Multiple Pendulum Research Articles

Experimental study and finite element analysis on impact resistance of wind turbine steel truss foundation

In recent years, steel truss structures have been widely used in civil engineering and wind turbines. In addition to bearing vertical loads, they are also vulnerable to impact accidental loads. If the foundation structure undergoes large deformation, it will lead to the overall instability and collapse of the structure. In this paper, the impact load resistance of steel truss foundation of wind turbine under eccentric mass is taken as the research object, and the combination of pendulum impact test and finite element simulation is adopted. The impact resistance of steel truss foundation was analyzed from the parameters of failure characteristics, impact response and energy dissipation capacity through multiple pendulum impact tests. The results show that after two impacts the stiffness of the impact leg is significantly reduced and even fracture at its root. The branch tube near the point of impact are squeezed and bent, and the deformation and acceleration peak of the impact position are the largest. The denting of the impact leg is the main energy dissipation mechanism. These data provide an experimental basis for improving the impact resistance of full-scale wind turbine structures.

Experimental and numerical study on impact resistance of offshore wind turbine CFDST jacket

With the development of large-scale offshore wind turbines (OWT), the jacket foundation widely used to offshore wind turbines are at significant risk of collision damage from vessels and sea ice. In order to improve the impact resistance of hollow steel tube jacket foundation, a new type of concrete-filled double-skin steel tube (CFDST) jacket foundation is proposed. From the analysis of impact force, displacement, acceleration and energy consumption, the peak strain and acceleration of the jacket leg at the impact position are reduced by 57.1 % and 50.4 %, respectively. The deformation mechanism and failure mode of the two foundations under multiple impact loads were obtained by combining LS-DYNA software and test. After two impacts the hollow steel tube jacket was seriously dented and damaged in bending. However, after multiple pendulum impacts on a concrete-filled double-skin steel tube, only the supporting struts were bent, the indentation depth and displacement peak of the jacket leg are reduced by 76.9 % and 72.2 %, respectively, and the entire structure showed excellent impact resistance, overall bending stiffness and energy dissipation capacity. Finally, the experimental basis and theoretical analysis are provided for the application of concrete-filled double-skin steel tube jacket in large offshore wind turbines and the resistance to impact loads.

Multiple pendulum and nonuniform distribution of average kinetic energy.

Multiple pendulums are investigated numerically and analytically to clarify the nonuniformity of average kinetic energies of particles. The nonuniformity is attributed to the system having holonomic constraints and it is consistent with the generalized principle of the equipartition of energy. With the use of explicit expression for Hamiltonian of a multiple pendulum, approximate expressions for temporal and statistical average of kinetic energies are obtained, where the average energies are expressed in terms of masses of particles. In a typical case, the average kinetic energy is large for particles near the end of the pendulum and small for those near the root. Moreover, the exact analytic expressions for the average kinetic energy of the particles are obtained for a double pendulum.

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.

Theoretical and experimental research on damping performance of suspended multiple mass pendulums damper

Conventional suspended mass pendulum damper (SMPD) include suspended single mass pendulum damper (SSMPD), tandem suspended multiple mass pendulums damper without springs (TSMP), and parallel suspended multiple mass pendulums damper without springs (PSMP), these types of SMPD have the advantages such as clear mechanism, simple construction, and good damping effect, but they have a narrow tuning frequency band and the pendulum length needs to be set too long for low-frequency super tall structures and too short for high-frequency tall structures, limiting their application in practical engineering. Because of this, the paper proposes connecting the spring to the mass pendulum, forming a tandem suspended multiple mass pendulums damper with spring (TSMP-S) and a parallel suspended multiple mass pendulums damper with spring (PSMP-S), and establishing structural system dynamics equations and performing the dynamic characterization of the TSMP-S and PSMP-S, obtain their frequency resolved solutions to explain the damping mechanism. In order to study the vibration-damping performance of these two types of SMPD, through numerical simulations to compare the damping effect of the conventional SMPD, TSMP-S, and PSMP-S on the controlled structures, and a series of shaking table tests on controlled and uncontrolled structures to verify the effectiveness of SSMPD, TSMP, PSMP, TSMP-S and PSMP-S for damping control of controlled structures under conventional earthquake and earthquake at four types of sites. Numerical simulations and shaking table test results in both show that the TSMP-S has the best damping effect, with the advantages of increasing the tuning frequency band, achieving multi-order vibration control, and flexible adjustment of the pendulum length, which is suitable for actual engineering.

