A method for two-dimensional stiffness modeling and tire deformation characteristics analysis of high aspect ratio sidewalls

Abstract

The present invention discloses a method for modeling the two-dimensional stiffness of a large aspect ratio sidewall and analyzing the deformation characteristics of a tire. Based on a flexible ring model based on a nonlinear elastic basis, a flexible ring complete tire model based on an elastic basis that considers the tire rotation effect is first established, and a simplified form based on different assumptions is given; secondly, based on the beam deformation theory under the action of inflation preload, a two-dimensional sidewall stiffness analytical model of a heavy-duty tire is established, and the modal characteristics of heavy-duty tires with different inflation pressures are predicted; finally, a grounding spring is introduced in the grounding area, and the grounding characteristics of the flexible ring tire model are analyzed using force balance conditions, and radial and tangential grounding imprint forces corresponding to different tire flattening amounts are given, and a tire stiffness experiment is performed. The present invention can update the sidewall stiffness value in real time, and can be used for sidewall structure design and tire vibration rapid design.

Description

Technical Field

[0001] The present invention relates to the field of vehicle engineering technology, and in particular to a method for modeling a large aspect ratio sidewall two-dimensional stiffness and analyzing tire deformation characteristics. Background Art

[0002] As the only element in contact with the ground during the driving of special vehicles, the tire transmits most of the forces to the whole vehicle, and also alleviates the impact and vibration caused by the unevenness of the road surface. Therefore, the characteristics of the tire affect the performance indicators of the whole vehicle, such as power, smoothness, braking and handling stability. Therefore, an accurate tire dynamics model becomes critical, especially the tire dynamics model based on structural flexibility. As a typical representative of the tire structure model, the flexible carcass tire model simplifies the tire into a flexible carcass on an elastic basis. The elastic basis is distributed along the circumferential radial and tangential directions to simulate the sidewall-carcass forces and inflation preload effects. How to accurately describe the road-rim vibration transmission characteristics caused by the vibration of the carcass and sidewall flexible structures has always been the focus and difficulty in tire dynamics research. Domestic and foreign scholars have studied different carcass flexibility modeling methods and proposed flexible carcass models from one dimension to three dimensions, including string models, beam models, plate models, ring models, shell models and distributed mass models based on elastic foundations;

[0003] The sidewall is the connection and force transmission component between the tire body and the rim. Its mechanical characteristics and vibration characteristics affect the transmission characteristics between the tire body and the rim. Domestic and foreign scholars have also proposed one-dimensional to three-dimensional sidewall stiffness models. Zhou F. [established a mechanical characteristic model of the one-dimensional radial stiffness of the sidewall; Liu Z. H. combined the one-dimensional radial sidewall stiffness model, considered the inertial mass and segmented stiffness of the sidewall, and established a large flat ratio heavy-duty tire flexible beam model based on a discrete elastic foundation. In addition, based on this model, considering the bending stiffness between the sidewall units, a tire model based on an elastic continuous foundation was established to study the tire model of the flexible tire body and the continuous foundation, and to study the coupled vibration problem of the flexible tire body and the continuous sidewall curved beam. Pacejia first introduced springs with radial and tangential stiffness to characterize the two-dimensional stiffness characteristics of the sidewall, established a flexible ring tire model with a two-dimensional elastic foundation, and explored the vibration characteristics of the tire; Gong established a ring model based on an elastic foundation based on radial and tangential springs, and studied the influence of different sidewall stiffness and inflation pressure on the vibration characteristics of the tire; Liu Z.H. established a two-dimensional sidewall stiffness model based on an elastic continuous foundation considering the bending stiffness of the sidewall, and studied the coupled vibration characteristics of the carcass ring and the continuous sidewall ring. Noga S. expanded the sidewall stiffness from two dimensions to three dimensions, and based on radial and tangential springs, considered the lateral stiffness outside the tire surface, and established a circular ring vibration model considering the three-dimensional sidewall stiffness characteristics.

[0004] The existing technology is based on the Euler beam model based on elastic foundation, and conducts research from two aspects: radial stiffness modeling of large aspect ratio sidewall curved beam and tire dynamics modeling based on analytical elastic foundation. However, as a typical circular ring structure, the carcass of heavy-duty tire has a coupling relationship between the radial deformation and tangential deformation of the carcass point. Therefore, it is necessary to consider the coupling relationship between the radial deformation and tangential deformation of the carcass ring, establish a flexible ring two-dimensional tire low-frequency dynamic model based on analytical elastic foundation, and establish analytical expressions for the radial and tangential two-dimensional sidewall stiffness. The tire-rim system is equivalent to a system composed of two-dimensional springs, flexible rings and discs. The radial and tangential springs connect the flexible rings and discs, as shown in the attached figure. Figure 1 The disk simulates the rim mass, the flexible ring represents the belt and the carcass connected to the crown, and the two-dimensional linear spring represents the sidewall and inflation effect.

[0005] The traditional sidewall model with linear fixed stiffness cannot solve the stiffness change caused by the deformation of the tire during the grounding process. Therefore, the present invention considers the coupling relationship between the radial deformation and tangential deformation of the carcass ring, proposes and establishes a radial and tangential analytical stiffness model for the sidewall, and combines the carcass flexible ring model to iterate and calculate the stiffness change of the sidewall caused by deformation in real time during the process of solving the tire grounding characteristics, so as to achieve high-precision solution of tire grounding. Summary of the invention

[0006] In view of the above-mentioned problems, the present invention proposes a method for modeling the two-dimensional stiffness of a large aspect ratio sidewall and analyzing the tire deformation characteristics, establishes a low-frequency dynamic model of the tire and analytical expressions of the radial and tangential two-dimensional sidewall stiffnesses, uses a grounding spring to solve the grounding, and uses the sidewall analytical model during the grounding iterative solution process to update the sidewall stiffness value in real time.

