Pendulum Research Articles

Nonlinear analysis of compound pendulum model

Abstract This paper investigates the relationships among various nonlinear physi cal equations, with a particular focus on the standard form of the ϕ^4 equation. Based on the theoretical framework of the Jacobi elliptic function, an exact solution for the ϕ^4 equation is derived. A key innovation of this work is the discovery of the consistency between the ϕ^4 equation and the motion equation of the compound pendulum. By utilizing this correspondence, an exact solution for the compound pendulum equation is obtained, grounded in the Jacobi elliptic function theory. Compared to numerical methods, this solution provides higher accuracy and has the potential to be applied to
more complex nonlinear physical systems. This model can also be applied in areas such as vibration damping in building materials, mechanical system analysis, and spacecraft control.

Dynamics of N-elastically longitudinal coupled rotating pendulums with smooth and discontinuous nonlinearities: Generation of chaotic bursting with many orbits as a solution

The dynamics of a mechanical network, consisting of a certain number (N) of cells, in which each unit cell consists of the irrational coupling of a simple pendulum and inclined mass-spring oscillator, is investigated. The single cell was recently introduced by Ning Han and Qingjie Cao, and it has strong irrational nonlinearity, exhibiting smooth and discontinuous characteristics due to the geometry configuration. The obtained network has a large degree of freedom and consequently has richer dynamics as compared to a single one. The set of discrete equations of this network is found, while the first equation is driven by an external sinusoidal force; the variables here are the angles between the vertical direction and the directions of mass in each unit cell. By using the non-driven version of this set of equations, the equilibrium points are found as well as their stability using the Jacobian matrix. Then, by using the multiple time scale method, the solutions of the set of the network equations are found for weak amplitude oscillations of the system, while the large amplitude solutions are found numerically, showing the ability of the network to transform the shape of some solutions at the input to other forms as the signals evolve in the network. One notices the generation of simple chaotic signals as well as the train of impulse signals. What is important here is the generation of the chaotic bursting with many orbits by the system, which are the simultaneous oscillations around several fixed points. For the case of numerous cells, the system can be used as a waveguide; this is why the set of the equation of the network is reduced to the extended complex Ginzburg Landau (ECGL) equation, with complex nonlinear dispersion and nonlocality terms, using the Tanuiti's reduction perturbation methods. The dissipative dark compacton and peakon as solutions for the ECGL equation are then found. The input-output matching of the network is next investigated via the transfer function of the system near the equilibrium points. The inverse Fourier transform of the transfer function multiplied by the Fourier transform of the input sinusoidal force gives the solution of the network equation for low amplitude oscillations. Finally, the supratransmission condition is studied, leading to the fact that the input sinusoidal signal with frequency nearly inside the forbidden gap zone gradually transforms into the train of chaotic bursting.

Classification and analysis of simple pendulum using artificial neural network approach

We describe an artificial neural network (ANN) for analyzing damped oscillations in a simple pendulum system by using a machine learning (ML) algorithm. We have first shown how to construct a simple ANN consisting of three layers-input, hidden and output, with each layer being composed of neurons representing a relevant feature of the oscillating pendulum. The train and test datasets for the ANN have been taken from the experimental data collected by using the methodology of a previously communicated work. A ML optimization algorithm called stochastic gradient descent has been employed in the neural network to predict the type of pendulum according to the values of the mass, size and damping coefficient of the pendulum.

Semiclassical perturbations of single-degree–of–freedom Hamiltonian systems I: Separatrix splitting

We study semiclassical perturbations of single-degree-of-freedom Hamiltonian systems possessing hyperbolic saddles with homoclinic orbits, and provide a sufficient condition for the separatrices to split, using a Melnikov-type approach. The semiclassical systems give approximations of the expectation values of the positions and momenta to the semiclassical Schrödinger equations with Gaussian wave packets as the initial conditions. The occurrence of separatrix splitting explains a mechanism for the existence of trajectories to cross the separatrices on the classical phase plane in the expectation value dynamics. Such separatrix splitting does not occur in standard systems of Hagedorn and Heller for the semiclassical Gaussian wave packet dynamics as well as in the classical systems. We illustrate our theory for the potential of a simple pendulum and give numerical computations for the stable and unstable manifolds in the semiclassical system as well as solutions crossing the separatrices.

