Nonlinear Pendulum Research Articles

New implicit time integration schemes for structural dynamics combining high frequency damping and high second order accuracy

The time integration schemes, GA-23 and GA-234, recently proposed by the authors for first order problems, are extended to solve second-order problems in structural dynamics. The resulting methods maintain unconditional stability and user-controlled high-frequency damping. They offer improved accuracy and exhibit less numerical damping in the low-frequency regime, outperforming the well-known generalised-α method. When the high-frequency damping is maximised the new schemes can be cast in the format of backward difference formulae, offering more accurate alternatives to the standard second order formula. The effectiveness of the new time integration schemes is validated through a number of numerical examples, including a linear elastic cantilever beam, a nonlinear spring pendulum, and wave propagation on a string.

Adaptive fuzzy inverse optimal output feedback decentralised prescribed performance control for interconnected nonlinear systems

This paper investigates an adaptive fuzzy inverse optimal output feedback decentralised prescribed performance control problem for interconnected systems. Firstly, the fuzzy logic systems (FLSs) are utilised to model unknown nonlinear functions in each subsystem and then an auxiliary system is constructed. Based on the auxiliary system, a fuzzy decentralised state observer is formulated to estimate unmeasurable states. Secondly, by applying the adaptive backstepping design technique and the inverse optimal control theory, an adaptive fuzzy inverse optimal output feedback decentralised controller is ultimately designed. It is proved that the proposed adaptive fuzzy inverse optimal decentralised controller not only minimises the cost function but also ensures that all variables of the control systems are semi-globally ultimately bounded and that the tracking errors converge to a small residual set with the prescribed performance bound. Finally, the effectiveness of the proposed control approach is verified by a nonlinear inverted pendulum large-scale system.

Stability and bifurcation analysis of a 2DOF dynamical system with piezoelectric device and feedback control

This study aims to demonstrate the behaviors of a two degree-of-freedom (DOF) dynamical system consisting of attached mass to a nonlinear damped harmonic spring pendulum with a piezoelectric device. Such a system is influenced by a parametric excitation force on the direction of the spring’s elongation and an operating moment at the supported point. A negative-velocity-feedback (NVF) controller is inserted into the main system to reduce the undesired vibrations that affect the system’s efficiency, especially at the resonance state. The equations of motion (EOM) are derived by using Lagrangian equations. Through the use of the multiple-scales-strategy (MSS), approximate solutions (AS) are investigated up to the third order. The accuracy of the AS is verified by comparing them to the obtained numerical solutions (NS) through the fourth-order Runge-Kutta Method (RK-4). The study delves into resonance cases and solvability conditions to provide the modulation equations (ME). Graphical representations showing the time histories of the obtained solutions and frequency responses are presented utilizing Wolfram Mathematica 13.2 in addition to MATLAB software. Additionally, discusses the bifurcation diagrams, Poincaré maps, and Lyapunov exponent spectrums to show the various behavior patterns of the system. To convert vibrating motion into electrical power, a piezoelectric sensor is connected to the dynamical model, which is just one of the energy harvesting (EH) technologies with extensive applications in the commercial, industrial, aerospace, automotive, and medical industries. Moreover, the time histories of the obtained solutions with and without control are analyzed graphically. Finally, resonance curves are used to discuss stability analysis and steady-state solutions.

Review Paper on Vibration Damping System Using Pendulum

Abstract: This paper explores the implementation and effectiveness of vibration damping systems utilizing pendulums, focusing on their ability to mitigate unwanted oscillations in various structures. Pendulum-based dampers operate on the principles of inertia and resonance, allowing them to counteract vibrations causedby dynamic loads, seismic events, and machinery operations. The study examines different configurations, including tuned mass dampers (TMDs) and nonlinear pendulum systems, highlighting their design, functionality,and specific applications in fields such as civil engineering, aerospace, and automotive industries. Through analytical modeling and experimental validation, the paper demonstrates the effectiveness of pendulum dampers in enhancing structural stability and occupant comfort. The results indicate significant reductions in vibration amplitude, showcasing the potential of these systems to improve the longevity andsafety of infrastructure. Furthermore, the discussion addresses the advantages of pendulum-based systems over conventional damping methods, emphasizing their adaptability and efficiency.In conclusion, this study underscores the importance of pendulum vibration damping systems as a viable solution for contemporary engineering challenges, paving the way for future research and development in vibration control technologies.

