Positive motion belt with elastic teeth
FIELD OF THE INVENTION 
     The present invention relates to a positive motion belt with elastic teeth of improved form. 
     BACKGROUND OF THE INVENTION 
     A positive motion belt with elastic teeth (hereinafter called the belt) of conventional type 1 has teeth 2 of all-addendum form, lacking part of the tooth tip and tooth root. The tooth surface 3 is straight, having a pressure angle α o . Reference numeral 4 designates the root fillet curve where the tooth root joins the bottom land 5. At the back of the bottom land 5 is laid a load carrying member 6, so that the pitch line 7 runs through the middle of said load carrying member 6. The tooth surface 3, bottom land 5 and top land 8 of the tooth 2 are covered, as required, with canvas or other material. On the inside of the tooth 2 and the back 9 of the belt 1 are filled rubber or rubber-like elastic material. 
     When the belt 1 is placed on wheels 40 and 40&#39; as illustrated in FIG. 2, individual pitch circles 45 and 45&#39; of the wheels 40 and 40&#39; must overlap with the pitch line 7 of the belt 1. Therefore, the pitch circles 45 and 45&#39; of the wheels 40 and 40&#39; exist outside their addendum circles 44 and 44&#39;. Consequently, their teeth 41 and 41&#39; assume the all-addendum form. The belt 1 is manufactured by winding its material around an external gear 11 having teeth 12 shown in FIG. 3. The tooth space 13 and the tooth surface 14 of the external gear 11 form a tooth 2 and a tooth surface 3 respectively, of the belt. To make a belt having a straight-sided teeth, the tooth surface 14 of the external gear 11 must be straight-lined. The external gear 11 having such straight-sided teeth can be cut by the so-called single cutting machine. However, this tooth cutting method has such shortcomings as follows: 
     I. Since tooth cutting must be effected while driving one tooth after another, tooth pitch error is liable to be introduced. 
     II. To permit the tip fillet curve 15 of the external gear 11, that forms the root fillet curve 4 of the belt 1, to smoothly join the outside diameter, it is necessary to use cutting machinery, workmanship and material of great accuracy for the manufacture of said external gear, which necessitates much time and labor. 
     III. A gear cutting operation according to this method is inefficient. 
     The above-described shortcomings can be obviated by generating the external gear 11 by topping, utilizing a rack-type tool. However, the form of the tooth of the gear generated by the rack-type tool with straight-sided teeth is not straight-sided but involute. Therefore, the tooth form of the rack-type tool must be designed with a special curve in order to obtain a gear with straight-sided teeth by the generating process. In addition, the tooth form of the tool must be changed when the number of gear teeth changes. That is, a different rack-type tool is required for every different number of teeth of the external gear. This means that a great number of costly rack-type tools must be prepared. The manufacture, maintenance and keeping of such tools will cost a great deal, which, as a consequence, makes the generation of an external gear with straight-sided teeth by the use of the rack-type tool very expensive. 
     The present invention eliminates the aforesaid shortcomings, as described hereafter with reference to the accompanying drawings. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a side elevation of a conventional positive motion belt with straight-sided elastic teeth. 
     FIG. 2 is a side elevation of the same positive motion belt engagingly wound over wheels. 
     FIG. 3 illustrates an external gear. 
     FIG. 4 illustrates a rack-type tool with a straight-sided tooth having a pressure angle αo. 
     FIG. 5 shows the tooth form of an external gear generated by the rack-type tool shown in FIG. 4. 
     FIG. 6 shows the tooth form of an external gear generated by the rack-type tool with a straight-sided tooth having a pressure angle αo + Δα. 
     FIG. 7 illustrates a rack-type tool with a partly curved, straight-sided tooth. 
     FIG. 8 schematically illustrates a difference in the tooth tip space of the belt, which results from a difference in angle between the pressure angle of the rack-type tool and the half core angle of the external gear. 
     FIG. 9 shows a positive motion belt with a concave-surfaced elastic tooth, formed by the external gear having the tooth form shown in FIG. 6. 
     FIG. 10 illustrates a conventional engagement of a tooth of an involute-toothed wheel with a tooth of a belt having straight-sided teeth. 
     FIG. 11 illustrates an engagement of a tooth of an involute-toothed wheel with a tooth of a belt having concave-sided teeth, according to this invention. 
    
