Double Pendulum Research Articles

A generalized framework of neural networks for Hamiltonian systems

When solving Hamiltonian systems using numerical integrators, preserving the symplectic structure may be crucial for many problems. At the same time, solving chaotic or stiff problems requires integrators to approximate the trajectories with extreme precision. So, integrating Hamilton's equations to a level of scientific reliability such that the answer can be used for scientific interpretation, may be computationally expensive. However, a neural network can be a viable alternative to numerical integrators, offering high-fidelity solutions orders of magnitudes faster.To understand whether it is also important to preserve the symplecticity when neural networks are used, we analyze three well-known neural network architectures that are including the symplectic structure inside the neural network's topology. Between these neural network architectures many similarities can be found. This allows us to formulate a new, generalized framework for these architectures. In the generalized framework Symplectic Recurrent Neural Networks, SympNets and HénonNets are included as special cases. Additionally, this new framework enables us to find novel neural network topologies by transitioning between the established ones.We compare new Generalized Hamiltonian Neural Networks (GHNNs) against the already established SympNets, HénonNets and physics-unaware multilayer perceptrons. This comparison is performed with data for a pendulum, a double pendulum and a gravitational 3-body problem. In order to achieve a fair comparison, the hyperparameters of the different neural networks are chosen such that the prediction speeds of all four architectures are the same during inference. A special focus lies on the capability of the neural networks to generalize outside the training data. The GHNNs outperform all other neural network architectures for the problems considered.

A hovering bubble with a spontaneous horizontal oscillation

Active matters, characterized by multi-mode motions, have been emerging for both engineering and biological applications. Generally, active objects rely on the symmetry-broken structures, compositions, or interfacial activities through a physical or chemical approach. Here, we report an active bubble spontaneously hovering with a horizontal oscillation at the solid/liquid interface by impacting a stationary laser beam into a liquid through a transparent solid cover. This spontaneous oscillation mode of the bubble synchronizes with that of the interfacial temperature and hydrodynamical flow. A physical mechanism is proposed, and the scaling analysis of the oscillation frequency agrees well with experiments in various liquids under different laser powers. Additionally, the bubble trajectory rotates azimuthally, arising from the symmetry breaking of the vortex pair accompanying the oscillation. Moreover, the double pendulum of oscillation bubbles has been demonstrated, achieving a preferable oscillation direction in a controllable way. These findings would not only advance our understanding of active matters but also shed light on the bubble-mediated technological applications, such as microrobots and drug deliveries.

Chaos: From celestial mechanics to climate

Abstract Chaos is unpredictable behaviour arising in systems governed by deterministic laws. Since its ‘discovery’ in the 1960s by Edward Lorenz in his studies of meteorological systems, chaos has become an important way of understanding seemingly random behaviour in many other areas. These include the double pendulum, the climate, El Niño and electrical oscillators. Understanding chaos allows the limits of the predictability of a system to be determined. However, despite this recent activity this paper will show that the theory of chaos has a rich history and chaotic behaviour was originally identified in the 1890s by Henri Poincaré in his investigations of the three-body problem in celestial mechanics. But remarkably this paper will also show that chaotic behaviour can be found in such a simple problem as multiplication by 10 and it is closely linked to the question (studied by the ancient Greeks) of whether a number is rational or irrational.

Correction to: Vibrational and stability analysis of planar double pendulum dynamics near resonance

Correction to: Vibrational and stability analysis of planar double pendulum dynamics near resonance

Leg compliance is required to explain the ground reaction force patterns and speed ranges in different gaits.

Two simple models, vaulting over stiff legs and rebounding over compliant legs, are employed to describe the mechanics of legged locomotion. It is agreed that compliant legs are necessary for describing running and that legs are compliant while walking. Despite this agreement, stiff legs continue to be employed to model walking. Here, we show that leg compliance is necessary to model walking and, in the process, identify the principles that underpin two important features of legged locomotion: First, at the same speed, step length, and stance duration, multiple gaits that differ in the number of leg contraction cycles are possible. Among them, humans and other animals choose a gait with M-shaped vertical ground reaction forces because it is energetically favored. Second, the transition from walking to running occurs because of the inability to redirect the vertical component of the velocity during the double stance phase. Additionally, we also examine the limits of double spring-loaded pendulum (DSLIP) as a quantitative model for locomotion, and conclude that DSLIP is limited as a model for walking. However, insights gleaned from the analytical treatment of DSLIP are general and will inform the construction of more accurate models of walking.

