Patent Publication Number: US-2023160462-A1

Title: Flexible external gear, wave reducer, and robot

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     The present application claims priority under 35 U.S.C. § 119 to Japanese Patent Application No. 2021-189298, filed on Nov. 22, 2021, the entire contents of which are hereby incorporated herein by reference. 
     1. FIELD OF THE INVENTION 
     The present disclosure relates to a flexible external gear, a wave reducer, and a robot. 
     2. BACKGROUND 
     Conventionally, a wave gear device including a flexible external gear and an internal gear is known. This type of wave gear device is mainly used as a reducer. 
     A conventional strain wave gear device includes a cup-shaped strain gear having external teeth, a ring gear having internal teeth, and a wave generator that causes relative rotation between the strain gear and the ring gear. The strain gear is formed of a strain gear blank having a uniform thickness of 0.015 D to 0.03 D (D: inner diameter of strain gear). The strain gear has a cylindrical portion in which external teeth are formed around an opening end, and an end portion including a diaphragm having a thickness half the thickness of the cylindrical portion. 
     The opening end of the strain gear comes into contact with the wave generator and is deformed into an elliptical shape. This causes the external teeth of the strain gear to engage with the internal teeth of the ring gear along each side of the elliptical major axis. Here, the number of external teeth of the strain gear and the number of internal teeth of the ring gear are different from each other. Due to this, the strain gear and the ring gear move relative to each other by rotation of the wave generator. However, in the conventional strain gear, it is conceivable to be difficult to favorably bend the cylindrical portion. 
     SUMMARY 
     An example embodiment of the present disclosure is a flexible external gear including a tubular portion extending in a direction including a component in a central axis direction, and a diaphragm portion extending in a direction including a radial component from one axial end portion of the tubular portion. The tubular portion includes a first portion on one axial side, the first portion having flexibility, and a second portion on another axial side relative to the first portion. The second portion includes external teeth protruding radially outward and arrayed in a circumferential direction, a maximum value of a thickness of the diaphragm portion is equal to or less than twice a length from a radially outer end of the external teeth to a radially inner surface of the second portion, and a minimum value of a thickness of the first portion is equal to or less than half the maximum value of the thickness of the diaphragm portion. 
     The above and other elements, features, steps, characteristics and advantages of the present disclosure will become more apparent from the following detailed description of the example embodiments with reference to the attached drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG.  1    is a schematic view of a robot according to a preferred embodiment of the present invention. 
         FIG.  2    is a longitudinal cross-sectional view of a wave reducer according to a preferred embodiment of the present invention. 
         FIG.  3    is a transverse cross-sectional view of the wave reducer. 
         FIG.  4    is a partial longitudinal cross-sectional view of a flexible external gear according to a preferred embodiment of the present invention. 
         FIG.  5    is a partial longitudinal cross-sectional view of a flexible external gear according to a modification of a preferred embodiment of the present invention. 
         FIG.  6    is an enlarged view of a portion of  FIG.  3   . 
     
    
    
     DETAILED DESCRIPTION 
     Hereinafter, example embodiments of the present application will be described with reference to the drawings. 
       FIG.  1    is a schematic view of a robot  100  equipped with a wave reducer  1  according to one example embodiment. For example, the robot  100  is a so-called industrial robot that performs works such as conveyance, processing, and assembly of components in a manufacturing line of an industrial product. As illustrated in  FIG.  1   , the robot  100  includes the wave reducer  1 . In the present example embodiment, the robot  100  includes a base frame  101 , an arm  102 , a motor  103 , and the wave reducer  1 . This allows the wave reducer  1  equipped on the robot  100  to favorably bend an entire tubular portion  21  including a first tubular portion  211  described later. 
     The arm  102  is rotatably supported with respect to the base frame  101 . The motor  103  and the wave reducer  1  are incorporated in a joint portion between the base frame  101  and the arm  102 . When a drive current is supplied to the motor  103 , a rotational motion is output from the motor  103 . The rotational motion output from the motor  103  is decelerated by the wave reducer  1  and transmitted to the arm  102 . Due to this, the arm  102  rotates with respect to the base frame  101  at a speed after deceleration. 
     Next, the entire structure of the wave reducer  1  will be described. 
