Non-geodesic trajectory design method of iterative friction coefficient

Abstract

The invention discloses a non-geodesic trajectory design method of an iterative friction coefficient, and belongs to the technical field of material forming, a winding trajectory is obtained by solving a non-geodesic equation through a value of an initial condition, and then whether a phase difference of a winding trajectory initial position of a winding cycle meets a target phase difference is calculated; and the phase difference is adjusted by iterating a friction coefficient, so that a winding track is shifted to a certain extent in space on the basis of closing, a first basic cycle is obtained, and then expanded winding is performed, so that a continuous and complete winding effect can be formed. According to the method, the optimal friction coefficient is automatically determined through the numerical optimization technology, the winding track can be well closed and uniformly distributed while the reasonable friction coefficient is ensured, and then the product quality can be improved.

Description

Technical Field

[0001] This invention belongs to the field of material forming technology, specifically a non-geodesic trajectory design method for iterative friction coefficient. Background Technology

[0002] Fiber-reinforced composite materials are now widely used in aerospace, pressure vessels, hydrogen storage cylinders, pipelines, and energy equipment. Fiber winding technology is an important technology for manufacturing fiber-reinforced composite materials. It typically uses a winding machine to drive the mandrel and fiber guiding mechanism to move relative to each other, so that continuous fibers are wound on the surface of the mandrel according to a predetermined trajectory while impregnated with resin. The composite material product is then obtained after curing.

[0003] In fiber winding technology, geodesic winding is the earliest applied winding method. Geodesic winding refers to winding the fiber along the shortest path between two points on the mandrel surface. This method can avoid fiber slippage and bridging, the trajectory calculation is relatively simple, and the technology is relatively mature. However, the fiber trajectory of geodesic winding is determined by the mandrel geometry and initial winding conditions, making it difficult to optimize the design of the trajectory and meet the requirements of certain special structures.

[0004] To improve the designability of fiber placement, a non-geodesic winding technique has been proposed. This technique utilizes the friction between the fiber and the mandrel to deviate the fiber from the geodesic path, thereby optimizing the fiber trajectory and improving structural performance. However, slippage is prone to occur during actual winding. In the non-geodesic winding design process, if the friction coefficient is not properly selected, it can easily lead to difficulties in closing the winding cycle, and the uniformity of fiber distribution on the mandrel surface cannot be guaranteed, thus affecting the overall quality and performance stability of the product.

[0005] Currently, there are two main methods for obtaining the coefficient of friction between fibers and mandrels: one is to directly measure it experimentally and derive the maximum slip coefficient by combining the force balance relationship and geometric conditions; the other is to design a mandrel of a specific shape, so that its radius gradually changes along the axial direction, and the winding angle remains constant at 90° during the winding process. As the radius of the mandrel decreases, the slip coefficient gradually increases. The maximum slip coefficient is determined by recording the instant position of fiber slippage using high-speed photography equipment.

[0006] However, the above methods still have certain shortcomings. For example, the method of directly measuring the coefficient of friction often only provides results suitable for theoretical verification or research on influencing factors due to differences between experimental conditions and actual winding conditions, and it is difficult to accurately reflect the friction state in the actual winding process. On the other hand, the method of using a special mandrel for sliding wire testing requires high-precision design and fabrication of the mandrel, resulting in a long test preparation cycle and high costs.

[0007] Therefore, existing technologies usually require extensive experimental verification to determine the friction coefficient during fiber winding, which is not only time-consuming and labor-intensive, but also results in mostly fixed friction coefficients that are difficult to adapt to the effects of different fiber materials, core mold surface conditions and process parameters. Summary of the Invention

[0008] The purpose of this invention is to provide a non-geodesic trajectory design method for iterative friction coefficients to solve the problems mentioned in the prior art.

[0009] A method for designing non-geodesic trajectories with iterative friction coefficients is provided, comprising the following steps: S1: Construct the error function according to the following formula: F(μ) = φ n -φ target n is a natural number; Where φ n φ represents the actual landing phase of the non-geodetic equation f(μ). target For the target phase; S2: Set two initial friction coefficients μ0 and μ1; S3: The friction coefficient is iterated using the following formula: μ n =μ n-1 -F(μ) n-1 ) / k n-1 , n≥2; Where k n-1 Let the error function F(μ) be in the interval [μ] n-2 μ n-1 The slope of the chord on the chord; S4: Solve for μ n In the actual landing point phase φ of the non-geodetic equation f(μ) n Judgment error |φ n -φ target | Check if the convergence condition is met. If not, proceed to step S3.