Partial Lagrangian for Efficient Extension and Reconstruction of Multi-DoF Systems and Efficient Analysis Using Automatic Differentiation

In the fields of control engineering and robotics, either the Lagrange or Newton–Euler method is generally used to analyze and design systems using equations of motion. Although the Lagrange method can obtain analytical solutions, it is difficult to handle in multi-degree-of-freedom systems because the computational complexity increases explosively as the number of degrees of freedom increases. Conversely, the Newton–Euler method requires less computation even for multi-degree-of-freedom systems, but it cannot obtain an analytical solution. Therefore, we propose a partial Lagrange method that can handle the Lagrange equation efficiently even for multi-degree-of-freedom systems by using a divide-and-conquer approach. The proposed method can easily handle system extensions and system reconstructions, such as changes to intermediate links, for multi-degree-of-freedom serial link manipulators. In addition, the proposed method facilitates the derivation of the equations of motion-by-hand calculations, and when combined with an analysis algorithm using automatic differentiation, it can easily realize motion analysis and control the simulation of multi-degree-of-freedom models. Using multiple pendulums as examples, we confirm the effectiveness of system expansion and system reconstruction with the partial Lagrangians. The derivation of their equations of motion and the results of motion analysis by simulation and motion control experiments are presented. The system extensions and reconstructions proposed herein can be used simultaneously with conventional analytical methods, allowing manual derivations of equations of motion and numerical computer simulations to be performed more efficiently.

Submerged marine towed instrument array: A theoretical investigation using Lagrange mechanics

Towed instrument arrays (TIA) measure physical data in the ocean surface boundary layer (OSBL). The TIA consists of n probes mounted on a cable and towed behind a vessel. The comparably low interpolation errors of the two-dimensional results vastly enrich research on ocean energy dissipation.Here, we develop a new theoretical framework considering the mounted probes and their effects on the dynamics of the TIA in analogy to multiple pendulums on a moving suspension point. The dynamics are induced by external velocity-dependent drag- and coordinate-dependent depressor forces.We show that our method of including nonlinear drag forces is consistent and that our discrete approach is capable of computing continuous solutions in the limit n≫1. Hence, the proposed method unifies earlier approaches and is tested against several analytical and known numerical solutions. The phase space for the case n=1 is similar to that of a damped harmonic oscillator. A typical timescale estimates the equilibrium state of the dynamical system. We provide evidence of our method by comparing the results with real measurement data.Based on the theoretical investigations, test cases, and the comparison with real data, our method is a powerful tool, suitable for campaign planning, instrument design, and post-processing purposes.

Experimental study on effects of U-shape dampers on earthquake responses of a base-isolated LNG inner tank

Base-isolated liquefied natural gas (LNG) storage tanks are prone to violent liquid sloshing and large isolator displacements when subjected to near-fault earthquakes, which are characterized by long-period velocity pulses and large-amplitude displacements. Employing supplemental damping into isolation systems is an effective method to control the displacement of isolators. This paper aims to investigate effects of U-shape dampers on earthquake responses of an inner steel tank isolated by multiple friction pendulum bearings (MFPB). Two materials were selected to manufacture the U-shape dampers, i.e., Mg-Al alloy and mild steel. Hysteretic behaviors of MFPBs with and without U-shape dampers were first evaluated by quasi-static cyclic tests. Afterward, shaking table tests were conducted on a 1:25 scaled steel tank with different isolation systems. Results showed that the U-shape dampers could be used to effectively control the displacement of the isolators. Under the excitation of near-fault ground motions, the peak displacement of isolation bearings with Mg-Al alloy dampers decreased by 16 % ∼ 26 % and that with mild steel dampers decreased by 20 % ∼ 22 %. Meanwhile, adoption of U-shape dampers exhibited excellent reduction effects in both acceleration and hydrodynamic pressure of the tank. In comparison with MFPB, isolation bearings with U-shape dampers exhibited a more plumper hysteresis curve and higher energy dissipation capacity provided by sufficient slides of the bearings.

On the motion of a triple pendulum system under the influence of excitation force and torque

In this article, a nonlinear dynamical system with three degrees of freedom (DOF) consisting of multiple pendulums (MP) is investigated. The motion of this system is restricted to be in a vertical plane, in which its pivot point moves in a circular path with constant angular velocity, under the action of an external harmonic force and a moment acting perpendicular to the direction of the last arm of MP and at the suspension point respectively. Multiple scales technique (MST) among other perturbation methods is used to obtain the approximate solutions of the equations of motion up to the third approximation because it is authorizing to execute a specific analysis of the system behaviour and to realize the solvability conditions given the resonance cases. The stability of the considered dynamical model utilizing the nonlinear stability analysis approach is examined. The solutions diagrams and resonance curves are drawn to illustrate the extent of the effect of various parameters on the solutions. The importance of this work is due to its uses in human or robotic walking analysis.

