Abstract:
A worm gear mechanism ( 44 ) comprises a worm ( 70 ) and a worm wheel ( 80 ) meshed with the worm. In the teeth of a hob ( 90 ) used for the gear cutting of the worm wheel, at least the addendum surfaces ( 91   c ) are formed into arcuate shapes. The radial centers ( 93 ) of the arcs of the addendum surfaces are positioned nearer to a center line (WL′) of the hob than a pitch line ( 94 ) of the hob. The worm wheel is cut into a gear by the hob. The worm is formed into the same shape as the hob. The recess meshing length (L) of the worm gear mechanism is designed to be greater than the recess meshing length (Llim) of a conventional worm gear mechanism.

Description:
TECHNICAL FIELD 
     The present invention relates to a technique for improving a worm gear mechanism. 
     BACKGROUND ART 
     A worm gear mechanism is installed, for example, in a power steering device of a vehicle (see, for example, FIG. 14 in Patent Literature 1). 
     The worm gear mechanism as disclosed in Patent Literature 1 is provided with a worm coupled to an electric motor through a worm shaft, and a worm wheel configured to mesh with the worm. It is a transmission mechanism configured to boost and transmit auxiliary torque generated by the electric motor from the worm to the worm wheel. 
     In general, when the worm is rotated and force is applied in a direction of pushing the worm wheel, the worm receives reaction force from the worm wheel at a contact point of the worm and the worm wheel. It is preferred that strength of the worm gear mechanism be enhanced as it may contribute to extending a life of the worm gear mechanism. 
     PRIOR ART LITERATURE 
     Patent Literature 
     Patent Literature 1: JP 2010-270908 A 
     SUMMARY OF INVENTION 
     Technical Problem 
     It is therefore an object of the present invention to provide a technique for enhancing strength of a worm gear mechanism. 
     Solution to Problem 
     According to the present invention, in a worm gear mechanism including a worm and a worm wheel meshed with the worm, at least an addendum surface of a tooth of the worm is formed into an arc shape, and a center of a radius of an arc of the addendum surface is positioned nearer to a center line of the worm than a pitch line of the worm, the worm wheel is gear cut by a hob used in gear cutting of the worm wheel, at least an addendum surface of a tooth of the hob being formed into an arc shape, and a center of a radius of an arc of the addendum surface being positioned nearer to a center line of the hob than a pitch line of the hob, and a length of recess path of the worm gear mechanism, in which the worm is meshed with the worm wheel, is set to be larger than a length of recess path of the worm gear mechanism having an involute profile worm and an involute profile worm wheel. 
     Preferably, at least a tooth of the worm wheel includes a resin molded article. 
     Advantageous Effects of Invention 
     With the present invention, it is possible to decrease face pressure around a base circle. Furthermore, since undercutting of a tooth profile on a tooth bottom side of the base circle can be eliminated, it is possible to make the tooth bottom side of the base circle a meshing face. Accordingly, it is possible to increase a contact ratio without increasing a diameter of a tooth tip of the worm wheel, whereby strength of a worm gear mechanism can be enhanced. 
     Furthermore, since a resin worm wheel has a small elastic modulus, a tooth may be easily bent in the present invention. In a case where a plurality of teeth thereof simultaneously meshes with teeth of the worm, a shared load on the meshed teeth becomes larger as a meshing depth becomes lower. However, it is possible to secure a large contact area in a part where the meshing depth is low, whereby the face pressure can be reduced. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG. 1  is a schematic view of an electric power steering device installed with a worm gear mechanism according to the present invention. 
         FIG. 2  is a view of whole constitution of the electric power steering device in  FIG. 1 . 
         FIG. 3  is a sectional view taken along line  3 - 3  of  FIG. 2 . 
         FIG. 4  is a sectional view taken along line  4 - 4  of  FIG. 2 . 
         FIGS. 5( a ) and 5( b )  are views comparing a worm wheel illustrated in  FIG. 4  with a conventional worm wheel. 
         FIG. 6  is a view illustrating an improvement measure for the conventional worm wheel illustrated in  FIG. 5 . 
         FIG. 7  is a view for formulating a profile of dedendum of the conventional worm wheel illustrated in  FIG. 5 . 
         FIG. 8  is a modelled view for formulating the profile of dedendum of the conventional worm wheel illustrated in  FIG. 5 . 
         FIGS. 9( a )-9( c )  are views illustrating a trochoid curve when a tooth tip arc center point of a hob illustrated in  FIG. 6  is shifted (Shifted Trochoid). 
         FIG. 10  is a view illustrating an envelope of a negative shifted trochoid illustrated in  FIG. 9( a )  and a line of action thereof. 
         FIG. 11  is a view illustrating an envelope of a zero shifted trochoid illustrated in  FIG. 9( b )  and a line of action thereof. 
         FIG. 12  is a view illustrating an envelope of a positive shifted trochoid illustrated in  FIG. 9( c )  and a line of action thereof. 
         FIG. 13  is a view illustrating a reason of a problem in the conventional worm wheel illustrated in  FIG. 5( a ) . 
         FIG. 14  is an enlarged view of a principal part of a worm wheel in which a tooth profile of a tooth root is formed by the positive shifted trochoid illustrated in  FIG. 9( c ) . 
         FIG. 15  is a view illustrating meshing of a worm with a worm wheel using the tooth profile illustrated in  FIG. 14 . 
         FIG. 16  is a view illustrating a state in which a worm rectified based on the worm illustrated in  FIG. 15  is meshed with a worm wheel using the tooth profile illustrated in  FIG. 14 . 
         FIG. 17  is a graph comparing meshing of the worm wheel illustrated in  FIG. 14  with meshing of the conventional worm wheel. 
         FIGS. 18( a ) and ( b )  are views comparing the meshing illustrated in  FIG. 17  in detail. 
         FIGS. 19( a ) and ( b )  are views illustrating a test for verifying performance of the meshing of the worm wheel illustrated in  FIG. 14 . 
         FIG. 20  is a graph illustrating a result of the test illustrated in  FIG. 19 . 
         FIG. 21  is a view illustrating a problem in the tooth profile of the conventional worm wheel illustrated in  FIG. 5( a ) . 
         FIG. 22  is a view illustrating a movement of a hob during gear cutting of the worm wheel illustrated in  FIG. 21 . 
         FIG. 23  is a view illustrating a movement of a tooth tip of the hob illustrated in  FIG. 22 . 
         FIG. 24  is a view illustrating a measure for not forming a narrow part formed in the worm wheel illustrated in  FIG. 21 . 
         FIGS. 25( a ) and ( b )  are views comparing the worm wheel illustrated in  FIG. 21  with the worm wheel having no undercut illustrated in  FIG. 4 . 
         FIG. 26  is a view illustrating a profile of dedendum formed by a positive shifted trochoid illustrated in  FIG. 25( b ) . 
         FIG. 27  is a view supplementing the trochoid curve illustrated in  FIG. 26 . 
         FIG. 28  is a view further supplementing the trochoid curve illustrated in  FIG. 26 . 
         FIG. 29  is a view illustrating meshing of the worm wheel with the worm illustrated in  FIG. 25 . 
         FIG. 30  is a view illustrating a gear cutting tool (hob) for forming the worm wheel illustrated in  FIG. 25 . 
         FIG. 31  is a view illustrating a tooth profile a worm wheel formed by the hob illustrated in  FIG. 30 . 
         FIG. 32  is a view illustrating meshing of the conventional worm wheel (involute profile) illustrated in  FIG. 5( a ) . 
         FIG. 33  is a view illustrating meshing of the worm wheel according to the present invention illustrated in  FIG. 5( b ) . 
         FIG. 34  is an enlarged view of a part  34  in  FIG. 32 . 
         FIG. 35  is an enlarged view of a part  35  in  FIG. 33 . 
         FIG. 36  is a view illustrating rectification of a tooth profile of the worm wheel illustrated in  FIG. 35 . 
         FIG. 37  is a view illustrating a modification of the worm wheel illustrated in  FIG. 36 . 
         FIG. 38  is a view illustrating an amount of rectification necessary for a hob to form the worm wheel illustrated in  FIG. 35 . 
     
