The composite elbow is an asymmetric rotating body with a relatively complex structure that can be divided into three parts, as shown in Fig. 1: a torus section BC and two cylindrical sections AB and CD.

A schematic diagram of the composite elbow winding machine is presented in Fig. 2. The machine consists of an elbow mandrel bracket (1), elbow mandrel (2), carriage assembly (3), servo motor for swinging the carriage (4), servo motor for generating vertical movement of the elbow mandrel (5), counterweight to the elbow mandrel (6), servo motor for rotating the elbow mandrel (7), and machine bed (8). The winding process of the composite elbow can be described as follows. When the cylindrical sections AB and CD are wound, the elbow mandrel is driven by the servo motor (5) to produce vertical linear movement along the machine bed (8), and the carriage assembly (3) is driven by the servo motor (4) and rotates around the elbow mandrel to complete the helical winding process of the cylindrical section. For winding the torus section BC, the composite elbow mandrel (2) is driven by the servo motor (7) and swings around center O while the carriage assembly (3), driven by the servo motor (4), rotates around the elbow mandrel (2) to complete the winding process of the torus section.

Traditional composite elbow winding mandrels usually have cylindrical sections (AB and CD) of equal length on both ends. The length of the cylindrical sections should not be too long, so that the elbow mandrel can rotate about its centerline at both ends, controlled by either a 4-or 5-axis computer-controlled winding machine. However, the 3-axis composite elbow winding machine does not have any cylindrical length requirements on either end of the elbow mandrel and the winding process can be completed by a 3-axis CNC winding machine, which reduces the cost and is more efficient. In this kind of winding machine, when swing radius R of the torus changes, the distance between the carriage assembly and the bed can be adjusted by the screw adjustment mechanism to ensure the geometric center O’ of the composite elbow coincides with the rotation center of the carriage assembly. The elbow mandrel is arranged vertically to ensure the carriage assembly rotates on the horizontal plane. This makes it easy to place the winding fiber tows and deal with problems in the winding process, however, it is also easy to collect excess resin during the winding process. Therefore, a counterweight weight must be added to counteract the weight of the elbow mandrel, which presents a disadvantage.

Generally, the elbow can be regarded as a part of the torus. In elbow section BC, shown in Fig. 1, the corresponding parameter expression of the torus r (φ, θ) is [11]

(1) r(φ,θ)={(R+rcosφ)cosθ,(R+rcosφ)sinθ,rsinφ}

where R is the bending radius of the torus; r is the radius of the torus; and φ and θ denote the angular coordinates along the meridional and parallel direction of the torus, respectively, φ ∈ (0, 2π), θ ∈ (0, π/2).

Considering a microelement on the torus of the elbow section, as shown in Fig. 3, we obtain.

(2) dφds=sin αr

(3) dθds=cos αR+rcosφ

where α is the winding angle.

According to differential geometry [12], the geodesic curvature of the torus k_g is

(4) kg=dαds+sin⁡φcos⁡αR+rcos⁡φ

When k_g is zero, from Eq. (4), the geodesic equation of the torus is

(5) dαds=−sinφcosαR+rcosφ

From Eqs. (2), (3), and (5), the geometric relationships dθ/dφ and dα/dφ can be defined as follows:

(6) dθdφ=rcos α(R+rcos φ)sin⁡α

(7) dαdφ=−rsin φcos α(R+rcos φ)sin⁡α

Then, integrating Eq. (6) yields the geodesic equation of the torus:

(8) (R+rcosφ)cos α=C (a constant)

If we set the ratio of bending radius and elbow radius as n = R/r, Eq. (8) can be reduced to

(9) (n+cosφ)cos α=C(a constant)

From Eq. (9), it can be seen that the geodesic equation on the torus is only related to the ratio of bending radius and torus radius n and independent of elbow size. If the initial winding angle α₀ is defined as the winding angle at the minimum bending radius R − r (φ = 180°) of the torus, α_m is the winding angle at the maximum bending radius R + r (φ = 0°) of the torus. In a circle of the torus, the change in winding angle is α₀ → α_m → α₀ and the winding trajectory on the torus is periodically and repeatedly extended. Fig. 4 shows the process of modifying the winding angle based on different n values in a cycle.

