Anti-vibration optimization method for gear dangerous pitch diameter vibration mode identification based on Campbell graph

Abstract

The invention discloses an anti-vibration optimization method for gear dangerous pitch diameter vibration mode identification based on a Campbell graph, and the method comprises the steps: S10, building a parameterized finite element model of a gear, and defining a preset to-be-optimized structure size as a design variable; s20, performing multi-rotating-speed-point modal analysis based on the parameterized finite element model, and performing linear fitting on an analysis result to obtain an inherent frequency straight line representing the change of each-order modal inherent frequency along with the rotating speed and the slope of the inherent frequency straight line; s30, a slope threshold value a is set, and the dangerous pitch diameter vibration mode is automatically recognized by comparing the slope absolute value of the inherent frequency straight line of each order with the threshold value a; s40, solving the intersection point of the inherent frequency straight line identified as the dangerous pitch diameter vibration mode and the double meshing frequency excitation line so as to predict the dangerous pitch diameter resonance frequency; and S50, establishing and executing an anti-vibration optimization process taking gear mass minimization as a target. By means of the method, the problems of poor reliability and low efficiency during gear anti-vibration optimization design can be solved.

Description

Technical Field

[0001] This invention relates to the technical field of gear vibration damping optimization methods, specifically to a vibration damping optimization method based on Campbell diagram for identifying the dangerous pitch diameter vibration mode of gears. Background Technology

[0002] Gears, as core mechanical components for transmitting power and motion, are widely used in high-end equipment fields such as aerospace, automotive industry, and precision machine tools. With the increasing demands for lightweight and high power-to-weight ratio in equipment, gear structural designs are trending towards thinner walls and greater complexity. However, while lightweighting reduces system weight, it also significantly weakens the overall stiffness of the gears, making them more susceptible to severe vibrations due to dynamic excitation during operation, thus affecting the stability and service life of the transmission system.

[0003] In gear transmission systems, the periodic meshing of gear pairs is the primary source of vibration excitation. Resonance occurs when the meshing frequency or its harmonics coincide with a natural frequency of the gear. Based on extensive engineering failure analysis experience, among various vibration modes, nodal diameter traveling wave resonance (NDR) induced by the pitch diameter mode is the most destructive to gears. Its vibration amplitude is far greater than other modes, easily leading to fatigue cracks or even tooth breakage within a short period. Therefore, avoiding dangerous pitch diameter resonance (i.e., vibration avoidance) must be a key design constraint during the gear optimization design phase.

[0004] In the existing technology, a common vibration damping optimization method is as follows: First, in the initial design stage, the modal order corresponding to the dangerous pitch diameter mode is identified through finite element modal analysis. For example, it is assumed that the Nth mode is a dangerous 2-pitch diameter mode. Then, in the subsequent automated optimization process, the resonance frequency of the Nth mode is constrained, requiring it to fall outside the meshing frequency range corresponding to the gear's operating speed range.

[0005] However, the aforementioned existing technologies have the following serious drawbacks in practical applications:

[0006] During automated optimization iterations, as the critical structural dimensions of the gear (such as design variables like flange thickness and bore diameter) continuously change, the overall structure and stiffness distribution of the gear also change, often leading to a phenomenon known as "mode veering." Specifically, the Nth mode, initially identified as a dangerous pitch circle mode, may evolve into a non-dangerous pitch circle or torsional mode after a certain iteration; simultaneously, another previously safe mode (e.g., the Mth mode) may evolve into a new dangerous pitch circle mode. Because the constraints in existing technologies are rigidly applied to a fixed "Nth mode," once mode veering occurs, the constraint loses its ability to control the truly dangerous mode, resulting in constraint failure and the failure of the entire optimization process, making it impossible to obtain a reliable design that meets dynamic performance requirements.

[0007] Furthermore, existing technologies often require designers to manually observe the mode shape contour maps of each mode to make judgments when identifying dangerous vibration modes. This method is not only inefficient and highly subjective, but also completely unsuitable for integration into automated optimization processes that require hundreds of iterations.

[0008] In summary, existing technologies suffer from poor reliability and low efficiency when performing lightweight vibration damping optimization design for gears because they cannot effectively address the problem of mode change during the optimization process and the identification of dangerous modes is difficult to automate. Summary of the Invention

[0009] The purpose of this invention is to overcome the aforementioned defects or problems in the prior art and to provide a vibration damping optimization method based on Campbell diagram for identifying the dangerous pitch diameter vibration mode of gears, which can improve the problems of poor reliability and low efficiency in the existing gear vibration damping optimization design.

[0010] To achieve the above objectives, the present invention adopts the following technical solution:

[0011] Technical Solution 1: A vibration avoidance optimization method for identifying dangerous pitch diameter vibration modes of gears based on Campbell diagrams, characterized by the following steps: S10: Establishing a parametric finite element model of the gear, wherein the preset dimensions of the structure to be optimized are defined as design variables; S20: Performing multi-speed modal analysis based on the parametric finite element model, and linearly fitting the analysis results to obtain natural frequency lines and their slopes characterizing the natural frequencies of each mode as a function of rotational speed; S30: Setting a slope threshold 'a', and automatically identifying dangerous pitch diameter vibration modes by comparing the absolute values ​​of the slopes of the natural frequency lines of each mode with the threshold 'a'; S40: Solving for the intersection points of the natural frequency lines identified as dangerous pitch diameter vibration modes and the excitation line at one meshing frequency to predict the resonant frequency of the dangerous pitch diameter; S50: Establishing and executing a vibration avoidance optimization process aimed at minimizing gear mass, wherein the process adopts the following constraint method to address the vibration mode jump problem during the optimization process:

[0012] S51: Create a virtual resonant frequency set of size M with a fixed dimension, where M is greater than the total number of dangerous pitch diameter resonant frequencies that may appear in any iteration step during the optimization process; S52: In each iteration step, fill the virtual resonant frequency set with all the dangerous pitch diameter resonant frequencies predicted in that iteration step, and fill the remaining empty slots in the set with a preset maximum value; S53: Apply uniform vibration damping constraints to each element in the virtual resonant frequency set to obtain a gear design scheme that meets the vibration damping requirements.

