Abstract:
A homokinetic coupling is disclosed for coupling two rotatable shafts each having an axis of rotation such that the ratio of the speeds of rotation of said shafts about their respective axes is maintained constant even as said axes undergo relative time-varying re-orientations.

Description:
BACKGROUND OF THE INVENTION 
     Couplings used to connect shafts are required for many purposes. A brief list of examples could include frontwheel drives of automobiles, automatic assembly and processing machinery, certain machine tools, precision instruments, and automatic control devices. The literature dealing with this subject in general is extensive and a comprehensive list of references in which the subject is discussed in its various aspects may be found in a recent book by Dudita, Dudita Fl. Cuplaje mobile homocinetice. Editura Teknica, Bucharest, 1974, pp. 226-228. 
     To perform satisfactorily in many of the above and other applications, a coupling should be constructed in such a way that it readily permits changes in relative shaft orientation during operation and maintains the ratio of input to output shaft speed constant for all input shaft speeds and all relative shaft orientations lying in a certain range. Thus, a coupling should not only insure a constant speed ratio in various orientations, but also insure a constant speed ratio during a change from one relative orientation to another. 
     An example of a coupling in which the ratio of the speeds of rotation of a pair of shafts about their respective axes varies as the axes undergo relative time-varying reorientations is a Hooke coupling. Probably the most widely known use of the Hooke coupling is the use to which it is put as a universal joint which couples a pair of shaft members in the drive line of an automobile and other motor vehicles. 
     In many motor vehicles the axes of the shaft members coupled by the joint may be considered as being nearly collinear -- that is to say, an angle θ which the axis of one shaft member makes with the axis of the other shaft member is very small. Under these conditions, the ratio of the speeds of rotation, γ, of the shaft members about their respective axes is given, for a Hooke coupling, by the equation 
     
         γ ≈ 1 + (θ/ω) θ sin φ cos φ 
    
     where ω is the angular speed of one of the shafts, φ is the angular displacement of this shaft about its axis of rotation with respect to a given reference plane in which both shaft axes are fixed, and θ is the time derivative of θ. From the foregoing equation, it can be seen that sufficiently large values of θ can give rise to appreciable fluctuations in the speed ratio, γ. Such values of θ may be encountered when a vehicle traverses a bumpy road at relatively high speed. They may also be encountered in equipment using such couplings which is subject to high frequency vibrations such as equipment used in aircraft, ships, rockets and the like. 
     SUMMARY OF THE INVENTION 
     In view of the foregoing, a principal object of the present invention is a shaft coupling for coupling a pair of shafts D and N such that the ratio of the speeds of rotation of said shafts about their respective axes of rotation is maintained constant even as said axes undergo relative time-varying re-orientations. 
     A component of the coupling is a so-called differential mechanism. The mechanism is provided as a subsystem of the coupling and comprises a rigid housing, A. One of the pair of shafts, D and N - namely, D - which the coupling is intended to join is rotatably fitted by means of the housing and a plurality of gear members to a pair of internal coaxial shaft members, B and C, for bringing about a certain relationship between the angular speeds of the three shafts, B, C and D, in the reference frame A such that 
     
         .sup.A ω.sup.D  = - R.sub.1 (.sup.A ω.sup.B + .sup.A ω.sup.C) 
    
     where R 1  is a constant. The nomenclature  A  ω D ,  A  ω B  and  A  ω C  follow the generalized form  V  ω U  which is defined as the angular velocity of a body (shaft) U in a body (reference frame) V. 
     The shafts B and C are also rotatably supported by another rigid frame member, M, and, by means of another plurality of gears, are rotatably fitted to the other of the shafts, D and N - namely, N. Certain ones of the latter plurality of gears are provided to have predetermined pitch radii and to mesh for providing the constant speed ratio for the shafts D and N as described above. 
    
    
     DESCRIPTION OF THE DRAWINGS 
     The above and other objects, features and advantages of the present invention will become apparent from the following detailed description of the accompanying drawings in which 
     FIG. 1 is a diagram of coordinates and reference frames of a generalized coupling. 
     FIG. 2 is a diagrammatic representation of a Hooke coupling. 
     FIG. 3 is a cross-sectional view of a differential mechanism constituting a subsystem of the present invention. 
     FIG. 4 is a cross-sectional view of an embodiment of the present invention including the sybsystem of FIG. 3. 
    
