Dynamics modeling method for micro-gear with clearance of rotational pair

Abstract

The application discloses a kind of micro gear dynamics modeling methods containing rotation pair gap, belong to gear transmission system dynamics simulation technical field.The method is established by considering the calculation model of meshing stiffness of rotation pair gap based on the meshing principle of involute gear and potential energy method, deduces the calculation formula of important parameters such as gear meshing angle, side clearance, coincidence degree, and establishes the time-varying meshing stiffness calculation model of gear containing rotation pair gap;The application comprehensively considers the change of gear pair center distance caused by the existence of rotation pair gap, influences the meshing stiffness of gear, load distribution coefficient, meshing angle and side clearance etc., the change of these parameters will cause gear to present different nonlinear dynamic behavior, provide theoretical support for noise reduction and life extension of gear transmission system.

Description

Technical Field

[0001] This application belongs to the field of dynamic simulation technology of gear transmission systems, specifically involving a dynamic modeling method for micro gears with revolute joint clearance. Background Technology

[0002] Gears are among the most important power transmission components in aerospace, automotive, robotics, and agricultural machinery. Gear drives are primarily used to transmit torque and speed. Therefore, they are also a major source of vibration and noise in many devices. After prolonged operation, rotating pairs such as bearings in gear drives develop clearances due to wear, which exacerbates gear vibration and noise. Therefore, elucidating the correlation between rotating pair clearances and the nonlinear dynamics of gear drives is of great significance for suppressing vibration and extending the service life of gear transmission systems.

[0003] Currently, gear dynamics research primarily focuses on the impact of initial errors such as gear eccentricity, tooth profile error, and pitch error on the dynamic performance of gear transmission systems. Accumulated errors formed during gear operation gradually worsen gear vibration, thus accelerating the failure of the gear transmission system. Early research focused on the impact of cracks on the dynamic performance of gear systems, particularly the weakening effect of tooth root cracks on meshing stiffness. Some researchers mainly addressed the impact of accumulated wear, pitting, and tooth root cracks on the dynamic performance of gear pairs. However, bearings and other rotating pairs develop clearances due to wear after prolonged operation, increasing the center distance, backlash, and reducing the contact ratio of the gears. Few studies have analyzed the impact of these parameter changes on gear meshing characteristics and dynamic performance. Therefore, establishing a dynamic model of a gear transmission system with rotating pair clearances, and revealing the influence of changes in meshing angle, backlash, contact ratio, and meshing stiffness caused by rotating pair clearances on gear dynamic performance, should be considered. Summary of the Invention

[0004] The purpose of this application is to provide a method for modeling the dynamics of micro gears with rotating pair clearance, which solves the technical problem that existing gear transmission systems do not consider the impact of changes in meshing angle, backlash, contact ratio, and meshing stiffness caused by rotating pair clearance on gear dynamic performance.

[0005] To solve the above-mentioned technical problems, this application is implemented as follows:

[0006] In a first aspect, embodiments of this application provide a method for modeling the dynamics of a micro-gear containing the backlash of a rotating pair, including:

[0007] Considering the influence of the backlash of the rotating joint on the center distance of the gear transmission system, calculate the meshing angle of the gear pair considering the backlash of the rotating joint;

[0008] Considering the change in the gear pair meshing angle, calculate the length of the line of action;

[0009] Calculate the contact ratio of the gear pair, taking into account the variation in the length of the line of action;

[0010] Considering the change in center distance caused by the clearance of the rotating pair, calculate the half-tooth side clearance after the change in center distance;

[0011] Based on the potential energy method, the gear teeth are equivalent to cantilever beams with the tooth root circle fixed. According to the contact deformation, bending deformation, shear deformation and compression deformation generated by the gear and the deformation of the hub body, the Hertz contact stiffness, bending stiffness, shear stiffness, compression stiffness and tooth deformation stiffness of the gear are calculated.

[0012] Treating the gears as rigid rotors, each gear has 3 degrees of freedom, a six-degree-of-freedom gear dynamics model considering the backlash of the rotating joint is established.