A modal momentum method for the identification of structural weak links

In order to meet the demands of weak-link identification of mechanical structures in high-precision equipment, this paper introduces the concepts and equations of modal momentum and modal momentum difference ratio (MMDR), as well as an identification method of weak links based on modal momentum. In this method, the modal data of the mechanical structure is acquired first through experimental modal analysis or finite element analysis. The modal momentum of each measured point is calculated based on the modal data and transformed to the global coordinate system. Then, the MMDRs between the measured point with the largest magnitude of modal momentum and its adjacent measured points are calculated. Weak links are judged if the MMDR exceeds a predefined threshold. Twelve different multi-degrees-of-freedom systems incorporating weak links were designed and then identified using the proposed method. The accuracy of identification is up to 100% for these systems. The method was applied to a double pendulum angle milling head of a five-axis numerically controlled machine tool, and the result reveals that the side plate lacks bending stiffness.

Vibrational and stability analysis of planar double pendulum dynamics near resonance

AbstractThe focus of this paper is to examine the motion of a novel double pendulum (DP) system with two degrees of freedom (DOF). This system operates under specific constraints to follow a Lissajous curve, with its pivot point moving along this path in a plane. The nonlinear differential equations governing this system are derived using Lagrange's equations. Their analytical solutions (AS) are subsequently calculated using the multiple-scales method (MSM), which provides higher-order approximations. These solutions are considered new, as the traditional MSM has been applied to this novel system for the first time. Additionally, the accuracy of these solutions is validated through numerical results obtained using the fourth-order Runge–Kutta method. The solvability conditions and characteristic exponents are determined based on resonance cases. The Routh–Hurwitz criteria (RHC) are employed to assess the stability of the fixed points corresponding to the steady-state solutions. They are also used to demonstrate the frequency response curves. The nonlinear stability analysis is performed by examining the stability and instability ranges. Resonance curves and time history plots are presented to analyze the behavior of the system for specific parameter values. The investigation delves into a comprehensive analysis of bifurcation diagrams (BDs) and Lyapunov exponent spectra (LEs), aiming to uncover the various types of motion present within the system. Systematic examination of these charts reveals critical insights into transitions between stable, quasi-stable, and chaotic dynamical behaviors. This work has practical applications in various fields, such as robotics, pump compressors, rotor dynamics, and transportation devices. It can be used to study the vibrational motion of these systems.

Double friction pendulum-based isolation optimization for transformer-bushing systems

Transformers and their bushings are critical equipment pieces in power substations, which are often exposed to significantly high seismic risk. To enhance the seismic performance of the transformer-bushing system, isolation through double friction pendulum (DFP) bearings has been considered. However, the selection of appropriate isolation parameters lacks theoretical guidance and efficient analysis methods. A theoretical model for the transformer-bushing system isolated with DFP bearings has been proposed. Based on this model, isolation parameters that ensure acceptable displacement response while achieving optimal acceleration isolation efficiency can be determined. Taking a high-voltage transformer isolated by DFP bearings as an example, a set of isolation parameters is optimized and used to design a shaking table test. The results demonstrate that the proposed model exhibits reliable calculation accuracy for the seismic response of the transformer-bushing system, regardless of whether it is isolated or not, within the range of peak ground acceleration(PGA) of 0.1–0.6 g. Moreover, at 0.4 g, the experimental results indicate that the isolation efficiency remains consistent with the parameter optimization, approaching 50 %. This research contributes to the understanding of seismic resistance and isolation strategies for transformer-bushing systems, providing valuable insights for the selection and optimization of isolation parameters in practice.

