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Taking as a starting point the law of conservation of the total energy of the system, and introducing two basic state functions - the Lagrangian and the Rayleigh function, the general form of the equation of motion for any dynamic system with a finite number of degrees of freedom is derived. The theory is illustrated by considering the rotating - t...
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... The centre of mass of each of its three components is located at point 0 identified with the origin of any of the applied coordinate systems. When the gyroscope is placed on the northern hemi-sphere of the Earth, the axis r 3 of the outer frame is perpendicular to the idealized spherical surface of the Earth and directed out of its interior, see Fig. 7. The remaining two axes r 1 and r 2 of the orthogonal system of coordinates are directed as follows: r 1 -to the north and r 2 -to the west, forming in result the right-hand side system of coordinates. Assuming next that the an- gular velocity vector Ω 0 representing the rotation of the Earth around its kinematic axis is directed as ...
Context 2
... 7. The remaining two axes r 1 and r 2 of the orthogonal system of coordinates are directed as follows: r 1 -to the north and r 2 -to the west, forming in result the right-hand side system of coordinates. Assuming next that the an- gular velocity vector Ω 0 representing the rotation of the Earth around its kinematic axis is directed as shown in Fig. 7, then its components in r-coordinates take the following form: r 1 = Ω 0 cos β, r 2 = 0, r 3 = Ω 0 sin β, where β is the corresponding Latitude. Let us now consider the case when the inner frame is fixed at the right angle to the outer frame, so that ϑ = 0 (see Fig. 6), and the gyroscope reduces to the system with one degree of freedom ...
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Citations
... 17 Ładziński started with the law of conservation of the total energy of the system, introducing the Lagrangian and Rayleigh functions as two basic state functions, and derived the general form of the equation of motion for any dynamic system with a finite number of degrees of freedom. 18 Mahmood et al. utilized the Lagrangian equation to explain the expanded unstable rotary planer inverted pendulum arrangement. 19 Obviously, based on the results of existing relevant research and by applying the principles of energy conservation and the steady action principle (also called the principle of least action), 20 the kinetic and potential energy of the system can be formulated as a function of the generalized coordinates and their corresponding time derivatives. ...
Large-scale rotating mechanical equipment in the mining arena plays a pivotal role in mining production, where vibration issues directly influence production efficiency and safety. This Review aims to provide a comprehensive review of the latest advancements and methodologies related to the generation mechanisms, identification, and applications of vibrational characteristics in large-scale mining rotating mechanical equipment. Semi-autogenous mills, ball mills, and coal mills are selected as archetype equipment, and the Lagrangian motion equation is employed to unveil the generation mechanisms of vibrations and the embedded physical information in the signals of these machines. Initially, the research delves deeply into the acquisition, extraction, and identification of vibrational signal features, emphasizing that while mechanical vibration signals can reveal the internal operational state and fault information of machinery, there remains a need to enhance their capability to depict complex vibrational signals. Subsequently, this Review discusses in depth the studies focused on predicting the vibrational state of equipment by establishing accurate and reliable soft measurement models, pointing out that current models still have room for improvement in prediction accuracy and generalization capabilities. Conclusively, based on the elucidation of mechanical vibration mechanisms and the collation and outlook of the existing research study, the importance of on-site monitoring, deep learning, Internet of Things technology, and full lifecycle management is accentuated. To better support practical engineering applications, further exploration into the physical properties of vibrational signals and the mechanisms of mechanical vibrations is essential.
... Investigation of dynamic systems with three degrees of freedom is a very important problem for applied mechanics, microwave technique, quantum physics, communication theory [1][2][3][4][5][6][7][8][9][10]. Three coupled mechanical oscillators with varying masses or spring stiffnesses can be considered as the simplest example of a sixdimensional linear dynamic system. ...
... Such systems with constant parameters have been written in detail, for example, in [1][2][3][4][5][6][7][8][9][10]. However, linear parametric systems are least studied for today. ...
In this paper the \(6 \times 6\) fundamental matrix of a linear six - dimensional system with arbitrary piecewise-constant parameters is obtained in the analytical form in elementary functions for the first time. To write this matrix the new sign-function \(F_q^{(i,j)}\) is introduced and described here. This function determines the order of interaction of eigen-oscillations of intervals with constant parameters in a resulting equivalent oscillation. The fundamentally new concept of equivalent oscillations for six-dimensional systems is introduced for the first time too. The stability problem of these systems is considered and the stability conditions are obtained in the analytical form in original parameters of a linear six-dimensional system. The results obtained allow solving a number of problems in mechanics, electrical engineering, communication systems, optoelectronics, automatic control theory and information theory. It can be, for example, the problems of determining the stability of dynamical systems, the problems of synthesizing dynamical systems of various nature, etc.
In this paper, we consider the LoRa technology to expand sensor network coverage in smart sustainable cities. A model of a LoRa mesh network is proposed using the AODV protocol in packet routing. With a simulation model developed based on OMNET++, a series of computer experiments was carried out with changing various parameters. In the experiments results, the end-to-end delay and packet loss ratio were analyzed in the dependence on the number of nodes and packet size in the network. The simulation results show that the latency is relatively high in the LoRa mesh network, but it might be accepted for some applications.
