Abstract:
A system for efficient torque transmission, consisting of at least one driving member and at least one driven member, which uses a complementing spline profile having almost nil stress concentration which enhances the overall torque transmission capacity of a shaft with a given diameter.

Description:
CROSS-REFERENCES TO RELATED APPLICATIONS 
       [0001]    This application claims the benefit of PCT application PCT/IN2008/000865, filed on Dec. 24, 2008, and incorporated herein by reference. 
     
    
     FIELD OF THE INVENTION 
       [0002]    The present invention, in general, relates to the shape of the tooth-profile of a shaft used for torque transmission. More specifically, it relates to the geometrical design of the tooth of a shaft in a twin-screw extruder. The design ensures almost nil stress concentration on the tooth while transferring the torque from one rotating member to another in the twin-screw extruder. 
       BACKGROUND 
       [0003]    Torque transmitting systems are ubiquitous in all forms of mechanical devices. All torque transmitting systems must have a driving member and a driven member. Conventionally, torque is transmitted from the driving member to the driven member through means of key-ways, serrations, spline-profile teeth and the like. In case of spline-profile teeth the stress developed as a result of torque transmission, is not well distributed in the driving member and the driven member due to discontinuities in the spline-profile teeth. As a result, stress levels are high in certain regions of spline-profile teeth. This phenomenon is known as stress concentration. Stress concentration occurs at those regions of the tooth spline-profile teeth where there are discontinuities, making such regions susceptible to failure. This stress concentration limits the torque transmitting capacity of the torque transmitting system. 
         [0004]    In case of twin-screw extruders, driven members associated with twin-screw extruders will be hereinafter referred to as extruder elements and driving members associated with twin-screw extruders will be hereinafter referred to as extruder shafts. For better and efficient compounding of polymers in twin-screw extruders with a given extruder geometry, the extruder shafts need to be bigger, which results in thinner cross-section for the extruder elements. The problem of stress-concentration multiplies in case of thin walled driven members, i.e., extruder element having thin wall thickness. For extruder elements having wall thickness less than 2-3 mm, stress concentration dramatically reduces stress bearing capacity. This limits the torque transmitting capacity of the extruder elements, thereby reducing the efficiency of the twin-screw extruder. 
         [0005]    Efforts have been made to increase the torque transmitting capacity of the torque transmitting systems by designing various spline-profile teeth. Spline-profiles employed in the prior art include the straight-sided spline with polygonal tooth profile, the involute tooth profiled spline, and the cycloidal tooth profiled spline. All the profiles employed earlier have failed to effectively remove the stress concentration. Many of the spline-profiles disclosed in the prior art have discontinuities in the profile, which is the prime reason for stress concentration. Even the spline-profiles which have no discontinuities do not completely eliminate stress concentration. 
         [0006]    For example, U.S. Pat. No. 5,716,159 titled “Spline and spline for rotary elements of multi spindle extruder” discloses a system which uses an elliptical spline-profile for tooth of extruder elements of a multi spindle extruder. The spline-profile has a semi-elliptical profile for its crest and a similar semi-elliptical profile for the trough. However, in the multi spindle extruder disclosed in the patent, the point where two semi-elliptical parts meet, forms a discontinuity and may lead to stress concentration. This again limits the torque transmitting capacity of the shaft. 
         [0007]    It is therefore necessary to have a torque transmitting system which has stress-concentration free profile tooth so that there is improved torque transmitting capacity of the shaft for a given diameter. 
       SUMMARY 
       [0008]    The present invention discloses a tooth profile for improved torque transmitting capacity and for smooth transmission of torque from the driving member to the driven member in a torque transmitting system. 
         [0009]    The present invention is accomplished by a spline-profile tooth having a torque transmitting system comprising a driving member having stress-concentration-free-profile teeth on its outer surface. The system further has a driven member having stress-concentration-free-profile teeth conjugate to the stress-concentration-free-profile teeth of the driving member for transmitting torque therebetween. The stress-concentration-free-profile teeth of the driving member and the driven member have a tooth profile defined in a plane of r-Θ polar coordinate system by the mathematical expression for a half curve being defined by 
         [0000]        r=h[Θ/β− 1/2π(sin (2πΘ/β))], in a defined area of 0≦Θ≦β,
 
         [0010]    wherein, r is the radial distance from the final base surface on which the tooth profile is generated; 
         [0011]    Θ is a rotational angle varying from 0 to β; 
         [0012]    h is a constant denoting the depth of the tooth; 
         [0013]    βis a constant equal to (π/z); and 
         [0014]    z is the number of teeth. 
         [0015]    The complete tooth profile is obtained by combining the half curve with its mirror image about a radial axis passing through central axis of the driving member and one of the two extreme points of the half curve. 
     