Eksik Tahrikli Döner Ters Sarkaç Sisteminin Yukarı Yükseltilmesi için Enerji Tabanlı Doğrusal Olmayan Kontrol Algoritması ve Deneysel Doğrulaması

Bu çalışmada, döner ters sarkaç sistemi hareketli koluyla sınırlı bir bölge içinde, salınımı kontrol edilerek ve sarkaç enerjisi artırılarak, kararlı denge noktasından kararsız denge noktasına çıkarılması hedeflenmektedir. Döner ters sarkaç sisteminin doğrusal olmayan modeli ve denge noktalarındaki doğrusal modelleri verildikten sonra sarkaç yukarı yükseltme algoritması tanıtılmaktadır. Yukarı yükseltme algoritması, sarkaç salınırken belli noktalarda kol yardımıyla sarkaca enerji yüklemesine dayanmaktadır. Burada kol ivmelendirilmesi sabit bir ivme ile hareket ettirilmektedir. Yukarı yükseltme işlemindeki algoritma, sarkacın hızının en fazla olduğu yere yani kararlı denge noktasına ilerlerken ve periyodunu tamamlarken devreye girmektedir. Bu analizler hesaplanmış sınırlar dikkate alınarak yapılmaktadır. Sarkacın yukarı yükselmesi gerçekleştiğinde doğrusal model üzerinden elde edilmiş temel bir tam durum geri beslemeli kontrolcü devreye girmektedir ve sarkacı kararsız denge noktasında tutmaktadır. Böylece, anahtarlamalı bir kontrol yapısı elde edilmektedir. Önerilen kontrol algoritmasının doğrulaması için önerilen algoritma bir döner sarkacı üzerinde gerçek zamanlı olarak test edilmiştir. Farklı kol ivmelendirme değerleri ile elde edilen karşılaştırmalı sonuçlar sunulmuştur. Kol ivmesi ±32 rad/s2’den ±64 rad/s2’ye çıkarıldığında sarkacın yukarı yükselme süresi yaklaşık 9 s’den 5 s’ye düşmektedir. Bu sonuçlar, önerilen algoritmanın deneysel başarısını göstermektedir.

Hierarchical Three-Body Problem at High Eccentricities = Simple Pendulum II: Octupole including Brown’s Hamiltonian

Abstract The very long-term evolution of the hierarchical restricted three-body problem with a massive perturber is analyzed analytically in the high eccentricity regime. Perturbations on the time scale of the outer orbit can accumulate over long timescales and be comparable to the effect of the octupole term. These perturbations are described by Brown’s Hamiltonian - having different forms in the literature. We show that at the high eccentricity regime - the effect of Brown’s Hamiltonian is an azimuthal precesssion of the eccentricity vector and can be solved analytically. In fact, the dynamics are equivalent to a simple pendulum model allowing an explicit flip criterion.

Hierarchical three-body problem at high eccentricities = simple pendulum I: octupole

ABSTRACT The gradual evolution of the restricted hierarchical three body problem is analysed analytically, focusing on conditions of Kozai–Lidov cycles that may lead to orbital flips from prograde to retrograde motion due to the octupole (third order) term which are associated with extremely high eccentricities. We revisit the approach described by Katz, Dong and Malhotra (2011) and show that for most initial conditions, to an excellent approximation, the analytical derivation can be greatly simplified and reduces to a simple pendulum model allowing an explicit flip criterion. The resulting flip criterion is much simpler than the previous one but the latter is still needed in a small fraction of phase space. We identify a logical error in the earlier derivation but clarify why it does not affect the final results.

