BACKGROUND AND SUMMARY 
     The present invention relates to vehicle transmissions, and more particularly to a system for preventing gear hopout in tooth clutches that are subjected to misalignment due to forces acting on rotating parts they connect. 
     BACKGROUND OF THE INVENTION 
     Tooth clutches are frequently used in stepped vehicle transmissions to engage and disengage the gears. A tooth clutch can rotatably connect a main part with a substantially coaxial connectable part. Normally, an engaging sleeve is used as an interconnecting member between these two parts. This engaging sleeve is often rotatably fixed but axially moveable with respect to said main part by means of, for instance, splines. On the engaging sleeve there are clutch teeth at the end that faces the connectable part. These clutch teeth need to be compatible with corresponding clutch teeth on the connectable part. These two sets of clutch teeth can be brought into mesh with each other by moving the engaging sleeve in axial direction towards the connectable part. 
     In double-acting tooth clutches, there are clutch teeth at both ends of the engaging sleeve. Thereby, the engaging sleeve can connect the main part to either a first or a second connectable part. These connectable parts must have clutch teeth that are compatible with the clutch teeth at the corresponding end of the engaging sleeve. 
     Some tooth clutches comprising a main part, an engaging sleeve and connectable parts can be seen in U.S. Pat. No. 2,070,140, U.S. Pat. No. 3,137,376, DE-4319135A1 and U.S. Pat. No. 6,422,105. 
     In heavy road vehicles, such as heavy trucks, transmissions of range compound type are often used. In such a transmission, a main section, having several selectable gears, is connected in series with a range section. There are two gears in the range section; one low-range gear with a large speed reduction and one high-range gear with no speed reduction, normally referred to as a direct gear. In practice, the range section doubles the number of gears in the main section. A typical state-of-the-art heavy truck transmission of range compound type is shown in FIG. 1 in WO-2004069621, featuring a main section 2 and a range section 3. 
     Range sections are often embodied as a planetary arrangement that is combined with a double-acting tooth clutch. Due to the design of the planetary arrangement, the main part of the tooth clutch may be fixedly connected to the engaging sleeve and move axially with the sleeve. In such cases, the main part usually is the ring gear of the planetary arrangement. A typical example is shown in U.S. Pat. No. 4,667,538, where the engaging sleeve 18 is fixedly connected to the ring gear 14. In some embodiments, the engaging sleeve is integrated in the ring gear, for example as shown in EP-0916872 (FIG. 3, items 56 and 58) and, more advanced, in U.S. Pat. No. 5,083,993 (FIG. 1, item 24). 
       FIG. 1   a  shows a longitudinal section of a simplified range section  101  of planetary type. The input shaft of the range section  101  is a main shaft  102  of a main section  103 . A transmission housing  104  rotatably supports the main shaft  102  by means of a bearing  105  and a bearing  105   b  in the main section  103 . There are external spline teeth  106  at the end of the main shaft  102 . The spline teeth  106  are meshing with internal spline teeth  107  of a sun gearwheel  108 . External gear teeth  109  of the sun gearwheel  108  are in mesh with external gear teeth  110  of a planet gearwheel  111 . A planet axle  112  rotatably supports the planet gearwheel  111  to a planet carrier  113  that is shown integral with an output shaft  114 . A number of identical planet gearwheels are located with substantially equal spacing along the periphery of the planet carrier  113 . An output bearing  115  rotatably supports the output shaft  114  to the transmission housing  104 . The external gear teeth  110  of the planet gearwheel  111  also mesh with internal gear teeth  116  of a ring gearwheel  117 . In the position shown in FIG. Ia, internal direct clutch teeth  118  of the ring gearwheel  117  mesh with external clutch teeth  119  of a direct engaging ring  120 . Internal spline teeth  121  of the direct engaging ring  120  are in mesh with the external spline teeth  106  of the main shaft  102 . Hence, in  FIG. 1   a  the ring gearwheel  117  is rotationally connected to the main shaft  102  by means of the direct engaging ring  120 . Thereby, the planet gearwheel  111  cannot move in peripheral direction relative to the main shaft  102 . The result is that the main shaft  102 , the output shaft  114  and the parts in between will rotate in a unison way, that is, with the same speed. This represents the direct high-range gear. In  FIG. 1   b , the ring gearwheel  117  has been moved to the right in comparison with  FIG. 1   a . Thereby, the direct clutch teeth  118  are no longer in mesh with the external clutch teeth  119  of the direct engaging ring  120 . Instead, internal reduction clutch teeth  122  of the ring gearwheel  117  have been brought into mesh with external clutch teeth  123  of a stationary engaging ring  124  that is fixedly connected to the transmission housing  104 . Thereby, the ring gearwheel  117  will not rotate when in the position of  FIG. 1   b . The result will be the low-range reduction gear; the output shaft  114  will rotate slower than the main shaft  102 . 
