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This paper conducts a comparative analysis of the global dynamics of a harmonically excited oscillator with geometrical nonlinearities. Static analysis of the oscillatory system shows that adjusting the horizontal distance ratio from 1 to 0 can lead to single, double and quadruple well configurations successively. Intra-well and inter-well resonant responses are deduced analytically. Qualitative and quantitative results both reveal that the oscillator displays the stiffness–softening characteristic in cases of double and quadruple wells and the stiffness–hardening characteristic in the case of a single well. The initial-sensitive phenomenon jump is performed via fractal basins of attraction. Complex dynamical behaviors, including higher-order periodic responses and chaos, are also exhibited. The results demonstrate that the oscillator with a double or quadruple well configuration can achieve the inter-well response with large displacement, thus confirming its desirability in engineering applications of geometrically nonlinear oscillators. [ABSTRACT FROM AUTHOR]

This study presents multistability mechanisms that can enhance the energy harvesting performance of a piezoelectric energy harvester (PEH) with geometrical nonlinearities. To configure triple potential wells, static bifurcation diagrams in the structural parameter plane are depicted. On this basis, the key structural parameter is considered, of which three reasonable values are then chosen for comparing and evaluating the performances of the triple-well PEH under them. Then, intra-well responses and the corresponding voltages of the system are investigated qualitatively. A preliminary analysis of the suitable energy-harvesting conditions is carried out, which is then validated by numerical simulations of the evolution of coexisting attractors and their basins of attraction with variations in the excitation level and frequency. It follows that, under a low-level ambient excitation, the intra-well responses around the trivial equilibrium dominate the energy-harvesting performance. When the level of the environmental excitation is very low, which one of the three values of the key structural parameter is the best for improving the performance of the PEH system depends on the range of the excitation frequency; when the excitation level increases sufficiently to induce inter-well responses, the maximum one is the best for improving the performance of the PEH. The findings provide valuable insights for researchers working in the structure optimization and practical applications of geometrically nonlinear PEHs. [ABSTRACT FROM AUTHOR]

This study's objective is an irrationally nonlinear oscillating system, whose bifurcations and consequent multi-stability under the circumstances of single potential well and double potential wells are investigated in detail to further reveal the mechanism of the transition of resonance and its utilization. First, static bifurcations of its nondimensional system are discussed. It is found that variations of two structural parameters can induce different numbers and natures of potential wells. Next, the cases of mono-potential wells and double wells are explored. The forms and stabilities of the resonant responses within each potential well and the inter-well resonant responses are discussed via different theoretical methods. The results show that the natural frequencies and trends of frequency responses in the cases of mono- and double-potential wells are totally different; as a result of the saddle-node bifurcations of resonant solutions, raising the excitation level or frequency can lead to the coexistence of bistable responses within each well and cause an inter-well periodic response. Moreover, in addition to verifying the accuracy of the theoretical prediction, numerical results considering the disturbance of initial conditions are presented to detect complicated dynamical behaviors such as jump between coexisting resonant responses, intra-well period-two responses and chaos. The results herein provide a theoretical foundation for designing and utilizing the multi-stable behaviors of irrationally nonlinear oscillators. [ABSTRACT FROM AUTHOR]

This paper focuses on global dynamic behaviors of a bistable piezoelectric cantilever energy harvester with a tip magnet and a single external permanent magnet at the near side. The initial distance between the magnetic tip mass and the external magnet is altered as a key parameter for the enhancement of the energy harvesting performance. To begin with, the dynamical model is established, and the equilibria as well as potential wells of its non-dimensional system are discussed. Three different values of the initial distance are selected to configure double potential wells. Next, the saddle-node bifurcation of periodic solutions in the neighborhood of the nontrivial equilibria is investigated via the method of multiple scales. To verify the validity of the prediction, coexisting attractors and their fractal basins of attraction are presented by employing the cell mapping approach. The best initial distance for vibration energy harvesting is determined. Then, the Melnikov method is utilized to discuss the threshold of the excitation amplitude for homoclinic bifurcation. And the triggered dynamic behaviors are depicted via numerical simulations. The results show that the increase of the excitation amplitude may lead to intra-well period-2 and period-3 attractors, inter-well periodic response, and chaos, which are advantageous for energy harvesting. This study possesses potential value in the optimization of the structural design of piezoelectric energy harvesters. [ABSTRACT FROM AUTHOR]