Distributed Control of Networked Nonlinear Systems via Interconnected Takagi–Sugeno Fuzzy Systems With Nonlinear Consequent

This article deals with the problem of designing nonlinear distributed control laws for continuous-time networked nonlinear heterogeneous systems with bounded sector nonlinear interconnections. The networked system is a combination of interconnected Takagi–Sugeno (TS) fuzzy systems with nonlinear consequent subject to state and control input constraints. The new sufficient conditions, taking into account state constraints and actuators saturation, are derived in terms of linear matrix inequalities. Further, it is shown that the closed-loop system can be made asymptotically stable while reducing the conservatism over decentralized and linear distributed control laws derived following the same method. At the same time, the maximization of the estimate of the domain of attraction (DoA) for the closed-loop system is pursued. Finally, two cases are presented to illustrate the effectiveness of the proposed design approach: 1) an electrical power system composed of two-machine subsystems and 2) a network of multiple inverted pendulums connected by nonlinear springs.

Seismic resistant design of highway bridge with multiple-variable frequency pendulum isolator

Multiple variable frequency pendulum isolator (MVFPI) has been recently developed as a superior alternative to the traditional friction pendulum bearing (FPB) especially for the seismic isolation in near-fault regions. The MVFPI is characterized by its variable frequency and self-adaptability, which are achieved by piecewise function of sliding surface and shape memory alloy (SMA). The objective of this study is to propose the design algorithm of the MVFPIs in highway bridge as an extension of the direct displacement-based design (DDBD) framework. The nonlinearities of the structural components are taken into account in the design procedure, and the corresponding damage states satisfy the two-stage design philosophy. The accuracy and robustness of the design procedure are verified by an isolated four-span highway bridge through nonlinear time history (NLTH) analyses. The analytical results indicate that the proposed design procedure can predict the profile of deck displacement and amplitude, as well as the damage states of the piers. From statistic aspect, the fragility analyses illustrate that the bridge isolated by MVFPIs exhibits better seismic performance than that of the bridge isolated by FPBs.

Metachronal motion of a thermally actuated double pendulum driven by self-propulsion caused by spontaneous asymmetrical heat transfer

A metachronal motion produces a net flow in a low Reynolds regime and is thus important for the development of artificial cilia. In this study, we report that a thermally actuated double pendulum exhibits a metachronal motion using a self-propulsion mechanism caused by spontaneous asymmetrical heat transfer. Specifically, by using a multiple pendulum structure in which three beams of different lengths are hung in parallel in order of length on a U-shaped nichrome heater, we demonstrate that phase differences can be produced among beams. Our device can be applied to biomedical microfluidic systems or microrobots in water.

A Hybrid Control Approach for the Swing Free Transportation of a Double Pendulum with a Quadrotor

In this article, a control strategy approach is proposed for a system consisting of a quadrotor transporting a double pendulum. In our case, we attempt to achieve a swing free transportation of the pendulum, while the quadrotor closely follows a specific trajectory. This dynamic system is highly nonlinear, therefore, the fulfillment of this complex task represents a demanding challenge. Moreover, achieving dampening of the double pendulum oscillations while following a precise trajectory are conflicting goals. We apply a proportional derivative (PD) and a model predictive control (MPC) controllers for this task. Transportation of a multiple pendulum with an aerial robot is a step forward in the state of art towards the study of the transportation of loads with complex dynamics. We provide the modeling of the quadrotor and the double pendulum. For MPC we define the cost function that has to be minimized to achieve optimal control. We report encouraging positive results on a simulated environmentcomparing the performance of our MPC-PD control circuit against a PD-PD configuration, achieving a three fold reduction of the double pendulum maximum swinging angle.

Fault-Tolerant Control of Multilayer Interconnected Nonlinear Systems: An Inclusion Principle Approach

This paper addresses the fault-tolerant control (FTC) issue for a class of multilayer interconnected nonlinear systems by using the inclusion principle. First, an overlapping decomposition method is proposed that expands the original interconnected system into a new one where each layer is not overlapped with each other. Second, for such an expanded system, the coupling effects among multiple layers are analyzed by using cyclic-small-gain theorem, and FTC schemes based on layer cooperation are further developed. Finally, the control designed for the expanded system is contracted back to the original interconnected system to achieve its FTC goal. An example of multiple pendulums is taken to illustrate the efficiency and applicability of the obtained theoretical results.