[0007] In order to achieve the above object, the technical solution adopted by the present invention is as follows:

[0008] A method for two-dimensional stiffness modeling of a high aspect ratio sidewall and tire deformation characteristics analysis, characterized in that it comprises the following steps:

[0009] Step 1: Based on the flexible ring complete tire model based on elastic foundation considering the tire rotation effect, a simplified form of the tire dynamics model based on the "carcass inextensibility assumption" is obtained, and based on this simplified form, a flexible ring two-dimensional tire low-frequency dynamics model based on analytical elastic foundation is established;

[0010] Step 2: Based on the Euler beam bending deformation theory, considering the inflation preload of the sidewall curved beam along the radial direction, a two-dimensional tire sidewall stiffness analytical model is established;

[0011] Step 3: Based on the flexible ring two-dimensional tire low-frequency dynamic model, the sidewall tangential bending stiffness is inversely calculated using the sidewall tangential stiffness under different inflation pressures;

[0012] Step 4: Introduce a grounding spring in the contact area, analyze the grounding characteristics of the flexible ring tire model constructed in step 1, and obtain the radial and tangential grounding imprint forces corresponding to different tire flattening amounts;

[0013] Step 5: The tire sidewall tangential bending stiffness and ground contact characteristics obtained in steps 3-4 are verified through tire stiffness experiments.

[0014] Furthermore, the specific operation steps of step 1 are:

[0015] Step 11: Establish a flexible ring complete tire model based on elastic foundation considering the tire rotation effect:

[0016]

[0017] and

[0018] Where E is the section modulus of the carcass ring beam, I is the section inertia of the carcass ring beam, w is the radial deformation of the carcass ring, v is the tangential deformation of the carcass ring, θ is the rotation angle of the carcass ring relative to the hub disk, x is the displacement of the hub disk in the X direction, y is the displacement of the hub disk in the Y direction, and k w is the radial spring stiffness, k v Tangential spring stiffness, P0 is the rated air pressure, ρ is the equivalent density of the ring beam, Ω is the angular velocity of the disk, q w is the radial distributed load, q v is the tangential distributed load, I w is the disk moment of inertia, γ is the disk rotation angle, T0 represents the tire carcass preload, and

[0019] Step 12: Based on the “carcass inextensibility assumption”, the simplified form of equation (13) is obtained:

[0020]

[0021] Step 13: Based on the simplified form obtained in step 12, assuming that the tire rim is fixedly supported and does not rotate, a new simplified form of equation (13) is obtained:

[0022]

[0023] Step 14: Substituting into formula (19), we get:

[0024]

[0025] Then, the tire modal resonance frequency based on the "carcass inextensibility assumption" is:

[0026]

[0027] The ± sign indicates that the tire has two resonant frequencies at each modal order, which are opposite to each other.

[0028] Furthermore, the specific steps of step 2 are:

[0029] Step 21: Taking into account the chord preload effect and structural deformation effect of the sidewall curved beam, the analytical formula for the radial sidewall stiffness of the heavy-duty tire is established according to formula (1):

[0030]

[0031] Among them, θ is half of the angle of the sidewall curved beam, E s h is the elastic modulus of the sidewall curved beam along the radial direction, s is the sidewall section thickness, λ is the Poisson's ratio of the sidewall curved beam, P0 is the inflation pressure, l s is the length of the sidewall curved beam;

[0032] Step 22: Based on the Euler beam bending deformation theory and considering the radial inflation preload of the sidewall curved beam, the tangential dynamic model of the sidewall curved beam is obtained.

[0033]

[0034] Among them, u tt is the tangential force f at the end point of the upper sidewall curved beam tt The displacement produced by the action; E t I t is the tangential bending stiffness of the sidewall curved beam;

[0035] Step 23: Ignore the radial distributed inertia force of the sidewall curved beam and simplify equation (2) to:

[0036]

[0037] in,

[0038] The general solution of formula (4) is:

[0039] u tt (θ)=C1e βθ +C2e -βθ +C3e iβθ +C4e -iβθ (5)

[0040] Among them, C1, C2, C3 and C4 are constant coefficients, which are related to the boundary conditions;

[0041] Then formula (5) can be transformed into:

[0042] u tt (θ)=C1cosβθ+C2sinβθ+C3coshβθ+C4sinhβθ (6)

[0043] Its related variables are transformed into:

[0044]

[0045] Step 24: Assuming that the intersection of the sidewall curved beam and the circumferential continuous beam is fixed, equation (7) must satisfy the following equation:

[0046]

[0047] Step 25: Solve equation (8) and obtain:

[0048]

[0049] Then, in the tangential force f tt Under the action, the displacement u generated by the end point of the side curved beam tt for:

[0050] u tt (θ) = C3 (cosh βθ c -cosβθ c )+C4(sinhβθ c -sinβθ c ) (10)

[0051] In formula (10):

[0052]

[0053]

[0054] The analytical formula of the tangential stiffness of the sidewall curved beam is:

[0055]

[0056] in, At the same time, R s = l s / θ c , and b t =1, then

[0057]

[0058] Furthermore, the specific steps of step 3 are:

[0059] Step 31: Considering only the + form of equation (21), transform equation (20) into:

[0060]

[0061] Step 32: Using formula (22) to identify the tire sidewall tangential stiffness under different inflation pressures;

[0062] Step 33: Using the tire dynamics model of formula (19), the sidewall tangential stiffness at different inflation pressures is used to inversely calculate the sidewall tangential bending stiffness E in formula (11) based on formula (21). t I t .

