract: Composite square tubes have gained increasing attention as energy absorbers due to their high specific energy absorption capacity and long stroke. One of the key important issues for producing filament-wound composite square tubes demands both windability and uniform coverage of winding patterns. Based on the analytic geometry, the spatial relation between the feed eye and the mandrel was outlined and the kinematic equations for coupling the motion of the mandrel and the feed eye were derived. Consequently, a design method for small-angle winding of composite square tubes was proposed, taking the non-slippage condition of winding trajectories into account. A periodically geodesic winding theory was presented and its winding error for various initial winding points was analyzed. The designed fiber patterns were then applied to the practical production of a composite square tube with small winding angles. The results show that the present design method for filament-wound square tubes is accurate and reliable. The obtained kinematic equations and motion laws of the feed eye and the mandrel satisfy the basic winding principle and manufacturability of filament-wound composite square tubes. The present method is able to provide a useful tool for design and production of composite square tubes. Keywords:
Composite materials; Filament winding; Square tube; Small-angle geodesic; Windability
*Corresponding author. Tel.: +86 551 62901137; Fax: +86 551 62901137 E-mail address: [email protected] (B. Zhang).
1. Introduction The major challenge for manufacturing square cross-sectional tubes for aircraft frame structures is to make them light-weight so as to reduce the operational cost and improve the overall system requirements to make them an attractive option for aerospace industries. Filament winding has been widely used in producing and manufacturing polymeric composite structures [1-5]. Through the relative motion between the mandrel and the feed eye, the fiber yarn winds to the mandrel in accordance with certain rules to create a composite component. Filament-wound products have been widely used in many industrial fields because of their high specific strength & stiffness, high production efficiency, excellent vibration and corrosion resistance, and good fatigue behavior. The winding parameters of the manufacturing process can be deliberately designed to maximize the fiber strength according to the stress conditions of products. Aerospace frames made of composite structures possess significant crashworthy characteristics accompanied by weight and cost effectiveness. Composite square tubes have become a key component of load-carrying structures and energy-absorbing members in aerospace, civil and automobile applications due to their high specific energy absorption capacity, long stroke, and high specific bending stiffness. Moreover, tubes with square cross-sections possess better compatibility for connecting with other frame structures. Various methods and theories have focused on energy absorption and crashworthy characteristics of composite square tubes. Zhang et al. [6] investigated both experimentally and numerically the applicability and effectiveness of adopting buckling initiator for axially loaded square tubes. Oshkovra et al. [7] carried out a comprehensive finite element simulation to evaluate the response and analyzing energy absorption capacity of natural silk/epoxy square tubes. Mamalis et al. [8] studied the compressive properties and crushing response of square carbon FRP tubes subjected to static axial compression and impact testing using the LS-DYNA3D
explicit finite element code. Kalhor [9] tested hybrid square tubes made from S2 glass/epoxy composites and 304 stainless steel with different fiber orientation, stacking sequence, and thickness under quasi-static loading. Palanivelu [10] presented an experimental investigation on the progressive deformation behavior of uni-directional pultruded composite square and circular cross sectional tubes subjected to an axial impact load. Shin et al. [11] investigated energy absorption capability of axial crush and bending collapse of aluminum/GFRP hybrid square tubes. Mamalis et al. [12] reported the crashworthy behavior of square frusta of fibreglass composite material subjected to axial compression at various strain rates; they [13] also evaluated the behaviour and crashworthiness characteristics of square composite tubes subjected to static and dynamic axial compression exerted by a hydraulic press and a drop-hammer, respectively. However, previous research has a major defect in that manufacturing of composite square tubes was based on unidirectional pultrusion process, and the fiber alignments in the transverse direction of the tube were thus considered negligible. Despite the fact that pultruded square tubes subjected to uniaxial tension/compression do show great performance and manufacturability, the absence of fiber stiffness/strength in the transverse direction of the tubes proved to significantly confine their load-carrying capacity. A typical example of this restriction is the limit for designing a composite square tube under transverse bending moment or shear forces. Hence, filament winding process has emerged as an attractive alternative to improve the transverse structural performance as well as to reduce the tube weight. Many researchers have so far concentrated on design and manufacturing of filament wound products. Zu et al. [14-15] developed non-geodesically overwound toroidal and domed pressure vessels; they also determined the optimal shapes for filament-wound bellow-shapes [16] and pressure vessels with unequal polar openings [17]. Vasiliev et al. [18] presented the optimality conditions for a pressure vessel and derived the optimal dome profiles for various
anisotropy parameters. Liang et al. [19] outlined the optimal dome contours based on the maximum shape factor and evaluated the effect of the dome depth on the structural performance. Teng et al. [20] investigated the effect of the fiber band width on the stability of the winding patterns and carried out the optimal design for an ellipsoid dome. Vafaeesefat et al. [21] presented a multi-level strategy for the optimization of composite pressure vessels with geodesic and ellipsoidal heads. Hojjati et al. [22] assessed the effect of mechanical properties of composite materials on the dome profiles and proved that the matrix properties have a major role in the dome design. Fukunaga et al. [23] obtained optimal meridian shapes using several failure criteria and outlined an analytic approach for the design of dome structures. Braun et al. [24] analyzed the process parameters of carbon fiber overwound seamless-aluminum-lined composite pressure vessels. Seereerem et al. [25] investigated the all-geodesic manufacture of a T-shaped form. Nevertheless, very few investigations have been devoted to design of filament-wound composite square cross-sectional shapes. Up to date a complete theory for designing and obtaining the winding patterns and the feed eye movement of the square tube remains scarce. Moreover, since the load in the transverse direction of the tube is generally much lower than that in the longitudinal direction, the small-angle fiber orientations referred to the longitudinal axis are required for achieving a rational and optimal stress distribution. The small-angle winding of square tubes is more likely to cause fiber slippage on the tube mandrel as compared to the winding of circular cross-sections due to non-rationally asymmetric cross-sectional shape of the square tube that leads to uneven motion trajectories of the feed eye and the mandrel [26]. Based on the elaboration of the relative motion of the feed eye and the contact points on the mandrel, we formulate a non-slippage-based design theory and successfully achieve the geodesic winding on the mandrel of the square tube. The kinematic equations of the feed eye and the mandrel are then derived for the whole winding circuit. In
addition, a special yarn-hanging device is installed to ensure that the small-angle fiber trajectories will return back at the end of the tube and smoothly transit to the next winding circuit. Finally, a composite square tube is produced using the small-angle winding to verify the feasibility and the accuracy of the present design method.
