Method for Calculating Transients in a Gas Pipeline

Abstaract. An analytical solution of a system of partial differential equations describing the unsteady isothermal flow of real gases in gas pipelines is considered. Such a problem arises when studying the regularity of alterations in the instantaneous values of pressure and gas flow in main gas pipelines, for example, during startups and shutdowns of large gas consumers. Meanwhile, transients are not necessarily of a pronounced oscillatory nature, despite the fact that they are described by periodic functions. In the course of the research, the task was set to obtain a mathematical model of the process taking into account the inertial term of the equation of motion, the neglect of which is possible only if the friction losses are exceeded by 3.5–4 times over the shock pressure. An important feature of the solution that have been found is its universality, which makes it possible to significantly reduce labor costs when using it to find partial solutions to practical problems that differ in boundary conditions. The boundary conditions of the first kind are given as an arbitrary function of both the gas flow rate and its pressure. The solution is based on the widely used method of separation of Fourier variables. In order to simplify the calculations, the original differential equation is transformed in such a way that the boundary conditions acquire the property of homogeneity. It has been determined that the requirements that the boundary conditions are equal to zero at the initial moment of time introduced into the solution make it possible to obtain a concise record of the obtained analytical model, but do not limit the area of the use of the model with a surge change in the gas flow rate or pressure. The obtained analytical model of unsteady gas flow makes it possible, without using the Duhamel integral, to find analytical solutions under more complex boundary conditions than the flow rate jump. At the same time, the solutions found completely coincide with the solutions based on the Duhamel integral, but in the course of the solution that we have found it is possible to avoid integration, which has a positive effect on the applicability of this approach in the practice of engineering calculations.

Frequency/Laplace Domain Methods for Computing Transient Responses of Fractional Oscillators

Although computing the transient response of fractional oscillators, characterized by second-order differential equations with fractional derivatives for the damping term, to external loadings has been studied, most existing methodologies have dealt with cases with either restricted fractional orders or simple external loadings. In this paper, considering complicated irregular loadings acting on oscillators with any fractional order between 0 and 1, efficient frequency/Laplace domain methods for getting transient responses are developed. The proposed methods are based on pole-residue operations. In the frequency domain approach, "artificial" poles located along the imaginary axis of complex plane are designated. In the Laplace domain approach, the "true" poles are extracted through two phases: (1) a discrete impulse response function (IRF) is produced by taking the inverse Fourier transform of the corresponding frequency response function (FRF) that is readily obtained from the exact TF, and (2) a complex exponential signal decomposition method, i.e., the Prony-SS method, is invoked to extract the poles and residues. Once the poles/residues of the system are known, those of the response can be determined by simple pole-residue operations. Sequentially, the response time history is readily obtained. Two fractional oscillators with rational and irrational derivatives, respectively, subjected to sinusoidal and complicated earthquake loading are presented to illustrate the procedure and verify the correctness of the proposed method. The verification is conducted by comparing the results from both the Laplace and the frequency domain approaches with those from the numerical Duhamel integral method.

SOLUTIONS OF THE PROBLEM OF RANDOM VIBRATIONS IN HEREDITARY-DEFORMABLE SYSTEMS USING IMPULSIVE FUNCTIONS

In recent years, probabilistic-statistical methods for studying various problems of mechanics of solid deformable bodies are being applied more than ever. The principal part of its field of application is the development of a general theory of strength and a hereditarily deformed solid. As known, the theory of random oscillations is increasingly being used in technology. Current research is aimed to present a numerical-analytical approach for studying the dynamic response of a hereditarily deformable system to unsteady input influences. It is established that the dynamic reaction of hereditarily deformable systems to an arbitrary form of random perturbations can also be represented as the Duhamel integral if the impulse transition function satisfies special Cauchy problems for the integro-differential equation (IMU). A study proposes an accurate analytical solution to the IMU of an impulsive transition function in existing weakly singular Rzhanitsyn-Koltunov nucleuses.