Rapid sloshing-free transport of liquids in arbitrarily shaped containers

AbstractWe present a model-based feedforward control strategy suitable for designing swift rest-to-rest maneuvers for liquids in arbitrarily shaped containers. We employ the commonly used equivalent pendulum model to represent the sloshing dynamics and suggest a novel parameter identification scheme suitable for arbitrary container shapes and any number of sloshing modes. By computing natural modes and fluid reaction forces and torques for imposed harmonic container motions via a finite element model, we obtain data for the identification scheme. A fitting procedure then yields highly accurate parameters for a physical pendulum model, where each pendulum represents one sloshing mode. We also provide a thorough analysis of parameter identifiability and guidelines for obtaining robust parameter estimates. The proposed feedforward control method uses a virtual tray pendulum on which we place the container (in the form of its equivalent pendulum model). Designing the virtual tray such that the fluid’s dominant sloshing mode cannot be excited by horizontally moving the tray pendulum pivot effectively zeros out any sloshing motion in this mode. We then exploit the flatness property of the resulting system to design rest-to-rest maneuvers where any residual sloshing motion (in higher modes) can be exactly stopped at the end of the maneuver. The effectiveness of the proposed method is demonstrated through extensive simulations and experimental results using a Martini cocktail glass, whose shape is challenging in terms of sloshing. The experimental results show the successful, accurate suppression of sloshing, validating the efficacy of the proposed concept.

Cronpes: time measurement device suitable for teaching experimental physics in Brazil

Abstract Several public schools and also universities in Brazil have few or none financial resources to acquire certain equipment for teaching physics. Nevertheless, it is extremely important that institutions, schools and universities offer laboratory conditions for the students, mainly in engineering courses, in order to get the first contact with physics concepts in practice. For this purposes, it was proposed to develop a simple construction, low-cost and easy maintenance photogate (Cronpes) device, so that teachers, technicians and even students from the basic and high education network can build them for their schools, institutions and universities, thus improving the quality of teaching physics. This study aims to implement a photogate Esp32 microcontroller based programmed in MicroPython with good resolution, accuracy and user friendly in order to apply in experiments like simple pendulum, spring oscillations and conservation of mechanical energy. The main focus of this research was to demonstrate, through the development of Cronpes, that we can create solutions for educational laboratories, in a viable, creative and inexpensive way.

Tracking the Motion of a Simple Pendulum with Tracker

Tracking the Motion of a Simple Pendulum with Tracker

How does the period of a pendulum vary with the number of strings?

This article serves as an extension of analysis of a simple pendulum, a typical example of simple harmonic motion and a common topic in high school physics syllabus. Since the analysis of a simple pendulum ignores the mass of the string in order to model the system as a point mass, a more complex pendulum system consisting of heavy strings and a pendulum bob is explored in this article. Based on theoretical analysis of the model, a relationship between the number of strings and the period of the pendulum is investigated. An experiment was also conducted on a pendulum system using copper wire, a glass bob, a stand, clamps, a clip and a stopwatch. To obtain more accurate results, time taken for five periods was measured three times for each number of strings. A graph is drawn based on data acquired from the experiment and the theoretically deduced relationship is therefore verified. Further study of the general expression for the period of an object that cannot be modelled as a point mass oscillating about a fixed axis is involved. Application of the principle that a change in the position of the center of mass of a pendulum causes a change in its period in Big Ben is also discussed along with a brief introduction of its pendulum structure.