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         [0016]      FIG. 1  shows a half tooth with stress-concentration-free-profile, according to an embodiment of the invention. 
           [0017]      FIG. 2  shows complete tooth with stress-concentration-free-profile, according to an embodiment of the invention. 
           [0018]      FIG. 3  shows a cross-sectional view of an extruder shaft with stress-concentration-free-profile on its outer surface, according to an embodiment of the invention. 
           [0019]      FIG. 4  shows the two engaging members of a torque transmitting system, according to an embodiment of the invention. 
           [0020]      FIG. 5  shows the schematic diagram of the forces acting on a spline-profile tooth, according to an embodiment of the invention. 
           [0021]      FIGS. 6A and 6B  depict the torque transmitting mechanism employed in a co-rotating twin-screw extruder according to an embodiment of the invention. 
       
    
    
     DETAILED DESCRIPTION 
       [0022]    The present invention will be discussed hereafter in detail with reference to accompanying drawings and embodiments. 
         [0023]    The present invention relates to a torque transmitting system which uses a particular tooth profile for transfer of torque from one rotating member to another. The tooth profile disclosed in this invention will be hereinafter referred to as stress-concentration-free-profile. 
         [0024]    According to an embodiment of the invention, torque transmitting system is a twin-screw extruder. 
         [0025]    However, it would be apparent to a person skilled in the art that the torque transmitting systems would include but not limited to single-screw extruders, multi-screw extruders, shaft-gear systems and gear-gear systems. 
         [0026]    The stress-concentration-free-tooth profile is such that the torque is transmitted from the driving member to the driven member with minimum stress concentrations on the spline tooth and hence provides resistance to the material failure of the driving member or driven member, thereby avoiding enormous economic damages and safety hazards. 
         [0027]      FIG. 1  shows half-tooth profile  102  having stress-concentration-free-profile, according to an embodiment of the invention. The stress-concentration-free-profile of the tooth is obtained by computer controlled metal cutting and machining methods (such as EDM or Wire Electro Discharge Method) on the surface of the driving member or the driven member. 
         [0028]    According to an embodiment of the invention, the cutting of stress-concentration-free-profile tooth is governed by following equation: 
         [0000]        r=h[Θ/β− 1/2π(sin (2πΘ/β))], in a defined area of 0≦Θ≦β,
 
         [0000]    where:
 
r is the radial distance from the base surface  106  on which the tooth profile is generated;
 
Θ is a rotational angle varying from 0 to β, Θ=0 starts at point  104  and sweeps to Θ=β at point  108 ;
 
h is a constant denoting the depth of the tooth;
 
β is a constant equal to (π/z); and
 
z is the number of teeth.
 
