Oscillations Of Pendulum Research Articles

Implementation of Small Term Reduction in Taylor Series in Analytical Physics Problem

This research is motivated by the low ability of students to apply mathematical solving methods to analytic physics. The purpose of this research is to show the implementation of small term reduction in Taylor series on analytic physics problems. The analytical physics material described in this study is the pendulum oscillation material, heat conduction and the perturbation quantum state function. This research method involves a mathematical derivation of the Taylor series formation. The epicenter of the idea of reducing the Taylor series in this study is the use of very small terms in the Taylor series to reduce terms with values reaching towards zero, namely the third term and the fourth term. This small term reduction method is spread into problems of mechanics, oscillations, heat, electricity, magnetism and quantum mechanics. The results of this study show that after using the small term reduction principle, the equations to be solved are like first-order and second-order differential equations. By solving these equations we get a variable physical quantity.

Internal flow characteristics of radial rotating oscillating heat pipe filled with acetone or water

The study investigates the internal flow dynamics of radial rotating oscillating heat pipes (RaRo-OHPs) to assess their potential applications in rotations. It examines influences of heat flux and centrifugal acceleration on thermal conditions and internal flow in RaRo-OHPs. Results show that both Bo and Fr rise significantly with increased centrifugal acceleration and heat flux, emphasizing radial rotation's impact on internal flow. With increment of heat flux and centrifugal acceleration, flow pattern analysis reveals shifts from slug/annular flow to churn flow/separation of vapor-liquid phases; motion modes progress from pendulum oscillation to circulation, then oscillation or single/double-side boiling. Visualizations underscore the importance of maintaining vapor slugs and liquid plugs for effective circulation flow and heat transfer. It is noted that under radial rotation, slug flow and the circulation of chains of vapor slugs and liquid plugs were still observed, achieving Bo of 8.26 for acetone and 5.10 for water as the working fluid. Compared with the static state, radial rotation can promote the internal flow, the velocity increases by 228% and 185% for RaRo-OHPs filled with acetone and water, respectively. The study emphasizes the capillary effect's role in enhancing internal flow under radial rotations, crucial for improving thermal performance.

Multiple scales method for analyzing a forced damped rotational pendulum oscillator with gallows

This study reports the analytical solution for a generalized rotational pendulum system with gallows and periodic excited forces. The multiple scales method (MSM) is applied to solve the proposed problem. Several types of rotational pendulum oscillators are studied and talked about in detail. These include the forced damped rotating pendulum oscillator with gallows, the damped standard simple pendulum oscillator, and the damped rotating pendulum oscillator without gallows. The MSM first-order approximations for all the cases mentioned are derived in detail. The obtained results are illustrated with concrete numerical examples. The first-order MSM approximations are compared to the fourth-order Runge–Kutta (RK4) numerical approximations. Additionally, the maximum error is estimated for the first-order approximations obtained through the MSM, compared to the numerical approximations obtained by the RK4 method. Furthermore, we conducted a comparative analysis of the outcomes obtained by the used method (MSM) and He-MSM to ascertain their respective levels of precision. The proposed method can be applied to analyze many strong nonlinear oscillatory equations.

One-degree-of-freedom galloping instability of a 3D bluff body pendulum at high Reynolds number

The cross-flow swinging dynamics of a cube pendulum is studied experimentally in a flow at high Reynolds numbers (Re∼105) with a low free-stream turbulence intensity. A galloping instability is observed and results in the exponential growth of the swinging motion. The onset of galloping is found to be very sensitive to the static yaw angle of the cube. Despite the 3D geometry of the cube, flow mechanisms similar to the case of a square cylinder appear to govern the onset of the instability. A quasi-steady linear model of the motion is assessed to predict the stability of the pendulum.For the lowest reduced velocity investigated (U∗=18.5), unsteady phenomena arise during the saturation phase of the pendulum oscillations. From the analysis of the unsteady loads and the pressure distribution on the faces of the cube, an unsteady phase delay between the wake and the pendulum dynamics is identified. It produces an energy loss in the pendulum motion which favors its saturation.