Dynamics of a Double Pendulum with Viscous Friction at the Hinges. II. Dissipative Vibration Modes and Optimization of the Damping Parameters

Dynamics of a Double Pendulum with Viscous Friction at the Hinges. II. Dissipative Vibration Modes and Optimization of the Damping Parameters

Flutter instability characteristics and mechanisms of Ziegler double pendulum with arbitrary masses, stiffness and damping

Flutter instability characteristics and mechanisms of Ziegler double pendulum with arbitrary masses, stiffness and damping

Mechanism of eccentricity influence on 3-DOF aerodynamic stability: New insights into instability evolution, energy harvesting, and vibration control

The inconsistency of mass center and elastic center, i.e., eccentricity, widely exists in various engineering structures and components, including iced conductors, wind turbine blades, airfoils, and cable-supported bridges, and can also be extended to the application of energy harvesting and vibration control. Although the eccentricity influence on aerodynamic stability has already been recognized as important, the underlying mechanism remains rather unclear. This study established a comprehensive framework of explicit eigenvalue solutions, offering in-depth insights into the aerodynamic stability mechanism in a vertical-horizontal-torsional 3-degree-of-freedom linear system under all possible frequency scenarios. These solutions elucidated the coupling mechanism of eccentricity, aerodynamic damping, aerodynamic stiffness, and structural frequencies. Remarkably straightforward criteria to predict the promoting/suppressing effect of eccentricity were proposed, consisting of eccentric angle and aerodynamic forces, which not only align with the traditional principles but also offer a broader application range in terms of structure types and wind speeds. Validations of the proposed solutions were performed for all frequency cases via numerical studies. For instability tendency evaluation, numerical studies on a thin plate showed that eccentricity has a significant impact due to the strong aerodynamic effect and large ratio of width/gyration radius, highlighting the importance of considering small eccentricity errors during experiments. Examination for energy harvesting revealed that a small eccentricity can dramatically enhance power generation efficiency, and the explicit solutions successfully explained the rules governing performance changes against frequency ratio, mass center position, rotation center position, etc. The analysis of a bundled conductor showed that the double pendulum can control galloping because it diminishes the eccentricity effect by shifting the eccentric angle. The proposed explicit solution framework provides a systematic view and new insights into the aerodynamic instability mechanism influenced by eccentricity, serving as a reference for the design and studies of various engineering structures, energy harvesting, and vibration control.

A new hybrid learning control system for robots based on spiking neural networks

This paper presents a new hybrid learning and control method that can tune their parameters based on reinforcement learning. In the new proposed method, nonlinear controllers are considered multi-input multi-output functions and then the functions are replaced with SNNs with reinforcement learning algorithms. Dopamine-modulated spike-timing-dependent plasticity (STDP) is used for reinforcement learning and manipulating the synaptic weights between the input and output of neuronal groups (for parameter adjustment). Details of the method are presented and some case studies are done on nonlinear controllers such as Fractional Order PID (FOPID) and Feedback Linearization. The structure and the dynamic equations for learning are presented, and the proposed algorithm is tested on robots and results are compared with other works. Moreover, to demonstrate the effectiveness of SNNFOPID, we conducted rigorous testing on a variety of systems including a two-wheel mobile robot, a double inverted pendulum, and a four-link manipulator robot. The results revealed impressively low errors of 0.01 m, 0.03 rad, and 0.03 rad for each system, respectively. The method is tested on another controller named Feedback Linearization, which provides acceptable results. Results show that the new method has better performance in terms of Integral Absolute Error (IAE) and is highly useful in hardware implementation due to its low energy consumption, high speed, and accuracy. The duration necessary for achieving full and stable proficiency in the control of various robotic systems using SNNFOPD, and SNNFL on an Asus Core i5 system within Simulink’s Simscape environment is as follows:– Two-link robot manipulator with SNNFOPID: 19.85656 hours– Two-link robot manipulator with SNNFL: 0.45828 hours– Double inverted pendulum with SNNFOPID: 3.455 hours– Mobile robot with SNNFOPID: 3.71948 hours– Four-link robot manipulator with SNNFOPID: 16.6789 hours.This method can be generalized to other controllers and systems like robots.

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.

Optimum design parameters for double concave friction pendulum system considering ground motion characteristics and friction heating

Optimum design parameters for double concave friction pendulum system considering ground motion characteristics and friction heating

Chaotic dynamics of a continuous and discrete generalized Ziegler pendulum

We present analytical and numerical results on integrability and transition to chaotic motion for a generalized Ziegler pendulum, a double pendulum subject to an angular elastic potential and a follower force. Several variants of the original dynamical system, including the presence of gravity and friction, are considered, in order to analyze whether the integrable cases are preserved or not in presence of further external forces, both potential and non-potential. Particular attention is devoted to the presence of dissipative forces, that are analyzed in two different formulations. Furthermore, a study of the discrete version is performed. The analysis of periodic points, that is presented up to period 3, suggests that the discrete map associated to the dynamical system has not dense sets of periodic points, so that the map would not be chaotic in the sense of Devaney for a choice of the parameters that corresponds to a general case of chaotic motion for the original system.