An adaptive time-step energy-preserving variational integrator for flexible multibody system dynamics

An adaptive time-step variational integrator for simulating flexible multibody system dynamics is proposed. The integrator can adapt the time-step based on the variation of the system's energy. The flexible components in the system can undergo large overall motions and large deformations and are modelled by elements of absolute nodal coordinate formulations. In addition, a three-stage Newton-Raphson iteration method is developed to accurately solve the nonlinear discrete Euler-Lagrange equations in each time-step. Finally, three dynamic examples are presented to validate performance of the proposed integrator. Numerical results indicate that the proposed three-stage method has fast convergence rate. For the nonlinear flexible double pendulum system and the slider-crank mechanism, compared with constant time-step integrators, the proposed integrator can preserve the system's total energy more accurately and lead to more accurate dynamic responses. For the contact problem, the proposed integrator can quickly change the time-step size based on the sudden changes of energy to precisely compute the contact force and dynamic responses. Moreover, the proposed integrator can exactly preserve the displacement constraints and the velocity constraints simultaneously. In addition, it is noted that the computation efficiency of the proposed integrator needs to be further improved.

Modeling and Simulation of Fractional PID Controller for Under-actuated Inverted Pendulum Mechanical System

The stabilization of the non-linear inverted pendulum system requires a robust control strategy, as this system is inherently unstable and sensitive to disturbances. This research utilizes Lagrangian mechanics, a powerful technique in analytical dynamics, to derive the mathematical representation of the system. By applying the principles of Lagrangian dynamics, we can accurately model the energies involved and derive the equations of motion that govern the pendulum’s behavior. Following this, state-space feedback is employed to determine the Proportional, Integral, and Derivative (PID) values essential for effective control. This control strategy is particularly useful due to its ability to minimize error over time and ensure stability. To further enhance the control process, a comprehensive mathematical model is developed to establish the transfer function that correlates the pendulum's angle with the displacement of the cart. This relationship is crucial for understanding how changes in the cart's position affect the pendulum's stability. To validate the proposed control law, extensive simulations are conducted, allowing for comparative analysis against an Integer Order Controller. These simulations not only highlight the effectiveness of the PID controller but also provide insights into the dynamic behavior of the system under various conditions. The results demonstrate significant improvements in settling time and overshoot, showcasing enhanced performance metrics for the selected objective functions. This research contributes to the broader field of control systems engineering, suggesting that advanced control strategies can effectively manage complex, non-linear systems.

Nonlinear pendulum metamaterial to realize an ultra-low-frequency field effect bandgap

A field effect transistor (FET) is an extensively employed component in large-scale integrated circuits to regulate circuit on or off by modulating its gate voltage. The mechanical counterpart of a FET is of great significance for advanced regulations of acoustic or elastic waves. Recently proposed active metamaterials are promising scenarios of FET based on the opening and closing of frequency bandgaps through active elements, but hindered by the structural complexity in practice. In this study, we propose and experimentally realize a nonlinear pendulum metamaterial (NPMM) as a mechanical ultra-low-frequency FET. By employing the perturbation method to investigate its dispersion relationship, we show that the NPMM possess a field effect bandgap. Namely, the bandgap can be opened or closed by controlling the excitation amplitude of the impinging wave, which is largely affected by the relative mass of the inner pendulum unit. As a verification, the transmission behavior of the NPMM is numerically studied and experimentally tested. It shows that, once the amplitude of the impinging wave exceeds a certain threshold, the disappearance of a bandgap will be observed with a sudden amplification of the output wave. The central frequency of the field effect bandgap is as low as 4 Hz, whereas the characteristic size of the unit cell is only 4 cm. We expect that the present work opens a new avenue for low-frequency wave regulation.

Neural Observer With Lyapunov Stability Guarantee for Uncertain Nonlinear Systems.

In this article, we propose a novel nonlinear observer based on neural networks (NNs), called neural observers, for observation tasks of linear time-invariant (LTI) systems and uncertain nonlinear systems. In particular, the neural observer designed for uncertain systems is inspired by the active disturbance rejection control, which can measure the uncertainty in real time. The stability analysis (e.g., exponential convergence rate) of LTI and uncertain nonlinear systems (involving neural observers) are presented and guaranteed, where it is shown that the observation problems can be solved only using the linear matrix inequalities (LMIs). Also, it is revealed that the observability and controllability of the system matrices are required to demonstrate the existence of solutions for LMIs. Finally, the effectiveness of neural observers is verified in three simulation cases, including the X-29A aircraft model, the nonlinear pendulum, and the four-wheel steering vehicle.

