A method for optimal cogging phase design of cogging phase tuning gear
Patent Information
- Application Number
- CN202411306016.5
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2024-09-18
- Publication Date
- 2026-10-23
AI Technical Summary
[0002]高速重载齿轮在传递动力的过程中，由于轮齿变形及啮合周期性等因素，整个传动系统不可避免地产生振动和噪声，进而影响传动系统的可靠性及疲劳寿命
[0056]本发明建立了错齿相位调谐齿轮的时变啮合刚度模型及平行级齿轮传动系统动力学模型，定量分析了错齿相位对传动系统振动响应的影响，推导了传动系统动力学响应与错齿相位的映射关系，获得了圆柱直齿轮最优错齿相位角度
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Abstract
Description
Technical Field
[0001] This invention belongs to the field of mechanical dynamics technology and is extensively related to design methods for time-varying meshing stiffness calculation, dynamic analysis, optimal tooth misalignment phase analysis, and gear structure optimization. Background Technology
[0002] During the power transmission process, high-speed heavy-duty gears inevitably generate vibration and noise due to factors such as tooth deformation and meshing periodicity, which in turn affects the reliability and fatigue life of the transmission system.
[0003] Therefore, how to reduce vibration and noise during the operation of high-speed, heavy-load gears has become a technical challenge that professionals in this field need to solve. Summary of the Invention
[0004] The purpose of this invention is to provide an optimal tooth phase design method for tooth phase tuning gears to solve the problems mentioned in the background art.
[0005] To achieve the above objectives, the technical solution provided by the present invention is as follows:
[0006] A method for designing the optimal tooth phase of a tooth-shifting tuned gear includes the following steps:
[0007] Step 1: Obtain the basic parameters of the cylindrical gear pair;
[0008] Step 2: Establish a calculation model for the time-varying meshing stiffness of the misaligned phase-tuned gear;
[0009] Step 3: Establish a dynamic model of a misaligned phase-tuned cylindrical gear pair;
[0010] Step 4: Calculate the time-varying meshing stiffness and dynamic response at key nodes for different tooth misalignment phase amounts;
[0011] Step 5: Select the optimal misalignment phase by taking the minimum dynamic meshing force fluctuation of the gear pair and the minimum vibration acceleration at key nodes as the evaluation criteria;
[0012] Furthermore, step 1 includes the following steps: Step 101: Obtain various geometric and physical parameters, including the number of teeth, module, pressure angle, misalignment phase, elastic modulus, and Poisson's ratio.
[0013] Furthermore, step 2 includes the following steps:
[0014] Step 201: Establish a calculation model for the time-varying meshing stiffness of a single-sided gear pair with arbitrary initial meshing phases. The expression for the time-varying meshing stiffness of a single-sided gear pair with arbitrary initial meshing phases is:
[0015]
[0016] In the formula, Δk(t) represents the average meshing stiffness over one cycle, and Δk(t) represents the variation in meshing stiffness, which changes periodically with time.
[0017]
[0018] Where, k min k max The minimum and maximum values of the spur gear meshing stiffness are given by α1, which is a coefficient related to the contact ratio, α1 = 1 - ε′. α , ε α ε′ represents the degree of overlap. α For ε α The decimal part.
[0019]
[0020] Where T is the meshing period.
[0021]
[0022] Where p is the phase when meshing occurs. Obviously, for a gear with staggered phase, p1 ≠ p2.
[0023] Further, step 202: establish a calculation model for the time-varying meshing stiffness of the misaligned phase-tuned gear;
[0024] The formula for calculating the time-varying meshing stiffness of a misaligned phase-tuned gear is as follows:
[0025]
[0026] Where, k (1) (t), k (2) (t) represents the meshing stiffness of the two meshing pairs of the staggered-tooth phase-tuned gear, which can be further simplified as follows:
[0027]
[0028] In the formula, Δk represents the average value of the meshing stiffness of the misaligned phase-tuned gear pair, and Δk represents the time-varying part of the meshing stiffness of the misaligned phase-tuned gear pair. The superscripts (1) and (2) represent the two meshing pairs, respectively.