[0010] As a further aspect of the present invention: in step S1, the target phase φ target Obtained from the following formula: φ target = (360° + B) degree P direction ) / N Skip; Among them B degree P represents the total phase angle of the yarn. direction is the direction coefficient, N is the number of tangent points, and Skip is the skipped tooth index.

[0011] As a further aspect of the present invention: the total phase angle B of the yarn belt degree Obtained from the following formula: B degree = (B effect / R) (180° / π) / Skip; B effect =B / cosα; Among them, B effect B is the effective yarn width, α is the actual yarn width, R is the winding angle, and Skip is the skip tooth indexing.

[0012] As a further aspect of the present invention: In step S3, the slope of the chord is obtained according to the chord-tangent method, specifically by the following formula: K n-1 =(F(μ) n-1 )-F(μ n-2 )) / (μ n-1 -μ n-2 ), n≥2.

[0013] As a further aspect of the present invention: in step S3, a design friction coefficient μ is set based on experience. design The two initial friction coefficients μ0 and μ1 are selected based on the following formula: μ0=max(0.01, μ design -μ interval ); μ1=min(0.99, μ design +μ interval ); Where, μ interval μ is the spacing gradient value. interval The range is [0.01, 0.50].

[0014] As a further aspect of the present invention: μ interval The range is [0.01, 0.10].

[0015] As a further aspect of the present invention: μ interval The range is [0.01, 0.05].

[0016] As a further aspect of the present invention: μ interval The range is [0.01, 0.02].

[0017] As a further aspect of the present invention: in step S3, a design friction coefficient μ is set based on experience. design The two initial friction coefficients μ0 and μ1 are selected based on the following formula: μ0=max(0.01, μ design -0.01); μ1=min(0.99, μ design +0.01).

[0018] As a further aspect of the present invention: boundary protection is performed based on the following formula after friction coefficient iteration: μ n =max(0.001,min(0.999,μ n )), n≥2.

[0019] As a further aspect of the present invention: when the variable values ​​output by the previous two iteration steps are the same, the iteration step is terminated and the friction coefficient located in the range of (0.001, 0.999) at the last iteration is output as the final value.

[0020] Compared with the prior art, the beneficial effects of the present invention are as follows: 1. This design method takes the target phase as the core of optimization and uses the landing phase of the winding trajectory as the basis for determining the friction coefficient iteration. It can automatically adjust the friction coefficient according to the actual winding conditions, achieve precise linear closure control, and avoid the complexity brought about by the trial and error method.

[0021] 2. By constructing an error function, a mathematical relationship is established between the friction coefficient and the actual landing point phase and the target phase, transforming trajectory optimization into a numerical optimization problem. The optimal friction coefficient is obtained automatically through numerical optimization techniques, ensuring that the friction coefficient is reasonable while the winding trajectory can also be well closed and evenly distributed.

[0022] 3. The chord-tangent method is adopted as the core of the iteration. The slope of the chord is determined by initial two points, and the friction coefficient is gradually corrected through iteration. This avoids the complexity of derivative calculation and ensures the convergence speed of the iteration. The iteration termination condition is whether the phase error is within the design accuracy range, ensuring that the final friction coefficient can achieve precise trajectory closure. Attached Figure Description

[0023] To more clearly illustrate the technical solutions in the embodiments of this drawing or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this drawing. For those skilled in the art, other drawings can be obtained based on the structures shown in these drawings without creative effort.

[0024] Figure 1 This is a flowchart of the non-geodesic trajectory design method of the present invention; Figure 2 A simulation diagram of the winding trajectory obtained using the non-geodetic trajectory design method of the present invention; Figure 3 for Figure 2A magnified view of a portion of region A in the middle. Detailed Implementation

[0025] To make the objectives, technical solutions, and advantages of this application clearer, the application is described and illustrated below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the scope of this application. All other embodiments obtained by those skilled in the art based on the embodiments provided in this application without inventive effort are within the scope of protection of this application.

[0026] Obviously, the accompanying drawings described below are merely some examples or embodiments of this application. Those skilled in the art can apply this application to other similar scenarios based on these drawings without any inventive effort. Furthermore, it is understood that although the efforts made in this development process may be complex and lengthy, for those skilled in the art related to the content disclosed in this application, any changes to design, manufacturing, or production based on the technical content disclosed in this application are merely conventional technical means and should not be construed as insufficient disclosure of the content of this application.

[0027] However, there may be instances where unnecessary detailed descriptions are omitted. For example, detailed descriptions of well-known matters or repetitive descriptions of essentially the same structures may be omitted. This is to avoid unnecessarily lengthy descriptions and to facilitate understanding by those skilled in the art. Furthermore, the accompanying drawings and the following description are provided to enable those skilled in the art to fully understand this application and are not intended to limit the subject matter of the claims.

