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In view of their unique shape morphing behaviour, dielectric elastomer-based minimum energy structures (DEMES) have received an increasing attention in the technology of electroactive soft transduction. Because several of them undergo a time-dependent motion during their operation, understanding their nonlinear dynamic behaviour is crucial to their effective design. Additionally, in the recent past, there has been a growing scientific interest in imparting anisotropy to the material behaviour of dielectric elastomers in view of ameliorating their actuation performance. Spurred with these ongoing efforts, this paper presents an analytical framework for investigating the nonlinear dynamic behaviour of aniso-visco-hyperelastic DEMES actuator with an elementary rectangular geometry. We use a rheological model comprising two Maxwell elements connected in parallel with two single spring elements for modelling the material behaviour of the DE membrane. The governing equations of motion for the underlying non-conservative system are then derived using the Euler–Lagrange equation. The proposed model is used for building insights into the attainable equilibrium states, periodicity of the response as well as the resonant behaviour of the DEMES actuator over a feasible range of anisotropy and viscosity parameters. Our results reveal that the DEMES with hyperelastic material properties exhibits a supercritical pitchfork bifurcation of equilibrium state which is further accelerated in terms of attained equilibrium angle due to membrane anisotropy. A significant enhancement in the equilibrium angle attained by the structure with the extent of membrane anisotropy parameter is observed, indicating a favourable impact of material anisotropy. Poincare maps and phase-portraits are presented for assessing the periodicity of the nonlinear oscillations. The frequency response of the actuator for a combined DC and AC load indicates an upsurge in the resonant frequency with an increase in anisotropy parameter. The underlying analytical model and the trends presented in this study can find their potential use in the design and development of the futuristic anisotropic DEMES actuators subjected to time-dependent actuation.

This paper proposes a method for detecting weak harmonic signals in power systems based on differential Duffing oscillator arrays. Firstly, a differential Duffing oscillator array is constructed, a differential Poincaré map is established, and the time series of the differential Poincaré map is converted into a pulse sequence. Secondly, two Duffing oscillators that exhibit obvious intermittent chaotic states are identified according to the number of pulses and pulse variance. Finally, the frequency of the harmonic signal is calculated based on the frequencies of these two intermittently chaotic Duffing oscillators, thereby detecting the harmonic signal with the corresponding frequency. Simulation results demonstrate that the proposed method can detect weak harmonic signals submerged in strong noise environments and accurately calculate the frequency of the measured harmonic signals, with the minimum detectable signal-to-noise ratio as low as −24.95dB.

Despite its importance and widespread applications, the use of the Poincare map has remained in its rudimentary stages since its proposition in the nineteenth century and there exists no systematic method to effectively obtain Poincare sections. Additionally, and due to its graphical structure, it has previously been very arduous to utilize Poincare maps for high dimensional systems, and two- and three-dimensional systems remain as its sole area of applicability. In this study, a novel systematic geometrical-statistical approach is proposed that is capable of obtaining the effective Poincare sections regardless of the attractor’s complexity and provides insight into the entirety of the attractor’s structure. The presented algorithm requires no prior knowledge of the attractor’s dynamics or geometry and can be employed without any involvement with the governing dynamical equations. Several classical systems such as the Van der Pol, Lorenz, and Rossler’s attractor are examined via the proposed algorithm and the results are presented and analyzed.

This work aims to investigate the stability, bifurcation, and vibration control of a discontinuous dynamical model simulating the nonlinear oscillation of a horizontally suspended nonlinear rotor system. A novel Proportional-Integral-Resonance-Control (PIRC) algorithm is introduced to dampen the rotor's vibrations. An 8-pole electromagnetic bearing is employed as an active actuator through which the PIRC control signals are applied in the form of eight electrical currents. These currents, in turn, generate controllable attractive forces that counteract the rotor's vibrations. Based on the presented control structure, the mathematical model of the closed-loop system, including the magneto-electromechanical coupling and rotor-actuator rub-impact force, is derived as a 2-DOF discontinuous dynamical system connected to two first-order filters. A closed-form solution using the multiple scales analysis (up to the second-order approximation) of the system model was obtained. Additionally, the autonomous dynamical system governing the evolution of oscillation amplitudes and modified phases was derived when the rub-impact was neglected. Based on this derived autonomous system, the bifurcation characteristics, vibration control, and system stability have been explored. The analytical results obtained are utilized as a heuristic tool to report the conditions under which the system will experience rub and/or impact forces. Then, the entire discontinuous model was numerically analyzed using bifurcation diagrams, zero–one chaotic test, poincaré maps, orbital plots, and instantaneous radial and lateral oscillations at the rub and/or impact conditions reported analytically. The principal findings demonstrate that the introduced analytical investigation can be used to predict accurately when the rotor-actuator will be subject to a rub-impact force. Furthermore, it is proven analytically that the system's damping coefficients are proportional to the cartesian product of the controller's feedback and control gains. Moreover, the numerical analysis also illustrates that the presence of a rotor-actuator rub-impact can lead to period-1, period-2, period-3, period-4, or quasiperiodic oscillations, depending on the rotation speed and/or disc eccentricity. Finally, it is proven that optimizing the controller gains can prevent the rotor-actuator rub-impact effect regardless of the rotor speed and disc eccentricity.

