Stress and rigidity analysis method under angular load of multi-waveform diaphragm disc

Abstract

The invention discloses a stress and rigidity analysis method under a multi-waveform diaphragm disc angular load, which comprises the steps of defining geometric and working condition parameters, constructing a segmented deflection model, establishing a non-axisymmetric stress analysis model and deducing an angular rigidity analysis formula. Deducing a deflection expression of the maximum deformation section by segmenting the corrugated molded line and setting continuous conditions of displacement, slope and curvature; establishing a bending-torsion coupling model through a cylindrical coordinate system, and obtaining radial, circumferential and tangential stress analytic expressions in combination with Hooke's law; and the total bending moment is solved by performing circumferential integration on the infinitesimal bending moment and the shearing force, and the angular stiffness is calculated, so that integrated analytic solution of the deflection, the stress and the stiffness is realized. According to the method, high-precision prediction of the angular mechanical behavior of the complex corrugated diaphragm disc is achieved, the calculation efficiency is high, the result is stable and consistent, a large amount of finite element iteration can be effectively replaced, and a reliable tool is provided for structural optimization and engineering design of the diaphragm disc coupler.

Description

Technical Field

[0001] This invention belongs to the field of mechanical transmission component design and mechanical performance analysis technology, specifically to the mechanical characteristic analysis technology of multi-waveform diaphragm disk, a core component of flexible couplings. It is particularly suitable for stress distribution prediction and angular stiffness calculation of multi-waveform diaphragm disk under angular loads in high power density and high speed operating conditions (such as aero-engine, high-speed pump, and compressor transmission systems). Background Technology

[0002] Flexible couplings, as key connecting components in rotating machinery transmission systems, are widely used in aero-engines, high-speed compressors, high-speed centrifugal pumps, and other high-speed, high-power-density equipment. Their main function is to transmit torque and compensate for relative displacements between two shafts caused by installation errors, thermal deformation, or dynamic vibrations, including radial misalignment, axial runout, and angular deviation. With the continuous development of aerospace, energy equipment, and high-speed transmission devices towards higher power, higher speeds, and lighter weights, flexible couplings, while ensuring transmission efficiency and reliability, also need to possess higher angular compensation capabilities, higher torque carrying capacity, and longer fatigue life, placing more stringent requirements on their mechanical properties.

[0003] The most commonly used flexible couplings in current industrial applications mainly include diaphragm couplings and diaphragm disc couplings. Traditional diaphragm couplings typically employ a multi-layered, stacked diaphragm structure, achieving displacement compensation through in-plane stretching and out-of-plane bending of the diaphragm. While planar diaphragm structures offer advantages such as simple processing and convenient installation, their out-of-plane bending capacity is limited by the diaphragm's thickness, diameter, and material properties. Under large angular deviations, localized stress concentrations easily occur, leading to a decrease in fatigue life. With the continuous increase in transmission system power and speed, traditional planar diaphragms approach material limits in terms of stress and deformation under high torque and large angular compensation conditions, making it difficult to meet the lightweight and high load-bearing requirements of high-end equipment.

[0004] Diaphragm disc couplings, as another common type of high-performance flexible coupling, improve torque transmission capacity and compensation performance to some extent by introducing an annular diaphragm disc structure. However, traditional flat diaphragm disc structures still have inherent design limitations: the effective load-bearing area is mainly concentrated near the inner and outer connecting rings of the diaphragm disc, making it difficult to further increase the torque that can be transmitted per unit radial dimension (torque-to-diameter ratio); at the same time, the diaphragm disc has a single shape, limiting its elastic deformation capacity, and significant stress concentration may still occur under large angular deviation conditions, resulting in a bottleneck in fatigue life. In addition, in practical applications, the stress distribution of flat diaphragm discs is highly dependent on geometric dimensions and assembly deviations, and its sensitivity to manufacturing and assembly precision limits the space for structural optimization.

[0005] To overcome the limitations of diaphragm-disc couplings in terms of compensation capacity and load-bearing efficiency, existing technologies have proposed various structural improvement schemes, such as changing the diaphragm thickness, increasing the number of diaphragm discs, and adopting local chamfering or equal-thickness zone structures, in order to improve stress concentration. However, most of these improvements are still based on traditional planar structures, and their improvement in compensation performance is limited. They still cannot meet higher performance requirements in terms of structural lightweighting and stress distribution. At the same time, to adapt to high-speed operating conditions, some technologies have proposed using material reinforcement and new laminated structures, but overall, they still cannot avoid the inherent structural deficiencies of planar diaphragms or flat diaphragm discs when compensating at large angles.

[0006] In recent years, to adapt to high-load and high-deviation operating conditions, some studies have proposed using corrugated diaphragms, corrugated diaphragm disks, or other elastic elements with periodic structures to improve the out-of-plane bending characteristics of the diaphragm disks, enabling them to achieve greater elastic deformation capacity while maintaining constant radial dimensions. Related publicly available technologies generally employ periodic corrugations to enhance structural compliance and, to some extent, improve torque transmission capacity. However, the geometric complexity of these structures is significantly higher than that of traditional planar diaphragms. The presence of periodic corrugations causes the load-bearing path and stress distribution to exhibit significant non-axisymmetric characteristics, and the mechanical behavior no longer satisfies traditional circular symmetry conditions, posing considerable challenges to their mechanical analysis and design methods.

[0007] In existing patents and literature, mechanical studies on corrugated diaphragm or wave-shaped diaphragm disk structures mainly focus on axial load conditions, such as axial stiffness analysis, axial displacement compensation capability, and axial stress distribution. For example, some technologies establish axial stiffness models by simplifying the corrugations to equivalent flat plate structures; others improve the overall load-bearing capacity by stacking multiple diaphragms. However, under angular deviation, the most critical load condition in actual operation, the existing mechanical analysis methods are still insufficient. Especially for multi-wave-shaped diaphragm disks with multiple corrugation periods, they will produce complex three-dimensional bending deformation under angular loads, and the stiffness distribution at different circumferential positions of the corrugations is significantly uneven, resulting in large differences in stress levels and bending responses in different parts of the diaphragm disk. Traditional shell theory methods based on axisymmetric assumptions often cannot accurately describe these non-axisymmetric key mechanical characteristics.

[0008] To improve analytical accuracy, some publicly available techniques attempt to use the finite element method (FEM) for modeling and stress analysis of wave-shaped diaphragm disks. While the FEM can achieve high-precision stress and deformation predictions, it has significant drawbacks: its modeling process is complex, especially in high-stress areas such as corrugation inflection points and diaphragm disk edge transition zones, requiring dense mesh generation; the computation process demands substantial hardware resources and computation time, making it unsuitable for rapid design iterations and large-scale parameter optimization. Furthermore, the FEM calculation results are highly sensitive to modeling, mesh generation, and boundary settings, often requiring repeated model adjustments in engineering applications, resulting in low overall analysis efficiency.

