Method and system for motion analysis of a nutation reducer

Abstract

The present application relates to a kind of nutation reducer motion analysis method, the analysis object of this method includes outer bevel gear output type single-stage nutation reducer, inner bevel gear output type single-stage nutation reducer, unilateral double-stage nutation reducer and bilateral double-stage nutation reducer;The method comprises: according to the type of nutation reducer, input the relevant parameters of nutation reducer;The relative angular velocity and absolute angular velocity of the outer bevel gear or inner bevel gear rotating with input shaft are calculated;The angular velocity of the outer bevel gear or inner bevel gear rotating with input shaft in absolute coordinate system is calculated;The absolute angular velocity of output shaft or output gear is calculated;Output motion data, and extract the motion characteristics of the required nutation reducer.The method and system are beneficial to simply, quickly obtain the motion characteristics of nutation reducer.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of nutation reducer, in particular to a nutation reducer motion analysis method and system. BACKGROUND

[0002] Nutation reducer has the advantages of large transmission ratio, high carrying capacity, compact structure, etc., so it has wide application prospect in the field of industrial robots, automobile manufacturing, etc. However, nutation reducer combines the structural characteristics of bevel gear transmission and planetary gear transmission, and its transmission principle is more complex than that of traditional reducer, so its motion analysis is difficult. The motion analysis of nutation reducer is an important basis for subsequent structure design and performance analysis, so it is necessary to analyze the complex motion characteristics of nutation reducer. SUMMARY

[0003] The purpose of the present application is to provide a nutation reducer motion analysis method and system, which is beneficial to simply and quickly obtain the motion characteristics of nutation reducer.

[0004] To achieve the above purpose, the technical scheme adopted by the present application is: a nutation reducer motion analysis method, the analysis object of the method includes outer bevel gear output type single-stage nutation reducer, inner bevel gear output type single-stage nutation reducer, single-sided double-stage nutation reducer and double-sided double-stage nutation reducer; the method includes:

[0005] According to the type of nutation reducer, input the related parameters of nutation reducer;

[0006] Calculate the relative angular velocity and absolute angular velocity of the outer bevel gear or inner bevel gear rotating with the input shaft;

[0007] Calculate the angular velocity of the outer bevel gear or inner bevel gear in the absolute coordinate system;

[0008] Calculate the absolute angular velocity of the output shaft or output gear;

[0009] Output motion data and extract the required motion characteristics of nutation reducer.

[0010] Further, the outer bevel gear output type single-stage nutation reducer is a reducer with an inner bevel gear fixed and an outer bevel gear rotating with the input shaft; the outer bevel gear is connected with a constant speed output mechanism, and the constant speed output mechanism is used to transmit the rotating speed of the outer bevel gear to the output shaft at a constant speed.

[0011] Further, for the outer bevel gear output type single-stage nutation reducer, δ1 is the cone top angle of the outer bevel gear, δ2 is the cone top angle of the inner bevel gear, and θ is the nutation angle; according to the right-hand rule, the coordinate system is established, O f and O grespectively, whose coordinate origin is the apex of the outer and inner bevel gears; assuming the meshing point of the two gears is A, the centroid of the outer bevel gear is B, and the centroid of the inner bevel gear is C;

[0012] the absolute angular velocity of the outer bevel gear ω a1 is the combination of two motions; the first motion is the dependent angular velocity ω e1 of the outer bevel gear rotating with the input shaft, and the second motion is the relative angular velocity ω r1 of the outer bevel gear rotating relative to the input shaft, and the vector relationship can be obtained according to the angular velocity synthesis theorem:

[0013] ω a1 = ω e1 + ω r1 (1)

[0014] wherein,

[0015] According to the sine theorem, we can obtain:

[0016]

[0017] According to equations (1) and (2), the relative angular velocity and the absolute angular velocity of the outer bevel gear are:

[0018]

