COOPERATION OF CRANKSHAFT AND DRIVEN SHAFTS Engines like the single cylinder, the typical two cylinder in line, the typical four in line have a common disadvantage. The total kinetic energy of their pistons and connecting rods varies significantly during the cycle. In order to rotate the crankshaft at constant angular velocity while it drives connecting rods and pistons, an oscillating torque of high amplitude is necessary. Without this driving torque an intense oscillation of the angular velocity of the crankshaft is unavoidable, i.e. an angular deviation of the crankshaft from the point where it would be, if it were moved with its mean angular velocity, is observed.

The shafts driven by the crankshaft, like the primary shaft of the gearbox, the counterweight shaft, the electric generator shaft etc., have no reason to follow this strange rotation of the crankshaft. If we force them to follow this motion by inflexible gearing, e.g. the typical toothed gears, then there is an over loading of the system by inertia torque which oscillates between crankshaft and driven shafts, with all resulting disadvantages. FIG. 1 shows the basic parts on the table of a Fellow machine for manufacturing gears, by means of which we can modify its operation to manufacture toothed gears which when cooperate will give a desirable not linear relationship between their rotation angles.

FIG. 2 is relative to the geometrical construction of the vertical cut of a toothed gear Zo which cooperates with the given profile of another toothed gear Z _\ and achieves a given not linear transmission. In FIG. 3 is shown alone and complete the profile of a normal typical toothed gear with 35 uniform teeth and is also shown a set of intersecting curves which fonn a second toothed gear with 42 not uniform teeth. FIG. 4 shows the pitch circles of the gears of FIG. 3 in a proper radial scale. In FIG. 5 is shown a part of a toothed gear with 35 teeth and is also shown a set of intersecting curves which form a gear Z ₀ with 23 teeth. FIG. 6 shows the pitch circles of the gears of FIG. 5 in a proper radial scale. In FIG. 7 is shown a part of a toothed gear Zi with 35 uniform teeth. It is also shown a set of intersecting curves which form another toothed gear Z with 35 not uniform teeth. The mean transmission ratio is 1 :1. FIG. 8 shows the pitch circles of the gears of FIG. 7 in a proper radial scale. The problem which basically tries to solve the present application is the control, the reduction or even the elimination of the torsional inertia torque which oscillates between the crankshaft and the driven shafts, in the reciprocating piston engines. Neglecting the influence of the gas pressure, e.g. by the removal of the cylinder head, removing also the influence of the driven shafts, by proper unmeshing or remove, and regarding as negligible the energy loss due to

friction, we examine the rotary motion of the crankshaft. We give the crankshaft an initial push for rotation and then, leaving it alone, we register its rotary motion as it drives connecting rods and pistons. The crankshaft will perform a completely defined not uniform rotary motion so that during a complete rotation the total kinetic energy of it together with the driven connecting rods and pistons will remain constant. This variable rotary motion of the crankshaft may also be calculated arithmetically or may be defined analytically. Every other rotary motion creates variable total kinetic energy of the crankshaft and the driven connecting rods and pistons, and so in order to be realized it demands a driving torsional torque which every moment offers or absorbs the necessary amount of kinetic energy. A typical counterweight shaft tends to rotate with constant angular velocity and this is the only case it has a constant kinetic energy due to rotation. Any kind of oscillation of its angular velocity requires torsional driving torque. So to be the cooperation of the crankshaft with the counterweight shaft absolutely free of inertia torsional torque it is necessary the crankshaft to rotate with the rotary motion mentioned before, which insures constant total kinetic energy of the crankshaft, the connecting rods and the pistons, but it is also necessary the driven counterweight shaft to rotate with constant angular velocity which is the only motion that insures to the counterweight shaft constant kinetic energy during rotation. So it is necessary the ratio of the transmission from the crankshaft to the counterweight shaft to be not constant during a cycle of operation, but to variate with a specific way which permits the above motion. If this variable, during each cycle of operation, transmission ratio is achieved and we give an initial push for rotation to the counterweight shaft which is in cooperation with the crankshaft, and then release it, the counterweight shaft will continue its uniform rotation while the crankshaft will rotate with a not uniform rotation which insures constant total kinetic energy of the crankshaft and the driven connecting rods and pistons, and the transmission between counterweight shaft and crankshaft will be completely free from oscillating inertia torsional torque.