     A range shift actuator  125  accomplishes the axial displacement of the ring gearwheel  117 . A range shift rod  126  is being pushed or pulled in appropriate direction by the range shift actuator  125 . A range shift fork  127  is fixedly attached to the range shift rod  126 . The range shift fork  127  extends into a circumferential groove  128  on the ring gearwheel  117 . The range shift actuator  125  may be of one of several types, for instance hydraulic, pneumatic, electromagnetic or electromechanical. Normally, the range shift actuator  125  is only activated during a shift. When a shift has been completed, it will be deactivated. 
     In the range section of  FIG. 1   a  and  FIG. 1   b  the ring gearwheel can be regarded as a combined main part and engaging sleeve of a double-acting tooth clutch. Furthermore, it can be noted that only one bearing  115  supports the output shaft  114 . The planetary range section  101  provides another support. When torque is being transferred by gearwheels and clutch teeth of the planetary range section  101 , contact forces in the gear and tooth clutch meshes around the periphery will urge the parts to become substantially coaxial. When the planetary range section  101  is transferring no torque, it will still provide some support for the output shaft  114 , albeit with a lower degree of coaxiality between the parts. Thus, the gearwheels along with the tooth clutches of the planetary range section  101  will act as some kind of a second supporting bearing for the output shaft  114 . Then, in high range position, as shown in  FIG. 1   a , the main shaft  102  indirectly supports the output shaft  114 . 
     Tooth clutches are normally designed to be self-retaining in engaged state. This means that once the tooth clutch has been engaged, no external force is required to retain the tooth clutch in this engaged state. Different design solutions are used to achieve this self-retaining feature. One common design solution is to have the clutch teeth angled in order to create a nominal axial force that urges the sleeve to retain in engaged position when torque is being transferred in the tooth clutch. This solution is often referred to as back-taper design. An example is shown in U.S. Pat. No. 5,626,213. There, in FIG. 2 it can be seen that the clutch teeth flanks 21, 26 are angled α, θ with respect to the flanks 28 of the spline teeth 11 of the engaging sleeve 8. Thereby, the contact forces will urge the clutch teeth towards fully engaged position when torque is being transferred. Some other design solutions for self-retaining action can be seen in U.S. Pat. No. 2,070,140 and FR-2660723. 
     In most self-retaining tooth clutch designs at least one of the sets of clutch teeth is made by modifying a set of spline or gear teeth. Returning to U.S. Pat. No. 5,626,213, the angled back-tapered flanks 26 of the engaging sleeve 8 can be regarded as a slight modification of the flanks 28 of the internal spline teeth 11. Similarly, in  FIG. 1   a  and  FIG. 1   b  back taper on the clutch teeth  118  and  122  can be made by modifying the internal gear teeth  116  of the ring gearwheel  117 . A rolling operation is a rapid and very cost-effective method to embody such modifications. In a rolling operation the flanks of the spline or gear teeth of an engaging sleeve or gearwheel are deformed plastically by meshing with the teeth of a mating tool wheel under radial load and rotation. Unfortunately, the material volume that can be plastically deformed in a rolling operation is small. Hence, the back-taper angles (α, β in U.S. Pat. No. 5,626,213) that are feasible to achieve in a rolling operation are small, typically about 5 degrees. This is, however, sufficient for most applications of tooth clutches. 
     There are some applications where conventionally made back-tapered clutch teeth have been shown to have insufficient self-retaining action. One example is shown in  FIG. 2 , where, in comparison with  FIG. 1   a , a retarder unit  230  has been added to the range section  201 . The retarder unit  230  is an auxiliary brake that can be used in long down-hill slopes in order to reduce wear and prevent over-heating of the ordinary wheel brakes of the vehicle. The retarder unit is driven by a retarder shaft  231  that is rotatably connected to a retarder driven gearwheel  232 . In turn, the retarder driven gearwheel  232  meshes with a retarder driver gearwheel  233  that is rotationally connected to the output shaft  214  of the range section  201 . 