A dual-axis solar tracking system with a novel and simple structure was designed and constructed, as documented in this paper. The photoelectric method was utilized to perform the tracking. The solar radiation values of the designed system and a fixed panel system were theoretically estimated and compared, showing that the proposed system is more efficient in collecting solar energy than a fixed solar panel with a 30° tilted fixed surface facing south. The experimental results verified the validity of the prediction as well as the efficiency of the proposed solar tracking system. In a comparison of the data obtained from the measurements, 24.6% more energy was seen to have been obtained in the dual-axis solar tracking system compared to the fixed system. This study possesses potential value in small- and medium-sized photovoltaic applications. [ABSTRACT FROM AUTHOR]

Jump and pull-in instability are common nonlinear dynamic behaviors leading to the loss of the performance reliability and structural safety of electrostatic micro gyroscopes. To achieve a better understanding of these initial-sensitive phenomena, the dynamics of a micro gyroscope system considering the nonlinearities of the stiffness and electrostatic forces are explored from a global perspective. Static and dynamic analyses of the system are performed to estimate the threshold of the detecting voltage for static pull-in, and dynamic responses are analyzed in the driving and detecting modes for the case of primary resonance and 1:1 internal resonance. The results show that, when the driving voltage frequency is a bit higher than the natural frequency, a high amplitude of the driving AC voltage may induce the coexistence of bistable periodic responses due to saddle-node bifurcation of the periodic solution. Basins of attraction of bistable attractors provide evidence that disturbance of the initial conditions can trigger a jump between bistable attractors. Moreover, the Melnikov method is applied to discuss the condition for pull-in instability, which can be ascribed to heteroclinic bifurcation. The validity of the prediction is verified using the sequences of safe basins and unsafe zones for dynamic pull-in. It follows that pull-in instability can be caused and aggravated by the increase in the amplitude of the driving AC voltage. [ABSTRACT FROM AUTHOR]

Coexisting attractors and the consequent jump in a harmonically excited smooth and discontinuous (SD) oscillator with double potential wells are studied in detail herein. The intra-well periodic solutions in the vicinity of the nontrivial equilibria and the inter-well periodic solutions are generated theoretically. Then, their stability and conditions for local bifurcation are discussed. Furthermore, the point mapping method is utilized to depict the fractal basins of attraction of the attractors intuitively. Complex hidden attractors, such as period-3 responses and chaos, are found. It follows that jumps among multiple attractors can be easily triggered by an increase in the excitation level or a small disturbance of the initial condition. The results offer an opportunity for a more comprehensive understanding and better utilization of the multistability characteristics of the SD oscillator. [ABSTRACT FROM AUTHOR]

The investigation of global bifurcation behaviors the vibrating structures of micro-electromechanical systems (MEMS) has received substantial attention. This paper considers the vibrating system of a typical bilateral MEMS resonator containing fractional functions and multiple potential wells. By introducing new variations, the Melnikov method is applied to derive the critical conditions for global bifurcations. By engaging in the fractal erosion of safe basin to depict the phenomenon pull-in instability intuitively, the point-mapping approach is used to present numerical simulations which are in close agreement with the analytical prediction, showing the validity of the analysis. It is found that chaos and pull-in instability, two initial-sensitive phenomena of MEMS resonators, can be due to homoclinic bifurcation and heteroclinic bifurcation, respectively. On this basis, two types of delayed feedback are proposed to control the complex dynamics successively. Their control mechanisms and effect are then studied. It follows that under a positive gain coefficient, delayed position feedback and delayed velocity feedback can both reduce pull-in instability; nevertheless, to suppress chaos, only the former can be effective. The results may have some potential value in broadening the application fields of global bifurcation theory and improving the performance reliability of capacitive MEMS devices. [ABSTRACT FROM AUTHOR]

Global dynamics of a piezoelectric energy harvester with tristable potential is investigated. The dynamical model of a cantilever beam energy harvester is considered; its static bifurcation is also discussed. Multiple intra-well attractors and their basins of attraction are presented to discuss the mechanism of multistability and its initial sensitivity. Moreover, the Melnikov method is applied to present the conditions for global bifurcations and the induced complex dynamics. The results show that the variation of coefficients of the polynomial may affect the number and shapes of potential wells, while the increase of the excitation amplitude may trigger multistability around one equilibrium, initial-sensitive jump, inter-well attractor and chaos. The results may provide some theoretical reference for increasing the working performance of energy harvesters. [ABSTRACT FROM AUTHOR]