Study on the variable length simple pendulum oscillation based on the relative mode transfer method.

In this study, we employed the principle of Relative Mode Transfer Method (RMTM) to establish a model for a single pendulum subjected to sudden changes in its length. An experimental platform for image processing was constructed to accurately track the position of a moving ball, enabling experimental verification of the pendulum's motion under specific operating conditions. The experimental data demonstrated excellent agreement with simulated numerical results, validating the effectiveness of the proposed methodology. Furthermore, we performed simulations of a double obstacle pendulum system, investigating the effects of different parameters, including obstacle pin positions, quantities, and initial release angles, on the pendulum's motion through numerical simulations. This research provides novel insights into addressing the challenges associated with abrupt and continuous changes in pendulum length.

Study of Tracker Software-assisted Spring Coupled Pendulum Oscillation

Oscillatory motions are an essential part of our physical world. These motions need to be studied and learned by students. However, complicated oscillatory motions, e.g.: spring coupled pendulum, are rarely taught to students because their analytical solutions (if they exist) are complicated Here, we use the Tracker software to bridge between the actual coupled pendulum oscillation and its analytical solution. This study aimed to determine the period and frequency of Tracker software-assisted spring coupled pendulum oscillations. The displacement of the pendulum is divided into in-phase and out-of-phase motions. Data collection was carried out by recording the oscillatory motions up to 10 oscillations. The data were obtained from the pendulum position with respect to time. The data analysis techniques in this study used two methods, namely: i) Tracker software and ii) analytical calculations. The results showed that the period and frequency obtained via the Tracker software were in accordance with the analytical calculations. The periods of the in-phase motion and analytical calculation were 1.42 s and 1.54 s, respectively. The periods of the out-of-phase motions and analytical calculation were 0.7 s and 0.55 s, respectively. The frequencies of the in-phase motion and analytical calculation were 0.70 Hz and 0.65 Hz, respectively. The frequencies of the out-of-phase motions and analytical calculation were 1.50 Hz and 1.83 Hz, respectively. The uncertainty values of the results were 7% to 22%. Hence, the spring coupled pendulum oscillations became tractable and can be compared to the analytical solution.

Використання моделі балістичного маятника в аналізі перехідних процесів кранових систем

A number of factors affecting the accuracy of experimental determination of small velocities of pendulum vibrations of the load on the rope of the crane hoisting mechanism, arising at start-up, braking or reversing of the crane system, have been analyzed. The rope is considered as an elastic thread. The influence of finite mass and bending elasticity of the viscus on the frequency of natural oscillations of the pendulum is theoretically analyzed. It is shown that the main factor influencing the natural frequency is the bending elasticity of the viscus thread, and the influence of the mass of the thread (rope) of the ballistic pendulum can be minimized. An exact analytical expression for the natural frequency of a real pendulum at arbitrary stiffness of the lightweight viscus is obtained. For small mass of cargo and speed of movement at impulse influence on the crane system (starting, braking, reversing) there are certain difficulties in taking into account the final mass and stiffness of the viscus for calculation of natural frequency of pendulum oscillations. In an ideal situation, the mass of the temple can be made too small compared to the mass of the pendulum (load and rope), but with a short length of rope increases the natural frequency of the pendulum, which reduces the sensitivity of the system to external impulsive influences, and increases the influence of the elasticity of the thread (rope) on this frequency. Therefore, in a real situation, the mass of the pendulum, its length (of the rope) and the stiffness of the temple should always be chosen and taken into account in accurate studies (and calculations) based on compromise considerations. In this study, the degree of influence of the above factors on the natural frequency of oscillation of a pendulum (i.e., essentially a weight on a rope) is theoretically evaluated in the small-parameter approximation. The results obtained in this study can be further used to refine and improve the existing engineering methods of calculation and analysis of transients (start-up, braking, reversing, etc.) of crane systems both at the stages of their design and in the modes of real operation to optimize the performance of the above systems (and mechanisms).

