A calculation method for the meshing stiffness of a micro gear pair

Abstract

The present invention discloses a method for calculating the meshing stiffness of a micro gear pair. Based on the involute gear meshing principle and the material mechanics potential energy method, a calculation model for the meshing stiffness of the micro gear pair is established, which accurately describes the contact state during the gear transmission process with errors and analyzes the dynamic variation relationship between the actual meshing point and the meshing stiffness; considering the changes in the gear pressure angle and the contact ratio, compared with the previous meshing models that regarded the pressure angle and the contact ratio as constants, the calculation of the meshing stiffness is more accurate, which has more practical significance for analyzing the vibration characteristics and dynamic response of gears; the meshing stiffness of the micro gear pair under perfect installation and the comprehensive influence of the center distance manufacturing error and the shaft hole clearance error is calculated. Through comparative analysis, the vibration and noise of the gear transmission system can be reduced, and the stability of the transmission system can be improved.

Description

Technical Field

[0001] The present invention relates to the technical field of gears, and particularly to a calculation method for the meshing stiffness of a micro gear pair.

Background Art

[0002] Gears are widely used in high-performance demand transmission fields such as aerospace, new energy vehicles, and robots due to their excellent performance in aspects such as speed change and torque. At present, gear transmission systems are gradually developing towards high-precision and high-performance directions. Developing a high-performance micro gear transmission system with high meshing accuracy and stable transmission is extremely important. Meshing stiffness is an important characterization of gear performance and directly affects the dynamic response of the gear system. It has a significant role in aspects such as optimal design, dynamics, vibration, and noise. The center distance of the micro gear transmission system itself is relatively small, and there is a gap between the gear and the intermediate shaft. Traditional gear transmission systems often do not consider the clearance fit between the gear and the shaft. In fact, due to the manufacturing error of the center distance and the existence of the shaft hole clearance, it will cause changes in the gear meshing position and key meshing parameters, change the time-varying meshing stiffness of the gear, strengthen gear vibration and noise, and thus directly affect the operating conditions of the transmission system. Studying the calculation method of the micro gear pair is an important prerequisite for improving the reliability and stability of the micro gear transmission system.

[0003] At present, the calculation methods of gear meshing stiffness mainly include the potential energy analytical method, the finite element method, and the high-precision equipment experiment method. Although the finite element method can accurately calculate the time-varying meshing stiffness of gears, the time-consuming modeling and fine preprocessing will result in a long cycle and low calculation efficiency. The experimental method for measuring the meshing stiffness of gears has high requirements for experimental equipment, and at the same time, the measurement is difficult and the cost is high. Due to the advantages of convenience and rapidity, the potential energy method of material mechanics has become a widely used method for calculating meshing stiffness in the field of gear transmission at home and abroad. However, most of the previous research methods regard the gear pressure angle and the contact ratio as constants in the gear meshing motion model. In fact, the changes in both the pressure angle and the contact ratio will affect the meshing stiffness of the gear. Therefore, they should be considered when studying the calculation method of the meshing stiffness of the micro gear pair. This method provides a useful reference for the dynamic design and optimization of the micro gear transmission system.

Summary of the Invention

[0004] The present invention discloses a calculation method for the meshing stiffness of a micro gear pair, which can effectively solve the technical problems involved in the background art.

[0005] To achieve the above object, the technical solution of the present invention is as follows:

[0006] A calculation method for the meshing stiffness of a micro gear pair, the calculation method comprising the following steps:

[0007] Step 1: Based on the involute gear meshing principle, establish a dynamic meshing motion model of gear transmission under perfect installation, and define the relevant meshing parameters and single- and double-tooth meshing boundary conditions during the gear meshing process;

[0008] Step 2: Consider the influence of center distance manufacturing error and shaft hole clearance on the position of the meshing point, analyze the changes in the contact ratio and pressure angle during the actual gear meshing process, and redefine the meshing parameters and single- and double-tooth meshing boundary conditions during the gear meshing process;

[0009] Step 3: Establish a gear tooth profile geometric model and list the tooth root transition curve equation:

[0010] x 1 =R 1 ×sin(Φ)-(a 1 / sinγ+r ρ )×cos(γ-Φ)

[0011] y 1 =R 1 ×cos(Φ)-(a 1 / sinγ+r ρ )×sin(γ-Φ)

[0012] where γ is the angle between the common normal and the tool machining pitch line, α 0 ≤γ≤π / 2; x 1 , y 1 is a function of γ, corresponding to the position of any point on the tooth root transition curve, the tool fillet radius r ρ =c 0 m / (1-sinα 0 ); the distance a 1 from the center of the tool tip fillet to the center line is a a0 =(h 0 +c ρ )×m-r 1 ; the distance b a0 from the center of the tool tip fillet to the center line of the tool tooth groove is b 0 =πm / 4+h ρ m×tanα 0 +r 1 cosα 1 ; the angular displacement Φ of the tool machining the tooth root transition curve segment is Φ=(a 1 / tanγ+b a0 ) / R 0 ; for a standard gear, h 0 =1; c 0

[0013] Step 4: Equivalent the tooth to a cantilever beam fixed at the root circle based on the potential energy method of material mechanics. Calculate the Hertz contact stiffness, bending stiffness, shear stiffness, compression stiffness, and tooth deformation stiffness of the gear according to the contact deformation, bending deformation, shear deformation, compression deformation generated by the gear, and the deformation of the hub matrix.

[0014] Step 5: Set the basic parameters of the gear, and use MATLAB software to calculate the time-varying mesh stiffness of the micro gear pair under the comprehensive influence of perfect installation, center distance manufacturing error, and shaft hole clearance error.

