In the former parts, the author applied the octahedral shearing stress-strain theory to the analysis of the plastic deformation of bent plate. However, the experiments on some metals, aluminium for instance, showed that it was better to employ the maximum shearing stress-strain theory. And so, the analytic calculation by means of that theory is described in this paper. Furthermore, the analysis of the bending of laminated plates of various metals is developed and it is shown theoretically that the thickness of the plate in the course of bending increases or decreases according to the mechanical properties of metals constructing that plate.

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This paper theoretically deals with a chatter vibration which occurs frequently when metals are being machined. An equation of the chatter has been introduced on the basis of the former experimental results that the fluctuation of horizontal cutting force lags behind the horizontal vibration of work during the chatter. By this equation it has been proved that the chatter is a self-excited vibration caused by this lag.

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Temperature rise of rolling bearing is determined by the frictional heat and the rate of heat transfer. In this paper are analysed these factors on forced lubrication using the data obtained from the measurements of the bearing temperature and frictional torque of the roller bearing N 307. Further, the measurement of the electrical resistance between the retainer and inner or outer race of the bearing shows that the lowest allowable oil viscosity at the operating bearing temperature is 10cp.

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Using the ball bearing 6307 instead of the roller bearing N 307, similar results to the first report have been obtained concerning the temperature rise, coefficient of friction, rate of heat transfer and the lowest allowable oil viscosity. The results of the grease lubrication which is antipodal to the forced lubrication in its nature are also added.

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Since the oil-whip was found by Newkirk and Taylor about 30 years ago, the effect of the oil film on this phenomenon has been more precisely studied by many people. It seems to us, however, that no satisfactory explanation for the phenomenon has been yet given, probably because of the inadequate assumptions they have made. In the present paper, the phenomenon is discussed as a problem of the dynamical stability of a rotating shaft with a journal bearing, considering the influences of the shaft motion on the pressure distribution of the bearing, and assuming that the pressure of oil in the unshaded region of Fig. 1 is zero. Then the equation of motion of the shaft is given by (13), and the Routh's criterion gives the stability chart as shown in Fig. 5 for this system, in which the shaded area represents the unstable region and the unshaded area the stable region. The symbols appearing in the stability chart are as follows : γ : radius of bearing b : width of bearing δ : radial clearance between shaft and bearing χ : eccentricity ratio F : transvers load upon the shaft m : (equivalent) mass of the rotor μ : coefficient of viscosity of the lubricant ω : angular velocity of the shaft. This chart shows that the system is always stable (the shaft can be always stably driven), if the deviation of the center of the shaft from that of the bearing is more than 70% of the bearing clearance, or if the vertical displacement of the shaft center from its lowest position is less than 50% of the clearance. This agrees with the practical experiences.

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The air-lubricated bearing which is of slight friction and of no temperature-rise by good cooling effect of air may by favourably applied to spinning spindles which usually rotate at a high speed of 8000∼12000rpm. In this research, as the 1st step of the attempt, the air-lubricated bearings are applied to spinning spindle without remarkable change in the design of the ordinary spinning spindle which is lubricated with oil. The performances of such air-lubricated spindles in ordinary operating condition are compared with those of the ordinary oil-lubricated spindles. As an experimental result, it was found that the frictional power of the air-lubricated spindle can be reduced to about 40% of that of the oil-lubricated spindle.

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Succeeding the research of the 1st report, the design of air-lubricated bearing of the spinning spindle has been changed and specialized, and the operating characteristics of the spindle is inspected, comparing the oil-lubricated spindle with the formerly designed air-lubricated spindle. As an experimental result, it is obtained that the frictional power is reduced to about 30% of that of the oil-lubricated spindles and about 50% of the formerly designed air-lubricated spindle. Such result shows an evident advance comparing with the result of the 1st report.

