The Van der Pol oscillator is well known for generating a self-sustained limit cycle in its unforced state; however, when an external periodic force is applied, its dynamical responses may deviate sharply away from this natural rhythm. The existing Lindstedt-Poincaré method becomes cumbersome in the presence of strong nonlinearity, damping, and external forces. To overcome this limitation, an alternative modified Lindstedt-Poincaré method is introduced to obtain the steady-state response of the Duffing-Van der Pol oscillator. Excellent accuracy is achieved by comparing the approximate results with those obtained using the Runge-Kutta fourth-order (RK4) method and the harmonic balance method (HBM). The proposed modified Lindstedt-Poincaré method offers a simple solution procedure and demonstrates broad applicability to a wide range of nonlinear oscillatory problems in applied sciences and engineering.
Ultra-low frequency vibration is crucial for high-end precision instruments and manufacturing equipment. Twisted and coiled polymer actuators (TCPAs) have the advantages of large driving force, high driving accuracy and excellent flexibility, making them highly promising for ultra-low frequency vibration control. This paper establishes a history-dependent forward and inverse neural-network modeling framework for TCPA force regulation and flexible-structure vibration control. The available experimental data are used to train and validate the model, while additional numerical simulations are introduced to evaluate cross-condition and frequency-domain performance. The plant model and the inverse model of the neural network are then established, after which the open-loop vibration control of a flexible cantilever beam is investigated. The combined numerical and existing experimental results show that the proposed model-based open-loop control can reduce the low-frequency vibration of the flexible beam within the evaluated range, providing a basis for further TCPA-based vibration-control studies.
This paper presents a systematic decoupling framework for strongly coupled two-degree-of-freedom (2-DOF) nonlinear vibration systems. Here, the coupling is classified as strong rather than weak because the cross-mode interaction terms in the governing equations (e.g., the x 2 y-, xy 2 -, and xy-type terms), enter with coefficients of the same order as the leading cubic stiffness nonlinearities, with no small bookkeeping parameter scaling them down as would be assumed in a weakly coupled (perturbative)formulation. The proposed approach assigns each mode its own independently stretched time domain, ξ = Ω 1 t and ζ = Ω 2 t, through a two-time-domain Lindstedt-Poincaré-type transformation. Combined with mode-specific weighted averaging, this decouples the governing equations into two analytically tractable subsystems, each carrying its own closed-form, amplitude-dependent frequency. The methodology extends El-Dib’s frequency formula to accommodate two-term coupled configurations, producing closed-form frequency-amplitude relationships that accurately characterize the nonlinear system dynamics. Analytical solutions exhibit close agreement with direct numerical simulations in terms of amplitude, phase, and oscillatory envelope; residual discrepancies are attributed solely to higher-order truncation effects. The analysis demonstrates that both inter-mode coupling and modal damping significantly influence the energy exchange between vibrational modes and the overall stability of the system. The proposed methodology is directly applicable to several nonlinear problems arising in fluid structure interaction, rotating machinery, MEMS resonators, and magnetic fluid interfaces, thereby providing a rigorous analytical platform that bridges nonlinear dynamical theory and quantitative engineering predictions.
This study develops classical and fractional variational models for a simple pendulum whose pivot undergoes prescribed vertical harmonic motion. The nonlinear classical equation of motion and its small-angle, Mathieu-type reduction are first derived from the Euler-Lagrange equation. Memory is then introduced through a dimension-preserving Caputo-type operator of order 0 < α ≤ 1 , and the corresponding left-right fractional Euler-Lagrange equation is obtained under the adopted endpoint convention. The small-angle, linearized fractional problem is recast as a coupled system consisting of a left-sided state equation and a right-sided auxiliary equation. Numerical solutions are computed over the finite interval 0 ≤ t ≤ 5 using an L1 discretization of the left- and right-sided Caputo derivatives. The discretization produces a coupled sparse linear boundary-value system that is solved directly for several values of α and the excitation frequency W . The results demonstrate that the finite-time response depends strongly on both parameters. For the selected conditions, the cases W < ω 0 , W = ω 0 , and W > ω 0 produce quantitatively and qualitatively different trajectories. Maximum amplification, RMS response, deviation from the corresponding α = 1 solution, and a Hamiltonian-like diagnostic are used to supplement the time histories and parametric trajectories. In addition, as α → 1 , the fractional equations and their numerical solutions approach the corresponding integer-order boundary-value problem. The formulation, therefore, provides a reproducible framework for studying the interaction between hereditary effects and prescribed base excitation.
