The free vibrations of the bar bent to L-type (both ends built-in) are discussed. The method used is to minimize Lagrangian which is expressed in terms of the unknown functions and the known and unknown boundary conditions which describe the vibrations. In the case that there are lateral, twisting and longitudinal vibrations at the same time, the unknown vibratory functions are obtained in general forms, and Lagrangian is expressed in the terms of the boundary conditions. Using the known boundary conditions and the conditions of continuity, Lagrangian is expressed in the terms of unkown constants. Then, Lagrangian is again minimized about unknown constants and these constants and the frequency equations are obtained. In the case of L-bar, the frequency equations of both vibrations in the plane of L-bar and perpendicular to that plane are obtained and the numerical calculations are carried out, and the comparison with the contents of papers already published are made.

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Theoretical analysis and experiments were carried out in connection with the speed of granular solid when repeating bath sliding and jumping movements on the oscillating trough in straight line motion. In this case, solid on the trough makes regular repetition of the process such as sliding movement in the plus and minus directions, jumping movement and relative rest, for which the relative displacement Δx_i is made clear. Travelling speed V is, consequently, concluded as [numerical formula], and by using the formula, numerical calculation is made possible. In order to find the coefficient of friction between solid and trough, which is necessitated for this analysis, we adopted a method different from conventional one. Our method is this : measuring by oscillograph the acceleration transmitted to solid from the trough. As for the measures of friction and travelling speed, we could investigate over a wider range by using the oscillating apparatus designed by us, which has various directions of throw. And as a result of the investigation, it has been found that the experimental value shows approximate accordance with the theoretical analysis.

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Succeeding to the 1st Report, this paper treats of the theoretical analysis and experimental studies concerning the travelling speed of granular solid on the oscillating trough in harmonic circular motion. As for the matter, we discuss the two cases : the travelling speed of granular solid both in the sliding movement and the sliding and jumping movements. After theoretical analysis based upon the same idea as the analysis of straight line motion, we give a formula for relative disp1acemant Δx_i between oecillating trough and solid. According to the said formula, travelling speed V of solid is given as [numerical formula] and numerical calculation is consequently made possible. Theoretical analysis is in approximate accordance with the results acquired from experiments, and the rationality of the analysis could be ascertained. Samples and measuring methods for the experiments are both the same as described in the 1st report.

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As the way to reduce an axial vibration caused by a ball bearing, the author worked out to equip the viscous damper and the vibration absorber with damping. By making use of the equipments it has been shown that the maximum amplitude of the vibration can be reduced to about one-third or one-fourth when the thrust load is zero. A subharmonic vibration of the order one-half caused by the passing balls in a ball bearing appears in the certain region of the rotational speed. The subharmonic vibration in the case where the vibration absorber is mounted has been analyeed. Some experiments agreed with the analytical results.

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In this report, the effect of Coulomb friction brought about from the packing gland of the damper on the free and transient vibrations of the above system is considered with respect to some numerical examples. The result shows (1) that a cycle of free vibration consits of two motions, one the motion when the friction is infinite and the other the motion with combined Coulomb friction and viscous damping, and (2) that the transient state while the system runs over a bump (shaped like a portion of sinusoidal curve) is regarded as a chain of fragmentary pieces of the two kinds of motion stated above. The existence of the motion when the damper is momentarily stuck-up gives a negative effect on the diminishing of the amplitude of the free-vibration of the system, but comparing the transient states with those of the system with no friction, a larger decrement of the amplitude of the initial response may occur in some cases according to the combination of the magnitude of friction, height of projection and the passing speed over it.

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The analysis on the forced vibration of the system with elastically supported viscous damper whose damping is accompanied by the Coulomb friction is a very complicated one, for the reason that its every cycle is composed of two different kinds of motion as stated in the 1st report and that seven boundary conditions are needed to solve it. In this paper, the author presents certain characters of the vibration of the system with optimum damping and Coulomb friction, the calculation of which is carried out by a digital computer. It became clear from the results of the analysis and the experiments that, while the Coulomb friction is smaller than the disturbing force, the amplitude seems to grow larger continuously from infinite damping curve to the optimum damping one as the frequency increase, but in the caee where the friction overcomes the disturbing force, the jumping phenomenon of the amplitude occurs at a fixed place corresponding to the amount of friction.