[0013] Technical Solution 2 based on Technical Solution 1: In step S30, the slope threshold a is less than the absolute value of the slope of the natural frequency line corresponding to the 1-node mode shape.

[0014] Technical Solution Three based on Technical Solution One: In step S53, the mathematical expression for the unified vibration damping constraint condition is: (S j -F L )·(S j -F H )>0; where S j F is the j-th element in the set of virtual resonant frequencies; L and F H These are the meshing frequencies corresponding to the lower and upper limits of the gear's operating speed range, respectively.

[0015] Technical solution four based on technical solution three: the lower limit F of the meshing frequency L and upper limit F H The calculation formulas are as follows: F L =z·N min / 60, F H =z·N max / 60; where z is the number of teeth of the gear, N min N is the lower limit of the operating speed. max This is the upper limit of the operating speed.

[0016] Technical Solution 5 based on Technical Solution 1: In step S40, the calculation formula for the excitation line at one meshing frequency is: F e = z·n / 60, where z is the number of teeth of the gear and n is the rotational speed.

[0017] Technical Solution Six based on Technical Solution One: Step S30 further includes: sorting all the natural frequency lines of all identified dangerous nodal vibration modes from low to high according to their natural frequency values ​​at zero rotation speed, and recording them sequentially as nodal 1, nodal 2, etc., according to the rule of grouping every two lines, until the nodal number of all nodal vibration modes is recorded.

[0018] Technical solution seven based on technical solution one: In step S10, the structural dimensions to be optimized as design variables include: wall thickness parameters at different positions of the wheel axle, parameters characterizing the radial length of the spokes at different positions, and parameters characterizing the axial thickness of the spokes.

[0019] Technical Solution 8, based on Technical Solution 1: The vibration damping optimization process is solved using the Pointer optimization algorithm.

[0020] Technical Solution Nine based on Technical Solution One: In step S20, the method for linearly fitting the analysis results is polynomial fitting based on the least squares method.

[0021] As can be seen from the above description of the present invention, compared with the prior art, the present invention has the following beneficial effects:

[0022] Technical Solution 1 provides a method for identifying and optimizing the vibration mode of dangerous pitch diameters of gears based on Campbell diagrams. This method constructs an automated and robust vibration optimization closed loop, which to some extent solves the problems of subjective and inefficient identification of dangerous vibration modes in gear optimization design, as well as the problem of optimization constraint failure caused by the "mode jump" phenomenon.

[0023] First, the technical solution of this invention establishes a front-end processing flow through steps S10 to S40 that can automatically, objectively, and accurately identify and predict dangerous resonance frequencies. Steps S10 and S20 provide a digital raw data foundation for the entire method by establishing a parameterized finite element model and performing multi-rotation point modal analysis. The most critical front-end processing step of this invention is S30, which utilizes the physical differences in the slopes of the straight lines of each natural frequency and sets a slope threshold 'a' to provide a reliable and real-time hazard source identification capability for the entire automated optimization process. In rotating machinery dynamics, axisymmetric structures such as gears exhibit frequency splitting in their pitch diameter mode shape due to the gyroscopic effect during rotation. A standing wave vibration in a stationary coordinate system decomposes into a forward traveling wave rotating in the same direction and a backward traveling wave rotating in the opposite direction in a rotating coordinate system. The natural frequencies of these two traveling waves change linearly with increasing rotational speed, one increasing and the other decreasing. Therefore, on a Campbell's diagram with rotational speed as the horizontal axis and frequency as the vertical axis, they appear as a pair of straight lines with approximately equal absolute slopes but opposite signs. Other modes, such as the nodal circle mode or the overall torsional mode, have natural frequencies that are much less affected by changes in rotational speed, and the slope of their corresponding natural frequency lines is close to zero. This invention utilizes this inherent difference in physical laws, transforming the existing problem of image-based judgment based on mode shape contour maps—which requires manual intervention by designers based on engineering experience—into a fully programmable, objective, and efficient automated identification step through a simple numerical comparison operation (step S30). This step is the foundation for the automation of the entire technical solution, ensuring that all dangerous nodal circle modes that need to be controlled under the current structure are accurately and completely captured in each iteration of the optimization process. Next, step S40 transforms the abstract "dangerous mode" into a specific "dangerous resonance frequency" value that can be used for mathematical constraints by solving for the intersection of the natural frequency lines of the identified dangerous modes and the excitation lines. Thus, the front-end processing flow of this invention solves the shortcomings of existing technologies, namely "low efficiency and lack of automation in manual identification," and prepares the necessary input data for subsequent robust constraint application.