[0013] Optionally, the calculation of the gear pair meshing angle considering the revolute joint clearance includes:

[0014] The gear pair meshing angle is calculated using equation (1):

[0015]

[0016] Where, r b1 and r b2 Z1 and Z2 are the base circle radii of the driving and driven gears, respectively; m is the gear module; z1 and z2 are the number of teeth of the driving and driven gears, respectively; d′ is the center distance including the clearance of the revolute joint; Δ d This refers to the clearance of the rotating pair.

[0017] Optionally, the calculation of the length of the engagement line includes:

[0018] The length of the meshing line is calculated using equation (2):

[0019] B1B2=r b1 (tanα a1 -tanα′)+r b2 (tanα a2 -tanα′) (23)

[0020] Where, α a1 and α a2 These represent the pressure angles of the tooth tip circles of the driving gear and the driven gear, respectively.

[0021] Optionally, the calculation of the overlap ratio of the gear pair includes:

[0022] The overlap ratio of the gear pair is calculated using equation (3):

[0023]

[0024] Where α is the gear pair meshing angle.

[0025] Optionally, the calculation of the half-tooth backlash after the center distance change includes:

[0026] The half-tooth backlash after the center distance change is calculated using equation (4):

[0027] b=b0+d′(invα′-invα)cosα′ (25)

[0028] Where b0 is the initial half-tooth side clearance.

[0029] Optionally, the calculation of the Hertzian contact stiffness, bending stiffness, shear stiffness, compressive stiffness, and tooth deformation stiffness of the gear includes:

[0030] When a pair of gear teeth engage, the total variable potential energy U is the contact variable potential energy U0. h Bending deformation U b Shear deformation U s and compression deformation U a and matrix change form energy U f The sum, that is:

[0031]

[0032] In equation (5), the subscripts i = 1, 2 represent the driving wheel and the driven wheel, respectively; λ i is the matrix stiffness correction coefficient, determined by the finite element method; F is the total contact force in the meshing teeth, k is the total meshing stiffness of a pair of meshing teeth, and the potential energies can be expressed as:

[0033]

[0034] Where, k h For contact stiffness, k a For compressive stiffness, k b For bending stiffness, k s For shear stiffness, k f For matrix stiffness;

[0035] In equation (6), the contact stiffness k h and matrix stiffness k f The formula for calculation is:

[0036]

[0037]

[0038] Where E, υ, and B are the elastic modulus, Poisson's ratio, and tooth width of the gear material, respectively; τ iis the angle between the force and the y-axis; u is the distance from the intersection of the line of meshing and the line of symmetry of the gear teeth to the root circle; Sf is the arc length corresponding to the entire tooth profile curve of the gear; L*, M*, P*, and Q* are four parameters related to the gear module and number of teeth; in equation (5), the compressive potential energy, bending potential energy, and shear potential energy can be expressed as:

[0039]

[0040]

[0041]

[0042] In the formula

[0043]

[0044] G is the shear modulus; x C x D The x-coordinates of the starting and ending points of the tooth root transition curve are respectively represented by the x-coordinates. i I is the horizontal distance from the current contact point i to the original point. x1 I x2 A x1 A x2 Let represent the area moment of inertia and cross-sectional area of ​​the pinion and gear, respectively; y1 and y2 in the formulas for calculating area and moment of inertia are the y-coordinates of the integration point on the tooth root transition curve and the involute tooth profile segment, respectively; therefore, the compressive stiffness k a Bending stiffness k b and shear stiffness k s The formula for calculation is:

[0045]

[0046]

[0047]

[0048] From equations (4) and (5), the formula for calculating the meshing stiffness of a single tooth can be obtained:

[0049]

[0050] The contact ratio of a standard spur gear is between 1 and 2, exhibiting both single-meshing and double-meshing regions; therefore, the total time-varying meshing stiffness can be expressed as:

[0051]

[0052] Optionally, each gear has 3 degrees of freedom, including:

[0053] The generalized coordinate matrix of the gear is q = [x py p θ p x g y g θ g ] T .