2. Kinematic equations of the feed eye with small-angle winding Assuming that the mandrel does not move and the feed eye spirally moves around the mandrel, we determine the kinematic equations of the feed eye using the theory of relative motion. 2.1 Spatial relations of the feed eye and the contact point Compared with the general cylindrical winding, filament-wound square tubes share a complex relationship to the feed eye and the contact points; thus, we build a 3D analysis model. The starting point is fixed on the vertex of the mandrel, and the initial yarn and mandrel surfaces are on the same plane. During winding, the mandrel rotates and the feed eye moves horizontally along the longitude of the mandrel. The trajectories of the yarn and the feed eye at different times are illustrated in Fig. 1.
2.2 Approximate circular method The so-called approximate circular method is here used for calculating the kinematic equations of the feed eye, as shown in Fig. 2. The square cross-sectional tube with length a can be approximately regarded as a circular tube of radius R, both of which have the same circumference. The length of a chord connecting the both intersections of the approximate circle and the square edge is a', making a central angle β. The following equations hold:
4a 2 R
(1)
a ' 2R sin
(2)
Since the fiber path has an angle α with respect to the parallel circle of the torus, the relation among geometric parameters dθ/dl, dφ/dl and α, can be given by (see Fig. 2): and the approximate circle winding angle can be calculated as: a' ) Z
(3)
2 R 180 tan
(4)
' arctan( Z
' arctan(
2 sin 90 tan
)
(5)
The relationship of φ and φ' is depicted in Fig. 3.
2.3 Calculation of geometry to prevent yarn slippage During the winding of square tubes, the winding angle calculated using the approximate circle method deviates from the true winding angle. Slippage becomes highly likely when the contact point transits from one mandrel plane to another. Geodesic winding is the most stable in the winding process. Therefore, to prevent yarn slippage, the triangular envelope for the projections of various fibers in the X-Y plane should comprise a series of similar triangles. As shown in Fig. 4, the feed eye moves from position 1, and the triangular envelope for the projections of the yarn in the X-Y plane is in the plane of the mandrel surface. When the feed eye reaches position 2, the former and the latter are not in the same plane. To avoid yarn slippage, the plane including the triangular envelope for the projections of the yarn in the X-Y plane should be folded to the plane of the mandrel surface while keeping a certain angle φ, as shown in Fig. 5. In the X-Y plane shown in Fig. 6, the relatively kinematic angle of the feed eye increases by 90° from β0 to β0'. At this point, the position of the feed eye corresponds to position 1. The following winding process is repeated until winding ends. We call this repetitive process a
cycle. In the cycle, Lt is the length of mandrel, a is the side length of the cross section, A is the distance of the feed eye to the center shaft of the mandrel, φ is the winding angle, and M(x0, y0) is the vertex of the mandrel. The distance of the feed eye ahead is given by
x x0 2 y y0 2 z z0 2 ctg x x0 2 y y0 2 ctg A2 x02 y02 2 Ax0 cos y0 sin
Z cos
(6)
3. Determination of the virtual initial point We previously calculated the kinematic equation for the starting point of the mandrel vertex. However, the mandrel vertex is not always the starting point during winding. The initial turn-around angle can be expressed as β0=arccos(a/2A), β0'=β0+90°. The relation between the turn-around angle and Y-coordinate of the feed eye is shown in Fig. 7. Y is the moving distance of the feed eye in winding machine coordinates, and X is the radian for the turn-around angle. If the starting points are not from M, the feed eye moves at the first plane of the mandrel surface and then enters into the circulation. Fig. 7 shows that the turn-around angle is not in proportional to the motion of the feed eye along the Y-coordinate; this indicates that the kinematic equations of the feed eye and the mandrel are varied for different initial winding point. Therefore, formulating a motion control theory for the feed eye for an arbitrary initial point is the key to the success of the winding. When winding begins at an initial point different from the mandrel vertex, the oppositely elongated line of geodesic trajectory and the line of the mandrel edge intersect at the virtual starting point. As shown in Fig. 8, the geodesic equations for the contact point starting from point M are equal to the equations after point M, which starts from point M'. We then achieve geodesic winding from any starting point by calculating the virtual starting point for different
actual starting points. We take a cycle as a unit and explore the starting point in a cycle. If the distance between the point M and the mandrel edge is ax , then the distance between the virtual starting point and the mandrel is L1 , which is expressed as L1=axctgφ. Similarly, when the yarns are wound back, the contact points start from the virtual starting point on the oppositely elongated line of the mandrel edge. The distance between the virtual starting point and the mandrel is L2 , which is expressed as L2=NL0-Lt-L1. N is a positive integer that meets the requirements of 0
tan 1 tan
ii N '
2
(7)
(8)
N' is a positive integer irrelevant to N and meets the requirement of 0≤β<π/2, and i is the number of cycles for the feed eye to make a round trip and has a maximum value. From the requirements of uniform coverage, we can derive the equation ibcosϕ=4a, where b is the width of the yarn. L1 ax ctg a
tan ctg 1 tan
(9)
Therefore, the mandrel turns a certain radian π/2 when the initial turn-around angle θi is certain, iθi is increased, and N' is changed. Then, we calculate the virtual starting point on the basis of the calculated value of β and realize the geodesic winding from any actual starting point. The flow-process diagram is shown in Fig. 10.