    
    
     DESCRIPTION OF EMBODIMENTS 
     An embodiment for carrying out the present invention is described below with reference to the attached drawings. 
     Embodiment 
     An example in which a worm gear mechanism according to an embodiment is installed in an electric power steering device and the electric power steering device is used in a vehicle is described. 
     As illustrated in  FIG. 1 , an electric power steering device  10  includes: a steering system  20  ranging from a steering wheel  21  of a vehicle to steered wheels  29  and  29  (for example, front wheels) of the vehicle; and an auxiliary torque mechanism  40  configured to apply auxiliary torque to the steering system  20 . 
     In the steering system  20 , the steering wheel  21  is coupled to a pinion shaft  24  through a steering shaft  22  and universal shaft couplings  23  and  23 , a rack shaft  26  is coupled to the pinion shaft  24  through a rack and pinion mechanism  25 , and the right and left steered wheels  29  and  29  are coupled to both ends of the rack shaft  26  through right and left tie rods  27  and  27  and knuckles  28  and  28 . 
     The rack and pinion mechanism  25  includes a pinion  31  formed in the pinion shaft  24  and a rack  32  formed in the rack shaft  26 . 
     With the steering system  20 , it is possible to steer the right and left steered wheels  29  and  29  through the rack and pinion mechanism  25  and the right and left tie rods  27  and  27  by a driver steering the steering wheel  21 . 
     The auxiliary torque mechanism  40  is a mechanism in which a steering torque sensor  41  detects steering torque of the steering system  20  applied to the steering wheel  21 . A controller  42  generates a control signal based on a torque detection signal of the steering torque sensor  41 . An electric motor  43  generates the auxiliary torque in accordance with the steering torque based on the control signal. The auxiliary torque is transmitted to the pinion shaft  24  through a worm gear mechanism  44 . Furthermore, the auxiliary torque is transmitted from the pinion shaft  24  to the rack and pinion mechanism  25  in the steering system  20 . 
     The steering torque sensor  41  detects the torque applied to the pinion shaft  24  and outputs it as the torque detection signal. It may be constituted, for example, by a magnetostriction torque sensor or a torsion bar suspension type torque sensor. 
     According to the electric power steering device  10 , it is possible to steer the steered wheels  29  and  29  through the rack shaft  26  by composite torque in which the auxiliary torque of the electric motor  43  is added to the steering torque by the driver. 
     As illustrated in  FIG. 2 , a housing  51  extends in a vehicle width direction (right and left direction in the figure), and slidably houses the rack shaft  26  in a shaft direction. In the rack shaft  26 , the tie rods  27  and  27  are coupled at both ends thereof projecting from the housing  51  in a longitudinal direction through ball joints  52  and  52 . 
     As illustrated in  FIG. 3 , the electric power steering device  10  houses the pinion shaft  24 , the rack and pinion mechanism  25 , the steering torque sensor  41 , and a worm gear mechanism  44  inside the housing  51 , and an upper portion opening of the housing  51  is covered with an upper portion cover portion  53 . The steering torque sensor  41  is attached to the upper portion cover portion  53 . 
     The housing  51  rotatably supports an upper portion  24   a , a longitudinal central portion  24   m , and a lower end portion  24   d  of the vertically extending pinion shaft  24  with three bearings (a first bearing  55 , a second bearing  56 , and a third bearing  57  from the top to the bottom in order). The electric motor  43  is further attached to it, and it is provided with a rack guide  60 . Rolling bearings are used as three bearings  55  to  57 . 
     The rack guide  60  is a rack pressurization unit including a guide portion  61 , which touches the rack shaft  26  from an opposite side of the rack  32 , and an adjustment bolt  63 , which pushes the guide portion  61  through a compression spring  62 . 
     As illustrated in  FIG. 4 , the electric motor  43  is attached to a side face of the housing  51 , and is provided with a horizontal motor shaft (output shaft)  43   a . The motor shaft  43   a  is extended inside the housing  51  and is coupled with a worm shaft  46  by a shaft coupling  45 . The housing  51  rotatably supports both end parts  46   a  and  46   b  of the horizontally extending worm shaft  46  through bearings  47  and  48  while limiting movement in a shaft direction. Both of two bearings  47  and  48  are rolling bearings. 
     The worm gear mechanism  44  is an auxiliary torque transmission mechanism, or a booster mechanism, transmitting the auxiliary torque generated by the electric motor  43  to the pinion shaft  24 . To be more specific, the worm gear mechanism  44  includes a worm  70  and a worm wheel  80 , which meshes with the worm  70 . Hereinafter, the worm wheel  80  is abbreviated as the “wheel  80 ”. Relative to a center line WL of the worm  70 , a center line CL of the wheel  80  is arranged at substantially a right angle. The center line CL of the wheel  80  is also the center line CL of the pinion shaft  24 . 
     The worm  70  is a metal product integrally formed with the worm shaft  46 , and it is, for example, a steel product such as a carbon steel material for mechanical structure (JTS-G-4051). The whole wheel  80  or at least a tooth  81  thereof is a resin product such as of nylon resin. Since the worm  70 , which is the metal product, is meshed with the wheel  80 , which is the resin product, it is possible to make meshing comparably smooth while further reducing noise. 
     A screw thread  71  (or, a tooth  71 ) of the worm  70  is set to be a single thread. On an outer periphery of the wheel  80 , a plurality of teeth  81  having an equal pitch on the entire periphery thereof is formed. The wheel  80  is attached such that relative movement in the shaft direction relative to the pinion shaft  24  is limited, while relative rotation thereof is also limited. For example, the wheel  80  is coupled by a serration or a spline in the rotational direction relative to the pinion shaft  24 , while it is attached by a snap circle in the shaft direction. By meshing the wheel  80  on a load side with the worm  70  on a drive side, it is possible to transmit torque from the worm  70  to the load through the wheel  80 . 
     Various performances are required for this worm gear mechanism  44 . For example, improvement of a contact ratio and enhancement of strength are listed among them. Details are described using the next drawing and after. 
     Firstly, a conventional worm gear mechanism  200  illustrated in  FIG. 21  is described. A tooth profile of a worm wheel  220  of the worm gear mechanism  200  is an involute profile having a tooth tip  221   a , a tooth bottom  221   c , a base circle  301 , and a pitch circle (meshing pitch circle)  302 . On an outer periphery side of the base circle  301 , a tooth thickness is W 2  at a part where a tooth  221  has the maximum thickness. Conventionally, to improve the contact ratio and to enhance the strength of the worm gear mechanism  200 , there has been known a method of increasing a tooth depth HT of the wheel  220  having the involute profile. 
     However, when the worm wheel  220  undergoes gear cutting by a hob, undercutting occurs to a tooth root  221   b . On a side nearer to a center of the wheel  220  than the base circle  301 , a tooth thickness is W 1  at a part where the tooth  221  has the minimum thickness. In this way, since the tooth root  221   b  has a narrow part, the tooth thickness W 1  is smaller than W 2 . As a result, bending strength of the tooth  221  is decreased. Furthermore, the tooth profile of the wheel  220  is a projected shape having a small curvature radius at a part around the base circle  301 . Since it is the projected shape having the small curvature radius, a contact area, which contacts with the worm, decreases. As a result, a meshing contact face pressure increases. That is, when the tooth depth FIT of the wheel  220  having the involute profile is increased, the bending strength and the face pressure strength tend to be decreased. 