[0024] Secondly, the core innovation of this invention lies in solving the deficiency of optimization constraint failure caused by "mode jump" in the prior art through the "large value assignment method" defined in step S50 and its specific constraint processing methods (S51, S52, S53). During automated optimization, the structural parameters of the gear continuously change within the design space, and its mass and stiffness distribution also dynamically change accordingly. This leads to changes in the energy and morphology of each mode, resulting in dynamic changes in the modal order and total number of dangerous pitch diameter modes between different iteration steps. This phenomenon means that the number and index position of the targets requiring constraint (i.e., dangerous resonance frequencies) are not fixed. For most optimization algorithms, the premise is that the dimension and structure of the constraint conditions remain constant throughout the optimization process. Therefore, this dynamically changing constraint target is a problem that the prior art cannot handle. The "large value assignment method" of this invention resolves this contradiction from a computational perspective through the following three synergistic sub-steps. Step S51 first creates a set of virtual resonant frequencies of fixed size, independent of the physical modal order. Mathematically, this set is a vector with a constant dimension of M, establishing a static and standardized constraint container for subsequent constraint application. Step S52 then performs a crucial mapping operation in each iteration step, filling this static constraint container in real time and sequentially with the variable number of dangerous resonant frequencies obtained from steps S30 and S40. This dynamic-to-static mapping process is key to solving the modal transition problem in this invention; it successfully transforms an uncertain, dynamic physical problem into a deterministic, static mathematical problem. Therefore, in step S53, the optimization algorithm only needs to apply uniform and form-invariant vibration avoidance constraints to this structurally constant, dimension-invariant constraint container. This design fundamentally removes the binding relationship between constraint conditions and specific modal orders. Regardless of how dangerous nodal mode shapes transition during optimization, as long as they are identified, they will be included in this container and effectively constrained. The empty spaces in the container not filled by actual resonant frequencies are occupied by a preset maximum value. This maximum value is set to a value far exceeding any actual resonant frequency, so it mathematically must satisfy the vibration avoidance constraint condition. As a neutral placeholder, it ensures that the dimension of the constraint vector is constant and does not cause any substantial interference to the gradient calculation and search direction of the optimization algorithm.

[0025] In summary, steps S10 to S50 in this technical solution constitute a logically rigorous and functionally interdependent overall technical solution. The automated real-time identification in step S30 is the data input foundation for the "large value method" in step S50. Without reliable identification, subsequent constraint application will lose its target. Conversely, without the robust constraint processing mechanism in step S50, the accurate identification result in step S30 will ultimately lose its engineering significance in iterative optimization due to "mode jumps." It is precisely this close coupling and synergistic effect between "accurate identification capability" and "robust constraint mechanism" that enables this invention to achieve a gear vibration damping optimization design with high reliability and high efficiency while ensuring the dynamic reliability of the structure.

[0026] In technical solution two, by further limiting the slope threshold 'a' to less than the absolute value of the slope of the natural frequency line corresponding to the 1-pitch-diameter mode shape, the completeness of the dangerous mode shape identification process and the safety of the method are further improved. In the gear pitch-diameter mode shape system, the 1-pitch-diameter mode shape, as the lowest-order mode shape, usually has the smallest rate of change of its natural frequency with rotational speed, meaning its absolute slope value is the smallest. This additional technical feature provides a clear and physically meaningful upper limit for setting the slope threshold 'a'. By ensuring that the threshold 'a' is lower than the absolute slope value of the least significant 1-pitch-diameter mode shape, this method can guarantee that all higher-order mode shapes with larger absolute slope values ​​(such as 2-pitch-diameter, 3-pitch-diameter, etc.) will also be included in the identification range. This limitation method logically ensures the completeness of the identification process and effectively avoids false negative judgments that may occur due to improper setting of the threshold 'a', i.e., missing a dangerous mode shape that should be constrained. Therefore, this additional technical feature improves the reliability and security of the entire optimization method, ensuring that the final design will not have potential resonance risks due to oversights in identification.

[0027] Technical Solution 3 provides a clear, calculable, and algorithm-friendly technical means for achieving the vibration isolation objective by giving a specific mathematical expression for the unified vibration isolation constraint condition. The "unified vibration isolation constraint condition" in the aforementioned technical solutions conceptually requires the frequency value to fall outside a certain safe range. The expression given in this technical solution (S...) j -F L )·(S j -F H The expression > 0 represents a precise mathematical transformation of this logical objective. This expression utilizes the relationship between the roots of a quadratic function and the sign changes of the function values ​​inside and outside the interval. When the variable S... j The value lies in the two F roots L and F H When S is in the range of zero, the value of the expression is negative or zero; while when S is in the range of zero, the value of the expression is negative or zero. jWhen the value of is outside the interval, the value of the expression is always positive. Therefore, this simple inequality constraint is functionally equivalent to S. j <F L or S j >F H The logical judgments in this solution are crucial. This transformation is significant in computational optimization because it avoids the use of discontinuous and non-smooth judgment statements such as if-else or logical OR in constraint evaluation of optimization algorithms. Such statements are detrimental to the convergence and stability of many modern optimization algorithms, especially gradient-based algorithms. The smooth mathematical expressions provided by this technical solution enable the efficient and stable calculation of constraints and their gradients, thereby improving the convergence speed and computational robustness of the entire optimization process.

[0028] In technical solution four, upper and lower limits F of the meshing frequency are given. L and F H The specific calculation formula directly links the mathematical constraint boundaries defined in Technical Solution 3 with the actual engineering application scenarios of gears, ensuring the practical engineering significance of the optimization objective. This technical solution connects these two boundary values ​​to the number of teeth z of the gear and the lower limit N of the operating speed range, respectively. min and upper limit N max The direct hook provides an accurate and practical physical basis for defining the safe range of vibration damping optimization. This ensures that the optimization process avoids the frequency range in which resonance is most likely to occur during actual gear operation. Therefore, this additional technical feature makes the entire optimization problem setting closer to actual engineering needs, ensuring that the final output design solution truly solves the resonance problem under specific working conditions and has direct application value.