This study investigates the (3+1)-dimensional extended Kadomtsev–Petviashvili equation using a symbolic–neural hybrid framework. First, the equation is transformed into its bilinear form, and the bilinear neural network method is employed to construct various analytical wave solutions, including bright, dark, periodic, and interaction waves. These solutions reveal the rich nonlinear behavior of the model and are illustrated through representative graphical visualizations. To further examine the dynamics of the obtained solutions, the governing equation is reduced to a nonlinear ordinary differential system via a wave transformation. The reduced system is analyzed using phase portraits, sensitivity analysis, and Lyapunov exponents to characterize its dynamical behavior. The results demonstrate transitions between stable, periodic, and chaotic states under different initial conditions and parameter perturbations. The proposed framework provides an effective approach for constructing analytical solutions and investigating their dynamical properties, offering new insights into higher-dimensional nonlinear wave phenomena.
A unified framework for analyzing nonlinear vibrations in a two-degree-of-freedom (2DOF) system comprising an inverted pendulum (IP) mounted on a cart is proposed. Unlike existing approaches that rely on direct numerical integration or restrictive small-angle assumptions, this work introduces a novel combination of the Optimized Equivalent Linearization Method (OELM) and El-Dib’s frequency formula to systematically transform the non-autonomous governing equations into an autonomous form via an equivalent periodic forcing function, a treatment not previously applied to this class of coupled nonlinear oscillators. The linearized equations are examined under both common-frequency and two-frequency response regimes, yielding closed-form stability criteria that show strong agreement with full numerical solutions. Key factors governing stability are the applied force amplitude, vibration amplitude, and friction coefficient; increasing friction promotes instability while higher vibration amplitudes enhance dynamic stability, a counterintuitive finding with direct design implications. The framework is validated through comprehensive time-history analysis, demonstrating minimal errors and sustained periodic behavior in the absence of chaotic motion. The results apply to a broad class of engineering systems. The inverted pendulum on a cart serves as a canonical model for self-balancing robots, Segway-type vehicles, and bipedal locomotion systems, where maintaining upright stability under external excitation is critical. The findings also inform the design of vibration isolation platforms, active suspension systems, and offshore structures subjected to wave-induced loading, where controlled nonlinear oscillations must be managed to prevent structural failure. The stability maps generated by this framework offer practical design guidelines for tuning damping and forcing parameters in such systems. The study thereby broadens the analytical treatment of nonlinear oscillatory systems and provides concrete guidance for engineers in robotics, structural dynamics, and mechanical control.
The article presents the use of selected machine learning algorithms for the detection, classification and assessment of the degree of wear of the main components of an axial multi-piston positive displacement pump. The first part of the article describes current work on diagnostic systems for positive displacement pumps. The authors then present their own diagnostic tests of the object (multi-piston pump), equipped with a damage models of its main components. Based on the obtained signal matrices, the statistical characteristics of the signals are calculated. The next step is to use available machine learning classifiers to detect the type of damaged pump component and assess its degree of wear. Then, using Matlab programme, models are developed to classify the wear condition of the pump. Using the adopted criteria, their effectiveness in recognising the modelled pump damage is assessed.
Composite materials are widely used in aerospace, automotive, and marine engineering because of their high strength-to-weight ratio and design flexibility. However, their long-term reliability is limited by fatigue damage, which develops gradually through matrix cracking, delamination, and fiber fracture under cyclic loading. Conventional fatigue-life prediction methods often fail to capture early degradation and typically require extensive experimental calibration. Vibration-based Structural Health Monitoring (SHM) offers a non-destructive alternative by exploiting the direct relationship between stiffness degradation and changes in dynamic properties such as natural frequencies, damping ratios, and mode shapes. This review synthesizes current research linking vibration response to fatigue damage in composite structures. Experimental methods, sensing technologies, and computational modeling approaches are examined within a unified framework. Particular attention is given to sandwich and auxetic configurations, environmental influences, and design optimization for improved fatigue resistance. The study also discusses emerging data-driven paradigms, including physics-guided machine learning and digital-twin concepts, which enable predictive maintenance rather than post-damage detection. By integrating mechanics-based understanding with monitoring and data analytics, this review provides a structured perspective on vibration-based integrity assessment and outlines research directions toward reliable in-service monitoring and extended operational life of advanced composite structures.
Despite the extensive literature on nonlinear pendulum dynamics, the analytical treatment of strongly coupled multi-degree-of-freedom systems remains a significant challenge, as classical linearization methods neglect the nonlinear interactions among coupled coordinates and thus fail to capture essential mode-coupling effects. To address this limitation, the present study develops an Extended Equivalent Linearization Approach (EELA) for a planar two-degree-of-freedom double pendulum, incorporating amplitude-dependent weighting functions that explicitly account for the coupling between the two oscillators. In parallel, a coupled Lindstedt-Poincaré (two-time domains) transformation is introduced, assigning independent strained time variables to each degree of freedom and reformulating the governing nonlinear equations into a unified coordinate framework. The combined analytical strategy yields closed-form amplitude-dependent frequency expressions and an explicit stability criterion showing that the stability boundaries are governed exclusively by the initial amplitude ratio and the mass ratio, independently of the individual link lengths. Validation against direct numerical integration confirms that the analytical solutions remain accurate for moderate-to-large oscillation amplitudes, well beyond the small-angle regime, with maximum absolute errors strictly bounded and free of secular growth throughout the considered time domain.