[0009] Therefore, there is an urgent need for an analytical method that can accurately predict the stress distribution and angular stiffness of multi-waveform diaphragm discs under angular loads and has high computational efficiency, in order to fill the gap in existing technology, provide a reliable theoretical tool for the rapid design and parameter optimization of waveform diaphragm disc couplings, and promote their further application in high-end equipment transmission systems. Summary of the Invention

[0010] This invention aims to solve the technical problems of complex deformation modes, non-axisymmetric stress distribution, and difficulty in efficiently and accurately obtaining angular stiffness of multi-waveform diaphragm disks under angular loads. Existing technologies relying on the finite element method are not only complex in modeling but also computationally time-consuming, making it difficult to support rapid design. This application discloses a stress and stiffness analysis method for multi-waveform diaphragm disks under angular loads. Based on plate and shell theory, it establishes a piecewise deflection model, non-axisymmetric strain and stress expressions, and a total bending moment-deflection angle relationship, achieving continuous analytical solutions for deflection, stress, and stiffness.

[0011] In view of this, the present invention provides a stress and stiffness analysis method for a multi-waveform diaphragm disk under angular load, comprising: Step 1: defining the geometric parameters and working parameters of the multi-waveform diaphragm disk, and establishing a set of geometric features based on the radial dimensions of the diaphragm disk, the corrugated profile, and the connecting hub and rim structure, for constructing a mechanical analysis model under angular load; Step 2: Based on the plate and shell theory, assume the deflection function of the maximum deformation section of the multi-waveform membrane disk under angular load, and divide the membrane disk into segments according to the periodicity of the corrugated structure. Combine the continuity conditions of the inner ring boundary, outer ring boundary and segment boundary to derive the analytical expression of the vertical deflection of the maximum deformation section. Step 3: Based on the cylindrical coordinate system, establish a non-axisymmetric bending-torsional coupling mechanical model for the multi-waveform diaphragm disk. Obtain analytical expressions for the radial stress, circumferential stress and tangential stress of the diaphragm disk under angular load through geometric equations and physical equations, and form a stress distribution prediction model under angular load. Step 4: Based on the moment balance and shear force balance relationships of the membrane disk micro-element, establish the total moment-deflection angle equation and derive the analytical formula for the angular stiffness of the membrane disk under angular load.

[0012] In some examples of this application, the geometric parameters defined in step one include: inner radius b of the profile, outer radius a of the profile, corrugation thickness h, corrugation amplitude H, hub thickness h1, rim thickness h2, hub transition fillet radius r2, rim transition fillet radius r3, inner radius r0 of the diaphragm disk, and outer radius r1 of the diaphragm disk; the operating parameters include the number of diaphragm disks m and the operating speed r.

[0013] In some examples of this application, when segmenting the corrugated profile, the corrugation is divided into at least two segments along the radial direction. Deflection assumption functions v1 and v2 are established for each segment, and the conditions of displacement continuity, slope continuity and curvature continuity are satisfied at the segment boundary.

[0014] In some examples of this application, in step two, after the piecewise deflection functions v1 and v2 of the first and second segments are established, they are respectively set The boundary conditions for the inner ring are: vertical deflection v1 = t, where t is the displacement of the inner ring, and bending moment M = 0; The piecewise boundary conditions are: v1=v2, dv1 / dr=dv2 / dr, d 2 v1 / dr 2 =d 2 v2 / dr 2 ; The outer boundary conditions are: vertical deflection v2=0, dv2 / dr=0; Concave-convex constraint points are set within the segmented intervals to ensure that the second derivative of the ripple segment is zero.

[0015] In some examples of this application, in step three, a cylindrical coordinate system with the Z-axis coinciding with the symmetry axis of the diaphragm disk is used to establish a non-axisymmetric mechanical model with radius r and circumferential rotation angle θ as variables. Based on the deflection function v(r), its spatial variation relationship, and the angle φ between the corrugation normal and the symmetry axis of the diaphragm disk, the geometric equations of radial strain εr, circumferential strain εθ, and tangential strain γrθ are derived so that the deflection curve, deformation gradient, and corrugation tilt angle can be used to characterize the non-axisymmetric deformation of the diaphragm disk under angular load.

[0016] In some examples of this application, in step three, the radial strain εr, circumferential strain εθ, and tangential strain γrθ are substituted into Hooke's law to establish physical equations, thereby converting them into radial stress σr, circumferential stress σθ, and tangential stress τrθ, respectively, to obtain stress expressions that can reflect the multi-directional mechanical response of the diaphragm disk under angular loads, wherein: Radial stress σr=E / (1-μ) 2 )・(εr+μεθ); Circumferential stress σθ=E / (1-μ) 2 )・(εθ+μσr); Shear stress τrθ=E / (2 (1+μ))・γrθ; E is the elastic modulus, and μ is Poisson's ratio.

[0017] In some examples of this application, in step four, the total bending moment M is obtained by circumferentially integrating the bending moment and shear force of the membrane disk element, and the angular stiffness K is calculated based on the relationship between the total bending moment and the angular deflection angle α. K=M / α.

[0018] In some examples of this application, the total bending moment M is obtained by circumferentially integrating the horizontal bending moment Mr, the circumferential bending moment Mθ, the torque Mrθ, and the shear force Qr of the membrane disk element, and establishing the overall static equation by combining the force balance relationship Qr=(Mr + ∂Mr / ∂r + Mθ) / r, so that the total bending moment M is obtained by superposition of ∫(0~2π) Mr・cosθ・R・dθ and ∫(0~2π) Qr・h・R・cosθ・dθ.

[0019] In some examples of this application, the analytical calculations in steps two through four are implemented using MATLAB programming.

[0020] In some examples of this application, the stress and stiffness analysis method of the multi-waveform diaphragm disk under angular load further includes step five, which uses numerical simulation to obtain reference data of the deflection, stress and stiffness of the multi-waveform diaphragm disk, and compares them with the analytical calculation results to verify the accuracy of the model.

[0021] Compared with existing technologies, the stress and stiffness analysis method for multi-waveform diaphragm disks under angular loads described in this invention has the following advantages: 1. This application constructs a set of geometric features based on real corrugated lines and hub flange structures, and establishes a segmented deflection analytical model by combining plate and shell theory. This enables the three-dimensional bending and local inverse bending behavior of multi-waveform membrane disks under angular loads to be fully expressed mathematically. By setting continuous conditions for displacement, slope, and curvature at the segment boundaries, the entire deflection surface maintains physical consistency and mathematical smoothness. At the same time, geometric equations reflecting the corrugation tilt effect and radial gradient change are introduced, so that the strain expression can include the non-axisymmetric deformation characteristics unique to corrugated structures. Furthermore, by combining Hooke's law, the strain is transformed into radial, circumferential, and tangential stresses to obtain an analytical expression that fully reflects the stress state of the membrane disk. This allows the overall method to analytically reproduce the complex mechanical behavior of multi-waveform membrane disks and significantly improves the ability to describe the deformation law and stress distribution characteristics of structures.

[0022] 2. This application uses the membrane disk micro-element as the force-bearing unit, establishes the balance relationship between horizontal bending moment, circumferential bending moment and shear force respectively, and derives the total bending moment expression by integrating the bending moment contribution in the entire circumferential direction. This allows the angular stiffness to be directly calculated from the total bending moment and angular deflection angle, which reflect the actual structural force characteristics. This avoids the stiffness deviation caused by neglecting the shear force contribution or using a simplified structural model in traditional theories. At the same time, the stiffness calculation process can accurately reflect the complex effects of bending moment coupling, corrugation periodicity and local bending softening in multi-wave structure. The final angular stiffness has a clear physical meaning and can truly reflect the overall bending resistance of the membrane disk in actual working conditions, thus forming a stiffness calculation model that combines theoretical rigor and engineering applicability.