For special curved surfaces, like the torus, not all geodesic winding can cover the entire surface. To avoid bridging and slippage, the initial winding angle α₀ should satisfy the following equation:

(10) α0≥arctan⁡rR−r

Substituting n = R/r into Eq. (10), we obtain

(11) α0≥arctan⁡1n−1

From Eq. (11), variation of the minimum initial winding angle α₀ with n in the torus section of the composite elbow is shown in Tab. 1.

According to Eq. (9), the geometric parameter n of the composite elbow and the initial winding angle α₀ can be obtained as

(12) cos⁡α=(n−1)cos α0n+cos φ

In the composite elbow winding machine shown in Fig. 2, when the torus section is wound, the elbow mandrel swings around its center of curvature O and the winding eye continuously rotates on the winding plane of the carriage assembly. Since the swing trajectory of the elbow mandrel around the center of curvature O is exactly same for the torus forming process, a circle can be obtained by truncating the torus through the winding plane of the center of curvature O and its center O’ coincides with the center of the circle formed by the rotation of the winding eye. The combination of the swinging elbow mandrel and rotating winding eye can be used to complete the filament winding process of the torus section. In the winding process, letting(n − 1)cos α₀ = a (a constant) and substituting this into Eq. (7), the fiber trajectory motion equation of the torus section can be obtained as

(13) f(φ)=dθdφ=a(n+cos φ)(n+cos φ)2−a2

In the winding process of the torus section, the geometric relationship between the rotation angle φ of the winding eye corresponding to different n values and swing angle θ is shown in Fig. 5.

As shown in Fig. 1, cylindrical sections AB and CD are located at opposite ends of the elbow mandrel. According to the basic principle of filament winding, the winding angle should reach exactly 90° when passing the open cylindrical ends of the elbow. This means that non-geodesic trajectories should be used for the cylindrical end sections [13]. Thus, the fiber trajectory equations can be expressed as [14]

(14) Δy=rλ(1sin⁡α′−1sin⁡α)

(15) Δφ=ln⁡[tan⁡(α2)]−ln⁡[tan⁡(α′2)]λ

where Δy is the displacement along the axial line of the cylindrical sections AB and CD,

r is the radius of AB and CD,

λ is the slippage coefficient between the fibers and the mandrel,

α′ is the initial winding angle of AB and CD,

α is the final winding angle of AB and CD, and

Δ φ is the rotation angle of the elbow mandrel.

From Fig. 5, we can see that swing angle θ of the elbow mandrel basically changes in line with changes in the rotation angle φ of the winding eye as the ratio of bending radius to torus radius n increases. Integrating Eq. (13) across one circle of the torus, we obtain the swing angle θ_circle of the elbow mandrel for one circle of rotation angle φ of the winding eye, as follows:

(16) θcircle=∫02πa(n+cos φ)(n+cos φ)2−a2dφ

For the elbow shown in Fig. 2, swing angle θ of the elbow mandrel is 90 degree. When the elbow mandrel swings 90 degree, the total rotation angle φ_BC of the elbow mandrel can be given by

(17) φBC=π/2θcircle×360∘

From the differential terms dφ/ds in Eqs. (2) and (12), the total fiber trajectory length Scircle in one circle of rotation angle φ of the winding eye can be expressed as

(18) Scircle=∫02πr(n+cos⁡ φ)(n+cos φ)2−a2dφ

The optimized winding angle α’ can be given by

(19) α′=arcsin2πrScircle

The optimized winding angle α’ can be used as the initial winding angle for cylindrical sections AB and CD, which can each be divided into two sections: a helical section and a transition section. From Eq. (13), the lengths of the transition sections of cylindrical sections AB and CD, denoted L_1T and L_2T, can be defined as

(20) L1T=L2T=rλ(1sin⁡α′−1)

The helical section lengths of cylindrical sections AB and CD, denoted L_1H and L_2H, can be given as

(21) L1H=LAB−L1T

(22) L2H=LCD−L2T

From Eq. (15), the rotation angle of the winding eye in the transition sections of cylindrical sections AB and CD, denoted ϕ1T and ϕ2T , can be given as

(23) φ1T=φ2T=−ln⁡[tan⁡(α′2)]πλ×180∘

The rotation angles of the winding eye in the helical sections of cylindrical sections AB and CD are denoted φ1H and φ2H , and can be given as follows:

(24) φ1H=(L1−L1T)×tgα′πr×180∘

(25) φ2H=(L2−L2T)×tgα′πr×180∘

The total rotation angle φeye of the winding eye as the elbow mandrel swings back and forth is