     As illustrated in  FIG. 22 , the tooth profile of the wheel  220  is molded by a hob (hob cutter)  230 . A pitch center  231 Ce of a tooth  231  of the hob  230  is positioned at a part of a line  312  (pitch height  312 ). A locus of the pitch center  231 Ce is illustrated by a line  311 . As it is clear from the locus  311 , the tooth  231  of the hob  230  moves so as to roll on a pitch circle  302  of the wheel  220 , and creates (or molds) the tooth profile of the wheel  220 . At this time, a tooth tip  231   a  of the hob  230  gouges out a part lower than the base circle  301  of a tooth surface (a face close to the tooth bottom) of the wheel  220 . 
     As illustrated in  FIG. 23 , a corner portion of the tooth tip  231   a  of the tooth  231  of the hob  230  is shaped into an arc having a predetermined small curvature radius. A locus of a center  231   b  of an arc at a corner portion of the tooth tip  231   a  is denoted by a line  313 . The present inventors have gained knowledge that a face of the tooth root  221   b  is formed into a recessed shape (narrow part shape) by the center  231   b  of the arc moving so as to make a circle. That is, the locus  313  of the center  231   b  of the arc at the corner portion of the tooth tip  231   a  makes a circle at a position on the center side of the base circle  301 , whereby it is considered as a factor of causing an undercut phenomenon (undercutting). A length from the line  312  to the center  231   b  of a bent shape is h. The length h is referred to as an arm length. 
     In order to prevent this undercut phenomenon (undercutting) from occurring, as illustrated in  FIG. 24 , it is possible to decrease a tooth depth HTh of the hob  230 . By decreasing the tooth depth HTh, the locus  313  of the center  231   b  of the arc at the corner portion of the tooth tip  231   a  (a circle made by the center  231   b  of the arc) becomes small on a side nearer to the center of the wheel  220  than the base circle  301 . Accordingly, the undercut phenomenon (undercutting) is more unlikely to be caused. However, when the tooth depth HTh of the hob  230  is decreased, the tooth depth HT of the wheel  220  is also decreased. 
     Back to  FIG. 23 , the locus  313  of the center  231   b  of the arc at the corner portion of the tooth tip  231   a  of the tooth  231  of the hob  230  is considered again. The locus  313  forms a “negative shifted trochoid curve”. The present inventors have considered that the reason why the locus  313  forms the negative shifted trochoid curve is because the center  231   b  of the arc at the corner portion exists on the side nearer to the tooth tip  231   a  than the pitch height  312 . That is, by the locus  313  forming the negative shifted trochoid curve, the corner portion of the tooth tip  231   a  of the hob  230  performs a gear cutting action so as to form a circle on the side nearer to the center than the base circle  301 . A face of the tooth root  221   b  is undercut by the corner portion of the tooth tip  231   a  of the hob  230 . As a result, undercutting occurs on the face of the tooth root  221   b.    
     Next, the conventional wheel  220  illustrated in  FIG. 22  is compared with the wheel  80  according to the embodiment illustrated in  FIG. 4  based on  FIGS. 25( a ) and 25( b ) . 
       FIG. 25( a )  is a schematic view illustrating the pitch circle  302  of the conventional wheel  220  illustrated in  FIG. 22  and the tooth profile of the tooth  231  of the hob  230  for performing gear cutting on the wheel  220 . The tooth  231  of the hob  230  is illustrated to be larger than the wheel  220 . The tooth profile of the tooth  231  of the conventional hob  230  is an involute profile, and the corner portion of the tooth tip  231   a  is formed into an arc shape. The center  231   b  of the arc is positioned on the tooth tip  231   a  side relative to the pitch line  312  of the hob  230  (on a tooth root side of a tooth of the wheel  220 ). In this case, the locus  313  of the center  231   b  forms the negative shifted trochoid curve. 
       FIG. 25( b )  is a schematic view illustrating a pitch circle  112  of the wheel  80  according to the embodiment illustrated in  FIG. 4 , and a tooth profile of a tooth  91  of a hob  90  for performing gear cutting of the wheel  80 . The tooth  91  of the hob  90  is illustrated to be larger than the wheel  80 . The tooth profile of the tooth  91  of the hob  90  according to the embodiment is the involute profile. Note, however, that an addendum surface  91   c  of the tooth  91  is rectified into an arc shape having a large curvature radius. This is to prevent the locus  313  of a center  93  of the arc of the addendum surface  91   c  (hereinafter, referred to as an “addendum surface center  93 ”) from becoming the negative shifted trochoid. Specifically, the center  93  of the addendum surface  91   c  is positioned nearer to a center line (axis line) WL′ of the hob  230  than is a pitch line  94 . Accordingly, the locus  313  of the addendum surface center  93  forms a positive shifted trochoid. That is, it is possible to suppress a conventional locus, which forms a circle, by making it the positive shifted trochoid. 
     The above descriptions can be summarized as below. The tooth  91  of the hob  90  according to the embodiment is formed such that at least the addendum surface  91 C has an arc shape with a large radius of curvature. The center  93  of the radius of the arc of the addendum surface  91 C is positioned nearer to the center line (axis line) WL of the hob  90  than is the pitch line  94  of the hob  90 . 
     It is preferred that the worm  70 , which meshes with the wheel  80 , be formed into a shape similar to that of the hob  90 . That is, at least an addendum surface  71   c  of the tooth  71  of the worm  70  is formed into an arc shape. A center  73  of the radius of the arc of the addendum surface  71   c  is positioned nearer to the center line (axis line) WL of the worm  70  than a pitch line  74  of the worm  70 . 
     As illustrated in  FIG. 26 , the locus of the center  93  of an addendum surface of the wheel  80  according to the embodiment forms the positive shifted trochoid curve as denoted by the line  313 . The tooth  81  of the wheel  80  formed by the hob  90 , which moves along the line  313  (locus  313 ), is formed into a shape having no undercutting in a tooth root  81   c . A reference numeral  81   a  denotes a tooth tip of the tooth  81  of the wheel  80 . A reference numeral  81   b  denotes a tooth bottom of the tooth  81 . 
     Here, the trochoid curve, which is a principle of the present invention, is supplemented based on  FIGS. 27 and 28 . Firstly, with reference to  FIG. 27 , for a rolling circle  402   a  rolling on a fixed circle  401 , a locus of a point (having coordinates X and Y) inside the rolling circle  402   a , or the trochoid curve, is obtained as below. Note that the fixed circle  401  is a circle assuming a pitch circle of the wheel  80 . Reference numeral  402   a  denotes a rolling circle rolling on the fixed circle  401 . Reference numeral  402   b  denotes a rolling circle that has rolled on the fixed circle  401  for a predetermined distance from  402   a . A line  403  denotes a locus of the point (X, Y) inside the rolling circle  402 . That is, the line  403  is the trochoid curve. A reference numeral P 1  denotes a point on the line  403  in the rolling circle  402   a , and is referred to as an arm tip. A reference numeral h is a length (arm length) from the point P 1  to the fixed circle  401 . 
     With reference to  FIG. 27 , for the rolling circle  402 , which rolls on the fixed circle  401 , the locus of the point (X, Y) inside the rolling circle  402   b , or the trochoid curve, is obtained as below. 
     