4. Design of winding patterns The winding patterns reflect the alignment law of the continuous fiber trajectories on the mandrel surface. We set the mandrel turn-around angle with the round trip of the feed eye as the symbol of the winding patterns. Continuous fibers start at a certain point on the mandrel and wind for several cycles before finally returning to the starting point. We call this winding process a complete cycle. A complete cycle has to satisfy the following conditions: (1) The location of two sequential tangent points is adjacent. The polar location has one tangent point before the appearance of the tangent point adjacent to the location of the starting point. We call this winding pattern the single tangent point. (2) The location of two sequential tangent points is segregative. The polar location has two or more tangent points before the appearance of a tangent point adjacent to the location of the starting point. This is called the multi-tangent point. Each bundle of yarn corresponds to a tangent point on the circumference of the polar location. Yarns are uniformly distributed piece by piece when every tangent point equally divides the rotated angle of the mandrel and the staggered distance for the yarns in the cylinder corresponding to the adjacent tangent points is equal to the width of the yarn. The rotation angle of a round trip of the feed eye can be expressed as:
n (
K N '') 360 n n
(10)
where n is the tangent point, N'' represents a non-negative integer, K is the integer inside the scope of 1 to n-1, K/n is a simple proper fraction, and Δθ is the turn-around degree for the yarn sheet distance in the cylinder corresponding to the adjacent tangent points. According to Eq. (10), the values of n, K, N, and θn corresponding to various winding patterns are shown in Table 1.
Table 1. Winding patterns for S0, n, K, N, and θn. n
K
0
1
2
3
4
5
6
7
8
N 1
1
360°
720°
1080°
1440°
1800°
2160°
2520°
2880°
3240°
2
1
180°
540°
900°
1260°
1620°
1980°
2340°
2700°
3060°
1
120°
480°
840°
1200°
1560°
1920°
2280°
2640°
3000°
2
240°
600°
960°
1320°
1680°
2040°
2400°
2760°
3120°
1
90°
450°
810°
1170°
1530°
1890°
2250°
2610°
2970°
3
270°
630°
990°
1350°
1710°
2070°
2430°
2790°
3150°
1
72°
432°
792°
1152°
1512°
1872°
2232°
2592°
2952°
2
144°
504°
864°
1224°
1584°
1944°
2304°
2664°
3024°
3
216°
576°
936°
1296°
1656°
2016°
2376°
2736°
3096°
4
288°
648°
1008°
1368°
1728°
2088°
2448°
2808°
3168°
3
4
5
After determining the tangent points, we assign the mandrel an initial angle θi to meet the requirements of uniform coverage in a complete cycle. 1 1 (11) 2 2 2 where θL is the one-way turn-around angle for the virtual length of the tube. When the
n 2(i L ) , i n L n N
filament-wound patterns are certain, we select the proper value of θn and calculate the initial turn-around angle θi. Finally, we obtain the virtual starting point and finish the winding.
5. Simulations of winding patterns on a square tube The results of winding deviate from the truth because of winding error. Thus, simulations of theoretical winding patterns are necessary to provide a theoretical guidance for the actual winding and a reference for the final result of winding. With the aid of MATLAB that can calculate mathematical equations and graphic simulations, we build an entity model of the
square tube in the dimensional coordinate. Then, according to the requirements of the winding parameters, we calculate the linear equations on the model of the square tube. Finally, we formulate the linear equations of the trajectory on the square tube. On this account, we explore the theoretical winding patterns of the square tube in geodesic winding. The simulation of the winding patterns on a mandrel of the square tube is shown in Fig. 11.
6. Production for filament-wound composite square tubes The numerical control system of the winding device SINUMERIK 810D is here used to produce a composite square cross-sectional shape. We control the rotation of the spindle, the linear motion of the winding trolley, and the arm movement of the feed eye by writing procedure codes. From the system settings of the winding machine, X stands for the rotated radian of the spindle, Y stands for the linear kinematic distance of the winding trolley, and Z stands for the arm-moving distance of the feed eye. The winding trolley moves a distance ΔY when the spindle rotates a radian Δx. A deliberately designed and produced fiber-hanging device was installed at the ends of the square mandrel, in order to ensure a smooth transition to the subsequent wound circuit when the fibers return back at the ends of the mandrel. At the beginning of winding, the mandrel rotates at a degree of θi and the yarn is fixed on the rod of the mandrel. The whole winding process has five tangent points, and the yarns entirely cover the mandrel after the feed eye completes 19 round trips. The winding condition is shown in Fig. 12.
