Spur arc gear engagement pair, and design method

Abstract

The present invention relates to the technical field of gear designs. Particularly disclosed are a spur arc gear engagement pair, and a design method. The spur arc gear engagement pair comprises two spur gears engaging with each other, wherein the two spur gears are spur arc gears, and the contact mode of the two spur arc gears is point contact. The design method comprises the following steps: first, determining a virtual auxiliary involute gear; then, selecting a smooth curve on the virtual auxiliary involute gear to serve as an engagement trajectory line Γ1; next, by means of coordinate transformation, solving a pairing engagement curve Γ2 engaging with the engagement trajectory line Γ1; next, by means of sweeping a normal section tooth profile Γ2(t2) along the pairing engagement curve Γ2, constructing a tooth surface Σ2 of a spur arc gear II; and by means of sweeping a normal section tooth profile Γ1(t1) along the engagement trajectory line Γ1, constructing a tooth surface Σ1 of a spur arc gear I. The present solution solves the problem in the prior art of arc gears failing to achieve spur engagement; and the designed gear pair has a low sliding rate and has adaptability to axis errors and center distance changes.

Description

A spur gear meshing pair and design method Technical Field

[0001] The present invention relates to the technical field of gear design, and in particular to a spur circular arc gear meshing pair and a design method thereof. Background Art

[0002] Gears, as basic mechanical components that can continuously mesh and transmit motion and power, largely determine the functions of mechanical manufacturing equipment. Therefore, designs that enhance gear performance are also very beneficial and have high engineering practical value.

[0003] In existing applications, helical arc gears are widely used due to their advantages such as reliable transmission, ability to withstand high loads and linear speeds, and high transmission efficiency. However, such helical arc gears have problems such as high manufacturing cost, high requirements for center distance, and poor adaptability to axis errors.

[0004] Existing spur gears have the advantages of simpler manufacturing, lower cost, and lower requirements on center distance compared to helical gears. However, existing circular arc gears cannot be designed and applied to spur gears due to their special structure.

[0005] Summary of the Invention

[0006] The present invention aims to provide a spur gear meshing pair and a design method thereof, so as to solve the problem in the prior art that circular arc gears cannot achieve spur gear meshing.

[0007] In order to achieve the above object, the present invention adopts the following technical solutions:

[0008] A spur gear meshing pair comprises two mutually meshing spur gears, both of which are spur gears, and the contact mode of the two spur gears is point contact.

[0009] A design method for a spur gear meshing pair includes the following steps:

[0010] Step 1: Determine a virtual auxiliary involute gear;

[0011] Step 2: Select the smooth curve on the virtual auxiliary involute gear as the meshing trajectory line Γ1;

[0012] Step 3: Based on the relative positions of the axes of the gear pairs, construct the spatial coordinate system of the point-meshing spur gear pair, solve the motion velocity and normal between the meshing trajectory lines, and solve the paired meshing curve Γ2 that meshes with the meshing trajectory line Γ1 through coordinate transformation;

[0013] Step 4: By dividing the normal section tooth profile Γ 2 (t2) Sweep along the mating meshing curve Γ2 to construct the tooth surface Σ2 of the spur circular arc gear 2;

[0014] Step 5: By dividing the normal section tooth profile Γ 1 (t1) Sweep along the meshing trajectory line Γ1 to construct the tooth surface Σ1 of the spur gear 1.

[0015] Preferably, as an improvement, the tooth surface equation of the virtual auxiliary involute gear is as follows:

[0016] Among them, r is the base circle radius, θ is the involute rolling angle, and b is a variable with a value of [0, W B ],W B is the tooth width length;

[0017] The equation of the meshing trajectory Γ1 is as follows:

[0018] Among them, θ A ,θ B They are respectively the meshing starting point A(z Σ1 =0) and meshing end point B(z Σ1 =W B ) is the involute rolling angle at , r is the pitch circle radius.

[0019] Preferably, as an improvement, the step three is specifically as follows:

[0020] Establish the coordinate system S1 and coordinate system S2 of the point meshing spur gear pair, and seek the transformation relationship between different coordinate systems between the gear pairs. The transformation relationship from coordinate system S1 to coordinate system S2 is:

[0021] here:

[0022] The virtual auxiliary involute gear rotates around the z0 axis by an angle of φ1, and the spur gear rotates around the z p The shaft rotates through an angle of φ2.

