An approximate solution is presented for the problem of the elastic instability of a thin circular plate subject to a pair of locally distributed forces applied on the periphery. The Ritz's method is used and the problem is solved for the two cases where the edge is clamped or simply supported. In each case above cited, buckling loads and deflection surfaces for various values of the load-width are calculated and the results are graphed and discussed.

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In order to estimate the buckling load of a column, the vibration method has been developed by many investigators. In this paper, the author investigated the method of estimating the axial load and the buckling load of a bar using the third mode of vibration, which has an advantage of eliminating the procedure of superposing several known loads upon the column.

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The stress distribution which is produced in a rotating circular disc has ever been discussed not only by Stodola, but also by many other investigators, However, as to the stress in a rotating tangentially-stepped circular disc, it seems to be very few discussed theoretically or experimentally. Recently, the authors constructed a measuring apparatus with wire strain gage providing a new electric connecting device, with which they measured stresses produced in some rotating circular discs. Now, they tried to apply the same device to the case of rotating stepped discs and to compare the obtained results with those of theoretically calculated values. According to the results of calculation, the strain distribution varies its form as the variation of the area of stepped parts. So far as this area is kept constant, the strain distribution is not strongly affected by the number of steps. The largest stress in the disc occurs always among the circumferential stresses in the thin part. The experimental results coincide with these theoretical values fairly well.

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In this paper, as a two dimensional problem of elasticity, the problem of the stress distribution in a hollow elliptical cylinder was solved under the condition of two concentrated internal loads. As an example of the present analysis, the values of stresses at any point in a thin cylinder of which the cross section is bounded by two confocal ellipses under the condition of two equal and opposite concentrated internal loads acting at the two crossing points of the inner ellipse and the major axis were obtained numerically and the state of stress at any point in the cylinder was clarified. As another similar example, the values of stresses at any point in a thick cylinder of which the cross section is bounded by two confocal ellipses under the same condition of loading above mentioned were obtained numerically and the state of stress at any point in the cylinder was also clarified.

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The specimens of plate beams having a circular arc projection on one side or on either side symmetrically were prepared by an epoxy plate and the stress distribution under an uniform bending moment or the tensile load was studied by photo-elastic experiment. The results are as follows : The maximum stress always occurs near the root of the projection and the stress concentration factor at the maximum stress position in the plate beam having an arc projection on one side is larger than that having an arc projection on either side symmetrically, and then the factor under the tensile load is larger than that under the uniform bending moment. The factor α is affected by the height ΔH and the length θ of the arc projection, the radiue of curvature r at the root of the projection and the original breadth H of the plate beam. The smaller r/H, θ or the larger ΔH/H is, the larger α is obtained, and α increases suddenly when r/H is smaller than about 0.1.

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The threshold repeated stress (nominal) to develop crack at the root of a notch is different from the threshold one (nominal) to spread the developed crack. At the branch point these two kinds of threshold stresses become equal to each other. In steel, branch point has close relation with the yield point of the material. In this paper it is shown that, although annealed copper shows no definite yield point in tension, branch point can be observed in rotary bending test of notched copper specimens. At the branch point the fatigue limit expressed by the maximum stress at the root of notch is equal to the stress at which tangent modulus of fatigued material begins to decrease sharply. It is suggested that severely deformed zone at the root of notch, appeared before fatigue crack is formed there, plays an important role in the mechanism of fatigue crack propagation.

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None has been known how to choose the shaft diameter of bolts and the bolt fastening forces caused by nuts, for the purpose of getting the strongest bolt connection under repeated load. The authors were able to solve the above problems on the assumptions that bolts should neither break down by fatigue nor the stresses of bolts should exceed yield points. An example was carried out about the calculation of the bolt strength, and the most proper fastening force was determined for a given bolt and external repeated load. It was also cleared by this example that tension bolts have never more strength than those of normal bolts. A method for graphical determination of the strength of bolts under repeated load was also stated.

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There has been some difficulties in explaining compression endurance limit diagram, by preceeding fatigue criteria. The authors propose a new criterion and show that this can explain the compression endurance limit diagram. In this criterion they consider that the material is fatigued by the stress amplitude of the repeated shearing stress and the strength is decreased in proportion to that stress amplitude. Then the failure occurs when the material can not bear the normal or shearing stress working there. The formar is the separating failure, and in this case the separating strength is weakened by the coexisting shearing stress. The latter is the sliding failure and the shearing strength is weakened by the tensile normal stress working there, while it is strengthened by the compressive normal stress. The fracture occurs in the plane where these total effects become maximum.