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The study on this problem has not been hitherto conducted on a shaft with perfect semi-circular groove but with the groove of shape almost similar to circular arc. So the present author has studied the undevoloped side of the problem and found its solution. As a result of an experiment a graph therefor has been completed, which is useful for finding the relation between the maximum shearing stress and the twisting moment.

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The authors have investigated the shearing stress on each side and the torsional angle in the thin-walled hollow rectangular bar with a hole of various sizes in a longer side subjected to torsion. The resu1ts obtained were as follow : (1) The meas shearing stress through the thickness calculated by the usual formula is far larger than the measured shearing stress in the case of short hole. (2) By combining Fig.3 or 4 with the calculated shearing stress, it has been found that the shearing stress of each side, having a hole of any size, can be obtained by caluculation. (3) The torsional angle steeply increases with the hole length. But in the case of short hole, the torsional angle hardly differs from that of the case without the hole.

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The large deflections of a circular-arc cantilever spring of trapezoidal profile subjected to a concentrated force at the free end in the lateral direction have been studied. A solution of the nonlinear beam equation is obtained in the form of power series. However, the power series used yields poor convergency as the shape of spring approaches the triangular profile and it diverges lastly. An approximate solution is also derived from the nonlinear beam equation after approximated appropriately. In comparison between results calculated from the seriesform and approximate solutions respectively, it has been seen that the approximate one gives practically satisfactory results. Numerical results due to both the series-form and approximate solution are also presented.

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In the present paper, slip behavior in α-brass before and after yielding is investigated by means of etch-pits technique which can reveal the faint slip lines as dislocation rows. By this method, it has been found that the stress concentrations at the inclusions and the edge corner of the test piece activate nearby dislocation sources to cause the preparatory slips, and that the degree of the constraint for slip deformations imposed by the neighbouring crystals is one of the important factors determining whether the preparatory slip can propagate into undeformed regions or not. The slip deformations at the lower yield stress propagate over the test piece under these conditions.

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In order to clarify the influence of the speed of stress cycles on the fatigue strength under corrosion, the following three kinds of corrosion fatigue tests were carried out on the carbon steels : (1) the ordinary corrosion fatigue tests for the three kinds of frequency, by rotating-beam bending, (2) the corrosion fatigue tests in which the repeated stressing under corrosion and the pure corrosion without repeated stressing are alternatively applied, by rotating-beam bending, and (3) the corrosion fatigue tests of the two kinds of frequency, by repeated impact. These test results were summarized and discussed as follows : (1) The influence of speed of stress cycles on the fatigue strength under corrosion should be determined by both the time rate of deformation and the time of corrosion. (2) The coefficient of fatigue effect which determines the factors dominating failure under corrosion-fatigue was proposed as the following expression : k_f=-(d(logt))/(d(logN)) The reciprocal of this coefficient 1/k_f was termed as the coefficient of corrosion effect. The va1ue of k_f depends not only upon the kind of material but upon the applied stress level.

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The charactaristics of the stress and strain amplitudes accompanied by the strain and stress cycling of constant amplitudes were investigated during the all life of test. Their phenomena are devided into five stages. If stress or strain amplitude has a value enough to fracture within 1000 cycles, the first and the second stages vanish. The fatigue strength of constant stress amp1itude cycling is a little larger than the fatigue strength of the constant strain amplitude cycling. This may be explained by the phenomena of double repeated stress in two levels. The change of stressstrain hysterisis loop accompanied by cyclings is different from the changes of stress amplitude ; although the dynamic modulus of elasticity does not change from about half life to fracture, the form of hysterisis loop changes and the area of loop decreases there.

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Torsional fatigue tests on 0.16%C carbon steel are made at room temperature, 350°C and 500°C. The strengths of thin walled tubular specimens where the stresses are considered to be distributed uniformly over the cross section are compared with those of solid specimens to study the effect of stress gradient on the fatigue strength. On the nominal stress basis, higher strengths are obtained in solid specimens at elevated temperatures as well as at room temperature. On the true stress basis, no difference of strength is found at room temperature. While the solid specimens are rather stronger than the tubular specimens at elevated temperatures. These results may be explained by the gradient of true stress which is smaller at room temperature than at elevated temperatures. The temperature rise of specimens during tests and the shapes of failure are also discussed.