[0063] Furthermore, the specific steps of step 4 are:

[0064] Step 41: Define the point from the non-touchdown area to the touchdown area as the front touchdown point, and the point from the touchdown area to the non-touchdown area as the rear touchdown point. The angles between the front touchdown point and the rear touchdown point and the Y axis are φ f and φ r , the angle between any point in the grounding area and the Y axis is φ, and φ f ≤φ≤φ r ;

[0065] Step 42: introducing a grounding spring in the grounding area, the grounding range is [-a~a], the grounding length is 2a, and the grounding pressure distribution in the grounding area is q(φ);

[0066] Step 43: The grounding imprint force of the grounding area is calculated based on the deformation of the grounding spring and is:

[0067] q(φ)=k2·w tread (φ)

[0068] =k2·[w(φ)+d0-(R+h)(1-cosφ)] (twenty four)

[0070] Where k2 represents the ground spring stiffness of the grounding area;

[0071] Step 44: During the tire rolling process, the force between the tire and the road is distributed along the tire contact area. Considering that the deformation of any point of the tire should be along the footprint in the tire contact area, the deformation of any point φ is expressed as:

[0072]

[0073] Formula (25) is simplified to:

[0074]

[0075] in,

[0076]

[0077] Indicates angle;

[0078] Step 45: Based on equation (26), the known contact angle φ of the flexible carcass is obtained: f and φ r Solve the equation for the grounding state under the conditions:

[0079]

[0080] Where, α={α1,α2,…,α n}, β={β1,β2,…,β n}; A, B, C and D matrices are n×n matrices, E and F matrices are n×1 matrices, and they are all tire contact angles φ f and φ r Function of

[0081] Step 44: Calculate the radial and tangential contact imprint forces corresponding to different tire flattening amounts using equation (29).

[0082] Furthermore, the specific operation steps of step 44 are:

[0083] Step 441: estimating the tire's rear contact angle. Since the front contact angle and the rear contact angle are opposite to each other under steady-state conditions, the deformation coefficients α and β corresponding to the rear contact angle can be calculated to obtain the tire body deformation at the front and rear contact angles.

[0084] Step 442: Calculate the ground contact force at the front and rear ground contact angles, and determine the value range of the ground contact angle according to the ground contact force at the ground contact angle;

[0085] Step 443: Taking the ground contact mark force at the ground contact angle as 0 as the judgment condition, the ground contact angle of the flexible carcass is solved by the dichotomy method to determine the given flattening amount d0 or the loading force F z Flexible carcass deformation coefficients α and β under conditions;

[0086] Step 444: Substitute the obtained deformation coefficients α and β into equation (29) to solve the contact force at the front and rear contact angles.

[0087] The beneficial effects of the present invention are:

[0088] The present invention proposes a two-dimensional stiffness modeling and tire deformation characteristic analysis method for a large aspect ratio sidewall. Based on a flexible ring model based on a nonlinear elastic foundation, the ground contact modeling research of a heavy-duty tire is carried out, and the stiffness characteristics of the tire are analyzed. First, a flexible ring complete tire model based on an elastic foundation considering the tire rotation effect is established, and a simplified form based on different assumptions is given; secondly, based on the beam deformation theory under the action of inflation preload, a two-dimensional sidewall stiffness analytical model of a heavy-duty tire is established, and the modal characteristics of heavy-duty tires with different inflation pressures are predicted; finally, a ground spring is introduced in the ground contact area, and the ground contact characteristics of the flexible ring tire model are analyzed using the force balance condition, and the radial and tangential ground contact imprint forces corresponding to different tire flattening amounts are given, and a tire stiffness experimental study is carried out. The simulation experiment shows that the flexible ring tire model based on a nonlinear elastic foundation can accurately characterize the ground contact imprint, stiffness characteristics and rolling asymmetric characteristics of the tire, and the analytical results are well matched with the experimental results. This solves the problem that in the process of solving the tire ground contact characteristics, the traditional method uses a constant stiffness sidewall model, and the stiffness changes caused by the sidewall deformation, resulting in inaccurate solution of the ground contact characteristics. BRIEF DESCRIPTION OF THE DRAWINGS

[0089] Figure 1 It is a flexible ring tire model based on elastic foundation;

[0090] Figure 2 It is a schematic diagram of the sidewall units distributed along the circumferential direction of a heavy-load tire;

[0091] Figure 3 This is the tangential force analysis diagram of the sidewall curved beam;

[0092] Figure 4 (a)-(b) are the tire vibration modal test principle and the test results based on the least squares complex exponential method for different inflation pressures;

[0093] Figure 5 Analytical stiffness of the sidewall curved beam under different inflation pressures;

[0094] Figure 6 (a)-(e) are the prediction results of the tire low-frequency ring mode under different inflation pressures;

[0095] Figure 7 This is a geometric diagram of the ground contact deformation of the tire flexible carcass;

[0096] Figure 8 It is a schematic diagram of the deformation relationship between the tire carcass and the ground spring;

[0097] Fig. 9 (a)-(b) are the flow charts of flexible ring grounding analysis based on linear elastic foundation and flexible ring grounding analysis based on nonlinear elastic foundation, respectively;

[0098] Fig.10 (a)-(f) are the tire contact force at different flattening amounts;

[0099] Fig.11 A heavy-duty tire mechanical vibration test system constructed in the embodiment;

[0100] Fig.12 It is the contact footprint diagram of heavy-load tires with different loading forces;

[0101] Fig.13 (a)-(c) are the comparison results of the simulation and experiment of the stiffness characteristics of heavy-duty tires (standard inflation pressure)

[0102] Fig.14 Comparison results between simulation and experiment on stiffness characteristics of heavy-load tires at different inflation pressures. DETAILED DESCRIPTION

[0103] In order to enable those skilled in the art to better understand the technical solution of the present invention, the technical solution of the present invention is further described below in conjunction with the accompanying drawings and embodiments.

