The authors analyzed the axial motion of the horizontal rotor supported by a ball bearing and a journal bearing. As the analytical results, it is shown that the ball bearing possesses the spring constant, varying according to the arrangement of the balls, So in some regions the motion becomes unstable and the subharmonic vibration of order 1/2 appears. The statical experiments in order to determine the stiffness of the ball bearing and the dynamical experiments to measure the subharmonic vibrations of order 1/2 were carried out for the two ball bearings, having 3 and 6 balls, respectively. The unstable regions where the subharmonic vibration arose agreed with the analytical results. At the boundaries of the stable and the unstable, the subharmonic vibration and the forced vibration caused by the inclination of the groove of the inner ring built up a beat. The period and the phase of the beat agreed with the analytical results.

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In this paper, steady, forced vibrations in an unsymmetrical piecewise-linear system of one-degree-of-freedom, whose characteristics for restoring and damping force are composed of two distinct linear regions, are analysed by means of a new method utilizing appropriate Fourier series expansion. Main features of this method consist in the following procedures : 1) Linearizing the original nonlinear equation of motion by assumedly expanding the non-linear part of the restoring and damping force into a Fourier series with the same period as of the given exciting force. 2) Obtaining the formal solution of thus linearized equation by deeming the above-mentioned nonlinear part as if it were an exciting force from without. This solution contains certain unknown coefficients of the Fourier expansion above assumed. 3) Determining these unknown coefficients from the conditions that the above obtained formal solution satisfies the given piecewise-linear characteristics of the system. A certain process of convergency improvement by means of series transformation may prove highly effective in this step.

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Results of analysis, by means of the Fourier series method introduced in the first report, of steady, forced vibrations in an unsymmetrical piecewise-linear system, whose characteristics are composed of two distinct linear regions in relation to not only the restoring force but also to the damping force, are compared with those of the analog-computer experiments performed for the above mentioned system with the help of the auxiliary apparatus composed of electronic limiters and a voltage comparator etc. Main results of the comparison are as follows : 1) Experimental and theoretical values coincide with each other up to satisfactory degrees for the amplitude versus gap diagram and for the waveforms of displacements as well as of velocities. 2) Improved 1st approximation is sufficiently accurate for most engineering purposes in the case of the nonlinearity and frequency range studied by the author and of the staying time ratio less than 0.6, so that the minute details of the waveforms of displacements and velocities are fully reproduced by this step of approximate calculation. 3) To calculate waveforms of displacements and velocities, we utilized their newly obtained expressions which, as the result of summing up of the infinite series for them, include the coefficients of the transformed series only and are of finite terms.

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By further developing the hitherto utilized dynamical model of the typical ultrasonic carving machine, we considered it as an unsymmetrical piecewise-linear system containing a solid horn as a continuous body. And by applying the analytical method introduced by the author in the 1st report to the solving of the forced vibrations of this system, we deduced theoretically various working characteristics of the ultrasonic carving machine. Some of them explain the experimental results only qualitatively, but some do even quantitatively. These are 1) Existence of the optimun feeding force, 2) That this optimum feeding force is realized when the contact time ratio ranges from 0.2 to 0.4, 3) That the optimum feeding force is directly proportional to the magneto-strictive exciting force, 4) Maximum impulsive force versus feeding force characteristics and phase lag angle versus feeding force characteristics, 5) That the optimum exciting frequency is slightly higher than the resonance frequency, 6) Oscillation jumping phenomena peculiar to nonlinear resonance.

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Among the statistical methods for obtaining the endurance limit in the fatigue test, "probit" technique and "staircase method" are well known. When the fatigue test is performed by these methods, a great number of specimens are required. In this report the fatigue test by staircase method with small samples is presented and the accuracy of this method is discussed. The fatigue strength S at fixed stress cycle N is obtained by this method. Such values of S at several values of N are plotted in the S-N graph, and then the S-N curve can be drawn. First, the distribution of the sample mean obtained by this method is considered, where the size of sample is 4. The influence of the step size on the variance of the sample mean is discussed. Next certain experiments were performed on 0.35% carbon steel. The values of S at five values of N were obtained by this method on each one of the 4 specimens. The S-N curve drawn by this procedure was found to coincide with that drawn by other method.