[0029]
[0030] In the formula, This represents the initial phase of meshing for a staggered-tooth phase-tuned gear. Further simplification:
[0031]
[0032] in,
[0033]
[0034] Furthermore, step 3 includes the following steps:
[0035] Step 301: Establish the shaft segment element model, meshing element model, and bearing support element model. Based on key factors such as the diameter, support position, and meshing position of the drive shaft, and adhering to the principle of consistent natural frequencies and mode shapes, the drive shaft is equivalent to a shaft segment element, whose dynamic model is as follows:
[0036]
[0037] In the formula, M ij K ij G ij and C ij These are the mass, stiffness, gyroscope, and damping matrices of the Timoshenko beam element, respectively, X. ij This represents the displacement column vector of the two nodes of the shaft segment element.
[0038] The meshing relationship of the staggered-tooth phase-tuned gear can be equivalently represented as two base cylinders connected by a spring-damped element. Therefore, the kinematic differential equation of the meshing element can be written as follows:
[0039]
[0040] In the formula, M sp C sp K sp Let F represent the meshing mass matrix, meshing damping matrix, and meshing stiffness matrix of the meshing unit, respectively; spe q represents external load excitation such as error. sp This represents the displacement column vector of the meshing node.
[0041] The bearing is equivalent to a spring-damped element model, meaning that the bearing node i on the drive shaft is directly connected to the housing through the bearing stiffness and damping. The dynamic equation of the bearing-housing connection element can be written as follows:
[0042]
[0043] In the formula, M bi C bi K bi These represent the mass matrix, damping matrix, and stiffness matrix of the support element, respectively.
[0044] Further, in step 302: Based on the geometry of the transmission system, considering the time-varying meshing stiffness and meshing error of the staggered-tooth phase-tuned gear pair, and according to the node coupling principle, the mass, stiffness, and damping matrices at corresponding nodes are superimposed to establish a dynamic model of the parallel-axis staggered-tooth phase-tuned gear transmission system. The dynamic differential equation of the staggered-tooth phase-tuned gear transmission system can be written as follows:
[0045]
[0046] In the formula, M z C z K z Let F represent the combined mass matrix, damping matrix, and stiffness matrix of the staggered-tooth phase-tuned gear transmission system, respectively; Ω represents the gear speed, and F represents the gear rotation speed. z F e These represent the load on the transmission system and the load caused by meshing error, respectively. z This represents the displacement matrix of the global nodes.
[0047] Further, in step 303: large-scale analytical calculations are performed using the precise integration method to transform the second-order differential equation into a first-order differential equation by reducing its order, and then the corresponding non-homogeneous equation is solved.
[0048] Further, step 4: calculate the time-varying meshing stiffness and dynamic response at key nodes for different misalignment phase amounts.
[0049] Calculate the vibration acceleration at the bearing support and the dynamic meshing force of the gear pair in the parallel gear transmission system when different values of the tooth misalignment phase are selected, and determine their root mean square (RMS) values. The expression for the dynamic meshing force is as follows:
[0050]
[0051] In the formula, k sp c sp These represent meshing stiffness and meshing damping, respectively, and δ represents the component of the nodal vibration displacement in the meshing line direction.
[0052] The calculation methods for dynamic meshing force and root mean square acceleration are as follows:
[0053]
[0054] Further, in step 5: establish the mapping relationship between the misaligned tooth phase and the vibration response of the transmission system, and select the optimal misaligned tooth phase by taking the minimum dynamic meshing force fluctuation of the gear pair and the minimum vibration acceleration at key nodes as the evaluation index.