The same happens while the crankshaft drives the shaft of an electric generator but also while the crankshaft drives the primary shaft of the gearbox. In these cases the driven shaft tends to rotate with a substantially constant angular velocity, during each rotation, when it is rotated free of driving torque.

From the above description it is obvious that there exist a characteristic rotary motion of the crankshaft which maintains the total kinetic energy of the crankshaft and of the connecting rods and pistons it drives, constant. Said motion, according to its definition, does not demand torsional driving

torque. This motion is described by the function which relates the angular offset me(f) of the crankshaft from the angle fin which the crankshaft would be if it were rotated with its mean angular velocity. With the term mean angular velocity of a rotating shaft we mean here the constant angular velocity with which if the crankshaft were rotated it would need the same time for a complete rotation. The angular offset me(f) is independent from the mean frequency of rotation of the crankshaft, which is easy to be proved since the kinetic energy of the crankshaft, the kinetic energy of the driven connecting rods as well as the kinetic energy of the driven pistons are all functions of the rotation angle of the crankshaft times the square of its mean rotation speed. If we call fj the rotation angle of the crankshaft and f the angle in which the crankshaft would be if it were rotated with constant angular velocity equal to its mean angular velocity, then: me(f) = f! - f (1) The typical driven shafts have the tendency to rotate with constant angular velocity. This exactly happens with the countershaft, the main shaft of an electric generator and the primary shaft of the gearbox in the typical engines used in vehicles. So in the case of driven shaft which has the tendency to rotate with constant angular velocity, the transmission between the shaft and the crankshaft has to be such that when the driven shaft rotates with constant angular velocity, the crankshaft is forced through the transmission to rotate with the characteristic rotation described by the relationship (1). The Fellow machine is a widely used machine for manufacturing toothed gears. It uses a cutting tool of the form of a toothed gear. The cutting tool and the wheel to be formed are both rotating in a way so that the variation of the rotation angle of the cutting tool to the variation of the rotation angle of the wheel to be formed is constant and equal to the transmission ratio. Furthermore the cutting tool reciprocates in relation to the wheel to be formed, removing, from the wheel to be formed, material which has to be removed so that it will not spoil the future cooperation of the toothed gears. The principle is that the creation of the toothed gear is based substantially on the representation of the cooperation of the toothed gear to be formed with a toothed gear which is similar to the cutting tool. On a Fellow machine we locate a cutting tool of the same form as the toothed gear which is secured on the shaft which is driven by the crankshaft. This toothed gear may be even a typical toothed gear with involute teeth. The wheel Z, according FIG. 1, which will be secured to the crankshaft is located for formation on the Fellow machine. The machine is set to cut the wheel as if it were a typical toothed gear, that is the machine forces the table L, where normally is secured the wheel to be formed, to an angular rotation so that the variation of the rotation angle of the table L to the variation of the