[0023] 3. This application transforms the piecewise deflection coefficient solution, stress expression derivation, and angular stiffness calculation into a MATLAB-executable symbolic computation, linear algebra solution, and numerical integration process. This enables the entire mechanical analysis process to be executed automatically within a unified code framework, significantly reducing errors caused by manual derivation and greatly improving the efficiency of parameter modification and model iteration. At the same time, the analytical model can be verified through finite element simulation results, ensuring that the model has the reliability and consistency required for engineering applications based on theoretical interpretability. This method is highly suitable for the rapid design, structural optimization, stress verification, and life prediction of high-performance diaphragm couplings, effectively reducing the design cycle and dependence on finite element software and manual experience, thereby improving the efficiency and accuracy of overall engineering analysis. Attached Figure Description

[0024] Figure 1 This is a geometric shape diagram of the multi-waveform diaphragm disk described in an embodiment of the present invention; Figure 2 This is a schematic diagram of the maximum deformation cross-sectional angle of the multi-waveform diaphragm disk according to an embodiment of the present invention; Figure 3 This is a schematic diagram of the corrugation segmentation of the multi-waveform diaphragm disk according to an embodiment of the present invention; Figure 4 This is a cross-sectional micro-element analysis diagram of the multi-waveform diaphragm disk described in an embodiment of the present invention; Figure 5 This is a force analysis diagram of the micro-element of the multi-waveform diaphragm disk described in the embodiment of the present invention; Figure 6 This is a comparison diagram of the vertical deflection curves of the multi-waveform diaphragm disk described in the embodiments of the present invention; Figure 7 This is a graph showing the horizontal and circumferential stresses relative to x of the multi-waveform diaphragm disk under angular load, calculated analytically using an analytical method according to an embodiment of the present invention. Figure 8The curves of horizontal stress and circumferential stress relative to x under angular load are obtained by Ansys finite element analysis of the multi-waveform diaphragm disk described in this embodiment of the invention. The markings in the diagram are as follows: 1-Outer ring bolt holes; 2-Sine wave pattern; 3-Inner ring spline. Detailed Implementation

[0025] The technical solutions of the embodiments of this application will be clearly described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of this application. All other embodiments obtained by those skilled in the art based on the embodiments of this application are within the scope of protection of this application.

[0026] It should be noted that all directional and positional terms used in this invention, such as "up," "down," "left," "right," "front," "back," "vertical," "horizontal," "inner," "outer," "top," "lower," "lateral," "longitudinal," and "center," are only used to explain the relative positional relationships and connections between components in a specific state (as shown in the accompanying drawings). They are merely for the convenience of describing the invention and do not require the invention to be constructed and operated in a specific orientation; therefore, they should not be construed as limitations on the invention. Furthermore, descriptions involving "first," "second," etc., are for descriptive purposes only and should not be construed as indicating or implying their relative importance or implicitly specifying the number of technical features indicated.

[0027] In the description of this invention, unless otherwise explicitly specified and limited, the terms "installation," "connection," and "linking" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; and they can refer to the internal communication between two components. Those skilled in the art can understand the specific meaning of the above terms in this invention based on the specific circumstances.

[0028] In the description of this specification, the references to terms such as "one embodiment," "some embodiments," "illustrative embodiment," "example," "specific example," or "some examples," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of the invention. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples.

[0029] This application discloses a method for stress and stiffness analysis under angular loads on a multi-waveform diaphragm disk, including: Step 1: Define the geometric and operating parameters of the multi-waveform diaphragm disk, and establish a set of geometric features based on the radial dimensions of the diaphragm disk, the corrugated profile, and the connecting hub and rim structure, which will be used to construct a mechanical analysis model under angular loads; Step 2: Based on the plate and shell theory, assume the deflection function of the maximum deformation section of the multi-waveform membrane disk under angular load, and divide the membrane disk into segments according to the periodicity of the corrugated structure. Combine the continuity conditions of the inner ring boundary, outer ring boundary and segment boundary to derive the analytical expression of the vertical deflection of the maximum deformation section. Step 3: Based on the cylindrical coordinate system, establish a non-axisymmetric bending-torsional coupling mechanical model for the multi-waveform diaphragm disk. Obtain analytical expressions for the radial stress, circumferential stress and tangential stress of the diaphragm disk under angular load through geometric equations and physical equations, and form a stress distribution prediction model under angular load. Step 4: Based on the moment balance and shear force balance relationships of the membrane disk micro-element, establish the total moment-deflection angle equation and derive the analytical formula for the angular stiffness of the membrane disk under angular load.

[0030] This application discloses a method for stress and stiffness analysis of multi-waveform membrane disks under angular loads. In the structural analysis of multi-waveform membrane disks, it first establishes a unified set of geometric features using geometric parameters and working conditions as inputs. This gives the analyzed object clear boundaries and structural representation, providing a stable input basis for subsequent calculations and reducing repeated corrections caused by unclear parameters in traditional modeling. Based on plate and shell theory, a deflection function is constructed for the maximum deformation section under angular loads. According to the periodicity of the corrugated structure, the membrane disk is divided into mechanical intervals composed of multiple continuous segments. By setting boundary constraints between the inner and outer rings and the continuity conditions of displacement and derivatives between segments, a solvable continuous deformation model is formed. This yields the analytical form of the vertical deflection of the maximum deformation section. This not only significantly improves the fitting degree to the actual deformation law but also transforms the originally complex three-dimensional corrugated structure into a solvable segmented continuous model, thereby improving the efficiency and accuracy of deformation analysis. Based on this, A non-axisymmetric mechanical model with coupled bending and torsion is constructed using a cylindrical coordinate system, enabling the complex mechanical behavior of the membrane disk under angular loads to be expressed within a unified mathematical framework. The displacement and strain fields are correlated through geometric equations, and physical equations are formed by combining material constitutive relations. Analytical expressions for radial stress, circumferential stress, and tangential stress are derived, allowing for stress distribution prediction under angular loads. This avoids the time-consuming problem of relying solely on finite element simulations and enables rapid stress distribution prediction. Finally, using the membrane disk micro-element as the research object, the relationship between bending moment balance and shear force balance under angular loads is analyzed to establish a mathematical equation between the total bending moment and the angular deflection angle. Furthermore, an analytical expression for angular stiffness is obtained, providing quantitative results for the membrane disk's angular stiffness without relying on extensive numerical iterations. This achieves a continuous solution process from deformation to stress to stiffness, forming a complete mechanical analysis system under angular loads.

[0031] The stress and stiffness analysis method for multi-waveform diaphragm disks under angular loads disclosed in this application establishes a complete system covering deformation analysis, stress calculation, and stiffness solution, thereby improving the overall calculation efficiency, analysis accuracy, and parameter controllability of the mechanical properties of multi-waveform diaphragm disks under angular loads. It is particularly suitable for rapid design verification and structural optimization scenarios of high-performance diaphragm disk couplings.