(26) φeye=2φ1T+2φ1H+2φBC+2φ2H+2φ2T

For asymmetric complex parts such as composite elbows, use of the multiple tangent point method for filament winding will lead to fiber bridging on both dome ends. Therefore, the single tangent winding method is usually adopted. If the initial winding angle α₀ is given in a winding loop, the sum of the winding eye rotation angle in the torus section and the cylindrical sections are decided. As long as the dwell angle θ_S in the cylindrical sections is appropriately adjusted on both ends, the total rotation angle of the winding eye in a loop will be an integer multiple of 360° to obtain the single-tangent point winding process. The solution of ϕs is

(27) φs={[int(φeye360o)+1]×360o−φeye}÷2

The composite elbow is an asymmetric component and the winding angle is a function of the rotation angle of the winding eye. Among the latitude lines corresponding to the surface of the torus, φ = 0 is the longest latitude line, called the outer arc. During the winding process, if the fiber bands evenly cover the outer arc surface, they must also cover the entire torus surface. Assuming the swing angle increment of the elbow mandrel after one rotation circle of the winding eye is θcircle . From Eq. (12), winding angle α_m corresponding to φ = 0° is

(28) αm=arcos(n−1)cos α0n+1

The projection of a fiber band of width w onto the outer arc is w∕sinα_m. The number of fiber bands N_p required to cover the outer arc is

(29) Np=int(R+r)θcirclesin αmw+1

To obtain a fiber band offset, the rotation angle increment φw of the winding eye is

(30) φw=360ow2πrcos α′

With continuous advances in materials science, computer technology, modern control technology, artificial intelligence, and other related technologies, the traditional manufacturing industry is constantly evolving towards intelligent manufacturing. In recent years, due to continuous progress and improvements in motion control technology, motion control systems have become a mature technology and occupy an important position in the automation industry as independent industrial automation control products. Applications include decision-making in production processes, integrated manufacturing, friendly human-machine, and energy-saving and environmentally friendly processing. Indeed, intelligent manufacturing has become an important trend in factory automation, in which motion control technology also plays a major role, and will beat the core of advanced intelligent manufacturing in the future [15–20].

The filament winding machine proposed in this paper is a low-cost system with only three motion control axes, including swing motion of the elbow mandrel, linear motion of the elbow mandrel, and rotation of the mandrel. Three Panasonic servo motors of the A5 series ere used to produce the swing and linear movements of the elbow mandrel and rotation of the winding eye. To control the 3-axis motion of the machinein space, a PMAC-PC104 motion control card (fifth-generation, Delta Tau, USA) was selected. The motion control card uses advanced digital signal processing (DSP) technology, including Motorola 56K series, which is the most powerful motion controller in the world and an outstanding representation of current open CNC system controllers. The fast and accurate calculation ability is transformed into high-precision and fast motion trajectory calculation and control, which can be transformed into precise multi-axis motion trajectories via simple motion programs. The high performance, flexibility, ease of use, and low cost of the controller provide a complete technical solution for realizing the winding patterns of elbows. Once the 3-axis coordinate information is defined, the absolute coordinate mode or relative coordinate mode can be selected. In this study, the relative coordinate mode was used. To cover the surface of the composite elbow, N_p loops were required. The relationship between the motion of the elbow mandrel and the winding eye for the three motion axes of the PMAC-PC104 motion control card in each loopare shown in Tab. 2.

Based on the previous winding motion trajectory analysis, Advantech’s industrial control computer, PMAC motion control card and VC++ language were used to design a CAM system for composite elbow filament winding. The control system only requires knowing the length of cylindrical sections AB and CD, ratio n of bending radius R and radius r of the torus, initial winding angle α₀, fiber band width w, as inputs to wind the composite elbow.

Take the winding process of the composite elbow as an example, the length of cylindrical sections AB and CD were 400 mm and 150 mm, respectively, the ratio n was 10, the radius of the torus r was 22 mm, the initial winding angle α₀ was 51°, the fiber band width w was 5 mm, and the number of tangent points was 1. The steps of the filament-winding process for the elbow are as follows: (1) Enter the parameters in the parameter dialog box; (2) Calculate the stable fiber path to evenly cover the surface of the composite elbow; (3) Generate a control data file based on the 3-axis filament winding machine and download the file to the programmable multi-axis controller (PMAC) board program storage area; (4) Start the winding process of the composite elbow. The composite elbow filament winding pattern is shown in Fig. 6.