[0054] Optionally, establishing a six-degree-of-freedom gear dynamics model considering the backlash of the revolute joint includes:

[0055] The dynamic transmission error of gears is defined as:

[0056] δ d =Vq (34)

[0057] V=[sinα′cosα′r b1 -sinα′-cosα′r b2 (35)

[0058] The shaft frequency error and meshing frequency error of a gear are expressed as follows:

[0059]

[0060] In equation (15), f m f is the meshing frequency. s Let e1 and e2 be the meshing frequency error and shaft frequency error amplitude, respectively, and φ1 and φ2 be the phase angles; then the deformation of the gear contact is:

[0061] δ m =δ d -e s (37)

[0062] After considering backlash and static transmission error es, the contact force between the gears is:

[0063]

[0064] In equation (17), k m and c m These represent gear meshing stiffness and damping, respectively; b represents half-tooth backlash; γ m0 and γ m1 The sign function, used to determine gear engagement or disengagement, can be represented as:

[0065]

[0066]

[0067] Therefore, the gear dynamics equations considering backlash and static transmission error can be expressed as:

[0068]

[0069] In equation (20),

[0070] M = diag[M1 m1 J1 m2 m2 J2]

[0071] C=-γ m0 c m V T V

[0072] K = -γ m0 k m V T V

[0073]

[0074] F0 = [0 0 T1 0 0 T2] T (42)

[0075] In this application embodiment, a meshing stiffness calculation model considering the revolute joint clearance was established based on the involute gear meshing principle and the potential energy method. The calculation formulas for important parameters such as gear meshing angle, tooth flank clearance, and contact ratio were derived, and a time-varying meshing stiffness calculation model for gears containing revolute joint clearance was established. This application comprehensively considers the impact of the existence of revolute joint clearance on the change of the center distance of the gear pair, which affects the meshing stiffness, load distribution coefficient, meshing angle, and tooth flank clearance of the gear. Changes in these parameters will lead to different nonlinear dynamic behaviors of the gear, providing theoretical support for noise reduction and life extension of gear transmission systems. Attached Figure Description

[0076] Figure 1 This is a flowchart of the method provided in the embodiments of this application;

[0077] Figure 2 This is a schematic diagram of the involute gear meshing principle provided in the embodiments of this application;

[0078] Figure 3 This is an involute tooth profile and force diagram provided in an embodiment of this application;

[0079] Figure 4 This is a nonlinear dynamic model of the gear transmission system provided in the embodiments of this application. Detailed Implementation

[0080] The technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, not all embodiments. Based on the embodiments of this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0081] The terms "first," "second," etc., used in the specification and claims of this application are used to distinguish similar objects and not to describe a specific order or sequence. It should be understood that such use of data can be interchanged where appropriate so that embodiments of this application can be implemented in orders other than those illustrated or described herein, and the objects distinguished by "first," "second," etc., are generally of the same class and the number of objects is not limited; for example, a first object can be one or more. Furthermore, in the specification and claims, "and / or" indicates at least one of the connected objects, and the character " / " generally indicates that the preceding and following objects are in an "or" relationship.

[0082] The following description, in conjunction with the accompanying drawings, details the method for modeling the dynamics of micro gears with revolute joint clearance provided in this application through specific embodiments and application scenarios.

[0083] Please see Figure 1 This application provides a method for modeling the dynamics of a micro-gear containing revolute joint clearance, comprising:

[0084] Step S101: Calculate the gear pair meshing angle considering the backlash of the rotating pair;

[0085] Step S102: Calculate the length of the line of engagement;

[0086] Step S103: Calculate the overlap ratio of the gear pair;

[0087] Step S104: Calculate the half-tooth side clearance after the center distance changes;

[0088] Step S105: Based on the potential energy method, the gear teeth are equivalent to a cantilever beam with the tooth root circle fixed. Based on the contact deformation, bending deformation, shear deformation and compression deformation generated by the gear and the deformation of the hub body, the Hertz contact stiffness, bending stiffness, shear stiffness, compression stiffness and gear tooth deformation stiffness of the gear are calculated.

[0089] Step S106: Treat the gears as rigid rotors, with each gear having 3 degrees of freedom, and establish a six-degree-of-freedom gear dynamics model that considers the backlash of the rotating joint.

[0090] In step S101, when calculating the gear pair meshing angle considering the backlash of the rotating pair, it is necessary to consider the influence of the backlash of the rotating pair on the center distance of the gear transmission system.

