Forced liquid vibrations in prismatic tanks under vertical and horizontal loads

The method of studying forced vibrations of a liquid in rigid prismatic tanks partially filed by a liquid is offered. It is supposed that the liquid is an ideal and incompressible one, and its motion, caused by the action of external influences, is irrotational. In these assumptions, there exists a velocity potential that satisfies the Laplace equation. The boundary value problem for this potential is formulated. On the wetted surfaces of the tank the non-penetration conditions are chosen. On the free surface of the liquid, the kinematic and static conditions are specified. The static condition consists in the equality of pressure on the free surface to atmospheric one. The liquid pressure is determined from the Cauchy-Lagrange integral. To formulate the kinematic condition, an additional unknown function is introduced, which describes the motion of the free surface. The kinematic condition is the equality of the velocity of the liquid, which is described by the velocity potential, and the velocity of the free surface itself. These modes of free vibrations are used as a system of basic functions in solving problems of forced fluid vibrations in reservoirs. Unknown functions are presented as series of the basic functions. The coefficients of these series are generalized coordinates. Periodic excitation forces acting in the vertical and horizontal directions are considered. If vertical excitation is studied, this leads to appearance of additional acceleration. Here we obtain a system of unbounded differential equations of the Mathieu type. This allows us to investigate the phenomena of parametric resonance. The effect of parametrical resonance is considered when the vertical excitation frequency is equal to double own frequency of liquid vibrations Dependences of change in the level of free surface via time under both separate and mutual action of horizontal and, vertical forces of are obtained. The phase portraits of a dynamic system with indication of resonances are presented. The method allows us to carry out the adjustment of undesired excitation frequencies at the design stage at reservoir producing in order to prevent the loss of stability.

Forced fluid fluctuations in cylindrical reservoirs under vertical excitation

The shell and shell structures containing various types of liquid fillers can be exposed to intense dynamic effects during the exploitation. In order to analyze the strength of structures in these conditions, it is necessary to take into account nonlinear phenomena in fluid motion, since the application of linear equations does not provide an adequate assessment for the determination of the pressure and amplitude of the splashing. In this paper, a study of fluid fluctuations in a rigid cylindrical reservoir partially filled with the liquid under condition of vertical agitation has been carried out. The systems of differential equations that correspond to the linear and nonlinear formulation of the problem are presented. The fluid is believed to be perfect and incompressible, and its movement induced by external influences is non-vortex. Under these conditions there is a velocity potential that satisfies the Laplace equation. The conditions of non-leakage on the wetted surfaces of the shell are chosen as the boundary conditions for solving the boundary value problem. The kinematic and static conditions are specified on a free surface. The static condition consists in the equality of pressure on the liquid surface with atmospheric pressure. The pressure is determined from the Cauchy-Lagrange integral. In this case the linearization of the Cauchy-Lagrange integral leads to the linear formulation of the problem. Quadratic components are taken into account for the nonlinear formulation. To formulate the kinematic condition an additional unknown function describing the motion of the free surface is introduced. The kinematic condition is the equality of the liquid velocity described by the velocity potential and the velocity of the free surface itself. If there is a vertical agitation, an additional acceleration will be present. Therefore for the linear formulation we obtain a system of unbounded differential equations, each of which is the equation of Mathieu. This allows us to investigate the phenomena of parametric resonance. When analyzing differential equations which occur in case of a nonlinear problem, it has been found that the solutions of such equations depend essentially on the initial conditions. The phase portraits of a dynamic system with indication of resonances are presented. A numerical analysis of the differential equation corresponding to nonlinear formulation has been carried out.

Generalized Robertson-Walker Space-Time Admitting Evolving Null Horizons Related to a Black Hole Event Horizon

A new technique is used to study a family of time-dependent null horizons, called “Evolving Null Horizons” (ENHs), of generalized Robertson-Walker (GRW) space-time (M¯,g¯) such that the metric g¯ satisfies a kinematic condition. This work is different from our early papers on the same issue where we used (1+n)-splitting space-time but only some special subcases of GRW space-time have this formalism. Also, in contrast to previous work, we have proved that each member of ENHs is totally umbilical in (M¯,g¯). Finally, we show that there exists an ENH which is always a null horizon evolving into a black hole event horizon and suggest some open problems.

Numerical Simulation of Sloshing in 2D Rectangular Tanks Based on the Prediction of Free Surface

A finite difference method for analyzing 2D nonlinear sloshing waves in a tank has been developed based on the potential flow theory. Afterσ-transformation, the free surface is predicted by the kinematic condition, and nonlinear terms are approximated; the governing equation and boundary conditions are discretized to linear equations in the iterative process of time. Simulations of standing waves and sloshing in horizontally excited tanks are presented. The results are compared with analytical and numerical solutions in other literatures, which demonstrate the effectiveness and accuracy of this numerical method. The beating phenomenon of sloshing in the tank with different aspect ratios is studied. The relationship between sloshing force and aspect ratio under the same external excitation is also discussed.

Autodriver Algorithm for Autonomous Vehicle

In a recent research a superior algorithm has been introduced to design an autodriver to keep an autonomous vehicle on a given road using independent four-wheel-steering (4WS) system. The kinematic condition of steering sets the steer angles such that the kinematic center of rotation of the vehicle be controlled on a two dimensional space. The vehicle however, will turn about an actual point that is not necessarily at the road curvature center; because of the road, tire characteristics, and dynamics of the moving vehicle. The algorithm shows how the position of the dynamic turning center can be controlled by adjusting the steer angles such that it coincides with the road curvature center. Such a vehicle will move on the desired road autonomously. In this paper, the required facilities to apply the algorithm experimentally will be discussed.

Four-Wheel Steering Autonomous Vehicles

Employing an independent four-wheel-steering (4WS) system, we introduce an autodriver algorithm to keep an autonomous vehicle on a given road. The kinematic condition of steering can be used to set the steer angles such that the kinematic center of rotation is at a desired point. The road and tire characteristics, along with the dynamics of a moving vehicle cause the vehicle to turn about an actual point that is not necessarily at the road curvature center. The position of the dynamic turning center can be controlled by adjusting the steer angles such that it coincides with the road curvature center. Such a vehicle will move on the desired road autonomously.