ANALYSIS OF DYNAMICS AND OPTIMISATION OF TRANSIENT MOTION MODES OF HOISTING MECHANISMS OF OVERHEAD URBAN CRANES

The research of operating modes of hoisting mechanisms and machines, in particular, overhead and city-building cranes is carried out. Based on dynamic analysis of operating modes of mechanisms of the mentioned machines the character of load occurrence in their elastic elements (ropes) is established. The synthesis of optimum transient modes of operation (start/braking) of city-building and overhead cranes is carried out. The models of L.D. Landau and N.A. Lobov for physical and mechanical analysis of motion of the specified machines are used, on the basis of which with attraction of methods of classical calculus of variations the laws of motion of a cargo and a cargo carriage of similar cranes which optimize (minimize) inevitably arising pendulum oscillations of a cargo on a rope (as in the period of start-up of the crane, and at its braking, up to full stop) are received analytically. The evaluation of motion modes in the work is carried out by using a criterion with sub-integral functions in the form of "energy" of accelerations of higher orders (third, fourth, etc.). Exactly such criteria are used in the selection of modes of motion of lifting mechanisms of cranes of bridge/bridge-building type, which have in their composition elastic elements (ropes). The above approach makes it possible to achieve continuity of functions of velocities and accelerations of various links of the system, which leads to a significant reduction in their oscillations. The analysis of modes of motion of the system, which is described by the main coordinate of motion and elastic coordinates, allowed one to establish the regularity of using the proposed criterion to minimize the elastic vibrations of individual links (ropes) of the crane hoisting mechanism. The results of this study can be further used to clarify and improve existing engineering methods for calculating loads in flexible/elastic elements (ropes) of bridge and city-building cranes both in the design of hoisting mechanisms and in the modes of their real operation when mechatronic control systems are used in controlling the movement of the load carriage.

SIMULATION OF OSCILLATIONS OF WATER MOLECULES USING THE MODEL OF A TWO-FREQUENCY PENDULUM IN THE JAHN-TELLER POTENTIAL

The interaction potential for water molecules is considered, which corresponds to the presence of bends in hydrogen bonds in it. To explain the nature of this phenomenon, the theory of the Jahn-Teller effect (JTE) is applied. A model of the Jahn–Teller potential (JTP) was constructed, which has a minimum at a certain angle Θ. To simulate oscillations of water molecules in JTP using a pendulum, a correction of the angular potential U1 in the potential of directed forces (PDF) of intermolecular interaction is proposed using a wavelet type additive ΔU1 = c cos(mΘ)/exp(sΘ2). The parameters for the wavelet are selected based on the magnitude of the bending of hydrogen bonds in water (m=s=8, c=0…0.1). Modeling of rotational oscillations of molecules in JTP was carried out using the model of a two-frequency pendulum, which takes into account the ratio of the moments of inertia of the molecule (pendulum) along the axes k=3 and the PDF index n=8 in JTP Un =U1n. New types of pendulum oscillations in the JTP are determined in comparison with the original PDF (c=0) and their features. It is established that several types of oscillations are observed for this potential. These are new: sector, rotation of the pendulum in the potential trough – disordered or ordered, as well as types for PDF: two-frequency independent oscillations (IO) along the axes and ellipse-like oscillations (ELO) at one frequency. Oscillations in the JTP chute are observed for the main range of initial velocities inside an elliptical ring compressed along the Y axis, and only for the ELO, in an elliptical region elongated along the Y axis, as in the PDF. The methodology for calculating and analyzing the oscillation parameters of a two-frequency pendulum has been improved. The boundaries of oscillation types are determined for given parameters of the potential and a number of initial data. The difference between the patterns of oscillations is established for cases when the initial displacement of the pendulum is greater or less than the position of the minimum of the JTP. It is shown that the velocities at the boundary of the transition from the rotation of the pendulum in the potential trough to the IO correlate with the magnitude of the JTP maximum on the axis of the pendulum. The presence of stable transverse vibrations in the JTP for the case of protons of water molecules can apparently be considered as a new degree of freedom for vibrations of water molecules, which can lead to an explanation of the large contribution of rotational vibration modes to its heat capacity.

