An optimization design method for non-pneumatic tires based on Bezier curves

Abstract

The present invention discloses an optimization design method for non-pneumatic tires based on Bezier curves, which relates to the technical field of wheel design and solves the technical problem of poor performance of non-pneumatic tires. The key point of the technical solution is to construct spoke geometry based on Bezier curves, overcoming the problem of discontinuous curvature of traditional straight line segments and circular arcs. In practical applications, only the coordinates of the control points need to be stored, and the description of the spoke structure can be achieved using the control points and Bezier curve parameter expressions. The overall structure is smooth, the algorithm time complexity and control complexity are low, and the optimization efficiency is high. At the same time, the requirements of free modes, fatigue performance, impact performance, thermodynamic performance, wear resistance, aerodynamic performance, lightweight performance, etc. of non-pneumatic wheels are comprehensively considered and met, and the design and optimization of the performance of a single discipline are no longer limited to improve the performance and life of the tire.

Description

Technical Field

[0001] The present application relates to the technical field of wheel design, and in particular to an optimization design method for a non-pneumatic tire based on a Bezier curve. Background Art

[0002] In recent years, with the sustained and rapid development of my country's economy, the number of motor vehicles has continued to increase, which has brought unprecedented prosperity to the road transportation industry, but also brought serious road safety hazards. According to statistics, the proportion of highway traffic accidents caused by tires is as high as 46%. In response to this phenomenon, non-pneumatic tires have become an inevitable development trend, solving key problems in traditional pneumatic tires such as tire deflation, easy wear of rubber materials, and poor temperature adaptability.

[0003] The tire test system is a large-scale, high-speed test equipment. In order to ensure that the non-pneumatic tire with good performance is obtained, it is necessary to ensure that the test system can work reliably. The structural system must have good static and dynamic characteristics at the same time. Its performance directly affects the safety and reliability of the tire test system under high speed and high load. The basic designable parameters of the tire structure are: spoke thickness, the angle and number of basic structural units in the spokes and the matching relationship between the three, the thickness of the shear ring and the rubber tread, etc. The spoke structure is diverse, and its number, inclination and shape are highly designable. Traditional tire design only considers the strength requirements of the tire, and the function is relatively single. In terms of parametric design, the curvature of the spokes changes discontinuously. The traditional curve fitting method has a high-order polynomial interpolation method, which uses the polynomial to approximate a function and calculate the corresponding function value. As the number of polynomials increases, the required data will increase, and the prediction will be more accurate. However, as the number of interpolation increases, the interpolation result deviates more from the original function, and errors such as deformation and Runge occur.

[0004] For engineering applications, how to perform parametric modeling on non-pneumatic tires and conduct multidisciplinary and multi-objective optimization design to further improve their performance remains an urgent problem to be solved. Summary of the invention

[0005] The present application provides an optimization design method for a non-pneumatic tire based on a Bezier curve, the technical purpose of which is to perform parametric modeling and multidisciplinary and multi-objective optimization on the non-pneumatic tire to improve its performance.

[0006] The above technical objectives of this application are achieved through the following technical solutions:

[0007] An optimization design method for a non-pneumatic tire based on a Bezier curve, comprising:

[0008] S1: parametric modeling of the non-pneumatic tire is performed using Bezier curves to obtain a parametric model, and the design variables and constraints of the parametric model are defined under five working conditions; a finite element model of the non-pneumatic tire under five working conditions is constructed using the structure provided by the parametric model, and the performance indicators of the non-pneumatic tire are analyzed using the finite element model to obtain comprehensive performance requirements; wherein the performance indicators include free modes, fatigue performance, impact performance, aerodynamic performance, wear resistance, vibration performance, thermodynamic performance, and lightweight performance;

[0009] S2: uniformly randomly sampling is performed within the value range of the design variables by optimizing the Latin hypercube design to select a first sample set, and the Kriging approximation model of the performance index is fitted by the first sample set; a second sample set is selected within the value range of the design variables by central composite design, and the prediction accuracy of the Kriging approximation model is tested by the second sample set until the prediction accuracy of the Kriging approximation model meets the requirements;