     When the retarder unit  230  is in operation, gear mesh forces  240  will act on the retarder driver gearwheel  233 . These forces will tend to misalign the output shaft  214 . Normally, engine braking is used simultaneously with retarder operation. Thereby, torque will be transferred by the range section, and there will be contact forces in the gear meshes and between the clutch teeth of the range section. These contact forces will urge the parts of the range section towards a substantially coaxial state, as was described earlier. Hence, the contact forces will counteract the tendency of the gear mesh forces on the retarder driver gearwheel  233  to misalign the output shaft  214 . 
     Some retarder operating conditions have shown to cause problems in a planetary range section as in  FIG. 2 . One example is when there is a relatively large braking action in the retarder unit  230  and a relatively small engine braking action. This is illustrated schematically in  FIG. 2 . The retarder gear mesh force  240  tends to misalign the output shaft  214  in clockwise sense in the view of  FIG. 2 . However, the retarder gear mesh force  240  is balanced by a planet gear mesh force  241  that acts on the gear teeth  210  of a planet gearwheel  211  in the gear mesh with the internal teeth  216  of the ring gearwheel  217 . The counter force to the planet gear mesh force  241  is the ring gear mesh force  242  that acts on the ring gearwheel  217 . In turn, the ring gear mesh force  242  is balanced by a ring clutch mesh force  243  in the mesh between the clutch teeth  218  of the ring gearwheel  217  and the clutch teeth  219  of the direct engaging ring  220 . 
     The ring mesh force  242  and the ring clutch force  243  compose a force couple that tends to misalign the ring gearwheel  217  in counter-clockwise sense as is indicated in  FIG. 2 . Thereby, an axial gap  244  will result between the clutch teeth  218  of the ring gearwheel  217  and the clutch teeth  219  of the direct engaging ring  220 . Hence, during rotation there will be an urge for relative motion in axial direction between the clutch teeth  218  of the ring gearwheel  217  and the clutch teeth  219  of the direct engaging ring  220 . This urge for relative motion may turn into an unstable state if the friction between the contacting clutch teeth is large and the self-retaining action from for instance back taper is insufficient. Then, the clutch teeth  218  of the ring gearwheel  217  will be fed out of engagement with the mating clutch teeth  219  of the direct engaging ring  220 . Thereby, no torque can be transferred by the range section  201 , and, consequently, no engine braking is possible. 
     Another example is shown in  FIG. 3 ; a splitter unit  350  of a gearbox. An engaging sleeve  351  can rotationally connect an input shaft  352  to either of a first gearwheel  353  and a second gearwheel  354 . Each of gearwheels  353  and  354  is in mesh with a mating gearwheel that is rotationally fixed to a countershaft (not shown). The second gearwheel  354  is rotatably supported on a main shaft  355  by means of bearings  356 . The main shaft  355  is supported in one end by a gearbox housing (not shown) by a symbolically shown bearing  357 . The other end of the main shaft  355  is supported by the input shaft  352  by a taper roller bearing  358 . The input shaft  352 , in turn, is supported directly or indirectly by the gearbox housing by two symbolically shown bearings  359  and  360 . 
     In  FIG. 3 , the engaging sleeve  351  is positioned to rotationally connect the input shaft  352  and the second gearwheel  354 . Thereby, torque can be transferred from the input shaft  352  to the second gearwheel  354  and on to the mating gearwheel on the countershaft. Then, gear mesh forces  361  would act on the second gearwheel  354 . 
     In operation, there might be an axial gap in the taper roller bearing  358 . This axial gap could be the result of for instance thermal expansion and axial force components in gear meshes. In a taper roller bearing, an axial gap always corresponds to a radial gap. In the splitter unit  350  such a radial gap would decrease the radial support and allow a misalignment of the main shaft  355 . Then, that misalignment would be counteracted by contact forces between the spline and clutch teeth of the input shaft  352 , engaging sleeve  351  and second gearwheel  354 . This is similar to what has been described above for planetary range sections. For the second gearwheel  354 , the gear mesh force  361  would then be balanced by a