[0015] As a preferred improvement of the present invention, Step 1 specifically includes:

[0016] The theoretical line of action N 1 N 2 is tangent to the base circles of the driving gear and the driven gear, and the tangent points are N 1 、N 2 respectively; the driving gear rotates clockwise and continuously transmits power to the driven gear. The target tooth pair starts meshing at the intersection point B 2 of the addendum circle of the driven gear and the line of action. During the power transmission process, the position of the meshing point will always move along the theoretical line of action N 1 N 2 ; during the meshing motion, the driving gear and the driven gear always remain in contact along N 1 N 2 . The meshing point on the tooth profile of the driven gear moves from the tooth tip to the tooth root, while the meshing point on the driving gear moves from the tooth root to the tooth tip; when the previous pair of tooth profiles reaches the upper limit point C 1 of the single meshing zone, the next pair of tooth profiles simultaneously enters point B 2 ; when the meshing point of the previous pair of tooth profiles reaches point B 1 , meshing separation occurs, and at this time, the next pair of tooth profiles moves to the lower limit point C 2 of the single meshing zone;

[0017] where R b1 、R 1 and R b2 、R 2 are the base circles and pitch circles of the pinion and the gear respectively; P is the pitch point of the gear, and O 1 、O 2 represent the geometric centers of the gears respectively; ω 1 、ω 2 are the rotational frequencies of the pinion and the gear respectively; α s is the pitch circle pressure angle and also the actual pressure angle of the gear, and α a1 is the addendum circle pressure angle. When a pair of gears is in a perfectly installed state, the actual pressure angle α s is equal to the theoretical pressure angle α 0 . The actual center distance of the gear pair is:

[0018]

[0019] Among them, O 1 and O 2 respectively represent the geometric centers of the gears, m is the module of the gear, Z 1 and Z 2 are the number of teeth of the small gear and the large gear, Δδ 1 is the center distance deviation caused during the gear manufacturing process. Under perfect installation, Δδ 1 = 0;

[0020] The actual pressure angle α s of the gear pair is:

[0021]

[0022] Among them, α 0 is the theoretical pressure angle;

[0023] Under perfect installation, the target tooth pair starts to mesh and contact from point B 2 , and starts to mesh and separate from point B 1 . At this time, the actual meshing line B 1 B 2 is:

[0024] B 1 B 2 = PB 1 + PB 2

[0025] Among them

[0026] PB 1 = R b1 × (tanα a1 - tanα s )

[0027] PB 2 = R b2 × (tanα a2 - tanα s )

[0028] Among them, P is the pitch point of the gear, R b1 and R b2 are the base circles of the small gear and the large gear respectively, α s is the pressure angle of the pitch circle and also the actual pressure angle of the gear, α a1 and α a2 are the addendum circle pressure angles of the small gear and the large gear respectively;

[0029] During the actual meshing process of gears, the contact ratio of standard spur gears usually ranges between 1 and 2, with single-tooth meshing and double-tooth meshing existing. When the driving gear rotates by an angle, that is, when the meshing motion moves from position B 2 to position C 1 , the previous pair of teeth completes the meshing process from double-tooth meshing to single-tooth meshing; when the target tooth pair starts meshing at the initial contact point B 2 , set the initial rotation angle of the gear Then the actual meshing line of the driving gear during the meshing process can be expressed as:

[0030] B 1 B 2 = N 1 B 1 - N 1 B 2

[0031] The angular displacements of a pair of meshing teeth in the single-tooth meshing area and the double-tooth meshing area can be expressed as:

[0032] θ 1-double ∈ [0, B 2 C 2 / R b1 ∪ [B 2 C 1 / R b1 , (N 1 B 1 - N 1 B 2 ) / R b1

[0033] θ 1-single ∈ [B 2 C 2 / R b1 , B 2 C 1 / R b1

[0034] where R b1 , R b2 are the base circles of the driving gear and the driven gear respectively.

[0035] As a preferred improvement of the present invention, step two specifically includes:

[0036] During the gear meshing motion, there is a center distance deviation Δδ caused by the fit clearance between the holes of gear 2 and gear 3 and the fixed shaft. 2 , assuming that gear 1 and gear 4 are in a perfect installation state, O 1 , O 2 , O 3 and O 4 are the rotation centers of each stage of gears respectively, O21 is the actual geometric center of Gear 2 and Gear 3, O 21 ′ is the projection of O 21 on the Y-axis, Δδ x = Δδ 2 cosθ, Δδ y = Δδ 2 sinθ, where θ corresponds to the angular displacement of the gear; during the power transmission process, as the angular displacement continuously changes, Gears 2 and 3 on the same transmission shaft will shift along the resultant force direction of the meshing force between the first-stage gear and the second-stage gear; due to the continuous change of the center distance between the gears during the meshing cycle, the actual center distances between the first-stage gear and the second-stage gear can be respectively expressed by the following calculation formulas:

[0037]

[0038]

[0039] where α s1 、α s1 ′、α s2 、α s2 ′ are respectively the theoretical pressure angle and the actual pressure angle at the starting contact position of the first-stage gear and the second-stage gear; N 1 、N 1 ′、N 2 、N 2 ′、N 3 、N 3 ′、N 4 、N 4 ′ are respectively the initial position and the ending position of the theoretical meshing line of the first-stage gear and the second-stage gear, corresponding to the geometric center position O 2 ′ or O 3 ′ and O 2 ″ or O 3 ″;

[0040] The actual phase difference generated by two continuously meshing tooth pairs considering the shaft hole clearance error is:

[0041] θ d = 2π / N 1 +(N 1 B 2 / r b1 -N 1 ′B 2 ′ / r b1 )

[0042] where N 1 、N 1′ corresponds to two different initial state moments under normal installation. When there is clearance in the shaft-hole revolute pair of the gear transmission system, the center distance of the gear changes with the change of angular displacement at all times, and the actual pressure angle α of the gear s1 ′ is a function of the actual center distance O 1 O 2′ When the center distance changes, the actual meshing line during the gear meshing process can be expressed as:

[0043] B 1 ′B 2 ′ = P′B 1 ′ + P′B 2 ′

[0044] The contact ratio of the gear will change with the change of the gear center distance at all times. The single-tooth meshing time and the double-tooth meshing time will both change compared with the perfect installation. At this time, the angular displacements of a pair of meshing tooth pairs in the single-tooth meshing area and the double-tooth meshing area can be respectively expressed as:

[0045]

[0046] As a preferred improvement of the present invention, step four specifically includes:

[0047] In the calculation of the meshing stiffness by the potential energy method, the tooth is equivalent to a cantilever beam fixed at the root circle. Under the action of the force F, the tooth will produce contact deformation, bending deformation, shear deformation and compression deformation. In addition, the hub matrix will also produce deformation; when a pair of teeth enters meshing, its contact deformation potential energy U h 、bending deformation U b 、shear deformation U s and compression deformation U a and the matrix deformation potential energy U f are:

[0048]

[0049] where F is the total contact force in the meshing teeth. The total deformation potential energy of a single pair of meshing teeth is composed of the sum of the contact deformation potential energy, bending deformation, shear deformation, compression deformation and matrix deformation potential energy, and can be:

[0050]

[0051] where k represents the total meshing stiffness of a pair of meshing tooth pairs, and the subscripts 1 and 2 respectively represent the driving wheel and the driven wheel. The calculation formula for the meshing stiffness of a single pair of tooth pairs is:

[0052]

[0053] The contact ratio of standard spur gears is between 1 and 2, and there is single-tooth meshing and double-tooth meshing. Therefore, the total time-varying meshing stiffness is:

[0054]

[0055] where k m is the total time-varying mesh stiffness during the meshing process of a pair of gear teeth, j represents the number of meshing gear teeth within the same time, and the calculation formulas for the Hertz contact stiffness and the gear tooth deformation stiffness are respectively:

[0056]

[0057]

[0058] where E, B, and v respectively represent Young's modulus, tooth width, and Poisson's ratio; α p is the pressure angle at the current meshing position P, u f is the distance from the intersection point of the line of action and the tooth symmetry line to the root circle, S f is the arc length corresponding to the entire tooth profile curve of the gear, and L*, M*, P*, Q* are 4 parameters related to the gear module and number of teeth;

[0059] Based on beam theory, the axial compression energy, bending energy, and shear energy of a gear tooth composed of an involute and a root fillet curve segment can be calculated by the following formulas:

[0060]

[0061]

[0062]

[0063] where F b = F cosα p , F a = F sinα p , x p represents the distance from the current contact point P to the tooth center line, y p represents the horizontal distance from the current contact point P to the original point, represents the shear modulus, y 1 , y 2 represent the horizontal coordinates of any point on the fillet curve and the involute, y c , y b respectively represent the abscissas of the starting point and the ending point of the root fillet curve, I y1 , I y2 , A y1 , A y2 respectively represent the area moment of inertia and the cross-sectional area of the pinion and the gear; the torque M 1 = F b (y b - y 1 ) - Fa x b and M 2 = F b (y b - y 2 ) - F a x b indicates the bending effect of F a 、F b in the transition curve and involute portions;

[0064] According to the characteristics of the involute and transition curve, x b 、y b 、x 1 、x 2 、I y1 、I y2 、A y1 、A y2 are expressed as functions of the angles γ, τ, α p The axial compression, bending, and shear stiffness based on the equation are as follows:

[0065]

[0066]

[0067]

[0068] where and can be expressed as:

[0069]

[0070]

[0071] The beneficial effects of the present invention are as follows:

[0072] 1. The present invention establishes a meshing stiffness calculation model for a micro gear pair based on the involute gear meshing principle and the material mechanics potential energy method, accurately describes the contact state during gear transmission with errors, and analyzes the dynamic change relationship between the actual meshing point and meshing stiffness;

[0073] 2. The present invention takes into account the changes in the gear pressure angle and contact ratio. Compared with the previous meshing models that regarded the pressure angle and contact ratio as constants, the calculation of meshing stiffness is more accurate, which has more practical significance for analyzing the vibration characteristics and dynamic response of gears;

[0074] 3. The present invention calculates the meshing stiffness of the micro gear pair under perfect installation and the comprehensive influence of center distance manufacturing error and shaft hole clearance error. Through comparative analysis, the vibration and noise of the gear transmission system can be reduced, and the stability of the transmission system can be improved.

BRIEF DESCRIPTION OF THE DRAWINGS

[0075] To more clearly illustrate the technical solutions in the embodiments of the present invention, the following will briefly introduce the drawings required for the description of the embodiments. Obviously, the drawings in the following description are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can be obtained based on these drawings, where:

[0076] Figure 1 is the gear meshing motion model diagram of the present invention;

[0077] Figure 2 is the meshing position diagram of any point in the gear transmission of the present invention;

[0078] Figure 3 is the partial structure diagram of the micro gear transmission system of the present invention;

[0079] Figure 4 is the gear meshing process diagram under the manufacturing error of the center distance of the present invention;

[0080] Figure 5 is the diagram of the change in the meshing position of two-stage gears under the shaft hole clearance error of the present invention;

[0081] Figure 6 is the gear meshing process diagram under the shaft hole clearance error of the present invention;

[0082] Figure 7 is the geometric model diagram of the involute spur gear tooth profile of the present invention;

[0083] Figure 8 is the time-varying meshing stiffness diagram of the first-stage gear of the present invention under no error and comprehensive error;

[0084] Figure 9 is the time-varying meshing stiffness diagram of the second-stage gear of the present invention under no error and comprehensive error.

DETAILED DESCRIPTION OF THE EMBODIMENTS

[0085] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in combination with the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, rather than all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts belong to the scope of protection of the present invention.