    
     DETAILED DESCRIPTION 
     Referring to FIG. 1, the most general coupling of the kind to be considered can be discussed by reference to FIG. 1, where S and S&#39; represent portions of the two shafts to be connected by a coupling (not shown); X and X&#39; are respectively the axes of S and S&#39;, while Y and Y&#39; are lines respectively perpendicular to X and X&#39; and lying in the plane determined by X and X&#39;; and Z designates a reference frame in which X and Y are fixed, while Z&#39; is a reference frame in which X&#39; and Y&#39; are fixed. 
     Suppose now that  Z  ω S  and  Z   &#39;  ω S   &#39;  denote respectively the angular velocity of S in Z and the angular velocity of S&#39; in Z&#39;. Then  Z  ω S  and  Z   &#39;  ω S   &#39;  are necessarily respectively parallel to X and X&#39;, and, if n and n&#39; are unit vectors respectively parallel to X and X&#39; (see FIG. 1), one can write 
     
         .sup.Z ω.sup.S =  ωn , .sup.Z.sup.&#39; ω.sup.S.sup.&#39; = ω&#39;n&#39;                                                (1) 
    
     where ω and ω&#39; are certain scalars, called respectively the angular speed of S in Z and the angular speed of S&#39; in Z&#39;; and γ, defined as 
     
         γ = ω&#39;/ω                                 (2) 
    
     can be termed the speed ratio of the system formed by S, S&#39;, and the coupling that connects S and S&#39;. 
     In principle, ω&#39;, and hence γ, can depend on not only ω, but also the angle θ (see FIG. 1), the time-derivative θ of θ, and the rotation angle φ of S in Z, that is the angle between a line that is fixed in S and perpendicular to X, such as the line L in FIG. 1, and a line that is fixed in Z and perpendicular to X, such as line Y. This sort of dependence arises, for example, when S and S&#39; are connected by a Hooke coupling. 
     Referring to FIG. 2, there is shown a Hooke coupling. Specifically, as will be shown presently, γ for this system is given by ##EQU1## 
     Clearly, couplings could be classified in terms of the functional dependence of γ on ω, θ, θ, and φ; and wide acceptance of such a classification scheme would facilitate communication. A modest first step in this direction is to define as a &#34;constant speed ratio coupling for shafts with time-varying orientations&#34;, a coupling such that γ is a constant, that is, does not depend on ω, θ, θ, and φ. This is the sense in which the phrase is used in the present application. 
     Throughout the sequel, the angular velocity of a body U in a reference frame V (or, equivalently, relative to a body V) is denoted by  V  ω U . Using the addition theorem for angular velocities, one can thus write, for any coupling, 
     
         .sup.S ω.sup.S.sup.&#39; = .sup.S ω.sup.Z + .sup.Z ω.sup.Z.sup.&#39; + .sup.Z.sup.&#39; ω.sup.S.sup.&#39;    (4) 
    
     applied to Hooke&#39;s joint, this theorem yields 
     
         .sup.S ω.sup.S.sup.&#39; = .sup.S ω.sup.C + .sup.C ω.sup.S.sup.&#39;                                       (5) 
    
     where C designates the coupling member of the joint (see FIG. 2). Furthermore, in accordance with FIG. 2, 
     
         .sup.z ω.sup.z.sup.&#39; = θ n × n&#39;/sin θ(6) 
    
     and, if λ and λ&#39; are unit vectors directed as shown, 
     
         .sup.S ω.sup.C =  sλ, .sup.C ω.sup.S.sup.&#39;  = s&#39;λ&#39;(7) 
    
     where s and s&#39; are certain scalars. Equating the right-hand members of Eqs. (4) and (5), and using Eqs. (1), (6), and (7), one thus obtains 
     
         -ωn + θ n × n&#39;/sin θ + ω&#39;n&#39; = s λ + s&#39;λ&#39;                                               (8) 
    
     from which it follows by scalar multiplication with λ × λ&#39; that 
     
         -ωn. (λ × λ&#39;) + ω&#39;n&#39;.(λ × λ&#39;) + θ(n × n&#39;) . (λ × λ&#39;)/sin θ = 0                                               (9) 
    
     Moreover, λ&#39; = ±λ × n&#39;, since λ&#39; is perpendicular to both λ and n&#39;. Consequently, 
     
         λ × λ&#39; = ±λ × (λ × n&#39;) = ±(λ.n&#39;λ - n&#39;)                            (10) 
    
     so that 
     
         n.(λ  × λ&#39;) = ±n.n&#39; = ± cos θ(11) 
    
     
         (n × n&#39;) . (λ × λ&#39;) = ±λ.n&#39;(n × n&#39;) . λ = ±sin θ sin φ cos φ      (12) 
    
     
         n&#39; . (λ × λ&#39;) = ±(λ.n&#39;).sup.2 ± 1 = ±(1 - sin.sup.2 θ cos.sup.2 φ)                      (13) 
    
     which makes it possible to re-write Eq. (9) as 
     
         ωcos θ + θ sin θ sin φ cos φ -ω&#39;(1 - sin.sup.2 θ cos.sup.2 φ) = 0                    (14) 
    