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Succeeding the previous reports, the new-designed spindle with a radial air-lubricated bearing and a thrust bearing made of a steel ball has been investigated. Such design of thrust bearing is taken owing to the fact that the thrust air-lubricated bearings with conical surface are not sufficient to reduce the frictional power as described in the previous reports. But the design of the upper radial bearing is left as same as it was in the 2nd report, because it had the good characteristic. As an exprimental result, it is clarified that the new spindle is superior to the other spindles especially in the range of high speed rotation, and its frictional power at 12, 000rpm is as small as about 25% of the ordinary oil-lubricated spindle, 60% of the air-lubricated spindle in the 1st report and 80% of the spindle in the 2nd report. And moreover, by measuring the flowing quantity of air and calculating the power required to compress air, the total power of the new spindle, that is the sum of the frictional and air-compressing power, is shown to be less than the power of the ordinary oil-lubricated spindle. This fact shows that the air-lubricated bearing is practically useful for the spinning spindles.

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The purpose of this study is to secure all the reversible gears in ex-or internal gearing. In view of the fact that the theory of this kind hitherto on this question was lacking in examining fully the intervals of parameters used in them, it seems that these gears have not been examined to the minutest details and in the ways of discarding the useless points. Now, to attain this purpose, I will treat of the problem to decide these gears by giving the number of teeth in contact "n", the pressure angle "α", and the gear ratio "ε". Then it follows as a necessary consequence that we will meet the well-known problem on maximum or minimum number of teeth. It is noticeable that we cannot completely solve the problem to decide these gears by giving the number of teeth of both gears z₁ and z₂, till the above-mentioned problem is solved. In this report I will deal with the case of usual external gearing. And I will state the study for internal gearing in the second report.

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In this paper I will treat of the process to decide all the reversible gears in internal gearing by giving the number of teeth in contact "n", the gear ratio "ε" and the pressure angle "α" or by giving the number of teeth of both gears z₁ and z₂. The courses of the study are alike the preceding report, save the argument of the trochoidal interference. With n≥1, ε>1, given, we can produce the trochoidal interference in internal gearing, if we prefer the properly small positive number α as the pressure angle. Therefore, we cannot evade the argument of the trochoidal interference in the study of internal gearing.

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The pan-angular intermeshing with unintersecting contact lines at both sides of tooth is the most general case in gearing of a pair of cylindrical toothed gears, but the ortho-angular intermeshing with intersecting contact lines is only a special case. Unadoptability of usual theory or equations based on ortho-angular intermeshing condition to gear shaving which is an application of direct contacting pan-angular intermeshing made the author solve involutehelicoid to clear up the pan-angular intermeshing theory including fundamental equations that an arbitrary pair of cylindrical gears can intermesh perfectly and correctly with free axle cross angle in a certain range with straight contact lines of equal pressure angles and equal distances from common perpendicular of gear axles and that a relation such as equality of helical angles, intersection of contact lines or an evaluation of any one of axle cross angle, center distance and contact line distance between gears determines their intermeshing condition uniquely, making accurate numerical calculations possible. The problem of eccentric error in gear shaving is analyzed by this theory, for example. Applications of the fundamental and derived equations to shaving and to hobbing are discussed. Result of numerical analysis of the actual shaving data is shown.

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To make clear the space image of meshing characters and to contribute to manufacturing of hypoid gears, the writer introduced a general idea of "gear mediator", giving teeth to which we can generate a pair of hypoid gears like by a rack in spur gears and a crown gear in bevel gears. Relating to it the writer treated the following in this paper : (1) The character of the mediator in general. (2) Specific slidings of the hypoid gears generated by the mediator. (3) The limiting cases in meshing of the hypoid gears generated by the mediator.

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Relating to the peculiarities of hypoid gears the writer made it clear that : (1) The surface of meshing (the plane in spur gears and the sphere in bevel gears) does not exist in hypoid gears. (2) There are infinite pitch lines between a pair of hypoid gears. The sliding lines on a tooth surface can be thought to be the intersections between the tooth surface and the pitch surfaces. (3) Relating to the pressure angle, we can settle the tooth normal in order to get an adequate progress of the line of contact on it. (4) The domain of "recession" is defined in the form that it is the domain in which the frictional forces between the tooth surfaces make the positive drive to the follower.

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