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This paper is succeeded by No.2 and No.4 reports of the same title in which propagation of fatigue cracks on large smooth surface and fillet specimens has been dealt with. In this paper, the author has studied the following two problems : (1) The fatigue strength of large smooth specimens having a diameter of 130 mm with fine cracks artificially produced beforehand by sea water corrosion. (2) The influence of size on the fatigue strength of cracked specimens and propagation speed of fatigue cracks. The test results have revealed that the fatigue limit of steel specimens, 130 mm in diameter with fine corrosion cracks on the surface is about ±11.0 kg/mm². This limit coincides with the result assumed by studying the crack propagation of specimens, 120 mm in diameter. Analytical investigation of the propagation speed of fatigue cracks of 60 φmm and 20 φmm specimens has shown that the fatigue limit of these specimens are assumed to be about ±12.0 kg/mm² and ±15.0 kg/mm² respectively, when fine cracks may previously exist on their surfaces. From the above results, it may be stated that the fatigue limit of smooth specimens having fine cracks on their surface will tend to increase as the diameter deccreases. On the other hand, from the analytical studies on the shape of shelly patterns on the fracture surface, the propagation of fatigue crack may be given approximately by the following formulae in connection with the diameter of specimen : [numerical formula] Where : υ_c is propagation speed of fatigue crack μ/c, υσ_n : stress on the edge line of fracture surface kg/mm², σ_n : stress on the specimen having no creck, υ : stress magnifying factor due to crack, σ₀ : fatigue limit of cracked specimen kg/mm², σ₈ : yielding point of the material kg/mm², K and k : constants varying according to the properties of material and range of υσ_n, d : diameter of specimen mm.

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It is usually accepted that steel parts surface hardened by induction heating generally exhibitless total distortion. However, the induction hardening of small-sized gears is a matter of recent years and we have as yet very little information on the results concerning the systematic measurements of the distortions of them. In this investigation spur gears of 10 diametral pitch, were induction hardened with various heating conditions, and distortions, generated by thermal changes as well as by quenching transformations, were measured in detail. In the 1st report, the results of measurements of distortions, caused by thermal changes only, were described. Materials of gears were pure iron. Items of measurements were as follows : outer diameter, inner diameter, width, whole depth, radius or root circle, thickness of tooth, undulation of tooth trace. and tooth profile.

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Spur gears of 10 diametral pitch were induction hardened with various heating conditions, and distortions, generated by thermal changes only, were shown in the 1st report. In the 2nd report, results of measurements of distortions, caused by quenching transformations as well as by thermal changes, were described, Materials of gears were 0.45%C steel. Items of measurements were as follows : outer diameter, inner diameter, width, whole depth, radius of root circle, thickness of tooth, undulation of tooth trace, undulation of end surface of gear, tooth profile, eccentricity, out of roundness, normal pitch and error measured with Carl Mahr type meshing test machine. Further, the influences of key way on eccentricity, out of roundness and normal pitch were also measured.

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This is the 2nd report, which is continued from the 1st report. In this report, we have studied on some etching solutions for the strain figures of steel and for ferrite. The former is macroetching and the latter is the micro-etching of ferrite Structure. As a result, we have established some methods of etching for the strain figures of steel, for A₁ transformation band and for the crystals which constitute the ferrite structures of pure iron and hypo-eutectoid steel. Each crysatal of ferrite structure is etched in different colors and has lines which run parallel and have equal pitch. Some samples of etching are illustrated with photographs.

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The depth and the degree of work hardening of the surface layers of tensile specimen of high Mu-Cr steel finished under several conditions (turning, grinding, polishing and electro-polishing) were studied experimentally as a function of depth of cut. Then, the behavior of these surface layers under the tensile load test was observed through a microscope. The conclusion of these studies are as follows : (i) The depth of surface layer and the degree of work hardening (we defined tentatively H_w as the representation of this degree) are relative to the depth of cut at finishing condition. (ii) The work hardening to these surface layers affects the local contraction of specimen and under some finishing condition invites the surface cracking. (iii) The notch effect caused by cracking and the stress concentration in the surface layer are the reason for the influences ascribed above.