rotation angle of the cutting tool is always constant and equal to the ratio of the number of teeth of the cutting tool to the number of teeth of the gear Z which we want to foπn. Until here the procedure is identical to what happen in the Fellow machine while the manufacturing of typical toothed gears. The difference is that we keep the wheel to be formed on the table L so that it can rotate in relation to L, and we impose to the wheel to be formed an angular offset me(f) relating to the table L, where f is the imposed angle from the Fellow machine to the table L, and me(f) is a function of the f defined according the previous description, so that to achieve the desirable improvement in the cooperation of driving and driven shafts. That is the rotation of the wheel Z to be formed is a combination of one rotation at angle f together with the holding table L, and of one additional rotation at an angle me(f) relating to the holding table L. We considered before, that in relation to the rotation angle f of the table L, the wheel to be formed presents an angular offset me(f) from the table L. There are many ways to realize a similar demand and we will describe one of them. The arm AC is articulated at the end A of a proper extension L' of the table L of the Fellow machine, according FIG. 1. The arm BC is articulated at the end B of a proper extension G' of the chock G. On the chock G is secured the wheel Z to be formed. The chock G as we said is articulated on the table L, at the center of rotation O, permitting the relevant rotation of the chock G, so permitting the relevant rotation of the wheel Z which is secured to the chock G, relating to the table L. The above two arms AC and BC are articulated at the point C. The point C follows, as the table L rotates, a specific path D, by means of a proper extension which is moving inside a corresponding groove of the immovable slate E. As the table L is rotated, the arm AC articulated to the table L at the point A, has its point C moving along the path D. Due to the cooperation of the table L and of the chock G through the articulation at the point C of the two arms AC and BC, the chock G is forced to a rotation which presents a specific angular offset me(f) with respect to the table L, depending on the rotation angle f of the table L and on the form of the path D. So it is enough the groove on the slate E to be made properly so that for each rotation angle f of the table L the corresponding angle of rotation of the chock G, in relation to the table L, to be me(f). Note that if the difference of the lengths of the OA and OB is small enough compared to the lengths of the arms AC and BC, then increased accuracy is achieved because for a small variation of the angle between the OA and OB, that is for a small variation of the angle between the chock G and the table L, it is necessary a significant radial offset of the point C. The above can similarly be applied in the case of helical gears. In FIG. 3 is shown alone and complete the vertical cut of a typical toothed

gear Z _\ with 35 teeth, it is also shown a set of intersecting curves which form a second toothed gear Z ₀ with 42 not uniform teeth. The intersecting curves are the successive positions, relating to the wheel to be foπned, of the profile of the toothed gear Z _\ as it acts like the cutting tool of the modified Fellow machine described previously, during the formation of the gear Z ₀ with the 42 teeth. When the toothed gear Zi is rotated uniformly intermeshed to the toothed gear Zo, the last one is forced to perform a not uniform rotation presenting, by the position where it would be if it were rotated with the 35/42 of the rotation speed of the gear Zi, an angular offset of sinusoidal form with frequency twice as much as the frequency of the rotation of the toothed gear with the 42 teeth and with a variation of -1.6 to +1.6 degrees. During its cooperation with the uniformly rotated toothed gear Zi, the angular velocity of the toothed gear Zo presents a maximum value which is 1 1% greater than its minimum value, difference which in general exceeds the necessary offset of the angular velocity of the crankshaft of typical engines, in case the proposed method is to be applied. O ₀ and Oi are the rotation centers of the Zo and Zi gears. Co and Ci are the pitch circles of gear Z ₀ and gear Zi respectively. The curve is a closed curve but to close it has to travel around the center Oi of the gear Zj three complete times. So the radius of the pitch circle of the Zi gear is a multiple-valued function of the angle around the gear Z _λ . This means that selecting on the Zi a start and a direction to measure the angles around the gear, then, there are angles between 0 and 360 degrees for which the radius of the pitch circle has more than one valid values. The Bo and Bj curves, drawn with dashed line, are two circles with centers at Oo and Oi respectively and represent the mean pitch circles, that is the pitch circles in case the transmission ratio is constant and equal to the ratio of the teeth of the two gears Zo and Z _\ . In FIG. 5 the two gears Z ₀ and Zi are not shown in cooperation. To cooperate these two gears Zo and Zi, the distance between their centers Oo and 0 _\ must be equal to the sum of the radii of the B ₀ and Bi circles.

In FIG. 4 are shown the pitch circles of the gears Zo and Zi of FIG. 3 in a proper radial scale. In this scale the radial offset of the Co and Ci pitch circles from the radius of the corresponding mean pitch circles B ₀ and Bi is equal to ten times the actual radial offset shown in FIG. 3, in order to be the details easily observed.