[0091] Combined Figure 2 As shown, the calculation of the gear pair meshing angle considering the revolute joint clearance includes:

[0092] The gear pair meshing angle is calculated using equation (1):

[0093]

[0094] Where, r b1 and rb2 Z1 and Z2 are the base circle radii of the driving and driven gears, respectively; m is the gear module; z1 and z2 are the number of teeth of the driving and driven gears, respectively; d′ is the center distance including the clearance of the revolute joint; Δ d This refers to the clearance of the rotating pair.

[0095] In step S102, when calculating the length of the line of engagement, the change in the gear pair meshing angle needs to be considered.

[0096] The calculation of the length of the engagement line includes:

[0097] The length of the meshing line is calculated using equation (2):

[0098] B1B2=r b1 (tanα a1 -tanα′)+r b2 *(tanα a2 -tanα′) (44)

[0099] Where, α a1 and α a2 These represent the pressure angles of the tooth tip circles of the driving gear and the driven gear, respectively.

[0100] In step S103, when calculating the overlap ratio of the gear pair, the change in the length of the meshing line needs to be considered.

[0101] The calculation of the overlap ratio of the gear pair includes:

[0102] The overlap ratio of the gear pair is calculated using equation (3):

[0103]

[0104] Where α is the gear pair meshing angle.

[0105] In step S104, when calculating the half-tooth side clearance after the center distance change, it is necessary to consider the center distance change caused by the rotating pair clearance.

[0106] The calculation of the half-tooth side clearance after the center distance change includes:

[0107] The half-tooth backlash after the center distance change is calculated using equation (4):

[0108] b=b0+d′(invα′-invα)cosα′ (46)

[0109] Where b0 is the initial half-tooth side clearance.

[0110] In step S105, the calculation of the Hertzian contact stiffness, bending stiffness, shear stiffness, compressive stiffness, and tooth deformation stiffness of the gear includes:

[0111] The geometric profile and force diagram of the involute spur gear are shown below. Figure 3 Curve AB represents the addendum circle, curve BC represents the involute curve, and curve CD represents the root transition curve. In the potential energy method for calculating meshing stiffness, the gear teeth are equivalent to cantilever beams fixed at the root circle. Under the action of force F, the gear teeth will undergo contact deformation, bending deformation, shear deformation, and compressive deformation (along the x-direction). In addition, the hub body will also deform.

[0112] When a pair of gear teeth engage, the total variable potential energy U is the contact variable potential energy U0. h Bending deformation U b Shear deformation U s and compression deformation U a and matrix change form energy U f The sum, that is:

[0113]

[0114] In equation (5), the subscripts i = 1, 2 represent the driving wheel and the driven wheel, respectively; λ i The matrix stiffness correction coefficient is determined by the finite element method; F is the total contact force in the meshing teeth; k is the total meshing stiffness of a pair of meshing teeth; the potential energies can be expressed as:

[0115]

[0116] Where, k h For contact stiffness, k a For compressive stiffness, k b For bending stiffness, k s For shear stiffness, k f For matrix stiffness;

[0117] In equation (6), the contact stiffness k h and matrix stiffness k f The formula for calculation is:

[0118]

[0119]

[0120] Where E, υ, and B are the elastic modulus, Poisson's ratio, and tooth width of the gear material, respectively; τ iis the angle between the force and the y-axis; u is the distance from the intersection of the line of meshing and the line of symmetry of the gear teeth to the root circle; Sf is the arc length corresponding to the entire tooth profile curve of the gear; L*, M*, P*, and Q* are four parameters related to the gear module and number of teeth; in equation (5), the compressive potential energy, bending potential energy, and shear potential energy can be expressed as:

[0121]

[0122]

[0123]

[0124] In the formula,

[0125]

[0126] G is the shear modulus, x C x D The x-coordinates of the starting and ending points of the tooth root transition curve are respectively represented by the x-coordinates. i I is the horizontal distance from the current contact point i to the original point. x1 I x2 A x1 A x2 Let represent the area moment of inertia and cross-sectional area of ​​the pinion and gear, respectively; y1 and y2 in the formulas for calculating area and moment of inertia are the y-coordinates of the integration point on the tooth root transition curve and the involute tooth profile segment, respectively; therefore, the compressive stiffness k a Bending stiffness k b and shear stiffness k s The formula for calculation is:

[0127]

[0128]

[0129]

[0130] From equations (4) and (5), the formula for calculating the meshing stiffness of a single tooth can be obtained:

[0131]

[0132] The contact ratio of a standard spur gear is between 1 and 2, exhibiting both single-meshing and double-meshing regions; therefore, the total time-varying meshing stiffness can be expressed as:

[0133]

[0134] In step S106, the gear contact model can be equivalent to the contact between two basic cylinders connected by a spring-damping system. In the gear contact model, the gear body is regarded as a rigid rotor. Each gear has 3 degrees of freedom, that is, the generalized coordinate matrix is ​​q = [x p y p θ p x g y g θ g ] T .