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As the fatigue of metals is directly related with the variation of their crystal imperfection, X-ray method is the most adequate for the investigation of fatigue. In the present paper, the fatigue process is studied by observing the half-value breadth of intensity distribution curves taken from the X-ray patterns, which were obtained from the specimens of steel stressed by alternate torsion and rotary bending. It was found that during the fatigue process the change of half-value breadth was directly related with the change of micro-stresses. Results are obtained as follows : 1) The half-value breadth changes remarkedly with the repetition of stresses. 2) The change of half-value breadth complies with the mechanism of fatigue that consists of the accumlation of micro-stresses. 3) On a view of the half-value breadth variation during stress repetition, it is considered that the fatigue process consists of the three sequent stages which follows; a) A stage in which the half-value breadth changes markedly with stress repetition.-The primary stage of fatigue. b) A stage in which the half-value breadth changes steadily with stress repetition.-The secondary stage of fatigue. c) A stage in which the half-value breadth sharply varies until fracture occurs.-The tertiary stage of fatigue. 4) At the end of the secondary stage of fatigue, the half-value breadth has a constant value irrespective of the magnitude of alternate stresses. 5) The relation between b/B and log (n/N), where b and B are the current and the inintial value of half-value breadth, and n and N are the current and the ultimate number of stress repetition respectively, is represented by a straight line irrespective of magnitude of alternate stresses applied. This relation offers possibility of the non-destructive detection of fatigue damage.

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Dynamic creep and rupture tests were carried out with annealed specimens of a 0.15 percentcarbon steel under the combined static and alternating stress of stress ratios (alternating stress/mean stress) more than unity. The test temperature was 450°C. The results of the present study together with those obtained in the previous study for the stress ratios less than unity were analysed in a view of the author's theory, which aims to relate the strength under dynamic stress to the static creep strength. The conclusion obtained is summarized as follows : 1. The forms of the dynamic creep curves were similar to those of the static creep curves even for the case of the stress ratio more than unity. 2. The prediction of dynamic creep rupture strength from the static creep rupture data together with the fatigue data under reversed stress was established over the entire range of stress ratio from zero (static stress) to infinity (reversed stress). This is based on the idea that the damages of material under varying stress are composed of the creep damage previously discussed and the fatigue damage caused by the dislocation oscillation under alternating stress. 3. The equivalent static stress, which was defined previously to predict the dynamic creep strength from the static creep data in the stage of transient creep, is applicable to the case of creep including the steady state component in addition to the transient component and also to the case of rupture. 4. The influence of the dislocation oscillation under alternating stress on the dynamic creep was also discuseed

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Creep of mild steel was studied at a temperature of 450°C under unsymmetrical periodic stresses such that in a stress cycle the period under stresses higher than the mean stress was shorter than the period under stresses lower than the mean stress. It was found that in this case the "mechanical equation of state in solid" predicted a smaller deformation as compared with the one obtained by experiment, while it had been applicable without any appreciable error in the case of a symmetrical stress wave. It was also ascertained that the smaller the ratio of the loading time of the higher stress to that of the lower stress and the shorter the period of a stress cycle, the more the tendency was accentuated, although the lower stress affected little the total deformation under a given stress wave provided that the higher stress was constant.

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In the preceding paper, it was clarifyed that the deformation of mild steel subjected to an unsymmetrical stress wave could not be predicted by the "mechanical equation of state in solid", while the deformation under the symmetrical stress wave could be predicted by it. The purpose of this paper is to establish a method of predicting the deformation under unsymmetrical periodic stresses as well as the deformation under symmetrical periodic stresses. The deformation was predicted by the authors' theory of deformation, which explained the behavior of material at any instant from the view of the integrated effect of work hardening and recovery, which the material had experienced before. It is concluded that the prediction by this method is considerably well in any case of rectangular stress wave.