[0023] Calculate the relative motion velocity and determine the expression in the fixed coordinate system S0, that is: v (12) =-(ω1+ω2)yi+[(ω1+ω2)x-aω2]j

[0024] Where i, j, k represent the unit vectors of coordinate axes x, y, and z respectively, ω1 is the modulus of the angular velocity of the auxiliary involute gear, and ω2 is the modulus of the angular velocity of the second spur gear.

[0025] Determine the normal direction of the meshing trajectory line Γ1 as:

[0026] According to the meshing equation n·v (12)=0, the meshing equation can be expressed as: 21 an y1 cosφ1-i 21 an x1 sinφ1=(1+i 21 )(y1n x1 -x1n y1 )

[0027] Determine the mating meshing curve Γ2. According to the selected meshing trajectory line Γ1, the coordinate transformation and meshing equation are used to obtain the equation of the mating meshing curve Γ2:

[0028] where i 21 is the transmission ratio of the gear pair.

[0029] Preferably, as an improvement, the step 4 is specifically as follows:

[0030] The tooth surface Σ2 equation is: Σ2=M c Γ 2 (t2)

[0031] here:

[0032] in, is the principal normal vector of surface Σ2 at this point, is the tangent vector of the mating meshing curve Γ2 on the surface Σ2 at the contact point, Can be Get; here Γ 2 (t2) is the normal section tooth profile equation of the spur gear 2, which is as follows:

[0033] Where t2 is a variable and k2 is the tooth profile radius.

[0034] Preferably, as an improvement, in step 5, the equation of the tooth surface Σ1 is: Σ1=M d Γ 1 (t1)

[0035] here:

[0036] in, is the principal normal vector of surface Σ1 at the contact point, is the tangent vector of the mating meshing curve Γ4 on the surface Σ1 at the contact point, Can be get;

[0037] Here Γ 1(t1) is the normal section tooth profile equation of the spur circular arc gear 1, which is as follows:

[0038] Where t1 is a variable and k1 is the tooth profile radius.

[0039] Preferably, as an improvement, the meshing trajectory line Γ1 needs to cross the pitch line when it is selected.

[0040] The principles and advantages of this solution are:

[0041] 1. The present invention proposes a spur gear meshing pair and a design method thereof. First, the equation of a virtual auxiliary involute gear surface is determined; a smooth curve is selected as the meshing trajectory line Γ1, and its equation is determined; secondly, a spatial coordinate system of the point-meshing spur gear pair is established, the transformation relationship between different coordinate systems between the gear pairs is sought, and the transformation relationship from coordinate system S1 to coordinate system S2 is determined; the relative speed is calculated, and the normal vector of the meshing trajectory line Γ1 is determined; based on the equation of the meshing trajectory line Γ1 in the fixed coordinate system S0 and the meshing equation, the equation of the mating meshing curve Γ2 is determined; then, the normal section tooth profile Γ is obtained. 2 (t2) Sweep along the mating meshing curve Γ2 to construct the tooth surface of the spur gear 2, and use the normal section tooth profile Γ 1 (t1) Sweep along the meshing trajectory line Γ1 to construct the tooth surface of the spur circular arc gear 1.

[0042] 2. In the prior art, circular arc gears cannot be designed as spur gears, primarily due to the lack of end face contact. The key to the circular arc gears of the present invention being able to be designed as spur gears lies in the selected meshing trajectory Γ1. On the one hand, because the projection of this meshing trajectory Γ1 onto the gear end face is an involute, it possesses the characteristic of an involute's strong adaptability to center distances. On the other hand, the meshing trajectory is a curve that crosses the pitch line, resulting in a low degree of contact. Therefore, the spur circular arc gear pair designed in the present invention meshes from the tooth root to the tooth tip along the meshing trajectory, achieving both axial contact and end face contact, thus enabling the application of circular arc gears in spur gear meshing.