[0028] Please see Figure 1 As shown in the embodiment of the present invention, a non-geodesic trajectory design method for iterative friction coefficient includes the following steps: Step 1: Collect key winding-related parameters, including mandrel geometry parameters, yarn parameters, winding process parameters, and process constraint parameters. Derive core derived parameters from the basic parameters. Combine the 360° basic circumferential coverage value and bandwidth angle increment adjustment term to calculate the total phase angle, ensuring complete circumferential coverage of the trajectory and accurate offset of adjacent yarns. Based on parameters such as the total phase angle, number of tangent points, and skip tooth indexing, calculate the target phase φ using a practical non-geodesic mathematical model. target .

[0029] Step 2: Based on the actual landing point phase φ n Phase φ with the target target The difference is used to construct an error function: F(μ) = φ n -φ target .

[0030] Step 3: Set two initial friction coefficients, μ0 and μ1. Calculate the error between the actual impact point phase and the target phase based on the error function: F(μ0) = φ0 - φ target ; F (μ1) = φ1 - φ target ; The chord slope k1 is calculated based on the error function value of the two initial friction coefficients and the difference in friction coefficients.

[0031] Step 4: Derive the new friction coefficient μ2 using the chord slope, the formula is: μ2 = μ1 - F(μ1) / k1; Substituting the corrected μ2 into the non-geodetic differential equation, we solve for the corresponding actual impact phase φ2. We then calculate the absolute error between φ2 and the target phase: |φ2 - φ target If the error is less than the preset convergence tolerance threshold, the iteration is considered to have converged, and μ2 is output as the optimal friction coefficient; if the error does not meet the threshold, the iteration continues.

[0032] Step 5: Calculate the previous friction coefficient μ n-2 μ n-1 As the new initial point, calculate the new chord slope k. n-1 k n-1 For the error function in [μ n-2 μ n-1 The slope of the chord in the interval. Calculate the new coefficient of friction μ using the following formula. n : μ n =μ n-1 -F(μ n-1 ) / k n-1 , n≥3; Solve for μ n The corresponding actual landing phase φ n Calculate the phase error and perform convergence determination again. Repeat the above operation until the phase error meets the standard or the number of iterations reaches the preset maximum value (to avoid infinite iteration), and finally output the optimal friction coefficient.

[0033] Step 6: Generate a winding trajectory that meets the process requirements using the optimal friction coefficient. Verify the fiber coverage of the base trajectory to confirm there are no overlaps, gaps, slippage, or gaps. Using the base cycle trajectory as a reference, extend the yarn of subsequent winding cycles closely following the previous cycle trajectory with a similar line pattern to generate a continuous multi-layered non-geodesic winding trajectory.

[0034] It should be noted that traditional methods use empirical or fixed values ​​for the friction coefficient, which cannot be adapted to the landing phase based on actual process parameters. This is the root cause of poor trajectory closure and uneven fiber distribution. This invention uses the landing phase as the optimization target and criterion for iterative friction coefficient selection, transforming the selection of the friction coefficient from empirical trial and error into scientific calculation aimed at achieving precise phase.

[0035] Furthermore, the landing phase of the winding trajectory used in this invention is the angular phase difference between the starting point and the endpoint position on the circumference of the mandrel's pole hole after the yarn completes a full winding cycle. If the landing phase matches the target value, the yarn can accurately connect to the next cycle after completing one cycle, achieving linear closure, and ensuring uniform coverage without overlap or gaps between adjacent yarns. If the landing phase deviation is too large, winding misalignment will occur, resulting in the winding line not being able to close, local fiber accumulation or gaps, directly affecting the structural strength and quality of the composite material product.

[0036] In one specific embodiment, in step 3, the initial friction coefficients μ0 and μ1 are substituted into the non-geodesic differential equations and solved by numerical integration (such as the fourth-order Runge-Kutta method or the DOP853 method) to obtain the winding trajectory of a complete winding cycle for each. The actual landing point phases φ0 and φ1 of the two trajectories are extracted. The actual landing point phase is taken as the circumferential angle of the last point at the end of the winding cycle, converted into an angle value, and normalized to the range [0, 360°).

[0037] Specifically, the generation of non-geodesic winding trajectories is determined by non-geodesic differential equations, and the friction coefficient is the core parameter of these equations. Different friction coefficients will yield different winding trajectories, corresponding to different actual landing point phases.