[0104] 1. Two-dimensional analytical stiffness of high aspect ratio sidewall curved beam

[0105] The high aspect ratio sidewall curved beam is a typical structural feature of heavy-duty tires. It connects the flexible carcass and the rim structure, and has the functions of transmitting force and displacement in the radial and tangential directions. Its two-dimensional mechanical properties affect the vibration characteristics of the tire. The present invention studies the radial and tangential mechanical properties of the high aspect ratio sidewall curved beam. The radial analytical stiffness of the sidewall in the 2D tire model has been established in the prior art "A method for modeling heavy-duty tires based on analytical sidewall stiffness" with patent number 202010621194.2. Referring to formula (1), the chord preload effect and structural deformation effect of the sidewall curved beam are taken into account, and the analytical formula for the radial sidewall stiffness of the heavy-duty tire is established as follows:

[0106]

[0107] Where θ is half of the curvature angle of the sidewall curved beam, Es is the elastic modulus of the sidewall curved beam in the radial direction, and h s is the sidewall section thickness, λ is the Poisson's ratio of the sidewall curved beam, P0 is the inflation pressure, l s is the length of the sidewall curved beam;

[0108] Attached Figure 2 is a schematic diagram of the sidewall unit distributed along the circumference. In order to analyze the stiffness of the sidewall curved beam, it is assumed that the sidewall curved beam is subjected to the tangential force f tt effect.

[0109] Based on the Euler beam bending deformation theory, the force analysis is carried out by considering the radial inflation preload of the sidewall curved beam. Figure 3 As shown, referring to the Euler beam modeling method including preload effect, the tangential dynamic model of the sidewall curved beam is shown in formula (2).

[0110]

[0111] Among them, u tt is the tangential force f at the end point of the upper sidewall curved beam tt The displacement produced by the action; E t I t is the tangential bending stiffness of the sidewall curved beam;

[0112] Ignoring the influence of the radially distributed inertia force of the sidewall curved beam, when analyzing the sidewall tangential stiffness, equation (2) is simplified to:

[0113]

[0114] Formula (3) is a fourth-order partial differential equation, which requires at least four boundary conditions to solve the equation. tt By solving the problem, we can know that the parameters related to deformation include: tt , and

[0115] Then equation (3) can be simplified as:

[0116]

[0117] in,

[0118] The general solution of formula (4) is:

[0119] u tt (θ)=C1e βθ +C2e -βθ +C3e iβθ +C4e -iβθ (5)

[0120] Among them, C1, C2, C3 and C4 are constant coefficients and are related to the boundary conditions.

[0121] Formula (5) is transformed into:

[0122] u tt (θ)=C1cosβθ+C2sinβθ+C3coshβθ+C4sinhβθ (6)

[0123] Then its related variables are transformed into:

[0124]

[0125] Assume that the intersection of the sidewall curved beam and the circumferential continuous beam is fixed, then the following boundary conditions must be met:

[0126] (1) Deformation u at the intersection tt (0) and corner All are 0;

[0127] (2) When θ = θ c (θ c =2θ0) at the end point, there exists an external force f tt Shear force at phase equilibrium

[0128] (3) When θ = θ c End point, corner

[0129] Substituting equation (7) into the boundary conditions, the following equation is satisfied:

[0130]

[0131]

[0132] By solving, we get:

[0133]

[0134] Then, in the tangential force f tt Under the action, the displacement u generated by the end point of the side curved beam tt for:

[0135] u tt (θ) = C3 (cosh βθ c -cosβθ c )+C4(sinhβθ c -sinβθ c ) (10)

[0137] Where:

[0138]

[0139]

[0140] The tangential stiffness of the sidewall curved beam is:

[0141]

[0142] Among them, the dimensionless term At the same time, R s = l s / θ c , and b t =1,

[0143] Then β is transformed into:

[0144]

[0145] Equations (11) and (12) show that the tangential stiffness of the sidewall curved beam is related to the tire inflation pressure, geometry and structural parameters, and can be used to analyze the influence of different parameters on the sidewall tangential stiffness.

[0146] 2. Flexible ring tire model based on elastic foundation and simplified form analysis

[0147] There are many studies on the vibration characteristics of circular rings based on elastic foundations in the prior art, such as the flexible ring tire model based on elastic foundation shown in formula (13), where: The preload force is generated by the inflation pressure and the centrifugal force of the ring.

[0148]

[0149] Taking the carcass ring as the research object, we can get:

[0150]

[0151] Where E is the section modulus of the carcass ring beam, I is the section inertia of the carcass ring beam, w is the radial deformation of the carcass ring, v is the tangential deformation of the carcass ring, θ is the rotation angle of the carcass ring relative to the hub disk, x is the displacement of the hub disk in the X direction, y is the displacement of the hub disk in the Y direction, and k w is the radial spring stiffness, k v Tangential spring stiffness, P0 is the rated air pressure, ρ is the equivalent density of the ring beam, Ω is the angular velocity of the disk, q w is the radial distributed load, q v is the tangential distributed load, I w is the disk’s moment of inertia, γ is the disk’s rotation angle, and

[0152] Then, the inflation pressure and the rotation effect produce the initial stress of the tire body for:

[0153]

[0154] Formula (13) is a complete tire model, including the radial and tangential vibration equations of the tire ring and the vertical, horizontal and rotational vibration equations of the rim disk, and taking the rim degree of freedom and rotation effect into account. When analyzing the tire vibration characteristics, different constraint conditions can be simplified.