[0043] 3. The gear pair constructed by the present invention is a point-contact spur arc gear. Each tooth of the spur arc gear pair is thick at both ends and narrow in the middle in the tooth width direction, which ensures that the two gears are always in point-contact meshing during the meshing process, so that the designed spur arc gear has good adaptability to changes in center distance. When an axis error occurs, that is, when parallel axes become staggered axes or intersecting axes, the line contact characteristics of the existing technology will cause the involute gear to become point contact, resulting in abnormal meshing of edge contact, which will lead to gear failure. The design of the contact track of the present invention is selected on the auxiliary involute gear. The projection of the contact track on the end face is a section of involute, so the contact track has the adaptability of the involute to changes in center distance, and the point contact characteristics make the gear still in point contact when an axis error occurs. The contact point position only has a slight movement at the theoretical contact point, so abnormal meshing will not occur.

[0044] 4. In addition, the point contact spur arc gear pair designed in the present invention also has the characteristic of low slip rate. This is because the slip rate of point contact is inherently lower than that of line contact gears. In addition, the present invention uses virtual auxiliary involute gears to design spur arc gears, and when selecting the meshing trajectory line Γ1, a curve that crosses the pitch line is specifically used, so that the meshing trajectory line Γ1 is closer to the pitch circle, further reducing the slip rate.

[0045] In summary, the present invention provides a spur-tooth circular arc gear pair, which breaks the conventional practice of designing circular arc gears only on helical gears. At the same time, the spur-tooth circular arc gear pair designed by the present invention not only has the advantages of circular arc helical gears, but also can solve the problem that existing circular arc gears cannot achieve spur tooth meshing. In addition, the present invention does not have high requirements for center distance, and also has the advantages of low slip rate, adaptability to axis error and center distance changes, etc. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] FIG1 is a schematic diagram of a spatial coordinate system provided by the present invention.

[0047] FIG2 is a schematic diagram of selecting the meshing trajectory line Γ1 provided by the present invention.

[0048] FIG3 is a schematic diagram of a spatial coordinate system of relative motion speed provided by the present invention.

[0049] FIG4 is a schematic diagram showing the relationship between the meshing trajectory line Γ1 and the mating meshing curve Γ2 and the virtual involute gear tooth surface provided by the present invention.

[0050] FIG5 is a schematic structural diagram of the second tooth surface Σ2 of the spur circular arc gear swept out on the mating meshing curve Γ2 of the present invention.

[0051] FIG6 is a schematic diagram of a tooth profile provided by the present invention.

[0052] FIG7 is a schematic diagram of a spur circular arc gear provided by the present invention.

[0053] FIG8 is a schematic diagram of the meshing of the spur circular arc gear pair provided by the present invention.

[0054] FIG9 is a schematic diagram of a partial three-dimensional structure of a spur circular arc gear provided by the present invention having only one tooth.

[0055] FIG10 is a top view of FIG9 .

[0056] FIG11 is a front cross-sectional view of FIG9 (the cross-sectional view is located at the middle of the teeth). DETAILED DESCRIPTION

[0057] The following is further described in detail through specific implementation methods:

[0058] The embodiment is basically as shown in Figures 1 to 11.

[0059] The present invention provides a spur circular arc gear meshing pair, comprising a first spur circular arc gear and a second spur circular arc gear meshing with each other, wherein the contact mode of the two spur circular arc gears is point contact.

[0060] The present invention provides a method for designing a spur gear meshing pair, comprising the following steps:

[0061] Step 1: Determine a virtual auxiliary involute gear;

[0062] Step 2: Select the smooth curve on the virtual auxiliary involute gear as the meshing trajectory line Γ1;

[0063] Step 3: Based on the relative positions of the axes of the gear pairs, construct the spatial coordinate system of the point-meshing spur gear pair, solve the motion velocity and normal between the meshing trajectory lines, and solve the paired meshing curve Γ2 that meshes with the meshing trajectory line Γ1 through coordinate transformation;

[0064] Step 4: By dividing the normal section tooth profile Γ 2 (t2) Sweep along the mating meshing curve Γ2 to construct the tooth surface Σ2 of the spur circular arc gear 2;

[0065] Step 5: By dividing the normal section tooth profile Γ 1 (t1) Sweep along the meshing trajectory line Γ1 to construct the tooth surface Σ1 of the spur gear 1.

[0066] The more specific design steps for steps 1 to 5 above are as follows:

[0067] i) Determine a virtual auxiliary involute gear. The tooth surface equation of the virtual auxiliary involute gear is:

[0068] Among them, r is the base circle radius, θ is the involute rolling angle, and b is a variable with a value of [0, W B ],W B is the tooth width length.