[0086] In addition, the technical solutions between the various embodiments of the present invention can be combined with each other, but it must be based on the fact that those of ordinary skill in the art can implement it. When the combination of technical solutions results in contradictions or cannot be implemented, it should be considered that such a combination of technical solutions does not exist and is not within the scope of protection required by the present invention.

[0087] The present invention provides a method for calculating the meshing stiffness of a micro gear pair, and the calculation method includes the following steps:

[0088] Step 1: Based on the involute gear meshing principle, establish a dynamic meshing motion model of gear transmission under perfect installation, and define relevant meshing parameters and single- and double-tooth meshing boundary conditions during the gear meshing process;

[0089] Specifically, as shown in Figure 1 the theoretical meshing line N 1 N 2 is tangent to the base circles of the pinion 1 (the driving wheel) and the gear 2 (the driven wheel), and the tangent points are N 1 , N 2 . The pinion 1 (the driving wheel) rotates clockwise and continuously transmits power to the gear 2 (the driven wheel). The target tooth pair starts meshing from the intersection point B 2 of the addendum circle of the gear 2 (the driven wheel) and the meshing line. During the power transmission process, the position of the meshing point will always move along the theoretical meshing line N 1 N 2 direction. During the meshing motion process, the driving wheel and the driven wheel always remain in contact with each other along N 1 N 2 . The meshing point on the tooth profile of the driven wheel moves from the tooth tip to the tooth root, while the meshing point on the driving wheel moves from the tooth root to the tooth tip. When the previous tooth profile reaches the upper boundary point C 1 of the single meshing area, the next tooth profile simultaneously enters point B 2 . When the meshing point of the previous tooth profile reaches point B 1 , meshing separation occurs, and at this time, the next tooth profile moves to the lower boundary point C 2 of the single meshing area.

[0090] Among them, R b1 , R 1 and R b2 , R 2 are the base circles and pitch circles of the pinion and the gear respectively; P is the pitch point of the gear, and O 1 , O 2 represent the geometric centers of the gears respectively; ω 1 , ω 2 are the rotational frequencies of the pinion and the gear respectively; α s is the pitch circle pressure angle and also the actual pressure angle of the gear, and α a1 is the addendum circle pressure angle. When a pair of gears is in a state of perfect installation, the actual pressure angle α s is equal to the theoretical pressure angle α 0 . The actual center distance of the gear pair is:

[0091]

[0092] Among them, m is the module of the gear, and Z 1 , Z 2 are the number of teeth of the pinion and the gear, and Δδ 1 is the center distance deviation caused during the gear manufacturing process. Under perfect installation, Δδ 1 = 0. The actual pressure angle α s of the gear pair is:

[0093]

[0094] Under perfect installation, the target tooth pair starts meshing and contacting from point B 2 , and starts meshing and separating from point B 1 . At this time, the actual meshing line of the gear meshing motion is B 1 B 2 and is:

[0095] B 1 B 2 = PB 1 + PB 2

[0096] Where:

[0097] PB 1 = r b1 × (tanα a1 - tanα s )

[0098] PB 2 = r b1 × (tanα a2 - tanα s )

[0099] During the actual meshing process of the gears, the contact ratio of standard spur gears usually ranges between 1 and 2, and there is single-tooth meshing and double-tooth meshing. When the pinion rotates an angle, that is, when the meshing motion reaches position C 2 from position B 1 , the previous pair of teeth completes the meshing process from double-tooth meshing to single-tooth meshing. When the target tooth pair starts meshing at the initial contact point B 2 , setting the initial rotation angle of the gear, then the actual meshing line of the pinion during the meshing process can be expressed as:

[0100] B 1 B 2 = N 1 B 1 - N 1 B 2

[0101] The angular displacements of a pair of meshing teeth in the single-tooth meshing zone and the double-tooth meshing zone can be expressed as:

[0102] θ 1-double ∈[0, B 2 C 2 / R b1 ∪ [B 2 C 1 / R b1 , (N 1 B 1 -N 1 B 2 ) / R b1

[0103] θ 1-single ∈[B 2 C 2 / R b1 , B 2 C 1 / R b1

[0104] The relevant parameters are shown in Table 1.

[0105] Table 1

[0106]

[0107] Figure 3 Describes the contact position of any point on the involute curve during the meshing process of spur gears. According to the formation characteristics of the spur gear involute, the coordinate positions of the current meshing point P in the coordinate systems of the pinion and the gear are shown in Table 2.

[0108] Table 2

[0109]

[0110] Among them, the half-tooth angle θ b1 = π / 2Z 1 + invα 0 , θ b2 = π / 2Z 2 + invα 0 , x p1 , x p2 , y p1 , y p2 respectively represent the coordinates of the current contact point P relative to the pinion and the gear during the gear meshing process. β 1 , β 2 respectively represent the pressure angles of the circles at the current contact point P of the pinion and the gear during the gear meshing process.

[0111] ​​Step 2: Consider the influence of center distance manufacturing error and shaft-hole clearance on the position of the meshing point, analyze the changes in the contact ratio and pressure angle during the actual meshing process of the gears, and redefine the meshing parameters and single- and double-tooth meshing boundary conditions during the gear meshing process;

[0112] Specifically, as gear transmission gradually develops towards high speed and heavy load, micro-gear transmission systems are often designed with two or more stages of transmission. In a micro-miniature gear transmission system, the intermediate shaft is fixed without bearing support, and there is a certain clearance between the gear hole and the intermediate shaft, as Figure 3 shown. During the meshing process of a two-stage gear transmission system, the driving wheel of the first-stage gear and the driven wheel of the second-stage gear are fixed, and the driven wheel of the first-stage gear and the driving wheel of the second-stage gear are located on the same transmission shaft. The meshing transmission process of two-stage spur gears with center distance manufacturing error is as Figure 4 shown. An increase in the gear center distance will reduce the actual meshing line during the meshing process, shorten the double-tooth meshing time, and extend the single-tooth meshing time, resulting in the meshing point separating earlier than the theoretical moment and the meshing of the next pair of teeth being delayed, and vice versa.