[0038] Therefore, this invention establishes a mathematical relationship between the friction coefficient and the actual landing point phase and the target phase by constructing an error function. This transforms the physical problem of designing a non-geodesic winding trajectory that meets the closure requirement into a numerical optimization problem of finding the optimal friction coefficient μ that makes the error function approach zero. This allows the trajectory deviation to be calculated quantitatively rather than qualitatively, providing a clear numerical basis for subsequent iterative optimization. Since the error function F(μ) is a nonlinear function of friction coefficient-phase error, and the derivative of F(μ) cannot be directly solved analytically (non-geodesic differential equations have no explicit solutions), the slope of the chord is used instead of the derivative.

[0039] The chord-tangent method is a numerical iterative optimization method that does not require calculating the derivative of a function. This invention applies the chord-tangent method to the iterative solution of the friction coefficient. Two physically reasonable initial friction coefficients, μ0 and μ1, are set and substituted into the non-geodetic equations to solve for the corresponding actual impact point phases φ0 and φ1, thereby obtaining the chord slope. This slope is the average rate of change of the error function in this interval and can be used as a numerical approximation of the derivative of the error function.

[0040] Compared to the complexity of directly calculating derivatives, the chord-tangent method avoids complex derivative calculations for non-geodetic differential equations, significantly reducing the difficulty of engineering calculations and making it more suitable for practical industrial applications. Specifically, the chord-tangent method iteratively corrects the friction coefficient to approach its optimal value. Based on the chord slope, a new friction coefficient is calculated using an iterative formula. This new coefficient is then substituted into the non-geodetic equation to solve for the new actual impact phase, completing one iteration. Each iteration dynamically adjusts the correction magnitude based on the previous calculation results, continuously bringing the friction coefficient closer to the optimal value that makes the error function approach zero.

[0041] In one embodiment, the slope of the first line is calculated using the following formula.

[0042] K1=(F(μ1)-F(μ0)) / (μ1-μ0); By using the error function values ​​corresponding to the two initial friction coefficients, the average rate of change of the error function in that interval is calculated and used as a numerical approximation of the derivative. This slope directly determines the direction and magnitude of the friction coefficient correction in the first iteration. That is, the sign of the slope reflects the trend of the error function changing with the friction coefficient (if the slope is positive, an increase in the friction coefficient will increase the actual landing point phase), and the absolute value of the slope reflects the rate of change, providing a scientific mathematical basis for the first iteration and avoiding blind correction.

[0043] The following formula is used to perform iterative calculation of the line slope, that is, to dynamically calculate the chord slope using the friction coefficient and error value of the first two iterations, instead of the fixed slope; K n-1 =(F(μ) n-1 )-F(μ n-2 )) / (μ n-1 -μ n-2 ), n≥3; The dynamic slope can adapt to the changing characteristics of the error function in different friction coefficient ranges (the error function is not linear, and the rate of change is different in different ranges), so that the correction direction and magnitude of each subsequent iteration are in line with the actual situation of the current friction coefficient range, ensuring the convergence and accuracy of the iteration, and avoiding iteration divergence or slow convergence caused by a fixed slope.

[0044] Furthermore, the secant method (chord-tangent method) used in this invention is a derivative-free variant of Newton's iterative method. The core of its convergence lies in approximating the tangent of the error function with a chord, gradually approaching the zero point of the error function through linear interpolation, even when F(μ) = 0. At this point, μ is the optimal friction coefficient. By continuously correcting μ, F(μ) gradually approaches 0 from its initial non-zero value, achieving phase error convergence.

[0045] Specifically, in step 4, the first iteration of the friction coefficient is completed using the following formula: μ2 = μ1 - F(μ1) / k; If the convergence condition is not met in the first iteration, the friction coefficient is iterated multiple times using the following formula: μ n =μ n-1 -F(μ) n-1 ) / k n-1 , n≥3; The essence of the iterative formula is to find the analytical solution of the zero point of the error function under linear approximation. Here, -F(μ1) / k1 is the friction coefficient adjustment amount for the first iteration, -F(μ n-1 ) / k n-1 This is the adjustment amount for the friction coefficient in subsequent iterations. The core of the adjustment amount lies in using the error function in μ. n The tangent line at a given point approximates the function curve, and the intersection of the tangent line and the x-axis is the next iteration point.

[0046] The adjustment amount is the core of convergence. The negative sign of the adjustment amount determines the direction of adjustment, and the absolute value determines the magnitude of adjustment. Both work together to ensure that μ moves towards the optimal value. The adaptive nature of the adjustment magnitude makes the iteration process fast at first and then slow down. In the early stage, it quickly reduces large errors, and in the later stage, it achieves fine convergence, ensuring both efficiency and accuracy.