[0019] In order to obtain the angular velocity of the outer bevel gear along the three coordinate axes in the global fixed coordinate system, first, the absolute angular velocity ω a1 of the outer bevel gear is decomposed into the angular velocity along the three coordinate axes in the moving coordinate system, and the three angular velocities are:

[0020]

[0021] The moving coordinate system is coincident with the global fixed coordinate system at the beginning, and the moving coordinate system rotates with the rotation of the input shaft; when the input shaft rotates an angle , the coordinate transformation matrix between the moving coordinate system and the global fixed coordinate system is:

[0022]

[0023] Combined with equations (4) and (5), the angular velocity of the outer bevel gear along the three coordinate axes in the global fixed coordinate system is:

[0024]

[0025] wherein, is the rotation angle of the input shaft,

[0026] In the transmission process of the outer bevel gear output type single-stage nutation reducer, the movement of the outer bevel gear is decomposed into two steps to combine the movement of the output shaft; on the one hand, when the constant velocity mechanism transmits the relative angular velocity ω r1 of the outer bevel gear to the output shaft ω r5 , the instantaneous relative angular velocities of the two shafts are equal;

[0027] ω r5 = ω r1 (7)

[0028] On the other hand, when the constant velocity mechanism transmits the relative angular velocity ω e1 of the outer bevel gear to the output shaft ω e5 , the two angular velocities are equal;

[0029] ω e5 = ω e1 (8)

[0030] Therefore, according to the speed composition theorem, the absolute angular velocity of the output shaft is:

[0031] ω a5 = ω e5 + ω r5 (9)

[0032] Combined with equations (3), (7), (8) and (9), the absolute angular velocity of the output shaft is:

[0033]

[0034] Further, the inner bevel gear output type single-stage nutation reducer is a reducer in which the outer bevel gear is restricted in circumferential rotation by a pin shaft and can swing with the rotation of the input shaft, the outer bevel gear is engaged with the inner bevel gear while swinging, and finally the inner bevel gear rotates.

[0035] Further, for the inner bevel gear output type single-stage nutation reducer, δ1 is the cone top angle of the outer bevel gear, δ2 is the cone top angle of the inner bevel gear, z1 is the number of teeth of the outer bevel gear, z2 is the number of teeth of the inner bevel gear, and θ is the nutation angle.

[0036] Since the pin shaft restricts the circumferential rotation of the outer bevel gear, the absolute rotation speed of the outer bevel gear is perpendicular to the Z axis; according to the sine theorem, the relative angular velocity and the absolute angular velocity of the outer bevel gear are:

[0037]

[0038] The angular velocities of the outer bevel gear along the three coordinate axes in the global fixed coordinate system are:

[0039]

[0040] When the input shaft rotates one revolution, the outer bevel gear oscillates one revolution. After the outer bevel gear meshes with the inner bevel gear for one revolution, the inner bevel gear rotates (z2-z1) teeth. One revolution of the input shaft is equivalent to the outer bevel gear rotating z1 teeth. Therefore, the inner bevel gear...

[0041] The rotational speed is:

[0042]

[0043] Furthermore, the single-sided double-stage nutation reducer is a double-stage nutation reducer composed of two pairs of internal meshing bevel gear pairs, with the meshing points on the bevel gear pairs on the same side.

[0044] Furthermore, for a single-sided two-stage nutation reducer, since its first and second external bevel gears are fixedly connected, their motion characteristics are the same; let δ1 be the cone apex angle of the first external bevel gear, δ2 bevel angle of the first internal bevel gear, δ3 bevel angle of the second external bevel gear, δ4 bevel angle of the second internal bevel gear, z3 be the number of teeth of the second external bevel gear, z4 be the number of teeth of the second internal bevel gear, and θ be the nutation angle;

[0045] The relative angular velocity and absolute angular velocity of the first and second external bevel gears are:

[0046]

[0047] The angular velocities of the first and second external bevel gears along the three coordinate axes in the globally fixed coordinate system are:

[0048]

[0049] When the second-stage internal meshing bevel gear pair increases the relative angular velocity ω of the first and second external bevel gears... r13 Passed to

[0050] On the second internal bevel gear ω r4 Since the ratio of rotational speeds is the reciprocal of the ratio of the number of teeth, the relative angular velocity of the second internal bevel gear is...