Impact of a Multiple Pendulum with a Non-Linear Contact Force

This article presents a method to solve the impact of a kinematic chain in terms of a non-linear contact force. The nonlinear contact force has different expressions for elastic compression, elasto-plastic compression, and elastic restitution. Lagrange equations of motion are used to obtain the non-linear equations of motion with friction for the collision period. The kinetic energy during the impact is compared with the pre-impact kinetic energy. During the impact of a double pendulum the kinetic energy of the non-impacting link is increasing and the total kinetic energy of the impacting link is decreasing.

Comment on \u201cHyperchaos in constrained Hamiltonian system and its control\u201d by J. Li, H. Wu and F. Mei

The aim of this comment is to show that discovery of hyperchaos in three systems investigated in Li et al. (Nonlinear Dyn 94(3):1703–1720, 2018) is not correct. It is justified both theoretically and numerically. Corrected calculations of Lyapunov exponents and corresponding bifurcation diagram are given. Examples of hyperchaotic Hamiltonian multiple pendulum systems are presented.

Federated Reinforcement Learning for Training Control Policies on Multiple IoT Devices.

Reinforcement learning has recently been studied in various fields and also used to optimally control IoT devices supporting the expansion of Internet connection beyond the usual standard devices. In this paper, we try to allow multiple reinforcement learning agents to learn optimal control policy on their own IoT devices of the same type but with slightly different dynamics. For such multiple IoT devices, there is no guarantee that an agent who interacts only with one IoT device and learns the optimal control policy will also control another IoT device well. Therefore, we may need to apply independent reinforcement learning to each IoT device individually, which requires a costly or time-consuming effort. To solve this problem, we propose a new federated reinforcement learning architecture where each agent working on its independent IoT device shares their learning experience (i.e., the gradient of loss function) with each other, and transfers a mature policy model parameters into other agents. They accelerate its learning process by using mature parameters. We incorporate the actor–critic proximal policy optimization (Actor–Critic PPO) algorithm into each agent in the proposed collaborative architecture and propose an efficient procedure for the gradient sharing and the model transfer. Using multiple rotary inverted pendulum devices interconnected via a network switch, we demonstrate that the proposed federated reinforcement learning scheme can effectively facilitate the learning process for multiple IoT devices and that the learning speed can be faster if more agents are involved.

Design of fixed-time synchronization algorithm with applications

Fixed-time synchronization problem for a class of leader–follower multi-agent systems with second-order nonlinearity is studied in this article. A new fixed-time synchronization control algorithm is developed by effectively combining homogeneous system theory, Lyapunov stability theory, and fixed-time/finite-time control technology. The leader–follower multi-agent system is considered to achieve fixed-time synchronization control. Finally, numerical simulations including coordination control multiple pendulum robot systems and electric power networks are carried out to verify the control performance of the control strategy.

Cone complementarity approach for dynamic analysis of multiple pendulums

The multibody system dynamics approach allows describing equations of motion for a dynamic system in a straightforward manner. This approach can be applied to a wide variety of applications that consist of interconnected components which may be rigid or deformable. Even though there are a number of applications in multibody dynamics, the contact description within multibody dynamics still remains challenging. A user of the multibody approach may face the problem of thousands or millions of contacts between particles and bodies. The objective of this article is to demonstrate a computationally straightforward approach for a planar case with multiple contacts. To this end, this article introduces a planar approach based on the cone complementarity problem and applies it to a practical problem of granular chains.

The motion characteristics of the double‐pendulum system with skew walls

Many components of machines and other technological devices (chains and bodies hanging on the ropes) can be modeled as multiple pendulums situated in tubes or holes of limited space. The investigation of motion of such systems represents the substance of many real technological problems; therefore, it stirred up motivation to perform research on vibration of a double pendulum situated between two skew rigid walls. The analyzed system was set into motion by the horizontal movement of its suspending. The results of the simulations show that the system can exhibit both the regular and chaotic movements depending on the excitation frequency. The development of the computational model, which is applicable for more complex investigations, and learning more on the motion character of a double pendulum, the movement of which is limited by skew walls, are the principal contributions of the presented article.