ANALISIS KORELASI ANTARA SUDUT SIMPANGAN DAN WAKTU RATAAN PADA BANDUL FISIS DENGAN LATIN SQUARE DESIGN

This study aims to examine the relationship between the angle of deviation and the average time length of a physical pendulum. This study uses primary data, namely practicum report data in a physics laboratory. Data collection was conducted under closed conditions with control variables (shaft length, number of oscillations, and radius of gyration) and dependent variables (angle of deviation and average time). The analysis technique used is a statistical approach technique, namely Latin Square Design analysis. The findings of this study indicate that the variables of deviation angle and average time on the physical pendulum have a strong and unidirectional correlation relationship. This result is evident from the value of F0 > Fcritical and the value of Pr (> P) > 0, 05(significant level) where both of these prove that there is a strong reason to reject H0 which means that there is at least 1 difference in the effect of the angle of deviation on the average time. It shows that the angle of deviation affects the average time of the physical pendulum

Design and Development of a Teaching–Learning Sequence about Deterministic Chaos Using Tracker Software

In this paper, we present the design, development, and pilot implementation of a Teaching–Learning Sequence (TLS) about the physics of deterministic chaos. The main aim of the activities is to let students become aware of two key ideas about deterministic chaos: (1) the role of initial conditions and (2) the graphical representation in a momentum–position graph. To do so, the TLS is based on the observation and analysis of the trajectory of the free end of a double pendulum through the modeling software Tracker. In particular, the Tracker-based activities help students understand that, by modifying the well-known simple pendulum dynamic system into a double pendulum, long-time-scale predictability is lost, and a completely new behavior appears. The TLS was pilot tested in a remote teaching setting with about 70 Italian high school students (16–17 years old). The pretest analysis shows that before participating in the activities, students held typical misconceptions about chaotic behavior. Analysis of the written responses collected during and after implementation shows that the proposed activities allowed students to grasp the two key ideas about nondeterministic chaos. A possible integration of the TLS with an online simulation is finally discussed.

Stabilisation of the swing pattern of an anisotropic simple pendulum

The suppression of the effects of anisotropy on a pendulum by use of a rotating mount was initially envisaged by Léon Foucault, based on his observations of the vibrations of a rod clamped in a lathe. However, the method seems to never have been tried due to the practical difficulties involved. We report a computational study of the stabilisation of the swing pattern of a simple pendulum, showing anisotropic behaviour in a static configuration, by rotation of the system mount. When the mount is static, for most initial conditions the swing patterns quickly evolves into unstable, complex Lissajous-like patterns. When the pendulum mount is rotated faster than the pendulum frequency effects of anisotropy are suppressed, and the swing pattern stabilises to that of an isotropic 3D simple pendulum. Suppression of mount anisotropy influence occurs for relatively low rotation rates. We also study swing evolution in the presence of random variations in the orientation of the mount principal axes. The use of computational techniques confirms Foucault’s original observations and hypothesis and provides an interesting avenue for students to engage meaningfully with this historically important and inspiring experiment in a novel and challenging manner.

Exploring the Possibility of Automatic Scoring for Arrow Responses of Simple Pendulum Items Using a Image Contours

Exploring the Possibility of Automatic Scoring for Arrow Responses of Simple Pendulum Items Using a Image Contours

Investigating the drag impact on pendulum through numerical models and the stability analysis

As a traditional physical model, the pendulum plays a crucial role in the dynamics. However, most research focuses on the ideal pendulum structure which deviates from reality. In order to study the pendulum in practical situations, this paper not only investigates the ideal model but also the simple pendulum motion with drag. Firstly, this work derived analytical and numerical solutions of the simple pendulum motion equations, which come from physical phenomena, and compared these two solutions. Moreover, this study investigated the error of this numerical model and accomplished further research on demonstrating the conditions to meet the stability of the model.

