單桁翼に於ける彈性的屈曲と捩りの聯成振動

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The object of this investigation is to find mathematically and experimentally the periods of coupled free flexural and torsional vibrations of an elastic monospar wing, and to obtain some general criteria concerning both the stiffness of a wing resisting against coupled vibrations and the relation of the period between the coupled and uncoupled vibrations. Although the vibration problem of a wing has already received attentions from many authors and different attempts to get its solutions have been made by them, it seems still worth while to investigate further the above described vibration in some way, owing to its importance in resonance conditions of vibration, to reexamine several points which are not adapted to practical purpose or to establish new theories. We assumed, for convenience, that mass is concentrated at the point of every rib. Although the method of the present study is never difficult, it is however extremely troublesome to carry out the complete numerical calculations. The equations of motion in this case are expressed by [numerical formula] the significance of every notation being omitted. If we write we obtain [numerical formula] The boundary conditions are such that [numerical formula] where n is the number of ribs,I_1, I_2,……, the sum of the moments of inertia of cross sections of each spar, m_n the mass at each rib, I_n the length between adjoining ribs, α_n the distance between the centre of gravity and the spar axis, J_n the polar moment of inertia of cross section of each spar, G_n, E_n the torsional rigidity and the Young's modulus of each spar respectively, and I_<An> the mass moment of inertia of each rib about the elastic axis. From (2) and (3) we obtain the equations of deflection curve and frequency of vibrations of the wing. We calculate three cases of conditions, namely that the wing has one, two, and three ribs respectively. No approximation is made to the solution of the equations. From a certain theoretical reason it is possible to form empirical equations for the fundamental free vibrations as follows: [numerical formula] The periods of torsional vibrations increase linearly as the number of ribs increases, while those of flexural vibrations increase roughly as the square of the number of ribs. In order to ascertain whether it is possible or not to extrapolate the equations (4), (5), and (6) to a general case we studied the problem of coupled torsional and flexural vibrations of a uniform clamped free bar with mass distributed excentrically. The equations of motion of coupled torsional and flexural vibrations of a uniform clamped free bar are expressed by [numerical formula] where m'is the mass per unit length of a bar, I_A' the mass (per unit length) moment of inertia about the elastic axis α the distance between the centre of gravity and elastic axis, I, J the moment of inertia of cross section and polar moment of inertia of cross section respectively, and E, G the Young's modulus and the torsional rigidity of a bar respectively. We solve these equations by means of the method of parterbation and get approximately the new formulae for determing the periods of coupled vibrations of a bar. The periods of the fundamental vibrations can be obtained from the following new formulae [numerical formula] where [numerical formula] , k is the radius of gyration about the centre of gravity, T_b, T_t the period of flexure and torsion of coupled vibrations of a bar respectively, and L the total length of a bar. The result shows that the periods of vibrations derived from (8) are in a fairly good coincidence with those obtained from (4), (5), and (6). The calculated curves showing this fact are plotted in Figs. 7 (a), (b), and (c). By means of model experiments we studied the coupled flexural and torsional vibrations, and confirmed whether or not the periods of such vibrations are capable of being expressed by equation (8). From the results of the experiments nevertheless shows. that the periods of coupled vibrations may be expressed by the new formulae in (8).
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單桁翼に於ける彈性的屈曲と捩りの聯成振動

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