7. Conclusions In this paper, we study the winding patterns of composite square tubes with small winding angles. Considering the relatively kinematic principle, studying the movement of the feed eye, and analyzing the geometrical relationship of the feed eye and the mandrel, we meet the
requirements of non-slippage winding and implement the feed eye movement of geodesic winding. Using the segmented cycle method and turning the center angle by 90° as a cycle, we obtain the uniform equation of the feed eye motion for the small-angle winding-composite square tubes. In addition, we achieve the winding uniformity of the composite square tubes by controlling the NC winding machine. The results show almost no slippage during winding. At the end of the winding process, the yarn uniformly spreads over the whole mandrel. However, this paper does not cover the thickness uniformity of the square sections because of the incomplete theory regarding the measure and control of section thickness during winding. This topic will be the focus of our follow-up work. The geodesic calculation of composite tubes in this paper shows the correct practice of theory. The results enrich the geodesic calculation theory of the traditional surface for filament-wound production. It also provides a reference for the geodesic calculation theory of the profiled products with filament winding. In the future, the structural performance (stiffness & strength) of the pultruded square tubes and filament-wound square tubes will be evaluated and compared to each other, in order to demonstrate that the filament winding process leads to preferable performance of composite square tubes subjected to transversely distributed loads.
Acknowledgements This research is supported by the National Natural Science Foundation of China (Grant No. 11302168).
References [1] Wilson B. Filament winding: the jump from aerospace to commercial frame. SAMPE Journal, 1997, 33(3): 25.
[2] Wang R, Jiao W, Liu W, Yang F. Dome thickness prediction of composite pressure vessels by a cubic spline function and finite element analysis. Polym Polym Compos 2011, 19: 227-34. [3] Wang R, Jiao W, Yang F. A new method for predicting dome thickness of composite pressure vessels. J Reinf Plast Compos 2010; 29: 3345-52. [4] Zu L, Koussios S, Beukers A. Integral design for filament-wound composite pressure vessels. Polym Polym Compos 2011; 19: 413-420. [5] Zu L, Koussios S, Beukers A. Optimal cross sections for filament-wound toroidal hydrogen storage vessels based on continuum lamination theory. Int J Hydrogen Energy 2010; 35: 10419-29. [6] Zhang XW, Su H, Yu TX. Energy absorption of an axially crushed square tube with a buckling initiator. International Journal of Impact Engineering 2009; 36: 402-417. [7] Oshkovr SA, Taher ST, Oshkour AA, et al. Finite element modelling of axially crushed silk/epoxy composite square tubes. Compos Struct 2013 ; 95: 411-418. [8] Mamalis AG, Manolakos DE, Ioannidis MB, et al. Crashworthy characteristics of axially statically compressed thin-walled square CFRP composite tubes: experimental. Compos Struct 2004; 63: 347-360. [9] Kalhor R, Case SW. The effect of FRP thickness on energy absorption of metal-FRP square tubes subjected to axial compressive loading. Compos Struct 2015; 130: 44-50. [10] Palanivelu S, Van Paepegem W, Degrieck J, et al. Experimental study on the axial crushing behaviour of pultruded composite tubes. Polymer testing 2010; 29: 224-234. [11] Shin KC, Lee JJ, Kim KH, et al. Axial crush and bending collapse of an aluminum/GFRP hybrid square tube and its energy absorption capability. Compos Struct 2002; 57: 279-287. [12] Mamalis A G, Manolakos D E, Demosthenous G A, et al. Energy absorption capability of
fibreglass composite square frusta subjected to static and dynamic axial collapse. Thin-Walled Structures, 1996, 25(4): 269-295. [13] Mamalis A G, Manolakos D E, Demosthenous G A, et al. The static and dynamic axial crumbling of thin-walled fibreglass composite square tubes. Composites Part B: Engineering, 1997, 28(4): 439-451. [14] Zu L, Koussios S, Beukers A. Design of filament-wound toroidal hydrogen storage vessels based on non-geodesic fiber trajectories. Int J Hydrogen Energy 2010; 35: 660-70. [15] Zu L, Koussios S, Beukers A. Design of filament-wound domes based on the continuum theory and non-geodesic roving trajectories. Compos Part A: Appl Sci Manuf 2010; 41: 1312-20. [16] Zu L, Koussios S, Beukers A. Shape optimization of filament wound articulated pressure vessels based on non-geodesic trajectories. Compos Struct 2010; 92: 339-46. [17] Zu L, Koussios S, Beukers A. Design of filament-wound isotensoid pressure vessels with unequal polar openings. Compos Struct 2010; 92: 2307-13. [18] Vasiliev VV, Krikanov AA. New generation of filament-wound composite pressure vessels for commercial applications. Compos Struct 2003; 62(3): 449-59. [19] Liang CC, Chen HW, Wang CH. Optimum design of dome contour for filament-wound composite pressure vessels based on a shape factor. Compos Struct 2002; 58(4): 469-82. [20] Teng TL, Yu CM, Wu YY. Optimal Design of Filament-Wound Composite Pressure Vessels. Mech Compos Mater 2005; 41: 333-40. [21] Vafaeesefat A, Khani A. Head Shape and Winding Angle Optimization of Composite Pressure Vessels Based on a Multi-level Strategy. Appl Compos Mater 2007, 14 (5-6): 379-91. [22] Hojjati M, Safavi AV, Hoa SV. Design of domes for polymeric composite pressure
vessels. Compos Eng 1995; 5(1): 51-59. [23] Fukunaga H, Uemura M. Optimum design of helically wound composite pressure vessels. Compos Struct 1983; 1(1): 31-49. [24] Braun CA, Haddock RC. Manufacturing process controls for high reliability carbon filament wound seamless aluminum lined composite pressure vessels. AIAA, SAE, ASME, and ASEE, Joint Propulsion Conference and Exhibit, 28th, Nashville, TN, July 6-8, 1992. 9 [25] Seereerem S, Wen JTY. An all-Geodesic algorithm for filament winding of a T-shaped form. IEEE Transactions on Industrial Electronics 1991; 38: 484-490. [26] Mazumdar SK, Hoa SV. Analytical models for low cost manufacturing of composite components by filament winding, part I: direct kinematics. Journal of Composite Materials 1995, 29(11): 1515-1541.