In this work, the propagation of kinetic Alfvén solitary waves (KASWs) and kinetic Alfvén cnoidal waves (KACWs) is investigated in a strongly magnetized, low- β electron-ion plasma with ion pressure anisotropy and superthermal electrons, motivated by conditions in Saturn’s magnetospheric environment. Starting from a two-fluid description with a κ -distributed electron population and Chew-Goldberger-Low (CGL)-type ion pressure anisotropy, the reductive perturbation method is used to derive a planar Korteweg-de Vries (KdV) equation governing small-amplitude kinetic Alfvén structures. The model is then generalized to a time-fractional KdV equation by replacing the first-order temporal derivative with a Caputo fractional derivative to incorporate temporal nonlocality and memory effects in the plasma response. The resulting fractional evolution problem is treated using the Tantawy technique, which yields rapidly convergent analytical approximations for both fractional KASWs and fractional KACWs. The analysis shows that, for the low- β anisotropic conditions considered, only compressive kinetic Alfvén structures are supported in the sub-Alfvénic regime, and that superthermality, ion pressure anisotropy, plasma beta, and propagation obliqueness jointly control the amplitudes and widths of the solitary and cnoidal structures. Increasing the superthermal index reduces the deviation from Maxwellian behavior and produces stronger, broader compressive pulses, while larger ion pressure anisotropy and plasma beta mainly broaden the profiles with only minor changes in amplitude. The fractional order modulates the temporal development and localization of the nonlinear structures: reducing the order weakens the peak amplitude and slows the formation of sharp kinetic Alfvén fronts, reflecting the enhanced memory inherent to fractional dynamics. These results may assist in the interpretation of kinetic Alfvén activity in low- β space and astrophysical plasmas with ion pressure anisotropy and superthermal particles, such as Saturn’s magnetosphere and the near-Earth magnetosheath.
Off-road vehicle drivers and passengers experience severe discomfort due to uneven road surfaces. In order to lessen this effect, the suspension system—which permits relative motion between the wheels and the vehicle frame—is essential. The suspension system guarantees road grip, handling stability, and efficient shock absorption in addition to improving ride comfort. In order to attain vibration stability under road disturbances like bumps, this study develops a small model of an off-road vehicle and designs appropriate controllers. With Integral Square Error (ISE), Integral Absolute Error (IAE), and Integral Time Absolute Error (ITAE) as objective functions, two hybrid control strategies are used: a Hybrid Ant Colony Optimization–Pattern Search (ACO-PS) algorithm-based PID controller and a Hybrid Model Predictive Control–Proportional Integral Derivative (MPC-PID) controller. Error indices, settling time, and peak overshoot are used to assess the system’s performance. In comparison to the Hybrid MPC-PID controller, the suggested Hybrid ACO-PS–based PID controller using ISE as the goal function provides a 3.3% faster settling time and 95% less vibration. In comparison to current controller designs, the suggested controller performs better overall.
Integrating renewable energy sources (RESs) into isolated microgrids introduces major load frequency control (LFC) challenges due to reduced system inertia, nonlinear dynamics, stochastic power fluctuations, and communication delays. Conventional PID-based controllers often exhibit limited robustness and poor adaptability under rapidly varying operating conditions. Although fuzzy logic controllers (FLCs) improve disturbance handling and adaptive response, they may suffer from limited tuning precision, while standalone fractional-order controllers (FOCs) usually involve complex parameter adjustment and reduced flexibility under highly dynamic disturbances. To overcome these limitations, this study proposes a novel hybrid LFC framework that combines an FLC with a fractional-order TIλDμ controller, whose parameters are optimally tuned using the Harris hawk’s optimization (HHO). The proposed hybrid structure exploits the adaptive decision-making capability of FLC together with the superior damping characteristics and dynamic flexibility of FOC. In addition, a hydrogen fuel cell (HFC) is integrated as a dynamically controlled energy storage unit to provide fast corrective power support during transient disturbances. The proposed controller adapts in real time to system nonlinearities, fluctuating renewable generation, communication delays, and varying load conditions without relying on highly detailed mathematical models. Its performance was evaluated under seven operating scenarios, including single-step load changes, sequential disturbances, random load variations, cyclic oscillations, solar intermittency, and delayed response conditions. Comparative results demonstrate that the proposed HHO-optimized fuzzy fractional controller consistently outperforms PID, fuzzy-based, and FO-(PD–PI) controllers in terms of overshoot reduction, settling time, steady-state accuracy, and damping performance. Under single-step disturbance conditions, the proposed controller achieved the lowest overshoot of 0.0012 pu and the minimum ITAE value of 0.0043 pu·s, while maintaining stable and well-damped responses under solar intermittency and communication delay scenarios. These results confirm the robustness, adaptability, and effectiveness of the proposed control strategy.
