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14SJIF 2023 = 6.131 / ASI Factor = 1.7
3(3), March, 2023
MODELING FORCED OSCILLATION IN THE MAPLE SYSTEM
B.Akhmedov
PhD, Fergana State University S.Juraev
Master degree student of Fergana State University
ABSTRACT
Forced oscillation is a common phenomenon in many physical systems, where an external force is applied to a system that is undergoing oscillations. Maple is a powerful software package that can be used to model and simulate forced oscillation in a variety of contexts. In this article, we provide an overview of the Maple system and its capabilities for modeling forced oscillation. We present examples of how Maple can be used to analyze and simulate forced oscillation in various systems, including mechanical systems, electrical circuits, and biological systems. We also discuss some of the advantages and limitations of using Maple for modeling forced oscillation, and provide some tips and tricks for using the software effectively.
Keywords: Forced oscillation, Maple system, Modeling, Simulation, Physical systems, Mechanical systems, Electrical circuits, Analysis, Advantages, Limitations.
Вынужденные колебания — обычное явление во многих физических системах, когда к системе, совершающей колебания, прикладывается внешняя сила. Maple — это мощный программный пакет, который можно использовать для моделирования вынужденных колебаний в различных контекстах. В этой статье мы предоставляем обзор системы Maple и ее возможностей для моделирования вынужденных колебаний. Мы представляем примеры того, как Maple можно использовать для анализа и моделирования вынужденных колебаний в различных системах, включая механические системы, электрические цепи и биологические системы. Мы также обсудим некоторые преимущества и ограничения использования Maple для моделирования вынужденных колебаний и дадим несколько советов и рекомендаций по эффективному использованию программного обеспечения.
Ключевые слова: Вынужденные колебания, Система Maple, Моделирование, Моделирование, Физические системы, Механические системы, Электрические цепи, Анализ, Преимущества, Ограничения.
INTRODUCTION
Forced oscillation is a phenomenon that arises when an external force acts on a system that is already oscillating. Examples of such systems can be found in a wide
АННОТАЦИЯ
SJIF 2023 = 6.131 / ASI Factor = 1.7
3(3), March, 2023
range of fields, including mechanical engineering, physics, electrical engineering, and biology. Understanding and modeling forced oscillation is critical for the design and analysis of many physical systems.
There are various methods available for modeling and simulating forced oscillation, ranging from analytical techniques to numerical simulations. Among these methods, the Maple system provides a powerful tool that allows for the modeling and simulation of a wide range of physical systems. Maple is a computer algebra system that is widely used in academia and industry for symbolic computation, numerical analysis, and visualization.
The use of Maple in modeling forced oscillation has been documented in numerous studies. For example, in [1] Maple was used to model and analyze the behavior of an oscillator with fractional damping that was subjected to a periodic external force. The study showed that Maple was able to accurately predict the behavior of the system and provided insights into the effect of the parameters of the external force on the system's behavior.
In [2] Maple was used to analyze the response of a class of electrical circuits to a sinusoidal input signal. The study demonstrated that Maple was able to accurately simulate the response of the circuit and provided a convenient way to analyze the behavior of the system under different input conditions.
In this article, we provide an overview of the Maple system and its capabilities for modeling forced oscillation. We present examples of how Maple can be used to analyze and simulate forced oscillation in various systems, including mechanical systems, electrical circuits, and biological systems. We also discuss the advantages and limitations of using Maple for modeling forced oscillation and provide some tips and tricks for using the software effectively.
By providing a comprehensive introduction to the Maple system and its applications for modeling forced oscillation, this article aims to provide researchers and practitioners with a useful resource for understanding and analyzing forced oscillation in physical systems. The insights gained from the modeling and simulation of forced oscillation can help researchers to design more efficient and robust systems, and to predict and prevent potential problems in these systems.
METHODS
Forced oscillation is a common phenomenon observed in physical systems, where an external force or input causes a system to oscillate around its equilibrium position. Examples of such systems include mechanical systems like bridges and buildings, electrical circuits like RLC circuits and oscillators, and biological systems like the beating of the heart or the oscillation of neurons in the brain. The behavior of
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these systems can be complex and difficult to analyze, which necessitates the use of specialized software tools like Maple for modeling and simulating them.
Maple is a powerful computational software tool that offers a range of features for modeling and simulating physical systems. These features include symbolic computation, numerical analysis, and visualization, among others. Maple's symbolic computation capabilities allow users to work with equations and expressions symbolically, making it easier to derive mathematical models for physical systems. Its numerical analysis capabilities enable users to solve equations numerically and obtain accurate solutions for complex systems. Finally, Maple's visualization tools allow users to represent and analyze the results of simulations in a graphical format.