[0047] The subsequent iterative formula uses k n-1 The dynamic updated chord slope (k) instead of a fixed k is used to adapt to the nonlinear characteristics of the error function and ensure continuous convergence of the iteration. Specifically, in practice, the error function F(μ) is not strictly linear (the relationship between μ and phase error is affected by the nonlinearity of the non-geodesic equation). If the initial slope k is used throughout the process, the excessive deviation of the linear approximation in the later stages of the iteration will lead to errors in the adjustment direction and amplitude, or even divergence.

[0048] Secondly, k n-1 The new slope is calculated using μ and F(μ) from the previous iteration. Each iteration uses the two most recent points to calculate the slope, adapting to the changing characteristics of F(μ) within the current μ interval. Even if F(μ) is nonlinear, it can be approximated as linear in a small local range, ensuring the correctness of the adjustment direction and magnitude. As the number of iterations increases, μ gets closer to the optimal value, and the local nonlinearity of F(μ) weakens. n-1The closer to the true derivative, the more precise the adjustment, ultimately achieving convergence where F(μ) approaches 0.

[0049] In one embodiment, in step S1, the target phase φ target Obtained from the following formula: Φ target = (360° + B) degree P direction ) / N Skip; Among them B degree P represents the total phase angle of the yarn. direction Here, is the yarn offset coefficient, N is the number of tangent points, and Skip is the skipped tooth index. The core function of this formula is to construct a quantifiable and calculable method for determining the target phase based on key parameters of the winding process.

[0050] Specifically, the 360° in the formula is the basic circumferential angle, ensuring that the winding trajectory completes full circumferential coverage. This is the basic geometric requirement for winding the mandrel surface and avoids gaps in fiber coverage in the circumferential direction. B degree Derived from yarn width and mandrel radius, it is the conversion amount of linear yarn width to circumferential angle, reflecting the actual coverage angle of the yarn on the circumference of the mandrel's pole hole, and determining the basic offset scale of adjacent yarns. P direction The direction coefficient (usually +1 or -1) corresponding to the winding direction (front cut, back cut) determines the circumferential direction of yarn offset, adapting to the tangential requirements of different winding processes and avoiding yarn overlap. N is the number of fiber tangent points on the circumference of the pole hole in a standard wiring cycle, determining the distribution ratio of the target phase among the tangent points. Skip determines the circumferential skip step size of the yarn in each cycle.

[0051] The essence of phase is the positional deviation between the end point and the beginning point of the fiber winding cycle on the circumference of the mandrel's pole hole. This invention chooses angle as the expression of phase, rather than length, radian, coordinates, or other forms. This is the optimal choice based on the axisymmetric geometric characteristics of the mandrel, the circular motion law of the winding process, and the actual needs of engineering calculations and equipment execution.

[0052] Specifically, non-geodetic winding mandrels are mostly axisymmetric curved surfaces (such as cylinders and end caps of gas cylinders and pressure vessels). The core motion of fiber winding is the circular rotation around the mandrel axis and the linear movement along the axis, while the circumference of the polar hole is the key reference surface for the trajectory landing point. The positional deviation of the polar hole circumference is most intuitively described by angle, which can directly reflect the rotational offset of the fiber landing point on the circumference. Compared with arc length, no additional conversion is needed to reflect the relative position of the offset.

[0053] In addition, the calculation of the target phase requires the integration of multiple parameters such as yarn width, core mold radius, and number of cutting points, and involves the conversion from linear dimensions to circumferential dimensions. Using angle expression can greatly simplify the calculation process.

[0054] Furthermore, the total phase angle B of the yarn belt degree Obtained from the following formula: B degree = (B effect / R) (180° / π) / Skip; B effect =B / cosα; Among them, B effect B is the effective yarn width, α is the actual yarn width, R is the winding angle, and Skip is the skip tooth indexing.

[0055] Specifically, actual yarn width is a directly measurable basic parameter, representing the actual physical width of the yarn. Total yarn width = number of yarns. Single-bundle yarn width. In non-geodetic winding, the yarn tape is not wound along the tangential direction of the mandrel circumference, but rather obliquely wound at a certain designed winding angle on the axisymmetric curved surface of the mandrel. In this case, the actual physical width of the yarn tape will have a projection effect on the circumferential coverage direction of the mandrel curved surface. The effective yarn width is the width of the yarn tape that is actually effectively covered on the mandrel curved surface, and it is a corrected parameter that conforms to the physical reality of winding.