[0069] ii) The smooth curve on the virtual auxiliary involute gear is selected as the meshing trajectory line Γ1. The equation of the meshing trajectory line Γ1 is as follows:

[0070] Among them, θ A ,θ B They are respectively the meshing starting point A(z Σ1 =0) and meshing end point B(z Σ1 =W B ) is the involute rolling angle at , r is the pitch circle radius.

[0071] In MATLAB, the meshing trajectory line Γ1 is drawn, and a curve crossing the pitch circle is selected as the meshing trajectory line Γ1, so that one end of the meshing trajectory line Γ1 is close to the tooth root and the other end is close to the tooth top, as shown in Figure 2.

[0072] iii) Establish the spatial coordinate system of the point-meshing spur gear pair, as shown in Figure 1, and find the transformation relationship between different coordinate systems of the gear pair. The transformation relationship from coordinate system S1 to coordinate system S2 is:

[0073] The transformation matrix from coordinate system S1 to coordinate system S2 is:

[0074] The virtual auxiliary involute gear rotates around the z0 axis by an angle of φ1, and the spur gear rotates around the z p The shaft rotates through an angle of φ2.

[0075] Step iii) should include: at the starting position, the moving coordinate systems S1 (O1-x1, y1, z1) and S2 (O2-x2, y2, z2) are respectively aligned with S0, S p The auxiliary involute gear rotates around the z0 axis at a uniform angular velocity ω1, and the spur gear rotates around the z0 axis at a uniform angular velocity ω2. p The positive direction of ω1 is the same as the positive direction of z0 axis, and the positive direction of ω2 is the same as the positive direction of z p After a period of time from the starting position, the coordinate systems S1 and S2 move to the position shown in the figure. The auxiliary involute gear rotates around the z0 axis by an angle of φ1, and the spur gear rotates around the z0 axis by an angle of φ1. p The axis rotates through an angle of φ2. Based on the relationship in the figure, the transformation relationship between each coordinate system can be listed.

[0076] iv) Calculate the relative motion velocity according to Figure 3 and determine the expression in the fixed coordinate system S0, namely: (12)=-(ω1+ω2)yi+[(ω1+ω2)x-aω2]j

[0077] Where i, j, k represent the unit vectors of coordinate axes x, y, and z respectively, ω1 is the modulus of the angular velocity of the virtual involute gear, and ω2 is the modulus of the angular velocity of the spur gear.

[0078] v) Determine the normal direction of the meshing trajectory line Γ1, which is:

[0079] vi) According to the meshing equation n·v (12) =0, the meshing equation can be expressed as: 21 an y1 cosφ1-i 21 an x1 sinφ1=(1+i 21 )(y1n x1 -x1n y1 )

[0080] vi) The specific calculation process includes: point P is an arbitrary contact point on the paired meshing curve gear Γ2, i, j, k represent the unit vectors of the coordinate axes x, y, z respectively, the relative motion speed of the paired meshing curve Γ2 at the contact point is unique in the direction of space, but the expression form in different coordinate systems is different. As shown in Figure 3, the angular velocity of the auxiliary involute gear and the spur arc gear 2 in the coordinate system are respectively expressed by vectors: ω (1) =ω1k1

[0081] ω1 is the modulus of the angular velocity of the auxiliary involute gear, and ω2 is the modulus of the angular velocity of the spur gear.

[0082] As shown in Figure 3, r1 represents the radial vector of point P in the moving coordinate system S1,

[0083] Where i1, j1, k1 represent the unit vectors of the coordinate axes x1, y1, and z1 respectively. r2=O2P=O2O1+O1P=(x1-acosφ1)i1+(asinφ1+y1)j1+z1k1

[0084] In the moving coordinate system S1, the relative motion velocity vector Equal to P on the auxiliary involute gear (1) Velocity vector of a point With the straight arc gear two on P (2) Velocity vector of a point The difference is

[0085] Velocity Vector for

[0086] Velocity Vector for

[0087] Combining the relevant expressions we can get:

[0088] Then, from the formula n·v (12) =0, the meshing equation can be expressed as: 21 an y1 cosφ1-i 21 an x1 sinφ1=(1+i 21 )(y1n x1 -x1n y1 )

[0089] vii) Determine the mating meshing curve Γ2. Based on the selected meshing trajectory Γ1, the coordinate transformation and meshing equation are used to obtain the equation of the mating meshing curve Γ2:

[0090] where i 21 is the transmission ratio of the gear pair, and φ2=i 21 φ1.