[0029] In technical solution five, by clarifying the calculation formula for the excitation line at one meshing frequency, the prediction step S40 for the dangerous resonance frequency has an accurate and reasonable physical model basis, thereby improving the accuracy and effectiveness of resonance frequency prediction. The vibration excitation source of a gear system includes the fundamental frequency and its harmonics. In most gear transmission applications, the excitation force energy generated by the fundamental frequency of the gear meshing (i.e., one meshing frequency) is the most concentrated and is the most significant factor causing system resonance. This technical solution clarifies the excitation line as the one meshing frequency line, ensuring that the solution for the resonance point is targeted at the most important excitation source. This allows the entire analysis process to focus on the most likely and most harmful resonance scenario, avoiding unnecessary computational complexity caused by considering lower-energy, less influential higher-order harmonics. Therefore, this additional technical feature ensures the accuracy and specificity of resonance frequency prediction, making the entire vibration damping optimization method more efficient and targeted.

[0030] In technical solution six, a sub-step for systematically sorting and recording nodal-diameter vibration modes is added to the identification step S30, facilitating the monitoring of the optimization process, the analysis of results, and the engineering understanding. Although this recording step is not a computationally necessary step for automating the optimization process, it provides designers with a clear and physically meaningful perspective to observe and understand the optimization process. By categorizing the identified nodal-diameter vibration modes according to the number of nodal-diameters (1 nodal, 2 nodal, etc.), designers can intuitively track the evolution trend of the resonant frequencies of specific types of critical vibration modes (e.g., the 2-nodal-diameter mode, which has the greatest impact on vibration patterns) during optimization iterations. This structured data output plays a crucial supporting role in judging the correctness of the optimization direction, diagnosing potential problems, and ultimately verifying the optimization results in engineering, thereby significantly improving the engineering applicability of the present invention.

[0031] In Technical Solution Seven, by specifically limiting the design variables to key structural parameters such as axle wall thickness, spoke radial length, and axial thickness, the search range of the optimization algorithm is focused on the structural regions that have the most significant impact on the gear's dynamic characteristics and mass. The mass of a gear is primarily determined by the volume of its solid components, while its natural frequency (i.e., a function of the stiffness-to-mass ratio) is extremely sensitive to structural shape and dimensional distribution. The parameters listed in this technical solution are the key variables controlling the geometry of the gear spokes and axle; changes to these parameters can most directly and effectively adjust the gear's mass and stiffness distribution. Using these parameters as design variables ensures that each iteration of the optimization algorithm produces the most effective adjustment, avoiding wasting computational resources on parameters with minimal impact on dynamic characteristics. Therefore, this additional technical feature makes the optimization process more efficient, converges faster, and achieves significant weight reduction and vibration damping effects within a relatively small range of structural modifications.

[0032] Technical Solution 8 specifies the use of the Pointer optimization algorithm, further improving the efficiency of optimization and the likelihood of obtaining a better solution. Gear optimization design problems are typically highly nonlinear and non-convex complex problems, with numerous local optima potentially existing in their design space. Traditional single optimization algorithms (such as simple gradient methods or genetic algorithms) easily get trapped in local solutions that are not globally optimal. The Pointer algorithm, as an advanced hybrid intelligent optimization strategy, integrates a set of complementary optimization algorithms (such as the linear simplex method, sequential quadratic programming, and genetic algorithms), and can automatically switch and fuse different search strategies based on the real-time characteristics of the problem. This adaptive optimization mechanism allows it to utilize gradient information for rapid local searches while also escaping local extrema through global search strategies. Therefore, using the Pointer algorithm significantly increases the probability of finding a better design, resulting in gears with lower quality and better performance.

[0033] In technical solution nine, by explicitly employing polynomial fitting based on the least squares method, a mathematically rigorous and reliable method is provided for data processing in step S20, thus ensuring the accuracy of the data source for the entire technical solution. The accuracy of all subsequent judgments and calculations in the invention highly depends on the accuracy of the natural frequency line fitted in step S20. The least squares method, by minimizing the sum of squares of errors between data points and the fitted curve, can find the linear relationship that best represents the trend of the original data. Using this method, it is ensured that the discrete data points obtained from finite element modal analysis are transformed into a mathematically optimal linear model, thereby minimizing the errors introduced in the data processing stage and providing a solid and reliable data foundation for subsequent accurate calculation of the line slope and prediction of resonance points. Attached Figure Description

[0034] To more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings used in the following description of the embodiments are briefly introduced. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0035] Figure 1 This is a schematic diagram of the structure of the parameterized model of the gear spokes in an embodiment of the present invention;

[0036] Figure 2 This is a table of analytical data for the slope of the natural frequency line and the corresponding vibration type of each mode of the initial gear in the embodiments of the present invention;

[0037] Figure 3 This is the Campbell diagram obtained after automatic identification of dangerous nodal vibration modes in an embodiment of the present invention;

[0038] Figure 4 This is a comparative schematic diagram of mode shape contour maps used to illustrate the mode shape jump phenomenon in an embodiment of the present invention;

[0039] Figure 5 This is a convergence history curve of the optimization process in an embodiment of the present invention;

[0040] Figure 6 This is a comparison table of the gear mass and the number of pitch diameter resonance points in the working speed range before and after gear optimization in the embodiments of the present invention;

[0041] Figure 7 This is a comparison diagram of the Campbell diagrams for optimizing the front gear in an embodiment of the present invention;

[0042] Figure 8 This is a comparison diagram of the Campbell diagram of the optimized gear in an embodiment of the present invention. Detailed Implementation

[0043] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are preferred embodiments of the present invention and should not be considered as excluding other embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention.