    
     DETAILED DESCRIPTION 
     Referring to FIG. 5, the form of a tooth 28 of an external gear 11 cut by a rack-type tool 16 with straight-sided teeth having a pressure angle α o  (as shown in FIG. 4) assumes an involute curve 19 that has a pressure angle α o  on a pitch circle 18. When considering the tooth form of the tool and the external gear, chamfering at the edges of the tooth tip and the corners of tooth root as well as the radius of fillet are excepted. Assuming that the involute curve 19 and the pitch circle 18 intersect at point 20, the length of a circular arc 20-20&#39; is equal to the virtual thickness lo of the tooth 2 of the belt 1 on the pitch line 7. The virtual thickness lo is a constant value that is determined by belt size expressed by pitch length between adjacent teeth of a belt or the like. There arises a deflection δ 1  k between points 22 and 23 where the tangent 21 of the involute curve 19 at point 20 and the involute curve 19 intersect the outside diameter 25 of the external gear 11, and a deflection δ 1  f between points 26 and 27 where said tangent 21 and involute curve 19 intersect the bottom land 24 of the external gear 11. The absolute values δ 1  k and δ 1  f become smaller with an increase in the number of teeth in the external gear 11; that is, δ 1  k and δ 1  f become zero when the number of teeth becomes infinite. Assuming now that δ 1  k and δ 1  f of an external gear having a minimum number of teeth are Dk and Df, the values of δ 1  k and δ 1  f become smaller than the absolute values of Dk and Df as the number of teeth of the external gear increases. δ 1  k and δ 1  f are calculated from the following equations: ##EQU1## where 
     d = 1/2 of the difference between the outside diameter 25 of the external gear 11 and the diameter of the pitch circle 18 
     Dp = The diameter of the pitch circle 18 of the external gear 11 
     h = The depth of the tooth space 13 of the external gear 11 
     αo = The pressure angle of the rack-type tool 16 used for tooth cutting 
     It must be additionally noted that, within the range of the practically usable number of teeth on the external gear 11, a tooth surface 28 of the external gear 11 becomes totally consistent with the involute curve 19, owing to the tooth form conditions, including a pressure angle αo, of the tool. 
     In FIG. 6, a tooth form 30 of an external gear 11&#39; cut by a rack-type tool having straight-sided teeth with a pressure angle αo + Δα coincides with an involute curve 29. There arise a deflection δ&#39; 1  k between points 22 and 31 where the tangent 21 of the involute curve 19 and the involute curve 29 intersect the outside diameter 25 of the external gear 11&#39;, and a deflection δ&#39; 1  f between points 26 and 32 where said tangent 21 and involute curve 29 intersect the bottom land 24 of the external gear 11&#39;. δ&#39; 1  k and δ&#39; 1  f are calculated from the following equations, with the extent of each deflection being adjustable by Δα: ##EQU2## 
     The involute curve 29 increases its curvature and deviates more and more from the tangent 21 of the involute curve 19, as it approaches the bottom land 24 of the external gear 11&#39;. It is therefore effective, for making small the value of δ&#39; 1  f as well as the curvature at the root of the tooth form 30 of the external gear 11&#39;, to form the greater part of the tooth surface of a rack-type tool 33 shown in FIG. 7 with a straight line 34 and the remaining small part, for instance, about one-third from the tip of the tooth, with a curve 35. To be more precise, the tooth form of the external gear 11&#39; generated by the straight-lined portion 34 of the rack-type tool 33 forms the involute curve 29, whereas the tooth form generated by the curved portion 35 presents a curve deflected in such a direction as to relatively narrow the width 32-32&#39; of the bottom land 24, compared with the tooth form generated by an all straight-sided rack-type tool. The curved portion 35 of the tool 33 may be substituted by an arc of a circle. Assuming that the maximum values which said δ&#39; 1  k can take is D&#39;k  and that said δ&#39; 1  f can take is D&#39;f, the maximum amount of relief ε at the tooth tip of the tool 33 must be expressed by ε = D&#39;f - D&#39;k. 
     If the angle formed between points 20 and 20&#39; on the pitch circle 18 of the external gear 11 in FIG. 5 is 20, the pressure angle of the involute curve 19 on the pitch circle 18 is αo. Therefore, the angle of intersection of the tangents 21 and 21&#39; of the involute curve 19 and 19&#39; becomes 2(αo + θ), which is twice as great as the pressure angle αo of the tool, plus 20. θ is calculated from the following equation: ##EQU3## 
     In FIG. 8, let us consider two straight-sided tooth forms 36 and 37, having a pressure angle αo and αo + θ, respectively, at point 20 on the pitch circle 18. Then there arise deflections δ 2  k and δ 2  f between points 38 and 23 and points 39 and 27 where said tooth forms 36 and 37 intersect the outside diameter 25 and the bottom land 24. δ 2  k  and δ 2  f  are calculated from the following equations: 
     
         δ.sub.2 k = d {tan (αo + θ) - tanαo} 
    
     
         δ.sub.2 f = (d + h) {tan (αo + θ) - tanαo} 
    
     Assuming that the aforesaid δ 2  k  and δ 2  f  become δ&#39; 2  k  and δ&#39; 2  f  when the pressure angle αo is changed to αo + Δα, δ&#39; 2  k and δ&#39; 2  f are calculated from the following calculation: 
     
         δ&#39;.sub.2 k = d {tan (αo + Δα + θ) - tan (αo + Δα)} 
    
     
         δ&#39;.sub.2 f = (d + h) {tan (αo + Δα + θ) - tan (αo + Δα)} 
    
     Then, let us consider that: 
     