[0113] The existence of the rotational pair clearance causes the rotational center of the gear to be inconsistent with the geometric center, changing the contact mode of the tooth surfaces of the meshing gears and causing the position of the contact trace to deviate from the ideal state. As Figure 5 shown, during the meshing movement of the gears, there is a center distance deviation Δδ caused by the fit clearance between the holes of gear 2 and gear 3 and the fixed shaft 2 . Assuming that gear 1 and gear 4 are in a perfect installation state, O 1 , O 2 (O 3 ) and O 4 are the rotational centers of each stage of the gears respectively, O 21 is the actual geometric center of gear 2 and gear 3, O 21 ′ is the projection of O 21 on the Y-axis, Δδ x = Δδ 2 cosθ, Δδ y = Δδ 2 sinθ. Where θ corresponds to the angular displacement of the gear. During the power transmission process, as the angular displacement continuously changes, gears 2 and 3 located on the same transmission shaft will shift along the resultant force direction of the meshing forces of the first-stage gear and the second-stage gear. Due to the continuous change of the gear center distance during the meshing cycle, the actual center distances of the first-stage gear and the second-stage gear can be expressed by the following calculation formulas respectively:

[0114]

[0115]

[0116] The meshing transmission process of a two-stage spur gear with shaft-hole clearance error is as follows Figure 6 as shown, where α s1 、α s1 ′、α s2 、α s2 ′ are the theoretical pressure angle and the actual pressure angle at the starting contact position of the first-stage gear and the second-stage gear respectively; N 1 、N 1 ′、N 2 、N 2 ′、N 3 、N 3 ′、N 4 、N 4 ′ are the initial position and the ending position of the theoretical meshing line of the first-stage gear and the second-stage gear respectively, corresponding to the geometric center positions O 2 ′(O 3 ′) and O 2 ″(O 3 ″).

[0117] It can be seen through Figure 6 that taking the first-stage gear as an example, assuming that when the gear pair is at the theoretical pressure angle of α s1 , the first pair of teeth just enters the actual meshing line B 1 for meshing. If the shaft-hole clearance error is not considered, the theoretical pressure angle α s1 is equal to the actual pressure angle α s1 ′. That is, when the gear pair just turns through an angle Φ = 2π / Z 1 corresponding to the same actual pressure angle α s1 at the initial position B 1 of the actual meshing line, the next pair of teeth will start meshing. However, due to the existence of the clearance of the shaft-hole rotating pair, the theoretical pressure angle corresponding to the meshing of the second pair of teeth is not α s1 but the actual pressure angle α s1 ′. The actual phase difference generated by two continuously meshing tooth pairs considering the shaft-hole clearance error is:

[0118] θ d = 2π / N 1 +(N 1 B 2 / r b1 -N 1 ′B 2 ′ / r b1 )

[0119] where N 1 、N 1′ corresponds to two different initial state moments under normal installation. When there is clearance in the shaft-hole revolute pair of the gear transmission system, the center distance of the gear changes with the change of angular displacement at all times, and the actual pressure angle α of the gear s1 ′ is a function of the actual center distance O 1 O 2′ When the center distance changes, the actual meshing line in the gear meshing process can be expressed as:

[0120] B 1 ′B 2 ′ = P′B 1 ′ + P′B 2 ′

[0121] The changes of relevant parameters are shown in Table 4.

[0122] Table 4

[0123]

[0124] It can be seen from Table 4 that the contact ratio of the gear changes with the change of the gear center distance at all times. The single-tooth meshing time and the double-tooth meshing time both change compared with the perfect installation. At this time, the angular displacements of a pair of meshing tooth pairs in the single-tooth meshing area and the double-tooth meshing area can be respectively expressed as:

[0125] θ 1-double ∈[0, B 2 ′C 2 ′ / R b1 ∪ [B 2 ′C 1 ′ / R b1 , (N 1 ′B 1 ′ - N 1 ′B 2 ′) / R b1

[0126] θ 1-single ∈[B 2 ′C 2 ′ / R b1 , B 2 ′C 1 ′ / R b1

[0127] Step 3: Establish the gear tooth profile geometric model and list the tooth root transition curve equation:

[0128] Specifically, the tooth profile curve of the spur gear consists of four parts: the tooth tip curve AB, the involute curve BC, the tooth root transition curve CD, and the tooth root curve DE. From Figure 7 ​​As shown, xoy is the coordinate system of the spur gear tooth, O is the geometric center of the gear, y is the axis representing the horizontal direction, x is the axis perpendicular to y, and the point y c and y p correspond to the abscissas of the meshing point C and the meshing point P of the current tooth pair respectively. The variables x and y correspond to the ordinate and abscissa of any point on the tooth profile line. R b 、R f 、r c correspond to the base circle radius, root circle radius, and involute starting circle radius respectively; τ corresponds to the pressure angle at the involute starting circle, τ i corresponds to the pressure angle at any meshing point position, τ p corresponds to the pressure angle at the meshing point. θ b1 is the half tooth angle on the base circle.

