Динаміка осцилятора з жорсткою характеристикою пружності при дії силового імпульса
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Дата
2018
ORCID
DOI
10.20998/2078-9130.2018.33.151227
Науковий ступінь
Рівень дисертації
Шифр та назва спеціальності
Рада захисту
Установа захисту
Науковий керівник
Члени комітету
Видавець
НТУ "ХПІ"
Анотація
Розглянуто рух осцилятора з показником нелінійності 3/2 при дії ступінчастого та прямокутного імпульсів. Побудовано аналітичний розв'язок нелінійного диференціального рівняння другого порядку, де для розрахунку переміщень задіяно періодичні Ateb-функції та еліптичний косинус Якобі. Встановлено, що при навантаженні осцилятора миттєво прикладеною сталою силою коефіцієнт динамічності дорівнює (2,5)2/3. При дії на осцилятор прямокутного силового імпульсу коефіцієнт динамічності залежить від тривалості імпульсу, але не перевершує (2,5)2/3. Визначено такі тривалості, за яких розвантажений осцилятор має найбільшу та найменшу амплітуди коливань. Для спрощення розрахунків, з використанням одержаних розв'язків задачі Коші, складено таблиці, задіяних спеціальних функцій. Наведено приклади розрахунків, які підтверджують вірогідність виведених формул.
The motion of the oscillator with a non-linearity of the restoring force equal to 3/2 is considered under the action of a stepped (instantaneously applied constant force) and a rectangular pulse of finite duration of action. An analytic solution of a second-order nonlinear differential equation is constructed, where periodic Ateb-functions and Jacobi elliptic cosine are used to calculate the displacements. The two forms of the analytical solution obtained are equivalent and establish a connection between the used Ateb and elliptic functions. Unlike well-known works devoted to free and forced harmonic oscillations, the oscillator motion caused by impulse dynamic loading is considered here. The aim of the paper is to construct compact formulas for calculating the displacements of a nonlinear system with a rigid elastic characteristic under its nonstationary oscillations. To achieve this goal, integral representations of the above special functions are used, with the subsequent application of their tables. In addition to the known tables of Jacobi functions, it is also suggested to use the tables of Ateb-functions compiled in the work. The dynamic factor of the system is determined. It is established that when the oscillator is loaded instantly with a constant force, the dynamic coefficient is equal to (2,5)2/3 < 2, which is characteristic of systems with a rigid characteristic of elasticity. It is smaller than for linear systems. When the rectangular force pulse is applied to the oscillator, the dynamic coefficient depends on the pulse duration and does not exceed (2,5)2/3. As a result of the study, such lengths of the rectangular pulse are determined, at which the unloaded oscillator has the largest or smallest, equal to zero, amplitudes of the oscillations. In this case, the unloaded oscillator stops moving, that is, it goes into a state of rest. The conditions for achieving extreme amplitudes depend not only on the duration (width), but also on the height of the rectangular pulse, which is characteristic of nonlinear systems. To simplify the calculations, using the constructed analytic solutions of the Cauchy problem, tables of involved special functions are compiled. Examples of calculations are given that confirm the reliability of the derived formulas and illustrate the possibilities of the theory presented. Conclusions in the work are the result of theoretical analysis and calculations.
The motion of the oscillator with a non-linearity of the restoring force equal to 3/2 is considered under the action of a stepped (instantaneously applied constant force) and a rectangular pulse of finite duration of action. An analytic solution of a second-order nonlinear differential equation is constructed, where periodic Ateb-functions and Jacobi elliptic cosine are used to calculate the displacements. The two forms of the analytical solution obtained are equivalent and establish a connection between the used Ateb and elliptic functions. Unlike well-known works devoted to free and forced harmonic oscillations, the oscillator motion caused by impulse dynamic loading is considered here. The aim of the paper is to construct compact formulas for calculating the displacements of a nonlinear system with a rigid elastic characteristic under its nonstationary oscillations. To achieve this goal, integral representations of the above special functions are used, with the subsequent application of their tables. In addition to the known tables of Jacobi functions, it is also suggested to use the tables of Ateb-functions compiled in the work. The dynamic factor of the system is determined. It is established that when the oscillator is loaded instantly with a constant force, the dynamic coefficient is equal to (2,5)2/3 < 2, which is characteristic of systems with a rigid characteristic of elasticity. It is smaller than for linear systems. When the rectangular force pulse is applied to the oscillator, the dynamic coefficient depends on the pulse duration and does not exceed (2,5)2/3. As a result of the study, such lengths of the rectangular pulse are determined, at which the unloaded oscillator has the largest or smallest, equal to zero, amplitudes of the oscillations. In this case, the unloaded oscillator stops moving, that is, it goes into a state of rest. The conditions for achieving extreme amplitudes depend not only on the duration (width), but also on the height of the rectangular pulse, which is characteristic of nonlinear systems. To simplify the calculations, using the constructed analytic solutions of the Cauchy problem, tables of involved special functions are compiled. Examples of calculations are given that confirm the reliability of the derived formulas and illustrate the possibilities of the theory presented. Conclusions in the work are the result of theoretical analysis and calculations.
Опис
Ключові слова
нелінійний осцилятор, жорстка характеристика пружності, імпульсне навантаження, коефіцієнт динамічності, періодичні Ateb-функції, еліптичний косинус, nonlinear oscillator, rigid elasticity characteristic, impulse loading, dynamic coefficient, periodic Ateb-functions, elliptic cosine, integral representations, tables of special functions
Бібліографічний опис
Ольшанський В. П. Динаміка осцилятора з жорсткою характеристикою пружності при дії силового імпульса / В. П. Ольшанський, С. В. Ольшанський // Вісник Національного технічного університету "ХПІ". Сер. : Динаміка і міцність машин = Bulletin of the National Technical University "KhPI". Ser. : Dynamics and Strength of Machines : зб. наук. пр. – Харків : НТУ "ХПІ", 2018. – № 33 (1309). – С. 37-42.