[0135] Combined Figure 4 As shown, the establishment of a six-degree-of-freedom gear dynamics model considering the backlash of the revolute joint includes:

[0136] The dynamic transmission error of gears reflects the ability of gears to resist deformation in the meshing direction when transmitting torque, and is a key indicator for evaluating the vibration performance of gear systems. It is defined as follows:

[0137] δ d =Vq (55)

[0138] V=[sinα′ cosα′ r b1 -sinα′-cosα′ r b2 (56)

[0139] Gears will generate errors during manufacturing and installation, mainly including shaft frequency error and meshing frequency error. The shaft frequency error and meshing frequency error of gears are expressed as follows:

[0140]

[0141] In equation (15), f m f is the meshing frequency. s Let e1 and e2 be the meshing frequency error and shaft frequency error amplitude, respectively, and φ1 and φ2 be the phase angles; then the deformation of the gear contact is:

[0142] δ m =δ d -e s (58)

[0143] Considering tooth flank clearance and static transmission error e s Afterwards, the contact force between the gears is:

[0144]

[0145] In equation (17), k m and c m These represent gear meshing stiffness and damping, respectively; b represents half-tooth backlash; γ m0 and γ m1The sign function, used to determine gear engagement or disengagement, can be represented as:

[0146]

[0147]

[0148] Therefore, the gear dynamics equations considering backlash and static transmission error can be expressed as:

[0149]

[0150] In equation (20),

[0151] M = diag[m1 m1 J1 m2 m2 J2]

[0152] C=-γ m0 c m V T V

[0153] K = -γ m0 k m V T V

[0154]

[0155] F0 = [0 0 T1 0 0 T2] T (63)

[0156] It should be noted that, in this document, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes that element. Furthermore, it should be noted that the scope of the methods and apparatuses in the embodiments of this application is not limited to performing functions in the order shown or discussed, but may also include performing functions substantially simultaneously or in the reverse order, depending on the functions involved. For example, the described methods may be performed in a different order than described, and various steps may be added, omitted, or combined. Additionally, features described with reference to certain examples may be combined in other examples.

[0157] The embodiments of this application have been described above with reference to the accompanying drawings. However, 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 modeling the dynamics of a micro gear containing the backlash of a revolute pair, characterized in that, include: Considering the effect of the revolute joint clearance on the center distance, calculate the gear pair meshing angle considering the revolute joint clearance, including: The gear pair meshing angle is calculated using equation (1): （1） in, and These are the base circle radii of the driving and driven gears, respectively. m The gear module; and These are the number of teeth on the driving gear and the driven gear, respectively. The center distance including the clearance of the rotating joint. For the clearance of the rotating joint; Based on the calculated gear pair meshing angle, the length of the line of action is recalculated, including: The length of the meshing line is calculated using equation (2): （2） in, and These represent the pressure angles of the addendum circles of the driving and driven gears, respectively. Based on the calculated line of engagement length, the contact ratio of the gear pair is recalculated, including: The overlap ratio of the gear pair is calculated using equation (3): （3） in, The gear pair meshing angle; The half-tooth backlash after the center distance change is calculated, wherein the calculation of the half-tooth backlash is based on the following physical mechanism: taking the change in the center distance of the revolute joint clearance as input, and calculating it through a relationship including the changed center distance and the changed meshing angle, wherein the calculation of the half-tooth backlash after the center distance change includes: The half-tooth backlash after the center distance change is calculated using equation (4): （4） in, This represents the initial half-tooth side clearance. Based on the potential energy method, the gear teeth are equivalent to cantilever beams with fixed root circles. According to the contact deformation, bending deformation, shear deformation, and compression deformation of the gear, as well as the deformation of the hub body, the Hertzian contact stiffness, bending stiffness, shear stiffness, compression stiffness and gear tooth deformation stiffness of the gear are calculated. The recalculated overlap ratio is used to determine whether the gear is in the single tooth meshing area or the double tooth meshing area, so as to synthesize the total time-varying meshing stiffness. Treating the gears as rigid rotors, each gear has 3 degrees of freedom, a six-degree-of-freedom gear dynamics model is established that can reflect the dynamic influence of the revolute pair clearance on the system matrix. In the six-degree-of-freedom gear dynamics model, the recalculated gear pair meshing angle is used as a basis. Reconstruct the gear dynamic transmission error vector V, wherein the dynamic transmission error vector V is defined as: Furthermore, the damping matrix of the system is constructed based on the reconstructed V vector. and stiffness matrix ,in , For symbolic functions, k m and c m These are the gear meshing stiffness and damping, respectively; the calculated half-tooth backlash after the center distance change is used to reconstruct the contact force model between gears.