         2(δ&#39;.sub.1 k + δ&#39;.sub.2 k) = 2δ&#39;k 
    
     
         2(δ&#39;.sub.1 f + δ&#39;.sub.2 f) = 2δ&#39;f 
    
     From the above, δ&#39;k and δ&#39;f are defined as follows: 2δ&#39;k shows a difference between tooth thickness l 1  at the root of the tooth of the belt 1 with a straight-sided tooth 2 having a virtual tooth thickness lo and pressure angle αo on the pitch line 7 and tooth space 31-31&#39; on the outside diameter 25 of the external gear 11&#39; cut by a rack-type straight-sided tooth-cutting tool having a pressure angle αo + Δα. 2δ&#39;f shows a difference between tooth thickness l 2  at the tip of the tooth of the same belt and tooth space 32-32&#39; at the bottom land 24 of the external gear 11&#39;. 
     With the assumption that 2δ&#39;k - 2δ&#39;f -2δ&#39;, 2δ&#39; is defined as showing a dimensional error permissible for an ideal tooth space (31-31&#39; - 2h tanαo) at the bottom land 24 of the external gear 11&#39;, which determines the tooth tip thickness of the belt to be cut thereby, with respect to the tooth space 31-31&#39; on the outside diameter 25 of said external gear 11&#39;. 
     The thickness l&#39; 1  and l&#39; 2  at the tooth root and tip of the belt 46 (FIG. 9) cut by the external gear 11&#39; having the aforementioned 2δ&#39;k and 2δ&#39;f differ with the number of teeth in the external gear 11&#39;. At the same time, the tooth surface 47 of the belt 46 made by the external gear 11&#39; forms a concave involute curve, because the external gear 11&#39; has a convex involute tooth form 30. 
     The belt 1 and the toothed wheels 40 and 40&#39; are considered to engage with each other as in the case of a rack and a pinion. If a belt 46 is considered a rack, the aforesaid 2δ&#39;k is a value affecting the back-lash existing between the tooth 48 of the belt 46 and the teeth 41 and 41&#39; of the toothed wheels 40 and 40&#39;. Also, 2δ&#39; calculated from equation 2δ&#39;k - 2δ&#39;f = 2δ&#39; and the concave curve on the tooth surface 47 of the belt 46 are considered tooth form errors. 
     The aforementioned δ&#39;k and δ&#39;f were calculated with regard to the belt tooth forms of every pitch specified in British Standards 4548-1970, Specifications for Synchronous Belt Drives, assuming that Δα is approximately one degree. For the belts having great pitch dimensions, the tooth form of external gears cut by a partly curved, straight-sided rack-type tool were considered. It was then found out about the tooth form of every pitch that the amount of change in δ&#39;k (that is, the range of values which δ&#39;k can take) with the number of teeth in the external gear was small, and all such values were positive. It was also found that the amount of change in δ&#39; with the number of teeth in the external gear was small too, and, furthermore, the value of δ&#39; could be kept within a range between a very small positive value to a small negative value or between zero and a small negative value, depending on how ε in FIG. 7 was determined. The positive δ&#39; for the tooth tip space of the belt 46 becomes the amount of interference in the engaging mechanism of rack and pinion, while the negative δ&#39; becomes that of relief. In the engagement of the belt 46 and the toothed wheels 40 and 40&#39;, the presence of a very small positive δ&#39; does not constitute an obstacle to such engagement, because the tooth 48 of the belt 46, made of rubber-like elastic material, can permit some extent of transformation different from gears of hard metal. If the presence of such positive δ&#39; is undesirable, it can be changed to negative regardless of the number of teeth of the belt, by changing the value of Δα or ε of the tool 33 shown in FIG. 7, or the interrelationship therebetween. 
     When the above-described belt 46, which differs, in tooth tip thickness slightly from the base dimention and has a concave tooth surface 47, is placed over the wheels 40 and 40&#39; and moved, the belt smoothly engages the wheels. 
     The moment the teeth 2 and 48 of the belts 1 and 46, respectively, engage with the wheels 40 and 40&#39;, the tooth surfaces 3 and 47 of the belts 1 and 46 receive from the tooth surfaces 42 and 43 reactive force originated from power transmission. 
     For the transmission of power, it is more advantageous to engage the belt 46, having the concave tooth surface 47, of this invention with the wheels having the convex tooth surface 42 and 43, than to do so with the belt 1 with the straight-sided teeth 3, since the former produces less contact stress between both tooth surfaces (see FIGS. 10 and 11). Its superiority in durability has also been experimentally proven. 
     As described above, this invention permits cutting of external gears for the manufacture of timing belts having various numbers of teeth of the same pitch, using one rack-type gear-cutting tool. This means that one tool is enough for each pitch, thus eliminating the necessity of preparing many costly rack-type tools and facilitating the maintenance and keeping of such tools. Because the tooth form of this rack-type tool is straight-sided or partly curved, it can be more easily manufactured than a rack-type tool with totally curved teeth. Also the belt is designed with concave teeth so as to properly engage with convex-surfaced wheel teeth, which decreases contact stress during the transmission of power between both teeth and increases the durability of the teeth or the belt. These are the examples of remarkable results expected from this invention. 
     Although a particular preferred embodiment of the invention has been disclosed in detail for illustrative purposes, it will be recognized that variations or modifications of the disclosed apparatus, including the rearrangement of parts, lie within the scope of the present invention.