[0129] The root transition curve CD is formed by the arc segment trajectory of the tool tip during the manufacturing process and directly depends on the shape of the tool. When the tool tip shape is an ordinary fillet, the transition curve equation can be expressed as:

[0130] x 1 =R 1 ×sin(Φ)-(a 1 / sinγ+r ρ )×cos(γ-Φ)

[0131] y 1 =R 1 ×cos(Φ)-(a 1 / sinγ+r ρ )×sin(γ-Φ)

[0132] where α 0 ≤λ≤π / 2, x 1 , y 1 is a function of γ, corresponding to the position of any point on the root transition curve. The tool fillet radius r ρ =c 0 m / (1-sinα 0 ); The distance a from the center of the tool tip fillet to the center line is a 1 =(h a0 +c 0 )×m-r ρ ; The distance b from the center of the tool tip fillet to the center line of the tool tooth groove is b 1 =πm / 4+h a0 m×tanα 0 +r ρ cosα 0 ; The angular displacement Φ of the tool for machining the root transition curve segment is Φ=(a 1 / tanγ+b 1 ) / R1 ; For standard gears, h a0 = 1; c 0 = 0.25; α 0 = 20.

[0133] Step Four: Based on the potential energy method of material mechanics, the tooth is equivalent to a cantilever beam fixed at the root circle. According to the contact deformation, bending deformation, shear deformation, compression deformation generated by the gear, and the deformation of the hub matrix, calculate the Hertz contact stiffness, bending stiffness, shear stiffness, compression stiffness, and tooth deformation stiffness of the gear;

[0134] Specifically, in the calculation of the meshing stiffness by the potential energy method, the tooth is equivalent to a cantilever beam fixed at the root circle. Under the action of force F, the tooth will generate contact deformation, bending deformation, shear deformation, and compression deformation (along the y direction). In addition, the hub matrix will also generate deformation. When a pair of teeth enters meshing, its contact deformation potential energy Uh, bending deformation Ub, shear deformation Us, compression deformation Ua, and matrix deformation potential energy Uf are:

[0135]

[0136] Among them, F is the total contact force in the meshing teeth. The total deformation potential energy of a single pair of meshing teeth is composed of the sum of the contact deformation potential energy, bending deformation, shear deformation, compression deformation, and matrix deformation potential energy, and can be:

[0137]

[0138] Among them, k represents the total meshing stiffness of a pair of meshing teeth. Subscripts 1 and 2 represent the pinion and the gear respectively. The calculation formula for the meshing stiffness of a single pair of teeth is:

[0139]

[0140] The contact ratio of a standard spur gear is between 1 and 2, and there is single-tooth meshing and double-tooth meshing. Therefore, the total time-varying meshing stiffness is:

[0141]

[0142] Among them, k m is the total time-varying meshing stiffness during the meshing process of a pair of teeth. j represents the number of meshing teeth within the same time. The calculation formulas for the Hertz contact stiffness and the tooth deformation stiffness are respectively:

[0143]

[0144]

[0145] Among them, E, B, and v represent Young's modulus, tooth width, and Poisson's ratio respectively. α pis the pressure angle at the current meshing position P, u f is the distance from the intersection of the meshing line and the tooth symmetry line to the root circle, S f is the arc length corresponding to the entire tooth profile curve of the gear, and L*, M*, P*, Q* are 4 parameters related to the gear module and number of teeth;

[0146] Based on beam theory, the axial compression energy, bending energy, and shear energy of a gear tooth composed of an involute and a root transition curve segment can be calculated by the following formula:

[0147]

[0148]

[0149]

[0150] where F b = F cosα p , F a = F sinα p , x p represents the distance from the current contact point P to the tooth center line, y p represents the distance from the current contact point P to the original point in the horizontal direction, represents the shear modulus, y 1 , y 2 represent the horizontal coordinates of any point on the transition curve and the involute, y c , y b respectively represent the abscissas of the starting point and the ending point of the root transition curve, I y1 , I y2 , A y1 , A y2 respectively represent the area moment of inertia and cross-sectional area of the pinion and the gear; the torque M 1 = F b (y b - y 1 ) - F a x b and M 2 = F b (y b - y 2 ) - F a x b indicate the bending effects of F a , F b in the transition curve and involute parts.

[0151] For the convenience of calculation, angular displacement is adopted in the calculation. According to the characteristics of the involute and the transition curve, x b , y b , x 1 , x2 , I y1 , I y2 , A y1 , A y2 are expressed as functions of angles γ, τ, α p . Based on the equations, the axial compression, bending, and shear stiffness are as follows:

[0152]

[0153]

[0154]

[0155] The relevant parameters are shown in Table 5.

[0156] Table 5

[0157]

[0158] where and can be expressed as:

[0159]

[0160]

[0161] Step 5: Set the basic parameters of the gear, and use MATLAB software to calculate the time-varying mesh stiffness of the micro gear pair under the combined influence of perfect installation, center distance manufacturing error, and shaft hole clearance error.

[0162] Specifically, set the basic parameters of the micro gear pair as shown in Table 6. Use MATALB software to calculate the time-varying mesh stiffness of the two-stage gear during the transmission of power from the first-stage gear to the second-stage gear under perfect installation, with a center distance manufacturing error Δδ 1 of -0.2 mm and a shaft hole clearance Δδ 2 of 0.2 mm. The results are shown in Figure 8 , Figure 9 . The results show that compared with perfect installation, the mesh stiffness of the micro gear pair considering the center distance manufacturing error, shaft hole clearance error, and their combined influence changes significantly and cannot be ignored in the analysis of gear vibration, noise, and dynamic response.

[0163] Table 6

[0164]

[0165] The beneficial effects of the present invention are as follows:

[0166] 1. The present invention establishes a meshing stiffness calculation model for a micro gear pair based on the involute gear meshing principle and the material mechanics potential energy method, accurately describes the contact state during the gear transmission process with errors, and analyzes the dynamic change relationship between the actual meshing point and the meshing stiffness;

[0167] 2. The present invention takes into account the changes in the gear pressure angle and the contact ratio. Compared with the previous meshing models that regarded the pressure angle and the contact ratio as constants, the calculation of the meshing stiffness is more accurate, which has more practical significance for analyzing the vibration characteristics and dynamic response of gears;

[0168] 3. The present invention calculates the meshing stiffness of the micro gear pair under perfect installation and under the comprehensive influence of the center distance manufacturing error and the shaft hole clearance error. Through comparative analysis, the vibration and noise of the gear transmission system can be reduced, and the stability of the transmission system can be improved.