[0055] The technical effects and advantages of this invention are as follows:
[0056] This invention establishes a time-varying meshing stiffness model for misaligned phase tuned gears and a dynamic model for a parallel stage gear transmission system. It quantitatively analyzes the influence of misaligned phase on the vibration response of the transmission system, derives the mapping relationship between the dynamic response of the transmission system and the misaligned phase, and obtains the optimal misaligned phase angle for cylindrical spur gears. Attached Figure Description
[0057] Figure 1 A schematic diagram of time-varying meshing of a single pair of spur gears with arbitrary meshing phases;
[0058] Figure 2 A time-varying meshing stiffness curve of a single spur gear pair with arbitrary meshing phase;
[0059] Figure 3 For the time-varying meshing stiffness curve of spur gears that takes into account the misaligned tooth phase;
[0060] Figure 4 For a pair of spur gears considering the continuous phase of the misalignment;
[0061] Figure 5 This represents the mapping relationship between the dynamic meshing force and the misalignment phase of a gear with a staggered tooth phase.
[0062] Figure 6 This represents the mapping relationship between the vibration acceleration at the bearing node of the misaligned gear and the misaligned phase. Detailed Implementation
[0063] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all 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.
[0064] Example 1
[0065] Step 1: First, obtain the basic parameters of the cylindrical gear pair, including the number of teeth, module, pressure angle, misalignment phase, elastic modulus, Poisson's ratio, and other geometric and physical parameters.
[0066] Step 2: Without considering the misalignment phase, the time-varying meshing stiffness of a single meshing pair can be obtained through empirical formulas. For a spur gear pair, its meshing stiffness curve must be a rectangular wave that varies with time, such as... Figure 1 As shown. It can be written as the sum of the average meshing stiffness and its variation.
[0067]
[0068] In the formula, Let be the average meshing stiffness over one period, and Δk(t) be the variation in meshing stiffness, which changes periodically with time. This can be obtained from the geometric relationship of rectangular waves.
[0069]
[0070] Where, k min k max The minimum and maximum values of the spur gear meshing stiffness are given by α1, which is a coefficient related to the contact ratio, α1 = 1 - ε′. α , ε′ α =mod(ε α ,1), ε α The degree of overlap is represented by ; mod(x,y) is the modulo function.
[0071] To facilitate the derivation of the calculation formula, the change in meshing stiffness Δk(t) is written in the form of a Fourier series related to the meshing period T.
[0072]
[0073] in,
[0074]
[0075] Where p is the phase when meshing occurs. Obviously, for the two meshing pairs of staggered-tooth phase gears, p1 ≠ p2.
[0076] Step 3: Calculate the time-varying meshing stiffness of the misaligned phase-tuned gear.
[0077] For a staggered phase-tuned gear, its meshing stiffness can be considered as the average of the meshing stiffnesses of the two meshing pairs, that is,
[0078]
[0079] Where, k (1) (t), k (2) (t) represents the meshing stiffness of the two meshing pairs of the staggered-tooth phase-tuned gear, which can be further simplified as follows:
[0080]
[0081] In the formula, The average value of the meshing stiffness of the staggered-tooth phase-tuned gear pair is given, and Δk is the time-varying part of the meshing stiffness of the staggered-tooth phase-tuned gear. The superscripts (1) and (2) represent the two meshing pairs, respectively. Substituting the meshing stiffness of a single-sided meshing pair into the above formula yields the following:
[0082]
[0083] In the formula, This represents the initial phase of meshing for a staggered-tooth phase-tuned gear. Further simplification:
[0084]
[0085] in,
[0086]
[0087] Thus, a calculation model for the time-varying meshing stiffness of a staggered-tooth phase-tuned gear is established. For example... Figure 2 The figure shows a schematic diagram of the time-varying meshing stiffness curve of the misaligned phase tuned gear under different misaligned phase values.
[0088] Step 4: Establish the shaft segment unit model, meshing unit model, and bearing support unit model of the single-stage staggered tooth phase-tuned gear transmission system.