[0044] Unless otherwise expressly defined, the use of terms such as "first," "second," or "third" in the claims, description, and accompanying drawings of this invention is for distinguishing different objects and not for describing a specific order.

[0045] Unless otherwise expressly defined, in the claims, description, and accompanying drawings of this invention, the use of directional terms such as "center," "lateral," "longitudinal," "horizontal," "vertical," "top," "bottom," "inner," "outer," "upper," "lower," "front," "rear," "left," "right," "clockwise," and "counterclockwise" to indicate orientation or positional relationships is based on the orientation and positional relationships shown in the accompanying drawings and is only for the convenience of describing the invention and simplifying the description, and is not intended to indicate or imply that the device or element referred to must have a specific orientation or be constructed and operated in a specific orientation, and therefore should not be construed as limiting the specific scope of protection of this invention.

[0046] Unless otherwise expressly defined, the terms "fixed connection" or "fixed connection" used in the claims, description and drawings of this invention should be interpreted broadly to refer to any connection in which there is no displacement or relative rotation relationship between the two parties, including non-removable fixed connection, detachable fixed connection, integral connection and fixed connection by other means or components.

[0047] In the claims, description and accompanying drawings of this invention, the terms "comprising," "having," and variations thereof are used to mean "including but not limited to."

[0048] Example

[0049] This invention relates to a vibration damping optimization method based on Campbell's diagram for identifying the critical pitch diameter vibration mode of gears, which includes the following steps:

[0050] S10: Establish a parametric finite element model of the gear, wherein the preset structural dimensions to be optimized are defined as design variables;

[0051] S20: Based on the parametric finite element model, perform multi-rotation point modal analysis and linearly fit the analysis results to obtain the natural frequency line and its slope characterizing the natural frequency of each mode as a function of rotational speed.

[0052] S30: Set a slope threshold a, and automatically identify dangerous nodal mode shapes by comparing the absolute value of the slope of the natural frequency line of each order with the threshold a;

[0053] S40: Solve for the intersection of the natural frequency line of the identified dangerous nodal mode and the excitation line of one meshing frequency to predict the resonant frequency of the dangerous nodal mode.

[0054] S50: Establish and execute a vibration damping optimization process with the goal of minimizing gear mass. To address the mode shape jump problem during the optimization process, the process employs the following constraint method:

[0055] S51: Create a set of virtual resonant frequencies of size M with a fixed size, where M is greater than the total number of dangerous nodal resonant frequencies that may occur in any iteration step during the optimization process;

[0056] S52: In each iteration step, all the dangerous nodal resonance frequencies predicted in that iteration step are sequentially filled into the virtual resonance frequency set, and the remaining empty slots in the set are filled with a preset maximum value.

[0057] S53: Apply uniform vibration damping constraints to each element in the set of virtual resonant frequencies to obtain a gear design scheme that meets the vibration damping requirements.

[0058] Specifically, steps S10 to S50 above together constitute a complete automated vibration damping optimization process, the implementation details of which are as follows:

[0059] In step S10, the process of establishing a parametric finite element model first involves creating a precise 3D geometric model of the gear in 3D CAD software (such as CATIA, UG, SolidWorks, etc.). The core of parametric modeling lies in not using fixed numerical values ​​to define the model's geometric dimensions, but rather defining the key structural dimensions that need optimization (such as...) Figure 1The spoke thickness, aperture, etc., shown are defined as variables or parameters. This parametric CAD model is then imported into professional finite element analysis (FEA) software (such as ANSYS, Abaqus, etc.). In the FEA software, the model is first assigned realistic material properties, including elastic modulus, Poisson's ratio, and density. Next, the geometric model is meshed, that is, discretized into a large number of simple elements (such as hexahedral or tetrahedral elements) for numerical calculation. Then, boundary conditions are applied, for example, a fixed constraint is applied to the mounting hole at the center of the gear to simulate its mounting state on the axle. At this point, a parametric finite element model associated with CAD parameters, capable of automatically updating geometry and performing mechanical analysis, is completed.

[0060] In step S20, multi-rotation point modal analysis is performed to obtain the dynamic characteristics of the gear in rotation. Modal analysis itself is a method for calculating the natural frequencies and mode shapes of a structure. "Multiple rotation points" refers to selecting multiple discrete rotational speeds within a relatively wide rotational speed range (e.g., 0 to 50,000 rpm) covering the gear's operating range, and performing a complete modal analysis at each speed point. This is done to account for the influence of the gyroscopic effect and stress stiffening effect generated during gear rotation on its natural frequencies. After the analysis, a series of natural frequency data points at different rotational speeds are obtained. Since natural frequency curves of different orders may intersect in higher-order modes or complex structures, a modal tracking algorithm (such as an algorithm based on the modal confidence criterion MAC) is needed to correctly associate and identify data points belonging to the same order of modes. Finally, for the data point set of each tracked mode, a linear fitting based on the least squares method is used to obtain the natural frequency line that best represents its changing trend.

[0061] In step S30, a slope threshold 'a' is set and automatically identified. The physical basis for this is that different vibration modes respond differently to rotational effects. The pitch radius mode is a mode with vibration nodes along the radial line of the gear. Its vibration mode undergoes frequency splitting during rotation, exhibiting a significant linear change in natural frequency with rotational speed. Therefore, its slope in the Campbell's diagram has a large absolute value. In contrast, the pitch circle mode or torsional mode, etc., are less affected by rotational effects, and their slopes are close to zero. By setting an appropriate threshold 'a', the pitch radius mode can be efficiently and objectively distinguished from other modes through simple numerical comparison.