List of Figures Fig.1. Spatial model of the feed eye and the mandrel. Fig.2. Approximate circle in the X-Y plane. Fig.3. Deviation of the approximate circular method. Fig.4. Geometrical relations of various fiber positions and the mandrel. Fig.5. Triangular envelope for the projections of various fibers in the X-Y plane. Fig.6. Geometrical relations of the X-Y plane and the mandrel. Fig.7. Y-coordinates of the feed eye with turn-around angles of the mandrel. Fig.8. Virtual starting point corresponding to the actual starting point. Fig.9. Relationship between the turn-around angle β and the length ax. Fig. 10. Flow diagram of finding the virtual starting point. Fig. 11. Simulation of winding patterns for a square cross-sectional tube: (a) a square tube; (b) fiber trajectories at the junction of the adjacent mandrel edges; (c) Uniform coverage patterns on the mandrel. Fig. 12. Winding process of a square tube with small-angle geodesics: (a) feed eye and the mandrel at the beginning of winding; (b) fiber trajectories for two winding circuits; (c) uniform and full coverage.
S0263-8223(18)30469-0 https://doi.org/10.1016/j.compstruct.2018.02.069 COST 9419
To appear in:
Composite Structures
Received Date: Revised Date: Accepted Date:
31 January 2018 13 February 2018 22 February 2018
Please cite this article as: Zu, L., Xu, H., Zhang, B., Li, D., Zi, B., Zhang, B., Design and production of filamentwound composite square tubes, Composite Structures (2018), doi: https://doi.org/10.1016/j.compstruct.2018.02.069
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Design and production of filament-wound composite square tubes Lei Zu a, Hui Xu a, Bing Zhang a,*, Debao Li a, Bin Zi a, Bingzhan Zhang b a
School of Mechanical Engineering, Hefei University of Technology, Hefei 230009, China b
School of Automotive and Transportation Engineering, Hefei University of Technology, Hefei 230009, China
Abstract: Composite square tubes have gained increasing attention as energy absorbers due to their high specific energy absorption capacity and long stroke. One of the key important issues for producing filament-wound composite square tubes demands both windability and uniform coverage of winding patterns. Based on the analytic geometry, the spatial relation between the feed eye and the mandrel was outlined and the kinematic equations for coupling the motion of the mandrel and the feed eye were derived. Consequently, a design method for small-angle winding of composite square tubes was proposed, taking the non-slippage condition of winding trajectories into account. A periodically geodesic winding theory was presented and its winding error for various initial winding points was analyzed. The designed fiber patterns were then applied to the practical production of a composite square tube with small winding angles. The results show that the present design method for filament-wound square tubes is accurate and reliable. The obtained kinematic equations and motion laws of the feed eye and the mandrel satisfy the basic winding principle and manufacturability of filament-wound composite square tubes. The present method is able to provide a useful tool for design and production of composite square tubes. Keywords:
Composite materials; Filament winding; Square tube; Small-angle geodesic; Windability
*Corresponding author. Tel.: +86 551 62901137; Fax: +86 551 62901137 E-mail address: [email protected] (B. Zhang).