[0169] Although the embodiments of the present invention have been disclosed as above, they are not limited to the applications listed in the specification and the embodiments. It can be fully applied to various fields suitable for the present invention. For those familiar with the field, additional modifications can be easily made. Therefore, without departing from the general concept defined by the claims and the equivalent scope, the present invention is not limited to the specific details and the examples shown and described here.

Claims

1. A calculation method for the meshing stiffness of a micro gear pair, characterized in that, the calculation method comprises the following steps: Step 1: Based on the involute gear meshing principle, establish a dynamic meshing motion model of gear transmission under perfect installation, and define the relevant meshing parameters and single- and double-tooth meshing boundary conditions during the gear meshing process; Step 2: Considering the influence of center distance manufacturing error and shaft-hole clearance on the position of the meshing point, analyze the changes in the contact ratio and pressure angle during the actual meshing process of the gears, and redefine the meshing parameters and single- and double-tooth meshing boundary conditions during the gear meshing process, specifically including: During the gear meshing motion, there is a center distance deviation Δδ caused by the fit clearance between the holes of Gear 2 and Gear 3 and the fixed shaft. 2 , assuming that Gear 1 and Gear 4 are in a perfect installation state, O 1 , O 2 , O 3 and O 4 are the rotation centers of each stage of gears respectively, O 21 is the actual geometric center of Gear 2 and Gear 3, O 21 ′ is the projection of O 21 on the Y-axis, and θ corresponds to the angular displacement of the gear; due to the continuous change of the center distance of the gears during the meshing period, the actual center distances between the first-stage gear and the second-stage gear can be expressed by the following calculation formulas respectively: The actual phase difference generated by two continuously meshing tooth pairs considering the shaft-hole clearance error is: θ d = 2π / N 1 +(N 1 B 2 / r b1 -N 1 ′B 2 ′ / r b1 ) Where N 1 、N 1 ′ corresponds to two different initial states under normal installation. When there is a shaft-hole rotation pair clearance in the gear transmission system, the gear center distance changes all the time according to the change of angular displacement. The actual pressure angle α of the gear s1 ′ is about the actual center distance O 1 O 2′ As a function of , when the center distance changes, the actual meshing line during the gear meshing process can be expressed as: B 1 ′B 2 ′ = P′B 1 ′ + P′B 2 ′ The contact ratio of the gear will change with the momentary change of the gear center distance. The single-tooth meshing time and double-tooth meshing time will both change compared with the perfect installation. At this time, the angular displacements of a pair of meshing tooth pairs in the single-tooth meshing area and double-tooth meshing area can be respectively expressed as: θ 1-double ∈ [0, B 2 ′C 2 ′ / R b1 ∪ [B 2 ′C 1 ′ / R b1 , (N 1 ′B 1 ′ - N 1 ′B 2 ′) / R b1 ; θ 1-single ∈ [B 2 ′C 2 ′ / R b1 , B 2 ′C 1 ′ / R b1 ​ Step 3: Establish a gear tooth profile geometric model and list the tooth root transition curve equation: x 1 = R 1 × sin(Φ) - (a 1 / sinγ + r ρ ) × cos(γ - Φ) y 1 = R 1 × cos(Φ) - (a 1 / sinγ + r ρ ) × sin(γ - Φ) where γ is the angle between the common normal and the tool machining pitch line, and α 0 ≤γ≤π / 2; x 1 , y 1 is a function of γ, corresponding to the position of any point on the root fillet curve. The tool fillet radius r ρ = c 0 m / (1 - sinα 0 ); The distance a from the center of the tool tip fillet to the center line is a 1 = (h a0 + c 0 ) × m - r ρ ; The distance b from the center of the tool tip fillet to the center line of the tool tooth groove is b 1 = πm / 4 + h a0 m × tanα 0 + r ρ cosα 0 ; The angular displacement Φ of the tool for machining the root fillet curve section is Φ = (a 1 / tanγ + b 1 ) / R 1 ; For a standard gear, h a0 = 1; c 0 = 0.25; α 0 = 20; Step 4: Based on the material mechanics potential energy method, equivalent the tooth to a cantilever beam fixed at the root circle. According to the contact deformation, bending deformation, shear deformation, compression deformation generated by the gear and the deformation of the hub matrix, calculate the Hertz contact stiffness, bending stiffness, shear stiffness, compression stiffness and tooth deformation stiffness of the gear; Step 5: Set the basic parameters of the gear, and use MATLAB software to calculate the time-varying meshing stiffness of the micro gear pair under perfect installation and the comprehensive influence of center distance manufacturing error and shaft-hole clearance error.

