On axisymmetric parametric oscillations of a fluid in a cylindrical vessel
Fluid, gas and plasma mechanics
Bauman Moscow State Technical University, MSTU, 5, 2-nd Baumanskaya, Moscow, 105005, Russia
The authors considered the case of axisymmetric parametric oscillations of an ideal fluid filling the cylindrical vessel. They obtained the differential equation of parametric oscillation damping with account for the main parametric resonance, as well as numerical values of the variables comprising this equation.
It was shown that could initiate small oscillations of the system, with the corresponding values of the modulation index. The value of the modulation index is determined by Van der Pol method. The problem was considered in the linear formulation, with fluid movements were assumed small. The velocity potential satisfies the Laplace equation in a cylindrical domain. On the wetted surface of the vessel the conditions of impermeability were met, while on the free surface of the liquid the linearized boundary condition was met. The velocity potential, in accordance with the Fourier method, was represented as a series of the hyperbolic functions, Bessel functions of 1st order and the time factor.
The equation for the parametric oscillation was obtained for the time factor from the equation for the velocity potential. The kinetic energy of the mechanical system was determined by the gradient of the velocity potential. The reduced inertia coefficient of the system was calculated by the Green’s function method. The viscous resistance force in the fluid was assumed proportional to the first degree of speed. The fluid velocity was defined as the gradient of the velocity potential. Then we compose the Rayleigh dissipative function. Expression for the potential energy was composed. We composed the Lagrange equation of the II-nd order with account for the dissipative forces, due to the presence of viscous friction in the mechanical system under consideration.
The resulting differential equation is a second order differential equation with a small parameter. The damping coefficient implies experimental determination. We indicate references to the papers, which present the numerical values of the damping coefficient. Further, the damping coefficient was assumed as known. The averaging was performed, and the border region of parametric resonance instability was sett. The conclusion on the existence of the small oscillations of the respective values of the modulation index was made.
Keywords:parametric oscillation, fluid, damping, modulation index
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