Double Pendulum System Research Articles

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.

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

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.

Assessment and measurement of ground motion risk levels for structures isolated by double concave friction pendulum system

Determining the risk level of ground motions (GM) for structures is crucial for their damage control and resilience enhancement, while the assessment and measurement criteria of GM risk levels for structures isolated by double concave friction pendulum system (DCFPS) remain unclear. Through statistical analysis of the response distribution of DCFPS with various design parameters under 1013 GM records, this study determines an energy dissipation efficacy of 99 %, a floor velocity of 0.35 m/s, a floor velocity of 0.60 m/s, and the displacement capacity of DCFPS as the response criteria for assessing four performance states of efficacy, habitability, functionality, and safety, respectively. In addition, the value of GM intensity in PGA, PGV, PGD, and the equivalent velocity of seismic input energy (VE) for measuring the four GM risk levels corresponding to the four performance states are obtained. Furthermore, a classification of GMs based on earthquake magnitude and fault distance is proposed according to their risk to DCFPS. It is found that the intensity value of each GM risk level varies for different design parameters of DCFPS but fluctuate within a small range, however, great differences exist when using different intensity measures (IM) in classifying GM risk levels. Therefore, based on the correlation between IMs and DCFPS responses, the conservatism of PGA, PGV, PGD and VE in assessing the risk of GMs for DCFPS are evaluated.

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.

Dynamics Analysis and Control of a Two-Link Manipulator

This article develops a practicable, efficient, and robust PID controller for the traditional double pendulum system. Utilizing the Lagrangian method, the equations of motion for the two-link robot manipulator are initially derived. The system of ordinary differential equations for this nonlinearity describes these equations. As closed-form solutions for the equations of motion are absent, we approximate the solution of the initial-value problem. Securing precise user-defined positions while controlling the motion of the two-link robot manipulator proves to be a formidable challenge due to its non-linear behavior. The primary objective is to achieve the intended position of the robot manipulator by implementing the computed torque control method. Once the equation of motion has been derived, MATLAB is utilized to represent the control simulation. Several computational simulations are employed to validate the controller performance. Specifically, we implement a PID controller to simulate the balancing of the two links on a mobile robot at any given angle, including inverted.

Improved dynamic sliding mode control for plate hoisting of cable crane under wind load

In order to quickly eliminate the swing angle of plate hoisting under wind load, an improved dynamic sliding mode control method is proposed in this paper. Firstly, the plate hoisting model under wind load is simplified into a double pendulum system, and the dynamic model of a double pendulum cable crane with wind disturbance is established. To enhance the coupling relationship between trolley displacement and double swing angle, a new coupling state vector for the double pendulum system is defined. On this basis, a dynamic sliding surface is constructed and then an integral function of the system’s discontinuous term is established, essentially ensuring the continuity of the sliding surface and reducing the system’s chattering characteristics. Finally, the stability of the system is rigorously proved by Lyapunov theorem and Barbalat lemma. The simulation and experimental results show that the controller proposed in this paper has a good effect on the anti-swing problem of plate hoisting under wind load.

Non-quantum chirality and periodic islands in the driven double pendulum system

The paper presents a systematic investigation of the dynamic complexities exhibited by a driven double pendulum system. By utilizing isospike diagrams in the frequency–amplitude control parameter space, a wide range of novel dynamical behaviors are explored. In addition to the well-known shrimp-shaped period island, the study uncovers various unique self-organizations, including the quint point where five distinct stable oscillatory phases coalesce, the eye of chaos characterized by a nested ring structure resulting from forward and reverse cascades, and innovative non-quantum chiral structures. The observed chiral structures in the double pendulum system exist in more diverse and generalized structure, challenging the notion that ring structures are necessary for their emergence. Furthermore, the study reports the experimental observation of a newly generated parabolic phase near the quint point, validating the phenomena observed through numerical calculations in experiments. These findings not only advance our understanding of the control parameter space topology for driven double pendulums but also lay the foundation for the exploration of innovative control strategies.