Coupled oscillations of the Wilberforce pendulum unveiled by smartphones

The Wilberforce pendulum illustrates important properties of coupled oscillators including normal modes and beat phenomena. When helical spring is attached to a mass to create the Wilberforce pendulum, the longitudinal and torsional oscillations are coupled. A Wilberforce can be constructed simply from a standard laboratory spring, and a smartphone's accelerometer and gyroscope can be used to monitor the oscillations. We show that the resulting time-series data match theoretical predictions, and we share the procedures for observing both normal modes and beats.

Optimization of the Movement of a Cylindrical Vibration-Driven Robot in a Viscous Fluid, Induced by Pendulum Oscillations of the Internal Mass

Optimization of the Movement of a Cylindrical Vibration-Driven Robot in a Viscous Fluid, Induced by Pendulum Oscillations of the Internal Mass

IMPROVEMENT OF THE GENERALIZED FORCE CRITERION OPTIMIZATION OF OVERHEAD CRANES MOVEMENT MODES

During the operation of overhead cranes, pendulum oscillations of the payload are often observed, which causes uneven movement of these cranes, their trolleys, loads on the ropes and power elements of the cranes, which, in turn, create various inconveniences during their operation, reduce the reliability of the functioning of both the crane as a whole and its individual elements. It is clear that all of these factors must be taken into account in the refined calculations of cranes (especially in the modes of their optimal (with the minimum driving force required for this) start/braking). The paper uses a standard methodology and scheme for calculating pendulum oscillations of a payload on the cables of an overhead crane, which are usually carried out within a two-mass model of a crane system. Further refinements and improvements to the above methodology have been made on the basis of a well-founded generalized force criterion for the quality of crane movement. The dependencies describing the law of motion of the crane system and the law of change in time of the applied driving force during the startup/braking stages were obtained, which satisfy the above-mentioned power criterion and ensure high-quality (smooth) movement of the system during its startup or braking. The law of motion of the crane rotation mechanism (crane drive) at its stopping, as well as the law of motion of the cargo at its lifting by the corresponding crane mechanism and sharp braking, at which the dynamic loads in the drive and in the crane rope, respectively, are minimized, is established. The results obtained in the work can be further used to clarify and improve the existing engineering methods for calculating the start-up modes of overhead cranes both at the stages of their design and in the modes of real operation.

On the Orbital Stability of Pendulum Periodic Motions of a Heavy Rigid Body with a Fixed Point, the Main Moments of Inertia of which are in the Ratio 1 : 4 : 1

The motion of a heavy rigid body with a fixed point in a uniform gravitational field is considered. It is assumed that the main moments of inertia of the body for the fixed point satisfy the condition of D.N. Goryachev–S.A. Chaplygin, i.e., they are in the ratio 1 : 4 : 1. In contrast to the integrable case of D.N. Goryachev–S.A. Chaplygin, no additional restrictions are imposed on the position of the center of mass of the body. The problem of orbital stability of pendulum periodic motions of the body is investigated. In the neighborhood of periodic motions, local variables are introduced and equations of perturbed motion are obtained. On the basis of a linear analysis of stability, the orbital instability of pendulum rotations for all values of the parameters has been concluded. It has been established that, depending on the values of the parameters, pendulum oscillations can be both orbitally unstable and orbitally stable in a linear approximation. For pendulum oscillations that are stable in the linear approximation, based on the methods of KAM theory, a nonlinear analysis is performed and rigorous conclusions about the orbital stability are obtained.