[0010] S3: Based on the constructed Kriging approximation model, the multi-objective optimization design of non-pneumatic tires is carried out through the second generation non-inferior sorting genetic algorithm (NSGA-II), and the Pareto optimal solution set is obtained;

[0011] S4: Under the condition of satisfying the comprehensive performance requirements of the tire, a compromise solution with the minimum vehicle mass is selected from the Pareto optimal solution set to determine the tire optimization solution;

[0012] The design variables of the parameterized model include the horizontal coordinate DV1 of the end control point, the vertical coordinate DV2 of the end control point, the horizontal coordinate DV3 of the P1 control point, the vertical coordinate DV4 of the P1 control point, the horizontal coordinate DV5 of the P2 control point, the vertical coordinate DV6 of the P2 control point, the array period DV7, the thin side spoke thickness DV8, and the thick side spoke thickness DV9;

[0013] The constraints of the design variables include: DV1 2 +DV2 2 =317 2 mm 2 ;DV3≤DV5≤DV1;DV4≤DV6≤DV2;DV3 2 +DV4 2 ≤DV5 2 +DV6 2 ≤DV1 2 +DV2 2 ;8≤DV7≤24;4mm≤DV8≤DV9≤10mm.

[0014] Further, in step S1, the parameterized model includes the centerline distribution and thickness distribution of the spokes, and the centerline distribution is obtained by a third-order Bezier curve, which is expressed as:

[0015]

[0016] Where x(t) represents the horizontal coordinate of the midline; y(t) represents the vertical coordinate of the midline; P xi represents the horizontal coordinate of the ith control point; P yi represents the ordinate of the ith control point; t represents the standardized variable of the movement time;

[0017] The thickness distribution is expressed as: Y(t) = a + (ba)t;

[0018] Where Y(t) represents the thickness of the spoke; a represents the thickness of the thinner side of the spoke; b represents the thickness of the thicker side of the spoke;

[0019] The Bezier curve is represented as:

[0020]

[0021] Where, B(t) represents the Bezier curve, n represents the order of the Bezier curve; P i Represents the coordinate vector of the i-th control point; 0 represents the starting control point; 1 represents the ending control point.

[0022] Furthermore, in step S1, the five working conditions include a modal test condition, a bending fatigue test condition, a radial fatigue test condition, a hammer impact test condition at 13° to the spoke, and a hammer impact test condition at 90° to the spoke.

[0023] Furthermore, in the step S1, the parameters for analyzing the free modes of the non-pneumatic tire using the finite element model include modal damping, modal frequency and modal vibration shape.

[0024] Furthermore, in the step S1, the fatigue performance of the non-pneumatic tire is analyzed by the finite element model, including: calculating the obtained bending stress value by a bending fatigue calculation model to generate a bending fatigue damage value; calculating the obtained radial stress value by a radial fatigue calculation model to generate a radial fatigue damage value; calculating the Von Mises stress and node displacement offset of the tire at the same time; calculating the bending and radial fatigue life of the tire and the radial fatigue life safety factor based on the fatigue cumulative damage theory, and analyzing the relationship between different performance parameters and the design parameters of the non-pneumatic tire.

[0025] Furthermore, in step S1, the impact performance of the non-pneumatic tire is analyzed through the finite element model, including: through the finite element model of the impact test condition of the hammer facing the spoke at 13° and the finite element model of the impact test condition of the hammer facing the spoke at 90°, respectively analyzing the unit distribution of the effective plastic strain of the tire under the hammer facing the spoke, and analyzing the influence mechanism of the design variables and various impact performance indicators of the tire.