[0032] In a preferred embodiment of this application, the geometric parameters defined in step one include: inner radius b of the profile, outer radius a of the profile, corrugation thickness h, corrugation amplitude H, hub thickness h1, rim thickness h2, hub transition fillet radius r2, rim transition fillet radius r3, inner radius r0 of the diaphragm disk, and outer radius r1 of the diaphragm disk; the operating parameters include the number of diaphragm disks m and the operating speed r. In step one of this application, through the aforementioned further limitations, its working principle is as follows: by using the inner radius b and outer radius a of the profile as the boundary of the corrugated profile, and the corrugation thickness h and corrugation amplitude H as the main control quantities of corrugation compliance, and by using the hub thickness h1, rim thickness h2, hub transition fillet radius r2, and rim transition fillet radius r3 as structural parameters of the force flow transmission area at the connection point, all key parts of the diaphragm disk are represented in a specific quantitative way. This allows the entire mechanical analysis model to express the deformation mode, stress distribution, and stiffness level in a parametric manner. Simultaneously, the inner radius r0 and outer radius r1 of the diaphragm disk are used to construct the overall size range, giving the model a unified structural scale basis. The number of diaphragm disk pieces m in the operating parameters allows the model to describe the difference between a single diaphragm disk and a multi-piece stacked structure within the same theoretical system, while the operating speed r... The inclusion of this method enables the model to reflect the additional stress and stiffness changes brought about by the centrifugal field, making the entire mechanical analysis model applicable not only to static calculations but also to actual operating conditions. Furthermore, by ignoring local structures such as bolt holes and splines that have little impact on the mechanical behavior of the corrugated region during geometric modeling, the model can focus on the deformation and stress-dominant areas of the corrugated body. This avoids allocating computational resources to local micro-features that do not affect the overall mechanical trend, thereby reducing the complexity of the mathematical model and simplifying the subsequent deflection function setting, segmented region division, and boundary condition solution process. This makes the overall analysis more efficient, stable, and easier to match with the analytical solution process, and ensures that the core mechanical behavior does not exhibit unnecessary complexity due to local details. This further improves the solvability, stability, and computational efficiency of the overall model, enabling this method to achieve higher structural matching and analytical reliability in engineering design with lower computational costs.

[0033] Specifically, in the example of step one of this application, the geometric parameters and operating parameters of the multi-waveform diaphragm disk are clearly defined, and the interference of local structures such as bolt holes and splines on the corrugated mechanical behavior is ignored. The parameters are as follows (unit: mm, rotational speed unit: r / min): • Inner radius of profile b = 104, outer radius of profile a = 162; • Corrugated line thickness h=0.5, corrugated line amplitude H=3.5; • Hub thickness h1=1.2, rim thickness h2=1.2; • Hub transition fillet radius r2=2, rim transition fillet radius r3=2; • The inner radius of the diaphragm disk is r0 = 89, and the outer radius of the diaphragm disk is r1 = 182; The number of membrane disks in the membrane disk assembly is m=15, and the operating speed is 5200.

[0034] In a preferred embodiment of this application, in step two, when segmenting the corrugated line, the corrugation is divided into at least two segments radially, and deflection assumption functions v1 and v2 are established for each segment, satisfying the conditions of displacement continuity, slope continuity, and curvature continuity at the segment boundaries. In step two, this application further divides the corrugated line radially into at least two segments, allowing different segments to establish their own deflection assumption functions based on the differences in mechanical constraints at their locations. The inner ring segment is closer to the hub, and its deformation is more significantly controlled by the support and constraints of the hub structure, while the outer ring segment is closer to the rim, exhibiting greater deformation freedom and different stress transition modes. Therefore, segmentation allows the form and coefficients of the deflection function to better match the actual constraints corresponding to the segments, making the model more accurate in a local range. Furthermore, by setting displacement continuity conditions at the segment boundaries, the actual deformation of the two deflection functions at the boundaries is ensured to be consistent; by setting slope continuity conditions, the actual deformation of the two deflection functions at the boundaries is ensured to be consistent. The deflection direction transitions smoothly at the boundary; curvature continuity conditions are set to ensure that the degree of bending remains consistent at the boundary, making the piecewise model conform to the actual physical characteristics of continuous structures; in this way, the entire deflection model reflects the deformation characteristics of multi-waveform structures in different regions in the form of a combination of piecewise functions, while ensuring continuity and smoothness in the global scope, so that the analytical derivation process can be carried out mathematically stably, and subsequent stress calculation and stiffness analysis can be derived based on the continuous and differentiable deflection expression, reducing stress overshoot or stiffness anomalies caused by boundary discontinuities, so that the entire deflection model has both local adaptability and global consistency, thereby achieving accurate simulation of the deformation law of multi-waveform membrane disks under angular loads.

[0035] In a preferred embodiment of this application, in step two, after the piecewise deflection functions v1 and v2 of the first and second segments are established, respectively, the following settings are made: The boundary conditions for the inner ring are: vertical deflection v1 = t, where t is the displacement of the inner ring, and bending moment M = 0; The piecewise boundary conditions are: v1=v2, dv1 / dr=dv2 / dr, d 2 v1 / dr 2 =d 2 v2 / dr 2 ; The outer boundary conditions are: vertical deflection v2=0, dv2 / dr=0; Concave-convex constraint points are set within the segmented intervals to ensure that the second derivative of the ripple segment is zero.

[0036] In step two, this application introduces inner boundary conditions, segmented boundary continuity conditions, outer boundary conditions, and concave-convex constraint points to ensure that the segmented deflection functions v1 and v2 mathematically conform to the actual deformation process of the multi-waveform membrane disk under angular loads. The inner boundary conditions v1=t and M=0 reflect the displacement relationship of the inner ring directly controlled by the deflection angle α and the actual support state without additional bending moments, ensuring that the inner ring's state as the deformation starting point is consistent with reality. The displacement continuity, slope continuity, and curvature continuity conditions of the segmented boundaries are mathematical expressions of the membrane disk as a continuous solid structure, ensuring a natural connection between the deformation direction and bending degree in the transition region. Essentially, this ensures the continuity of the functions on the zeroth, first, and second derivatives, giving the overall deflection model a smooth surface-like continuity and avoiding false stress peaks caused by mathematical discontinuities. The outer boundary conditions v2=0 and dv2 / dr=0... This simulates the actual constraint state after the rim is fixed, allowing the model to converge naturally on the outer ring and avoiding non-physical outer ring warping; convexity constraint d 2 v2 / dr 2 Setting =0 mathematically defines the inflection point, enabling the deflection function to reproduce the mechanical properties of the change in the bending direction of the corrugations. This allows the model to not only match the overall structure but also reflect the true mechanical performance of the micro-corrugations. By uniformly substituting all conditions into the piecewise function to solve for the coefficients, the final analytical expression of deflection can reflect the mechanical differences in different radial sections while ensuring the overall uniformity and continuity of the model. This makes the entire deflection derivation process fully conform to the actual deformation law of the plate and shell structure under angular bending.