         [0029]    The origin of the r-Θ cylindrical co-ordinate system lies at the central axis of the driving member. 
         [0000]    Here, base surface  106  is the outer surface of the driving member. 
         [0030]    The equation r=h[Θ/β−1/2π(sin(2πΘ/β))] hereinafter will be referred to as the stress-concentration-free-profile equation. The stress-concentration-free-profile equation has been determined using numerical analysis methods such that the resultant tooth profile provides a minimal stress concentration factor. However, it must be evident to a person skilled in the art that the name ‘stress-concentration-free-profile equation’ is not limited to above mathematical formulae only. The term would be applicable for any mathematical equation which is continuous and has continuous first, second and third order derivatives. Thus, any minor modifications in the mathematical equation r=h[Θ/β−1/2π(sin(2πΘ/β))] for reducing the stress-concentration factor would fall within the scope of this invention. 
         [0031]    A person skilled in art will be able to appreciate the fact that the stress-concentration-free-profile tooth can be designed on the surface of the driven member using the stress-concentration-free-profile equation. Further, the tooth profile thus generated on the surface of the driven member will be conjugate to the stress-concentration-free-profile of the tooth of the driving member. 
         [0032]    According to an embodiment of the invention, the stress-concentration-free-profile equation is employed to design external gear teeth of a flexispline member in a harmonic gear speed reducer. (The Harmonic gear speed reducer is a type of torque transmitting system, having flexispline as one of the engaging members. The flexispline is a thin-walled flexible cup with external splines on its lip.) The external gear teeth of the flexispline mesh with internal teeth of a circular rigid ring. The circular rigid ring is a circular thick walled rigid ring machined with internal teeth. According to an embodiment of the invention, the internal teeth of the circular rigid ring can also be designed employing stress-concentration-free-profile equation. 
         [0033]    According to an embodiment of the invention, the stress-concentration-free-profile equation is employed to design the tooth profile of the extruder shaft and the extruder elements of a twin-screw extruder system. The stress-concentration-free-profile equation defining the tooth profile is such that the first four derivatives of the equation with respect to the angle Θ are continuous at all values of Θ from 0 to β excluding the endpoints (Θ=0 and Θ=β). The first derivative implies the rate of change of radial distance r with respect to angle Θ and each further derivative is the rate of change of the previous derivative with respect to angle Θ. The continuity of the derivatives implies a smooth and continuous half-tooth profile  102 . This smooth and continuous half-tooth profile  102  ensures smooth and jerk free torque transmission between the two engaged members. 
         [0034]      FIG. 2  shows the complete tooth profile  200  obtained by joining the half-tooth profile  102  with its mirror-image  202  about radial axis  204 . Radial axis  204  passes through the central axis of the driving member and point  108 . 
         [0035]    The first derivative of the stress-concentration-free-profile equation at its extreme points i.e., at Θ=0 and βis zero. Further, the mirror image of the stress-concentration-free-profile has first derivative equal to zero at its extreme points. Since the first derivative of resultant complete tooth-profile is continuous at the end points it ensures smooth (without discontinuities) profile throughout the spline. 
         [0036]      FIG. 3  shows the stress-concentration-free-profile  302  on driving member  304 , according to an embodiment of the invention. The complete spline profile is obtained by the stress-concentration-free-profile equation wherein the continuous line of half-tooth profile  102  is joined with its mirror-image  202  about radial axis  204  end to end, and the pattern hence obtained is repeated over the entire periphery of driving member  304 . 
         [0037]      FIG. 4  shows the two engaging members of the torque transmitting system according to an embodiment of the invention. The two engaging members are the driving member and the driven member of a twin-screw extruder. According to the embodiment of the invention, the driving member is extruder shaft  402  (which is similar to driving member  304  shown in  FIG. 3 ) of the twin-screw extruder and driven member is extruder element  404 . The tooth profile of extruder shaft  402  is governed by the stress-concentration-free-profile equation as shown in  FIG. 4 , while extruder element  404  has a tooth profile conjugating the stress-concentration-free-profile of the extruder shaft  402 . The conjugate tooth profile is governed by the stress-concentration-free-profile equation but is machined on the inner surface of extruder element  404 . Further, extruder shaft  402  has a plurality of extruder elements  404  mounted on it. Extruder shaft  402  transmits the torque to the plurality of extruder elements  404 . The stress-concentration-free-profile teeth corresponding to extruder shaft  402  and extruder element  404  inter-meshes and torque is transmitted therebetween. 
         [0038]      FIG. 5  shows the schematic diagram of the forces acting on spline-profile tooth  502 . Spline-profile tooth  502  is subjected to stresses in bending, shear, compressive and contact loads. Forces acting on spline-profile tooth  502  can be resolved into tangential force component FT  504  and the radial force component FR  506  as shown in  FIG. 5 . The stress concentration factor is found to be maximum at roots  508  and  514  of the spline-profile tooth  502 . 
         [0039]    Radial force component FR  506  induces compressive stress  512 . The stress due to radial force component FR  506  can be determined by the following equation: 
         [0000]      σ R=FR  /( L*tB )   (1)
 
         [0000]    where, L is the length of the tooth perpendicular to the plane of the paper; and tB is the tooth thickness at the base of the tooth. 
         [0040]    Tangential force component FT  504  induces a compressive stress at root  514  and tensile stresses  510  at root  508  of the spline-profile tooth  502 . The stress due to the tangential force component  504  can be determined by the following equation: 
         [0000]      σ T= (6 *FR*L*KF )/( L*tB 2)   (2)
 
         [0000]    For the determination of the stress concentration factor following formula by Dolan and Broghamer was used: 
         [0000]        KF=H+{ ( tB/rf ) N. ( tB/L ) M }   (3)
 
         [0000]    where: 
         [0000]        H= 0.331−0.436Φ;   (4a)
 
         [0000]        M= 0.261−0.545Φ;   (4b)
 
         [0000]        N= 0.324−0.494Φ;   (4c)
 