[0026] Furthermore, in step S3, the model of the multi-objective optimization design is expressed as:

[0027] Find:x=DV=(DV1,DV2,...,DV7) T ;

[0028] Minimize:m(x);

[0029]

[0030] Where x represents the design variable; m(x) represents the mass of the non-pneumatic tire in the multi-objective optimization design (kg); σ bd (x) represents the maximum VonMises stress of the spoke under the bending fatigue test condition (MPa); σ rd (x) represents the maximum Von Mises stress of the spoke under radial fatigue test conditions (MPa); σ d0 represents the yield stress of the spoke material (polyurethane) (MPa); f1(x) represents the first-order modal frequency of the tire (Hz); f 10 represents the initial first-order modal frequency (Hz); D b (x) represents the maximum displacement of the tire node under the bending fatigue test condition (mm); D b0 Indicates the allowable value of the tire node under bending fatigue test conditions (mm); C r (x) represents the flexibility of the spoke under radial fatigue test conditions (N.mm); C r0 Indicates the allowable value of the spoke under radial fatigue test conditions (N.mm); N b (x) represents the dynamic bending fatigue life of the tire (10,000 times); N b0 Indicates the allowable value of tire dynamic bending fatigue life (10,000 times); N r (x) represents the dynamic radial fatigue life of the tire (10,000 times); N r0 Indicates the allowable value of tire dynamic radial fatigue life (10,000 times); S N (x) represents the safety factor of the tire dynamic radial fatigue life; S N0 Indicates the allowable value of tire dynamic radial fatigue life; ε wd (x) represents the maximum effective plastic strain of the spoke when the hammer is directly facing the spoke at a 13° impact test condition; εd0 It represents the allowable strain of the spoke material (polyurethane); εfd(x) represents the maximum effective plastic strain of the spoke when the hammer impacts the spoke at 90°.

[0031] Furthermore, the initial value of DV1 is 250 mm; the initial value of DV2 is 195 mm; the initial value of DV3 is 83.3 mm; the initial value of DV4 is 65 mm; the initial value of DV5 is 166.7 mm; the initial value of DV6 is 130 mm; the initial value of DV7 is 24; the initial value of DV8 is 5 mm; and the initial value of DV9 is 5 mm.

[0032] The beneficial effects of this application are:

[0033] (1) The spoke geometry is constructed based on Bezier curves, overcoming the problem of discontinuous curvature of traditional straight line segments and circular arcs. In practical applications, only the coordinates of the control points need to be stored, and the control points and Bezier curve parameter expressions can be used to describe the spoke structure. The overall structure is smooth, the algorithm time complexity and control complexity are low, and the optimization efficiency is high.

[0034] (2) Comprehensively consider and meet the requirements of non-pneumatic wheel modal frequency, fatigue performance, impact performance, aerodynamics, vibration performance, thermodynamic performance, wear resistance, lightweight performance, etc., no longer limited to the design and optimization of single subject performance, and improve the performance and life of the tire.

[0035] (3) The surrogate model method is used in combination with the multi-objective optimization algorithm to perform multidisciplinary and multi-objective optimization design on non-pneumatic wheels. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] Figure 1 A flowchart of the method described in this application;

[0037] Figure 2 The schematic diagram of parametric modeling of petal-shaped non-pneumatic tire based on Bezier curve;

[0038] Figure 3 Schematic diagram of the DOE sampling and approximate model fitting platform. DETAILED DESCRIPTION

[0039] The technical solution of the present application will be described in detail below with reference to the accompanying drawings.

[0040] like Figure 1 As shown, an optimization design method for a non-pneumatic tire based on a Bezier curve comprises:

[0041] S1: parametric modeling of the non-pneumatic tire is performed using Bezier curves to obtain a parametric model, and the design variables and constraints of the parametric model are defined under five working conditions; a finite element model of the non-pneumatic tire under five working conditions is constructed using the structure provided by the parametric model, and the performance indicators of the non-pneumatic tire are analyzed using the finite element model to obtain comprehensive performance requirements; wherein the performance indicators include free modes, fatigue performance, impact performance, aerodynamic performance, wear resistance, vibration performance, thermodynamic performance, lightweight performance, etc.

[0042] Specifically, the spoke control original points P0, P1, P2, and P3 are input, and the spoke curve is fitted using the four control points, i.e., the third-order Bezier curve, to construct the spoke centerline. By changing the position of the control points, the spoke centerline can be changed arbitrarily, such as Figure 2 As shown. The Bezier curve is defined by the following formula:

[0043]

[0044] Where, B(t) represents the Bezier curve, n represents the order of the Bezier curve; P i Represents the coordinate vector of the i-th control point; 0 represents the starting control point; 1 represents the ending control point; t represents the standardized variable of the motion time.