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In the 1st report, the deflections which are observed at the time of the heat indication test of turbine shafts were classified into five basic types A, B, C, D and C'. In this report, certain results of the aforesaid model test using subsize shafts which were given some asymmetry in the matallurgical properties by the heat treatment and of basic experimental studies are mentioned as follows : (1) The model shafts normalized through eccentric cooling showed the type C deflection convexly to the side of faster cooling. (2) The model shafts tempered with eccentric heating showed the type C deflection convexly to the side of tempering at higher temperature. And in these two cases, no deflections other than type C were observed. (3) The model shaft with extremely eccentric segregation did not show any deflection. In addition, by analysing physical properties of the actual shaft forgings showing type C deflection and conducting some high precision measurements, it has been revealed that the cause of type C deflection is asymmetric distribution of Mo dissolved in the matrix of the forgings.

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In the indentation hardness, the measured hardness number is affected by some effects related to time such as the rate of denting of the indentor, the time of load application and the time from the beginning of unloading to the reading of the hardness number. The effects are considered remarkable in such testing as the Vickers and the Rockwell hardness tests which use sharp indentors. Then it cannot be expected to measure hardness numbers correctly, unless the testing is done under unified condition of these factors. The author has carried out a series of studies on various factors that affect the Rockwell hardness. In this report, as the first step of the research, the effects related to time in the testing were considered to establish the basis of testing method. As a result, dividing, construction and treatement of these factors have been made clear, and then a weak point in the hitherto treatement of the rate of denting has been pointed out. Then, need of reexamination of the provision in the standard of testing method provided for in each country has been presented. And a proposal of a method of unifying the rate of dentiong of the indentor has been also added here.

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The author has proposed the new measuring method with Rockwell hardness which is measured by indenter with a conical angle different from 120° or by the larger initial load than 10 kg. The conversion formulae between these hardnesses and Vickers hardness H_v and the equations of errors were induced theoretically and checked experimentally. These new hardnesses have many merits. For examples, using the 1arger conical angle than 120°, the hardness range of 60<H_v<1000 can be expressed by a single indenter continuously and the thin sheets are measured without employing special superficial tester. The smaller conical angle than 120° is effective to raise the sensitiveness of measurement for very hard specimens. For rough surfaces, we can get the reliable results which have very small scattering ranges by larger initial load.

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It is observed that the human behavior is much more complex and reasonable than that of conventional servomechenisms, when one controls his own action or drives an automobile. The author thinks that it is very useful for the improvement of the performance of servomechenisms to introduce such characteristics into the construction of servomechanisms. Prediction is a characteristic of the human behavior, but it is usually difficult to introduce prediction into servomechanism construction just as it is. The author introduced prediction into servomechanisms so far as servomechanisms are constructed easily. The author investigated three kinds of prediction servomechanism ; the first is error-prediction-servomechanisms, the second is the prediction servomechanisms in which manipulating vartable is computed continuously, and the third is the prediction servomechanisms in which the following velocity varies automatically. It is confirmed that these prediction control servomechanisms have more superior peformance than the conventional servomechanisms.

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Finite-settling-time systems which are nonminimal have some constants which can be assigned arbitrarily. Though these systems have not the shortest settling-time such as minimal systems have, this opens up the possibility of selecting the extra system constants with a view of optimizing the system under the integrated-square error criterion. In this paper a general procedure is discussed to synthesize such nonminimal finite-settling-time systems under this criterion without considering hidden oscillations. The fundamental formulas determining the optimum type of overall pulse transfer functions and calculating the minimum value of the integrated-square error are obtained. It is also shown that the case of more than two inputs can be reducd to that of one input. In examples, suggestion is given of determining the length of the settling-time.

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In recent years, the non-stationary random processes have attracted considerable attentions in the field of adaptive or self-optimizing control, but little seems to have been done for those cases. In this paper, a very simple but fundamental case of non-stationary random process of response of nonlinear control systems is treated. That is, the analytical study on the evaluation of the response of nonlinear control systems subjected to a suddenly applied stationary random signal is described The two methods are established here by introducing some extensive method of the equivalent linearization technique. A graphical method for evaluating the non-stationary random response is proposed and such procedures are explained by typical examples in detail.

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In the present paper, the author verifies analitically and experimentally that "summmed and differential harmonic" oscillations can take place in multiple-degree-of-freedom systems with symmetrical nonlinear spring characteristics, and that only summed type can appear and differential type cannot occur. This "summed and differential harmonic" oscillation consisits of two vibrations with frequencies ω_i and ω_j which are almost equal to the natural frequenies p_i and p_j respectively, and it appears at the frequency ω of the external force which satisfies the relation [numerical formula]. Obtained response curves by experiments are those of hard spring type with jump phenomena.

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