Effect of Physics Inquiry Learning with Low-cost Experiment Tool on Students’ Problem-solving Skills and Self-efficacy

The low problem-solving skills and self-efficacy of students in physics learning significantly impact their overall learning processes and outcomes. In order to address this issue, an inquiry learning model was implemented, along with the low-cost experiment tool. This learning model is designed to be valid and effective in enhancing students' problem-solving skills and self-efficacy. The research followed a non-equivalent control group design and involved 66 students to ensure the development of a high-quality learning model that meets the required standards and criteria. By utilising learning instruments, such as students' response questionnaires, problem-solving pre- and post-tests, and self-efficacy questionnaires, the focus of the physics topic in this research was the simple mathematic pendulum. The validity test results show that the learning instrument has high validity and reliability criteria, making it feasible to use in physics learning. Furthermore, the statistical effectiveness test results show that this learning model is effective in improving students' problem-solving skills and self-efficacy. As students engage in hands-on investigations and collaborate on meaningful projects, they are prompted to apply critical thinking and decision-making skills, directly contributing to the development of effective problem-solving strategies. The intrinsic motivation derived from successfully navigating the challenges of inquiry learning, coupled with the personal relevance of the content, nurtures students' confidence in their abilities, ultimately strengthening their self-efficacy as independent and capable learners. Research implies the development of innovative learning models, effectively enhancing students' PSS and SE.

Analysis of simple pendulum with uncertain differential equation

Uncertain differential equation is a type of differential equation involving uncertain processes. This paper analyzes a simple pendulum system with a varying drag coefficient by the tool of uncertain differential equation, and derives the uncertain simple pendulum equation. Afterwards, the numerical method of solving the uncertain simple pendulum equation and its parameter estimation method are given in this paper. Finally, a real-world example is provided to illustrate the uncertain simple pendulum equation.

Magnetic Control of a Simple Pendulum with a Moving Pivot Point

Magnetic Control of a Simple Pendulum with a Moving Pivot Point

Ground-based adaptive horsemanship lessons for veterans with post-traumatic stress disorder: a randomized controlled pilot study.

Equine-assisted services (EAS) has received attention as a potential treatment strategy for post-traumatic stress disorder (PTSD), as existing literature indicates that symptoms may decrease following EAS. Relatively little is known about the mechanisms at play during lessons and if physiological measures are impacted. The objectives of this pilot study were to 1) explore the effects of adaptive horsemanship (AH) lessons on symptoms of PTSD, hormone concentrations, and social motor synchrony; 2) determine if physiological changes occur as veterans interact with horses; and 3) explore if the interaction between veteran and horse changes over the 8-week session. Veterans with PTSD were randomly assigned to control (CON, n = 3) or AH (n = 6) groups for an 8-week period (clinical trial; NCT04850573; clinicaltrials.gov). Veterans completed the PTSD Checklist (PCL-5) and Brief Symptom Inventory (BSI) at pre-, post-, and 2- and 6-month follow-up time points. They also completed a social motor synchrony test (pendulum swinging) and blood draw at pre- and post-time points. In weeks 1, 4, and 8, blood samples were drawn at 0 min, 3 min, 5 min, 25 min, and 30 min during the 30-min AH lessons. Veterans completed the Human-Animal Interaction Scale (HAIS) after each lesson. Blood samples were assayed for plasma cortisol, epinephrine, norepinephrine, and oxytocin. Data were analyzed with repeated measure ANOVAs. Changes in PTSD symptoms from pre- to post-time point were analyzed with paired t-tests. Changes in PCL-5 scores tended to differ (p = 0.0989), and global BSI scores differed (p = 0.0266) between AH (-11.5 ± 5.5, mean ± SE; -0.5 ± 0.2) and CON (5.3 ± 5.4; 0.4 ± 0.2) groups. Social motor synchrony and hormone concentrations did not differ between groups or time points (p > 0.05). Cortisol, norepinephrine, and oxytocin concentrations did not differ across sessions (p > 0.05). Epinephrine concentrations tended (p = 0.0744) to decrease from week 1 to 4 of sessions. HAIS scores increased (p ≥ 0.0437) in week 3 and remained elevated as compared to week 1. Participant recruitment was the greatest challenge. These preliminary results agree with the literature suggesting that EAS can reduce symptoms of PTSD.