Dynamical interaction between double pendulum and nonlinear sloshing in two cranes carrying a liquid tank

Dynamical interaction between double pendulum and nonlinear sloshing in two cranes carrying a liquid tank

Hybrid State Estimation: Integrating Physics-Informed Neural Networks with Adaptive UKF for Dynamic Systems

In this paper, we present a novel approach to state estimation in dynamic systems by combining Physics-Informed Neural Networks (PINNs) with an adaptive Unscented Kalman Filter (UKF). Recognizing the limitations of traditional state estimation methods, we refine the PINN architecture with hybrid loss functions and Monte Carlo Dropout for enhanced uncertainty estimation. The Unscented Kalman Filter is augmented with an adaptive noise covariance mechanism and incorporates model parameters into the state vector to improve adaptability. We further validate this hybrid framework by integrating the enhanced PINN with the UKF for a seamless state prediction pipeline, demonstrating significant improvements in accuracy and robustness. Our experimental results show a marked enhancement in state estimation fidelity for both position and velocity tracking, supported by uncertainty quantification via Bayesian inference and Monte Carlo Dropout. We further extend the simulation and present evaluations on a double pendulum system and state estimation on a quadcopter drone. This comprehensive solution is poised to advance the state-of-the-art in dynamic system estimation, providing unparalleled performance across control theory, machine learning, and numerical optimization domains.

Dynamics and non-integrability of the double spring pendulum

This paper investigates the dynamics and integrability of the double spring pendulum, which has great importance in studying nonlinear dynamics, chaos, and bifurcations. Being a Hamiltonian system with three degrees of freedom, its analysis presents a significant challenge. To gain insight into the system’s dynamics, we employ various numerical methods, including Lyapunov exponents spectra, phase-parametric diagrams, and Poincaré cross-sections. The novelty of our work lies in the integration of these three numerical methods into one powerful tool. We provide a comprehensive understanding of the system’s dynamics by identifying parameter values or initial conditions that lead to hyper-chaotic, chaotic, quasi-periodic, and periodic motion, which is a novel contribution in the context of Hamiltonian systems. In the absence of gravitational potential, the system exhibits S1 symmetry, and the presence of an additional first integral was identified using Lyapunov exponents diagrams. We demonstrate the effective utilization of Lyapunov exponents as a potential indicator of first integrals and integrable dynamics. The numerical analysis is complemented by an analytical proof regarding the non-integrability of the system. This proof relies on the analysis of properties of the differential Galois group of variational equations along specific solutions of the system. To facilitate this analysis, we utilized a newly developed extension of the Kovacic algorithm specifically designed for fourth-order differential equations. Overall, our study sheds light on the intricate dynamics and integrability of the double spring pendulum, offering new insights and methodologies for further research in this field.

Nonlinear dynamics of a double pendulum energy harvesting device for continuously rotating systems

During operation, rotating systems develop a significant amount of kinetic energy that can be used for energy harvesting applications. However, this energy is hardly harnessed for the rotating body itself. In this work, it is proposed that a pendulum device connected to a DC generator can be an effective way to use part of the kinetic energy from continuously rotating devices. A double pendulum harvester mounted on a rotating body is experimentally and analytically analyzed. Different rotation speeds are used to evaluate the pendulum dynamics, along with the harvested energy. The results indicate that the system response can be classified into three distinct dynamic regimes based on the rotational speed. An analytical model is derived and used to analyze these regimes under different excitation conditions. It is experimentally shown that, at constant angular velocity, the double pendulum device can reach a maximum harvested power of 9.5 mW at 90 rpm. The analytical results prove that multiple period doubling bifurcations are observed as the rotation speed of the disk in slowly increased using a chirp type signal. Alike to experimental observations, chaotic-type response is detected by the analytical model, at rotation speeds similar to those observed experimentally.

Observer-controller tuning approach for double pendulum with genetic algorithm and neural network

Observer-controller tuning approach for double pendulum with genetic algorithm and neural network