[0051] The degree is:

[0052] ω r4 =ω r13 (z3 / z4)=ω r13 (sinδ3 / sinδ4) (16)

[0053] When the second-stage internal meshing bevel gear pair pulls the first and second external bevel gears together at an angular velocity ω... e13 Passed to

[0054] On the second internal bevel gear ω e4 At that time, the two rotate at the same speed;

[0055] The absolute angular velocity of the output shaft of the single-sided two-stage nutation reducer is:

[0056]

[0057] Further, the double-sided two-stage nutation reducer is a kind of two pairs of internal meshing bevel gear pairs, and the meshing points of the bevel gear pairs are on different sides of the double-stage nutation reducer.

[0058] Further, for the double-sided two-stage nutation reducer, δ1 is the cone top angle of the third outer bevel gear, δ2 is the cone top angle of the third inner bevel gear, δ3 is the cone top angle of the fourth inner bevel gear, δ4 is the cone top angle of the fourth outer bevel gear, and θ is the nutation angle.

[0059] The vector relationship of the absolute angular velocity, the relative angular velocity and the relative angular velocity of the third inner bevel gear can be obtained according to the sine theorem:

[0060]

[0061] Solving the above formula, the relative angular velocity and the absolute angular velocity of the third inner bevel gear and the fourth inner bevel gear are:

[0062]

[0063] The angular velocities of the third inner bevel gear and the fourth inner bevel gear of the double-sided two-stage nutation reducer along the three coordinate axes in the global fixed coordinate system of the absolute coordinate system are:

[0064]

[0065] The absolute angular velocity of the output shaft of the double-sided two-stage nutation reducer is:

[0066]

[0067] The application also provides a motion analysis system of the nutation reducer, which comprises a memory, a processor and computer program instructions stored in the memory and capable of being executed by the processor, and when the processor executes the computer program instructions, the above-mentioned method steps can be realized.

[0068] Compared with the prior art, the application has the following beneficial effects: a motion analysis method and system of the nutation reducer are provided, the objects that can be analyzed by the method include an outer bevel gear output type single-stage nutation reducer, an inner bevel gear output type single-stage nutation reducer, a single-sided two-stage nutation reducer and a double-sided two-stage nutation reducer, and most types of nutation reducers are included; through the method, the motion characteristics of the required nutation reducer can be obtained intuitively, accurately and quickly. BRIEF DESCRIPTION OF DRAWINGS

[0069] Figure 1The method of the embodiment of the present application realizes the flow chart.

[0070] Figure 2 The structural diagram of four types of nutation reducers in the embodiment of the present application.

[0071] Figure 3 The motion analysis diagram of the single-stage nutation reducer of the external bevel gear output type in the embodiment of the present application.

[0072] Figure 4 The motion analysis diagram of the single-stage nutation reducer of the internal bevel gear output type in the embodiment of the present application.

[0073] Figure 5 The motion analysis diagram of the double-side double-stage nutation reducer in the embodiment of the present application. DETAILED DESCRIPTION

[0074] The present application will be further described below in conjunction with the drawings and embodiments.

[0075] It should be noted that the following detailed description is exemplary and is intended to provide further explanation of the present application. Unless otherwise indicated, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which the present application pertains.

[0076] It should be noted that the terms used herein are only for the purpose of describing the specific embodiments and are not intended to limit the exemplary embodiments according to the present application. As used herein, the singular form is intended to include the plural form unless the context clearly indicates otherwise, and it should also be understood that when the terms "comprise" and / or "include" are used in the specification, there is a presence of a feature, step, operation, device, component, and / or combinations thereof.

[0077] As shown in FIG. 1, the present embodiment provides a motion analysis method of a nutation reducer, which comprises: Figure 1

[0078] 1) According to the type of the nutation reducer, input the relevant parameters of the nutation reducer.