For the geometrical construction of the profile of the toothed gear Z ₀ which is secured to the crankshaft, given the profile and the initial position, relating to the intercenter line O ₀ Oι, of the toothed gear Zi which is secured to the driven shaft, and given the function fo(f i ) that defines the rotation angle fo of the toothed gear Z ₀ when the toothed gear Zi is rotated at an angle fi, are followed the next steps, according to the FIG. 2, where K is the profile of

the one side of one tooth of the Zj toothed gear in its initial position, and O ₀ , Oi are the centers of rotation of the toothed gears Zo and Z _\ respectively. We calculate the derivative dfo(fι)/dfι of the function which relates the fo with the fi . This derivative is the instant transmission ratio when the rotation angle of the Zi toothed gear is f] . For fi rotation angle of the Zj we define on the intercenter line OoOi the corresponding rolling center R based on the above instant transmission ratio, that is we select the point R of the intercenter line O ₀ Oι so that the ratio of the lengths OiR and O ₀ R is equal to the derivative df ₀ (fι)/df] . We rotate the profile K from its initial position at an angle fj, so results the curve K', and from the point R we draw a noπnal line to the K'. The trace C of this normal line, as the basic law of the gearing demands, is one contact point of the profile of the toothed gear Z ₀ with the profile of the toothed gear Zj, when the last one is rotated at an angle fj . We define the rotation angle fo of the toothed gear Zo which corresponds to the present angle of rotation ϊ of the toothed gear Zi, we rotate the trace C at an angle f ₀ around the O ₀ and so we define its place Co when the profile of the Zo is at its initial position. We show the geometrical definition of a point of the profile of the toothed gear Z ₀ which is one of the contact points of the profiles of the two toothed gears when the Zj is rotated at an angle fi . Repeating the previous procedure for all the teeth of the Zi and for all the values of the fj angle we defined die complete profile of the toothed gear Zo. As happens in the case of the typical toothed gears, similarly in the case of the proposed toothed gears there are some limits beyond which appears footcutting which reduces the strength of the teeth and in extreme cases results in abnormal cooperation. Such limits are the minimum number of the teeth as well as the extreme values of the angular offset me(f) and its derivative.

In FIG. 5 the Zi gear with 35 uniform teeth is only partially shown. The gear Zo has 23 not uniform teeth. In this case the angular offset me(f), used as previously described to form the gear Z ₀ from the Zj gear, has a Fourier analysis with significant components of first, second and third order. That is why the teeth of the Zo gear as well as the Co pitch circle are so unsymmetric. In FIG. 6 are shown in a proper radial scale, the same with that used in FIG. 4, the pitch circles C ₀ and d of the Zo and Zi gears of FIG. 5, respectively. The B ₀ and Bi are the mean pitch circles of Zo and Z _\ gears respectively. The radius of the pitch circle is a multiple- valued function of the angle around the gear Z ₃ , having, for each value of the angle around the Zi gear, twenty three corresponding values. That is, the d curve in FIG. 6 travels around Zi gear twenty three times before it closes. In FIG 7. is partially shown the Z _\ gear with 35 teeth. It is also shown a set of intersecting curves which foπn the gear Zo with 35 teeth too. The angular

