a result, these equations are strongly coupled, which has to be taken into account when simulating UCFTs. Due to equations nonlinearity usually it is impossible to find an analytical solution of such a complicated problem, hence numerical methods are employed.The motion of MMOs as of solid bodies in the water flow has been well studied [2,3]. The MMO mathematical models include mathematical models of their hulls, hydrodynamic drags, propulsive complexes, bearing surfaces, and other elements. As a rule, it is possible to develop an MMO mathematical model in the form of ordinary differential equations system, despite complex dependencies between parameters of model elements. This allows applying effective methods of numerical solution of differential equations to the MMO simulation.The flexible tether modeling is a more challenging problem, as the FT is an object with distributed parameters which is described by a system of nonlinear differential equations with partial derivatives. Two approaches were evolved for modeling FT dynamics [4]. These are segmental and lumped-mass-spring (LMS) methods. For the former the FT is modeled as a continuous system and resulting partial differential equations are solved numerically by finite difference or any suitable approximation method. In the LMS approach the cable is modeled as mass points joined together by massless elastic elements of finite length, which makes it possible to express the FT model in the form of ordinary differential equations. All the forces along the cable are assumed to be concentrated at the mass points. IntroductionUnderwater complexes with flexible tethers (UCFT) consist of two types of elements: those with lumped and distributed parameters. The former include marine mobile objects (MMO): surface vessels, submarines, remotely operated vehicles (ROV) and others. The latter include flexible tethers (FT): umbilical cables, towing cables, anchor chains, etc. An example of a two-linked UCFT is shown in Fig. 1 [1]. Fig. 1. Two-linked underwater complex with flexible tethersMathematical modeling is one of the main methods of the UCFT studies. In this regard, the UCFT motion simulators should include corresponding MMO and FT mathematical models. It is known that their motion equations are non-linear and their dynamic behaviors are mutually dependent. As APPLIED MECHANICS DEVELOPMENT OF THE MATHEMATICAL MODELING METHOD FOR DYNAMICS OF THE FLEXIBLE TETHER AS AN ELEMENT OF THE UNDERWATER COMPLEX О . B l i n t s o vPhD, Assistant Professor Department of Information Security Lviv Polytechnic National University S. Bandery str., 12, Lviv, Ukraine, 79000 E-mail: energybox@mail.ruРозроблено метод математичного моделю-вання динаміки гнучкого зв'язку (ГЗ) на основі автоматичного контролю осьового руху його еле-ментів. Синтезовано регулятор відстаней між елементами ГЗ як складова математичної моде-лі. Запропоновано спосіб моделювання ГЗ зі змін-ною довжиною. Показано ефективність розробле-ного методу у порівнянні з методом зосереджених мас та еластичних зв'язків при мо...
The definition of a new class of control objects is proposed. It is an underwater complex with flexible tethers (UCFT) for which there is the need to automate motion control under uncertainty and nonstationarity of own parameters and external disturbances. Classification of marine mobile objects and characteristics of the flexible tethers as UCFT elements is given. The basic UCFTs configurations that are used in the implementation of advanced underwater technologies are revealed. They include single-, double- and three-linked structures with surface or underwater support vessels and self-propelled or towed underwater vehicles. The role of mathematical modeling in tasks of motion control automation is shown. The tasks of UCFT mathematical modeling are formulated for synthesis and study of its automatic control systems. Generalized structures of mathematical models of UCFT basic elements are proposed as the basis for the creation of simulating complex to study the dynamics of its motion. The tasks of UCFT identification as a control object are formulated. Their consistent solution will help to obtain a UCFT mathematical model. The basic requirements for UCFT automatic motion control systems are determined. Their satisfaction will ensure implementation of selected underwater technology. Areas of development of synthesis methods of UCFT automatic control systems are highlighted.