[0062] In step S40, predicting the pitch diameter resonance frequency involves finding the intersection of two straight lines. One line is the natural frequency line of the dangerous pitch diameter mode shape obtained in step S30, and the other is the excitation line at one meshing frequency, which represents the main external excitation frequency generated during gear meshing. The intersection of these two lines physically signifies that the excitation frequency equals the system's natural frequency, i.e., resonance has occurred. The rotational speed and frequency corresponding to this intersection are the resonant rotational speed and resonant frequency.

[0063] In step S50, the core of establishing and executing the vibration avoidance optimization process lies in using the "large value method" to address mode shape jumps. As mentioned earlier, the order and number of dangerous nodal-diameter modes change dynamically during the optimization process. Therefore, step S51 creates a virtual resonant frequency set of fixed size, whose dimensions remain constant throughout the optimization process. In step S52, at each iteration, all currently identified dangerous resonant frequencies of varying size are mapped and filled into this fixed-size set. For any unfilled spaces in the set, a maximum value much larger than any actual resonant frequency (e.g., 10) is used. 9 Step S53 then applies a uniform vibration avoidance constraint to this set with a constant dimension. In this way, a dynamically changing constraint problem is transformed into a static problem, allowing the optimization algorithm to proceed stably and reliably.

[0064] In step S30, the slope threshold 'a' is set to be less than the absolute value of the slope of the natural frequency line corresponding to the 1-pitch-diameter mode shape. Specifically, this setting provides a clear technical criterion for selecting the threshold 'a', ensuring the completeness of the identification. Among the various pitch-diameter modes of gears, the 1-pitch-diameter mode has the fewest pitch lines and its frequency splitting effect is relatively weakest. Therefore, the absolute value of the slope of the natural frequency line corresponding to it on the Campbell's diagram is usually the smallest among all pitch-diameter modes. Setting the threshold 'a' to be smaller than this minimum value logically ensures that any other pitch-diameter modes (such as 2-pitch, 3-pitch, etc.) with a larger (i.e., more significant) absolute slope value than the 1-pitch-diameter mode shape will also be identified. This avoids missing any dangerous modes due to improper threshold selection, thereby improving the safety of the entire optimization method.

[0065] In step S53, the mathematical expression for the unified vibration damping constraint condition is: (S j -F L )·(S j -F H )>0; where S j F is the j-th element in the set of virtual resonant frequencies; L and F HThese are the meshing frequencies corresponding to the lower and upper limits of the gear's operating speed range, respectively. Specifically, this mathematical expression is a precise mathematical transformation of the logical objective that "the resonance frequency must fall outside the safe range." It utilizes the property of quadratic functions: for the two roots x1 and x2 of a quadratic equation (x-x1)(x-x2) = 0, when the value of the variable x is outside the interval [x1, x2], the function value (x-x1)(x-x2) is always positive. Therefore, the inequality (S... j -F L )·(S j -F H The expression > 0 is functionally equivalent to the logical judgment "S". j <F L or S j >F H The advantage of this mathematical form is that it is a continuous and differentiable function, which ensures the computational stability and convergence efficiency of many modern optimization algorithms (especially gradient-based algorithms).

[0066] The lower limit F of the meshing frequency L and upper limit F H The calculation formulas are as follows: F L =z·N min / 60, F H =z·N max / 60; where z is the number of teeth of the gear, N min N is the lower limit of the operating speed. max This represents the upper limit of the operating speed. Specifically, these two formulas relate the constraint boundaries to the actual operating parameters of the gear. The gear meshing frequency refers to the number of times the teeth mesh per unit time; its value is equal to the gear speed (rpm) multiplied by the number of teeth, then divided by 60 to convert to Hertz (Hz). Therefore, F... L and F H These represent the primary excitation frequencies generated by gear meshing at the lowest and highest operating speeds, respectively. Confining the dangerous resonant frequency outside this frequency range determined by actual operating conditions is a fundamental requirement for effective vibration isolation.

[0067] In step S40, the calculation formula for the excitation line at one meshing frequency is: F e= z·n / 60, where z is the number of teeth on the gear and n is the rotational speed. Specifically, this formula defines the excitation line in the Campbell diagram. Although the excitation source in a gear transmission system includes the fundamental frequency and its harmonics, in most cases, the fundamental frequency (i.e., one meshing frequency) contains the most concentrated energy and is the primary factor causing resonance. Setting the excitation line as one meshing frequency line is a standard and efficient practice in engineering, focusing on the most critical resonance scenarios, thus ensuring the effectiveness of the analysis while avoiding unnecessary computational complexity.

[0068] Step S30 further includes: sorting the natural frequency lines of all identified dangerous nodal-diameter vibration modes according to their natural frequency values ​​at zero rotational speed from low to high, and recording them sequentially as nodal-diameter 1, nodal-diameter 2, etc., according to the rule of grouping every two lines, until the nodal-diameter number of all nodal-diameter vibration modes is recorded. Specifically, this step provides a physically meaningful annotation for the results of automated analysis. Nodal-diameter vibration modes always appear in pairs (forward traveling wave and backward traveling wave), and their natural frequencies at zero rotational speed (or the average frequency after splitting) increase with the increase of the nodal-diameter number. Therefore, through this sorting and grouping recording method, the abstract modal orders identified by the program can be mapped to the commonly used physical concepts of "nodal-diameter 1", "nodal-diameter 2", etc. in engineering. This allows designers to clearly track the changes of specific types of dangerous vibration modes during the optimization process, providing convenience for result analysis and engineering verification.