1. Introduction The major challenge for manufacturing square cross-sectional tubes for aircraft frame structures is to make them light-weight so as to reduce the operational cost and improve the overall system requirements to make them an attractive option for aerospace industries. Filament winding has been widely used in producing and manufacturing polymeric composite structures [1-5]. Through the relative motion between the mandrel and the feed eye, the fiber yarn winds to the mandrel in accordance with certain rules to create a composite component. Filament-wound products have been widely used in many industrial fields because of their high specific strength & stiffness, high production efficiency, excellent vibration and corrosion resistance, and good fatigue behavior. The winding parameters of the manufacturing process can be deliberately designed to maximize the fiber strength according to the stress conditions of products. Aerospace frames made of composite structures possess significant crashworthy characteristics accompanied by weight and cost effectiveness. Composite square tubes have become a key component of load-carrying structures and energy-absorbing members in aerospace, civil and automobile applications due to their high specific energy absorption capacity, long stroke, and high specific bending stiffness. Moreover, tubes with square cross-sections possess better compatibility for connecting with other frame structures. Various methods and theories have focused on energy absorption and crashworthy characteristics of composite square tubes. Zhang et al. [6] investigated both experimentally and numerically the applicability and effectiveness of adopting buckling initiator for axially loaded square tubes. Oshkovra et al. [7] carried out a comprehensive finite element simulation to evaluate the response and analyzing energy absorption capacity of natural silk/epoxy square tubes. Mamalis et al. [8] studied the compressive properties and crushing response of square carbon FRP tubes subjected to static axial compression and impact testing using the LS-DYNA3D
explicit finite element code. Kalhor [9] tested hybrid square tubes made from S2 glass/epoxy composites and 304 stainless steel with different fiber orientation, stacking sequence, and thickness under quasi-static loading. Palanivelu [10] presented an experimental investigation on the progressive deformation behavior of uni-directional pultruded composite square and circular cross sectional tubes subjected to an axial impact load. Shin et al. [11] investigated energy absorption capability of axial crush and bending collapse of aluminum/GFRP hybrid square tubes. Mamalis et al. [12] reported the crashworthy behavior of square frusta of fibreglass composite material subjected to axial compression at various strain rates; they [13] also evaluated the behaviour and crashworthiness characteristics of square composite tubes subjected to static and dynamic axial compression exerted by a hydraulic press and a drop-hammer, respectively. However, previous research has a major defect in that manufacturing of composite square tubes was based on unidirectional pultrusion process, and the fiber alignments in the transverse direction of the tube were thus considered negligible. Despite the fact that pultruded square tubes subjected to uniaxial tension/compression do show great performance and manufacturability, the absence of fiber stiffness/strength in the transverse direction of the tubes proved to significantly confine their load-carrying capacity. A typical example of this restriction is the limit for designing a composite square tube under transverse bending moment or shear forces. Hence, filament winding process has emerged as an attractive alternative to improve the transverse structural performance as well as to reduce the tube weight. Many researchers have so far concentrated on design and manufacturing of filament wound products. Zu et al. [14-15] developed non-geodesically overwound toroidal and domed pressure vessels; they also determined the optimal shapes for filament-wound bellow-shapes [16] and pressure vessels with unequal polar openings [17]. Vasiliev et al. [18] presented the optimality conditions for a pressure vessel and derived the optimal dome profiles for various
anisotropy parameters. Liang et al. [19] outlined the optimal dome contours based on the maximum shape factor and evaluated the effect of the dome depth on the structural performance. Teng et al. [20] investigated the effect of the fiber band width on the stability of the winding patterns and carried out the optimal design for an ellipsoid dome. Vafaeesefat et al. [21] presented a multi-level strategy for the optimization of composite pressure vessels with geodesic and ellipsoidal heads. Hojjati et al. [22] assessed the effect of mechanical properties of composite materials on the dome profiles and proved that the matrix properties have a major role in the dome design. Fukunaga et al. [23] obtained optimal meridian shapes using several failure criteria and outlined an analytic approach for the design of dome structures. Braun et al. [24] analyzed the process parameters of carbon fiber overwound seamless-aluminum-lined composite pressure vessels. Seereerem et al. [25] investigated the all-geodesic manufacture of a T-shaped form. Nevertheless, very few investigations have been devoted to design of filament-wound composite square cross-sectional shapes. Up to date a complete theory for designing and obtaining the winding patterns and the feed eye movement of the square tube remains scarce. Moreover, since the load in the transverse direction of the tube is generally much lower than that in the longitudinal direction, the small-angle fiber orientations referred to the longitudinal axis are required for achieving a rational and optimal stress distribution. The small-angle winding of square tubes is more likely to cause fiber slippage on the tube mandrel as compared to the winding of circular cross-sections due to non-rationally asymmetric cross-sectional shape of the square tube that leads to uneven motion trajectories of the feed eye and the mandrel [26]. Based on the elaboration of the relative motion of the feed eye and the contact points on the mandrel, we formulate a non-slippage-based design theory and successfully achieve the geodesic winding on the mandrel of the square tube. The kinematic equations of the feed eye and the mandrel are then derived for the whole winding circuit. In
addition, a special yarn-hanging device is installed to ensure that the small-angle fiber trajectories will return back at the end of the tube and smoothly transit to the next winding circuit. Finally, a composite square tube is produced using the small-angle winding to verify the feasibility and the accuracy of the present design method.
2. Kinematic equations of the feed eye with small-angle winding Assuming that the mandrel does not move and the feed eye spirally moves around the mandrel, we determine the kinematic equations of the feed eye using the theory of relative motion. 2.1 Spatial relations of the feed eye and the contact point Compared with the general cylindrical winding, filament-wound square tubes share a complex relationship to the feed eye and the contact points; thus, we build a 3D analysis model. The starting point is fixed on the vertex of the mandrel, and the initial yarn and mandrel surfaces are on the same plane. During winding, the mandrel rotates and the feed eye moves horizontally along the longitude of the mandrel. The trajectories of the yarn and the feed eye at different times are illustrated in Fig. 1.
2.2 Approximate circular method The so-called approximate circular method is here used for calculating the kinematic equations of the feed eye, as shown in Fig. 2. The square cross-sectional tube with length a can be approximately regarded as a circular tube of radius R, both of which have the same circumference. The length of a chord connecting the both intersections of the approximate circle and the square edge is a', making a central angle β. The following equations hold:
4a 2 R
(1)
a ' 2R sin
(2)
Since the fiber path has an angle α with respect to the parallel circle of the torus, the relation among geometric parameters dθ/dl, dφ/dl and α, can be given by (see Fig. 2): and the approximate circle winding angle can be calculated as: a' ) Z
(3)
2 R 180 tan
(4)
' arctan( Z
' arctan(
2 sin 90 tan
)
(5)
The relationship of φ and φ' is depicted in Fig. 3.