[0045] The spoke shape is constructed by the midline distribution and thickness distribution. The coordinates of any point on the midline (x m ,y m ) and its corresponding thickness distribution Y t After being determined, the coordinates of the point on the spoke surface (x h ,y h ) and the coordinates of the point on the lower surface (x l ,y l ) can be determined. In other words, when the midline distribution and thickness distribution are determined, the complete spoke structure can be determined through the operation of symmetry and center rotation array. The midline is expressed by the third-order Bezier curve construction as:

[0046]

[0047] Where x(t) represents the horizontal coordinate of the midline; y(t) represents the vertical coordinate of the midline; P xi represents the horizontal coordinate of the ith control point; P yi Represents the ordinate of the i-th control point.

[0048] The spoke thickness adopts a gradient thickness, and the thickness distribution is expressed as: Y(t)=a+(ba)t;

[0049] Where Y(t) represents the thickness of the spoke; a represents the thickness of the thinner side of the spoke; and b represents the thickness of the thicker side of the spoke.

[0050] As a specific embodiment, under five working conditions, nine design variables of the parameterized model including control point coordinates, array period and spoke thickness are defined, denoted as DV1, DV2, ..., DV9. The meaning, initial value and constraint conditions of each design variable are shown in Table 1.

[0051] Table 1 Initial values ​​and ranges of design variables

[0052]

[0053] Then, the finite element model of the petal-type non-pneumatic tire under five working conditions was established, including modal test condition, bending fatigue test condition, radial fatigue test condition, impact test condition with the hammer facing the spoke at 13° and impact test condition with the hammer facing the spoke at 90°.

[0054] The parameters of free modal analysis include modal damping, modal frequency and modal vibration shape. Through finite element analysis and experiments, the first three frequencies and vibration shapes of petal-shaped non-pneumatic tires with different structures are compared and analyzed, and the rules are summarized.

[0055] For fatigue performance research, multiple bending stress values ​​obtained are calculated through a bending fatigue calculation model to generate bending fatigue damage values; multiple radial stress values ​​obtained are calculated through a radial fatigue calculation model to generate radial fatigue damage values; at the same time, performance parameters such as the tire's Von Mises stress and node displacement offset are calculated; based on the fatigue cumulative damage theory, the tire's bending and radial fatigue life and radial fatigue life safety factor are calculated, and the relationship between different performance parameters and the design parameters of the non-pneumatic tire is studied.

[0056] For the impact performance study, the finite element model of the impact test condition with the hammer facing the spoke at 13° and the finite element model of the impact test condition with the hammer facing the spoke at 90° were used to analyze the unit distribution of the effective plastic strain of the tire when the hammer is facing the spoke, and the influence mechanism of the design variables and various impact performance indicators of the wheel was studied.

[0057] S2: A first sample set is selected by uniform random sampling within the value range of the design variables through optimized Latin hypercube design, and the Kriging approximation model of the performance index is fitted through the first sample set; a second sample set is selected within the value range of the design variables through central composite design, and the prediction accuracy of the Kriging approximation model is tested through the second sample set until the prediction accuracy of the Kriging approximation model meets the requirements.

[0058] As a specific embodiment, the first sample set selects 250 sample points in total from the design variables, and the second sample set selects another 30 sample points from the design variables.

[0059] S3: In the multidisciplinary optimization software Isight, based on the constructed Kriging approximation model, the multi-objective optimization design of non-pneumatic tires was carried out through the second-generation non-inferior sorting genetic algorithm, and the Pareto optimal solution set was obtained.