2. A calculation method for the meshing stiffness of a micro gear pair according to claim 1, characterized in that: Step 1 specifically includes: Theoretical engagement line N 1 N 2 is tangent to the base circles of the driving gear and the driven gear, and the tangent points are N 1 and N 2 respectively; the driving gear rotates clockwise and continuously transmits power to the driven gear, and the target tooth pair starts to engage at the intersection point B 2 of the addendum circle of the driven gear and the engagement line. During the power transmission process, the position of the engagement point will always move along the theoretical engagement line N 1 N 2 in the direction; during the engagement movement, the driving gear and the driven gear always remain in contact with each other along N 1 N 2 ; the engagement point on the tooth profile of the driven gear moves from the tooth tip to the tooth root, while the engagement point on the driving gear moves from the tooth root to the tooth tip; when the previous pair of tooth profiles reaches the upper boundary point C 1 of the single engagement zone, the next pair of tooth profiles simultaneously enters point B 2 ; when the engagement point of the previous pair of tooth profiles reaches point B 1 , the engagement separation occurs, and at this time, the next pair of tooth profiles moves to the lower boundary point C 2 of the single engagement zone; where R b1 , R 1 and R b2 , R 2 are respectively the base circle and the pitch circle of the pinion and the gear; P is the pitch point of the gear, O 1 , O 2 respectively represent the geometric centers of the gears; ω 1 , ω 2 are respectively the rotational frequencies of the pinion and the gear; α s is the pitch circle pressure angle and also the actual pressure angle of the gear, α a1 is the addendum circle pressure angle. When a pair of gears is in a perfectly installed state, the actual pressure angle α s is equal to the theoretical pressure angle α 0 , and the actual center distance of the gear pair is: Among them, O 1 and O 2 respectively represent the geometric centers of the gears, m is the module of the gear, Z 1 and Z 2 are the number of teeth of the small gear and the large gear, Δδ 1 is the center distance deviation caused during the gear manufacturing process. Under perfect installation, Δδ 1 = 0; The actual pressure angle α of the gear pair s is as follows: where α 0 is the theoretical pressure angle; When the target gear pair is perfectly installed, it starts to mesh and contact from point B 2 and starts to disengage from point B 1 . At this time, the actual meshing line of the gear meshing motion is B 1 B 2 which is: B 1 B 2 = PB 1 + PB 2 where PB 1 = R b1 × (tanα a1 - tanα s ) PB 2 = R b2 ×(tanα a2 - tanα s ) Among them, P is the pitch point of the gear, and R b1 , R b2 are the base circles of the pinion and the gear respectively, and α s is the pressure angle of the pitch circle and also the actual pressure angle of the gear. α a1 , α a2 are the addendum circle pressure angles of the pinion and the gear respectively; During the actual meshing process of gears, the contact ratio of standard spur gears is between 1 and 2, with single-tooth meshing and double-tooth meshing existing. When the driving gear rotates by an angle, that is, when the meshing motion reaches from position B 2 to position C 1 , the previous pair of teeth completes the meshing process from double-tooth meshing to single-tooth meshing; when the target tooth pair starts meshing at the initial contact point B 2 , setting the initial rotation angle of the gear then the actual meshing line of the driving gear during the meshing process can be expressed as: B 1 B 2 = N 1 B 1 -N 1 B 2 the angular displacements of a pair of meshing tooth pairs in the single-tooth meshing area and double-tooth meshing area can be expressed as: θ 1-double ∈ [0, B 2 C 2 / R b1 ∪ [B 2 C 1 / R b1 , (N 1 B 1 -N 1 B 2 ) / R b1 ​ θ 1-single ∈ [B 2 C 2 / R b1 , B 2 C 1 / R b1 ​ wherein, R b1 and R b2 are the base circles of the driving wheel and the driven wheel, respectively.

3. A calculation method for the meshing stiffness of a micro gear pair according to claim 1, characterized in that: Step 4 specifically includes: In the calculation of meshing stiffness by the potential energy method, the tooth is equivalent to a cantilever beam fixed at the root circle. Under the action of force F, the tooth will produce contact deformation, bending deformation, shear deformation and compression deformation. In addition, the hub matrix will also produce deformation; when a pair of teeth enter meshing, their contact deformation potential energy U h , bending deformation U b , shear deformation U s and compression deformation U a and the matrix deformation potential energy U f are as follows: where F is the total contact force in the meshing teeth. The total deformation potential energy of a single pair of meshing teeth is composed of the sum of the contact deformation potential energy, bending deformation, shear deformation, compression deformation and matrix deformation potential energy, and can be: where k represents the total meshing stiffness of a pair of meshing tooth pairs. The subscripts 1 and 2 respectively represent the driving gear and the driven gear. The calculation formula for the meshing stiffness of a single pair of tooth pairs is: The contact ratio of a standard spur gear is between 1 and 2, and there is single-tooth meshing and double-tooth meshing. Therefore, the total time-varying meshing stiffness is: where k m is the total time-varying mesh stiffness during the meshing process of a pair of gear teeth, j represents the number of meshing gear teeth in the same time, and the calculation formulas for the Hertz contact stiffness and the gear tooth deformation stiffness are respectively: where E, B, and v represent Young's modulus, tooth width, and Poisson's ratio, respectively; α p is the pressure angle at the current meshing position P, u f is the distance from the intersection of the meshing line and the tooth symmetry line to the root circle, S f is the arc length corresponding to the entire tooth profile curve of the gear, and L*, M*, P*, and Q* are four parameters related to the gear module and number of teeth; Based on beam theory, the axial compression energy, bending energy and shear energy of the gear tooth composed of the involute and tooth root transition curve segments can be calculated by the following formula: where F b = F cosα p and F a = F sinα p where x p represents the distance between the current contact point P and the center line of the tooth, and y p represents the distance in the horizontal direction from the current contact point P to the original point. represents the shear modulus, and y 1 , y 2 represent the horizontal coordinates of any point on the transition curve and the involute, and y c , y b represent the abscissas of the starting point and the ending point of the root transition curve respectively, I y1 , I y2 , A y1 , A y2 represent the area moments of inertia and the cross-sectional areas of the pinion and the gear respectively; the torque M 1 = F b (y b - y 1 ) - F a x b and M 2 = F b (y b - y 2 ) - F a x b indicate the bending effects of F a , F b in the transition curve and involute parts; According to the characteristics of the involute and the transition curve, x b , y b , x 1 , x 2 , I y1 , I y2 , A y1 , A y2 are expressed as functions of the angles γ, τ, α p . The axial compression, bending, and shear stiffness based on the equations are as follows: Among them and can be expressed as:

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