[0091] The mating meshing curve is shown in Figure 4.

[0092] viii) by dividing the normal section tooth profile Γ 2 (t2) Sweep along the mating meshing curve Γ2 to construct the tooth surface Σ2 of the spur gear 2. The equation of the tooth surface Σ2 is as follows: Σ2=M c Γ 2 (t2)

[0093] here:

[0094] in, is the principal normal vector of surface Σ2 at this point, is the tangent vector of the mating meshing curve Γ2 on the surface Σ2 at the contact point, Can be get;

[0095] Here Γ 2 (t2) is the normal section tooth profile equation of the spur circular arc gear 2, which is as follows:

[0096] Where t2 is a variable and k2 is the tooth profile radius.

[0097] ix) By dividing the normal section tooth profile Γ1 (t1) Sweep along the meshing trajectory Γ1 to construct the tooth surface Σ1 of the spur gear. The equation of the tooth surface Σ1 is: Σ1=M d Γ 1 (t1)

[0098] here:

[0099] in, is the principal normal vector of surface Σ1 at the contact point, is the tangent vector of the mating meshing curve Γ4 on the surface Σ1 at the contact point, Can be get;

[0100] Here Γ 1 (t1) is the normal section tooth profile equation of the spur circular arc gear 1, which is as follows:

[0101] Where t1 is a variable and k1 is the tooth profile radius.

[0102] The gear tooth profile obtained according to the above design is shown in FIG6 , which has the characteristics of being narrow in the middle and wide at both ends along the width direction of the gear.

[0103] Furthermore, the tooth profile of the spur circular arc gear 2 is stretched in the UG software to obtain the spur circular arc gear 2, as shown in Figure 7;

[0104] After constructing spur arc gear 2 using the auxiliary involute gear, the tooth surface ∑1 of spur arc gear 1 was constructed in MATLAB using the same method used to construct spur arc gear 2, forming the tooth profile of spur arc gear 1. Finally, through simulation using UG software, spur arc gears 1 and 2 were obtained, as shown in Figure 8.

[0105] The tooth-shaped structure in this embodiment is shown in FIG9 to FIG11.

[0106] The specific parameters of the spur gear in this embodiment are shown in Table 1 below:

[0107] The present invention proposes a spur gear meshing pair and a design method thereof. The two gears are spur gears. The meshing pair is formed by a spur gear 1 and a spur gear 2. The contact mode between the two gears is point contact, which has the advantages of low slip rate and high adaptability to axis errors. The spur gear meshing pair is obtained by selecting a smooth meshing trajectory line Γ1 on the auxiliary involute gear tooth surface, and then calculating the mating meshing curve Γ2 belonging to the spur gear tooth surface. Then, the meshing curve Γ2 is obtained by calculating the normal section tooth profile Γ1. 2(t2) Sweep along the mating meshing curve Γ2 to construct the tooth surface Σ2 of the spur gear 2, and pass the normal section tooth profile Γ 1 (t1) Sweep along the meshing trajectory line Γ1 to construct the tooth surface Σ1 of the spur arc gear 1, completing the design of the spur arc gear pair. The two constructed gears are transmitted by point meshing. When the gear pair meshes, it moves from the tooth root along the contact trajectory to the tooth top, and the projection of the moving position on the end face is an involute. Therefore, there is both axial overlap and end face overlap, meeting the requirements of spur gear meshing and solving the problem of spur gear meshing that cannot be achieved by existing helical arc gear meshing pairs. The spur arc gear pair designed in this embodiment not only has the advantages of circular arc helical gears, but also has the advantages of low center distance requirements, low slip rate, and adaptability to axis errors and center distance changes.

[0108] The above is only an embodiment of the present invention, and the common knowledge such as the specific technical solutions and / or characteristics in the solution are not described in detail here. It should be pointed out that for those skilled in the art, without departing from the technical solution of the present invention, several variations and improvements can be made, which should also be regarded as the scope of protection of the present invention, and these will not affect the effect of the implementation of the present invention and the practicality of the patent. The scope of protection required by this application shall be based on the content of its claims, and the specific implementation methods and other records in the description can be used to interpret the content of the claims.