[0079] 2) Calculate the relative angular velocity and absolute angular velocity of the external bevel gear or internal bevel gear rotating with the input shaft.

[0080] 3) Calculate the angular velocity of the external bevel gear or internal bevel gear rotating with the input shaft in the absolute coordinate system.

[0081] 4) Calculate the absolute angular velocity of the output shaft or output gear.

[0082] 5) Output the motion data and extract the required motion characteristics of the nutation reducer.

[0083] ​The method analyzes the object as shown in Figure 2 Figure 2 (a) the outer bevel gear output type single stage nutation reducer, Figure 2 (b) the inner bevel gear output type single stage nutation reducer, Figure 2 (c) the single-sided double-stage nutation reducer, and Figure 2 (d) the double-sided double-stage nutation reducer. Wherein:

[0084] The outer bevel gear output type single stage nutation reducer is a kind of reducer in which the outer bevel gear is fixed and the outer bevel gear can rotate with the input shaft; the outer bevel gear is connected with the constant speed output mechanism, and the constant speed output mechanism is used to transmit the rotation speed of the outer bevel gear to the output shaft at a constant speed.

[0085] The inner bevel gear output type single stage nutation reducer is a kind of reducer in which the outer bevel gear is limited in the circumferential rotation by the pin shaft and the outer bevel gear can swing with the input shaft; the outer bevel gear is engaged with the inner bevel gear while swinging, and finally makes the inner bevel gear rotate.

[0086] The single-sided double-stage nutation reducer is a kind of double-stage nutation reducer composed of two pairs of inner meshing bevel gear pairs, and the meshing points on the bevel gear pairs are on the same side.

[0087] The double-sided double-stage nutation reducer is a kind of double-stage nutation reducer composed of two pairs of inner meshing bevel gear pairs, and the meshing points on the bevel gear pairs are on different sides.

[0088] The method analyzes the motion characteristics of the nutation reducer of the type researched according to the implementation sequence of the flowchart of Figure 1 .

[0089] In the embodiment, the kinematics calculation method of the outer bevel gear output type single stage nutation reducer is as shown in Figure 3 (a); wherein δ1 is the cone top angle of the outer bevel gear, δ2 is the cone top angle of the inner bevel gear, and θ is the nutation angle. According to the right-hand rule, the coordinate system is established, O f and O g respectively represent the global fixed coordinate system and the mobile coordinate system, and the coordinate origins are the cone top points of the outer bevel gear and the inner bevel gear, as shown in Figure 3 (a). It is assumed that the meshing point of the two gears is A, the mass center point of the outer bevel gear is B, and the mass center point of the inner bevel gear is C.

[0090] The absolute angular velocity ω a1 of the outer bevel gear is the combination of two kinds of motions. The first motion is the dependent angular velocity ω e1 of the outer bevel gear rotating with the input shaft, and the second motion is the relative angular velocity ω r1 of the outer bevel gear rotating relative to the input shaft, as shown in Figure 3 ​(a) shown, the vector relationship can be obtained according to the angular velocity composition theorem

[0091] ω a1 = ω e1 + ω r1 (1)

[0092] wherein,

[0093] According to the sine theorem, it can be obtained that

[0094]

[0095] According to equations (1) and (2), the relative angular velocity and the absolute angular velocity of the external bevel gear can be obtained:

[0096]

[0097] In order to obtain the angular velocity of the external bevel gear along the three coordinate axes in the global fixed coordinate system, first, the absolute angular velocity ω a1 of the external bevel gear is decomposed into the angular velocity along the three coordinate axes in the moving coordinate system, as shown in Figure 3 (a), the three angular

[0098] velocities are

[0099]

[0100] The moving coordinate system is coincident with the global fixed coordinate system at the beginning, and the moving coordinate system rotates with the rotation of the input shaft. When the input shaft rotates an angle , the positional relationship between the moving coordinate system and the global fixed coordinate system is as shown in Figure 3 (a). The

[0101] transformation matrix is

[0102]