2.3 Calculation of geometry to prevent yarn slippage During the winding of square tubes, the winding angle calculated using the approximate circle method deviates from the true winding angle. Slippage becomes highly likely when the contact point transits from one mandrel plane to another. Geodesic winding is the most stable in the winding process. Therefore, to prevent yarn slippage, the triangular envelope for the projections of various fibers in the X-Y plane should comprise a series of similar triangles. As shown in Fig. 4, the feed eye moves from position 1, and the triangular envelope for the projections of the yarn in the X-Y plane is in the plane of the mandrel surface. When the feed eye reaches position 2, the former and the latter are not in the same plane. To avoid yarn slippage, the plane including the triangular envelope for the projections of the yarn in the X-Y plane should be folded to the plane of the mandrel surface while keeping a certain angle φ, as shown in Fig. 5. In the X-Y plane shown in Fig. 6, the relatively kinematic angle of the feed eye increases by 90° from β0 to β0'. At this point, the position of the feed eye corresponds to position 1. The following winding process is repeated until winding ends. We call this repetitive process a
cycle. In the cycle, Lt is the length of mandrel, a is the side length of the cross section, A is the distance of the feed eye to the center shaft of the mandrel, φ is the winding angle, and M(x0, y0) is the vertex of the mandrel. The distance of the feed eye ahead is given by
x x0 2 y y0 2 z z0 2 ctg x x0 2 y y0 2 ctg A2 x02 y02 2 Ax0 cos y0 sin
Z cos
(6)
3. Determination of the virtual initial point We previously calculated the kinematic equation for the starting point of the mandrel vertex. However, the mandrel vertex is not always the starting point during winding. The initial turn-around angle can be expressed as β0=arccos(a/2A), β0'=β0+90°. The relation between the turn-around angle and Y-coordinate of the feed eye is shown in Fig. 7. Y is the moving distance of the feed eye in winding machine coordinates, and X is the radian for the turn-around angle. If the starting points are not from M, the feed eye moves at the first plane of the mandrel surface and then enters into the circulation. Fig. 7 shows that the turn-around angle is not in proportional to the motion of the feed eye along the Y-coordinate; this indicates that the kinematic equations of the feed eye and the mandrel are varied for different initial winding point. Therefore, formulating a motion control theory for the feed eye for an arbitrary initial point is the key to the success of the winding. When winding begins at an initial point different from the mandrel vertex, the oppositely elongated line of geodesic trajectory and the line of the mandrel edge intersect at the virtual starting point. As shown in Fig. 8, the geodesic equations for the contact point starting from point M are equal to the equations after point M, which starts from point M'. We then achieve geodesic winding from any starting point by calculating the virtual starting point for different
actual starting points. We take a cycle as a unit and explore the starting point in a cycle. If the distance between the point M and the mandrel edge is ax , then the distance between the virtual starting point and the mandrel is L1 , which is expressed as L1=axctgφ. Similarly, when the yarns are wound back, the contact points start from the virtual starting point on the oppositely elongated line of the mandrel edge. The distance between the virtual starting point and the mandrel is L2 , which is expressed as L2=NL0-Lt-L1. N is a positive integer that meets the requirements of 0
tan 1 tan
ii N '
2
(7)
(8)
N' is a positive integer irrelevant to N and meets the requirement of 0≤β<π/2, and i is the number of cycles for the feed eye to make a round trip and has a maximum value. From the requirements of uniform coverage, we can derive the equation ibcosϕ=4a, where b is the width of the yarn. L1 ax ctg a
tan ctg 1 tan
(9)
Therefore, the mandrel turns a certain radian π/2 when the initial turn-around angle θi is certain, iθi is increased, and N' is changed. Then, we calculate the virtual starting point on the basis of the calculated value of β and realize the geodesic winding from any actual starting point. The flow-process diagram is shown in Fig. 10.
4. Design of winding patterns The winding patterns reflect the alignment law of the continuous fiber trajectories on the mandrel surface. We set the mandrel turn-around angle with the round trip of the feed eye as the symbol of the winding patterns. Continuous fibers start at a certain point on the mandrel and wind for several cycles before finally returning to the starting point. We call this winding process a complete cycle. A complete cycle has to satisfy the following conditions: (1) The location of two sequential tangent points is adjacent. The polar location has one tangent point before the appearance of the tangent point adjacent to the location of the starting point. We call this winding pattern the single tangent point. (2) The location of two sequential tangent points is segregative. The polar location has two or more tangent points before the appearance of a tangent point adjacent to the location of the starting point. This is called the multi-tangent point. Each bundle of yarn corresponds to a tangent point on the circumference of the polar location. Yarns are uniformly distributed piece by piece when every tangent point equally divides the rotated angle of the mandrel and the staggered distance for the yarns in the cylinder corresponding to the adjacent tangent points is equal to the width of the yarn. The rotation angle of a round trip of the feed eye can be expressed as:
n (
K N '') 360 n n
(10)
where n is the tangent point, N'' represents a non-negative integer, K is the integer inside the scope of 1 to n-1, K/n is a simple proper fraction, and Δθ is the turn-around degree for the yarn sheet distance in the cylinder corresponding to the adjacent tangent points. According to Eq. (10), the values of n, K, N, and θn corresponding to various winding patterns are shown in Table 1.