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The pulsating multiple repeated stress in this paper means a pulsating stress that changes its direction with some intervals periodically. The investigation about the fatigue strength when materials subject to such pulsating stress has not been reported so far. In this report, experiments was made on the fatigue strength when above-mentioned torsional stress was applied to the specimen of 0.34% carbon steel, and the results was compared with the fatigue strength under ordinary pulsating torsional stress in which the direction did not change. In conclusion, the difference between the fatigue life under such pulsating stress and that under ordinary one was hardly observed, while the endurance limit for the former stress was brought slightly down from that for the latter stress, because of the change of stress direction. In addition, some considerations were made on the behaviour of fatigue deformation or fatigue creep during the tests both under such pulsating stress and the ordinary one.

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Elastic hysteresis, in this paper, means the deviation from the linear elastic line in the stress-strain diagram. Under alternating stress near and above the endurance limit, some magnitude of plastic strain takes place together with the ordinary elastic one in the material. As a result, so-called hysteresis loops are obtained in the stress-strain diagram. The plastic strain in this sense had been considered so far to play an important role for fatigue phenomena of metals, and a number of literatures had been issued discussing the fatigue mechanism relating to the hysteresis loop and the plastic strain, mentioned above. Nevertheless, the magnitude of this plastic strain is so small that it is considerably difficult to be detected during the fatigue test especially in direct stress near the endurance limit. But, it is still now interesting subject to clarify the relation between the fatigue strength and the elastic hysteresis. Recently, we have constructed an equipment to magnify the total elongation of the specimen up to 2000 times, whose principle is same as that of the Martens mirror extensometer, using two rhombuses with a mirror. With this equipment, it is possible to detect the strain of the order 10^-6. The performance test proved the good and stable accuracy. The specimens of 0.21% C-steel were tested under direct alternating stress with zero mean stress with Haigh Fatigue Testing Machine at room temperature, and the total strain was measured using the above equipment during the fatigue test. The plastic strain was calculated by deducing the linear elastic strain from the measured total strain. Test results showed that, the plastic strain increased in most cases at very early stage of fatigue, then decreased and became constant at a certain level. Furthermore, the plot of the plastic strain at the constant level against the number of stress cycles to failure in both logarithmic scales indicates the good linear relation.

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Anisotropy of the sheet caused by rolling is one of the important factors which should be considered in machine design as well as plastic working and its theoretical consideration has been given in some books such as R. Hill's "The Mathematical Theory of Plasticity". In Hill's method, however, the anisotropy is not given as the function of strain histories. Recently Prof. Yoshimura has proposed "The Strain History Theory of Plasticity" by which this functional relation can be successfully explained. Therefore, it is considered that the anisotropy of rolled plates can be satisfactory studied by this theory. In this paper the deformation caused by rolling is approximated by pure shear and the resulting anisotropy is analized by this theory.

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The object of the present work hereby described is to obtain some information about the stresses in reinforced plastic bars. Part one of tests was carried out under bending on a bar with a composite reinforced semi-circular notch. The bars were of epoxy resin reinforced with duralumin. The reinforcements of the semicircular notches are of the concentric, crescent, and arch-types. The relations between the maximum shear stress and breadth of bars are obtained. The most suitable reinforcements are discussed. Part two of tests was carried out under central bending on a composite bar with a hole. The bars were of epoxy resin stiffened with duralumin strips. The relations between the stress-concentration factors of the stiffened models and radii of holes ars obtained.

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In this paper, the stress concentration factor of a shaft with a semicircular groove under tension is calculated by a new method. In this method axial uniform stress and the stresses applied to an imaginary cylindrical surface, imagined over the groove, are superposed so that the surface tractions at sevaral points on the groove disappear. The stress concentration factor thus obtained is in close agreement with the value calculated from Neuber's formula.