         [0041]    rf is the radius of the tooth fillet; 
         [0042]    tB is the tooth thickness at the base of the tooth; and 
         [0043]    Φ is the pressure angle (expressed in radians) 
         [0044]    The calculations below show comparison of the stress concentration factor for the stress-concentration-free-profile with that of the involute spline profile. 
         [0000]    Data for involute spline-profile:
 
tB=1.85 mm
 
       L=0.8 mm 
     Φ=30O 
       [0045]    rf=0.128 mm
 
Data for stress-concentration-free-profile (according to an embodiment):
 
tB=1.85 mm
 
       L=0.8 mm 
     Φ=40O 
       [0046]    rf=0.40 mm
 
Using the equations 3, 4a, 4b, 4c and the data of the profiles, the stress-concentration factors are computed as:
 
K F  (involute spline-profile)=1.28
 
K F  (stress-concentration-free-profile)=0.9
 
         [0047]    From the calculations shown above, it is clear that the stress concentration factor at roots  508  and  514  of a stress-concentration-free-profile tooth is very less compared to that of involute-spline profile tooth. Hence, switching from involute to stress-concentration-free-profile will ensure higher torque transmission capacity with same wall thicknesses of the torque transmitting members. In other words, for the same torque transmission capacity as the involute spline-profile torque transmitters, thinner walled torque transmitting members can be used in stress-concentration-free-profiles torque transmitters. Further, using thin walled torque transmission members will reduce the weight of the torque transmitter. The light weight torque transmitters are highly desirable in various applications in aerospace and automobile engineering. According to an embodiment of the invention, stress-concentration-free-profile teeth can be employed to design light weight gearboxes. The light weight gearboxes 
         [0048]      FIGS. 6A and 6B  depict the torque transmitting mechanism employed in a co-rotating twin-screw extruder according to an embodiment of the invention.  FIG. 6A  shows a cross-sectional view of a co-rotating twin-screw extruder. The twin-screw extruder system comprises barrel  604 . Barrel  604  is made of two parallel intersecting bores. Each bore consists of two extruder shafts  402 . The material to be extruded in twin-screw extruder is called charge. The charge to be extruded is put in barrel  604  and is subjected to processes of melting, mixing, conveying and kneading. 
         [0049]    The charge offers very high resistance against motion that extruder element  404  tries to induce on it. In order to overcome this resistance offered by the charge, extruder shaft  402  has to transmit more torque to each extruder element  404 . The high torque thus generated, causes stresses on the teeth of extruder shaft  402  and extruder element  404 . 
         [0050]    The torque transmitting capacity of extruder shaft  402  depends on the material property and the stress concentration factor, which in-turn depends on the geometry of the material and the loading pattern. The maximum permissible stress of the material of extruder shaft  402  puts a limit on maximum loading that extruder shaft  402  can bear without material failure. Loading pattern in the case of a twin-screw extruder is proportional to the amount of charge loaded into barrel  604 . Hence, for a given rotational speed of extruder shaft  402 , the maximum permissible stress limits the amount of charge being processed. This further limits the final extruded output. Using stress-concentration-free-profile teeth, the charge-filled volume of the barrel  604  can be increased, as greater loading is permitted due to increased stress bearing capacity. Hence, the extruded output increases substantially. 
         [0051]    According to an embodiment, the stress-concentration-free-profile teeth on extruder shaft  402  are manufactured by spark erosion method and stress-concentration-free-profile teeth of extruder element  404  are manufactured by WEDM (Wire Electrical Discharge Machining) method. 
         [0052]    Spark erosion method also referred to as Electrical Discharge Machining (or EDM) is used to manufacture extruder shaft  402 . EDM is a standard method of removing material by a series of rapidly recurring electric arcing discharges between an electrode (the cutting tool) and a work-piece, in the presence of an electric field. The EDM cutting tool is guided along the desired path very close to the work but it does not touch the piece. Consecutive sparks produce a series of micro-craters on the work-piece and remove material along the cutting path by melting and vaporization. 
         [0053]    EDM enables cutting of intricate contours or cavities in extremely hard material to very close tolerances, without the need for heat treatment to soften and re-harden them. 
         [0054]    Furthermore, WEDM is particular type of EDM in which wire is used as an electrode, with the wire being constantly fed from a spool. It is a generation process which is far more controllable and easier to operate. 
         [0055]    While the invention has been illustrated and described with reference to the certain embodiments thereof, it shall be appreciated by those ordinarily skilled in the art that various modifications can be made in the formalities and the details without departure from the spirit and scope of the invention.