[0056] The effective yarn width in the circumference of the mandrel is considered as the arc length on the circumference of the pole hole. By reversing the arc length formula, the central angle corresponding to this arc length is calculated, which is suitable for angle-type target phase calculation and conforms to the operating habits of engineers and the unit requirements of the CNC system of the winding machine. As a dimensionless relative quantity, the angle does not need to consider the influence of the absolute size of the mandrel radius on the offset ratio. For example, with the same yarn width on mandrels of different radii, the angle offset can directly reflect the relative coverage ratio, while the arc length cannot be directly reflected.

[0057] In one embodiment, the chord-tangent method, as a derivative-free numerical iterative method, directly determines the convergence speed and even convergence of the iteration by selecting the initial value. If the initial value is not selected properly, such as the distance between the two points being too large or too small, exceeding the physical range, or having no effective gradient, it will lead to meaningless calculation of the chord slope, incorrect iteration direction, slow convergence, or even complete divergence.

[0058] Therefore, in step S3, a design friction coefficient μ is set based on experience. design The two initial friction coefficients μ0 and μ1 are selected based on the following formula: μ0=max(0.01, μ design -μ interval ); μ1=min(0.99, μdesign +μ interval ); Where, μ interval μ is the spacing gradient value. interval The range is [0.01, 0.50].

[0059] Where, μ design These are engineering experience values, determined by technicians based on engineering practices such as conventional non-geodetic winding processes, fiber-core material combinations, and surface conditions. They serve as initial reference values ​​that closely reflect actual working conditions. μ interval The fixed deviation value and gradient design value are selected by technicians based on the complexity of the working conditions, the nonlinearity of the error function, and the convergence accuracy requirements. This determines the spacing gradient between the two initial values. It is a small deviation, verified in engineering, that provides an effective initial gradient for the tangent method, ensuring clear distinction between the two points while avoiding excessive deviation that could distort the initial chord slope. The max and min functions are extreme value constraint operators. The max operator ensures that μ0 is not lower than the physical lower limit, and the min operator ensures that μ1 is not higher than the physical upper limit. These dual operators ensure that the two initial values ​​strictly fall within the effective range of [0.01, 0.99], avoiding values ​​that have no engineering or physical significance.

[0060] Set the initial value to μ design By ensuring that the initial values ​​are within the neighborhood of the optimal friction coefficient from the outset, we can prevent the initial values ​​from being far from the optimal value, which would require a large number of iterations to approximate the optimal value, thus improving the iteration convergence efficiency from the source. If the distance between the two points μ0 and μ1 exceeds the neighborhood of the optimal value, the nonlinear characteristics of the error function in this interval are significant, and the slope of the chord cannot accurately approximate the actual rate of change near the optimal value, leading to errors in the initial iteration direction and amplitude, slow convergence, or even divergence.

[0061] Within this gradient range, it ensures that the two points have a clear distinction and that the effective chord slope can be calculated. Furthermore, because the spacing is small, the error function can be approximated as linear in this interval. The chord slope can accurately reflect the rate of change of the error function near the optimal value, allowing it to approach the optimal value precisely with fewer iterations.

[0062] In general, the friction coefficient μ is designed based on engineering experience. design Anchoring the optimal value neighborhood to the benchmark, through an adjustable deviation interval μ interval The system achieves flexible customization of dual initial gradients to adapt to the nonlinear characteristics of error functions under different working conditions. Then, it uses the max and min dual extreme value operators to apply hard constraints on physical boundaries, so that the generated dual initial friction coefficients meet the requirements of being close to the optimal value, adapting the gradient to the working conditions, and falling within the physical effective range.

[0063] In one specific embodiment, the two initial friction coefficients μ0 and μ1 are selected based on the following formula: μ0=max(0.01, μdesign -0.01); μ1=min(0.99, μ design +0.01).

[0064] Set μ interval This tiny gradient, the chord slope, can accurately reflect the rate of change of the error function near the optimal value, allowing the first iteration to accurately approximate the optimal value.

[0065] In one embodiment, boundary protection is performed based on the following formula after friction coefficient iteration: μ n =max(0.001,min(0.999,μ n )), n≥2.

[0066] This clause employs nested extremum operators to implement dual upper and lower bound constraints on the friction coefficient generated in each iteration. First, the upper bound is limited by the inner min operator, and then the lower bound is limited by the outer max operator, ultimately ensuring that all friction coefficients after iterations strictly fall within the physically valid range of [0.001, 0.999]. This boundary value is slightly wider than the initial constraint boundary [0.01, 0.99], which on the one hand conforms to the required numerical accuracy of the friction coefficient for non-geodesic winding, and on the other hand reserves tolerance for minor fluctuations in numerical iteration.