Experimental and numerical investigations on adaptive stiffness double friction pendulum systems for seismic protection of bridges

This study examines the design, development, and characterization of double friction pendulum systems with variable sliding surface geometry. The proposed isolator has a spherical surface with a small radius within an inner sliding range while different geometric functions are used to describe the sliding surface in the outer range. As a result of variable curvature used as sliding surface, the isolators exhibit an adaptive stiffness behavior with increasing displacement amplitude. First, the analytical equations that describe the characteristics of the proposed isolators were derived. Then, a large-scale prototype adaptive isolator that employs a logarithmic function for the sliding surface geometry in the outer range was fabricated. The response of the prototype isolator was characterized under increasing amplitude reversible cyclic loads considering two different levels of vertical isolator loads. A high-fidelity finite element model was developed to capture the response of the prototype isolator. The results obtained from the experimental tests and numerical simulations were used to verify the analytical equations derived for the proposed isolators. Next, a typical multi-span bridge structure was modeled with adaptive stiffness isolators and its response under bi-directional ground motions records were analyzed. In contrast to a conventional double friction pendulum bearing, the suggested isolators yield significantly reduced base shear forces while exhibiting greater isolator displacements. However, the proposed isolators retain the capability to effectively curtail excessive displacements, ensuring the prevention of collapse.

Friction modelling and the use of a physics-informed neural network for estimating frictional torque characteristics

This paper presents an exploration of friction modeling encompassing theoretical and practical aspects, utilizing a planar or 2D contact system. Various white-box friction models, including static and dynamic variants, are introduced, highlighting the superior capability of dynamic models in comprehensively capturing friction effects, substantiated through numerical simulation. Practical aspects of friction measurement and data-driven friction modeling are elucidated. The discourse extends to the development of grey-box and black-box friction models. A significant contribution lies in the proposition of a physics-informed neural network-based friction modeling approach, presenting it as an advanced and preferable alternative for friction estimation. To exemplify its efficacy, a case study of a torsion-based frictional contact scenario, employing Physics-Informed Neural Network (PINN) and the Nelder–Mead (N–M) algorithm for concurrent dynamics and friction model identification, was examined. Experimental data from a double torsion pendulum system, characterized by discontinuous dynamics, is employed for training. Results demonstrate the PINN’s superiority, providing more accurate representation of stick–slip phases at the contact zone and exhibiting faster performance compared to the N–M algorithm. The paper concludes by deliberating on challenges, prospects, and future directions in friction modeling.

Experimental Evaluation and Validation of Pressure Distributions in Ice–Structure Collisions Using a Pendulum Apparatus

This study introduced an experimental setup designed to predict collision pressure exerted on a structure due to ice impacts. The primary aim was to forecast the pressure distribution and structural behavior following an ice collision on a laboratory scale. The apparatus comprises a double-pendulum system situated within a cold chamber, facilitating the prediction of collision forces by varying both the collision energy and velocity. Experiments were conducted at collision velocities of 5, 7, and 10 knots, representative of conditions in Arctic regions. By altering these velocities, we measured both the strain experienced by a steel plate and the pressure due to the ice collision. The pressure distribution on the steel plate surface was also recorded by using pressure-sensitive film. This setup effectively captured the pressure distribution across the collision contact area. The measured pressure and force were compared with that of previous studies to determine the validity of the ice collision force measurement. This methodology successfully captures the relationship between the increasing collision speed and force, providing valuable insights into ice-induced forces and structural strain. It offers an effective measurement technique beneficial for designing ice-resistant structures. In conclusion, this paper enhances our understanding of ice–structure interactions, particularly in the domains of collision force and strain metrics.

An experimental demonstration of level attraction with coupled pendulums

We have experimentally demonstrated dissipative coupling in a double pendulum system through observation, which shows three distinctly different patterns of motion over the accessible parameter space. The described dissipative coupling apparatus is easy to manufacture and budget-friendly. The theoretical calculations are also suitable for the undergraduate level. Our experiment can serve as a novel demonstration for ubiquitous dynamic coupling effects encountered in many disparate physical systems. Unlike the well-known spring-coupled pendulums, our experiment employs Lenz's effect to couple the pendulums through electromagnetic damping, which, to the best of our knowledge, has not been demonstrated in the classroom. Our pendulums exhibit level attraction behaviour between two modes, induced by the dissipative coupling. This stands in contrast to the traditionally taught concept of level repulsion (avoided crossing) with spring-coupled pendulums. This experiment showcases distinctly different time domain dynamics of the dissipatively coupled pendulums over the parameter space, characterized by different oscillation patterns, damping rates, and relative phase between the two pendulums, which is a valuable lesson elucidating the dynamics of synchronization in linear systems for undergraduate students.