[0089] To consider the flexibility of the shaft segments in the dynamic analysis of the transmission system, the transmission shaft is divided into different segments based on key factors such as shaft diameter, support location, and meshing position. Based on the principle of consistent natural frequencies and mode shapes, each shaft segment is equivalent to a standard beam element model. For the Timoshenko beam element, its dynamic model is as follows:
[0090]
[0091] In the formula, M ij K ij G ij and C ij These are the mass, stiffness, gyroscope, and damping matrices of the Timoshenko beam element, respectively, X. ij This represents the displacement column vector of the two nodes of the shaft segment element.
[0092] For a gear connection between two drive shafts, it can be equivalently represented as a spring-damped element along the common tangent direction at the base circles of the two gears, and its dynamic equation can be written as follows:
[0093]
[0094] In the formula, M sp C sp K sp Let F represent the meshing mass matrix, meshing damping matrix, and meshing stiffness matrix of the meshing unit, respectively; spe q represents external load excitation such as error. sp This represents the displacement column vector of the meshing node.
[0095] The drive shaft of the transmission system is connected to the housing via bearings. The bearings can also be considered equivalent to spring-damped elements; that is, the bearing node i on the drive shaft is directly connected to the housing through the bearing stiffness and damping. The dynamic equation of the bearing-housing connection element can be written as follows:
[0096]
[0097] In the formula, M bi C bi K bi Let X represent the mass matrix, damping matrix, and stiffness matrix of the support element, respectively. bi This represents the displacement column vector of bearing node i.
[0098] Step 5: Establish a system-level dynamic model of the parallel gear transmission.
[0099] Based on the geometry of the transmission system and following the principle of node coupling, the mass, stiffness, and damping matrices at corresponding nodes are superimposed to establish a dynamic model of the parallel-axis staggered-tooth phase-tuned gear transmission system, such as... Figure 3 As shown, the dynamic differential equation of the staggered-tooth phase-tuned gear transmission system can be written as follows:
[0100]
[0101] In the formula, M z C z K z Let F represent the combined mass matrix, damping matrix, and stiffness matrix of the staggered-tooth phase-tuned gear transmission system, respectively; Ω represents the gear speed, and F represents the gear rotation speed. z F e These represent the load on the transmission system and the load caused by meshing error, respectively. z This represents the displacement matrix of the global nodes.
[0102] Since the dynamic model of the transmission system has many degrees of freedom, traditional calculation methods would be very time-consuming. Therefore, the precise integration method is used to transform the second-order differential equation into a first-order differential equation by reducing its order, and then solve the corresponding non-homogeneous equation.
[0103] The dynamic meshing force of the gear meshing pair and the vibration acceleration parameters at the bearing joint are obtained. To quantitatively evaluate the vibration response of the transmission system, the dynamic meshing force and vibration acceleration are converted into root mean square (RMS) form. The RMS calculation method is as follows:
[0104]
[0105] Step 6: Establish the mapping relationship between the misaligned tooth phase and the vibration response of the transmission system. Calculate the vibration response of the transmission system at different misaligned tooth phase values, and establish the mapping relationship between the vibration response and the misaligned tooth phase, such as... Figure 4 and Figure 5 As shown, the optimal misalignment phase is selected based on the minimum fluctuation of dynamic meshing force of the gear pair and the minimum vibration acceleration at key nodes.
[0106] Example 2
[0107] Taking the gear parameters shown in Table 2 as an example, the time-varying meshing stiffness of a single gear pair is first calculated, where the overlap ratio α1 = 1.69 and the minimum meshing stiffness k min =4.19×10 8 N / m, maximum meshing stiffness k max =6.24×10 8 N / m, the resulting curve is as follows Figure 1 As shown.
[0108] Table 1
[0109]
[0110] Based on the calculation method for the time-varying meshing stiffness of the staggered-tooth phase-tuned gear, the time-varying meshing stiffness curves of the staggered-tooth phase-tuned gear are obtained when the staggered tooth phase is p=0, p=0.2, p=0.4, p=0.6, p=0.8, and p=1.0, respectively. Figure 2 As shown.