gearwheel contact force  362  acting on the clutch teeth that are engaged with corresponding clutch teeth on the engaging sleeve  351 . The counter force to the gearwheel contact force  362  is a sleeve clutch contact force  363  that acts on the clutch teeth of the engaging sleeve  351 . For the engaging sleeve  351 , the sleeve clutch contact force  363  is balanced by a sleeve spline contact force  364 . Similar to  FIG. 2 , the sleeve clutch contact force  363  and the sleeve spline contact force  364  compose a force couple that urges to misalign the engaging sleeve  351 . Then, an axial gap  365  can be created between the engaging sleeve  351  and the second gearwheel  354 . During rotation, this might make the engaging ring  351  being fed out of engagement with the clutch teeth of the second gearwheel  354 , very similar to the ring gearwheel  217  in  FIG. 2 . 
     Some conclusions can be drawn from the analysis of the systems in  FIG. 2  and  FIG. 3 . In both cases there is a supported shaft ( 214 ,  355 ) that is supported radially by a supporting shaft ( 202 ,  352 ). A proper conventional radial support device between those shafts, such as a radial bearing ( 358 ), is either missing or insufficient under some conditions. Furthermore, the supported shaft is subjected to external forces ( 240 ,  361 ) that urge to misalign the supported shaft in relation to the supporting shaft. Those external forces can act directly on the supported shaft or via other parts, for instance a gearwheel ( 233 ,  354 ) that is fixed to or supported by the supported shaft. Finally, a tooth clutch with an engaging sleeve ( 217 ,  351 ) can selectably connect the supporting shaft for unison rotation with the supported shaft or a number of gearwheels ( 211 ,  354 ) that are radially supported by the supported shaft. Due to the combination of urge to misalign and an inadequate radial support device for the supported shaft, at least a part of the supporting action is accomplished by contact forces ( 241 ,  362 ) in the tooth clutch. These contact forces tend to misalign the engaging sleeve. Under certain conditions this misalignment might lead to gear hopout, that is, unwanted and uncontrolled disengagement of the tooth clutch. 
       FIG. 4  shows a peripheral section of the splines and clutch teeth of the tooth clutch of  FIG. 3 . The internal clutch teeth of the engaging sleeve  351  are back-tapered with the back-taper angle  481  with respect to the axis of rotation. The mating flanks of the clutch teeth of the second gearwheel  354  have a corresponding back-tapered shape. The spline teeth on the input shaft  352  are straight, that is, parallel with the axis of rotation. Thereby, the sleeve spline contact force  364  will have a spline normal component  364   n  that is perpendicular to the axis of rotation and, hence, in a true tangential direction. The sleeve clutch contact force  363  has a clutch normal component  363   n  that is perpendicular to the back-tapered flank of the internal clutch teeth of the engaging sleeve  351 . This clutch normal component  363   n  will be inclined with the back-taper angle  481  with respect to a true tangential direction. Due to this inclination, the clutch normal component  363   n  will have a component  363   p  that is parallel to the spline teeth of the engaging sleeve  351  and the input shaft  352  and, hence, perpendicular to the spline normal component  364   n . This parallel component  363   p  urges the engaging sleeve  351  towards fully engaged position and resists relative motion towards gear hopout. 
     In U.S. Pat. No. 6,066,062 a planetary range section 3 with back-tapered clutch teeth 67, 68, 69, 70 is shown. The gearwheels in the planetary range section 3 have helical gear teeth. In helical gear teeth, the gear mesh forces have axial components. For the ring gearwheel 56, that axial force component must, in general, be balanced by an axial force component in the contacting clutch teeth 67, 68, 69, 70. In addition, a back-taper action is required in order to prevent gear hopouts. Therefore, these clutch teeth have at least one helical flank. In order to handle this, the following general definition of back-taper angle can be used:
         i) The balanced helix angle of a set of clutch teeth of an axially moveable part of a tooth clutch is the helix angle that would, under friction-free conditions, result in no net axial force from the contact forces acting on gear, spline and clutch teeth of the axially moveable part when torque is being transferred via the set of clutch teeth.   ii) The back-taper angle of a set of contacting flanks of a set of clutch teeth of an axially moveable part of a tooth clutch is the difference between the actual helix angle for the set of contacting flanks and the balanced helix angle for the set of clutch teeth.       