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In his preceding paper, the present author made it clear that the concepts of strain and stress are not a priori ones, but should differ according as the deformation is elastic or plastic, and introduced, basing on it, a unifying theory of plasticity for small and finite deformations. The present theory is that for finite elastic deformation which is based on the contrasting concepts of strain and stress. The term "nonlinear" in the title is not concerned with the definition of strain, but with the elasticity law. In the first report are introduced the definition of strain and its increment legitimate for finite elastic deformation, the definition of actual stress for the purpose of describing equilibrium condition. With the exception of the nonlinear expression of strain, all these results are identical in form to the case of small deformation.

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As the successive items of the problem, there are deduced, in the second report, the energy principle, the concept of pseudo-stress valid for describing, together with the strain, finite elastic deformation and the existence of elastic potential, and are further introduced, basing on them, the virtual work principle, the Hamilton's principle, the principle of complementary energy and the nonlinear elasticity law. It is of great fortune for us that all of these results have the same form as for small deformation. Finally the validity of the present theory is confirmed by applying it to the case of simple shear deformation accompanied with rigid body rotation and showing that the principal axes of strain and pseudo-stress coincide with each other for such deformation.

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The object of the present paper is to report the fundamentals of an instantaneous photography that an electric spark in a high voltage is used as a light source of photoelastic stress patterns. The electric spark circuit with a subsidiary spark-gap is used in this experiment and the time of exposure of this circuit is set at about 2×1O^-6 sec. The best conditions of the exposure and contrast for the Fuji X-ray film are as follows : Deep red filter×2, F=2.8 ; main spark-gap vol. 46-49 K.V.P.P. Deep red filter×1, F=4.0 ; main spark-gap vol. 44-47 K.V.P.P. The main gap distance is set at 22 mm (64.2 K.V.-gap) and a starting voltage of 3 K.V.D.C. is taken in this case. As the examples photographed by this method, stress patterns of the impact of beams and the noched bers are shown. And then the behaviours of plastics under the dynamic stress state are discuseed by the use of Voigt and Maxwell models.

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A control system with large dead time is often found in industrial processes or in some kind of rolling works. Such loops are difficult to stabilize with the conventional controllers. The author presents a new method to stabilize such systems by introducing two kinds of feedback signals ; a signal obtained from system output and a signal from system intermediate point having no dead time effect. This paper shows the merits of the author's method by analytical frequency response calculations, analog computer responses and experimental results in a two-pass heat exchanger with large dead time.

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The optimum nonlinear control is applicable to the second order controlled systemes, even if there exists a dead time. In this case, the manipulated variable should be switched by four switching curves in the phase-plane, which however, results in the increase of the complexities of the controller. In the previous paper, the author proposed a control method-modified optimum nonlinear control-which reduces the complexities of the controller by replacing a switching curve with a switching line and modified a magnitude of the manipulated variable in the latter half of the response. This paper shows that the modified optimum control method is applicable to a system with a dead time and is very effective in reducing the complexities of the controller. The theoretical study and the analog computer study of this control method are also included.

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In the previous papers, we already discussed the cycling of relay servomechanism operating on sampled data. In this paper, however, we analyse them more precisely and find out many kinds of limit cycles which are stable. As it is determined only by the initial conditions, which limit cycle the system takes, the relations between these limit cycles and the initial conditions are hereby shown. It is also found out that the limit cycles change their amplitudes by the initial conditions and there exist the limit cycles which have asymmetrical amplitudes. Then the ranges of changeable amplitudes are calculated. Further, a method to decrease the limit cycle is suggested.

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In this paper the self excited rolling motion occasionally seen in a vehicle suspended by air springs has been treated. Investigation was made to see under what conditions this motion arises and how to design the control system of air springs to meet the situation. On an example of vehicle in which this motion had occurred, the amplitude and frequency of this motion were calculated.

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