[0155] A. Simplified form 1

[0156] The radial tire carcass is composed of composite materials such as steel wire and rubber, and the steel wire is along the meridian direction of the tire, and its circumferential tensile stiffness is large. In the 400Hz frequency range, the modal vibration mode of the GL073 radial tire is the bending vibration mode in the carcass surface, and the breathing vibration mode related to the carcass tensile stiffness does not appear in this frequency range, which verifies the validity of the "carcass inextensibility assumption". The "carcass inextensibility assumption" describes the characteristic that the circumferential length of the carcass ring neutral surface remains unchanged during the deformation process, which is expressed as:

[0157] v′+w=0 (16)

[0158] Substitute equation (16) into equation (13) (a and b), take the derivative of equation (13 (a)), and then add it to (13 (b)). Then, based on the "carcass inextensibility assumption", the flexible ring tire model (13) based on elastic foundation is transformed into:

[0159]

[0160] B. Simplified form 2

[0161] Furthermore, assuming that the tire rim is fixed and does not rotate, then x * =0,y * = 0 and γ = 0, and the “carcass inextensibility assumption”, then the model based on simplified form 1 is transformed into:

[0162]

[0163] Substituting equation (16) into equation (18), the equation is transformed into:

[0164]

[0165] Using the mechanical model based on the "carcass inextensibility assumption", Substituting into the dynamic equation (19), we get:

[0166]

[0167] Then, the tire modal resonance frequency based on the "carcass inextensibility assumption" is:

[0168]

[0169] Among them, the ± symbol indicates that the tire has two resonant frequencies at each modal order, which are opposite to each other. This is caused by the symmetrical structure of the tire. There are two vibration modes along the circumferential direction, clockwise and counterclockwise, where clockwise is + and counterclockwise is -. The present invention only discusses the modal frequencies corresponding to the vibration modes in a single direction.

[0170] There are unknown variables in the tire dynamics model based on the "carcass inextensibility assumption", including: bending stiffness EI, sidewall radial stiffness k w , sidewall tangential stiffness k v and carcass density ρA, where bending stiffness EI, sidewall radial stiffness k w and carcass density ρA have been determined in equations (20)-(21), so only the sidewall tangential stiffness k v The solution is shown in Table 1.

[0171] Considering only the + form of formula (21), formula (20) can be organized as follows:

[0172]

[0173] Table 1 Geometric and structural parameters of GL073A heavy-duty tire (including identified results)

[0174]

[0175] 3. Analysis of stiffness parameters of high aspect ratio tire sidewalls

[0176] Formula (11) represents the tangential analytical stiffness characteristics of the sidewall curved beam of a tire with a large aspect ratio, where the sidewall tangential bending stiffness E in the analytical formula is t I t As an unknown variable, the present invention adopts a tire model based on the “inextensibility assumption of the tire body” as shown in formula (19), takes the low-frequency ring mode of the heavy-duty tire as the identification objective function, and uses the sidewall tangential stiffness under different inflation pressures to inversely calculate E t I t .

[0177] A heavy-duty tire vibration modal test system was built. According to different inflation pressure conditions of heavy-duty tires, a mobile hammer method was used to sequentially excite 34 carcass points equally divided along the circumference to obtain the target transfer function. The driving point transfer function and the cross-point transfer function were calculated respectively, and the tire transfer characteristics under different inflation pressures were compared and analyzed. Figure 4 (a) is the tire vibration modal test principle, Figure 4 (b) The tire vibration modal frequency results based on the least squares complex exponential method for different inflation pressures, including: 0.3, 0.4, 0.5, 0.6, 0.7 and 0.8 MPa, are given.

[0178] The analysis shows that: (1) As the inflation pressure increases, the stiffness of the tire increases, resulting in an increase in the tire resonance frequency; (2) The effect of inflation pressure on tire vibration characteristics is reflected in two aspects, including: the axial preload of the tire beam caused by the inflation pressure and the changes in the radial and tangential stiffness of the sidewall caused by the inflation pressure.

[0179] Formula (22) is used to identify the sidewall tangential stiffness under different inflation pressures. The calculation results are as follows: Figure 5 As shown in the figure, the calculated radial and tangential stiffness of the tire sidewall are used to predict the low-frequency ring vibration mode of the tire under different inflation pressures. The results are shown in Figure 6 The results show that: (1) the radial stiffness of the tire sidewall is approximately linearly related to the inflation pressure, which is consistent with the conclusion of the radial stiffness of the tire sidewall; (2) Figure 6 (a)-(e) are comparison diagrams of the experimental modal results and analytical modal results of heavy-load tires under different inflation pressures. The comparison results show that the analytical modal results fit the experimental modal results well, and the error is guaranteed to be within 5%.

[0180] 4. Heavy-duty tire grounding solution and stiffness characteristics analysis

[0181] The forces between the tire and the road surface are all generated in the contact area. Therefore, it is of great practical value to carry out research on the ground contact solution of heavy-load tires, especially for heavy-load tires. During the deformation process of its flexible carcass in contact with the road surface, the large aspect ratio sidewall curved beam is also deformed, causing structural changes in the sidewall curved beam, resulting in changes in the stiffness of the sidewall curved beam. Therefore, it is necessary to take the nonlinear stiffness of the sidewall during the tire deformation process into account to solve the ground contact of the heavy-load tire; in addition, the ground contact characteristics of the tire determine the stiffness characteristics of the tire, and the stiffness characteristics of the tire are an important basis for tire selection, so solving the stiffness characteristics of the tire is very important.