[0103] Combined with equations (4) and (5), the angular velocity of the external bevel gear along the three coordinate axes in the global fixed coordinate system is

[0104]

[0105] In the formula, is the rotation angle of the input shaft,

[0106] As shown in Figure 3 (b), in the transmission process of the external bevel gear output type single-stage nutation reducer, the motion of the external bevel gear can be decomposed into two steps to combine the motion of the output shaft. On the one hand, when the constant velocity mechanism transmits the relative angular velocity ω r1 of the external bevel gear to the output shaft ωr5 Since the constant velocity mechanism has the advantage of synchronous transmission, it means that the instantaneous relative angular velocity of the two shafts is equal.

[0107] ω r5 = ω r1 (7)

[0108] On the other hand, when the constant velocity mechanism transmits the entrainment angular velocity ω e1 of the outer bevel gear to the output shaft ω e5 , the outer bevel gear and the output shaft can be regarded as the same component. Therefore, the two angular velocities are equal.

[0109] ω e5 = ω e1 (8)

[0110] Therefore, according to the speed composition theorem, the absolute angular velocity of the output shaft is

[0111] ω a5 = ω e5 + ω r5 (9)

[0112] Combining equations (3), (7), (8) and (9), the absolute angular velocity of the output shaft is

[0113]

[0114] In this embodiment, the kinematics calculation method of the inner bevel gear output type single-stage nutation reducer is as shown in Figure 4 ;

[0115] Where δ1 is the cone top angle of the outer bevel gear, δ2 is the cone top angle of the inner bevel gear, z1 is the number of teeth of the outer bevel gear, z2 is the number of teeth of the inner bevel gear, and θ is the nutation angle.

[0116] Since the pin shaft restricts the circumferential rotation of the outer bevel gear, the absolute speed of the outer bevel gear is perpendicular to the Z axis, and the vector relationship is as shown in Figure 4 According to the sine theorem, the relative angular velocity and the absolute angular velocity of the outer bevel gear are:

[0117]

[0118] The angular velocity of the outer bevel gear along the three coordinate axes in the global fixed coordinate system is

[0119]

[0120] When the input shaft rotates one circle, the outer bevel gear swings one circle, and after one week of engagement between the outer bevel gear and the inner bevel gear, the inner bevel gear rotates (z2-z1) teeth, and the input shaft rotates one circle, which is equivalent to the outer bevel gear rotating z1 teeth, so the rotation speed of the inner bevel gear is

[0121]

[0122] In this embodiment, the kinematic characteristics of the first-stage internal meshing bevel gear pair in the kinematic calculation method of the single-sided double-stage nutation reducer can be referred to the kinematic calculation method of the external bevel gear output type single-stage nutation reducer. The two methods share the same transmission principle in the first stage, which will not be elaborated further here. Since external bevel gear 1 and external bevel gear 3 are fixedly connected, their kinematic characteristics are the same. Wherein, δ1 is the cone apex angle of external bevel gear 1, δ2 is the cone apex angle of internal bevel gear 2, δ3 is the cone apex angle of external bevel gear 3, δ4 is the cone apex angle of internal bevel gear 4, z3 is the number of teeth of external bevel gear 3, z4 is the number of teeth of internal bevel gear 4, and θ is the nutation angle.

[0123] The relative angular velocity and absolute angular velocity of external bevel gear 1 and external bevel gear 3 are:

[0124]

[0125] The angular velocities of external bevel gear 1 and external bevel gear 3 along the three coordinate axes in the globally fixed coordinate system are:

[0126]

[0127] When the second-stage internal meshing bevel gear pair reduces the relative angular velocity ω of the external bevel gear 13... r13 Transmitted to internal bevel gear 4 ω r4 At that time, since the ratio of rotational speeds is the reciprocal of the ratio of the number of teeth, this means that the relative angular velocity of the internal bevel gear 4 is...