[0069] In step S10, the structural dimensions to be optimized as design variables include: wall thickness parameters at different positions of the axle, parameters characterizing the radial length of the spokes at different positions, and parameters characterizing the axial thickness of the spokes. Specifically, these parameters are selected as design variables because they are key geometric features that determine the mass and stiffness distribution of the gear spoke structure. The dynamic characteristics (natural frequency) of a gear directly depend on its stiffness-to-mass ratio. Changing the thickness, inner and outer diameters, and other dimensions of the spokes is the most direct and effective way to adjust the overall stiffness and mass of the gear, without affecting the tooth geometry that enables the gear to perform its transmission function. Focusing the optimization on these parameters ensures that each iteration of the optimization algorithm has the most significant impact on the dynamic characteristics of the gear, thereby improving optimization efficiency.

[0070] The vibration damping optimization process employs the Pointer optimization algorithm. Specifically, the Pointer algorithm is a hybrid intelligent optimization strategy that integrates multiple optimization algorithms with different characteristics (such as sequential quadratic programming and genetic algorithms), and can automatically select and switch algorithms based on the characteristics of the current optimization problem. Gear optimization problems typically exhibit high nonlinearity and multi-peak characteristics, making it easy for a single algorithm to get trapped in local optima. By combining local and global search capabilities, the Pointer algorithm can more effectively find the global optimum in a complex design space, thus increasing the probability of finding a lighter and higher-performance design solution.

[0071] In step S20, the method for linearly fitting the analysis results is polynomial fitting based on the least squares method. Specifically, the least squares method is a standard mathematical method used to find an optimal function model to describe the intrinsic laws from a set of data points containing measurement errors. In this invention, by linearly fitting the (rotational speed, frequency) data points obtained from modal analysis using this method, a "best-fit line" that minimizes the sum of the squares of the distances from all data points to this line can be found. This ensures that the slope of the line used in subsequent calculations can most accurately reflect the true trend of the data, thus providing a reliable data foundation for the accurate identification of dangerous vibration modes.

[0072] To further illustrate the advantages of the gear vibration damping optimization method provided in the embodiments of the present invention, the following specific examples are provided.

[0073] This example focuses on a bevel gear in an aero-engine, aiming to achieve lightweight design of its spoke structure while ensuring that no dangerous pitch diameter resonance occurs within the operating speed range. The gear has 36 teeth (z) and a rated operating speed of 30,000 rpm. The operating speed range to be tested is [75% to 107%] of the rated speed, i.e., [22,500 rpm to 32,100 rpm].

[0074] In this example, the vibration damping optimization of the aero-engine bevel gear is strictly followed according to the above steps S10 to S50.

[0075] S10: Establish the model. For example... Figure 1 As shown, a parametric finite element model of the bevel gear was established, with 10 key dimensions of the spokes (T1, T2, L1, L2, L3, L4, H1, H2, H3, H4) used as design variables. The material used was high-strength alloy steel with an elastic modulus of 206 GPa, a Poisson's ratio of 0.3, and a density of 7850 kg / m³. 3 The entire model was subjected to fixed constraints on its inner bore surface to simulate the actual installation state.

[0076] S20: Modal analysis. Within the rotational speed range of [0, 50000 rpm], multiple rotational speed points are selected at 5000 rpm intervals to perform modal analysis considering rotational effects, and the calculated first 20 modal results are linearly fitted.

[0077] S30: Identify mode shapes. For the initial design, the slopes of the natural frequencies obtained are as follows: Figure 2 As shown. In this embodiment, a slope threshold a is set. Figure 2 The data in the dataset identifies modes with an absolute slope greater than 0.01 as dangerous nodal-circle modes. For example, the first-order mode has a slope of -1.50e-2 and an absolute value of 0.015, which is greater than 0.01, thus it is identified as a nodal-circle mode. The third-order mode has a slope of -2.66e-6 and an absolute value much less than 0.01, thus it is identified as a non-dangerous mode. After judging all orders, the natural frequency lines of all identified nodal-circle modes are filtered out, forming a sequence like... Figure 3 The Campbell diagram shown contains only the critical nodal-diameter mode shape, clearly displaying the frequency lines from 1 nodal-diameter to 5 nodal-diameter.

[0078] S40: Predicted Resonance. Define the excitation line at one meshing frequency as F. e =36·n / 60 = 0.6n. Solve for the relationship between the excitation line and... Figure 3 The intersection points of the straight lines of each nodal radius vibration mode can be used to obtain the resonance frequencies of each nodal radius. Calculations show that within the operating speed range of [22500 rpm, 32100 rpm], the initial design has two critical resonance points, such as... Figure 7 The red intersection point falls within the working velocity range.

[0079] S50: Perform optimization. Establish an optimization model with the objective of minimizing mass. To demonstrate the necessity of this invention in solving the "mode jump" problem, such as... Figure 4 As shown, when the gear structure parameters change slightly, it can be clearly observed that the original 9th ​​and 10th order 4-pitch radial vibration modes jump to the 11th and 12th orders in the new structure, while the original 11th and 12th order oscillating vibration modes move to other positions. This clearly shows that the existing method of constraining fixed orders is completely unreliable and will inevitably lead to optimization failure. This invention uses the "assigning large values ​​method" for constraint. Based on the working speed range [22500rpm, 32100rpm], the meshing frequency range is calculated to be [13500Hz, 19260Hz]. A virtual resonant frequency set S of size M=12 is created, and for each element S in the set... j Apply uniform vibration damping constraints: (S) j -13500)·(S j-19260)>0. The Pointer optimization algorithm is used to solve this. After 766 automated iterative calculations, the optimization process converges, as follows: Figure 5 The optimized historical curve is shown.