A quasi-zero stiffness vibration isolator (QZSVI) is used in applications like precision instruments, aerospace, microelectronics manufacturing, and seismic isolation to protect sensitive equipment from low-frequency vibrations. Their key advantage lies in achieving near-zero stiffness, allowing for highly effective vibration attenuation while maintaining system stability. These passive systems are cost-effective and reliable, offering superior vibration isolation without the need for external power or active control. This work proposes the use of negative displacement, velocity, and cubic velocity feedback control techniques to enhance the QZSVI’s isolation performance. We found that the composite negative velocity and cubic velocity control (NVFC + NCVFC) is more effective with low cost compared to other types of controller (its effectiveness is about 94.8%). The approximate solutions (AS) of the controlling system of equations of motion (EOM) are acquired using a multiple-scales procedure (MSP) up to the second order, and it is subsequently validated numerically through the Runge–Kutta method (RKM) from the fourth-order. Modulation equations (ME) are obtained by exploring resonance instances and solvability conditions. Time history graphs and frequency response curves, generated via MATLAB and Wolfram Mathematica 13.2, are presented to analyze stability and steady-state solutions. It is investigated how altering the parameters affects the system amplitude. Poincaré maps, Lyapunov exponent spectra (LEs), and bifurcation diagrams are presented to illustrate the system’s diverse behavior patterns. Furthermore, the transmissibility of force, displacement, and acceleration is computed and displayed. A QZSVI minimizes low-frequency vibrations, making it ideal for precision applications in metrology, automotive, aerospace, civil engineering, medical equipment, and renewable energy. It achieves superior damping, ensuring high stability and precision.

Purpose The present work investigates and analyzes the mathematical modeling of a dynamical system attached to a piezoelectric device. It is well-established that piezoelectric transducers are effective energy harvesting devices commonly utilized in practical applications with mechanical systems. The structure of the dynamical model contains a damped Duffing oscillator acting as the major component, which is connected to an un-stretched pendulum and at the same time, to the piezoelectric harvester. Method Lagrange's equations are employed to deduce the governing equations of motion (EOM) based on the overall generalized coordinates characterizing the system. This model has been solved analytically using a perturbation technique known multiple-scales (MS) up to the third approximation. This indicates a level of complexity and detail in the model’s analysis. Moreover, the obtained solutions are compared with the numerical ones for more transparency and to highlight the accuracy of the approximate solutions. Results Detailed graphical figures have been executed to study the nonlinear stability analysis of the equations of modulation. Phase portrait diagrams, bifurcation ones, and spectrums of Lyapunov are exhibited to illustrate various types of systems’ behavior, complemented by Poincaré maps for further insight. Additionally, the varied ranges of the stabilities are explored and discussed. Applications The mechanical vibrations are converted to electricity due to the existence of the piezoelectric transducer that is connected to the dynamic model, which has wide uses and applications like crystal oscillators, medical ultrasound applications, gas igniters, and displacement transducers.