[0037] Specifically, in the example of step two of this application, under angular load, the maximum deformation section of the multi-waveform diaphragm disk is located in the angular deflection plane (deflection angle is α, as shown in Figure 2). The analytical expression of the deflection curve is derived using the "piecewise function assumption + boundary condition constraints": Corrugated segmentation: The axial section of the sinusoidal corrugation is divided into two segments. The first segment is the inner circle side with r∈[104,118.5] mm, and the second segment is the outer circle side with r∈[118.5,162] mm (including the second, third, and fourth segments in the original document, simplified using the periodicity of the structure). Deflection function assumption: First segment (r∈[104,118.5]): Let the vertical deflection function v1 satisfy the boundary constraint of the inner circle displacement t (t=104・tanα) of the membrane disk; Second segment (r∈[118.5,162]): Let the vertical deflection function be v2, and take the ripple frequency ω=108.331 (matching the ripple line period). Boundary conditions: Inner boundary (r=104mm): bending moment M=0, vertical deflection v1=t; Segment boundary (r=118.5mm): v1=v2, dv1 / dr=dv2 / dr, d 2 v1 / dr 2 =d 2 v2 / dr 2 (To ensure curve continuity); Outer boundary (r=162mm): v2=0, dv2 / dr=0; Second section of concave-convex constraint (r=133mm, r=147.5mm): d 2 v2 / dr 2 =0; Derivation of the analytical expression: Substitute the boundary conditions into the piecewise deflection function, solve for the function coefficients, and obtain the analytical expression of the vertical deflection curve at the section with the maximum deformation.

[0038] By setting the above, the deflection analytical expression obtained by solving the piecewise function not only satisfies the mathematical condition of overall continuity and smoothness, but also conforms to the real mechanical characteristics of the corrugated structure in bending, reverse bending and support transition, which significantly improves the accuracy of the final deflection analytical expression. This ensures that the subsequent stress and stiffness calculations are based on accurate and reliable deformation data, making the entire multi-waveform membrane disk analysis method more reliable and valuable in engineering design, structural optimization and life prediction.

[0039] In a preferred embodiment of this application, in step three, a cylindrical coordinate system with the Z-axis coinciding with the symmetry axis of the diaphragm disk is used to establish a non-axisymmetric mechanical model with radius r and circumferential rotation angle θ as variables. Based on the deflection function v(r), its spatial variation relationship, and the angle φ between the corrugation normal and the symmetry axis of the diaphragm disk, the geometric equations for radial strain εr, circumferential strain εθ, and tangential strain γrθ are derived to characterize the non-axisymmetric deformation of the diaphragm disk under angular load using the deflection curve, deformation gradient, and corrugation tilt angle. This application further specifies that by projecting the mechanical behavior of the multi-waveform membrane disk into a cylindrical coordinate system matching its structural shape, the radius r can represent the actual dimensional change of the membrane disk from the inner to the outer ring, the circumferential rotation angle θ can represent any circumferential position, and the Z-axis coincides with the axis of symmetry to ensure that bending, strain, and stress calculations are all based on the axis of symmetry, thereby simplifying the mechanical mapping relationship of the complex membrane disk structure in space. By deriving the radial, circumferential, and tangential strains based on the deflection function v(r), its spatial variation relationship, and the angle φ between the corrugation normal and the axis of symmetry of the membrane disk, the influence of membrane disk bending deformation, corrugation tilt angle, and radial variation trend on strain is comprehensively incorporated into the same geometric equation system. This makes the strain calculation closer to the real deformation mechanism of the corrugated structure under angular loads, avoiding the distortion caused by simplifying the corrugated structure to a planar structure. The resulting radial strain εr, circumferential strain εθ, and tangential strain γrθ... It can fully describe the deformation characteristics of corrugated structures under bending, torsion, and local rotation, allowing the model to be built layer by layer based on the actual structural deformation. It integrates geometric information, structural features, and spatial attitude into the strain expression, thereby reducing the calculation deviation caused by ignoring the corrugation tilt angle or deflection gradient in traditional models. It also ensures that the strain results accurately reflect the real response of the multi-waveform structure in all directions, enabling subsequent stress derivation, stiffness solution, and load distribution analysis to be completed on a basis that is closer to the real structural behavior. At the same time, it provides more accurate mechanical input for multi-waveform membrane disks under complex working conditions, making the overall analysis process more reliable and stable.

[0040] In a preferred embodiment of this application, in step three, the radial strain εr, circumferential strain εθ, and tangential strain γrθ are substituted into Hooke's law to establish physical equations, thereby converting them into radial stress σr, circumferential stress σθ, and tangential stress τrθ, respectively, to obtain stress expressions that can reflect the multi-directional mechanical response of the diaphragm disk under angular loads. This application, through the above further refinement, first obtains the strain of the diaphragm disk in different directions through geometric relationships, and considers these strains as the fundamental deformation variables generated by material micro-elements under external forces. Then, using the linear elastic law described by Hooke's law, the strain is proportionally converted in each direction using material constants such as elastic modulus and Poisson's ratio. Within the framework of three-dimensional elastic theory, the coupling effect of deformation in different directions is considered simultaneously, further converting it into radial stress σr and circumferential stress σθ. The addition of tangential stress τrθ transforms the previous method of obtaining stress values ​​based on empirical coefficients or simplified assumptions into a direct conversion method based on the elastic relationship of materials. This allows stress calculations to closely rely on the actual changing trends of strain data, eliminating omissions or deviations in stress expression due to structural complexity or varied corrugation morphology. Furthermore, it ensures mathematical continuity and consistency of stress in all directions, fully revealing the multi-directional mechanical behavior of the diaphragm disk under angular loads and maintaining linear characteristics consistent with material properties. Consequently, the obtained stress expression not only possesses physical interpretability but also connects with subsequent stress distribution equation stiffness derivation and steady-state response analysis, making the entire mechanical modeling process more standardized and reliable, and improving the model's adaptability to multi-waveform geometries and complex loading conditions.

[0041] Specifically, the stress analysis under angular load in this application is as follows: A non-axisymmetric mechanical model is established based on a cylindrical coordinate system (where the Z-axis coincides with the symmetry axis of the membrane disk, and the radius r and circumferential rotation angle θ are variables), and the stress distribution is derived: Derivation of geometric equations: Radial strain εr: derived from the arc length increment of bending deformation, εr=z・cosφ・dα / dr (z is the coordinate of the film disk thickness direction, φ is the angle between the corrugation normal and the Z axis, and α is the angle between the tangent of the infinitesimal element and the normal). Circular strain εθ: Considering torsional deformation, εθ = z・α / r + (∂w / ∂θ) / r (w is the normal displacement of the mid-surface); Shear strain γrθ: derived from the first derivative of deflection with respect to r and θ, γrθ = -2・cosφ・(∂w / ∂r∂θ); Derivation of the physical equation: Based on Hooke's law, substituting strain into the equation yields the stress expression: Radial stress σr=E / (1-μ) 2 )・(εr+μεθ); Circumferential stress σθ=E / (1-μ)2 )・(εθ+μσr); Shear stress τrθ = E / (2 (1+μ))・γrθ; (E is the elastic modulus, μ is Poisson's ratio) Bending stress calculation: Using the projection relationship between deflection w and vertical deflection v, w=v・cosφ, combined with the analytical expression of deflection, the bending stress on the upper surface of the membrane disk is obtained by substituting into the stress formula (taking z=h / 2).