     Hereinafter, a direction parallel to a central axis  9  of the wave reducer  1  is referred to as “axial”, a direction perpendicular to the central axis  9  of the wave reducer  1  is referred to as “radial”, and a direction along an arc about the central axis  9  of the wave reducer  1  is referred to as “circumferential”. The “parallel” mentioned above includes both “parallel” and “substantially parallel”. Moreover, the “perpendicular” mentioned above includes both “perpendicular” and “substantially perpendicular”. In the present application, in  FIGS.  2 ,  4 , and  5    described later, the shape and the positional relationship of each part will be described with the axial direction as a left-right direction, the left side as “one axial side”, and the right side as “other axial side”. 
       FIG.  2    is a longitudinal cross-sectional view of the wave reducer  1  according to one example embodiment.  FIG.  3    is a transverse cross-sectional view of the wave reducer  1  viewed from A-A position in  FIG.  2   . To avoid complication of the drawings, hatching that indicates a cross section is not illustrated in  FIG.  3   . As described above, the wave reducer  1  is equipped on the joint portion of the robot  100 , and decelerates and outputs rotational motion input from the motor  103 . More specifically, the wave reducer  1  is a device that, by using a differential between an internal gear  10  and a flexible external gear  20  described later, reduces a rotational motion at a first rotational speed obtained from the motor  103  to a second rotational speed lower than the first rotational speed. 
     As illustrated in  FIGS.  2  and  3   , the wave reducer  1  includes the internal gear  10 , the flexible external gear  20 , and a wave generator  30 . The wave reducer  1  of the present example embodiment further includes an outer ring  151 , an inner ring  152 , and an output unit  40 . 
     The wave reducer  1  is provided with an input member  104  for obtaining power from the motor  103 . The input member  104  extends in a tubular shape in the axial direction about the central axis  9 . An output shaft of the motor  103  is inserted radially inside the input member  104 . The input member  104  is fixed with the output shaft of the motor  103  so as to be relatively non-rotatable with respect to each other. Due to this, the input member  104  rotates at the first rotational speed about the central axis  9  together with a rotating portion of the motor  103 . The input member  104  may be the same member as the output shaft. 
     The internal gear  10  is an annular gear about the central axis  9 . The internal gear  10  is fixed to the base frame  101  of the robot  100 . The internal gear  10  is arranged to be coaxial with the central axis  9 . As described later, the flexible external gear  20  has a second tubular portion  212 . The internal gear  10  is arranged radially outside the second tubular portion  212 . The rigidity of the internal gear  10  is sufficiently higher than the rigidity of the tubular portion  21  described later of the flexible external gear  20 . For this reason, the internal gear  10  can be regarded as a substantially rigid body. The internal gear  10  has a plurality of internal teeth  11 . The plurality of internal teeth  11  protrude radially inward from a radially inner surface of the internal gear  10 . The plurality of internal teeth  11  are arrayed at a constant pitch in the circumferential direction on the inner surface of the internal gear  10 . 
     The internal gear  10  is provided with a plurality of through holes  110 . In the present example embodiment, the number of the through holes  110  is 8. Each of the eight through holes  110  penetrates the internal gear  10  in the axial direction. The eight through holes  110  are arranged at equal intervals in the circumferential direction about the central axis  9 . The internal gear  10  is fixed to the base frame  101  of the robot  100  by fastening screws (not illustrated) penetrating the respective eight through holes  110  to the base frame  101 . The internal gear  10  is provided with a plurality of screw holes  111 . Each of the plurality of screw holes  111  is recessed from the end surface on one axial side of the internal gear  10  toward the other axial side. The screw hole  111  may be a through hole. 
     The flexible external gear  20  is a bendable and deformable bottomed annular gear. As described later, the flexible external gear  20  is fixed to the arm  102  of the robot  100  via the output unit  40  and the inner ring  152 . The flexible external gear  20  is rotatably supported about the central axis  9 .  FIG.  4    is a partial longitudinal cross-sectional view in which a part of the flexible external gear  20  is enlarged. As illustrated in  FIGS.  2  to  4   , the flexible external gear  20  includes the tubular portion  21  and a diaphragm portion  22 . 
     The tubular portion  21  extends in a direction including a component in the central axis  9  direction. In the present example embodiment, the tubular portion  21  extends in a tubular shape in the axial direction about the central axis  9 . The tubular portion  21  is a flexible site that can be bent in the radial direction. In particular, the end portion on the other axial side of the tubular portion  21  (hereinafter referred to as “other axial end portion”) is a free end, and therefore it can be displaced in the radial direction more greatly than another part. The other axial end portion of the tubular portion  21  is positioned radially outside the wave generator  30  and radially inside the internal gear  10 . 