The van der Pol–Mathieu–Duffing oscillator (VMDO) finds applications across diverse fields due to its ability to model complex dynamic behaviors. Oscillators are versatile tools used in fields such as mechanics, biology, and electrical engineering. Its ability to model complex behaviors makes it an important subject of study for researchers and engineers studying nonlinear dynamic systems. A parametric forcing excited the VMDO under feedback control is investigated and discussed in the most severe resonance scenario. The multiple-scales-strategy (MSS) is utilized to obtain the approximate solution (AS). Furthermore, the AS is validated against the numerical solution (NS) obtained using the Runge–Kutta of fourth-order (RK-4) method. A negative-velocity-feedback (NVF) and negative-cubic-velocity-feedback (NCVF) controllers are integrated into the primary system to mitigate unwanted vibrations, which can significantly impact the system’s efficiency, particularly under resonance conditions. The stability analysis is thoroughly examined, and optimal feedback gains are selected to suppress amplitude peaks. A range of response curves is presented to illustrate and compare the effectiveness of the controllers. Bifurcation diagrams and Poincaré maps (PM) are utilized to investigate the system’s diverse motions, providing valuable insights into its complex behavior and the variations it undergoes under different conditions. The results are explained through the displayed curves and provide insights into the system’s dynamics. The VMDO is a versatile model in various scientific and engineering disciplines. Its ability to embody nonlinear dynamics makes it invaluable for studying the stability, resonance, and chaotic behavior of complex systems. As research continues, its applications may expand into new areas of technology.

This study introduces a novel approach to analyzing a four-degree-of-freedom (DoF) nonlinear system by leveraging advanced numerical and analytical techniques to comprehensively examine its dynamic behavior. The system’s nonlinear differential equations (DEs) are obtained through the application of Lagrange’s equations (LE). The solutions are obtained using the fourth-order Runge–Kutta method (4-RKM). The investigation involves analyzing the relationships between the angular solutions and their corresponding first-order derivatives, commonly referred to as phase plane analysis. The study aims to examine bifurcation diagrams and Lyapunov exponent spectra to reveal the various modes of motion within the system and visualize Poincaré maps. These tools are used to analyze a unique system configuration. Lastly, the conditions for solvability and the characteristic exponents are identified by examining resonance scenarios. The examination of resonance scenarios through characteristic exponents and solvability conditions, coupled with the application of Routh-Hurwitz criteria (RHC) for stability evaluation, provides an innovative framework for understanding frequency response and nonlinear stability across stable and unstable ranges. By exploring both theoretical and practical aspects of vibrational dynamics in applications like aviation, robotics, and underwater exploration, this work offers a significant advancement in analyzing complex systems, with wide-ranging implications for various engineering fields, including aerospace, structural mechanics, and energy harvesting.

In this article, the reduced differential transform method is introduced to solve the nonlinear fractional model of Tumor-Immune. The fractional derivatives are described in the Caputo sense. The solutions derived using this method are easy and very accurate. The model is given by its signal flow diagram. Moreover, a simulation of the system by the Simulink of MATLAB is given. The disease-free equilibrium and stability of the equilibrium point are calculated. Formulation of a fractional optimal control for the cancer model is calculated. In addition, to control the system, we propose a novel modification of its model. This modification is based on converting the model to a memristive one, which is a first time in the literature that such idea is used to control this type of diseases. Also, we study the system’s stability via the Lyapunov exponents and Poincare maps before and after control. Fractional order differential equations (FDEs) are commonly utilized to model systems that have memory, and exist in several physical phenomena, models in thermoelasticity field, and biological paradigms. FDEs have been utilized to model the realistic biphasic decline manner of elastic systems and infection of diseases with a slower rate of change. FDEs are more useful than integer-order in modeling sophisticated models that contain physical phenomena.

The occurrence of chaotic motion in a railway instability phenomenon known as wheel squeal is investigated. The analysis is motivated and applied to predicting the large amplitude friction excited oscillations of the coupled wheel and rail motion. The equations of motion reduce to two autonomous coupled nonlinear second-order systems. Instabilities of the wheel and rail motions are shown to be due to the friction coupling which at low amplitudes causes limit cycle behaviour via a Hopf bifurcation. When the amplitude grows large enough, full nonlinear creep oscillations are shown to occur causing oscillations about positive and negative sliding conditions. Chaos is shown to occur when the motion meanders and jumps between large positive and negative creep and is characterised by a Poincare map with fractal nature. The route to chaos is shown to be via quasiperiodicity. Based on this insight, necessary analytical conditions for wheel squeal chaos are developed and verified using numerical simulations over a range of angles of attack and wheel/rail contact angles. Conditions under which chaotic instability is more likely to occur are identified and discussed, including mode coupling and parametric excitation. The results may describe why some very loud occurrences of wheel squeal are not characterised by a pure tone.