Claims

1. A spur gear meshing pair, comprising two mutually meshing spur gears, characterized in that: Both spur gears are spur circular arc gears, and the contact mode of the two spur circular arc gears is point contact.

2. A design method for a spur gear meshing pair, characterized in that: The steps include: Step 1: Determine a virtual auxiliary involute gear; Step 2: Select the smooth curve on the virtual auxiliary involute gear as the meshing trajectory line Γ1; Step 3: According to the relative position of the axis of the gear pair, the spatial coordinate system of the point-meshing spur gear pair is constructed, the motion speed and normal between the meshing trajectory lines are solved, and the matching meshing curve Γ2 meshing with the meshing trajectory line Γ1 is solved by coordinate transformation; Step 4: By converting the normal section tooth profile Γ 2 (t2) Sweep along the mating meshing curve Γ2 to construct the tooth surface Σ2 of the spur gear 2; Step 5: By converting the normal section tooth profile Γ 1 (t1) The tooth surface Σ1 of the spur gear 1 is constructed by scanning along the meshing trajectory line Γ1.

3. The design method of a spur gear meshing pair according to claim 2, characterized in that: The tooth surface equation of the virtual auxiliary involute gear is as follows: Where r is the base circle radius, θ is the involute rolling angle, and b is a variable with a value of [0, W B ],W B is the tooth width length; The equation of the meshing trajectory Γ1 is as follows: Among them, θ A ,θ B are respectively the meshing starting point A(z Σ1 =0) and meshing end point B(z Σ1 =W B ) is the involute rolling angle at the point, and r is the pitch circle radius.

4. The design method of a spur gear meshing pair according to claim 3, characterized in that: The step three is as follows: establish the coordinate system S1 and the coordinate system S2 of the point meshing spur gear pair, and seek the transformation relationship between different coordinate systems between the gear pairs. The transformation relationship from the coordinate system S1 to the coordinate system S2 is: here The virtual auxiliary involute gear rotates around the z0 axis by an angle of φ1, and the spur gear rotates around the z p The shaft rotates through an angle of φ2. Calculate the relative motion speed and determine the expression in the fixed coordinate system S0, namely: v (12) =-(ω1+ω2)yi+[(ω1+ω2)x-aω2]j Where i, j, k represent the unit vectors of coordinate axes x, y, z respectively, ω1 is the modulus of the angular velocity of the auxiliary involute gear, ω2 is the modulus of the angular velocity of the second spur gear; Determine the normal direction of the meshing trajectory line Γ1 as: From the meshing equation n·v (12) =0, the meshing equation can be expressed as: -i 21 an y1 cosφ1-i 21 an x1 sinφ1=(1+i 21 )(y1n x1 -x1n y1 ) Determine the mating meshing curve Γ2. According to the selected meshing trajectory line Γ1, the coordinate transformation and meshing equation are combined to obtain the equation of the mating meshing curve Γ2: where i 21 is the transmission ratio of the gear pair.

5. The design method of a spur gear meshing pair according to claim 2, characterized in that: The step 4 is specifically as follows: The tooth surface Σ2 equation is: Σ2=M c C 2 (t2) here: in, is the principal normal vector of surface Σ2 at this point, is the tangent vector of the mating meshing curve Γ2 on the surface Σ2 at the contact point, Can be get; Here Γ 2 (t2) is the normal section tooth profile equation of the spur gear 2, which is as follows: Where t2 is a variable and k2 is the tooth profile radius.

6. The design method of a spur arc gear meshing pair according to claim 2, characterized in that: In step 5, the equation of tooth surface Σ1 is: Σ1=M d Γ 1 (t1) Here: in, is the principal normal vector of surface Σ1 at the contact point, is the tangent vector of the mating meshing curve Γ4 on the surface Σ1 at the contact point, Can be get; Here Γ 1 (t1) is the normal section tooth profile equation of spur gear 1, which is as follows: Where t1 is a variable and k1 is the tooth profile radius.

7. A method for designing a spur gear meshing pair according to any one of claims 2 to 6, characterized in that: The meshing trajectory line Γ1 needs to cross the pitch line when it is selected.

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