[0080] A comparative analysis of the gear design schemes before and after optimization was conducted, such as... Figure 6 , Figure 7 and Figure 8 As shown:

[0081] 1. Quality change: based on Figure 6 According to the data, the gear weight decreased from 231.77g before optimization to 216.21g after optimization, successfully achieving a weight reduction of 6.71%.

[0082] 2. Vibration damping effect: based on Figure 6 The data shows that the number of dangerous joint diameter resonance points in the operating speed range [22500rpm, 32100rpm] has been reduced from 2 before optimization to 0 after optimization, achieving 100% elimination of dangerous resonance points.

[0083] 3. Campbell Chart Comparison: Comparison Figure 7 The Campbell plot before optimization and Figure 8 The optimized version shown

[0084] The Campbell diagram clearly shows that the two resonance intersections that originally fell within the working speed range have been successfully moved out of that range, ensuring the dynamic safety of the gear throughout the entire working range.

[0085] The detailed numerical results of this example fully demonstrate that the method of the present invention can effectively solve the problem of mode shape transition and achieve significant weight reduction effect while ensuring dynamic safety.

[0086] The foregoing description of the specifications and embodiments is intended to explain the scope of protection of this invention, but does not constitute a limitation on the scope of protection of this invention. Modifications, equivalent substitutions, or other improvements to the embodiments of this invention or a portion thereof that can be obtained by those skilled in the art through logical analysis, reasoning, or limited experimentation, based on the teachings of this invention or the foregoing embodiments, in conjunction with common knowledge, general technical knowledge, and / or existing technology, should all be included within the scope of protection of this invention.

Claims

1. A vibration avoidance optimization method based on Campbell's diagram for identifying the dangerous pitch diameter vibration mode of gears, characterized in that, Includes the following steps: S10: Establish a parametric finite element model of the gear, wherein the preset structural dimensions to be optimized are defined as design variables; S20: Based on the parametric finite element model, perform multi-rotation point modal analysis and linearly fit the analysis results to obtain the natural frequency line and its slope characterizing the natural frequency of each mode as a function of rotational speed. S30: Set a slope threshold a, and automatically identify dangerous nodal mode shapes by comparing the absolute value of the slope of the natural frequency line of each order with the threshold a; S40: Solve for the intersection of the natural frequency line of the identified dangerous nodal mode and the excitation line of one meshing frequency to predict the resonant frequency of the dangerous nodal mode. S50: Establish and execute a vibration damping optimization process with the goal of minimizing gear mass. To address the mode shape jump problem during the optimization process, the process employs the following constraint method: S51: Create a set of virtual resonant frequencies of size M with a fixed size, where M is greater than the total number of dangerous nodal resonant frequencies that may occur in any iteration step during the optimization process; S52: In each iteration step, all the dangerous nodal resonance frequencies predicted in that iteration step are sequentially filled into the virtual resonance frequency set, and the remaining empty slots in the set are filled with a preset maximum value. S53: Apply uniform vibration damping constraints to each element in the set of virtual resonant frequencies to obtain a gear design scheme that meets the vibration damping requirements.

2. The vibration avoidance optimization method based on Campbell's diagram for identifying the dangerous pitch diameter vibration mode of gears, as described in claim 1, is characterized in that... In step S30, the slope threshold a is less than the absolute value of the slope of the natural frequency line corresponding to the 1-node mode shape.

3. The vibration avoidance optimization method based on Campbell's diagram for identifying the dangerous pitch diameter vibration mode of gears according to claim 1, characterized in that, In step S53, the mathematical expression for the unified vibration damping constraint condition is: (S j -F L )·(S j -F H )>0; where S j F is the j-th element in the set of virtual resonant frequencies; L and F H These are the meshing frequencies corresponding to the lower and upper limits of the gear's operating speed range, respectively.

4. The vibration avoidance optimization method based on Campbell's diagram for identifying the dangerous pitch diameter vibration mode of gears according to claim 3, characterized in that, The lower limit F of the meshing frequency L and upper limit F H The calculation formulas are as follows: F L =z·N min / 60, F H =z·N max / 60; where z is the number of teeth of the gear, N min N is the lower limit of the operating speed. max This is the upper limit of the operating speed.

5. The vibration avoidance optimization method based on Campbell's diagram for identifying the dangerous pitch diameter vibration mode of gears according to claim 1, characterized in that, In step S40, the calculation formula for the excitation line at one meshing frequency is: F e = z·n / 60, where z is the number of teeth of the gear and n is the rotational speed.

6. The vibration avoidance optimization method based on Campbell's diagram for identifying the dangerous pitch diameter vibration mode of gears according to claim 1, characterized in that, Step S30 further includes: sorting all the natural frequency lines of all identified dangerous nodal vibration modes in order of their natural frequency values ​​at zero speed from low to high, and recording them sequentially as nodal 1, nodal 2, etc., according to the rule of grouping every two lines, until the nodal number of all nodal vibration modes is recorded.

7. The vibration avoidance optimization method based on Campbell's diagram for identifying the dangerous pitch diameter vibration mode of gears according to claim 1, characterized in that, In step S10, the structural dimensions to be optimized as design variables include: wall thickness parameters at different positions of the wheel axle, parameters characterizing the radial length of the spokes at different positions, and parameters characterizing the axial thickness of the spokes.

8. The vibration avoidance optimization method based on Campbell's diagram for identifying the dangerous pitch diameter vibration mode of gears according to claim 1, characterized in that, The vibration damping optimization process is solved using the Pointer optimization algorithm.

9. The vibration avoidance optimization method based on Campbell's diagram for identifying the dangerous pitch diameter vibration mode of gears according to claim 1, characterized in that, In step S20, the method for linear fitting the analysis results is polynomial fitting based on the least squares method.

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