[0182] (1) Theoretical analysis of tire flexible ground contact

[0183] When the tire is under load, the tire will produce carcass flattening and tire contact footprint in the contact area. Define the XOY plane coordinate system and divide the tire into contact area and non-contact area according to the contact situation, such as Figure 7 shown.

[0184] Definition: The point from the non-touchdown area to the touchdown area is the front touchdown point, and the point from the touchdown area to the non-touchdown area is the rear touchdown point. The angles between the front touchdown point and the rear touchdown point and the Y axis are φ respectively. f and φ r , the angle between any point in the grounding area and the Y axis is φ, where φ f ≤φ≤φ r In order to characterize the flexibility of the carcass, a ground spring is introduced in the ground contact area, such as Figure 7 As shown, the grounding range is [-a~a], the grounding length is 2a, and the grounding pressure distribution in the grounding area is q(φ).

[0185] Attached Figure 8 is the relationship diagram between tire carcass deformation and ground spring deformation. Under the action of external force, the tire produces a flattening amount of vertical displacement d0, in which both the tire carcass and the ground spring are deformed, and the tire vertical deformation has the following relationship:

[0186] w tread (φ)=w(φ)+d0-(R+h)(1-cosφ) (23)

[0187] Among them, w(φ) is the deformation of the carcass in the contact area, w tread (φ) is the deformation of the grounding spring in the contact area, and h is the initial length of the grounding spring, i.e., the thickness of the tire body. Then, the grounding imprint force can be calculated based on the deformation of the grounding spring, which is:

[0188]

[0189] Where k2 represents the ground spring stiffness of the grounding area;

[0190] During the rolling process of the tire, the force between the tire and the road is distributed along the tire contact area. Therefore, the deformation of any point of the tire should take into account the imprint force along the tire contact area. f ≤φ≤φ r The integral of the imprint force inside, then, the deformation of any point φ can be expressed as:

[0191]

[0192] Formula (25) is simplified to:

[0193]

[0194] in,

[0195]

[0196] Indicates angle;

[0197] Substituting equation (23) into equation (26), we get n can be transformed into:

[0198]

[0199] Similarly, β n can be transformed into:

[0200]

[0201] Substituting equation (25) into the first term in equation (27) and equation (28), we can obtain i , β i The 2n×2n order equation system for (i=1,2,…,n) is:

[0202]

[0203] Where, α={α1,α2,…,α n}, β={β1,β2,…,β n}; A, B, C and D matrices are n×n matrices, E and F matrices are n×1 matrices, and they are all tire contact angles φ f and φ r function.

[0204] If the grounding angle φ is known f and φ r , it can be realized for α={α1,α2,…,α n} and β={β1,β2,…,β n} and put it into equation (23) to solve the tire deformation and the imprint force in the contact area. At the same time, the contact imprint force affects the contact angle. Therefore, it is necessary to determine the contact angle and contact imprint force through iterative solution under the condition that the resultant contact imprint force is equal to the external load force.

[0205] In addition, during the grounding process of the flexible carcass, the geometric structure of the high aspect ratio sidewall curved beam changes during the deformation process, and the tire carcass moves w(φ) along the radial direction of the tire. Then, the angle of the deformed sidewall curvature is 2θ′ and the deformed sidewall curvature is 1 / R s ′, the following geometric relationship exists:

[0206]

[0207] If w(φ) is known, then there are two variables θ′ and R s ′, the variables θ′ and R can be solved by solving the nonlinear equations. s ′. By combining the calculated θ′ and R s ′, and put it into equation (11), the radial analytical stiffness analysis of the sidewall with deformation ΔL0 can be realized.

[0208] (2) Analysis of the ground contact deformation of heavy-duty tires

[0209] Formula (29) gives the known contact angle φ of the flexible carcass: f and φ r The ground contact state under the condition is solved, but during the tire rolling process at any time, the ground contact angle φ f and φ r is an unknown parameter, so in order to calculate the ground contact force of the flexible carcass, it is necessary to f and φ r During the steady-state grounding process of the flexible carcass, the ground contact angle has the following relationship: f +φ r= 0, that is, the contact patch area is symmetrical about the projection point of the tire center in the contact area. Therefore, to solve the steady-state contact characteristics of the flexible carcass, only the contact angle φ needs to be solved. f and φ r Any item in can realize the n} and β={β1,β2,…,β n} and put it into equation (25) to solve the tire deformation and the imprint force in the contact area. At the same time, the contact force affects the contact angle. Therefore, it is necessary to determine the contact angle and contact force through iterative solution under the condition that the resultant force of the contact force is equal to the external load force. The resultant force of the vertical and horizontal imprint forces of the tire can be expressed as:

[0210]

[0211] Fig. 9 (a) A flowchart of the tire steady-state contact analysis is given. First, the tire's contact rear angle is estimated. Since the contact front angle and contact rear angle are opposite to each other under steady-state conditions, the deformation coefficients α and β corresponding to the contact rear angle can be calculated and substituted into formula (29) to obtain the tire deformation at the front and rear contact angles; Substituting into formula (29) can solve the contact imprint force at the front and rear contact angles, and determine the value range of the contact angle based on the control of the contact imprint force at the contact angle; Taking the contact imprint force at the contact angle as 0 as the judgment condition, the dichotomy method is used to solve the contact angle of the flexible tire body, and determine the given flattening amount d0 or loading force F z Flexible carcass deformation coefficients α and β under conditions.