[0042] In a preferred embodiment of this application, in step four, the total bending moment M is obtained by circumferentially integrating the bending moment and shear force of the membrane disk micro-element, and the angular stiffness K is calculated based on the relationship between the total bending moment and the angular deflection angle α. K=M / α.

[0043] In a preferred embodiment of this application, the total bending moment M is obtained by performing circumferential integration on the horizontal bending moment Mr, the circumferential bending moment Mθ, the torque Mrθ, and the shear force Qr of the membrane disk element, and establishing an overall static equation based on the force balance relationship Qr=(Mr + ∂Mr / ∂r + Mθ) / r, so that the total bending moment M is obtained by superposition of ∫(0~2π) Mr・cosθ・R・dθ and ∫(0~2π) Qr・h・R・cosθ・dθ.

[0044] This application, based on the aforementioned limitations, decomposes the bending mechanical response of the membrane disk under angular load into infinitesimal horizontal bending moment, circumferential bending moment, and shear force. The total bending moment M is then formed by the integral accumulation of these infinitesimal elements in the circumferential direction. Furthermore, the angular stiffness K is calculated using the proportional relationship between the total bending moment M and the angular deflection angle α. This transforms the overall stiffness parameter, which previously required multiple finite element iterations, into an analytical quantity that can be directly obtained from the accumulation of infinitesimal mechanical quantities. This shifts the overall stiffness evaluation from complex numerical solutions to a direct expression based on the actual force structure. The comprehensive introduction of infinitesimal bending moment and shear force ensures that the coupling effect caused by the membrane disk's corrugated structure is not overlooked in the calculation of the total bending moment, resulting in a more complete and accurate stiffness determination. The angular stiffness is defined using the formula K=M / α, allowing the stiffness value to directly reflect the overall bending capacity required for unit angular deformation. The core of this method lies in constructing a bottom-up stiffness expression system based on infinitesimal force elements, so that the contributions of bending moment and shear force can be truly reflected in the overall stiffness, avoiding oversimplification of corrugated structures or multi-directional load distribution, thereby making the obtained angular stiffness traceable and physically consistent, and finally forming a mechanical analysis framework that simultaneously possesses structural integrity, stress continuity and load interpretability.

[0045] Specifically, in the angular stiffness calculation of this application example, considering the total bending moment caused by the combined effects of bending stress and shear force, the analytical formula for angular stiffness is derived: Infinite element force analysis: Take the membrane disk infinitesimal element (as shown in Figure 5) and analyze the balance of horizontal bending moment Mr, circumferential bending moment Mθ, torque Mrθ and shear force Qr. The moment balance equation is established as: Mr + (∂Mr / ∂r)・r・dθ・dr + Mθ・dθ・dr - Qr・r・dθ・dr=0. Ignoring higher-order infinitesimals, we get Qr=(Mr + ∂Mr / ∂r + Mθ) / r; Total bending moment calculation: Let the total bending moment of the membrane disk be M, and the outer radius be R. Establish the static equilibrium equation: ∫(0~2π) Mr・cosθ・R・dθ + ∫(0~2π) Qr・h・R・cosθ・dθ=0, and solve for the total bending moment M. Derivation of the stiffness formula: Angular stiffness K = M / α (α is the angular deflection angle). Substituting the relationship between the total bending moment M and α, we obtain the analytical formula for angular stiffness.

[0046] In the preferred example of this application, the analytical calculations in steps two to four are implemented using MATLAB programming. Through the above setup, this application transcribes the entire process of deflection calculation, strain and stress derivation, and stiffness determination of the multi-waveform membrane disk under angular load into computer-executable mathematical expressions and logical instructions. The symbolic computation toolbox maintains the piecewise deflection function, boundary conditions, geometric equations, and Hooke's law in symbolic form for analytical derivation. The linear algebra solution module automatically obtains the unknown coefficients of the deflection function. The partial differential equation module performs symbolic differentiation on the deflection function to obtain radial, circumferential, and tangential strains, which are then substituted into the physical equations to form stress expressions in each direction. Finally, the numerical integration module integrates the circumferential interval from 0 to 2π to obtain the cumulative effect of bending moment and shear force along the entire circumferential direction, thereby obtaining the total bending moment M and finally using K=M / α. The formula for obtaining angular stiffness is used, and the entire process solidifies the mathematical derivation chain completely with code logic. The computer automatically executes each step of the calculation according to the predetermined order and formula, so that the substitution, differentiation, solution and integration of all variables are carried out within a unified logical framework. The entire mechanical analysis process forms a reusable, verifiable and scalable automated computing platform, thereby building a computer-driven continuous mechanical analysis system, reducing the entry cost of analytical calculation and the workload during model iteration, and improving the stability and efficiency of overall engineering analysis.

[0047] In a preferred embodiment of this application, the stress and stiffness analysis method for the multi-waveform membrane disk under angular load may further include step five: obtaining reference data on the deflection, stress, and stiffness of the multi-waveform membrane disk using numerical simulation, and comparing these data with analytical calculation results to verify the model accuracy. In the example of this application, the numerical simulation method uses finite element analysis software to establish a three-dimensional solid model of the multi-waveform membrane disk, and obtains deflection curves, stress distribution, and angular stiffness reference data by applying angular loads; the finite element analysis software may be ANSYS software.

[0048] In the example of this application, Figure 1 The inner ring spline (3), outer ring bolt hole (1), sinusoidal wave form profile (2), and center O are marked to show the core geometry of the multi-waveform diaphragm and to clarify the position of the inner / outer radius of the profile. Figure 2 The figures in Figure 1 and 2 are labeled with the maximum deformation section, angular deflection angle α, section position S, and Z-axis (axis of symmetry), used to illustrate the location and deflection pattern of the maximum deformation of the membrane disk under angular load; Figure 3 shows the intervals of the two corrugations (r=104~118.5mm, r=118.5~162mm), the segment boundary points, and the concave-convex constraint points (r=133mm, r=147.5mm), used to show the corrugation segmentation method and the application position of the boundary conditions; Figure 4 shows the micro-element dr, the angle φ between the corrugation normal and the Z-axis, the angle α between the micro-element tangent and the normal, and the neutral plane length dr / cosφ, used to illustrate the geometric relationship of the section micro-element and the derivation of the support strain; Figure 5 shows the micro-element dimensions (r, dθ, dr), horizontal bending moment Mr, circumferential bending moment Mθ, and torque Mrθ. The shear force Qr is used to illustrate the stress state of the micro-element and establish the support moment balance equation; the horizontal axis of Figure 6 is the radius r (unit: m), and the vertical axis is the vertical deflection Vm (unit: 10^-4), containing two curves (corresponding to the analytical method result and the finite element method result respectively), used to compare the deflection calculation results of the two methods and verify the accuracy of the analytical model; Specific Implementation Cases The following describes the implementation process of the present invention in detail with reference to specific parameters and steps: 1. Implementation Preparation Material parameters: The multi-waveform diaphragm disk is made of titanium alloy (elastic modulus E=110GPa, Poisson's ratio μ=0.33). Geometric and operating parameters: values ​​are taken from Table 1 (inner radius of profile b=104mm, outer radius of profile a=162mm, corrugation thickness h=0.5mm, operating speed 5200r / min); Tools required: MATLAB R2022b (for analytical calculations), ANSYS 2023R1 (for finite element verification).