Rigorous data‐driven computation of spectral properties of Koopman operators for dynamical systems

AbstractKoopman operators are infinite‐dimensional operators that globally linearize nonlinear dynamical systems, making their spectral information valuable for understanding dynamics. However, Koopman operators can have continuous spectra and infinite‐dimensional invariant subspaces, making computing their spectral information a considerable challenge. This paper describes data‐driven algorithms with rigorous convergence guarantees for computing spectral information of Koopman operators from trajectory data. We introduce residual dynamic mode decomposition (ResDMD), which provides the first scheme for computing the spectra and pseudospectra of general Koopman operators from snapshot data without spectral pollution. Using the resolvent operator and ResDMD, we compute smoothed approximations of spectral measures associated with general measure‐preserving dynamical systems. We prove explicit convergence theorems for our algorithms (including for general systems that are not measure‐preserving), which can achieve high‐order convergence even for chaotic systems when computing the density of the continuous spectrum and the discrete spectrum. Since our algorithms have error control, ResDMD allows aposteri verification of spectral quantities, Koopman mode decompositions, and learned dictionaries. We demonstrate our algorithms on the tent map, circle rotations, Gauss iterated map, nonlinear pendulum, double pendulum, and Lorenz system. Finally, we provide kernelized variants of our algorithms for dynamical systems with a high‐dimensional state space. This allows us to compute the spectral measure associated with the dynamics of a protein molecule with a 20,046‐dimensional state space and compute nonlinear Koopman modes with error bounds for turbulent flow past aerofoils with Reynolds number >105 that has a 295,122‐dimensional state space.

Dynamics of multiple pendulum system under a translating and tilting pivot

In this article, we study the dynamics of multiple pendulum systems under translation and tilt. The main application considered for such systems is inertial sensing for high-precision instrumentation. To emulate the translating multiple pendulum system, we attach the pivot point of the pendulum to a cart that is free to move in the horizontal plane. Similarly, the pivot point of the tilting pendulum system is attached to a platform that rotates, enabling tilting motion for the system. First, we approach the problem from a Lagrangian dynamics perspective for a double-pendulum system under translation and tilt and then extend the solutions to a system of n pendulums, each hanging one below the other. Then, the natural frequencies of the systems are derived. The behavior of the systems under translation and tilt is studied and compared with that of fixed pivot point multiple pendulum systems, using eigenvalue analysis to understand how the natural frequency fluctuates with changes in degrees of freedom, mass, length and stiffness.

Stabilization of Double Inverted Pendulum Systems Based on Hierarchical Sliding Mode Control Techniques

The double rotary inverted pendulum (DRIP) system belongs to the class of under‐actuated mechanical systems, and it is a highly nonlinear, unstable, and benchmark system to test the different control techniques. This paper successfully designed the nonlinear hierarchical sliding mode control (HSMC) techniques to stabilize the DRIP system. We compare the performance of these techniques numerically with each other and with the previously designed control technique. We propose an aggregated HSMC technique as it has a much shorter stabilization time than other designed techniques.

Explicit smooth/nonsmooth cosimulation using kinematic constraints

An explicit cosimulation scheme is developed to study the coupling of smooth and nonsmooth systems using kinematic constraints. Using the force-displacement decomposition, the coupling constraints are formulated at the velocity level, to preserve consistency with the impulse-momentum equations for frictional contacts in the nonsmooth solver, which however potentially leads to instability of the explicit cosimulation. To improve the stability of the cosimulation without affecting the format of the coupling constraints, guidelines for the modification of the prescribed motion are developed following the spirit of Baumgarte’s stabilization technique and the characteristics of the proposed integration scheme, which prescribes a combination of position, velocity, and acceleration to the constrained bodies. Using modified inputs, the stability of the cosimulation is tested using a rigidly connected two-mass oscillator model, which shows clear improvement compared to that with unaltered inputs. The performances of the cosimulation with modified inputs are further illustrated using a double-pendulum system and a complex flexible multibody system coupled with a particle damper. It follows that cosimulation results well agree with those obtained using monolithic simulation or simplified models, verifying the explicit smooth/nonsmooth cosimulation. The results also show a higher efficiency of the explicit cosimulation scheme, which requires much less computational time to obtain similar results, compared to the implicit smooth/nonsmooth cosimulation.

