U字管マノメータの液面の自由振動
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概要
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The problem of the oscillation of an incompressible fluid in an open-ended U-tube is frequently discussed in the textbooks of hydraulics, vibrations and automatic control systems etc., but their theoretical equations cannot explain the experimental results. On the other hand, many theoretical and experimental studies different from the statements in usual textbooks have been taken until now. However, it does not seem that the definite and final solution of this problem has been given. So a new attempt is presented here. In this paper, the momentum equations of the fluid column are established by applying the Navier-Stokes equation, and they are solved by applications of the Laplace transform. These solutions give the following results. 1. The free oscillation of the fluid in the U-tube can be explained by the Reynolds number /ν (put=W), where a is the internal radius of the U-tube, ν is the kinematic viscocity and ω_o =√ (g is the acceleration due to the gravity and l is the length of the fluid column). 2 If W= 3.435075 (put = W_c), the free oscillation of the fluid in the U-tube is critical damped and if W W_c, it is underda mped. 3 In the case of W > W_c, the free osclliation of the fluid in the U-tube consists of a fundamental underdamped motion and innumerable overdamped motions. But if we take the motion on and after decimal nine times as much as the period of the motion, the innumerable overdamped motions can be neglected and the motion can be regarded as a solution of the following second order differential equation with constant coefficients: x + 2ζω_nx + ω_n^2x = 0 The damping factor ζ and the undamped natural frequency η (=ω_n/ω_0) with respect to the dimensionless time ω_0t are given as follows: ζ = cos(arg.(s_0)), η = |s_0|/W where s_0 is the complex root of the characteristic equation s^2J_0(√) - W^2J_2(√) = 0 (It is proved that if W > W_c, the characteristic equation has one and only one pair of conjugate complex roots). Moreover an analytical solution for the fluid velocity is also given. In conclusion, the theory in this paper is verified by the experiments of the use of the distilled water and the mercury as the U-tube fluid.
- 1979-02-20