Application of modified Mickens iteration procedure to a pendulum and the motion of a mass attached to a stretched elastic wire

The paper deals with the application of a strong method called the modified Mickens iteration technique which is used for solving a strongly nonlinear system. The system describes the motion of a simple mathematical pendulum with a particle attached to it through a stretched wire. This model has great applications especially in the area of nonlinear vibrations and oscillation systems. The proposed method depends on determining the frequency and amplitude of the system through the modified Mickens iterative approach which is a modification of the regular Mickens approach. The preliminaries of the proposed technique are present and the application to the model is discussed. The method depends on the Mickens iteration approach which transforms the considered equation into a linear form and then is solving this equation result in the approximate solution. Some examples are given to validate and illustrate the effectiveness and convenience of the method. These results are compared with other relative techniques from the literature in terms of finding the frequency of the two examined models. The method produces more accurate results when compared to these methods and is considered a strong candidate for solving other nonlinear problems with applications in science and engineering.

MATHEMATICAL PENDULUM MODEL WITH MOBILE SUSPENSION POINT

Background. The new dynamic problem, which is posed and solved in this article, is a theoretical generalization of the well-known classical problem of free oscillations of a mathematical pendulum. In the proposed setting, it is new and relevant, and can be successfully used in such fields of technology as vibration protection, vibration isolation and seismic protection of high-rise flexible structures, long power lines, long-span bridges and other large-sized supporting objects. Objective. The aim of the work is to derive the equations of own oscillations of a new mathematical pendulum-absorber, to find a formula for determining the frequency of its small own oscillations and to establish those control parameters that allow you to tune the single-mass pendulum absorber to the frequency of the fundamental tone of the carrier object. Methods. To achieve this goal, we used the methods of analytical mechanics, namely, the Appel’s formalism, as well as the linearization of the obtained differential equations. Results. A mathematical model is constructed in the work that describes the own oscillations of a new-design mathematical pendulum with a movable (spring-loaded) suspension point with length L. The model is a system of differential equations obtained using the Appel’s formalism. Based on them, after linearization of nonlinear equations, a formula is established for the frequency of small own oscillations of a pendulum with a mobile suspension point. Conclusions. It is shown that the frequency of own oscillations of the new mathematical pendulum coincides with the frequency of own oscillations of the classical mathematical pendulum with an equivalent suspension length, which is equal to . In the case where the suspension point is fixed (k ® ¥), the frequency formula turns into a well-known formula for the frequency of small own oscillations of a classical mathematical pendulum . If the stiffness coefficient of elastic elements tends to zero (k ® 0), then the frequency w of the damper also tends to zero. An important structural feature of the proposed pendulum is noted, consisting in the fact that due to the appropriate choice of the three control parameters of the pendulum (k, L, m), its frequency in magnitude can be made any in the range from zero to .

Adiabatic invariants

This chapter addresses the adiabatic invariant for a mathematical pendulum, a model of a “gas” consisting of a single molecule in a piston, adiabatic approximation, and a simplified model of an ion H2+. The chapter also discusses the connection between the volume and the pressure of a gas consisting of particles inside an elastic cube, the adiabatic invariants for a charged anisotropic harmonic oscillator in a uniform magnetic field, a magnetic trap, and the action and angle variables for the simple systems.

Adiabatic invariants

This chapter addresses the adiabatic invariant for a mathematical pendulum, a model of a “gas” consisting of a single molecule in a piston, adiabatic approximation, and a simplified model of an ion H2+. The chapter also discusses the connection between the volume and the pressure of a gas consisting of particles inside an elastic cube, the adiabatic invariants for a charged anisotropic harmonic oscillator in a uniform magnetic field, a magnetic trap, and the action and angle variables for the simple systems.

Friction and Dynamics of Verge and Foliot: How the Invention of the Pendulum Made Clocks Much More Accurate

The tower clocks designed and built in Europe starting from the end of the 13th century employed the “verge and foliot escapement” mechanism. This mechanism provided a relatively low accuracy of time measurement. The introduction of the pendulum into the clock mechanism by Christiaan Huygens in 1658–1673 improved the accuracy by about 30 times. The improvement is attributed to the isochronicity of small linear vibrations of a mathematical pendulum. We develop a mathematical model of both mechanisms. Using scaling arguments, we show that the introduction of the pendulum resulted in accuracy improvement by approximately π/μ ≈ 30 times, where μ ≈ 0.1 is the coefficient of friction. Several historic clocks are discussed, as well as the implications of both mechanisms to the history of science and technology.

Improvement of Discrete-Mechanics-Type Time Integration Schemes by Utilizing Balance Relations in Integral form Together with Picard-Type Iterations

The search for efficient explicit time integration schemes is a relevant topic in the current literature on dynamic mechanical systems. In this paper, we describe a strategy of utilizing the balance relations of mechanics in their integral form, so-called general laws of balance, where the time-evolution of the integrands is approximated by established computational techniques of the discrete-mechanics-type. In a Picard-type iteration, the outcomes are used for repeating the procedure several times, leading to an increased accuracy. The advantages of the present explicit approach are discussed in the context of linear and nonlinear motions of the mathematical pendulum. We utilize the modern symbolic procedures to obtain the time integration formulae and compare the results of our methods with exact solutions and with the results of higher-order implicit methods and also with a recent explicit formulation from the literature.

Implementation Model Linear Regression, Neural Network and Support Vector Machine in Simple Mathematical Pendulum Experiments to Determine the Value of Gravity

machine in mathematical pendulum experiments to find the value of gravity. There were 4 data obtained from mathematical pendulum experiments which were then interpolated to obtain more data (13 data), then the data was used as training data for each model. Each model is tested to get a gravity value of 26 including training data, then compared with reference gravity values [17,18,19]. The results of the model Neural network proved to be the most accurate with an error value of 2.53%. The support vector machine model is the most accurate model with a standard deviation value of 0.03 and the error deviation of 0.058 is the smallest value of the three models in this paper.

Variational Iteration Method for the Solution of Differential Equation of Motion of the Mathematical Pendulum and Duffing-Harmonic Oscillator

In this work, the differential equation of motion of the undamped mathematical pendulum and Duffing-harmonic oscillator are discussed by using the variational iteration method. Additionally, common problems of pendulum are classified and Lagrange multipliers are obtained for each type of problem. Examples are given for illustration.