2. The method according to claim 1, characterized in that, The calculation of the gear's Hertzian contact stiffness, bending stiffness, shear stiffness, compressive stiffness, and tooth deformation stiffness includes: When a pair of gear teeth engage, the total variable potential energy... U To contact changing situation energy U h Bending deformation U b Shear deformation U s and compression deformation U a and matrix change energy U f The sum, that is: （5） In equation (5), the subscript i =1,2 represent the driving wheel and the driven wheel, respectively; The matrix stiffness correction factor is determined using the finite element method. This represents the total contact force in the meshing teeth. For the total meshing stiffness of a pair of meshing teeth, the potential energies can be expressed as: （6） in, For contact stiffness, To compress stiffness, For bending stiffness, For shear stiffness, For matrix stiffness; In equation (6), the contact stiffness k h and matrix stiffness k f The formula for calculation is: （7） （8） in, E , υ , B These are the elastic modulus, Poisson's ratio, and tooth width of the gear material, respectively. τ i For strength and y The included angle of the shaft; u is the distance from the intersection of the line of meshing and the line of symmetry of the gear teeth to the root circle; Sf is the arc length corresponding to the entire tooth profile curve of the gear; L M P Q These are four parameters related to the gear module and number of teeth; in equation (5), the compressive potential energy, bending potential energy, and shear potential energy can be expressed as: （9） In the formula , G Shear modulus , These represent the x-coordinates of the starting and ending points of the tooth root transition curve, respectively. For when front contact point The horizontal distance from the origin point. , , , Let A represent the area moment of inertia and cross-sectional area of ​​the pinion and gear, respectively; the formulas for calculating area and moment of inertia are as follows: , The points of integration are located at the root transition curve and the involute tooth profile segment, respectively. Coordinates; therefore, compressive stiffness k a Bending stiffness k b and shear stiffness k s The formula for calculation is: （10） From equations (4) and (5), the formula for calculating the meshing stiffness of a single tooth can be obtained: （11） The contact ratio of a standard spur gear is between 1 and 2, exhibiting both single-meshing and double-meshing regions; therefore, the total time-varying meshing stiffness can be expressed as: （12）。 3. The method according to claim 2, characterized in that, Each gear has 3 degrees of freedom, including: The generalized coordinate matrix of the gear is .

4. The method according to claim 3, characterized in that, A six-degree-of-freedom gear dynamics model is established to reflect the dynamic influence of the revolute joint backlash on the system matrix, including: The dynamic transmission error of gears is defined as: （13） The shaft frequency error and meshing frequency error of a gear are expressed as follows: （15） In equation (15), f m The meshing frequency, f s For shaft frequency, e 1 and e 2 represents the amplitude of meshing frequency error and shaft frequency error, respectively. φ 1 and φ 2 is the phase angle; then the deformation of the gear contact is: （16） Considering tooth flank clearance and static transmission error Afterwards, the contact force between the gears is: （17） In equation (17), k m and c m These are gear meshing stiffness and damping, respectively; b This is half-tooth side clearance; γ m0 and γ m1 The sign function, used to determine gear engagement or disengagement, can be represented as: （18） （19） Therefore, the gear dynamics equations considering backlash and static transmission error can be expressed as: （20） In equation (20), （21）。

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