Dynamics analysis of a gyrostat system with intermittent forcing

The dynamical characteristics of a gyrostat system with intermittent forcing are investigated, the main work and contributions are given as follows: (1) The gyrostat system with an intermittent forcing is studied, and its dynamical characteristics are investigated by the corresponding Lyapunov exponent spectrums and bifurcation diagrams with respect to the amplitude of intermittent forcing. The modified gyrostat system exists chaotic motion when the amplitude of intermittent forcing belongs to a certain interval, and it can be at a state of stable point or periodic motion by the design of amplitude. (2) The gyrostat system with multiple intermittent forcings is also investigated through the combination of Lyapunov exponent spectrums and bifurcation diagrams, and it behaves periodic motion or chaotic motion when the amplitude or forcing width is different. (3) By the selection of parameters in intermittent forcings, the modified gyrostat system is at a state of stable point, periodic motion or chaotic motion. Numerical simulations verify the feasibility and effectiveness of the modified gyrostat system.

Fuzzy Chaos Control of Fractional Order D-PMSG for Wind Turbine with Uncertain Parameters by State Feedback Design

To research the chaotic motion problem of the direct-drive permanent magnet synchronous generator (D-PMSG) for a wind turbine with uncertain parameters and fractional order characteristics, a control strategy established upon fuzzy state feedback is proposed. Firstly, according to the working mechanism of D-PMSG, the Lorenz nonlinear mathematical model is established by affine transformation and time transformation. Secondly, fractional order nonlinear systems (FONSs) are transformed into linear sub-model by Takagi–Sugeno (T-S) fuzzy model. Then, the fuzzy state feedback controller is designed through Parallel Distributed Compensation (PDC) control principle to suppress the chaotic motion. By applying the fractional Lyapunov stability theory (FLST), the sufficient conditions for Mittag–Leffler stability are formulated in the format of linear matrix inequalities (LMIs). Finally, the control performance and effectiveness of the proposed controller are demonstrated through numerical simulations, and the chaotic motions in D-PMSG can be eliminated quickly.

Chaotic Analysis of Fractional-Order Permanent Magnet Synchronous Generator for Wind Turbine

Fractional-order wind turbine is a strongly coupled non-linear dynamic system. It mainly studies the significant chaos characteristics such as the complex chaotic motion with fractional order varying. According to the mathematical model of the system, the fractional order Lorenz chaotic equation is established by linear affine transformation and time scale transformation. The theory of Lyapunov stability analysis is adopted to deeply study the development process of the system from stable operation to chaotic motion. The correctness of the chaos characteristics of the system is verified.

Molecular Dynamics Study of Collective Behavior of Carbon Nanotori in Columnar Phase

Supramolecular interaction of carbon nanotori in a columnar phase is described using the methods of classical molecular dynamics. The collective behavior and dynamic properties of toroidal molecules arising under the action of the van der Waals forces are studied. The conditions under which columnar structures based on molecular tori become unstable and rearrange into another structure are investigated. The reasons for the appearance of two types of directed rotational motion from the chaotic motion of molecules are discussed.

Non-Linear Dynamic Analysis of Drill String System with Fluid-Structure Interaction

In this research, the non-linear dynamics of the drill string system model, considering the influence of fluid—structure coupling and the effect of support stiffness, are investigated. Using Galerkin’s method, the equation of motion is discretized into a second-order differential equation. On the basis of an improved mathematical model, numerical simulation is carried out using the Runge—Kutta integration method. The effects of parameters, such as forcing frequency, perturbation amplitude, mass ratio and flow velocity, on the dynamic characteristics of the drill string system are studied under different support stiffness coefficients, in which bifurcation diagrams, waveforms, phase diagrams and Poincaré maps of the system are provided. The results indicate that there are various dynamic model behaviors for different parameter excitations, such as periodic, quasi-periodic, chaotic motion and jump discontinuity. The system changes from chaotic motion to periodic motion through inverse period-doubling bifurcation, and the support stiffness has a significant influence on the dynamic response of the drill string system. Through in-depth study of this problem, the dynamic characteristics of the drill string can be better understood theoretically, so as to provide a necessary theoretical reference for prevention measures and a reduction in the number of drilling accidents, while facilitating the optimization of the drilling process, and provide basis for understanding the rich and complex nonlinear dynamic characteristics of the deep-hole drill string system. The study can provide further understanding of the vibration characteristics of the drill string system.