[0049] ; Table 1 contains parameter names (such as inner radius of profile and corrugation thickness), program-defined symbols (such as b and h), parameter values ​​and units, providing a unified parameter basis for analytical calculations and finite element modeling.

[0050] 2. Implementation of Calculation of Deflection Curve at Maximum Deformation Section Piecewise function definition: First segment (r∈[104,118.5] mm): Let v1=A1・r 3 + B1・r 2 + C1・r + D1 (polynomial function, satisfying inner boundary constraints). Second segment (r∈[118.5,162] mm): Let v2=A2・sin (ωr) + B2・cos (ωr) + C2・r +D2 (sine function + polynomial, ω=108.331rad / mm, matching ripple frequency); Substitute the boundary conditions: When r=104mm, M=0 (d 2 v1 / dr 2 Substituting v1 with α = 0 and v1 = t = 0.454 mm (α = 0.25°, t = 104・tan0.25°), we get 2A1・104 + B1 = 0 and A1・104. 3 + B1・104 2 + C1・104 + D1=0.454; When r = 118.5 mm, v1 = v2, dv1 / dr = dv2 / dr, d 2 v1 / dr 2 =d 2 v2 / dr 2 Substituting the two functions, we get three equations; Substituting "when r=162mm, v2=0, dv2 / dr=0" into v2, we get A2・sin (ω・162) + B2・cos(ω・162) + C2・162 + D2=0, A2・ω・cos (ω・162) - B2・ω・sin (ω・162) + C2=0; When r=133mm and r=147.5mm, d 2 v2 / dr 2Substituting =0 into v2, we get -A2・ω 2 ・sin (ω・133) -B2・ω 2 ・cos (ω・133)=0, -A2・ω 2 ・sin (ω・147.5) - B2・ω 2 •cos (ω・147.5)=0; Solving for coefficients: Solve the above system of 10 linear equations using MATLAB to obtain A1, B1, C1, D1, A2, B2, C2, and D2, and determine the specific expressions for v1 and v2; Curve plotting: Use MATLAB's plot function to plot the deflection curve for r∈[104,162] mm.

[0051] 3. Implementation of stress analysis under angular loads Coordinate system establishment: Define a cylindrical coordinate system in MATLAB (r∈[104,162] mm, θ∈[0,2π] rad, z∈[-0.25,0.25] mm); Deflection conversion: Calculate w at each r using w = v・cosφ (φ is the corrugated normal angle, which is obtained from the corrugated line derivative as φ = arctan(dv / dr)); Strain calculation: Substitute εr=z・cosφ・dα / dr, εθ=z・α / r + (∂w / ∂θ) / r (θ direction cosine distribution, ∂w / ∂θ=-w_max・sinθ), γrθ=-2・cosφ・(∂w / ∂r∂θ) to calculate the strain; Stress calculation: Substitute into σr=E / (1-μ) 2 )・(εr+μεθ), σθ=E / (1-μ 2 )・(εθ+μσr), take z=0.25mm (upper surface), calculate radial and circumferential stress; Results visualization: Use MATLAB's mesh function to plot stress cloud diagrams, or the plot function to plot r-σ curves (as shown in Figures 7 and 8).

[0052] 4. Implementation of Angular Stiffness Calculation Bending moment calculation: From the bending stress, we get Mr=∫(h / 2~-h / 2)σr・z・dr and Mθ=∫(h / 2~-h / 2)σθ・z・dr. Substituting them into the expressions for σr and σθ, we get Mr and Mθ (both are functions of r and θ, with cosine distribution in the θ direction). Shear force calculation: Substitute Qr=(Mr + ∂Mr / ∂r + Mθ) / r to calculate Qr; Solving for the total bending moment: Integrating ∫(0~2π) Mr・cosθ・162・dθ + ∫(0~2π) Qr・0.5・162・cosθ・dθ=0, we get the total bending moment M=∫(0~2π) Mr・cosθ・162・dθ (absolute value); Stiffness calculation: K=M / α= M / 0.25°, so K=41.6N・m / °.

[0053] 5. Implementation of analytical model verification Finite element modeling: Establish a three-dimensional model of the multi-waveform membrane disk in ANSYS (according to the parameters in Table 1, ignoring bolt holes and splines), and divide it into tetrahedral meshes (mesh size 0.5mm, with finer meshes in the corrugated area); Load and constraint: Fix the inner ring of the diaphragm disk, and apply an angular deflection load (α=0.25°) to the outer ring. Results Extraction: After solving, the deflection curve of the maximum deformation section, the stress distribution on the upper surface, and the outer ring reaction moment (total bending moment M) are extracted. The finite element stiffness K_fe = M / 0.25° = 44.3 N·m / ° is calculated. Error calculation: Deflection error = |(v_ana - v_fe) / v_fe|×100% (maximum < 7%), stiffness error = |(K_ana - K_fe) / K_fe|×100% = 6.09%, verifying the effectiveness of the model.

[0054] When segmenting the corrugations, it is necessary to ensure the continuity (displacement, slope, curvature) at the segment boundaries; otherwise, the deflection curve will be distorted. When calculating strain in the θ direction, the cosine distribution of deflection must be considered (θ=0° is the direction of maximum deformation, θ=90° is the direction of minimum deformation), which conforms to the non-axisymmetric deformation characteristics; When modeling with finite element method, the mesh in the corrugated region needs to be refined (size ≤ h / 2) to avoid stress calculation errors caused by mesh coarseness.

[0055] This application uses ANSYS software to establish a three-dimensional solid model of a multi-waveform membrane disk, applies angular loads, and obtains the deflection curve, stress distribution, and angular stiffness using the finite element method. The analytical method results (calculated using MATLAB) are compared with the finite element results to verify the accuracy of the analytical model. The verification results are as follows: Figure 6 , Figure 7 , Figure 8 As shown, in Figure 6 In the middle, take angular compensation Then the displacement at the point of maximum deformation is Based on the above formula, the vertical deflection curve can be plotted using MATLAB software. Figure a shows a schematic diagram of the vertical deflection curve of the multi-waveform membrane disk using the analytical method, and Figure b shows a schematic diagram of the vertical deflection curve of the multi-waveform membrane disk using the finite element method. Figure 7 The stress diagrams shown are calculated using Matlab programming. Figure a represents the horizontal stress at t=0.454 mm, and Figure b represents the circumferential stress at t=0.454 mm. Figure 8 The stress diagrams obtained from Ansys finite element analysis are shown in Figure a. Figure a shows the horizontal stress under angular load obtained from Ansys analysis, and Figure b shows the circumferential stress under angular load obtained from Ansys analysis. The calculations show that the maximum relative error of deflection is < 7% and the stiffness deviation is < 7%.

[0056] The stress and stiffness analysis method for multi-waveform diaphragm disks under angular loads disclosed in this application has the following beneficial effects: 1. High model accuracy: The analytical solution for the maximum deformation section deflection shows a consistent trend with the finite element solution, with a maximum relative error of less than 7%, accurately reflecting the angular deformation law of the membrane disk. The analytical prediction of stress distribution is generally consistent with the finite element results (only the inner and outer ring boundaries of the corrugation differ due to edge effects), which can effectively identify stress concentration points and provide a reliable basis for fatigue life assessment. The analytical result of angular stiffness (41.6 N·m / °) deviates from the finite element result (44.3 N·m / °) by only 6.09%, which meets the accuracy requirements of engineering design.