Generalized N-rotor problems, synchronized subsystems, and associated solitons.

We consider systems of N particles interacting on the unit circle through 2π-periodic potentials. An example is the N-rotor problem that arises as the classical limit of coupled Josephson junctions and for various energies is known to have a wide range of behaviors such as global chaos and ergodicity, together with families of periodic solutions and transitions from order to chaos. We focus here on selected initial values for generalized systems in which the second order Euler-Lagrange equations reduce to first order equations, which we show by example can describe an ensemble of oscillators with associated emergent phenomena such as synchronization. A specific case is that of the Kuramoto model with well-known synchronization properties. We further demonstrate the relation of these models to field theories in 1+1 dimensions that allow static kink solitons satisfying first order Bogomolny equations, well-known in soliton physics, which correspond to the first order equations of the generalized N-rotor models. For the nonlinear pendulum, for example, the first order equations define the separatrix in the phase portrait of the system and correspond to kink solitons in the sine-Gordon equation.

Method for determining the Lyapunov exponent of a continuous model using the monodrome matrix

The Lyapunov exponent is a measure of the sensitivity of a dynamic system to any changes and disturbances. It is therefore applicable in many fields of science, including physics, chemistry, economics, psychology, biology, medicine, and technology. Therefore, there is a need to have effective methods for determining the value of this exponent. In particular, it concerns the definition of its sign. The positive value of the Lyapunov exponent confirms the sensitivity of the system, especially when the system is chaotic. A negative value indicates the stability of the dynamic system being tested. The numerical value of the Lyapunov exponent indicates the degree of sensitivity of the dynamic system under study. Although the mathematical definition of the Lyapunov exponent is clear and simple, in practice determining its value is not a trivial matter, especially for continuous models. This paper presents a method for determining the Lyapunov exponent for continuous models. This method is based on the monodromy matrix. The work, for example, presents research on the following models of physical systems: the Van der Pol model with external forcing, the nonlinear mathematical pendulum model with external forcing and the Lorenz weather model.

A high-performance dual-mode energy harvesting with nonlinear pendulum and speed-amplified mechanisms for low-frequency applications

Vibrations induced by human motions and vehicle operations feature low-frequency, broadband, and time-varying. However, enhancing the power density of energy harvesting for various low-frequency applications presents a significant challenge. This paper proposes a high-performance dual-mode energy harvester (DM-EH) incorporating coupled nonlinear pendulum and speed-amplified mechanisms with gears for the simultaneous scavenging vibration and rotation energy. The pendulum serves as the energy capture unit for detecting vibration energy while ensuring the effective operation of the generation unit in rotational environments. The gears amplify the speed of the excitation and enable frequency up-conversion, thereby enhancing the energy conversion. Experimental results substantiate the DM-EH's ability to extract power from vehicle operations at speeds ranging from 60 to 540 rpm, as well as from human motions at frequencies of 0.7–1.5 Hz with 30° amplitudes. In rotation mode, the prototype achieves a maximum average direct current power of 3.79 W, while in vibration mode, it attains 244 mW. The prototype successfully powers portable electronics and supports battery-free triaxial acceleration and temperature multi-sensors during human motions and railway simulation tests. This prototype showcases immense potential as a sustainable power source for portable electronics and self-powered monitoring applications.

Multilayer adaptive critic design with digital twin for data-driven optimal tracking control and industrial applications

In this paper, an optimal trajectory tracking control problem for general nonlinear systems is investigated. An adaptive critic control method with the digital twin (DT) theory is developed. Divergent from the existing tracking control methods, the advantages of adaptive dynamic programming (ADP) and the theory of DT are combined in this paper, and the novel multilayer artificial system structure is constructed. The action-critic structure is employed by each artificial system to obtain an approximate optimal control policy. The model network (MN) is built by using the actual input and output data sets of the controlled system, which means the dependence on the dynamics of the system is overcame. Then, the weights of the trained action network (AN) and MN are passed to the real system to realize the optimal tracking control. The feasibility of the algorithm is proved by theoretical analysis. Finally, the algorithm is applied to a simple nonlinear torsional pendulum system and an industrial wastewater treatment system (WWTS), and the effectiveness of the algorithm is verified. The algorithm effectively realizes the tracking control of nonlinear systems.