Chaotic motion control of a vibro-impact system based on data-driven control method

For the chaotic motion control of a vibro-impact system with clearance, the parameter feedback chaos control strategy based on the data-driven control method is presented in this article. The pseudo-partial-derivative is estimated on-line by using the input/output data of the controlled system so that the compact form dynamic linearization (CFDL) data model of the controlled system can be established. And then, the chaos controller is designed based on the CFDL data model of the controlled system. And the distance between two adjacent points on the Poincaré section is used as the judgment basis to guide the controller to output a small perturbation to adjust the damping coefficient of the controlled system, so the chaotic motion can be controlled to a periodic motion by dynamically and slightly adjusting the damping coefficient of the controlled system. In this method, the design of the controller is independent of the order of the controlled system and the structure of the mathematical model. Only the input/output data of the controlled system can be used to complete the design of the controller. In the simulation experiment, the effectiveness and feasibility of the proposed control method in this article are verified by simulation results.

Identification and Machine Learning Prediction of Nonlinear Behavior in a Robotic Arm System

In this study, the subject of investigation was the dynamic double pendulum crank mechanism used in a robotic arm. The arm is driven by a DC motor though the crank system and connected to a fixed side with a mount that includes a single spring and damping. Robotic arms are now widely used in industry, and the requirements for accuracy are stringent. There are many factors that can cause the induction of nonlinear or asymmetric behavior and even excite chaotic motion. In this study, bifurcation diagrams were used to analyze the dynamic response, including stable symmetric orbits and periodic and chaotic motions of the system under different damping and stiffness parameters. Behavior under different parameters was analyzed and verified by phase portraits, the maximum Lyapunov exponent, and Poincaré mapping. Firstly, to distinguish instability in the system, phase portraits and Poincaré maps were used for the identification of individual images, and the maximum Lyapunov exponents were used for prediction. GoogLeNet and ResNet-50 were used for image identification, and the results were compared using a convolutional neural network (CNN). This widens the convolutional layer and expands pooling to reduce network training time and thickening of the image; this deepens the network and strengthens performance. Secondly, the maximum Lyapunov exponent was used as the key index for the indication of chaos. Gaussian process regression (GPR) and the back propagation neural network (BPNN) were used with different amounts of data to quickly predict the maximum Lyapunov exponent under different parameters. The main finding of this study was that chaotic behavior occurs in the robotic arm system and can be more efficiently identified by ResNet-50 than by GoogLeNet; this was especially true for Poincaré map diagnosis. The results of GPR and BPNN model training on the three types of data show that GPR had a smaller error value, and the GPR-21 × 21 model was similar to the BPNN-51 × 51 model in terms of error and determination coefficient, showing that GPR prediction was better than that of BPNN. The results of this study allow the formation of a highly accurate prediction and identification model system for nonlinear and chaotic motion in robotic arms.

Chaotic Motion in the Breathing Circle Billiard

We consider the free motion of a point particle inside a circular billiard with periodically moving boundary, with the assumption that the collisions of the particle with the boundary are elastic so that the energy of the particle is not preserved. It is known that if the motion of the boundary is regular enough then the energy is bounded due to the existence of invariant curves in the phase space. We show that it is nevertheless possible that the motion of the particle is chaotic, also under regularity assumptions for the moving boundary. More precisely, we show that there exists a class of functions describing the motion of the boundary for which the billiard map has positive topological entropy. The proof relies on variational techniques based on the Aubry–Mather theory.