[0057] 2. High design efficiency: Compared to the finite element method (which has complex modeling and long calculation cycles), the analytical model of this invention can be quickly calculated using MATLAB programming (reducing the calculation time by more than 80%), and is suitable for the preliminary design and parameter optimization of waveform diaphragm couplings, supporting rapid iteration of design parameters.

[0058] 3. Strong applicability to engineering projects: The analytical model is based on plate and shell theory and fully considers the non-axisymmetric characteristics of the multi-waveform diaphragm disk. Its prediction results support the waveform diaphragm disk to achieve large angle compensation (α≥0.25°) and high torque transmission under high speed and high power conditions of 5200r / min, which meets the transmission requirements of high-end equipment such as aero engines and high-speed compressors. This provides standardized analytical tools for the design of high-performance waveform diaphragm couplings, reducing reliance on finite element software and lowering design costs and technical barriers.

[0059] The embodiments of this application have been described above with reference to the accompanying drawings. Unless otherwise specified, the embodiments and features in the embodiments of this application can be combined with each other. This application is not limited to the specific embodiments described above. The specific embodiments described above are merely illustrative and not restrictive. Those skilled in the art can make many other forms under the guidance of this application without departing from the spirit and scope of the claims, and all of these forms are within the protection scope of this application.

Claims

1. A method for stress and stiffness analysis of a multi-waveform diaphragm disk under angular load, characterized in that, include: Step 1: Define the geometric and operating parameters of the multi-waveform diaphragm disk, and establish a set of geometric features based on the radial dimensions of the diaphragm disk, the corrugated profile, and the connecting hub and rim structure, which will be used to construct a mechanical analysis model under angular loads; Step 2: Based on the plate and shell theory, assume the deflection function of the maximum deformation section of the multi-waveform membrane disk under angular load, and divide the membrane disk into segments according to the periodicity of the corrugated structure. Combine the continuity conditions of the inner ring boundary, outer ring boundary and segment boundary to derive the analytical expression of the vertical deflection of the maximum deformation section. Step 3: Based on the cylindrical coordinate system, establish a non-axisymmetric bending-torsional coupling mechanical model for the multi-waveform diaphragm disk. Obtain analytical expressions for the radial stress, circumferential stress and tangential stress of the diaphragm disk under angular load through geometric equations and physical equations, and form a stress distribution prediction model under angular load. Step 4: Based on the moment balance and shear force balance relationships of the membrane disk micro-element, establish the total moment-deflection angle equation and derive the analytical formula for the angular stiffness of the membrane disk under angular load.

2. The stress and stiffness analysis method for multi-waveform diaphragm disks under angular loads according to claim 1, characterized in that, In step one, the defined geometric parameters include: inner radius b of the profile, outer radius a of the profile, corrugation thickness h, corrugation amplitude H, hub thickness h1, rim thickness h2, hub transition fillet radius r2, rim transition fillet radius r3, inner radius r0 of the diaphragm disk, and outer radius r1 of the diaphragm disk; the operating parameters include the number of diaphragm disks m and the operating speed r.

3. The stress and stiffness analysis method for multi-waveform diaphragm disks under angular loads according to claim 1, characterized in that, In step two, when segmenting the corrugated line, the corrugation is divided into at least two segments along the radial direction. Deflection assumption functions v1 and v2 are established for each segment, and the conditions of displacement continuity, slope continuity and curvature continuity are satisfied at the segment boundary.

4. The stress and stiffness analysis method for multi-waveform diaphragm disks under angular loads according to claim 2, characterized in that, In step two, after establishing the piecewise deflection functions v1 and v2 for the first and second segments, respectively, the following settings are made: The boundary conditions for the inner ring are: vertical deflection v1 = t, where t is the displacement of the inner ring, and bending moment M = 0; The piecewise boundary conditions are: v1=v2, dv1 / dr=dv2 / dr, d 2 v1 / dr 2 =d 2 v2 / dr 2 ; The outer boundary conditions are: vertical deflection v2=0, dv2 / dr=0; Concave-convex constraint points are set within the segmented intervals to ensure that the second derivative of the ripple segment is zero.

5. The stress and stiffness analysis method for multi-waveform diaphragm disks under angular loads according to claim 1, characterized in that, In step three, a cylindrical coordinate system with the Z-axis coinciding with the symmetry axis of the membrane disk is used to establish a non-axisymmetric mechanical model with radius r and circumferential rotation angle θ as variables. Based on the deflection function v(r), its spatial variation relationship, and the angle φ between the corrugation normal and the symmetry axis of the membrane disk, the geometric equations of radial strain εr, circumferential strain εθ, and tangential strain γrθ are derived. The deflection curve, deformation gradient, and corrugation tilt angle are used to characterize the non-axisymmetric deformation of the membrane disk under angular load.

6. The stress and stiffness analysis method for multi-waveform diaphragm disks under angular loads according to claim 5, characterized in that, In step three, the radial strain εr, circumferential strain εθ, and tangential strain γrθ are substituted into Hooke's law to establish physical equations, thereby converting them into radial stress σr, circumferential stress σθ, and tangential stress τrθ, respectively, to obtain stress expressions that can reflect the multi-directional mechanical response of the diaphragm disk under angular loads, wherein: Radial stress σr = E / (1 - μ 2 )・(εr + μεθ); Circumferential stress σθ = E / (1 - μ 2 )·(εθ + μσr); Shear stress τrθ=E / (2 (1+μ))・γrθ; E is the elastic modulus, and μ is Poisson's ratio.

7. The method for stress and stiffness analysis under angular load of a multi-waveform diaphragm disk according to claim 1, characterized in that, In step four, the total bending moment M is obtained by circumferentially integrating the bending moment and shear force of the membrane disk element, and the angular stiffness K is calculated based on the relationship between the total bending moment and the angular deflection angle α. K=M / α.

8. The method for stress and stiffness analysis under angular load of a multi-waveform diaphragm disk according to claim 7, characterized in that, The total bending moment M is obtained by circumferentially integrating the horizontal bending moment Mr, the circumferential bending moment Mθ, the torque Mrθ, and the shear force Qr of the membrane disk element, and establishing the overall static equation by combining the force balance relationship Qr=(Mr + ∂Mr / ∂r + Mθ) / r, so that the total bending moment M is obtained by superposition of ∫(0~2π) Mr・cosθ・R・dθ and ∫(0~2π) Qr・h・R・cosθ・dθ.

9. The method for stress and stiffness analysis of a multi-waveform diaphragm disk under angular load according to any one of claims 1 to 8, characterized in that, The analytical calculations in steps two through four are implemented using MATLAB programming.

10. The method for stress and stiffness analysis under angular load of a multi-waveform diaphragm disk according to claim 9, characterized in that, The process also includes step five, which uses numerical simulation to obtain reference data on the deflection, stress, and stiffness of the multi-waveform membrane disk, and compares them with the analytical calculation results to verify the accuracy of the model.

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