[0060] Specifically, the model of the multi-objective optimization design is expressed as:

[0061] Find:x=DV=(DV1,DV2,...,DV7) T ;

[0062] Minimize:m(x);

[0063]

[0064] Where x represents the design variable; m(x) represents the mass of the non-pneumatic tire in the multi-objective optimization design; σ bd (x) represents the maximum VonMises stress of the spoke under the bending fatigue test condition; σ rd (x) represents the maximum Von Mises stress of the spoke under radial fatigue test conditions; σ d0 represents the yield stress of the spoke material; f1(x) represents the first-order modal frequency of the tire; f 10 represents the initial first-order modal frequency; D b (x) represents the maximum displacement of the tire node under the bending fatigue test condition; D b0 Indicates the allowable value of the tire node under the bending fatigue test condition; C r (x) represents the flexibility of the spoke under radial fatigue test conditions; C r0 Indicates the allowable value of the spoke under radial fatigue test conditions; N b (x) represents the dynamic bending fatigue life of the tire; N b0 Indicates the allowable value of tire dynamic bending fatigue life; N r (x) represents the dynamic radial fatigue life of the tire; N r0 Indicates the allowable value of tire dynamic radial fatigue life; S N (x) represents the safety factor of the tire dynamic radial fatigue life; S N0 Indicates the allowable value of tire dynamic radial fatigue life; ε wd (x) represents the maximum effective plastic strain of the spoke when the hammer is directly facing the spoke at a 13° impact test condition; ε d0 represents the allowable strain of the spoke material; ε fd(x) represents the maximum effective plastic strain of the spoke when the hammer impacts the spoke at 90°.

[0065] The calculation software and parameterized models of each working condition performance index are integrated into the Isight software of the multi-objective optimization design model to establish a DOE test sampling and approximate model fitting platform, such as Figure 3 As shown, the finite element model of each working condition is retrieved and the response value of each performance index in the DOE sampling space is calculated.

[0066] S4: Under the condition of satisfying the comprehensive performance requirements of the tire, a compromise solution with the smallest vehicle mass is selected from the Pareto optimal solution set to determine the tire optimization solution.

[0067] The above embodiments are preferred embodiments of the present invention, but the present invention is not limited to the above embodiments. Any obvious improvements, substitutions or modifications that can be made by those skilled in the art without departing from the essential content of the invention belong to the protection scope of the present invention.

Claims

1. An optimization design method for a non-pneumatic tire based on a Bezier curve, characterized in that: The steps include: S1: parametric modeling of non-pneumatic tires is performed using Bezier curves to obtain a parametric model, and design variables and constraints of the parametric model are defined under five working conditions; The finite element model of the non-pneumatic tire under five working conditions is constructed through the structure provided by the parameterized model, and the performance indicators of the non-pneumatic tire are analyzed through the finite element model to obtain comprehensive performance requirements; wherein the performance indicators include free mode, fatigue performance, impact performance, aerodynamic performance, wear resistance, vibration performance, thermodynamic performance, and lightweight performance; S2: uniformly randomly sampling is performed within the value range of the design variables by optimizing the Latin hypercube design to select a first sample set, and the Kriging approximation model of the performance index is fitted by the first sample set; a second sample set is selected within the value range of the design variables by central composite design, and the prediction accuracy of the Kriging approximation model is tested by the second sample set until the prediction accuracy of the Kriging approximation model meets the requirements; S3: Based on the constructed Kriging approximation model, the multi-objective optimization design of non-pneumatic tires is carried out through the second-generation non-inferior sorting genetic algorithm, and the Pareto optimal solution set is obtained; S4: Under the condition of satisfying the comprehensive performance requirements of the tire, a compromise solution with the minimum vehicle mass is selected from the Pareto optimal solution set to determine the tire optimization solution; The design variables of the parameterized model include the horizontal coordinate DV1 of the end control point, the vertical coordinate DV2 of the end control point, the horizontal coordinate DV3 of the P1 control point, the vertical coordinate DV4 of the P1 control point, the horizontal coordinate DV5 of the P2 control point, the vertical coordinate DV6 of the P2 control point, the array period DV7, the thin side spoke thickness DV8, and the thick side spoke thickness DV9; The constraints of the design variables include: DV1 2 +DV2 2 =317 2 mm 2 ;DV3≤DV5≤DV1;DV4≤DV6≤DV2;DV3 2 +DV4 2 ≤DV5 2 +DV6 2 ≤DV1 2 +DV2 2 ;8≤DV7≤24;4mm≤DV8≤DV9≤10mm; Wherein, in the step S1, the parameterized model includes the centerline distribution and thickness distribution of the spokes, and the centerline distribution is obtained by a third-order Bezier curve, which is expressed as: Where x(t) represents the horizontal coordinate of the midline; y(t) represents the vertical coordinate of the midline; p xi represents the horizontal coordinate of the ith control point; P yi represents the ordinate of the ith control point; t represents the standardized variable of the movement time; The thickness distribution is expressed as: Y(t) = a + (ba)t; Where Y(t) represents the thickness of the spoke; a represents the thickness of the thinner side of the spoke; b represents the thickness of the thicker side of the spoke; The Bezier curve is represented as: Where, B(t) represents the Bezier curve, n represents the order of the Bezier curve; P i Represents the coordinate vector of the i-th control point; 0 represents the starting control point; 1 represents the ending control point; In step S3, the model of the multi-objective optimization design is expressed as: Find:x=DV=(DV1,DV2,...,DV7) T ; Minimize:m(x); Where x represents the design variable; m(x) represents the mass of the non-pneumatic tire in the multi-objective optimization design; σ bd (x) represents the maximum Von Mises stress of the spoke under the bending fatigue test condition; σ rf (x) represents the maximum VonMises stress of the spoke under radial fatigue test conditions; σ d0 represents the yield stress of the spoke material; f1(x) represents the first-order modal frequency of the tire; f 10 represents the initial first-order modal frequency; D b (x) represents the maximum displacement of the tire node under the bending fatigue test condition; D b0 Indicates the allowable value of the tire node under the bending fatigue test condition; C r (x) represents the flexibility of the spoke under radial fatigue test conditions; C r0 Indicates the allowable value of the spoke under radial fatigue test conditions; N b (x) represents the dynamic bending fatigue life of the tire; N b0 Indicates the allowable value of tire dynamic bending fatigue life; N r (x) represents the dynamic radial fatigue life of the tire; N r0 Indicates the allowable value of tire dynamic radial fatigue life; S N (x) represents the safety factor of the tire dynamic radial fatigue life; S N0 Indicates the allowable value of tire dynamic radial fatigue life; ε wd (x) represents the maximum effective plastic strain of the spoke when the hammer is directly facing the spoke at a 13° impact test condition; ε d0 represents the allowable strain of the spoke material; ε fd (x) represents the maximum effective plastic strain of the spoke when the hammer impacts the spoke at 90°.