The Orbital Stability Analysis of Pendulum Oscillations of a Heavy Rigid Body with a Fixed Point Under the Goriachev–Chaplygin Condition

The Orbital Stability Analysis of Pendulum Oscillations of a Heavy Rigid Body with a Fixed Point Under the Goriachev–Chaplygin Condition

Motion of a parametrically driven damped coplanar double pendulum.

We present the results of linear stability of a damped coplanar double pendulum and its nonlinear motion, when the point of suspension is vibrated sinusoidally in the vertical direction with amplitude a and frequency ω. A double pendulum has two pairs of Floquet multipliers, which have been calculated for various driving parameters. We have considered the stability of a double pendulum when it is in any of its possible stationary states: (i) both pendulums are either vertically downward or upward and (ii) one pendulum is downward and other is upward. The damping is considered to be velocity-dependent, and the driving frequency is taken in a wide range. A double pendulum excited from its stable state shows both periodic and chaotic motion. The periodic motion about its pivot may be either oscillatory or rotational. The periodic swings of a driven double pendulum may be either harmonic or subharmonic for lower values of a. The limit cycles corresponding to the normal mode oscillations of a double pendulum of two equal masses are squeezed into a line in its configuration space. For unequal masses, the pendulum shows multi-period swings for smaller values of a and damping, while chaotic swings or rotational motion at relatively higher values of a. The parametric driving may lead to stabilization of a partially or fully inverted double pendulum.

The Krylov–Bogoliubov–Mitropolsky method for modeling a forced damped quadratic pendulum oscillator

In the present investigation, a quadratically forced damped pendulum-type equation is solved analytically using several effective and highly accurate approaches. Some different types of pendulum oscillators linked to the forced and damped terms, in addition to the power of the damping term, are analyzed. In the first part, the Krylov–Bogoliubov–Mitropolsky (KBM) technique and the ansatz method are carried out for analyzing the quadratically damped pendulum oscillator. In the second part, the two variants of the KBM technique are implemented for investigating the quadratically forced damped pendulum oscillator. Some symmetric approximations are derived and compared with the fourth-order Runge–Kutta numerical approximation. In addition, the maximum distance error is estimated in the whole study domain for ensuring that all obtained approximations are accurate. The obtained results are illustrated through concrete examples.

Application of Laplace-based variational iteration method to analyze generalized nonlinear oscillations in physical systems

In this paper, the variational iterative method (VIM) with the Laplace transform is utilized to solve the nonlinear problems of a simple pendulum and mass spring oscillator, which corresponds to the Duffing equation. Finding the Lagrange multiplier (LM) is a significant phase in the VIM, and variational theory is frequently employed for this purpose. This paper demonstrates how the Laplace transform can be utilized to locate the LM in a more efficient manner. The frequency obtained by Laplace-based VIM is the same as that defined in the already existing methods in the literature in order to ensure the clarity of the results. Numerous analytical techniques can be used to solve the Duffing equation, but we are the first to do it using a Laplace-based VIM and a distinctive LM. The fundamental results of my paper are that LM is also the same in the Elzaki transformation. In the vast majority of instances, Laplace-based VIM only requires one iteration to arrive at an answer with high precision and linearization, discretization or intensive computational work is required for this purpose. Comparing analytical results of VIM by Laplace transform to the built-in Simulink command in MATLAB which gives us the surety about the method’s applicability for solving nonlinear problems. Future work on the basic pendulum may examine the effects of nonlinearities and damping on its motion and the application of advanced control mechanisms to regulate its behavior. Future research on mass spring oscillators could examine the system’s response to random or harmonic input. The mass spring oscillator could also be used in vibration isolation to minimize vibrations from one building to another.