[0128] ω r4 =ω r13 (z3 / z4)=ω r13 (sinδ3 / sinδ4) (16)

[0129] When the second-stage internal meshing bevel gear pair pulls the first and second external bevel gears together at an angular velocity ω... e13 Passed to

[0130] On the second internal bevel gear ω e4 At that time, the two rotate at the same speed.

[0131] The absolute angular velocity of the output shaft of the single-sided two-stage nutation reducer is:

[0132]

[0133] In this embodiment, the kinematic characteristics of the first-stage internal meshing bevel gear pair in the kinematic calculation method of the dual-sided two-stage nutation reducer are similar to those of the external bevel gear output type single-stage nutation reducer, such as... Figure 5The vector relationship of absolute angular velocity, relative angular velocity and angular velocity of inner cone gear 2 is shown in Fig. 2, wherein δ1 is the cone top angle of outer cone gear 1, δ2 is the cone top angle of inner cone gear 2, δ3 is the cone top angle of inner cone gear 3, δ4 is the cone top angle of outer cone gear 4, and θ is the nutation angle.

[0134] The vector relationship of absolute angular velocity, relative angular velocity and angular velocity of inner cone gear 2 is shown in Fig. 2, wherein δ1 is the cone top angle of outer cone gear 1, δ2 is the cone top angle of inner cone gear 2, δ3 is the cone top angle of inner cone gear 3, δ4 is the cone top angle of outer cone gear 4, and θ is the nutation angle. Figure 5 The vector relationship of absolute angular velocity, relative angular velocity and angular velocity of inner cone gear 2 is shown in Fig. 2, wherein δ1 is the cone top angle of outer cone gear 1, δ2 is the cone top angle of inner cone gear 2, δ3 is the cone top angle of inner cone gear 3, δ4 is the cone top angle of outer cone gear 4, and θ is the nutation angle.

[0135] According to the sine theorem, the following equation can be obtained:

[0136]

[0137] Solving the above equation, the relative angular velocity and absolute angular velocity of inner cone gear 2 and inner cone gear 3 are as follows:

[0138]

[0139] Referring to the calculation method of outer cone gear output type single-stage nutation reducer, the angular velocities of inner cone gear 2 and inner cone gear 3 of the double-side double-stage nutation reducer along three coordinate axes in the absolute coordinate system globally fixed coordinate system are as follows:

[0140]

[0141] The kinematics calculation formula of the second-stage inner meshing cone gear pair refers to the single-side double-stage nutation reducer. After calculation, the absolute angular velocity of the output shaft of the double-side double-stage nutation reducer is as follows:

[0142]

[0143] Since there are various types of nutation reducers, the first step is to judge the type of the nutation reducer, the second step is to input the parameters of the studied nutation reducer, the third step is to calculate the required motion output data according to the above calculation formula, and the last step is to extract the required motion characteristics.

[0144] The application further provides a motion analysis system of a nutation reducer, which comprises a memory, a processor and computer program instructions stored in the memory and capable of being executed by the processor, and when the processor executes the computer program instructions, the above method steps can be realized.

[0145] Those skilled in the art should understand that the embodiments of the present application can be provided as a method, a system or a computer program product. Therefore, the present application can adopt a complete hardware embodiment, a complete software embodiment or an embodiment combining software and hardware aspects. Moreover, the present application can adopt a computer program product implemented on one or more computer usable storage media containing computer usable program codes (including but not limited to disk storage, CD-ROM, optical storage, etc.).

[0146] The computer program instructions can also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer-implemented process such that the instructions which execute on the computer or other programmable apparatus provide steps for implementing the functions specified in the flowchart block or blocks. Figure 1 one or more flow or blocks Figure 1 one or more flow or blocks

[0147] The computer program instructions can also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer-implemented process such that the instructions which execute on the computer or other programmable apparatus provide steps for implementing the functions specified in the flowchart block or blocks. Figure 1 one or more flow or blocks Figure 1 one or more flow or blocks

[0148] The computer program instructions can also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer-implemented process such that the instructions which execute on the computer or other programmable apparatus provide steps for implementing the functions specified in the flowchart block or blocks. Figure 1 one or more flow or blocks Figure 1 one or more flow or blocks

[0149] The above description is only preferred embodiments of the present application, not intended to limit other forms of the application. Any person familiar with the art can make changes or modifications to the above-mentioned technical content of the disclosure as equivalent embodiments. However, any simple modification, equivalent change and modification of the above embodiments without departing from the technical solution of the present application, according to the technical essence of the present application, still belongs to the protection scope of the technical solution of the present application.