       
         
           
             [ 
             
               Mathematical 
               ⁢ 
               
                   
               
               ⁢ 
               Formula 
               ⁢ 
               
                   
               
               ⁢ 
               1 
             
             ] 
           
         
       
       
         
           
             
               
                 
                   
                     
                       X 
                       ′ 
                     
                     = 
                     
                       
                         ( 
                         
                           r 
                           + 
                           
                             r 
                             p 
                           
                         
                         ) 
                       
                       ⁢ 
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       θ 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       Y 
                       ′ 
                     
                     = 
                     
                       
                         ( 
                         
                           r 
                           + 
                           
                             r 
                             p 
                           
                         
                         ) 
                       
                       ⁢ 
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       θ 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       X 
                       - 
                       
                         X 
                         ′ 
                       
                     
                     = 
                     
                       
                         - 
                         
                           ( 
                           
                             r 
                             - 
                             h 
                           
                           ) 
                         
                       
                       ⁢ 
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               r 
                               + 
                               
                                 r 
                                 p 
                               
                             
                             r 
                           
                           ⁢ 
                           θ 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       Y 
                       - 
                       
                         Y 
                         ′ 
                       
                     
                     = 
                     
                       
                         - 
                         
                           ( 
                           
                             r 
                             - 
                             h 
                           
                           ) 
                         
                       
                       ⁢ 
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               r 
                               + 
                               
                                 r 
                                 p 
                               
                             
                             r 
                           
                           ⁢ 
                           θ 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     Therefore, 
     
       
         
           
             
               
                 
                   
                     X 
                     = 
                     
                       
                         
                           ( 
                           
                             r 
                             + 
                             
                               r 
                               p 
                             
                           
                           ) 
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         θ 
                       
                       - 
                       
                         
                           ( 
                           
                             r 
                             - 
                             h 
                           
                           ) 
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           { 
                           
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   
                                     r 
                                     p 
                                   
                                   r 
                                 
                               
                               ) 
                             
                             ⁢ 
                             θ 
                           
                           } 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     Y 
                     = 
                     
                       
                         
                           ( 
                           
                             r 
                             + 
                             
                               r 
                               p 
                             
                           
                           ) 
                         
                         ⁢ 
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         θ 
                       
                       - 
                       
                         
                           ( 
                           
                             r 
                             - 
                             h 
                           
                           ) 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           { 
                           
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   
                                     r 
                                     p 
                                   
                                   r 
                                 
                               
                               ) 
                             
                             ⁢ 
                             θ 
                           
                           } 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     With reference to  FIG. 28 , a case in which a radius of the rolling circle (see the reference numeral  402  in  FIG. 27 ) forming the trochoid curve  403  is ∞ is considered.
 
 X′=r   p  sin θ
 
 Y′=r   p  cos θ
 
 X″−X′=r   p θ cos θ
 
 Y″−Y′=r   p θ sin θ
 
 X−X″=−h  sin θ
 
 Y−Y″=h  cos θ  [Mathematical Formula 2]
 
     Therefore,
 
 X =( r   p   +h )sin θ− r   p θ cos θ
 
 Y =( r   p   +h )cos θ+ r   p θ sin θ  (12)
 
     Here, reconsidering Formula (11), 
     
       
         
           
             
               
                 
                   X 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           r 
                           + 
                           
                             r 
                             p 
                           
                         
                         ) 
                       
                       ⁢ 
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       θ 
                     
                     - 
                     
                       
                         ( 
                         
                           r 
                           - 
                           h 
                         
                         ) 
                       
                       ⁢ 
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         { 
                         
                           
                             ( 
                             
                               1 
                               + 
                               
                                 
                                   r 
                                   p 
                                 
                                 r 
                               
                             
                             ) 
                           
                           ⁢ 
                           θ 
                         
                         } 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           r 
                           + 
                           
                             r 
                             p 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       θ 
                     
                     - 
                     
                       
                         ( 
                         
                           r 
                           - 
                           h 
                         
                         ) 
                       
                       ⁢ 
                       
                         { 
                         
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               · 
                               cos 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   
                                     r 
                                     p 
                                   
                                   r 
                                 
                                 ⁢ 
                                 θ 
                               
                               ) 
                             
                           
                           + 
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               · 
                               sin 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   
                                     r 
                                     p 
                                   
                                   r 
                                 
                                 ⁢ 
                                 θ 
                               
                               ) 
                             
                           
                         
                         } 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           r 
                           + 
                           
                             r 
                             p 
                           
                         
                         ) 
                       
                       ⁢ 
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       θ 
                     
                     - 
                     
                       
                         ( 
                         
                           r 
                           - 
                           h 
                         
                         ) 
                       
                       ⁢ 
                       
                         { 
                         
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                           
                           ⁢ 
                           
                               
                           
                           + 
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               · 
                               
                                 
                                   r 
                                   p 
                                 
                                 r 
                               
                             
                             ⁢ 
                             θ 
                           
                         
                         } 
                       
                     
                   
                 
               
             
           
         
       
     
     Substitution of +∞ for γ results in 
                     γ   p     γ     ⁢   θ     =   0     ,         
and then
 