Bright soliton propagation with loss and gain phenomena in transversely connected nonlinear pendulum pairs

In this work, we seek to investigate the dynamics of bright soliton in a chain of coupled pendulum pairs. After deriving the linear dispersion relation from the equation of the model, we find that among the obtained modes, the fast mode is the one on which we are going to be focused. Since the discrete simultaneous equation describing the dynamics of the model has not been extensively studied in the literature, we assume that the two lines of the model are proportional to each other. We use the rotating wave approximation method to derive a NLS equation governing the propagation of waves in the network. Depending on the choice of wave number, we deduce that the system supports bright and hole-soliton solutions. We use the obtained bright soliton as the initial condition for numerical computation, which demonstrates the significant role of the transverse coupling parameter in the system. That is, it affects the behavior of the forward-bright soliton generated in the system. The lattice allows gain and loss phenomena during the propagation of the waves.

Nonlinear double-mass pendulum for vibration-based energy harvesting

To enhance the performance of a vibration-based energy harvester, typical approaches employ frequency-matching strategies by either using nonlinear broadband or frequency-tunable harvesters. This study systematically analyzes the nonlinear dynamics and energy harvesting performance of a recently emerging tunable low-frequency vibration-based energy harvester, namely, a double-mass pendulum (DMP) energy harvester. This energy harvester can, to some extent, eliminate frequency dependence on pendulum length but exhibit vibration-amplitude-dependent softening nonlinearity. The natural frequency of the DMP structure is theoretically derived, showing several unique characteristics compared with the typical simple pendulum. The DMP energy harvester exhibits alternate single-period, multiple-period, and chaotic vibration behaviors with increase in excitation amplitudes. The analysis of gross output power indicates that the rotating motion, regardless of chaotic or periodic rolling motions, improves the energy harvesting performance in terms of power leap and broader bandwidth. Based on the parameter space analysis, the rotating motions usually occur at the shift-left locations of frequency ratios 1 and 2; a smaller damping ratio corresponds to a lower on-demand excitation amplitude for the rotating-motion occurrence. Numerical results confirm that the DMP is suitable for low-frequency energy harvesting scenarios, suggesting the realization of rotating motion for improving energy harvesting performance. Moreover, a shake table test was performed, and the experimental results validated the accuracy and effectiveness of the DMP modeling analysis. Practical issues related to DMP energy harvesters under different types of excitations are finally discussed. Although the analysis is for the DMP, the corresponding conclusions may shed light on other pendulum-type energy harvesters.

Stabilization control of rotary inverted pendulum using a novel EKF-based fuzzy adaptive sliding-mode controller: design and experimental validation

This article methodically develops an improved self-regulating fuzzy-adaptive Sliding Mode Controller (SMC) that strengthens the disturbance compensation capacity of the nonlinear rotary pendulum systems while effectively attenuating the chattering content and curbing the control energy consumption. The article contributes to augmenting the SMC with online adaptation tools to achieve the said objectives. It employs the conventional Gao’s power-rate reaching law as the baseline. The scaling gain and the power rate of the said reaching law are adaptively modulated via a pre-calibrated two-input state-error-driven fuzzy nonlinear function. Additionally, the sign function in the law is also replaced with an odd-symmetric nonlinear fuzzy function to address the hard limits imposed by the former. Finally, the membership functions of the fuzzy function are self-regulated using the Extended-Kalman-Filter to improve the compensator’s adaptability in handling the system’s rapidly changing control requirements under exogenous disturbances. The aforementioned propositions are verified by performing customized and reliable hardware-in-loop experiments on the Quanser single-link rotary pendulum platform. As compared to baseline SMC law, the proposed control procedure contributes a ∼45.2%, ∼48.5%, and ∼34.8% reduction in position-regulation errors, control energy consumption, and peak overshoots, respectively. The experimental assessment validates the proposed control system’s enhanced robustness and chattering-suppression capability.