     The tubular portion  21  includes the first tubular portion  211  and the second tubular portion  212 . The first tubular portion  211  is arranged on one axial side of the tubular portion  21  and has flexibility. The first tubular portion  211  is a radially bendable tubular site. 
     The second tubular portion  212  is arranged on the other axial side relative to the first tubular portion  211 . The second tubular portion  212  is positioned radially inside the internal gear  10 . The second tubular portion  212  has a plurality of external teeth  23 . The plurality of external teeth  23  are arrayed in the circumferential direction. Each of the plurality of external teeth  23  protrudes radially outward. The plurality of external teeth  23  are arrayed at a constant pitch along the circumferential direction. As described in detail later, an outer ring  323  of a flexible bearing  32  comes into contact with the inner peripheral surface of the second tubular portion  212 . Due to this, a part of the plurality of external teeth  23  and a part of the plurality of internal teeth  11  mesh with each other. That is, when the tubular portion  21  is pushed by the wave generator  30  from the radially inside, a part of the plurality of external teeth  23  and a part of the plurality of internal teeth  11  of the internal gear  10  mesh with each other. The number of the internal teeth  11  included in the internal gear  10  is slightly different from the number of the external teeth  23  included in the flexible external gear  20 . 
     The diaphragm portion  22  is a site extending in a direction including a radial component from an end portion on one axial side of the tubular portion  21  (hereinafter referred to as “one axial end portion”). That is, the diaphragm portion  22  extends in a direction including a radial component from one axial end portion of the tubular portion  21 . In the present example embodiment, the diaphragm portion  22  extends radially inward from one axial end portion of the tubular portion  21 . The diaphragm portion  22  extends annularly about the central axis  9 . The diaphragm portion  22  is a flat plate-like site that is less likely to bend than the tubular portion  21 . Since the diaphragm portion  22  includes such a structure, the flexible external gear  20  can be downsized in the radial direction. The diaphragm portion  22  is provided with a plurality of through holes  220 . Each of the plurality of through holes  220  penetrates the diaphragm portion  22  in the axial direction. The diaphragm portion  22  may extend radially outward from one axial end portion of the tubular portion  21 . 
     A thickness t(a) of the diaphragm portion  22  is substantially constant from a radially inner end to a radially outer end of the diaphragm portion  22 . In the present example embodiment, the thickness t(a) of the diaphragm portion  22  is a width in the axial direction. This makes it possible to easily manufacture, at the time of manufacturing the flexible external gear  20 , the diaphragm portion  22  as compared with the case where the thickness t(a) of the diaphragm portion  22  is not constant. 
     However, the thickness t(a) of the diaphragm portion  22  needs not be constant from the radially inner end to the radially outer end of the diaphragm portion  22 . For example, as illustrated in the modification of  FIG.  5   , a thick portion  25  having a larger axial thickness than the diaphragm portion  22  may be formed on the radially inside of the diaphragm portion  22 . The thickness of the diaphragm portion  22  in the axial direction may gradually increase toward the thick portion  25 . The plurality of through holes  220  may be provided in the thick portion  25 . 
     The tubular portion  21  further includes a connection portion  24 . The connection portion  24  extends in a direction having both axial and radial components. The connection portion  24  connects one axial end portion of the first tubular portion  211  and a radial end portion of the diaphragm portion  22 . In the present example embodiment, the connection portion  24  connects one axial end portion of the first tubular portion  211  and the radially outer end portion of the diaphragm portion  22 . 
     A more detailed structure of the flexible external gear  20  will be described later. 
     The wave generator  30  is a mechanism that generates periodical bending deformation in the tubular portion  21 . The wave generator  30  is arranged radially inside the second tubular portion  212 . The wave generator  30  of the present example embodiment includes a cam  31  and a flexible bearing  32 . The cam  31  and the flexible bearing  32  each extend annularly about the central axis  9 . The cam  31  is fixed to the outer surface of the input member  104  so as to be relatively non-rotatable with respect to each other, and is supported rotatably about the central axis  9 . The cam  31  of the present example embodiment has an elliptical cam profile. That is, a radially outer surface of the cam  31  has an elliptical shape when viewed in the axial direction, and has different outer diameters depending on the circumferential position. The flexible bearing  32  is a bearing that is bending deformable. The flexible bearing  32  is arranged between the radially outer surface of the cam  31  and the radially inner surface of the tubular portion  21  of the flexible external gear  20 . Accordingly, the cam  31  and the tubular portion  21  can rotate at different rotational speeds. 