[0212] In addition, during the deformation of the flexible carcass in contact with the road surface, the sidewall curved beam with a large aspect ratio changes in structure, resulting in a change in the stiffness of the sidewall curved beam. Fig. 9 (b) The steady-state ground contact flow chart of the tire considering the nonlinear stiffness of the sidewall is given.

[0213] Attached Fig.10 (a)-(f) are the vertical and horizontal imprint forces of heavy-duty tires under different flattening conditions d0. The tire contact imprint under different flattening conditions is solved as shown in the attached figure. Fig. 9 As shown in (b), the relationship between the flattening amount and the load force can be obtained.

[0214] Example

[0215] In order to further verify the effectiveness of this method, the tire ground contact characteristics are analyzed in combination with the tire stiffness experimental characteristics.

[0216] First, a heavy-duty tire mechanical vibration experimental system based on the MTS hydraulic servo excitation device was built. The mechanical part is as follows: Fig.11As shown, the electronic control part is a computer and RSSigEdit. The system mainly includes: the heavy-loaded tire to be tested, the tire fixing fixture (used to limit the vertical freedom of the tire and make the tire move vertically), the MTS hydraulic servo excitation device, the displacement sensor, the force sensor and the acceleration sensor, etc. The heavy-loaded tire is connected to the gantry and the excitation head equipment of the platform through the fixture. The MTS hydraulic servo excitation is the excitation source of the system, and the static excitation is applied; the hydraulic excitation end is equipped with displacement and force sensors, and the displacement and loading force signals of the heavy-loaded tire are sampled through the computer data acquisition system to realize the monitoring of the tire loading force and deformation. At the same time, the ink method is used to measure the grounding mark of the heavy-loaded tire, and the loading is performed at equal intervals, such as Fig.12 As shown, 1t, 2t, 3t, 4t, 5t and 6t are loaded respectively, the carcass is equivalent to a flexible ring, the influence of the lateral direction of the carcass on the tire ground contact characteristics is ignored, and the main analysis is conducted on the length of the tire ground contact mark.

[0217] Attached Fig.13 The relationship between the tire flattening amount in the range of 0 to 0.1 m and the loading force, mark length and contact angle is given. The comparison results show that:

[0218] (1) Within the small deformation range (0-0.02m), the stiffness characteristics of the flexible ring model based on the linear elastic foundation are not significantly different from those of the flexible ring model based on the nonlinear elastic foundation. However, as the deformation increases, the stiffness of the flexible ring model based on the nonlinear elastic foundation is lower than that of the flexible ring model based on the linear elastic foundation, which is consistent with the increase in the inward deformation displacement of the sidewall curved beam and the decrease in the radial stiffness of the sidewall curved beam.

[0219] (2) The stiffness curve and flattening-imprint length curve of the flexible ring tire model based on nonlinear elastic foundation fit well with the experimental results, which verifies the accuracy of the model and indicates that the nonlinear stiffness of the high-aspect-ratio sidewall curved beam must be taken into account when analyzing the ground contact characteristics of high-aspect-ratio tires.

[0220] The above shows and describes the basic principles, main features and advantages of the present invention. It should be understood by those skilled in the art that the present invention is not limited to the above embodiments, and the above embodiments and descriptions are only for explaining the principles of the present invention. Without departing from the spirit and scope of the present invention, the present invention may have various changes and improvements, which fall within the scope of the present invention to be protected. The scope of protection of the present invention is defined by the attached claims and their equivalents.

Claims

1. A method for modeling the two-dimensional stiffness of a high aspect ratio sidewall and analyzing tire deformation characteristics, characterized in that: The following steps are involved: Step 1: Based on the flexible ring complete tire model based on elastic foundation considering the tire rotation effect, a simplified form of the tire dynamics model based on the "carcass inextensibility assumption" is obtained, and based on this simplified form, a flexible ring two-dimensional tire low-frequency dynamics model based on analytical elastic foundation is established; Step 2: Based on the Euler beam bending deformation theory, considering the inflation preload of the sidewall curved beam along the radial direction, a two-dimensional tire sidewall stiffness analytical model is established; Step 3: Based on the flexible ring two-dimensional tire low-frequency dynamic model, the sidewall tangential bending stiffness is inversely calculated using the sidewall tangential stiffness under different inflation pressures; Step 4: Introduce a grounding spring in the contact area, analyze the grounding characteristics of the flexible ring tire model constructed in step 1, and obtain the radial and tangential grounding imprint forces corresponding to different tire flattening amounts; Step 5: The tire sidewall tangential bending stiffness and ground contact characteristics obtained in steps 3-4 are verified through tire stiffness experiments.

2. According to the method for two-dimensional stiffness modeling and tire deformation characteristics analysis of a high aspect ratio sidewall according to claim 1, the specific operation steps of step 1 are: Step 11: Establish a flexible ring complete tire model based on elastic foundation considering the tire rotation effect: and in, E is the section modulus of the carcass ring beam, I is the section inertia of the carcass ring beam, w is the radial deformation of the carcass ring, v is the tangential deformation of the carcass ring, θ is the rotation angle of the carcass ring relative to the hub disk, x is the displacement of the hub disk in the X direction, y is the displacement of the hub disk in the Y direction, k w is the radial spring stiffness, k v Tangential spring stiffness, P0 is the rated air pressure, ρ is the equivalent density of the ring beam, Ω is the angular velocity of the disk, q w is the radial distributed load, q v is the tangential distributed load, I w is the disk moment of inertia, γ is the disk rotation angle, T0 represents the tire carcass preload, and Step 12: Based on the "carcass inextensibility assumption", the simplified form of equation (13) is obtained: Step 13: Based on the simplified form obtained in step 12, assuming that the tire rim is fixedly supported and does not rotate, a new simplified form of equation (13) is obtained: Step 14: Substituting into formula (19), we get: Then, the tire modal resonance frequency based on the "carcass inextensibility assumption" is: The ± sign indicates that the tire has two resonant frequencies at each modal order, which are opposite to each other.