     For the clutch teeth of the engaging sleeve  351  in  FIG. 3  and  FIG. 4 , the spline teeth of the input shaft  352  are straight. Under hypothetic friction-free conditions, the sleeve spline contact force  364  would be perpendicular to the axis of rotation. Then, with a zero back-taper angle  481  the parallel component  363   p  would vanish, and the sleeve clutch contact force  363  would also be perpendicular to the axis of rotation. Hence, there would be no net axial force on the engaging sleeve  351  from the contact forces acting on its spline and clutch teeth. Thus, the balanced helix angle for the engaging sleeve  351  is zero. 
     For the planetary range section 3 in U.S. Pat. No. 6,066,062, the clutch teeth 67, 68, 69, 70 have a smaller diameter than the gear teeth of the ring gearwheel 56. Thereby, to fulfil torque equilibrium the contact forces on these clutch teeth will be larger than the corresponding mesh forces on the gear teeth. This implies that the balanced helix angle of the clutch teeth 67, 68 of the ring gearwheel 56 will be less than the helix angle of the gear teeth. 
     There are some known methods to prevent gear hopouts of the type described above. In general, radial support devices, such as bearings, have been introduced or improved in order to limit the possible misalignment of the supported shaft. In U.S. Pat. No. 5,839,319 a splitter unit similar to the one in  FIG. 3  is shown. However, a headset/fourth gear 74 (corresponding to the second gearwheel  354  in  FIG. 3 ) is not supported by a main shaft (corresponding to  355  in  FIG. 3 ) but by a spindle 62 that is rigidly secured to an input shaft 42 (corresponding to  352  in  FIG. 3 ). Thereby, a gear mesh force acting on the headset/fourth gear 74 will not cause any significant urge to misalign the main shaft. Hence, the tendency for gear hopout has been eliminated. However, the additional spindle will imply increased production cost. 
     U.S. Pat. No. 5,083,993 presents a planetary gear 1 that is similar to the planetary range section  101  in  FIG. 1   a . In order to reduce possible misalignment, a roller bearing has been included between a planet wheel carrier 9 (corresponding to the planet carrier  113  in  FIG. 1   a ) that is integral with an output shaft 3 (corresponding to  114  in  FIG. 1   a ) and a sun wheel 5 (corresponding to the sun gearwheel  108  in  FIG. 1   a ) that is arranged in a rotationally fixed manner on an input shaft 2 (corresponding to the main shaft  102  in  FIG. 1   a ). Thus, the roller bearing acts as a radial support device for a supported shaft, the output shaft 3, on a supporting shaft, the input shaft 2. However, the roller bearing will imply increased cost. 
     EP-239555B1 discloses a similar planetary gear 2. Therein, with the aid of a ball bearing 18 a clutch ring 16 supports a planet wheel keeper 10 that is fastened to a planet wheel carrier 11 which, in turn, is integrated with an output shaft 4. The clutch ring 16 is non-rotatably mounted on a sun wheel 7 that is non-rotatably mounted on an input shaft 3. In  FIG. 1   a  the equivalence would be an additional ball bearing between the engaging ring  120  and the part of the planet carrier  113  that is to the left of the planet gearwheel  111 . The additional ball bearing 18 will provide a radial support of the output shaft 4 and thereby reducing the possible misalignment. However, the ball bearing 18 will imply increased cost. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1   a  shows a planetary range section of prior art in a direct high-range gear. 
         FIG. 1   b  shows the planetary range section of  FIG. 1   a  in a low-range reduction gear. 
         FIG. 2  shows a planetary range section of prior art with a retarder unit, including forces and misalignments that may occur and may lead to gear hopout. 
         FIG. 3  shows a splitter unit in a gearbox of prior art, including forces and misalignments that may occur and may lead to gear hopout. 
         FIG. 4  shows a peripheral section of the teeth of the tooth clutch of the splitter unit of  FIG. 3 . 
         FIG. 5  shows a modified design of the clutch teeth of  FIG. 4  according to the invention. 
         FIG. 6  shows a modified design of the clutch teeth of  FIG. 2  according to an embodiment of the invention. 
         FIG. 7 ,  FIG. 8  and  FIG. 9  show variants of the clutch teeth of  FIG. 6  according to embodiments of the invention, said embodiments making it easy to detect wrong assembled parts. 
     