     The flexible bearing  32  has an inner ring  321 , a plurality of balls  322 , and an elastically deformable outer ring  323 . The inner ring  321  comes into contact with the radially outer surface of the cam  31 . The plurality of balls  322  are interposed between the inner ring  321  and the outer ring  323  and arrayed along the circumferential direction. The outer ring  323  elastically deforms (bending deforms) via the inner ring  321  and the balls  322  along the cam profile of the rotating cam  31 . The outer ring  323  comes into contact with the radially inner surface of the tubular portion  21  of the flexible external gear  20 . For this reason, the tubular portion  21  is deformed in an elliptical shape along the radially outer surface of the cam  31 . As a result, the external teeth  23  of the flexible external gear  20  and the internal teeth  11  of the internal gear  10  mesh with each other at two positions corresponding to both ends of the elliptical major axis. However, the external teeth  23  and the internal teeth  11  do not mesh with each other at another position in the circumferential direction. Thus, the ball bearing is used as the flexible bearing  32  of the present example embodiment. However, other types of bearings such as a roller bearing may be used instead of the ball bearing. 
     When the motor  103  is driven, the cam  31  rotates at the first rotational speed about the central axis  9  together with the rotating portion of the motor  103  and the input member  104 . Due to this, the elliptical major axis of the flexible external gear  20  also rotates at the first rotational speed. Then, the meshing position between the external tooth  23  and the internal tooth  11  also changes at the first rotational speed in the circumferential direction. As described above, the number of the internal teeth  11  of the internal gear  10  is slightly different from the number of the external teeth  23  of the flexible external gear  20 . Due to this difference in the number of teeth, the combination of meshing between the external teeth  23  and the internal teeth  11  slightly changes in the circumferential direction every rotation of the cam  31 . Here, as described above, the internal gear  10  is fixed to the base frame  101  of the robot  100  and does not rotate. As a result, the flexible external gear  20  rotates about the central axis  9  at the second rotational speed lower than the first rotational speed with respect to the internal gear  10  and the base frame  101 . 
     The outer ring  151  is a member that extends in an annular shape about the central axis  9 . Both the outer ring  151  and the inner ring  152  have high rigidity. The outer ring  151  is provided with a plurality of through holes  153 . Each of the plurality of through holes  153  penetrates the outer ring  151  in the axial direction. The outer ring  151  is fixed to the internal gear  10  by fastening a plurality of screws  154  respectively penetrating the plurality of through holes  153  to the plurality of screw holes  111  of the internal gear  10  adjacent to the other axial side of the outer ring  151 . 
     The inner ring  152  is arranged radially inside the outer ring  151 . The inner ring  152  is a member that extends in an annular shape about the central axis  9 . The arm  102  of the robot  100  is fixed to the inner ring  152 . The inner ring  152  has an outer diameter slightly smaller than the inner diameter of the outer ring  151 . The inner ring  152  is provided with a plurality of screw holes  155 . Each of the plurality of screw holes  155  is formed from the end surface on the other axial side of the inner ring  152  toward one axial side. 
     The inner ring  152  is rotatably connected to the outer ring  151  by a bearing  16 . As the bearing  16  of the present example embodiment, a cross roller bearing is used. As illustrated in  FIG.  2   , the bearing  16  has a plurality of cylindrical rollers  161  between the inner peripheral surface of the outer ring  151  and the outer peripheral surface of the inner ring  152 . The plurality of cylindrical rollers  161  are arranged with alternately changing orientations between an annular V groove provided on the inner peripheral surface of the outer ring  151  and an annular V groove provided on the outer peripheral surface of the inner ring  152 . Due to this, the outer ring  151  and the inner ring  152  are connected with high rigidity while allowing rotation of the inner ring  152  with respect to the outer ring  151 . Such a cross roller bearing can give sufficient rigidity in the axial direction and the radial direction even without being paired when used like a ball bearing. That is, use of the cross roller bearing can reduce the number of bearings provided in the wave reducer  1 . This makes it possible to reduce the weight of the bearing  16 , and suppress the axial dimension of the bearing  16 . 