Solutions of the Nonlinear Evolution Problems and their Applications

Abstract In this article, a well-known technique, the variational iterative method with the Laplace transform, is used to solve nonlinear evolution problems of a simple pendulum and mass spring oscillator, which represents the duffing equation. In the variational iteration method (VIM), finding the Lagrange multiplier is an important step, and the variational theory is often used for this purpose. This paper shows how the Laplace transform can be used to find the multiplier in a simpler way. This method gives an easy approach for scientists and engineers who deal with a wide range of nonlinear problems. Duffing equation is solved by different analytic methods, but we tackle this for the first time to solve the duffing equation and the nonlinear oscillator by using the Laplace-based VIM. In the majority of cases, Laplace variational iteration method (LVIM) just needs one iteration to attain high accuracy of the answer for linearization anddiscretization, or intensive computational work is needed. The convergence criteria of this method are efficient as compared with the VIM. Comparing the analytical VIM by Laplace transform with MATLAB’s built-in command Simulink that confirms the method’s suitability for solving nonlinear evolution problems will be helpful. In future, we will be able to find the solution of highly nonlinear oscillators.

К вопросу об устойчивости различных стационарных точек в задаче о малых колебаниях аэродинамического маятника в потоке среды

Построена математическая модель движения аэродинамического маятника в потоке среды. На основании теории квазистационарного обтекания пластинки средой получена система дифференциальных уравнений движения тела, в которую входят экспериментальные аэродинамические функции. Проведен переход к новым безразмерным переменным. Показано нарушение единственности решений уравнений движения. Проведен геометрический анализ областей неоднозначности в подпространстве фазовых переменных. Решены уравнения равновесия и найдены различные стационарные точки. Методом Раусса – Гурвица построены области устойчивости на плоскости различных безразмерных параметров. The analysis of the teleological concept of goal has shown that this concept has a dual nature: idealistic and materialistic. The article considers the principle of dualism as the key one. Therefore, from the point of view of cybernetics, the idealistic aspect of the goal is proposed to be interpreted as an idealized result, which is described by one or more mathematical logics of ML. The ML device serves as a tool for selecting the targeted behavior of a complex object or system S in accordance with the expert evaluation of EE. The materialistic aspect of the goal is understood as obtaining a real result RR through an object or system S. If by S we mean a team of robots configured to fulfill one or more goals, then a qualitative analysis of the behavior of such a system can be carried out using agent-based modeling. The robot collective model is a set of interconnected classes of intelligent agents. Any class of agents is characterized by an 11-digit set, which describes the ability of a class of agents to achieve their goals by adapting through training without finding the most effective actions. Agent-based modeling allows you to judge the degree of achievement of the goal.

ANALYTICAL APPROACH IN THE ANALYSIS AND OPTIMIZATION OF CRANES DYNAMIC LOADS DURING THEIR START-UP: SWINGING THE LOAD

During the operation of the cranes pendulum swings of the load are observed. They cause cranes or cargo carts uneven movement, additional loads on power ele-ments of cranes, create inconvenience during their operation. This must be taken into account during detailed cranes calculations. 
 In bridge, gantry and some other standard types of cranes which move along a rail path, load pendulum oscillations frequency relative to the crane is significantly lower than the elastic oscillations frequency of the crane metal structure and the movement mechanism transmission. Therefore, the load pendulum oscillations can be considered practically independent from the crane elastic oscillations and when cal-culating them, the metal structure and movement mechanism transmission can be taken as absolutely rigid. In this study to determine the dynamic loads acting on the metal structure and the movement mechanism transmission, the law of ropes tension hori-zontal component change, which arises as a result of the load pendulum oscillations, is given in the form of a known time function, which is determined according to the scheme of an absolutely rigid crane. This approach allows to reduce the order of mo-tion equations of the crane dynamic system by two units. In accordance with the above, the calculation of pendulum oscillations of the load on a ropes can be carried out ac-cording to the simplest two-mass scheme. This is exactly the approach implemented in the study. Motion conditions and equations of the “cargo cart − the load on a rope” system analytically determined, at which dynamic loads in the rope/elastic element have the smallest (minimum) values in the starting mode of the bridge-type crane.