## Vol. 12, No. 3 (1956)

##### 1956-02-05

1956, 1623 (3): 187-194.

Abstract +

The absorption spectra of cadmium sulphide single crystals were observed with natural light and polarized light, one piece of the crystals having a thickness of about 0.1 mm and another about 0.05mm.At room and liquid air temperatures, the absorptions spectra observed with natural light indicate that the wavelength of the long wavelength limit of CdS depends upon
the thickness and the temperature of the crystals.A change of the wavelength of the absorption edge which depended upon the orientation of the electric vector of the polarized light produced by a polaroid plate placed between the crystal and the slit of the spectrograph was observed.
At room and liquid air temperatures, the absorption spectra of the thinner CdS crystal with polarized light were observed .The spectrum at the liquid air temperature was a hydrogen
like series which consisted of seven members of strong and sharp lines located in the long wavelength side of the absorption edge of the crystal.
From the results of the wavelength measurements we found that the wave number of each spectral line fulfils the following relation: where k= 3 , 4 , 5 ,...9, and the line of k=2 was not seen in the spetrum. From the above emperical formula the series limit of the hydrogen-like series of CdS crystal after the trmperature correction was found to be 2.41 eV , which is in good agreement with the value of 2.42 eV obtained by othe methods. The reduced mass of the exciton in CdS crystal was found to be about 4.3 times the mass of electron. CdS and ZnS crystals have a similar strueture,
and the numerical values of their lattice constants are quite near. It could be infered that the full band of CdS crystal is pobably due to the S--ions and the observed hydrogen-like series of the excitons spectra is probably due to the excited state of S--ions.

1956, 1623 (3): 195-214.

Abstract +

A nuclear potential based on meson field theory is obtained in analogy with the derivation of Breit interaetion between two electrons, as given by Ivanenko and Sokolov.In this treatment the meson field is not quantized. All the four type of the current meson field and their mixture can be treated in the same way. The result has been further reduced by expansion in the v/c of the nucleon, the zeroth-order potental being identical with the static potential previously obtained by Mller and Rosenfeld . The first-order poential will vanish if there is only one coupling constant for each type of meson field , in which case the expansion has been carried to the second order of v/c , retardation effects being also included. Many-body forces naturally appear.

1956, 1623 (3): 215-245.

Abstract +

In this paper a triangular plate is considered as a portion of a rectangular plate by using the method of images in a special manner. Thus the expression for the deflection of this triangular plate is the same as that for the deflection of the rectangular plate, in spite of of the difference between the regions in which the expressions are defined. Since the solutions of the problems of a simply supported rectangular plate can be obtained without difficulty, the bending problem, as well as the problem of bending combining with tension or compression, of a 30° -60° -90° -
triangular plate is solved at once. Their results are expressed in the forms of trigonometric series, single or double, in this paper.
The bucking problem and the vibration problem of such a triangular plate are also discussed. The main results are as follows:
The smallest critical value of the compressive force per unit length is
Where D is the flexural rigidity of the plate and b is the length of the side opposite the 60° angle of the triangular plate.
The relation between the fundamental natural frequency w and the tensile (N＜0),or compressive (N＞0) force per unit length acting along the boundaries of the plate is
Where is the mass per unit area of the plate.
The method of images applied, in the same manner as mentioned above, to the torsion problem of a prismatical bar with 30°-60°-90°- triangular cross section gives more practical results than those obtained by other authors. The torsional rigidity numerically calculated coincides with that obtained by G.E.Hay in 1939.
In this paper a triangular plate is considered as a portion of a rectangular plate by using the method of images in a special manner. Thus the expression for the deflection of this triangular plate is the same as that for the deflection of triangular plate, in spite of the difference between the regions in which the expressions are defined. Since the solutions of the problems of a simply supported rectangular plate can be obtained without difficulty, the bending problem, as well as the problem of bending combining with tension or compression, of a 30 -60 -90 - triangular plate is solved at once. Their results are expressed in the forms of trigonometric series, single or double, in this paper.
The bucking problem and the vibration problem of such a triangular plate are also discussed. The main results are as follows:
The smallest critical value of the compressive force per unit length is
Where D is the flexural rigidity of the plate and b is the length of the side opposite the 60 angle of the triangular plate.
The relation between the fundamental natural frequency w and the tensile (N＜0),or compressive (N＞0) force per unit length acting along the boundaries of the plate is
Where is the mass per unit area of the plate.
The method of images applied, in the same manner as mentioned above, to the torsion problem of a prismatical bar with 30°-60°-90°- triangular cross section gives more practical results than those obtained by other authors. The torsional rigidity numerically calculated coincides with that obtained by G.E.Hay in 1939.

1956, 1623 (3): 246-260.

Abstract +

In this paper, an approximate method for detertmining the unstable regions of dynamic
stability of thin-walled beams is given. The beam is assumed to be underth the actions of concentrated longitudinal forces at both ends and of the type where P。=const, Pt(t) a periodic 2 and small parameter. The end conditions are arbitrary.By using trigonometric series or Galerkin's method satisfying the end conditions, the fundamental equations, based on Vlasof's theory, are reduced to a system of three ordinary linear differential equations (4) or (7) of 2nd order with periodic coefficients. Moreover, they can easily be transformed to canonical form
Therefore, their characteristic equations are reciprocal equations, with characteristic roots symmetrioally distributed with respect to the real axis and unit circle in a complex plane. The condition of boundary lines between stable and unstsable regions is taken as, that all of the characteristie roots have unit modulus (absolute value), but there exist equal roots. Expanding the characteristic exponentials in series of the small parameter ,this condition is represented by the following equation: where Wn/Wnk represent different frequencics of n-mode vibrations of the beam under the of a constant force P。, and W is the frequency of P1(t). When u-0,(27) and (28) become Hence, dynamic unstability would take place at the neighbourhoods of these critieal ratios, expresse By (29) and (30).
When the unstable regions of dynamic stability are desired, we use pertubation method to determine
For Practical use, it is sufficient to determine
only. The boundary lines can then approximately determined by the following equations: For illustrating this method, a simply supported beam of narrow rectangular cross-section、under the action of varying end moments (fig. 2) is considered.
The fundamental unstable regions, corresponding
to bending, torsional and “mixed” type of dynamic unstability
are calculated and shown in figs. 4, 3, 5.

1956, 1623 (3): 261-270.

Abstract +

In this paper, the problem of vibrations of thin-walled beams is investigated. The fundamental assumption is the same as that in a precceding paper [l], i.e. the non-deformability of contourlines of transverse cross-sections in its own Plane. Equations of vibrations, show that the vibrations of thin-walled beams generally take place as a combination of bending-torsional and longitudinal vibrations.Corresponding to a certain type of bending-torsional vibrations, instead of only one frequency as in the well-known Vlasof's theory, there are an infinite number of frequencies, corresponding to different types of longitudinal vibrations.The natural frequeneies ,as calculated by Vlasof's theory correspond to the lowest frequeneies in our theory, but the former are generally higher than the latter as should be expected.