 X =(γ p   +h )sin θ cos θ
 
     Similarly,
 
 Y =( r   p   +h )cos θ sin θ
 
     It is found that it corresponds with Formula (2). 
     This Formula is used in hob cutting. 
     Here, a description is given by comparing the conventional worm gear mechanism  200  with the worm gear mechanism  44  according to the embodiment.  FIG. 29  is a view illustrating a meshing state of the conventional worm gear mechanism  200 . The worm gear mechanism  200  includes a worm  210  and the worm wheel  220 . A tooth profile of each of teeth of the worm  210  and the wheel  220  is the involute profile. An intersection point of a tooth tip of the tooth of the worm  210  and the base circle  301  of the wheel  220  is a first intersection point P 11 . An intersection point, of a pitch line  332  of the worm  210  and the pitch circle  302  of the wheel  220  is a second intersection point P 12 . A straight line passing through the first intersection point P 11  and the second intersection point. P 12  is referred to as a meshing line  321 . An intersection point of the meshing line  321  and a tooth tip circle  305  of the wheel  220  is intersection point P 11  to the third intersection point P 13  is referred to as a “length of path of contact”. 
     The worm  210  and the wheel  220  can be meshed with each other in a range of the length of path of contact on the meshing line  321 . The base circle  301  of the wheel  220  having the involute profile is uniquely determined by a module, the number of teeth, and a twist angle. Therefore, a position of the third intersection point P 13  is uniquely determined as well. In order to make the length of path of contact longer, it is necessary to increase an outside diameter of the wheel  220 . Accordingly, there is a problem in that the worm gear mechanism  200  cannot be downsized. 
     Furthermore, in a case where the conventional worm gear mechanism  200  is used in an electric power steering device for a vehicle, a resin material is often used for the tooth  221  of the wheel  220 . In the wheel  220  using the resin material, elastic modulus of the material is small, whereby the tooth  221  is easily bent. In a case where a plurality of teeth  221  simultaneously meshes with each other, the lower a meshing depth is, the larger a shared load on the meshing tooth  221  is. That is, the load applied on each of the teeth  221  becomes larger. 
     Furthermore, in the involute profile, the curvature radius becomes smaller as it gets closer to the base circle  301 . A meshing face pressure around the base circle  301  is very large compared to the meshing face pressure around the pitch circle  302 . Accordingly, there is a problem in that it is difficult to extend the meshing line  321  nearer to a side of a wheel center than the base circle  301 . 
     A tooth profile of the tooth  231  of the conventional hob represented by an imaginary line in  FIG. 30  is an involute shape in which a meshing tooth surface is a straight line. In contrast, in a tooth profile of the tooth  91  of the hob  90  of the embodiment represented by a solid line in  FIG. 30 , a part of an addendum surface  910  is made to be thinner than that in the involute shape. Specifically, a part of the addendum surface  91   c  of the tooth  91  of the hob  90  according to the embodiment is decreased in tooth thickness to be a substantially arc shape abutting on an involute curve. 
     The tooth  221  of the conventional wheel  220  represented by an imaginary line in  FIG. 31  has been gear cut by the conventional hob  230  (see  FIG. 30 ). In the conventional tooth  221 , the undercut phenomenon (undercutting) occurs to a dedendum surface thereof. As a result, a tooth surface of the tooth  221  forms a remarkably projected shape around the base circle  301 . 
     In contrast, a tooth  81  of the wheel  80  of the embodiment represented by a solid line in  FIG. 31  is gear cut by the hob  90  of the embodiment. The tooth thickness of the tooth  91  of the hob  90  is thin. No undercut phenomenon (undercutting) occurs to a dedendum surface of the tooth  81  of the wheel  80 . As a result, a tooth surface of the tooth  81  does not form a projected shape around a base circle  111 . It is possible to decrease a face pressure acting on the tooth surface of the tooth  81 . 
       FIG. 32  is a view illustrating a meshing state of the conventional worm gear mechanism  200 , and it is illustrated in correspondence with the above-described  FIG. 29 . As denoted by an outlined arrow in  FIG. 34 , the worm  210  meshes with the wheel  220  around the base circle  301  (on the meshing line  321 ). 
       FIG. 33  is a view illustrating a meshing state of the worm gear mechanism  44  according to the embodiment, and it is illustrated in correspondence with the above-described  FIG. 29 . As denoted by an outlined arrow mark in  FIG. 35 , the worm  70  meshes with the wheel  80  at a position nearer to the tooth bottom than the base circle  111 . A reference numeral  121  denotes a meshing line of the wheel  80  and the worm  70 . 
       FIG. 36  is a view illustrating a change in the tooth profile of the tooth  81  of the wheel  80  by changing an amount of rectification of the tooth thickness of a tooth of the hob. The tooth  221  of the conventional wheel  220  represented by an imaginary line in  FIG. 36  has a recessed dedendum surface. That is, undercutting is caused to the dedendum surface. This is because a tooth  231  of the hob  230  (see  FIG. 23 ) is not rectified at all. 
     In contrast, in the embodiment, as illustrated in  FIG. 2500 , the tooth  91  of the hob  90  has been rectified. The tooth profile of the tooth  81  of the wheel  80  in a case where the amount of rectification of the tooth  91  is small is represented by a thin solid line in  FIG. 36 . A dedendum tooth thickness of the tooth  81  is larger than before. The tooth profile of the tooth  81  of the wheel  80  in a case where the amount of rectification of the tooth  91  is large is represented by a thick solid line in  FIG. 36 . The dedendum tooth thickness of the tooth  81  is even larger. 
     In this way, as the amount of rectification of the tooth  91  of the hob  90  becomes larger, the recess on the tooth  81  of the dedendum surface of the wheel  80  decreases, and the dedendum tooth thickness becomes larger as well. Furthermore, a curvature radius of the tooth surface of the tooth  81  around the base circle  111  becomes larger. That is, the tooth surface of the tooth  81  does not form a large projected shape around the base circle  111  as before. 
     The tooth  81  of the wheel  80  according to the above-described embodiment can also be a tooth  81 X of a wheel  80 X according to a modification illustrated in  FIG. 37 . In the tooth  81 X of the wheel  80 X according to the modification, at least a part of a dedendum tooth thickness is set to be larger than a dedendum tooth thickness of the tooth  221  of the conventional wheel Therefore, with the tooth  81 X according to the modification, it is possible to obtain an equal effect as the tooth  81  according to the embodiment. To be more specific, the tooth profile of the tooth  221  of the conventional wheel  220  is represented by an imaginary line in  FIG. 37 . The tooth profile of the tooth  81  of the wheel  80  according to the embodiment is represented by a thin solid line in  FIG. 37 . The tooth profile of the tooth  81 X of the wheel  80 X according to the modification is represented by a thick solid line in  FIG. 37 . 
     The tooth profile of the tooth  81 X according to the modification, for example, is formed to be an intermediate shape of the tooth profile of the conventional tooth  221  and the tooth profile of the tooth  81  according to the embodiment. For example, a dedendum height of the tooth  81  according to the embodiment is the same as a dedendum height of the conventional tooth  221 . However, the dedendum height of the tooth  81 X according to the modification is smaller than the dedendum height of the tooth  81  according to the embodiment. Furthermore, the dedendum tooth thickness of the tooth  81 X according to the modification is larger than the dedendum tooth thickness of the conventional tooth  221 , but is smaller than the dedendum tooth thickness of the tooth  81  according to the embodiment. Note, however, that there is no recess in the dedendum surface of the tooth  81 X according to the modification. 
     The tooth  81 X according to the modification has a special tooth profile, whereby it cannot be manufactured by a machine for creating an involute profile such as a hobbing machine; however, it can be directly created by injection molding using a metal mold or by milling. That is, in the embodiment, face pressure strength and bending strength of the tooth  81  is enhanced by an indirect method of creating the tooth  81  of the wheel  80  by a hob having a rectified tooth thickness. In contrast, in the modification, the tooth  81 X can be created directly to enhance the face pressure strength and the bending strength of the tooth  81 X. Accordingly, the tooth profile of the tooth  81 X to be obtained can be designed directly and finely. Therefore, it is possible to further improve the tooth  81  according to the embodiment. For example, it is possible to finely change a gear tooth depth, a curvature radius at a tooth bottom, and a tooth thickness. 
     In creating the tooth  81  of the wheel  80  according to the embodiment illustrated in the above-described  FIG. 26 , an amount of rectification δ of the tooth  91  of the hob  90  minimally required for not undercutting (not causing undercutting of) the tooth root  221   b , or a minimum amount of rectification δ, can be obtained as below (see  FIG. 38 ). That is, describing with reference to  FIG. 38 , the minimum amount of rectification δ of the tooth  91  of the hob  90  can be obtained by the following Formula (8). Note, however, that the tooth profile of the wheel  80  is based on an involute profile Tim. The wheel  80  rotates in a rotary moving direction Rr (clockwise direction Rr in the drawing). The tooth  91  of the hob  90  moves in parallel relative to a pitch line Lhp (moving direction Ds). The tooth profile of the tooth  91  of the hob  90  is denoted by a line Hc. An intersection point of an involute line of action Lia of the tooth  81  of the wheel  80  and the involute profile Tim of the tooth  81  is a cutting point Ps. A point where undercutting of the tooth  81  of the wheel  80  is started by the hob  90 , or an undercutting point, is denoted by Pr. 
     An intersection point of the involute line of action Lia of the tooth  81  of the wheel  80  and the pitch line Lhp of the tooth  91  of the hob  90  is denoted by Px. A straight line from a center CL of the wheel  80  to the intersection point PX is denoted by a standard line Lp. An intersection point of a tooth surface Th 1  of the hob  90  of the tooth  91  rectified for the minimum amount of rectification δ only and the base circle  111  of the tooth  81  of the wheel  80  is denoted by Py. A straight line passing through the center CL of the wheel  80  and the intersection point Py is denoted by a rectified standard line Lt. A tilt angle (rectification angle) of the rectified standard line Lt relative to the standard line Lp is denoted by θ. The rectification angle is larger than a pressure angle α of the tooth  81  of the wheel  80  (tooth  91  of the hob  90 ) as a condition (θ&gt;α). 
     m: a module of the wheel  80   
     Z: the number of teeth of the wheel  80   
     Rb: a radius of the base circle  111  of the wheel  80   
     Rp: a radius of the pitch circle  112  of the wheel  80   
     Rp−Rb·cos θ: height from the pitch line Lhp of the tooth  91  of the hob  90  to the intersection point (θ&gt;α). 
     