[0111] like Figure 3 As shown, based on the geometric parameters of the transmission system, such as shaft segment parameters, bearing support positions, and gear meshing positions, shaft segment element models, meshing element models, and bearing support element models are established. Furthermore, according to the node coupling principle, the mass, stiffness, and damping matrices at corresponding nodes are superimposed to establish a dynamic model of the parallel-shaft staggered-tooth phase-tuned gear transmission system.
[0112] Large-scale analytical calculations are performed using the refined integration method, transforming the second-order differential equations into first-order differential equations through order reduction, and then solving the corresponding non-homogeneous equations. The dynamic meshing force and vibration acceleration at key nodes are calculated for different tooth misalignment phase values, and their root mean square values are obtained. A mapping relationship between the dynamic results of the transmission system and the tooth misalignment phase is established, such as... Figure 4 and Figure 5 As shown.
[0113] The optimal misaligned tooth phase is selected based on the minimum fluctuation of the dynamic meshing force of the gear pair and the minimum vibration acceleration at the critical node. Clearly, for the gear parameters in Table 1, both the dynamic meshing force RMS and the vibration acceleration RMS reach their minimum values at p1 = 0.31 and p2 = 0.69. Furthermore, the mapping relationship between the dynamic meshing force of the gear pair, the vibration acceleration at the bearing node, and the misaligned tooth phase is symmetrical, with p1 = 0.5 as the axis of symmetry. Therefore, the optimal misaligned tooth phase is p1 = min(p1 = 0.31, p2 = 0.69) = 0.31, which corresponds to p = α1.
[0114] Finally, it should be noted that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments or make equivalent substitutions for some of the technical features. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.
Claims
1. A method for designing the optimal tooth misalignment phase of a tooth-tuning gear, characterized in that, Includes the following steps: Step 1: Obtain the basic parameters of the cylindrical gear pair; Step 2: Establish a calculation model for the time-varying meshing stiffness of the misaligned phase-tuned gear; Step 3: Establish a dynamic model of a misaligned phase-tuned cylindrical gear pair; Step 4: Calculate the time-varying meshing stiffness and dynamic response at key nodes for different tooth misalignment phase amounts; Step 5: Select the optimal misalignment phase by taking the minimum dynamic meshing force fluctuation of the gear pair and the minimum vibration acceleration at key nodes as the evaluation criteria; Step 2: Includes the following steps: Step 201: Establish a calculation model for the time-varying meshing stiffness of a single-sided gear pair with arbitrary initial meshing phase; Step 202: Establish a calculation model for the time-varying meshing stiffness of the misaligned phase-tuned gear: Step 3 includes the following steps: Step 301: Establish the shaft segment unit model, meshing unit model, bearing support unit model, and connection unit model of the single-stage gear transmission system; Step 302: Based on the principle of node coupling, establish a comprehensive dynamic model of the single-stage gear transmission system; Step 303: Perform large-scale analytical calculations using the precise integration method.
2. The optimal tooth-misalignment phase design method for a tooth-misalignment phase-tuned gear according to claim 1, characterized in that, Step 1: Obtain various geometric and physical parameters, including number of teeth, module, pressure angle, misalignment phase, elastic modulus, and Poisson's ratio.
3. The optimal tooth phase design method for a tooth-shifting phase tuned gear according to claim 1, characterized in that, Step 201: Establish a calculation model for the time-varying meshing stiffness of a single-sided gear pair with arbitrary initial meshing phases. The expression for the time-varying meshing stiffness of a single-sided gear pair with arbitrary initial meshing phases is: k(t) = k + Δk(t) In the formula, k is the average value, and Δk(t) is the fluctuation value. k=k min α1+k max (1-α1) Where, k min k max α1 represents the minimum and maximum values of the meshing stiffness of spur gears, and α1 is a coefficient related to the contact ratio. Where T is the meshing period. Where p is the phase when meshing occurs. Obviously, for a gear with staggered phase, p1 ≠ p2.