     The output unit  40  is a member for extracting power after deceleration. The output unit  40  extends in a cylindrical shape along the central axis  9 . As illustrated in  FIG.  2   , the other axial end portion of the output unit  40  is provided with an output flange portion  401  extending radially outward. The output flange portion  401  is provided with a plurality of through holes  400 . Each of the plurality of through holes  400  penetrates the output flange portion  401  in the axial direction. 
     As illustrated in  FIG.  2   , the diaphragm portion  22  of the flexible external gear  20  is arranged on one axial side of the output flange portion  401 . On one axial side of the diaphragm portion  22 , a washer  17  is interposed, and the inner ring  152  is further arranged. The number of the washers  17  to be arranged may be one or more. The washer  17  is not necessarily arranged. This makes it possible to easily adjust the axial positions of the flexible external gear  20  and the output unit  40  with respect to the inner ring  152 . 
     The washer  17  is provided with a plurality of through holes  170 . Each of the plurality of through holes  170  penetrates the washer  17  in the axial direction. The flexible external gear  20  and the output unit  40  are axially fixed to the inner ring  152  by fastening a plurality of screws  156  respectively penetrating the plurality of through holes  220  of the flexible external gear  20  and the plurality of through holes  400  of the output unit  40  to the plurality of screw holes  155  of the inner ring  152  via the plurality of through holes  170  of the washer  17 . Due to this, the inner ring  152 , the flexible external gear  20 , and the output unit  40  are coupled with one another so as to be relatively non-rotatable. 
     Here, as described above, the inner ring  152  is rotatably supported with respect to the outer ring  151  and the internal gear  10  via the bearing  16 . Due to this, the flexible external gear  20  fixed to the inner ring  152 , the output unit  40 , and the arm  102  of the robot  100  can rotate about the central axis  9  with respect to the base frame  101  to which the internal gear  10  is fixed. As a result, when the motor  103  is driven, the flexible external gear  20  and the arm  102  rotate about the central axis  9  at the second rotational speed lower than the first rotational speed that is the output of the motor  103 . 
     Next, a more detailed structure of the flexible external gear  20  will be described. In the following description, the thickness of the tubular portion  21  including the first tubular portion  211  and the second tubular portion  212  indicates the thickness in the normal direction with respect to the direction in which the tubular portion  21  extends when the tubular portion  21  is inclined with respect to the central axis  9 , and indicates the thickness in the radial direction when the tubular portion  21  is parallel to the central axis  9 . 
     The flexible external gear  20  can be molded into a final shape by, for example, press working on the basis of a plate-like material to create a cylindrical intermediate member, and then drawing or cutting. The external teeth  23  can be molded by rolling the intermediate member while pressing a roller against the intermediate member. In the present example embodiment, teeth along the axial direction, such as a spur gear, are formed as the external teeth  23 . A length t(d) from the radially outer end to the radially inner end of the external tooth  23  may vary depending on the axial and circumferential positions. A length t(c) from the radially outer end of the external teeth  23  to the radially inner surface of the second tubular portion  212  may also change depending on the axial and circumferential positions. 
     Stainless steel is used as the material for the flexible external gear  20  of the present example embodiment. However, steel having relatively low carbon content, aluminum, or the like may be used as the material for the flexible external gear  20 . 
       FIG.  6    is an enlarged view of a part of  FIG.  3   . As illustrated in  FIG.  6   , hereinafter, the two external teeth  23  adjacent in the circumferential direction in the flexible external gear  20  are referred to as “external tooth  23   k ” and “external tooth  23 ( k +1)” in order. The circumferential center of the “external tooth  23   k ” will be referred to as “circumferential center  23   o ”, and the circumferential center of the “external tooth  23 ( k +1)” will be referred to as “circumferential center  23 ( o +1)”. In the present example embodiment, the circumferential distance between the circumferential center  23   o  and the circumferential center  23 ( o +1), that is, a circumferential interval cp is larger than twice the value of the radially thinnest site in the thickness t(b) of the first tubular portion  211  illustrated in  FIGS.  4  and  5   . That is, in the external teeth  23   k  and  23 ( k +1) adjacent in the circumferential direction, the circumferential interval cp between the circumferential centers  23   o  and  23 ( o +1) of the external teeth  23   k  and  23 ( k +1) is larger than twice a minimum value min{t(b)} of the thickness t(b) of the first tubular portion  211 . Thus, in the present example embodiment, since the circumferential pitch between the adjacent external teeth  23   k  and  23 ( k +1) is long, the external teeth  23  can be more easily molded. 