Jump and excessive motion are undesirable phenomena in relative rotation systems, causing a loss of global integrity and reliability of the systems. In this work, a typical relative rotation system is considered in which jump, excessive motion, and their suppression via delayed feedback are investigated. The Method of Multiple Scales and the Melnikov method are applied to analyze critical conditions for bi-stability and initial-sensitive excessive motion, respectively. By introducing the fractal of basins of attraction and the erosion of the safe basin to depict jump and initial-sensitive excessive motion, respectively, the point mapping approach is used to present numerical simulations which are in agreement with the theoretical prediction, showing the validity of the analysis. It is found that jump between bistable attractors can be due to saddle–node bifurcation, while initial-sensitive excessive motion can be due to heteroclinic bifurcation. Under a positive coefficient of the gain, the types of delayed feedback can both be effective in reducing jump and initial-sensitive excessive motion. The results may provide some reference for the performance improvement of rotors and main bearings. [ABSTRACT FROM AUTHOR]

The vibrating system of a class of linkage-slider structure is considered, and its initial-sensitive dynamical behaviors such as safe jump, locking instability, and chaos are studied. First, static bifurcation of the dynamical system is discussed. Then, via analyzing the effect of the external excitation on the periodic solutions under primary resonance, it is found that the change of the excitation frequency may lead to bistability and safe jump. Furthermore, it follows from the investigation of the heteroclinic bifurcation that the increase of the external excitation amplitude may lead to locking instability, chaos, and static locking. The results have some potential values in the design of geometrically nonlinear oscillators. [ABSTRACT FROM AUTHOR]

Multiple attractors and their fractal basins of attraction can lead to the loss of global stability and integrity of Micro Electro Mechanical Systems (MEMS). In this paper, multistability of a class of electrostatic bilateral capacitive micro-resonator is researched in detail. First, the dynamical model is established and made dimensionless. Second, via the perturbating method and the numerical description of basins of attraction, the multiple periodic motions under primary resonance are discussed. It is found that the variation of AC voltage can induce safe jump of the micro resonator. In addition, with the increase of the amplitude of AC voltage, hidden attractors and chaos appear. The results may have some potential value in the design of MEMS devices. [ABSTRACT FROM AUTHOR]

In the military arms race, the focus of the defender is on selecting the optimal development plan of armaments, i.e., advantageous defense resources allocation. As an indispensable feature of armaments, the cooperative interactions between different armaments influence their effectiveness directly, but they have not been elaborated yet in the existing study. In this regard, a mathematical formulation, named operation loop, is proposed to investigate the cooperative interactions between armaments, sequentially facilitating the description of the development plans’ performance. Additionally, the defender's development plans of armaments depend much on the attacker's weapons, bringing the characteristics of multiobjective into this problem. Furthermore, the uncertainty about the development plans of the attacker has been taken into consideration as well. Therefore, the paper concentrates on formulating the resources allocation problem with incomplete information as a network design problem. By optimizing the network design using covariance matrix adaptation evolution strategy, an effective armaments development plan can be obtained. The feasibility is demonstrated by a case study. [ABSTRACT FROM AUTHOR]

Pull-in instability of the electrostatic microstructures is a common undesirable phenomenon which implies the loss of reliability of micro-electromechanical systems. Therefore, it is necessary to understand its mechanism and then reduce the phenomenon. In this work, pull-in instability of a typical electrostatic MEMS resonator is discussed in detail. Delayed position feedback and delayed velocity feedback are introduced to suppress pull-in instability, respectively. The thresholds of AC voltage for pull-in instability in the initial system and the controlled systems are obtained analytically by the Melnikov method. The theoretical predictions are in good agreement with the numerical results. It follows that pull-in instability of the MEMS resonator can be ascribed to the homoclinic bifurcation inducing by the AC and DC load. Furthermore, it is found that the controllers are both good strategies to reduce pull-in instability when their gains are positive. The delayed position feedback controller can work well only when the delay is very short and AC voltage is low, while the delayed velocity feedback will be effective under a much higher AC voltage and a wider delay range. [ABSTRACT FROM AUTHOR]