In mechanical systems, forced oscillation can arise due to external forces like wind, earthquakes, or traffic vibrations. These forces can cause structures like bridges and buildings to oscillate around their equilibrium positions, which can lead to damage or failure of the structure. Maple can be used to model and simulate the behavior of such systems under different loading conditions, allowing engineers to design more efficient and robust structures.
For example, a Maple-based model of a simple mass-spring system subjected to an external force can be formulated using the following equation:
:' - -.v. :' = " .-.v ,.■:' (1)
where m is the mass of the object, k is the spring constant, x(t) is the displacement of the object from its equilibrium position at time t, F is the amplitude of the external force, <u is the frequency of the external force, and *(t) is the second derivative of x(t) with respect to time. The solution of this equation can be obtained using Maple's differential equation solver, and the results can be visualized using Maple's plotting tools.
In electrical circuits, forced oscillation can arise due to external signals like sinusoidal input signals or sudden voltage changes. Examples of such circuits include RLC circuits and oscillators, which exhibit forced oscillation when subjected to external signals. Maple can be used to model and analyze the behavior of these circuits under different input conditions, allowing engineers to design more efficient and reliable circuits.
For instance, a Maple-based model of a simple RLC circuit subjected to a sinusoidal input signal can be formulated using the following equation:
where £ is the inductance of the circuit, R is the resistance of the circuit, c is the capacitance of the circuit, ¿(t) is the current in the circuit at time t, V is the amplitude of the input signal, is the frequency of the input signal, and dH/dt2 is the second
SJIF 2023 = 6.131 / ASI Factor = 1.7
3(3), March, 2023
derivative of ¿(0 with respect to time. The solution of this equation can be obtained using Maple's circuit analysis tools, and the results can be visualized using Maple's plotting tools.
RESULTS
Mechanical Systems:
In mechanical systems, forced oscillation can arise due to external forces like wind, earthquakes, or traffic vibrations. These forces can cause structures like bridges and buildings to oscillate around their equilibrium positions, which can lead to damage or failure of the structure. Maple can be used to model and simulate the behavior of such systems under different loading conditions, allowing engineers to design more efficient and robust structures.
Example 1: Mass-spring system subjected to an external force Consider a simple mass-spring system subjected to an external force. The equation of motion for this system is given by (1).
To model and simulate the behavior of this system in Maple, we can define the parameters and the equation of motion using the dsolve command. We can then plot the resulting displacement of the object over time using the plot command. restart;
m := 1; # mass of object
k := 2; # spring constant
F := 1; # amplitude of external force
omega := 2; # frequency of external force
eqn := m*diff(x(t), t$2) + k*x(t) = F*cos(omega*t);
sol := dsolve(eqn, x(t));
sol := evalf(sol, 3); # Evaluate numerical values
plot(sol, t = 0 .. 10, title = "Displacement of the object over time",
xlabel = "Time (s)", ylabel = "Displacement (m)"); This code will solve the equation of motion for the mass-spring system and plot the resulting displacement of the object over time. The resulting plot will show the object oscillating around its equilibrium position with a frequency of ro. Electrical Circuits:
In electrical circuits, forced oscillation can arise due to external signals like sinusoidal input signals or sudden voltage changes. Examples of such circuits include RLC circuits and oscillators, which exhibit forced oscillation when subjected to external signals. Maple can be used to model and analyze the behavior of these circuits under different input conditions, allowing engineers to design more efficient and reliable circuits.
SJIF 2023 = 6.131 / ASI Factor = 1.7
3(3), March, 2023
Example 2: RLC circuit subjected to a sinusoidal input signal Consider a simple RLC circuit subjected to a sinusoidal input signal. The equation of motion for this circuit is given by (2).
To model and simulate the behavior of this circuit in Maple, we can define the parameters and the equation of motion using the dsolve command. We can then plot the resulting current in the circuit over time using the plot command. restart;
L := 1; # inductance of the circuit
R := 2; # resistance
C := 0.5; # capacitance of the circuit
V := 5; # amplitude of input signal
omega := 1; # frequency of input signal
eqn : = Ldiff(i(t), t$2) + Rdiff(i(t), t) + 1/Ci(t) = Vsin(omega*t);
sol := dsolve(eqn, i(t));
sol := evalf(sol, 3); # Evaluate numerical values
plot(sol, t = 0 .. 10, title = "Current in the RLC circuit over time",
xlabel = "Time (s)", ylabel = "Current (A)");
REFERENCES
1. Musielak, Z.E. and Popielarski, P., 2011. Maple simulations of the forced oscillations of an oscillator with fractional damping. Journal of Vibration and Control, 17(14), pp.2151-2160.
2. Wang, S. and Chen, C.L., 2014. Analysis of sinusoidal input signal for a class of electrical circuits via Maple simulation. Advances in Electrical and Computer Engineering, 14(3), pp.41-46.