     Solving Eq. (14) for ω&#39; and substituting into Eq. (2), one arrives at Eq. (3). 
     Before leaving Eq. (3), it is worth pointing out that the term involving θ (which is sometimes ignored in the literature on this subject) can be physically important. Suppose, for example, that θ is very small, as in many automotive applications. Then Eq. (3) may be replaced with 
     
         γ ≈ 1 + (θ/ω) θ sin φ cos φ(15) 
    
     and it can be seen that sufficiently large values of θ can give rise to appreciable fluctuations in the speed ratio. Such values of θ may be encountered when a vehicle traverses a bumpy road at relatively high speed. 
     Referring to FIGS. 3 and 4, there is provided a coupling according to the present invention for coupling a pair of rotatable shafts D and N which correspond respectively to shafts S&#39; and S in FIGS. 1 and 2. The coupling to be described contains a so-called differential mechanism as a subsystem. FIG. 3 shows such a mechanism in schematic form. Its function is to bring about a certain relationship between the angular speeds of the shaft D and two coaxial shafts B and C rotatably supported in bores D&#39;, B&#39; and C&#39;, respectively, in a rigid body A. Specifically, if m and n&#39; are unit vectors fixed in a body A and perpendicular to each other, and if the axes of shafts B and C are parallel to m while that of shaft D is parallel to n&#39;, so that the angular velocities of B, C, and D in A can be expressed as 
     
         .sup.A ω.sup.B = .sup.A ω.sup.B m, .sup.A ω.sup.C = .sup.A ω.sup.C m,  .sup.A ω.sup.D = .sup.A ω.sup.D n&#39;(16) 
    
     then one can ensure that 
     
         .sup.A ω.sup.D = - R.sub.1 (.sup.A ω.sup.B + .sup.A ω.sup.C)                                            (17) 
    
     where R 1  is a constant (presently to be expressed as a ratio of two lengths). As shown in FIG. 3, this is accomplished by keying a pair of bevel gears E and F to B and C, respectively, and permitting these to engage a pair of bevel gears G and H which are free to rotate on pins G&#39; and H&#39; rotatably fixed in a casing I, this casing, in turn, being free to rotate about the common axis of B and C which projects through a pair of bores E&#39; and F&#39; in the casing. Furthermore, a bevel gear J is rigidly attached to I, and this meshes with a bevel gear K keyed to D. The constant R 1  is then given by the familiar relationship 
     
         R.sub.1 = j/2k                                             (18) 
    
     where j and k are the pitch radii of J and K, respectively. 
     FIG. 3 contains one more element of interest, namely a spur gear L that is rigidly attached to A and has a pitch radius l. This is not a part of a conventional differential mechanism, but it is required for the coupling of the present invention. 
     In FIG. 4, A, B, C, D, and L designate elements previously shown in FIG. 3. The elements interior to A in FIG. 3 are omitted for clarity in FIG. 4. In addition, there is provided a rigid member M which represents a carrier that supports the shafts B and C, as well as shaft N and a shaft O. Furthermore, there is provided a pair of bevel gears P and Q which are keyed to shafts N and B, respectively, and a plurality of spur gears R, S and T that are keyed to shafts C and O. P meshes with Q, R with S, and L with T. In its entirety, the coupling is thus formed by A, B, -- T, N and D being the elements corresponding to shafts S and S&#39;. (The axes of S and S&#39; (or N and D) are shown aligned with each other in FIG. 4 only for convenience of representation. They can, in fact, form an angle in excess of ninety degrees with each other.) 
     The bodies A and M can rotate relative to each other only about a line parallel to the unit vector m shown in FIG. 4; and N (or S) must rotate about a line fixed in M and parallel to n (see FIG. 4), while D (or S&#39;) is constrained to rotate about a line fixed in A and parallel to n&#39;. M and A can, therefore, be identified respectively with Z and Z&#39; of FIG. 1, while  M  ω N  and  A  ω D  play the roles of ω and ω&#39;, respectively, in Eqs. (1) if  M  ω N  is expressed as  M  ω N  =  M  ω N  n (see the third of Eqs. (16) for  A  ω D ). In accordance with Eq. (2), the speed ratio for the coupling is thus given by 
     
         γ = .sup.A ω.sup.D /.sup.M ω.sup.N       (19) 
    
     it will now be shown that γ is a constant, provided the pitch radii l, r, s, and t of gears L, R, S, and T, respectively, (see FIG. 4) satisfy the equations 
     
         ls/rt = 2                                                  (20) 
    
     and 
     
         r + s = t + l                                              (21) 
    
     The second of these ensures that R meshes with S when T meshes with L. As for the first, one may begin by observing that, if  M  ω B  is expressed as  M  ω B  =  M  ω B  m,  then 
     