offset me(f) used in this case is identical to that used in case of FIG. 5 and FIG. 6. The mean transmission ratio is 1 : 1 and so the mean pitch circles B ₀ and B] are two circles with equal radiuses. FIG. 8 is the representation of the pitch circles of the gears Zo and Zi of FIG. 7 in a proper radial scale, the same with that used in FIG. 4, thereby to be easily observed the details. In this case the pitch circles are both single- valued functions of the angle around the gears Z ₀ and Zj. A function is single-valued on a set S if it has just one value corresponding to each value in S. The gear Zi in FIG. 3, the gear Z\ in FIG. 5 and the gear Z _\ in FIG. 7 are deliberately identical to be shown that the pitch circles are curves depended on both cooperating gears, having radii which are either a multiple-valued function of the angle around the gear Z-, as happens in FIG. 3 and FIG. 5, or a single- valued function of the angle around the Zi gear, as happens in the case of FIG. 7. To see the application of the method on a typical single cylinder engine of a motorcycle having primary transmission with toothed gears, that is having a toothed gear Zo with zo teeth secured to its crankshaft, intermeshed with a toothed gear Z _\ with z _\ teeth driving the primary shaft of the gearbox through clutch, initially we calculate, either experimentally or arithmetically, the rotary motion of the crankshaft that keeps always constant the total kinetic energy of the crankshaft, of the connecting rod and of the piston. This motion is defined from the angular offset of the crankshaft from its position in case it were rotated with constant angular velocity equal to its mean angular velocity. Using the modified Fellow machine mentioned before, and using cutting tool of the same form to that of the toothed gear Zj, using if necessary as cutting tool a properly grinded and hardened gear like Zi, we manufacture according the previous description a toothed gear with Zo teeth. The angular offset we impose between the table L of the Fellow machine and the chock G on which is secured the wheel to be formed, when the table L is rotated at an angle f, is me(f) previously defined. When the Zo is completed, we secure it properly to the crankshaft. The problem of the elimination of the oscillating torsional inertia torque, which in every other case was conveyed from the crankshaft to the gearbox, is solved, the mean transmission ratio is equal to zo/zj and the distance from the crankshaft to the primary shaft of the gearbox is unchanged. The problem is solved by substantially substituting a single toothed gear. If the engine comprises also, as it is the typical case, a counterweight shaft of first order which is driven from the crankshaft by means of a pair of gears having the same number of teeth, we can following the same procedure as previously described to substitute the toothed gear which is secured to the crankshaft and drives the counterweight shaft with another toothed gear winch eliminates the

oscillating torsional inertia torque that fatigue this transmission. The above can similarly be applied to other arrangements like typical four in line, two in line etc. without any other complication. The described application of the method on a typical single cylinder engine leaves untouched the Zi gear which through clutch drives the primary shaft of the gearbox. This gear in general is a toothed gear with substantially uniform teeth and the problem of the elimination or the control of the torsional inertia torque is solved modifying just the Zo toothed gear secured to the crankshaft. The distance between the crankshaft and the primary shaft of the gearbox is unchanged. The mean transmission ratio of the primary transmission is also unchanged.

If the modification of the primary transmission of the above single cylinder engine, in the effort to achieve a desirable not linear relationship between the rotation angles of the Z ₀ and Z\, results in dissimilar teeth on the Zj toothed gear, then, in relation to the number of the teeth of Zo, the number of the teeth of the Z\ is restricted to a limited set of values. The me(f) in a single cylinder engine has minimum period a complete rotation of the crankshaft. So we must choose the number of teeth of the Zj equal to the product to an integer of the number of teeth of Zo, because in every other case, after a complete rotation of the Zi as it is intermeshed to the Z ₀ gear, the continuation of the cooperation of the two gears will be impossible. That is, if there exist dissimilarities between the teeth of the Z[ toothed gear then appear restrictions in the selection of the transmission ratio which in some cases is not a problem, like in the case the crankshaft is driving the counterweight shafts where the transmission ratio has to be an integer number, but in other cases may the permitted transmission ratios be not the desirable ones.

With the elimination of the oscillating inertia torsional torque conveyed from the crankshaft, beyond the improvement of the quality of operation, of the smoothness, of the roadholding and of the feeling, results also longer expected life of a number of important parts of the engine, like the crankshaft, the toothed gears and the bearings. The elimination of significant loads which were present continuously with high amplitude and high frequency, before the application of the above method, permits the reduction of the weight of significant parts as well as the reduction of the friction loss. It is also achieved a substantial reduction of the complication since we can remove the, before the application of the method, necessary elastic connections which were inserted between crankshaft and driven shafts.