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The failure of the condenser tube due to fatigue is caused in most cases by the vibration from neighbouring machines or repeated thermal expansion of the tube itself. The writers have conducted an experimental investigation recently to see the influence of the expansion in various degrees, annealing temperatures of tube ends and corrosion on the resistivity of the expanded condenser tube against fatigue. The results of the test are as given below : (1) The fatigue strength redrotion factor of an expanded condenser tube turned out to be about 25. (2) Condenser tubes annealed at 650°C and 750°C had smaller tensile strength and accordingly smaller fatigue strength when they were of unnotched type, but in case of an expanded tube specimen it showed nearly the same fatigue strength as compared with the one annealed at 550°C. (3) The variation in degree of expanding had scarcely any influence on the fatigue strength. (4) The failure due to fatigue initiated from the surface of the end part of the tube which is in contact with the head plate. This might be due to the fretting corrosion accelerated by minute relative slips between the outer and inner surfaces of the tube caused by expansion. Corrosion of the interior of the tube had little effect on the fatigue strength. (5) When the length of expanded part of tube is longer than that of the tube seat, the fillet radius at the edge of hole in head plate increases adding to the fatigue strength of the expanded tube.

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Although, as has already been reported, the resonance of synchronous backward precession can occur when the bearing pedestals deflect, this resonance can also appear when the pedestals have no flexibility and there is a loose fit of ball bearing in pedestal. Since there are errors in machining the outer ring of ball bearing as well as the inner surface of bearing box where neither are made exact circle, the degree of flt is not equal in all directions and the shaft has an unequal flexibility. Consequently the critical speeds of synchronous backward precession do occur. Similarly these critical speeds appear in the shaft supported by single-row radial ball bearings, because such a shaft has non-symmetrical, non-linear characteristics and unequal flexibility. Because the phenomenon of shaft whipping induced by dry friction takes place, it is undesirable to equip a gard ring to check an increase of deflection of the shaft at the critical speeds of synchronous backward precession.

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When single-row radial ball bearings are used the stiffness of a shaft has non-symmetrical, non-linear spring charactristics. In experiments using such a shaft mounting one disc, several critical speeds of peculiar modes of vibration occurred with a critical speed of sub-harmonic oscillation of order 1/2. These critical speeds consist of two vibrations having the same frequencies pi and pj as the natural frequencies of this system. It is remarkable that the absolute value of sum of or difference in two frequencies pi and pj is equal to the rotating speed of the shaft. In numerous experiments with several kinds of shafts, this relation between pi and pj always held at these critical speeds. For a system having linear spring characteristics or a system with one degree of freedom, this type of critical speed does not appear ; it occurs only in the system with multiple degrees of freedom and non-linearity.

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A rotating shaft which is driven through an universal joint such as the propeller shaft of an automobile sometimes shows bad vibrations at lower speeds than its ordinary critical speed. This report deals with the resonance phenomenon due to the angular velocity variation caused by the universal joint. This angular velocity variation occurs two times during one revolution of the shaft and theoretically gives an exciting force with the frequency three times as high as the rotating speed. Thus, when the shaft is driven at one third of the ordinary critical speed, the frequency of the exciting force caused by the angular velocity variation coincides with the ordinary critica1 speed and the natural frequency of the shaft, causing resonance vibration. This phenomenon is also confirmed by model tests.

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In this paper, free vibrations of the Greek-Crossed Bar, which is formed by two bars intersecting on their middle points at right angles, are studied analytically. And the normal modes of the vibrations and the characteristic equations for determining the natural frequencies are shown. Numerical calcurations are carried out in detail and the influences of the dimensions and the physical prperties of the two bars on the natural frequencies are also discussed.

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This paper deals with problems of a forced vibration characteristics of a elastically supported mechanical system with a shaft. Such a problem, for instance, occurs in a jet engine put on a wing of an aeroplane. Fundamental equations and their solutions are obtained. Numerical calculations and experiments for special cases are also given. The author shows how the forced vibration of such a system is influenced by the shaft within it and how the forced vibration of the shaft varies in response to conditions of the elastic supports. The shaft cross section is uniform.

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This paper deals with problems of a forced vibration of an elastically supported mechanical system with a shaft. The vibration is caused by forced displacements. Such a problem, for instance, occurs in a jet engine put on a wing of an aeroplane. Fundamental equations and their solution are obtained. Numerical calculations and experiments for special cases are also given. The author shows how the forced vibration of such a system is influenced by the shaft within it, and how the forced vibration of the shaft varies in response to conditions of the elastic supports. The shaft cross section is uniform.