Table 1. Winding patterns for S0, n, K, N, and θn. n
K
0
1
2
3
4
5
6
7
8
N 1
1
360°
720°
1080°
1440°
1800°
2160°
2520°
2880°
3240°
2
1
180°
540°
900°
1260°
1620°
1980°
2340°
2700°
3060°
1
120°
480°
840°
1200°
1560°
1920°
2280°
2640°
3000°
2
240°
600°
960°
1320°
1680°
2040°
2400°
2760°
3120°
1
90°
450°
810°
1170°
1530°
1890°
2250°
2610°
2970°
3
270°
630°
990°
1350°
1710°
2070°
2430°
2790°
3150°
1
72°
432°
792°
1152°
1512°
1872°
2232°
2592°
2952°
2
144°
504°
864°
1224°
1584°
1944°
2304°
2664°
3024°
3
216°
576°
936°
1296°
1656°
2016°
2376°
2736°
3096°
4
288°
648°
1008°
1368°
1728°
2088°
2448°
2808°
3168°
3
4
5
After determining the tangent points, we assign the mandrel an initial angle θi to meet the requirements of uniform coverage in a complete cycle. 1 1 (11) 2 2 2 where θL is the one-way turn-around angle for the virtual length of the tube. When the
n 2(i L ) , i n L n N
filament-wound patterns are certain, we select the proper value of θn and calculate the initial turn-around angle θi. Finally, we obtain the virtual starting point and finish the winding.
5. Simulations of winding patterns on a square tube The results of winding deviate from the truth because of winding error. Thus, simulations of theoretical winding patterns are necessary to provide a theoretical guidance for the actual winding and a reference for the final result of winding. With the aid of MATLAB that can calculate mathematical equations and graphic simulations, we build an entity model of the
square tube in the dimensional coordinate. Then, according to the requirements of the winding parameters, we calculate the linear equations on the model of the square tube. Finally, we formulate the linear equations of the trajectory on the square tube. On this account, we explore the theoretical winding patterns of the square tube in geodesic winding. The simulation of the winding patterns on a mandrel of the square tube is shown in Fig. 11.
6. Production for filament-wound composite square tubes The numerical control system of the winding device SINUMERIK 810D is here used to produce a composite square cross-sectional shape. We control the rotation of the spindle, the linear motion of the winding trolley, and the arm movement of the feed eye by writing procedure codes. From the system settings of the winding machine, X stands for the rotated radian of the spindle, Y stands for the linear kinematic distance of the winding trolley, and Z stands for the arm-moving distance of the feed eye. The winding trolley moves a distance ΔY when the spindle rotates a radian Δx. A deliberately designed and produced fiber-hanging device was installed at the ends of the square mandrel, in order to ensure a smooth transition to the subsequent wound circuit when the fibers return back at the ends of the mandrel. At the beginning of winding, the mandrel rotates at a degree of θi and the yarn is fixed on the rod of the mandrel. The whole winding process has five tangent points, and the yarns entirely cover the mandrel after the feed eye completes 19 round trips. The winding condition is shown in Fig. 12.
7. Conclusions In this paper, we study the winding patterns of composite square tubes with small winding angles. Considering the relatively kinematic principle, studying the movement of the feed eye, and analyzing the geometrical relationship of the feed eye and the mandrel, we meet the
requirements of non-slippage winding and implement the feed eye movement of geodesic winding. Using the segmented cycle method and turning the center angle by 90° as a cycle, we obtain the uniform equation of the feed eye motion for the small-angle winding-composite square tubes. In addition, we achieve the winding uniformity of the composite square tubes by controlling the NC winding machine. The results show almost no slippage during winding. At the end of the winding process, the yarn uniformly spreads over the whole mandrel. However, this paper does not cover the thickness uniformity of the square sections because of the incomplete theory regarding the measure and control of section thickness during winding. This topic will be the focus of our follow-up work. The geodesic calculation of composite tubes in this paper shows the correct practice of theory. The results enrich the geodesic calculation theory of the traditional surface for filament-wound production. It also provides a reference for the geodesic calculation theory of the profiled products with filament winding. In the future, the structural performance (stiffness & strength) of the pultruded square tubes and filament-wound square tubes will be evaluated and compared to each other, in order to demonstrate that the filament winding process leads to preferable performance of composite square tubes subjected to transversely distributed loads.
Acknowledgements This research is supported by the National Natural Science Foundation of China (Grant No. 11302168).
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List of Figures Fig.1. Spatial model of the feed eye and the mandrel. Fig.2. Approximate circle in the X-Y plane. Fig.3. Deviation of the approximate circular method. Fig.4. Geometrical relations of various fiber positions and the mandrel. Fig.5. Triangular envelope for the projections of various fibers in the X-Y plane. Fig.6. Geometrical relations of the X-Y plane and the mandrel. Fig.7. Y-coordinates of the feed eye with turn-around angles of the mandrel. Fig.8. Virtual starting point corresponding to the actual starting point. Fig.9. Relationship between the turn-around angle β and the length ax. Fig. 10. Flow diagram of finding the virtual starting point. Fig. 11. Simulation of winding patterns for a square cross-sectional tube: (a) a square tube; (b) fiber trajectories at the junction of the adjacent mandrel edges; (c) Uniform coverage patterns on the mandrel. Fig. 12. Winding process of a square tube with small-angle geodesics: (a) feed eye and the mandrel at the beginning of winding; (b) fiber trajectories for two winding circuits; (c) uniform and full coverage.