       
         
           
             
               
                 
                   [ 
                   
                     Mathematical 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     Formula 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     3 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   δ 
                   = 
                   
                     
                       
                         R 
                         b 
                       
                       ⁢ 
                       
                         { 
                         
                           
                             ( 
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 α 
                               
                               - 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 θ 
                               
                             
                             ) 
                           
                           - 
                           
                             tan 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               α 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     α 
                                   
                                   - 
                                   
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     θ 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         } 
                       
                     
                     + 
                     
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       m 
                       ⁢ 
                       
                         
                           
                             ( 
                             
                               θ 
                               - 
                               α 
                             
                             ) 
                           
                           ⁢ 
                           Z 
                         
                         
                           2 
                           ⁢ 
                           π 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     In this way, in this embodiment, an area in which the tooth profile is rectified by the hob  90 , which cuts an involute profile, is a tooth surface in a range where a height from the pitch line Lhp of the hob  90  toward a tooth tip direction, or a height from the pitch line Lhp of the tooth  91  of the hob  90  to the intersection point Py of the hob  90  is Rp−Rb·cos θ or above. Then, the tooth  91  is rectified in a direction of decreasing a tooth thickness thereof by the minimum amount of rectification δ or more at the intersection point Py where a height from the pitch line Lhp is Rp−Rb·cos θ. 
     The tooth surface of the tooth  91  rectified by the minimum amount of rectification δ is represented by the curve Th 1 . In this case, the involute line of action Lia of the tooth  81  of the wheel  80  is extended to a line of action L 1  on the base circle  111 . Furthermore, a tooth surface in a case where the amount of rectification of the tooth  91  is larger than the minimum amount of rectification δ is represented by a curve Th 2 . In this case, the involute line of action Lia of the tooth  81  of the wheel  80  is extended to a line of action L 2 , which is inside of the base circle  111 . In this way, it is possible to enhance the bending strength of the tooth  81  of the wheel  80 . 
     A theory on which the present invention is based is disclosed below. Note that a constituent element common with that in the above-described embodiment is denoted with the same reference numeral, and a description thereof is omitted. 
     A major problem in a conventional study has been a study of a tooth profile of an optimal worm for actively causing elastic deformation of a worm wheel. Accordingly, there has been still a room for improvement of a tooth profile of the worm wheel. 
     In a conventional designing method, in order to highly strengthen the worm wheel, a module and a twist angle are increased to geometrically improve a contact ratio. Using this method, it has been necessary to simply increase a diameter of the worm wheel in size. 
     In order to overcome this problem, the present inventors have tried to highly strengthen a small-sized worm wheel. In the present invention, the present inventors have further tried to downsize the worm wheel, and newly focused on a profile of dedendum of the worm wheel. As a result, they have reached an idea of improving the contact ratio by effectively meshing the worm even under a base circle of the worm wheel. In order to embody this, first, consideration has been given to a geometrical shape even under the base circle formed in actual processing. Based on the consideration, a theory of effectively meshing even under the base circle is referred to as the MUB (Meshing Under Base circle) theory. The MUB theory is proposed herein. 
     The tooth profile of the tooth  221  of the conventional worm wheel  220  illustrated in  FIG. 5( a )  is an involute profile. On a tooth surface of the tooth  221 , there is formed an undercut part U, which is undercut by the conventional hob  230 . 
     The tooth profile of the tooth  81  of the worm wheel  80  according to the embodiment illustrated in  FIG. 5( b )  is a new profile formed by the MUB theory. Based on the MUB theory, the worm wheel  80  (hereinafter, referred to as the wheel  80 ) is actually manufactured, and an effect of the MUB theory has been verified by measuring the length of path of contact. Knowledge obtained in a course of this study is reported herein. 
     Proposal of the MUB theory in which the worm is meshed even under the base circle of the wheel  80   
     In  FIG. 6 , a worm gear mechanism in which the conventional worm is meshed with a wheel is illustrated (Contact Line of Worm Tooth Tip Corner Radius, Contact Line of Involute Worm Wheel). Conventionally, the contact ratio has been improved by increasing the wheel  220  in size and by extending the meshing line  321  in a direction of the tooth tip  221   a . In the present invention, in reverse thinking, the present inventors have considered that if the meshing line  321  can be extended in a direction of the tooth root  221   b , it is possible to improve the contact ratio without increasing it in size. To realize this, studies have been made for a new tooth profile that favorably meshes even under the base circle  301 . 
     In order for the tooth  221  having an involute profile to mesh under the base circle  301 , it is necessary to undercut the wheel  220  by the hob (reference numeral  230  in  FIG. 5 ), and to mesh it with the worm having no bottom clearance (see  FIG. 16 ). However, by meshing simply in this method, the tooth tip  211   a  of the worm  210  only contacts the wheel  220 , whereby it is not possible to obtain an effective meshing. 
     Analysis of a conventional profile of dedendum not capable of meshing under the base circle 
     A locus of the hob cutter is illustrated in  FIG. 7 . As illustrated in  FIG. 7 , the profile of dedendum formed by the conventional involute hob  230  is formulated and analyzed. An ideal tooth profile that effectively meshes under the base circle  301  is sought after by analyzing. When the tooth  221  of the wheel  220  is fixed to an absolute coordinate system and is gear cut, the datum line  312  of the hob  230  rolls on a gear cutting pitch circle of the wheel  220  without slipping. A center  231 Ce of the tooth (edge)  231  of the hob  230  makes an epitrochoid curve  311 . An envelope, which is formed when the hob  230  moves along the epitrochoid curve  311 , forms the tooth profile of the tooth  221  of the wheel  220 . In particular, the shape of the tooth root  221   b  under the base circle  301  is formed by the tooth tip  231   a  of the hob  230 . 
     In the drawing, a working point (Hob Cutter Working Point) of the hob  230  is denoted by WP. A reference numeral  307  denotes an involute profile portion of the wheel  220 . A reference numeral  308  denotes a dedendum portion (Dedendum Formed by Corner Radius) of the wheel  220 . 
     In  FIG. 8 , an envelope of the hob tooth (Envelope of Hob Tooth Tip) is illustrated. As illustrated in  FIG. 8 , modeling for formulating a profile of dedendum has been performed. First, the line  313  made by a tooth tip arc center T of the hob  230  is obtained. Next, an envelope  314 , when a circle having a radius rh moves on the line  313 , is obtained. 
     When the datum line  312  of the hob  230  contacts a phase point B 1  of θ on the pitch circle  302  of the wheel  220 , a line segment A 1 B 1  is an unwound arc A 0 B 1 , and since both lengths are equal, coordinates (X, Y) of the tooth tip arc center T of the hob  230  can be expressed as Formulas (1) and (2) using θ as a variable. 
     [Mathematical Formula 4]
 
 X =( R   p   +h )sin θ− R   P θ cos θ  (1)
 
 Y =( R   p   +h )cos θ+ R   p θ sin θ  (2)
 
     Next, the envelope  314  of the tooth tip arc center is obtained. A point E on the envelope  314  is on a normal line  315  of the line  313  (trochoid curve), which passes through a point T. Since a distance TE corresponds to a hob tooth tip radius rh, it can be expressed as Formulas (3) to (5). 
     