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In our experiment, we investigated the amplitude dependency of vibration characteristics which appears remarkably in tread type stocks of rubber especially in small amplitude, using at the same time unbonded type of rubber in order to compare the results with the case in which bonded type rubber used. We obtained the results as follows : (1) Both bonded type and unbonded one show approximately the same amplitude dependency. (2) These are to be explained quantitatively by the theory proposed by Sawaragi. (3) No fundamental difference is found between the dynamical behaviours of two types of rubber, and these can be treated as materials having the same dynamical properties.

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The frequency (ω) of free vibration of a thin elastic cylindrical shell was obtained by a theoretical calculation. The effect of virtual mass of water filled inside (or outside) the shell, the effect of external (or internal) normal pressure, together with the effect of axial compreesive (or tensile) force are taken into account. It is inferred, from the results of calculation, that : -(a) The effect of axial force is rather small in comparison with the effect of virtual mass of water and that of normal pressure, so long as the shell whose length is larger than its diameter is concerned. (b) Under such a way of loading, in which the ratio of axial force to normal pressure is kept constant, the relation between the angular frequency ω and the pressure p becomes practically a straight line. It is suggested that, taking advantages of this fact, a method of non-destructive test for collapsing pressure of thin cylindrical shells (such as a pressure hull of a submarine) may be possible.

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Log-root-locus method is applicable to the distributed system whose transfer function is transcendental equation. The dead time and the distributed lag which are frequently encountered in the process control can be treated as the distributed line of Fig. 25 or Fig. 28 and Fig. 29 which lead to the known transfer functions e^-Ts and e^-√(Ts). Their phase loci and gain loci are depicted in Figs. 30 and 31 which are used to trace log-root-loci of systems comprising those terms. Figs. 32 and 33 are depicted as to the representative process with the dead time. Generally speaking, the dead time is apt to make higher order terms unstable. Figs. 35 and 36 are log-root-loci of systems comprising the distributed lag. Comparing them with Figs. 32 and 33, it is seen that the distributed lag effects rather conservatively to the system dynamics compared with aggravating effects of the dead time. The equivalent dead time system for the distributed lag e^-√(s) was chosen to have the same fundamental root at the same gain corstant. Fig. 37 contrasts each indicial responses of both systems which confirms conventional way of choosing the equivalent system.

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Pulse transfer function 〓(z) which is a z-transform of G(s) is studied on log z-plane. 〓(z) is generally a rational function of z as expressed by (15·8), so that its log-root-locus is traced following the same procedure as in the ordinary transfer function. In the synthesis of the pulsed feedback control system, adequate stability is specified on log z-plane as shown in Fig. 41. Log-root-loci of simple pulsed servos each lacking and comprising the dead time are depicted respectively in Figs. 43. and 44. Pulsed compensating network is most profitably discussed on log z-plane by pole and zero configuration. Sequence of transient response can be calculated by difference equation or by characteristic roots in the form of (17·9) and (17·10). In connection with the frequency response of the pulsed feedback system of Fig. 40, W^*'(jw) of (18·1) is seen to be equal to w'(jw) of (18·4). Frequency response of pulse transfer function is calculated in the same way as in the ordinary transfer function, and expressed as gain and phase diagrams as Fig. 47.

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This paper deals with the attachment of the Frequency Response Slide Rule. The slide rule which has been published in "Automatic Control" Vol.1 (1953) No.3 by K.IZAWA, is used for frequency response computation of real linear factors : (1+Ts), Ks and K/s and exponential factor : e^-Ls of such a transfer function as [numerical formula] The attachment is a transparent sheet upon which a logarithmic scale of hyperbolic sine function is graduated with two reference lines. The frequency response of the quadratic factor (1+2ζTs+T²s²) of the above transfer function is easily computed by the slide rule and the attachment, when the input frequency ω, 0^.001⦤ω⦤1000, the time constant T, 0^.01⦤T⦤100, and the relative damping ζ, 0^.005⦤ζ⦤2, are given. Three steps are necessary for the computation ; the first step is the setting of the cursor and the slide, the second and third steps are the settings of the attachment.

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In order to find out the frequency characteristice of pneumatic transmission lines used in automatic control systems, the authors have obtained the frequency transfer functions of pneumatic transmission lines through the theory of pressure transfer in slender pipes. Then, the actual frequency response was found with systems of various pipe lengths, pipe diameters and the attached air chambers. The authors found, as a result, that the frequency characteristics should be considered as a continuum, and also disclosed the effect of pipe length, pipe diameter, the volume of attached air chamber and the mean value of variable pressure. As a method of approximate solution, the authors proposed a solution by considering an equivalent linear lumped system with an equivalent dead time and obtained an approximate solution of frequency characteristics of a pneumatic transmission line with a closed end and an attached air chamber.

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