3. The method for two-dimensional stiffness modeling and tire deformation characteristic analysis of a high aspect ratio sidewall according to claim 1, characterized in that: The specific steps of step 2 are: Step 21: Taking into account the chord preload effect and structural deformation effect of the sidewall curved beam, the analytical formula for the radial sidewall stiffness of the heavy-duty tire is established according to formula (1): Among them, θ is half of the angle of the sidewall curved beam, E s h is the elastic modulus of the sidewall curved beam along the radial direction, s is the sidewall section thickness, λ is the Poisson's ratio of the sidewall curved beam, P0 is the inflation pressure, l s is the length of the sidewall curved beam; Step 22: Based on the Euler beam bending deformation theory and considering the radial inflation preload of the sidewall curved beam, the tangential dynamic model of the sidewall curved beam is obtained. Among them, u tt is the tangential force f at the end point of the upper sidewall curved beam tt The displacement produced by the action; E t I t is the tangential bending stiffness of the sidewall curved beam; Step 23: Ignore the radial distributed inertia force of the sidewall curved beam and simplify equation (2) to: in, The general solution of formula (4) is: you tt (θ)=C1e βθ +C2e -βθ +C3e iβθ +C4e -iβθ (5) Among them, C1, C2, C3 and C4 are constant coefficients, which are related to the boundary conditions; Then formula (5) can be transformed into: you tt (θ)=C1 cosβθ+C2 sinβθ+C3 coshβθ+C4 sinhβθ (6) Its related variables are transformed into: Step 24: Assuming that the intersection of the sidewall curved beam and the circumferential continuous beam is fixed, equation (7) must satisfy the following equation: Step 25: Solve equation (8) and obtain: Then, in the tangential force f tt Under the action, the displacement u generated by the end point of the side curved beam tt for: you tt (θ)=C3(coshβθ c -cosβθ c )+C4(sinhβθ c -sinβθ c ) (10) In formula (10): The analytical formula of the tangential stiffness of the sidewall curved beam is: in, At the same time, R s = l s / θ c , and b t =1, then 4. The method for two-dimensional stiffness modeling and tire deformation characteristic analysis of a high aspect ratio sidewall according to claim 1, characterized in that: The specific steps of step 3 are: Step 31: Considering only the + form of equation (21), transform equation (20) into: Step 32: Using formula (22) to identify the tire sidewall tangential stiffness under different inflation pressures; Step 33: Using the tire dynamics model of formula (19), the sidewall tangential stiffness at different inflation pressures is used to inversely calculate the sidewall tangential bending stiffness E in formula (11) based on formula (21). t I t .

5. The method for two-dimensional stiffness modeling and tire deformation characteristic analysis of a high aspect ratio sidewall according to claim 1, characterized in that: The specific steps of step 4 are: Step 41: Define the point from the non-touchdown area to the touchdown area as the front touchdown point, and the point from the touchdown area to the non-touchdown area as the rear touchdown point. The angles between the front touchdown point and the rear touchdown point and the Y axis are φ f and φ r , the angle between any point in the grounding area and the Y axis is φ, and φ f ≤φ≤φ r ; Step 42: introducing a grounding spring in the grounding area, the grounding range is [-a~a], the grounding length is 2a, and the grounding pressure distribution in the grounding area is q(φ); Step 43: The grounding imprint force of the grounding area is calculated based on the deformation of the grounding spring and is: q(φ)=k2·w tread (f) =k2·[w(φ)+d0-(R+h)(1-cosφ)] (24) Where k2 represents the ground spring stiffness of the grounding area; Step 44: During the tire rolling process, the force between the tire and the road is distributed along the tire contact area. Considering that the deformation of any point of the tire should be along the footprint in the tire contact area, the deformation of any point φ is expressed as: Formula (25) is simplified to: in, Indicates angle; Step 45: Based on equation (26), the known contact angle φ of the flexible carcass is obtained: f and φ r Solve the equation for the grounding state under the conditions: Where, α={α1,α2,…,α n }, β={β1,β2,…,β n }; A, B, C and D matrices are n×n matrices, E and F matrices are n×1 matrices, and they are all tire contact angles φ f and φ r Function of Step 44: Calculate the radial and tangential contact imprint forces corresponding to different tire flattening amounts using equation (29).

6. The method for two-dimensional stiffness modeling and tire deformation characteristic analysis of a high aspect ratio sidewall according to claim 5, characterized in that: The specific operation steps of step 44 are: Step 441: estimating the tire's rear contact angle. Since the front contact angle and the rear contact angle are opposite to each other under steady-state conditions, the deformation coefficients α and β corresponding to the rear contact angle can be calculated to obtain the tire body deformation at the front and rear contact angles. Step 442: Calculate the ground contact force at the front and rear ground contact angles, and determine the value range of the ground contact angle according to the ground contact force at the ground contact angle; Step 443: Taking the ground contact mark force at the ground contact angle as 0 as the judgment condition, the ground contact angle of the flexible carcass is solved by the dichotomy method to determine the given flattening amount d0 or the loading force F z Flexible carcass deformation coefficients α and β under conditions; Step 444: Substitute the obtained deformation coefficients α and β into equation (29) to solve the contact force at the front and rear contact angles.

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