Складено математичну модель підводного комплексу, яка містить нелінійні елементи гідродинамічної природи. Показано неможливість застосування методу оберненої динаміки для синтезу високоточних систем автоматичного керування такими об'єктами. Розроблено принцип декомпозиції для формування еталонної моделі, який дає змогу синтезувати системи керування підводними комплексами методом оберненої динаміки. Працездатність принципу декомпозиції перевірено синтезом системи керування підводним апаратом як складовою підводного комплексу Ключові слова: еталонна модель, метод оберненої динаміки, автоматичне керування, підводний комплекс Составлена математическая модель подводного комплекса, которая содержит нелинейные элементы гидродинамической природы. Показана невозможность применения метода обратной динамики для синтеза высокоточных систем автоматического управления такими объектами. Разработан принцип декомпозиции для формирования эталонной модели, который дает возможность синтезировать системы управления подводными комплексами методом обратной динамики. Работоспособность принципа декомпозиции проверена синтезом системы управления подводным аппаратом как составляющей подводного комплекса Ключевые слова: эталонная модель, метод обратной динамики, автоматическое управление, подводный комплекс
There is developed a specialized simulating complex for computer-aided study of the efficiency of automatic control systems for the underwater towed system «carrier ship -tether winch -tether -towed underwater vehicle». The complex includes mathematical models of the dynamics of a marine moving object, flexible connection and tether winch. The former is used to simulate the spatial motion dynamics of a towing vessel and a towed underwater vehicle as solid bodies in a water flow. Tether simulation employs a method of simulating a flexible connection with automatic control of the axial motion of its elements, which makes it possible to account for the dynamics of the change in the length of its released part. The model of the tether winch dynamics provides simulation of the tether release and gathering. The simulation complex enables studying the spatial motion of a towed underwater vehicle in the water column in relation to the towing vessel as a control object at the dynamic change of the length of the released part of the tether. Keywords: simulating complex; towed underwater vehicle; towing cable dynamics; spatial motion.Анотація. Розроблено спеціалізований моделюючий комплекс для комп'ютерного дослідження ефективності систем автоматичного керування підводною буксируваною системою у складі «судно-носій -кабельна лебід-ка -кабель-буксир -буксируваний підводний апарат». Моделюючий комплекс дає змогу досліджувати про-сторовий рух у водній товщі буксируваного підводного апарата відносно судна-буксирувальника як об'єкта керування при динамічній зміні довжини випущеної частини кабель-буксира. Ключові слова: моделюючий комплекс; буксируваний підводний апарат; динаміка кабель-буксира; просто-ровий рух.Аннотация. Разработан специализированный моделирующий комплекс для компьютерного исследования эффективности систем автоматического управления подводной буксируемой системой в составе «судно-носитель -кабельная лебедка -кабель-буксир -буксируемый подводный аппарат». Моделирующий ком-плекс позволяет исследовать пространственное движение в водной толще буксируемого подводного аппарата относительно судна-буксировщика как объекта управления при динамическом изменении длины выпущенной части кабель-буксира.
The towed underwater system is one of the fixed assets of the study of water areas. The effectiveness of its application depends on the characteristics laid at the design stage. The main task of the towed underwater vehicle (TUV) is the motion of technological equipment. Therefore, it is important to ensure the specified dynamic properties of the unit and automate the control of its motion. In the paper the typical forms of the unit are analyzed, the features of their control at small depths are set. TUV control is carried out in conditions of uncertainty. Therefore, the design of an automatic control system (ACS) for its motion is proposed to be carried out using the appropriate synthesis method – the method of minimizing local functionals. The control law contains integral components and, under the constraints of control actions, generates the problem of integral saturation. To eliminate the integral saturation in the work, the condition integration method is improved. On its basis, the control law and the structure of the controller of high dynamic accuracy of a second-order nonlinear object are synthesized. It is the basis for the synthesis of ACS controlled degrees of freedom of the underwater vehicle in conditions of uncertainty. Usually TUVs contain two degrees of mobility. Translational motions of the unit are generated by changing its angular orientation. The paper synthesizes TUV controllers of pitch and roll based on the control law of the second order. Each control signal of the unit can affect both the roll and the pitch of the unit, which leads to decrease in the quality of control in general. To coordinate the work of controllers, a method is proposed, which is based on adjusting the initial conditions of the controller with greater error. On its basis, the automatic control system of the rotational motion of the unit is synthesized. It provides high dynamic precision control of two-dimensional rotational motion of the unit in uncertainty and is the basis for the ACS synthesis of its translational motion in space.
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