[0067] On the other hand, substituting invalid friction coefficients into non-geodetic differential equations can easily lead to numerical anomalies, unsolvable problems, or distorted trajectories causing phase errors, ultimately resulting in iterative divergence and process interruption. This clause performs boundary checks on the friction coefficients before solving the equations, ensuring that the parameters substituted are always physically valid and numerically reasonable. This guarantees the continuity and accuracy of trajectory solving, phase calculation, and slope derivation, ensuring that the iteration always proceeds in the correct direction.

[0068] Furthermore, if the variable values ​​output by the previous two iterations are the same, the value is determined to be stable, and the iteration is terminated. The friction coefficient located in the range (0.001, 0.999) is output as the final value.

[0069] On the one hand, the chord-tangent method, based on numerical calculation iteration, is limited by the precision of computer floating-point numbers and the characteristics of numerical interpolation algorithms. It may encounter situations where the friction coefficient no longer changes, but the error between the actual landing point phase and the target phase still hasn't reached the preset precision threshold. Continuing the iteration in this case will not produce new correction results, but will only create a meaningless infinite loop, wasting computational resources. By supplementing with a termination mechanism, this numerical convergence state can be accurately identified, and the iteration can be terminated in a timely manner, fundamentally avoiding invalid infinite calculations and ensuring the computational efficiency of the iterative process.

[0070] On the other hand, the clause explicitly requires that the friction coefficient located in the range of (0.001, 0.999) be the final value. Combined with the boundary protection mechanism, it ensures that even if the value exceeds the boundary, the final output parameter is always within the physical effective range of non-geodetic winding and will not have invalid values. At the same time as termination, the engineering usability of the results is guaranteed, avoiding the output of parameters that have no practical significance due to premature termination. Moreover, the output data can provide a reference for engineering research.

[0071] Furthermore, if the variable values ​​output by the previous two iterations are different and have not converged, the value is determined to be unstable, and the iteration process continues to update the iteration variable.

[0072] Taking a typical gas cylinder core mold as an example, a non-geodetic winding method is adopted, and the friction coefficient is iterated by the chord-tangent method to finally obtain a trajectory that meets the requirements of linear closure.

[0073] The following is the specific implementation process: The non-geodetic differential equation used in this embodiment is: ; in: r(y) is the core mold radius function, obtained by spline interpolation; r y =dr / dy, r yy =d 2 r / dy 2 ; L is the angular momentum, which is determined by the initial conditions; μ is the coefficient of friction; For normal curvature, the expression is: .

[0074] Step 1: Core mold geometry and process parameters The core mold is a cylinder with a hemispherical head, and its specific dimensions are as follows: The cylindrical section has a length L = 800 mm, the initial radius of the mandrel R = 200 mm, and both the left and right end caps are elliptical in shape with an extension length L. ext =150mm, pole hole diameter d pole =100mm.

[0075] The yarn parameters are as follows: Yarn quantity = 4, single bundle yarn width = 5mm, total yarn width = 20mm.

[0076] The winding process parameters are as follows: Winding type: non-geodetic winding, design winding angle α=25°, number of tangent points N=3 (front tangent), skipped tooth indexing K=2, design friction coefficient μdesign =0.2.

[0077] Step 2: Target Phase Calculation Based on the above parameters: Effective yarn width = 20 / cos25° ≈ 22.07 mm; The bandwidth angle increment Δφ = 22.07 / 200×(180 / π) / 2≈3.16°; Total phase φ total =360° + 3.16 × (+1) = 363.16°; Target phase φ target =363.16 / 3×2=242.11°; Therefore, iterative steps are needed to bring the actual landing point phase close to 242.11°.

[0078] Step 3: Iterative Calculation Process The friction coefficient is iterated using the tangential method, with two initial values ​​set: μ0=0.19, μ1=0.21; Solve the non-geodesic equations separately, substituting the initial friction coefficients μ0 and μ1 into the non-geodesic differential equations. Use the fourth-order Runge-Kutta method to numerically integrate the non-geodesic differential equations to obtain the winding trajectory of a complete winding cycle for each trajectory. Extract the actual landing phases φ0 and φ1 of the two trajectories. The actual landing phase is taken as the circumferential angle of the last point at the end of the winding cycle, converted to an angle value, and normalized to the range [0, 360°). For μ0 = 0.19: φ0 = 238.54°; For μ1=0.21: φ1=244.78°.

[0079] First iteration: Calculate the error function value: F(μ0) = φ0 - φ target = -3.57°, F(μ1)=φ1-φ target = 2.67°; Chord slope k1=(F(μ1)-F(μ0)) / (μ1-μ0)=312.