A discrete adjoint gradient approach for equality and inequality constraints in dynamics

The optimization of multibody systems requires accurate and efficient methods for sensitivity analysis. The adjoint method is probably the most efficient way to analyze sensitivities, especially for optimization problems with numerous optimization variables. This paper discusses sensitivity analysis for dynamic systems in gradient-based optimization problems. A discrete adjoint gradient approach is presented to compute sensitivities of equality and inequality constraints in dynamic simulations. The constraints are combined with the dynamic system equations, and the sensitivities are computed straightforwardly by solving discrete adjoint algebraic equations. The computation of these discrete adjoint gradients can be easily adapted to deal with different time integrators. This paper demonstrates discrete adjoint gradients for two different time-integration schemes and highlights efficiency and easy applicability. The proposed approach is particularly suitable for problems involving large-scale models or high-dimensional optimization spaces, where the computational effort of computing gradients by finite differences can be enormous. Three examples are investigated to validate the proposed discrete adjoint gradient approach. The sensitivity analysis of an academic example discusses the role of discrete adjoint variables. The energy optimal control problem of a nonlinear spring pendulum is analyzed to discuss the efficiency of the proposed approach. In addition, a flexible multibody system is investigated in a combined optimal control and design optimization problem. The combined optimization provides the best possible mechanical structure regarding an optimal control problem within one optimization.

Neural Data–Enabled Predictive Control

Data–enabled predictive control (DeePC) for linear systems utilizes data matrices of recorded trajectories to directly predict new system trajectories, which is very appealing for real-life applications. In this paper we leverage the universal approximation properties of neural networks (NNs) to develop neural DeePC algorithms for nonlinear systems. Firstly, we point out that the outputs of the last hidden layer of a deep NN implicitly construct a basis in a so-called neural (feature) space, while the output linear layer performs affine interpolation in the neural space. As such, we can train of-line a deep NN using large data sets of trajectories to learn the neural basis and compute on-line a suitable affine interpolation using DeePC. Secondly, methods for guaranteeing consistency of neural DeePC and for reducing computational complexity are developed. Several neural DeePC formulations are illustrated on a nonlinear pendulum example.

Analysis of peakon-like soliton solutions: (3+1)-dimensional Fractional Klein-Gordon equation

<abstract><p>In this study, we investigate the fundamental properties of ($ 3+1 $)-$ D $ Fractional Klein-Gordon equation using the sophisticated techniques of Riccatti-Bornoulli sub-ODE approach with Backlund transformation. Using a more stringent criterion, our study reveals new soliton solutions that have peakon-like properties and unique cusp features. This research provides significant understanding of the dynamic behaviours and odd events related to these solutions. This work is important because it helps to elucidate the complex dynamics that exist within physical systems, which will benefit many different scientific fields. Our method is used to examine the existence and stability of compactons and kinks in the context of actual physical systems. Under a double-well on-site potential, these structures are made up of a network of connected nonlinear pendulums. Both $ 2D $ and contour plots produced by parameter changes provide as clear examples of the efficiency, simplicity, and conciseness of the computational method used. The results highlight how flexible this approach is, and demonstrate how symbolic calculations broaden its application to more complex events. This work offers a useful framework and studying intricate physical systems, as well as a flexible computational tool that may be used in a variety of scientific fields.</p></abstract>

Neural network adaptive PID-like coupling control for double pendulum cranes with time-varying input constraints

Industrial cranes commonly experience the double pendulum swing effect as a result of a number of reasons, including the magnitude of the cargo and the non-negligible hook mass. Nevertheless, the majority of control strategies for double-pendulum cranes, whether open-loop or closed-loop, are created using linearized dynamics and necessitate accurate knowledge of the model parameters, rendering them susceptible to parametric uncertainties and unmodeled disturbances. To solve these problems, a new neural network adaptive PID-like coupling control method for the nonlinear double pendulum cranes is proposed in this paper. By making full use of the designed composite signal and its integral action, oscillation suppression and precise positioning are realized. In addition, the problem of conventional PID parameter adjustment is solved successfully by determining the PID gains through a neural network adaptive algorithm, automatically. Moreover, it has been shown that the systematic error converges into a compact set around zero for a variety of time-varying actuator failures and saturation situations that can easily occur in practical engineering. Finally, the effectiveness of the proposed control method is verified through hardware experiments, and the comparison results prove the superior performance of the designed controller.

Adaptive stable backstepping controller based on support vector regression for nonlinear systems

In this paper, a novel adaptive stable backstepping controller(BSC) based on support vector regression (SVR) has been introduced for nonlinear dynamical systems. Stable BSC is designed over Lyapunov stability of the closed-loop system. The nonlinear system dynamics required to constitute the BSC architecture are identified via SVR. The prediction competency of SVR and the stable behavior of BSC are aggregated in this architecture for nonlinear systems. The performance evaluation of the proposed adaptive BSC has been examined on a nonlinear inverted pendulum(IP) and a nonlinear mass–spring–damper(NMSD) system. The acquired results provide a successful and stable BSC control performance for both nonlinear systems.