4. The optimal tooth phase design method for a tooth-shifting phase tuned gear according to claim 1, characterized in that, Step 202: Establish a calculation model for the time-varying meshing stiffness of the staggered-tooth phase-tuned gear; the calculation formula for the time-varying meshing stiffness of the staggered-tooth phase-tuned gear is as follows: Where, k (1) (t), k (2) (t) represents the meshing stiffness of the two meshing pairs of the staggered-tooth phase-tuned gear, which can be further simplified as follows: In the formula, Δk represents the average value of the meshing stiffness of the staggered-tooth phase-tuned gear pair, and Δk represents the time-varying part of the meshing stiffness of the staggered-tooth phase-tuned gear pair. The superscripts (1) and (2) represent the two meshing pairs respectively. In the formula, This represents the initial phase of meshing for a staggered-tooth phase-tuned gear. Further simplification: in, 5. The optimal tooth phase design method for a tooth-shifting phase tuned gear according to claim 1, characterized in that, Step 301: Establish the shaft segment unit model, meshing unit model, and bearing support unit model. Based on key factors such as the diameter, support position, and meshing position of the drive shaft, the drive shaft is equivalent to a shaft segment unit, and its dynamic model is as follows: In the formula, M ij K ij G ij and C ij These are the mass, stiffness, gyroscope, and damping matrix of the Timoshenko beam element; The meshing element model of the gear meshing pair is as follows: In the formula, M sp C sp K sp Let F represent the meshing mass matrix, meshing damping matrix, and meshing stiffness matrix of the meshing unit, respectively; spe This indicates external load excitation such as error; The bearing support element model is as follows: In the formula, M bi C bi K bi These represent the mass matrix, damping matrix, and stiffness matrix of the support element, respectively.
6. The optimal tooth misalignment phase design method for a tooth misalignment phase tuned gear according to claim 1, characterized in that, Step 302: Based on the principle of node coupling, establish a comprehensive dynamic model of the single-stage gear pair transmission system; the comprehensive model equations are: In the formula, M z C z K z Let F represent the combined mass matrix, damping matrix, and stiffness matrix of the staggered-tooth phase-tuned gear transmission system, respectively; Ω represents the gear speed, and F represents the gear rotation speed. z F e These represent the load on the transmission system and the load caused by meshing error, respectively. z This represents the displacement matrix of the global nodes.
7. The optimal tooth-misalignment phase design method for a tooth-misalignment phase tuned gear according to claim 1, characterized in that, Step 303: Perform large-scale analytical calculations using the precise integration method to transform the second-order differential equation into a first-order differential equation by reducing its order, and solve the corresponding non-homogeneous equation.
8. The optimal tooth-misalignment phase design method for a tooth-misalignment phase-tuned gear according to claim 1, characterized in that, Step 4: Calculate the time-varying meshing stiffness and dynamic response at key nodes for different tooth misalignment phase amounts; calculate the vibration acceleration at the bearing support and the dynamic meshing force of the gear pair for different values of tooth misalignment phase amount in the parallel gear transmission system, and obtain their root mean square values. The expression for the dynamic meshing force is: In the formula, k sp c sp These represent meshing stiffness and meshing damping, respectively, and δ represents the component of the nodal vibration displacement in the meshing line direction. The calculation methods for dynamic meshing force and root mean square acceleration are as follows:
9. The optimal tooth-misalignment phase design method for a tooth-misalignment phase-tuned gear according to claim 1, characterized in that, Step 5: Establish the mapping relationship between the misaligned tooth phase and the vibration response of the transmission system. The optimal misaligned tooth phase is selected based on the minimum dynamic meshing force fluctuation of the gear pair and the minimum vibration acceleration at key nodes.