    
    
     DETAILED DESCRIPTION 
     When studying  FIG. 3  and  FIG. 4  it can be noted that the misalignment of the engaging sleeve  351  will lead to a varying degree of engagement along the periphery with the clutch teeth of the second gearwheel  354 . The lowest degree of engagement is where the axial gap  365  is largest. During rotation, this varying degree of engagement will require a relative motion between the clutch and spline teeth in contact of the engaging sleeve  351 , input shaft  352  and second gearwheel  354 . In order to prevent gear hopout, it is vital that this relative motion takes place. Thereby, friction will occur between the teeth in contact. In  FIG. 4 , the friction component of the sleeve clutch force  363  is referred to as  363   f . Similarly, the friction component of the sleeve spline force  364  is referred to as  364   f . In order to enable the relative motion between the contacting teeth, substantially the parallel component  363   p  of the sleeve clutch force  363  must be larger than the sum of the friction forces  363   f  and  364   f:  
 
363 p&gt; 363 f+ 364 f  
 
     This assumes that the back-taper angle  481  is fairly small, for instance, less than 20 degrees. In  FIG. 4 , this inequality requirement does not seem to be fulfilled:
         i) When sliding, the friction component  363   f  of the sleeve clutch force  363  is equal to a coefficient of friction  363   mu  multiplied by the normal component  363   n:  
 
363 f= 363 mu* 363 n  
   ii) When sliding, the friction component  364   f  of the sleeve spline force  364  is equal to a coefficient of friction  364   mu  multiplied by the normal component  364   n:  
 
364 f= 364 mu* 364 n  
   iii) Between clutch and spline teeth in contact in a vehicle gearbox, the coefficient of friction typically has a value of about 0.1:
 
363mu=364mu=0.1
   iv) Torque equilibrium, again assuming a small back-taper angle  481 , requires that the normal component  363   n  of the sleeve clutch force  363  is substantially equal to the normal component  364   n  of the sleeve spline force  364 :
 
363n=364n
   v) Now, the right-hand side of the inequality can be written:
 
363 f+ 364 f= 0.1*363 n+ 0.1*364 n= 0.2*363 n  
   vi) The parallel component  363   p  of the sleeve clutch force  363  is equal to the normal component  363   n  multiplied by the tangent function of the back-taper angle  481 :
 
363 p= 363 n *tan(481)
   vii) For back-tapered clutch teeth manufactured by a rolling process, the back-taper angle is in general not larger than 5 degrees, as was stated earlier:
 
tan(481)=tan(5 degrees)&lt;0.09
   viii) Thus, for conventional prior art back-tapered clutch teeth, the parallel component  363   p  of the sleeve clutch force  363  is in general limited to:
 
363 p&lt; 0.09*363 n  
   ix) By comparing the results in steps v) and viii) it can be seen that the inequality requirement is indeed not fulfilled in  FIG. 4 ; 0.09 is not larger than 0.2. Thus, the back-taper angle  481  is not large enough to prevent gear hopout, that is, to enforce the relative motion between the clutch and spline teeth.       

     By eliminating the normal component  363   n , the inequality requirement for relative motion can be written:
 
tan(back-taper angle)&gt;(coefficient of friction between clutch teeth)+(coefficient of friction between spline teeth)
 
     Assuming a common coefficient of friction between both clutch and spline teeth, this can be simplified:
 
tan(back-taper angle)&gt;2*(coefficient of friction between clutch and spline teeth)
 
     Step iv) above assumes that the sleeve clutch force  363  acts on the same diameter as the sleeve spline force  364 . This is the case for the tooth clutch in  FIG. 3  and substantially the case for the corresponding forces in the planetary range section in  FIG. 2 . For the planetary range section 3 in U.S. Pat. No. 6,066,062, however, the back-tapered clutch teeth 67, 68, 69, 70 have a significantly smaller diameter than the inner teeth 57 of the sleeve 58. The inner teeth 57 of the sleeve 58 along with the teeth 65 of the planet gears 43 will act as splines for the axially moveable sleeve 58. In such a case, the inequality requirement will be:
 
tan(back-taper angle)&gt;(coefficient of friction between clutch teeth)+(coefficient of friction between spline teeth)*(pitch diameter of clutch teeth)/(pitch diameter of spline teeth)
 
     or, with a common coefficient of friction:
 
tan(back-taper angle)&gt;(coefficient of friction between clutch and spline teeth)* (1+(pitch diameter of clutch teeth)/(pitch diameter of spline teeth))
 