Experimental and Theoretical Analysis for Isolation Performance of New Combined Isolation Devices under Blast Loading

The strong shock and vibration effect caused by explosion may pose a serious threat to the surrounding environment and the safety of personnel and equipment. This also makes the problem of vibration isolation and absorption of the structures subjected to blast loading increasingly prominent. In this paper, three kinds of new combined isolation devices with high resistance are designed and manufactured, and the characteristic parameters such as natural vibration period, frequency, and damping ratio are obtained through drop hammer impact test. Based on the Duhamel integral principle, analytical solutions of dynamic response of the combined isolation devices under rectangular pulse blast loading are derived, and the calculation expressions of transmissibility and vibration isolation rate are proposed. Combined with the test results, the isolation performance of three kinds of combined isolation devices under blast loading is obtained by using the theoretical calculation formula, and the influencing factors of isolation performance are further analyzed parametrically. The research results provide a reference for the application of combined isolation devices in isolation and shock absorption of structures under blast loading.

Dynamic response of a 2-2 multi-layered cement-based piezoelectric composite under arbitrary mechanical load

Theoretical analyses of the dynamic properties of a 2-2 multi-layered cement-based piezoelectric composite are presented based on the theory of piezoelasticity and D’Alembert’s principle. An arbitrary mechanical load acts on the free end of the composite in the form of pressure. The Taylor series expansion method is introduced for the arbitrary mechanical load, and the theoretical solutions of the composite are obtained mainly based on the eigenfunction expansion method, Duhamel integral, and Laplace transform. Comparisons between the theoretical results, numerical results, and four related theoretical studies are presented, and good agreements are found. Furthermore, the theoretical expression of magnification factors of the composite under the harmonic load are obtained and analyzed. In addition to providing a theoretical basis for the design and experimentation of 2-2 cement-based piezoelectric composites under the arbitrary loading, the theoretical methods presented in the article could be extended to analyze the dynamic characteristics of any multilayered composite structure.

TO THE SOLUTION OF ISSUES OF NONSTATIONARY DIFFUSION IN LAYERED ENVIRONMENTS

The issues of nonstationary diffusion in layered structures are considered. When designing the devices for implementing mass transfer processes, it is necessary to take into account the properties of the substance and the nature of the processes. Design time reduces significantly and the efficiency of the devices is higher if a good physical model is built and a mathematical analysis with kinetics of the processes is applied. The difficulties of theoretical analysis and calculation of mass transfer are determined by the complexity of the transfer mechanism to and from the phase boundary. Therefore, simplified models of mass transfer processes are used in which the mass transfer mechanism is characterized by a combination of molecular and convective mass transfer. Many important practical problems involve the calculation of nonstationary diffusion (Fick's second law of diffusion) for a certain volume of substance (substances). For qualitative evaluation of processes, in the case of symmetry, volumetric issues can be considered as one-dimensional tasks, i.e. dependent on one coordinate. The general solution of the non-stationary diffusion equation for layered environments is proposed. The case of non-stationary boundary conditions of the third kind on the external surface and boundary conditions of the fourth kind conjugation for contiguous layers has been considered. The solution is obtained by separating the Fourier variables by the eigenfunctions of the problem using the Duhamel integral. The proposed solution is explicit and due to the recurrent form of the basic relations can be useful in numerical calculations

Dynamic Assessment of the Core Barrel During Loss of Coolant Accident

This article focuses on the dynamic behavior of the Pressurized Water Reactor (PWR) during the Loss Of Coolant Accident (LOCA) which cause the significant acoustic loads on the Core Shrouds. The finite element analysis of a PWR was performed to obtain the acoustic response to the LOCA event. We have performed dynamic stress and strain calculations in the frequency domain for the Core Barrel, according to classical shell theories. The Duhamel integral was used to calculate the transient response of a shell to the transient load caused by the water hammer event. The results obtained were used for fracture mechanics evaluations for flaws, which may occur between inservice inspections.

Transient processes in linear electric circuits (examples)

In the tutorial on specific examples of typical algorithms for calculating transients in linear circuits by classical and operator methods, as well as several examples of the application of the Duhamel integral and the method of state variables using the program MathCAD. A distinctive feature of the textbook is a detailed methodological analysis of all the examples, a versatile approach to solving problems and comparative analysis of the solutions. It is intended for students of higher technical educational institutions of electrotechnical and electric power specialties, can be used by students of colleges of this direction, conforms to the standard of fgos VPO of the third generation, can be recommended at two-level system of education at preparation of bachelors as the additional source of information.