[0080] Determine the direction of correction: K1=312>0 indicates that F(μ) increases as μ increases. The current F(μ1)=2.67°>0 indicates that μ1 is too large, causing the phase to exceed the target. Therefore, μ needs to be reduced to make F(μ) approach 0, and vice versa.

[0081] The new friction coefficient μ2 = μ1 - F(μ1) / k ≈ 0.2014; Solve for the actual landing phase corresponding to μ2: φ2 = 242.06°; Error |φ2-φ target |=0.05°<5°, the convergence condition is met, the iteration ends, and the optimal friction coefficient μ is obtained. opt =0.2014.

[0082] The trajectory and endpoint phase are obtained by numerical integration of the non-geodetic differential equation using the fourth-order Runge-Kutta method.

[0083] Step 4: Result Verification Please see Figure 2 and Figure 3 As shown, the final friction coefficient μ opt = 0.2014, landing point phase 242.06°, the error with the target phase 242.11° is only 0.05° (about 0.02%), which fully meets the engineering accuracy requirements (usually the allowable error is ≤5°).

[0084] Figure 2 and Figure 3 The diagram shows the winding trajectory of the first iteration (red curve) and the expanded multi-layer trajectory (yellow and green curves). It is evident that the yarn is evenly covered without overlap or gaps, resulting in good line closure. The results demonstrate that this method can achieve rapid convergence with fewer iterations, high accuracy in the friction coefficient, and minimal line closure error.

[0085] It should be noted that this application is not limited to the above-described embodiments. The above embodiments are merely examples, and any embodiments with the same structure and effect as the technical concept within the scope of this application are included in the technical scope of this application. Furthermore, various modifications that can be conceived by those skilled in the art to the embodiments, and other ways of constructing by combining some of the constituent elements of the embodiments, without departing from the spirit of this application, are also included in the scope of this application.

Claims

1. A non-geodesic trajectory design method for iterative friction coefficient, characterized in that, Includes the following steps: S1: Construct the error function according to the following formula: F(μ) = φ n -φ target n is a natural number; Where μ is the coefficient of friction, φ n φ represents the actual landing phase of the non-geodetic equation f(μ). target For the target phase; S2: Set two initial friction coefficients μ0 and μ1; S3: The friction coefficient is iterated using the following formula: m n =m n-1 -F(μ) n-1 ) / k n-1 ,n≥2; Where k n-1 Let the error function F(μ) be in the interval [μ] n-2 μ n-1 The slope of the chord on the chord; S4: Solve for μ n In the actual landing point phase φ of the non-geodetic equation f(μ) n Judgment error |φ n -φ target | Check if the convergence condition is met. If not, proceed to step S3.

2. The method for designing a non-geodesic trajectory of an iterative friction coefficient according to claim 1, characterized in that, In step S1, the target phase φ target Obtained from the following formula: φ target =(360°+B degree P direction ) / N Skip; Among them B degree P represents the total phase angle of the yarn. direction is the direction coefficient, N is the number of tangent points, and Skip is the skipped tooth index.

3. The method for designing a non-geodesic trajectory of an iterative friction coefficient according to claim 2, characterized in that, Total phase angle B of the yarn degree Obtained from the following formula: B degree =(B effect / R) (180° / π) / Skip; B effect =B / cosα; Among them, B effect B is the effective yarn width, α is the actual yarn width, R is the winding angle, and Skip is the skip tooth indexing.

4. The method for designing a non-geodesic trajectory of an iterative friction coefficient according to claim 1, characterized in that, In step S3, the slope of the chord is obtained according to the chord-tangent method, specifically by the following formula: K n-1 = (F (μ) n-1 )-F(μ n-2 )) / (μ n-1 -m n-2 ), n≥2.

5. The method for designing a non-geodesic trajectory of an iterative friction coefficient according to claim 1, characterized in that, In step S3, a design friction coefficient μ is set based on experience. design The two initial friction coefficients μ0 and μ1 are selected based on the following formula: μ0=max(0.01,μ design -m interval ); μ1=min(0.99,μ design +m interval ); Where, μ interval μ is the spacing gradient value. interval The range is [0.01, 0.50].

6. The method for designing a non-geodesic trajectory of an iterative friction coefficient according to claim 1, characterized in that, Boundary protection is performed based on the following formula after friction coefficient iteration: m n =max (0.001), min (0.999), μ n )), n≥2.

7. The method for designing a non-geodesic trajectory of an iterative friction coefficient according to claim 6, characterized in that, If the variable values ​​output by the previous two iterations are the same, the iteration step is terminated and the friction coefficient located in the last interval (0.001, 0.999) is output as the final value.

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