2. The optimization design method according to claim 1, characterized in that: In step S1, the five working conditions include a modal test condition, a bending fatigue test condition, a radial fatigue test condition, a hammer impact test condition at 13° to the spoke, and a hammer impact test condition at 90° to the spoke.

3. The optimization design method according to claim 2, characterized in that: In the step S1, the parameters for analyzing the free modes of the non-pneumatic tire using the finite element model include modal damping, modal frequency and modal vibration shape.

4. The optimization design method according to claim 3, characterized in that: In the step S1, the fatigue performance of the non-pneumatic tire is analyzed by the finite element model, including: calculating the obtained bending stress value by a bending fatigue calculation model to generate a bending fatigue damage value; calculating the obtained radial stress value by a radial fatigue calculation model to generate a radial fatigue damage value; calculating the Von Mises stress and node displacement offset of the tire at the same time; calculating the bending and radial fatigue life of the tire and the radial fatigue life safety factor based on the fatigue cumulative damage theory, and analyzing the relationship between different performance parameters and the design parameters of the non-pneumatic tire.

5. The optimization design method according to claim 4, characterized in that: In the step S1, the impact performance of the non-pneumatic tire is analyzed by the finite element model, including: using the finite element model of the impact test condition of the hammer facing the spoke at 13° and the finite element model of the impact test condition of the hammer facing the spoke at 90°, respectively analyzing the unit distribution of the effective plastic strain of the tire under the hammer facing the spoke, and analyzing the influence mechanism of the design variables and various impact performance indicators of the tire.

6. The optimization design method according to claim 5, characterized in that: The initial value of DV1 is 250mm; the initial value of DV2 is 195mm; the initial value of DV3 is 83.3mm; the initial value of DV4 is 65mm; the initial value of DV5 is 166.7mm; the initial value of DV6 is 130mm; the initial value of DV7 is 24; the initial value of DV8 is 5mm; the initial value of DV9 is 5mm.

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