Model based reduced-order observers for a class of mechatronic systems with mitigation of disturbances effects using the Attractive Ellipsoid Method

State-estimation of mechatronic systems is essential because, in addition to position and velocity, there are electric and magnetic signals that must be measured, and can lead to sophisticated and expensive instrumentation systems. This paper addresses the problem of state-estimation for underactuated mechanical systems whose active joints are driven by permanent magnet dc-motors. Luenberger-type structures are designed, where the measured state variables and the dynamic model are used to construct the unavailable ones, in presence of uncertainties and disturbances as inputs. To avoid the peaking phenomenon is one of the main challenges on state observation for nonlinear systems. Underactuated mechatronic systems are not immune to this problem, which is exacerbated when external disturbances are taken into account. The major purpose here is to use a robust reduced observer to estimate the unmeasured state variables. The Attractive Ellipsoid Method provides reliable estimation of unmeasured state signals because it reduces the influence of external disturbances on the estimate error signals. This ensures that the estimated signals converge to an invariant set of minimal size around the real ones. The first design is based on the quasi-linear representation that meets the observability condition. The second considers the entire nonlinear model, which necessitates the availability of the entire position vector. The major findings are demonstrated experimentally on a non-traditional mechatronic system: the linear double pendulum system whose last joint is constrained by two linear springs. This shows a significant improvement of the estimation response, concerning robust techniques based on the linear approximation, towards its application in designing an adaptive observer to provide state and parameters estimation simultaneously, which is essential in robust and adaptive control systems.

Image motion and experimental study of a 0.1″ space pointing measuring instrument for micro-vibration conditions

There exists an increasing need for Milli-Arc-Seconds (MAS) accuracy pointing measurement for current and future space systems. To meet the 0.1″ space pointing measurement accuracy requirements of spacecraft in future, the influence of spacecraft micro-vibration on a 0.1″ Space Pointing Measuring Instrument (SPMI) is studied. A Quasi-Zero Stiffness Device (QZSD) with adaptive adjustment and variable stroke was proposed. Then, a series of micro-vibration experiments of the SPMI were carried out. The influence of the micro-vibration generated by Guidance Navigation Control (GNC) attitude control components under different attitudes on the SPMI was analyzed. Point spread function of image motion in micro-vibration was also derived. Further, the changes of image motion under the micro-vibration environment were evaluated by extracting the gray centroid of the images, and the experiment processes and results are deeply discussed. The results show that the first-order frequency of the QZSD system is 0.114 Hz, and it is induced by a double pendulum system; the image motion of single flywheel spinning reached 0.015 pixels; whilst the image motion reached 0.03 pixels when three flywheels are combined spinning. These latest findings provide a beneficial theoretical and technical support for the development of spacecraft with 0.1″ pointing accuracy.

Flight Dynamics and Control of an Unmanned Helicopter with Underslung Double Pendulum

This paper presents systematic modeling and simulation of the dynamics and control of a carrier helicopter unmanned aerial vehicle (UAV) with a suspended net used for capturing target UAVs. The goal is to study the effect of key design parameters such as the weight of the net end rod, its length, and point of impact of the UAV on the swing angle of the underslung net. First, a net with an entangled target UAV is modeled as a planar double pendulum. Next, the coupled flight dynamic model of carrier UAV with underslung net with entangled target UAV is simulated and validated using the test data gathered for this purpose. From the simulations, it is identified that the length of the net and point of capture of the UAV on the net are critical parameters that govern the maximum swing angles of the double-pendulum system. Moving the point of impact of the target UAV by 10% of net length resulted in up to 23 and 43% reduction in swing angles of pendulums 1 and 2. The active swing suppression using proportional derivative (PD) controller is effective in attenuating the swing angles in the simulation for successful capture.

White Noise Tests on the LSTM Model Trained with Double Pendulum

This paper describes the intrinsic qualities of a simple double pendulum (DP), with a visual representation, a rigorous deduction of the Lagrangian equation, and a concrete factor analysis. LSTM model was utilized to simulate the double pendulum’s periodic and chaotic behaviors and evaluates the effectiveness of the model. The auto-correlation coefficients was calculated. Meanwhile, Box-Pierce test and Ljung-Box tests for various state-dependent time series were conducted to give various initial conditions to explore the DP system’s random characteristics. The research results are as follows: 1) Chaos did not lead to direct randomness; 2) seasonality could coexist with chaos; 3) the highly auto-regressive nature of DP’s time series data are found. Therefore, it can be concluded that the chaos in a double pendulum has particular patterns (such as the positive relationship with the likelihood of being a random white noise series) that could be further explored.