     In the present example embodiment, the circumferential distance between the circumferential center  23   o  and the circumferential center  23 ( o +1), that is, the circumferential interval cp is larger than the value of the axially thinnest site in the thickness t(a) of the diaphragm portion  22  illustrated in  FIGS.  4  and  5   . That is, in the external teeth  23   k  and  23 ( k +1) adjacent in the circumferential direction, the circumferential interval cp between the circumferential centers  23   o  and  23 ( o +1) of the external teeth  23   k  and  23 ( k +1) is larger than a minimum value min {t(a)} of the thickness t(a) of the diaphragm portion  22 . Thus, in the present example embodiment, since the circumferential pitch between the adjacent external teeth  23   k  and  23 ( k +1) is long, the external teeth  23  can be more easily molded. 
     In the present example embodiment, a maximum value max{t(d)} of the length t(d) from the radially outer end to the radially inner end of the external teeth  23  is larger than the minimum value min{t(b)} of the thickness t(b) of the first tubular portion  211 . In this manner, by making the maximum value max{t(d)} of the length t(d) from the radially outer end to the radially inner end of the external tooth  23  larger than the minimum value min{t(b)} of the thickness t(b) of the first tubular portion  211 , that is, providing the external teeth  23  to be long to some extent in the radial direction, the external teeth  23  and the internal teeth  11  of the internal gear  10  favorably mesh with each other. This enables the wave reducer  1  to transmit torque more accurately. 
     In the present example embodiment, a maximum value max{ct} of a tooth thickness ct of the external tooth  23  at a radial midpoint cm between the radially outer end and the radially inner end of the external teeth  23  is smaller than the minimum value min{t(b)} of the thickness t(b) of the first tubular portion  211 . Thus, by suppressing the tooth thickness ct of the external teeth  23 , the degree of freedom in molding the external teeth  23  can be improved. The meshing between the external teeth  23  and the internal teeth  11  of the internal gear  10  is further improved. The tooth thickness ct may vary depending on the axial direction. The maximum value max{ct} of the tooth thickness ct in the region where the tooth thickness ct is maximized in the axial direction is only required to be smaller than the minimum value min{t(b)} of the thickness t(b) of the first tubular portion  211 . 
     In the present example embodiment, a maximum value max{t(c)} of the length t(c) from the radially outer end of the external teeth  23  to the radially inner surface of the second tubular portion  212  is larger than the minimum value min{t(b)} of the thickness t(b) of the first tubular portion  211  and smaller than twice the minimum value min{t(b)} of the thickness t(b) of the first tubular portion  211 . Thus, by making the maximum value max{t(c)} of the length t(c) from the radially outer end of the external teeth  23  to the radially inner surface of the second tubular portion  212  larger than the minimum value min{t(b)} of the thickness t(b) of the first tubular portion  211 , it is possible to suppress the external teeth  23  from becoming excessively long in the radial direction by making the maximum value max{t(c)} smaller than twice the minimum value min{t(b)} of the thickness t(b) of the first tubular portion  211  while securing the radial length of the external teeth  23  for favorably meshing with the internal teeth  11  of the internal gear  10  to some extent. As a result, the rigidity of the external teeth  23  can be improved. The length t(c) from the radially outer end of the external teeth  23  to the radially inner surface of the second tubular portion  212  and the thickness t(b) of the first tubular portion  211  may have different values depending on the respective axial position. In that case, it is only required to compare the respective values in the region where the length t(c) from the radially outer end of the external teeth  23  to the radially inner surface of the second tubular portion  212  and the thickness t(b) of the first tubular portion  211  are maximized or minimized. 
     As illustrated in  FIG.  4   , in the present example embodiment, the thickness t(a) of the diaphragm portion  22  is substantially the same as the length t(c) from the radially outer end of the external teeth  23  to the radially inner surface of the second tubular portion  212 . This makes it possible to easily manufacture the flexible external gear  20  as compared with the case where the thickness t(a) of the diaphragm portion  22  is different from the length t(c) from the radially outer end of the external teeth  23  to the radially inner surface of the second tubular portion  212 . 