     According to the invention, the back-taper angle shall be large enough to fulfil the inequality requirement.  FIG. 5  shows a peripheral section of the clutch and spline teeth of a variant of the tooth clutch in  FIG. 3  that has been modified according to the invention. The clutch teeth of a modified second gearwheel  554  and the mating clutch teeth of a modified engaging sleeve  551  have a modified back-taper angle  581  that is significantly larger compared to the back-taper angle  481  in  FIG. 4 . Thereby, the parallel component  563   p  of the sleeve clutch force  563  is larger than the sum of the friction force  563   f  between the clutch teeth and the friction force  364   f  between the spline teeth. Then, relative motion between the parts involved is possible, and gear hopout is prevented. 
     Thus, when the back-taper angle  581  is large enough to fulfil the inequality requirement, the self-retaining ability of the tooth clutch is increased significantly. The parallel component  563   p  of the sleeve clutch force  563  is then large enough to be able to pull the engaging sleeve  551  towards fully engaged state. Thereby, gear hopout is prevented also for conditions of a misaligned engaging sleeve, for instance as shown in  FIG. 2  and  FIG. 3 . 
     For the case of equal pitch diameters of the clutch and spline teeth along with a coefficient of friction of 0.1, the inequality requirement is:
 
tan(back-taper angle)&gt;0.1+0.1=0.2
 
     This implies that the back-taper angle shall be at least 11.3 degrees. Furthermore, for an extreme case of the planetary range section 3 in U.S. Pat. No. 6,066,062, the pitch diameter of the inner teeth 57 could be twice as large as the pitch diameter of the clutch teeth 67, 68, 69, 70:
 