Claims

1. A method of motion analysis of a nutation reduction gear, characterized by, The method includes: According to the type of the nutation reducer, input the relevant parameters of the nutation reducer; Calculate the relative angular velocity and absolute angular velocity of the outer bevel gear or inner bevel gear rotating with the input shaft; Calculate the angular velocity of the outer bevel gear or inner bevel gear in the absolute coordinate system; Calculate the absolute angular velocity of the output shaft or output gear; Output the motion data and extract the required motion characteristics of the nutation reducer; The single-side double-stage nutation reducer is a double-stage nutation reducer composed of two pairs of inner meshing bevel gear pairs, and the meshing points on the bevel gear pairs are on the same side. For unilateral double-stage nutation reducer, since its first outer bevel gear and second outer bevel gear are fixedly connected, their motion characteristics are the same; let be the cone top angle of the first outer bevel gear, be the cone top angle of the first inner bevel gear, be the cone top angle of the second outer bevel gear, be the cone top angle of the second inner bevel gear, z3 be the number of teeth of the second outer bevel gear, z4 be the number of teeth of the second inner bevel gear, and θ be the nutation angle. The relative angular velocity and absolute angular velocity of the first outer bevel gear and the second outer bevel gear are: (14) The angular velocity of the first outer bevel gear and the second outer bevel gear along three coordinate axes in the global fixed coordinate system is: (15) When the second stage internal cone pair transmits the relative angular velocity of the first and second external cones to the second internal cone Since the ratio of the rotational speeds is the inverse of the ratio of the number of teeth, the relative angular velocity of the second internal cone is (16) When the second stage internal meshing bevel gear pair transmits the entrainment angular velocity of the first outer bevel gear and the second outer bevel gear to the second inner bevel gear , the rotational speeds of both are equal; The absolute angular velocity of the output shaft of the single-side double-stage nutation reducer is: (17)。 2. The method of kinematic analysis of a nutating speed reducer according to claim 1, wherein The outer bevel gear output type single-stage nutation reducer is a reducer in which the outer bevel gear can rotate with the input shaft, and the inner bevel gear is fixed.