     In the present example embodiment, the value of the axially thickest site in the thickness t(a) of the diaphragm portion  22  is equal to or less than twice the length t(c) from the radially outer end of the external teeth  23  to the radially inner surface of the second tubular portion  212 . That is, a maximum value max{t(a)} of the thickness t(a) of the diaphragm portion  22  is equal to or less than twice the length t(c) from the radially outer end of the external teeth  23  to the radially inner surface of the second tubular portion  212 . In the tubular portion  21  of the flexible external gear  20 , the value of the radially thinnest site in the thickness t(b) of the first tubular portion  211  positioned on the diaphragm portion  22  side is equal to or less than half the value of the axially thickest site in the thickness t(a) of the diaphragm portion  22 . That is, the minimum value min{t(b)} of the thickness t(b) of the first tubular portion  211  is equal to or less than half the maximum value max{t(a)} of the thickness t(a) of the diaphragm portion  22 . Thus, by suppressing the thickness t(b) of the first tubular portion  211 , the first tubular portion  211  can be favorably bent in the radial direction while securing the rigidity of the diaphragm portion  22  and the second tubular portion  212 . As a result, it is possible to favorably bend the entire tubular portion  21  including the first tubular portion  211 . Since the wave reducer  1  includes the flexible external gear  20 , it is possible to achieve the wave reducer  1  in which the first tubular portion  211  is favorably bent. 
     In the present example embodiment, the minimum value min{t(b)} of the thickness t(b) of the first tubular portion  211  is less than half the maximum value max{t(a)} of the thickness t(a) of the diaphragm portion  22 . This makes it possible to more favorably bend the first tubular portion  211  in the radial direction while securing rigidity of the diaphragm portion  22  and the second tubular portion  212 . Furthermore, in the present example embodiment, the minimum value min{t(b)} of the thickness t(b) of the first tubular portion  211  is equal to or less than half the minimum value min {t(a)} of the thickness t(a) of the diaphragm portion  22 . As a result, the first tubular portion  211  can be more favorably bent in the radial direction while securing rigidity of the diaphragm portion  22  and the second tubular portion  212 . 
     As described above, the connection portion  24  is further provided between the first tubular portion  211  and the diaphragm portion  22  of the flexible external gear  20  of the present example embodiment. The connection portion  24  extends in a direction having both axial and radial components while connecting one axial end portion of the first tubular portion  211  and the radially outer end portion of the diaphragm portion  22 . Due to this, the first tubular portion  211  and the diaphragm portion  22  can be firmly connected while securing both the flexibility of the first tubular portion  211  and the rigidity of the diaphragm portion  22 . 
     The connection portion  24  has an arc shape curved in a direction having both axial and radial components in a cross section along the central axis  9 . As illustrated in  FIGS.  2 ,  4   , and  5 , in the present example embodiment, the connection portion  24  is curved in an arc shape in a direction having both axial and radial components in a longitudinal cross section along the central axis  9 . The maximum value of the curvature radius of the connection portion  24  is equal to or less than 10 times the thickness t(b) of the first tubular portion  211 . Thus, by providing the connection portion  24  that curves with a high curvature, it is possible to relax stress concentration at the connection portion  24  while securing bending tendency of the tubular portion  21 . That is, since the connection portion  24  has the above-described configuration, for example, stress concentration at the connection portion  24  can be relaxed as compared with a case where the first tubular portion  211  and the diaphragm portion  22  are connected at a right angle. 
     While an example embodiment of the present disclosure has been described above, the present disclosure is not limited to the above example embodiment. The configurations of each member and each site may be appropriately combined or replaced without departing from the gist of the present disclosure. 
     Shapes of details of the flexible external gear, the wave reducer, and the robot may be different from the shapes illustrated in the drawings of the above example embodiment. 
     The present application can be applied to, for example, a flexible external gear, a wave reducer, and a robot. 
     Features of the above-described example embodiments and the modifications thereof may be combined appropriately as long as no conflict arises. 
     While example embodiments of the present disclosure have been described above, it is to be understood that variations and modifications will be apparent to those skilled in the art without departing from the scope and spirit of the present disclosure. The scope of the present disclosure, therefore, is to be determined solely by the following claims.