       
         
           
             
               
                 
                   [ 
                   
                     Mathematical 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     Formula 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     5 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       ⅆ 
                       X 
                     
                     
                       ⅆ 
                       θ 
                     
                   
                   = 
                   
                     
                       h 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       θ 
                     
                     + 
                     
                       
                         R 
                         p 
                       
                       ⁢ 
                       θ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       θ 
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ⅆ 
                       Y 
                     
                     
                       ⅆ 
                       θ 
                     
                   
                   = 
                   
                     
                       
                         - 
                         h 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       θ 
                     
                     + 
                     
                       
                         R 
                         p 
                       
                       ⁢ 
                       θ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       θ 
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           ( 
                           
                             
                               ⅆ 
                               X 
                             
                             
                               ⅆ 
                               θ 
                             
                           
                           ) 
                         
                         2 
                       
                       + 
                       
                         
                           ( 
                           
                             
                               ⅆ 
                               Y 
                             
                             
                               ⅆ 
                               θ 
                             
                           
                           ) 
                         
                         2 
                       
                     
                   
                   = 
                   
                     
                       
                         h 
                         2 
                       
                       + 
                       
                         
                           R 
                           p 
                           2 
                         
                         ⁢ 
                         
                           θ 
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     Accordingly, the point E (X′, Y′) on the envelope can be expressed as Formulas (6) and (7). 
     
       
         
           
             
               
                 
                   [ 
                   
                     Mathematical 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     Formula 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     6 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         
                           X 
                           ′ 
                         
                         = 
                           
                         ⁢ 
                         
                           X 
                           + 
                           
                             
                               
                                 r 
                                 h 
                               
                               
                                 
                                   
                                     h 
                                     2 
                                   
                                   + 
                                   
                                     
                                       R 
                                       p 
                                       2 
                                     
                                     ⁢ 
                                     
                                       θ 
                                       2 
                                     
                                   
                                 
                               
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   ⅆ 
                                   Y 
                                 
                                 
                                   ⅆ 
                                   θ 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               ( 
                               
                                 
                                   R 
                                   p 
                                 
                                 + 
                                 h 
                               
                               ) 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                           
                           - 
                           
                             
                               R 
                               p 
                             
                             ⁢ 
                             θ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                           
                           ⁢ 
                           
                               
                           
                           - 
                           
                             
                               r 
                               h 
                             
                             
                               
                                 
                                   h 
                                   2 
                                 
                                 + 
                                 
                                   
                                     R 
                                     p 
                                     2 
                                   
                                   ⁢ 
                                   
                                     θ 
                                     2 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           ( 
                           
                             
                               h 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                             
                             - 
                             
                               
                                 R 
                                 p 
                               
                               ⁢ 
                               θ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           Y 
                           ′ 
                         
                         = 
                           
                         ⁢ 
                         
                           Y 
                           + 
                           
                             
                               
                                 r 
                                 h 
                               
                               
                                 
                                   
                                     h 
                                     2 
                                   
                                   + 
                                   
                                     
                                       R 
                                       p 
                                       2 
                                     
                                     ⁢ 
                                     
                                       θ 
                                       2 
                                     
                                   
                                 
                               
                             
                             ⁢ 
                             
                               ( 
                               
                                 - 
                                 
                                   
                                     ⅆ 
                                     X 
                                   
                                   
                                     ⅆ 
                                     θ 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               ( 
                               
                                 
                                   R 
                                   p 
                                 
                                 + 
                                 h 
                               
                               ) 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                           
                           ⁢ 
                           
                               
                           
                           + 
                           
                             
                               R 
                               p 
                             
                             ⁢ 
                             θ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                           
                           - 
                           
                             
                               r 
                               h 
                             
                             
                               
                                 
                                   h 
                                   2 
                                 
                                 + 
                                 
                                   
                                     R 
                                     p 
                                     2 
                                   
                                   ⁢ 
                                   
                                     θ 
                                     2 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           ( 
                           
                             
                               h 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                             
                             + 
                             