3. A method of kinematic analysis of a nutating speed reducer according to claim 2, characterized in that, For the single-stage nutation reducer with external bevel gear output, δ1 is the cone top angle of the external bevel gear, δ2 is the cone top angle of the internal bevel gear, and θ is the nutation angle; according to the right-hand rule, a coordinate system is established, O f and O g are respectively represented as a global fixed coordinate system and a mobile coordinate system, and the coordinate origins are the cone top points of the external bevel gear and the internal bevel gear; it is assumed that the meshing point of the two gears is A, the mass center point of the external bevel gear is B, and the mass center point of the internal bevel gear is C; The absolute angular velocity of the outer bevel gear is the combination of two motions; the first motion is the dependent angular velocity of the outer bevel gear rotating with the input shaft , the second motion is the relative angular velocity of the outer bevel gear rotating with respect to the input shaft The vector relationship of which can be obtained according to the angular velocity composition theorem: (1) According to the sine theorem, the relative angular velocity and absolute angular velocity of the outer bevel gear are: (2) According to equations (1) and (2), the relative angular velocity and absolute angular velocity of the outer bevel gear are: (3) In order to obtain the angular velocities of the external bevel gear along three coordinate axes in the global fixed coordinate system, the absolute angular velocity of the external bevel gear is first decomposed into angular velocities along three coordinate axes in the moving coordinate system, the three angular velocities being: ​ (4) Initially, the mobile coordinate system coincides with the global fixed coordinate system, and the mobile coordinate system rotates with the rotation of the input shaft; when the input shaft rotates by an angle , the coordinate transformation matrix between the mobile coordinate system and the global fixed coordinate system is: (5) According to equations (4) and (5), the angular velocity of the outer bevel gear along three coordinate axes in the global fixed coordinate system is: (6) In the formula, is the rotation angle of the input shaft, ; In the transmission process of the single-stage nutation reducer of the outer bevel gear output type, the movement of the outer bevel gear is decomposed into two steps to combine the movement of the output shaft; on the one hand, when the constant velocity mechanism transmits the relative angular velocity of the outer bevel gear to the output shaft , the instantaneous relative angular velocity of the two shafts is equal; . (7) On the other hand, when the constant velocity mechanism transmits the entrainment angular velocity of the outer bevel gear to the output shaft at which the two angular velocities are equal;​ (8) Therefore, according to the velocity composition theorem, the absolute angular velocity of the output shaft is: (9) According to equations (3), (7), (8) and (9), the absolute angular velocity of the output shaft is: (10)。 4. The method of kinematic analysis of a nutating speed reducer according to claim 1, wherein The inner bevel gear output type single-stage nutation reducer is a reducer in which the outer bevel gear is restricted in circumferential rotation by a pin shaft, and the outer bevel gear can swing with the input shaft, and the outer bevel gear meshes with the inner bevel gear while swinging, finally making the inner bevel gear rotate.

5. A method of kinematic analysis of a nutating speed reducer according to claim 4, characterized in that, For the single-stage nutation reducer of the inner bevel gear output type, let be the cone top angle of the outer bevel gear, be the cone top angle of the inner bevel gear, z1 be the number of teeth of the outer bevel gear, z2 be the number of teeth of the inner bevel gear, and θ be the nutation angle; Since the pin shaft restricts the circumferential rotation of the outer bevel gear, the absolute speed of the outer bevel gear is perpendicular to the Z axis; according to the sine theorem, the relative angular velocity and absolute angular velocity of the outer bevel gear are: (11) The angular velocity of the outer bevel gear along three coordinate axes in the global fixed coordinate system is: (12) When the input shaft rotates one circle, the outer bevel gear swings one circle, and after one week of meshing between the outer bevel gear and the inner bevel gear, the inner bevel gear rotates (z2-z1) teeth, and the input shaft rotates one circle, which is equivalent to the outer bevel gear rotating z1 teeth, so the inner bevel gear rotates at a speed of: (13)。 6. The method of kinematic analysis of a nutating speed reducer according to claim 1, wherein The double-side double-stage nutation reducer is a double-stage nutation reducer composed of two pairs of inner meshing bevel gear pairs, and the meshing points on the bevel gear pairs are on different sides.

7. A method of kinematic analysis of a nutating speed reducer according to claim 6, characterized in that, For a double-sided double-stage nutation reducer, let be the cone top angle of the third outer bevel gear, be the cone top angle of the third inner bevel gear, be the cone top angle of the fourth inner bevel gear, be the cone top angle of the fourth outer bevel gear, and θ be the nutation angle. According to the sine theorem, the vector relationship between the absolute angular velocity, the dependent angular velocity and the relative angular velocity of the third inner bevel gear is: (18) Solving the above equation, the relative angular velocity and absolute angular velocity of the third inner bevel gear and the fourth inner bevel gear are: (19) The angular velocity of the third inner bevel gear and the fourth inner bevel gear along three coordinate axes in the absolute coordinate system global fixed coordinate system is: (20) The absolute angular velocity of the output shaft of the double-side double-stage nutation reducer is: (21)。 8. A motion analysis system for a nutating speed reducer, characterized by, A computer program product comprising a storage medium to store computer program instructions, which, when executed by a processor, implement the method steps of any one of claims 1-7. A computer program product comprising a storage medium to store computer program instructions, which, when executed by a processor, implement the method steps of any one of claims 1-7.

Images