         .sup.M ω.sup.N = - R.sub.2 .sup.M ω.sup.B      (22) 
    
     where R 2  is defined as 
     
         R.sub.2 = q/p                                              (23) 
    
     in which p and q are the pitch radii of the gears P and Q. Furthermore, using the addition theorem for angular velocities once again, one has, after expressing  A  ω B  and  A  ω M  as  A  ω B  =  A  ω B  m and  A  ω M  =  A  ω M  m, 
     
         .sup.A ω.sup.B = .sup.A ω.sup.M + .sup.M ω.sup.B(24) 
    
     or, after using Eq. (22) to eliminate  M  ω B , 
     
         .sup.a ω.sup.b = .sup.a ω.sup.m - .sup.m ω.sup.n /r.sub.2(25) 
    
     next, if R 3  is defined as 
     
         R.sub.3 = l/t                                              (26) 
    
     and  M  ω O  is expressed as  M  ω O  =  M  ω O  m, then 
     
         .sup.M ω.sup.O = - R.sub.3.sup.M ω.sup.A       (27) 
    
     or, since  M  ω A  = -  A  ω M , 
     
         .sup.m ω.sup.o = r.sub.3.sup.a ω.sup.m         (28) 
    
     similarly, with  M  ω C  expressed as  M  ω C  =  M  ω C   m, one can write 
     
         .sup.M ω.sup.C = - R.sub.4.sup.M ω.sup.O       (29) 
    
     where R 4  is defined as 
     
         R.sub.4 = s/r                                              (30) 
    
     so that, after elimination of  M  ω O  by use of Eq. (28), one has 
     
         .sup.M ω.sup.C = -R.sub.3 R.sub.4.sup.A ω.sup.M(31) 
    
     this, together with the addition theorem applied to A, M and C, that is, with 
     
         .sup.A ω.sup.C = .sup.A ω.sup.M + .sup.M ω.sup.C(32) 
    
     yields 
     
         .sup.A ω.sup.C = .sup.A ωM (1 - R.sub.3 R.sub.4)(33) 
    
     and, using this result together with Eq. (25) in Eq. (17), one can now express  A  ω D  as 
     
         .sup.A ω.sup.D = - R.sub.1 [.sup.A ω.sup.M (2 - R.sub.3 R.sub.4) - .sup.M ω.sup.N /R.sub.2 ]                (34) 
    
     consequently, if 2 - R 3  R 4  vanishes, which is the case whenever Eq. (20) is satisfied (see Eqs. (26) and (30) for R 3  and R 4 ), then 
     
         .sup.A ω.sup.D /.sup.M ω.sup.N = R.sub.1 /R.sub.2(35) 
    
     and substitution into Eq. (19) produces, in view of Eqs. (18) and (23), 
     
         γ = jp/2kq                                           (36) 
    
     so that γ is, indeed, a constant. Moreover, the speed ratio evidently can take on values lying in a wide range, for j, k, p and q each can be chosen with considerable latitude. 
     When the axis of D (FIG. 4) is nearly perpendicular to that of N, small changes in the orientation of the axis of D in any reference frame in which the axis of N is fixed are accompanied by minimal motions of other parts of the mechanism; and such changes can be made in any plane passing through the point of intersection of the axes of D and N. Hence the coupling may be expected to perform especially well under these circumstances. The only situation in which the axis of D does not possess complete freedom of movement is that depicted in FIG. 4, for the axis of D cannot move in the plane determined by the axes of B and N. This state of affairs should probably be avoided, since it tends to bring excessively large forces into play. However, even in this situation, the coupling performs in accordance with Eq. (36). 
     It is apparent from the foregoing that the coupling formed of elements A, . . . , T arranged as shown in FIGS. 3 and 4 has the constant speed ratio given by Eq. (36) whenever Eqs. (20) and (21) are satisfied and that the speed ratio is solely dependent on the pitch radii of bevel gears, whereas the constraint relations involve the pitch radii of spur gears. 
     While a specific embodiment of the invention is disclosed, it is understood that various other sizes, types and arrangements of the gears within the constraints prescribed herein may be employed. Likewise it is apparent that the gears and shafts employed may be held in rigid body structures having shapes other than those shown for rigid body members A, I and M. It is also understood that while only a coupling for shafts having intersecting axes of rotation is described, a constant speed ratio for shafts having non-intersecting axes of rotation is possible with the use of an appropriate arrangement of shafts and gears, without departing from the spirit and scope of the present invention. For example, the axis of shaft N can be placed in such a way that it never intersects the axis of shaft D. 
     Accordingly, it is intended that the scope of the invention should not be limited to the embodiment described but, rather, be determined by the claims hereinafter provided.