tan(back-taper angle)&gt;0.1+0.1*1/2=0.15
 
     This is equivalent to a back-taper angle of at least 8.5 degrees. This is still significantly more than the above mentioned 5 degrees that can be regarded as an approximate upper limit of the back-taper angle that can be achieved by a cost-efficient rolling operation. 
     Instead, less cost-efficient manufacturing methods, for instance cutting methods such as shaping, will have to be used for sets of clutch teeth that are made by modifying a set of spline or gear teeth. 
     In order to keep the costs down, it would be of advantage to keep the use of said less cost-efficient manufacturing methods to a minimum. This is addressed in an embodiment of the invention. Thereby, it is noted that in planetary range sections with a retarder, as in  FIG. 2 , the described risk for gear hopout occurs when there is simultaneous retarder and engine braking operation in the high-range direct position. Then, it would be sufficient to have a large back-taper angle on those flanks of the clutch teeth that are in contact during engine braking, only. The opposite flank of each clutch tooth could have a smaller back-taper angle. This could be made using a combination of manufacturing methods for sets of clutch teeth that are made by modifying a set of spline or gear teeth. As an example, a rolling operation could be used to make a small back-taper angle on both flanks of the clutch teeth. Then, a shaping operation could be used to form a large back-taper angle only on the flanks that are in contact with the mating clutch teeth during engine braking. 
       FIG. 6  shows a peripheral section of meshing clutch teeth in the high range direct position during retarder and engine braking operation of the planetary range section  201  of  FIG. 2 . In order to prevent gear hopout due to misalignment, the flanks of the clutch teeth that are in contact have a large back-taper angle according to the invention. The clutch teeth  218  of the ring gearwheel  217  have braking flanks  218   b  that during engine braking operation are in contact with corresponding braking flanks  219   b  on the clutch teeth  219  of the direct engaging ring  220 . Similarly, there are driving flanks  218   d  and  219   d  that are in contact during engine driving operation, that is, when the engine drives the vehicle. The braking flanks  218   b  and  219   b  have a large back-taper angle  691   b , and the driving flanks  218   d  and  219   d  have a normal, small, back-taper angle  691   d . The large back-taper angle  691   b  makes the parallel component  243   p  of the ring clutch mesh force  243  larger than the sum of the friction components  242   f  and  243   f  of the ring gear mesh force  242  and ring clutch mesh force  243 , respectively. Thereby, gear hopout is prevented during simultaneous retarder and engine braking operation. During engine driving operation, the retarder unit  230  is not in operation, and no gear mesh forces  240  will act on the retarder driver gearwheel  233  in  FIG. 2 . Hence, there will be no tendency to misalign the output shaft  214 , and no large back-taper angle is required on the driving flanks  218   d  and  219   d  for preventing gear hopout. 
     In general, a retarder unit  230  is optional and is only included on a minority of the gearboxes. Therefore, from a cost point of view it would not be of advantage to have a large back-taper angle  691   b  on the braking flanks  218   b  and  219   b  in all gearboxes. Instead, it would be better to have the large back-taper angle  691   b  in gearboxes with a retarder unit  230 , only. In gearboxes that do not have a retarder unit, both flanks of the clutch teeth  218  and  219  could have conventional, small, back-taper angles. However, if any of the braking flanks  218   b  and  219   b  in a gearbox with a retarder unit  230  would have a conventional, small, back-taper angle, the ability to prevent gear hopout is lost. Thus, it is important to, as soon as possible, discover an accidentally assembled ring gearwheel  117  or direct engaging ring  120  with a conventional, small, back-taper angle on the braking flanks in a gearbox with a retarder unit  230 . 
       FIG. 7  shows a modified design of the clutch teeth in  FIG. 6 . After a rolling operation, the clutch teeth  7180  on the ring gearwheel have a conventional, small, back-taper angle on both flanks  7180   b  and  718   d . In that condition, the ring gearwheel is equivalent to ring gearwheels  117  used in gearboxes that do not have a retarder unit  230 . The maximum pitch diameter tooth thickness  7180   t  of these rolled clutch teeth  7180  is similar to the pitch diameter tooth thickness  216   t  of the internal gear teeth  216  of the ring gearwheel. For gearboxes with a retarder unit  230 , the clutch teeth  718  of the ring gearwheel  217  have reduced maximum pitch diameter tooth thickness  718   t  compared to  FIG. 6 . In principle, more material has been removed by the additional cutting process when creating the braking flank  718   b , starting from clutch teeth  7180 . Moreover, the clutch teeth  719  of the direct engaging ring  220  have increased tooth thickness and, hence, decreased minimum pitch diameter tooth space width  719   w  compared to  FIG. 6 . The minimum pitch diameter tooth space width  719   w  is smaller than the maximum pitch diameter tooth thickness  7180   t  of the rolled clutch teeth  7180 . Thereby, a ring gearwheel with the thick, rolled clutch teeth  7180  with a conventional, small, back-taper angle on both flanks cannot mesh with a direct engaging ring  220  that has thick clutch teeth  719  with a large back-taper angle on the braking flanks  719   b . Thus, in a gearbox with a retarder unit, it easy to detect a ring gearwheel  117  made for a gearbox without a retarder unit. 
       FIG. 8  and  FIG. 9  show a further embodiment of the invention. That embodiment solves the problem of detecting a direct engaging ring  120  with a conventional, small, back-taper angle on the braking flanks in a gearbox with a retarder unit  230 . In  FIG. 8 , a longitudinal section is shown of a modified design of the clutch teeth  818  and  819  of the ring gearwheel  117  and direct engaging ring  120  for gearboxes without a retarder unit  230 , that is, with a conventional, small, back-taper angle on both flanks. The minimum inner tip diameter  818   i  of the ring gearwheel clutch teeth  818  and the maximum root diameter  819   r  of the engaging ring clutch teeth  819  are included. In order to allow meshing of the clutch teeth  818  and  819 , the minimum inner tip diameter  818   i  must be larger than the maximum root diameter  819   r . Similarly,  FIG. 9  shows a corresponding section of clutch teeth  918  and  919  of the ring gearwheel  217  and direct engaging ring  220  with a large back-taper angle on the braking flanks for gearboxes with a retarder unit  230 . Included are the minimum inner tip diameter  918   i  of the ring gearwheel clutch teeth  918  and the maximum root diameter  919   r  of the engaging ring clutch teeth  919 . Again, the minimum inner tip diameter  918   i  must be larger than the maximum root diameter  919   r  to enable meshing of the clutch teeth. Now, the maximum root diameter  819   r  is larger than the minimum inner tip diameter  918   i . Thereby, the clutch teeth  918  of a ring gearwheel  217  made for gearboxes with a retarder unit  230  cannot mesh with the clutch teeth  819  of a direct engaging ring  120  that is made for gearboxes without a retarder unit  230 . Thus, in a gearbox with a retarder unit, it easy to detect an accidentally assembled direct engaging ring  120  made for a gearbox without a retarder unit. 
     Although the present invention has been set forth with a certain degree of particularity, it is understood that various modifications, substitutions and rearrangements of the components are possible without departing from the spirit and scope of the invention as hereinafter claimed.
A system is provided for preventing gear hopout in a tooth clutch in a vehicle transmission, the tooth clutch including an engaging sleeve having sleeve clutch teeth. The tangent function for at least one of driving back-taper angle and braking back-taper angle is larger than the average value of clutch coefficient of friction and spline coefficient of friction multiplied by the sum of unity and the ratio of clutch teeth pitch diameter and spline teeth pitch diameter.