                               
                                 R 
                                 p 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     By using the above Formulas (6) and (7), a trochoid curve in a case where the tooth tip arc center point of the hob  230  is shifted is illustrated in  FIGS. 9( a )  to  9 (C). The trochoid curve changes with a shifting direction based on the datum line of the hob  230 . A negative shifted trochoid in  FIG. 9( a )  is a curve forming a circle. A zero shifted trochoid in  FIG. 9( b )  is a substantially V-shaped curve having a corner portion at an intersection point with the pitch circle  302 . A positive shifted trochoid in  FIG. 9( c )  is a substantially V-shaped curve having both a recessed shape and a low curvature projected shape. 
     Next, an envelope formed by each of the lines  313  is considered in  FIG. 10 , an envelope by the negative shifted trochoid and a line of action of a gear using the envelope as a tooth profile are illustrated (Meshing of Gears Formed by Negative Shifted Trochoid). In  FIG. 10 , a horizontal axis corresponds to a tooth thickness direction, and a vertical axis corresponds to a tooth tip direction. 
     A line of action  316  can be extended nearer to a center side than the base circle  301 . However, a pressure angle of a contact point. P 5  reaches 75 degrees (see P 6 ) and increases up to around 90 degrees. Accordingly, the worm is self-locked and becomes not rotatable (see SL). On the other hand, when a bottom clearance is provided in the worm to avoid this, it does not contact geometrically. In the drawing, PP denotes a pitch point. The pitch point is a point through which a normal line of a tooth surface at a gear meshing contact point always passes. A line  317  is a worm tooth profile (Worm Profile). 
     In  FIG. 11 , an envelope by the zero shifted trochoid and a line of action thereof is illustrated (Meshing of Gears Formed by Zero Shifted Trochoid). In  FIG. 11 , a horizontal axis corresponds to a tooth thickness direction, and a vertical axis corresponds to a tooth tip direction. 
     The envelope  314  under the base circle  301  has an arc shape Novikov tooth profile. Accordingly, a transverse contact ratio becomes less than 1, whereby it is not possible to satisfy an isokinetic, which is a mechanical condition of the gear. In order to transmit constant speed rotation, it is necessary to realize an overlap ratio of 1 or more by a multi-row worm, whereby the wheel is increased in size. In the drawing, MS denotes a simultaneous meshing (Meshing Simultaneously) area. 
     In  FIG. 12 , an envelope by the positive shifted trochoid and a line of action thereof is illustrated (Meshing Of Gears Formed by Positive Shifted Trochoid). In  FIG. 12 , a horizontal axis corresponds to a tooth thickness direction, and a vertical axis corresponds to a tooth tip direction. 
     The line of action  316  can be extended nearer to the center side than the base circle  301 . The normal line  315  of the envelope  314  at a contact point always passes through the pitch point PP, whereby it satisfies a mechanical condition of the gear, and it is possible to mesh effectively. 
     In  FIG. 13 , a profile of dedendum is illustrated. As illustrated in  FIGS. 12 and 13 , based on the above consideration, it is found that the conventional involute profile is unable to mesh under the base circle  301  because the profile of dedendum is formed by the negative shifted trochoid. 
     Proposal of the MUB Theory 
     Based on the above-described consideration, if the addendum tooth profile of a wheel can be formed by the positive shifted trochoid, it is possible to obtain a tooth profile that can effectively mesh even under the base circle. In order to achieve the positive shifted trochoid, a hob tooth tip (addendum) arc radius may be increased, and a center point of the arc may be shifted in a positive direction of the datum line of the hob. 
     In  FIG. 14 , a wheel having a dedendum tooth profile formed by the positive shifted trochoid is illustrated (MUB Profile of Worm Wheel). Meshing of a worm using the tooth profile is illustrated in  FIG. 15 . It has a conventional involute meshing line (add a reference numeral in the drawing) on a face of the wheel tooth tip  81   a , and on an addendum surface thereof, it is possible to mesh to under the base circle  111  along a line of action of the positive shifted trochoid. These two lines of action (add a reference numeral in the drawing) are linked smoothly, and the gear mechanical conditions are satisfied in all contact areas, whereby it is confirmed that effective meshing can be obtained by this new tooth profile. 
     As illustrated in  FIG. 15 , with the new tooth profile, a length of recess path can be extended from a conventional limit LOA to LOAmod, whereby it is indicated that the contact ratio can be improved from that of the involute profile. Here, the length of recess path refers to a length of path of contact from a pitch point to around a tooth tip of a worm. 
     This meshing theory by which it is possible to effectively mesh even under the base circle  111  is named the Meshing Under Base circle (MUB) theory. 
     Meshing Considering Elastic Deformation of the Wheel 
     Up to here, the wheel has been regarded as a rigid body in consideration. Based on the study so far, it is expected that the length of path of contact can be further extended considering elastic deformation of the wheel, whereby consideration is made in order to study this effect. 
     Considering the elastic deformation of the wheel, as in  FIG. 15 , a shared load concentrates on a tooth  81 A having a low meshing depth. Therefore, conventionally a tooth thickness of a worm tooth tip face is modified into a negative direction to distribute the concentrated load to other meshing teeth  81 B and  81 C. 
     Using the tooth  81 , it is possible to move an actual meshing line of action when a torque is applied in a direction of the pitch circle  112  of the wheel. The rectified worm meshed with the wheel  80  based on the MUB theory is illustrated in  FIG. 16 , and since a tilt of the meshing line of action  121  can be decreased, it is possible to further extend the length of path of contact from the result in the previous chapter, (see L). It is possible to design the contact ratio, which has been 2.2 before, to be 3.0 or above without increasing the wheel in size. 
     Next, a meshing contact area is considered. In  FIG. 17 , there is a comparison between the conventional involute profile and a tooth profile based on the MUB theory. A horizontal axis is a contact height of the wheel (Contact Height of Worm Wheel). A vertical axis is a contact height based on the pitch line (Contact Height Above Pitch Line). To the top of the drawing is an addendum direction, and to the bottom of the drawing is a dedendum direction. A line  341  connecting points plotted with black rhombuses denotes a conventional result. A line  342  connecting points plotted with white circles is a result by the MUB theory. A line  343  is a line indicating a meshing depth (Base Line) of the base circle. An area indicated with right down oblique lines is a contact area in both of the conventional and MUB theories. An area indicated with left down oblique lines is a contact area by the MUB theory only. By adopting a wheel by the MUB theory, it is possible to obtain the contact area in a wider area. 
     As illustrated in  FIG. 18( a ) , with the involute profile wheel  220 , it is difficult to increase the meshing area to under the base circle  301  due to undercutting. Since the undercutting is not caused in the wheel  80  based on the MUB theory as illustrated in  FIG. 18( b ) , it, is possible to favorably enlarge the meshing area to under the base circle  111 . 
     Verification of Test of Meshing by the MUB Theory 
     To verify meshing performance of the worm designed by the proposed MUB theory, after a change in meshing is calculated according to a worm phase, the worm  70  is actually manufactured and tooth bearing during meshing is verified. 
     As illustrated in  FIGS. 19( a ) and ( b ) , as the verification method, a blue paste BP is applied on a tooth surface of the worm  70 , which is meshed with the wheel  80 . Then, torque is applied to the worm  70 , and a shape of an area where the blue paste BP has been peel off is measured. 
     In  FIG. 19( a ) , the worm  70  on which the blue paste BP is applied is illustrated. In the drawing, S denotes a point where contact with the wheel  80  is started (Start Point of Mesh). A reference numeral E denotes a point where the contact with the wheel  80  is ended (End Point of Mesh). A reference numeral CAa denotes a part where the blue paste BP has been peeled off, or a part that has contacted the wheel  80  (Contact Area of Worm). 
     In  FIG. 19( b ) , there is illustrated the wheel  80  that has meshed with the worm  70 . In the drawing, CAb is a part where the blue paste BP has attached, or a part that has contacted the worm  70  (Contact. Area of Worm Wheel). It is found that the worm  70  has contacted nearer to the center side of the base circle  111 . 
     These verification results are illustrated in  FIG. 20 . In  FIG. 20 , a contact area of the worm is illustrated. A horizontal axis represents a rotation angle of the worm. A vertical axis represents a contact height. A rhombuses shape plotting indicates a measurement result in a case where the involute profile wheel is used (Actual Measurement of involute gear). A triangular shape plotting indicates a calculation value in a case where the wheel according to the embodiment is used (Calculated Point). A round shape plotting indicates a measurement result in a case where the wheel according to the embodiment is used (Actual Measurement). Outside the points plotted with the rhombuses shape and inside the points plotted with the round shape, or a hatched area, corresponds to an area in which the meshing area has been enlarged by the wheel according to the embodiment. 
     Based on a phase of the worm at which meshing geometrically starts, a worm rotation direction in a case where meshing progresses from a dedendum to a tooth tip direction is set as a positive direction. Note that a rotation angular velocity of the wheel is set to 1.0 rps, and input torque to the worm is set to 3.2 Nm in the verification. 
     Since the meshing area of the worm tooth surface corresponds to about 1080 degrees of the worm rotation phase, it is verified that the contact ratio becomes 3.0. The contact ratio is increased by 36% compared to 2.2 of the conventional tooth profile. Furthermore, it is confirmed that the meshing area of the wheel is favorably increased to under the base circle. 
     These substantially correspond with a result of theory consideration, whereby effect of the MUB theory can be verified. Accordingly, it is now possible to predict meshing of a worm gear mechanism designed based on the MUB theory. Therefore, the MUB theory is effective as a designing method of an electric power steering device (EPS) in which installation of a small-sized and high strength worm gear mechanism is required. 
     In order to downsize the wheel, there has been proposed the MUB theory in which the contact ratio is improved by effectively meshing even under the base circle, and an effect of the theory has been verified through a test. As a result, the following has become found. 
     It has been found that the profile of dedendum of the wheel can be categorized into three types according to a shifting direction of the tooth tip arc center point of the hob. The tooth profile formed by the negative shifted trochoid is self-locked, whereby it is found that it cannot mesh effectively under the base circle. The tooth profile formed by the zero shifted trochoid has an are tooth profile under the base circle, whereby a multi-row worm is necessary in order to satisfy the isokinetic of a gear, and it is found that the wheel is increased in size. By the MUB theory in which a dedendum tooth profile is formed by the positive shifted trochoid, it is possible to avoid increasing the wheel size and to effectively mesh under the base circle, with which it has not been possible to mesh with the involute profile. It has been proven that, by applying the MUB theory, it is possible to achieve a high contact ratio of 3.0 even with a single-row worm, which is generally considered to have a low contact ratio. 
     Descriptions have been given based on an example of installing the worm gear mechanism in an electric power steering device; however, it is also possible to install the worm gear mechanism in other apparatus, and it is not to be limited to the electric power steering device. 
     INDUSTRIAL APPLICABILITY 
     The worm gear mechanism according to the present invention is particularly suitable for use on an electric power steering device of a vehicle. 
     REFERENCE LIST 
       44  worm gear mechanism,  70  worm,  71  worm tooth,  71   c  worm addendum surface.  74  worm pitch line,  80  worm wheel,  90  hob,  91  hob tooth,  91   c  hob addendum surface,  93  hob addendum surface center,  94  hob pitch line, WL worm center line,  210  involute profile worm,  220  involute profile worm wheel,  200  conventional worm gear mechanism, WL′ hob center line, L length of recess path, Llim conventional length of recess path.