The nonlinear oscillations frequently occur in real-world challenges, such as mechanical structures and biological cycles, and can give rise to complex behaviour like bifurcations and chaos that are vital in understanding and predicting dynamical systems. Advances in analytical and numerical methods have enabled novel applications in fields including engineering, medicine, and materials science. We analyse oscillators with pronounced nonlinear features, incorporating both damping and restoring forces, using a blend of theoretical and computational approaches. Two examples are presented from diverse scientific and technical areas. The innovative methodology described significantly reduces computation time and resources when compared to conventional perturbation methods widely used in this field. Based on He's frequency formula, the proposed non-perturbative approach transforms weakly nonlinear oscillator of ordinary differential equation into linear one, allowing us to establish a new frequency corresponding to the linearized one. The theoretical outcomes are validated via numerical simulations using Mathematica Software, with results revealing excellent regularity between the two ordinary differential equations. An inclusive investigation of system stability can be conducted with this approach, expanding the capabilities beyond those available with previous techniques. Consequently, the non-perturbative approach offers a more practical and reliable framework via numerical solutions of weakly nonlinear oscillators. Moreover, it stands out as a flexible tool for use in applied research and engineering due to its flexibility to a variety of nonlinear situations. We also examine the impact of different parameters on stability, with results approving the approach's simplicity, efficiency, and reliability. Adjusting bifurcation parameters alters the structure of bifurcation diagrams and their associated Poincar & eacute; maps, and we further illustrate these effects by mapping the Lyapunov exponent curves.
To address the difficulty of suppressing low-frequency line spectra in underwater vehicle radiated noise and the narrow operating bandwidth of conventional linear vibration absorbers with fixed tuning frequencies, this paper proposes a flexible hinge-type nonlinear energy sink (NES). First, a double-slotted flexible hinge-type NES structure is presented. Static mechanical analysis shows that the structure exhibits cubic stiffness characteristics near the static equilibrium position. Second, the IHB method is employed to characterize the periodic responses of the NES system. The effects of NES mass, stiffness, and damping on the amplitude-frequency response and vibration suppression performance are then investigated in detail. Subsequently, local optimization is performed to determine the optimal damping and stiffness parameters of the NES. The optimized NES shows strong robustness against variations in excitation frequency and amplitude. Finally, a prototype of the flexible hinge-type NES is fabricated, and a vibration test platform is established for experimental validation. Both simulation and experimental results show that the NES reduces the first-order resonance peak near 5.00 Hz by approximately 8.17 dB, decreases the average line-spectrum level over 4.43-5.82 Hz by about 3.74 dB, and broadens the vibration absorption bandwidth by approximately 27.8%. Compared with an equivalent linear dynamic vibration absorber, the NES provides better broadband vibration absorption performance. In addition, the NES can shift the resonance frequency of the coupled system and induce a strong modulation response.
Ensuring the long-term safety and durability of civil infrastructure increasingly depends on data-driven structural health monitoring systems capable of objectively tracking changes in structural performance. However, their reliability is undermined by the evolving environmental conditions associated with climate change. Variations in temperature and other environmental factors can modify the dynamic response of a target structure, producing apparent changes in damage-sensitive features that may be misinterpreted as degradation. Therefore, distinguishing environmental influences from genuine structural deterioration is essential for a reliable assessment. This study presents an integrated framework for resilient structural monitoring that separates climate-induced variability from damage effects. The methodology combines automated operational modal analysis with Gaussian process regression to model the nonlinear dependence of natural frequencies on temperature and quantify uncertainty. Climate projections from the CH2018 model under Representative Concentration Pathway scenarios are incorporated to forecast the long-term evolution of modal properties. The approach is demonstrated on the case study of the Chillon Viaduct in Switzerland, where continuous monitoring revealed a consistent reduction in natural frequencies with increasing temperature, reflecting temperature-dependent material, revealing a consistent reduction in natural frequencies with increasing temperature, reflecting a combination of thermally driven mechanisms, including material property changes and structural boundary variations. Sparse Gaussian process models enabled efficient long-term forecasting of frequency variations while maintaining predictive accuracy. Cointegration analysis was used to remove temperature-driven trends, isolating residuals sensitive to structural damage. Simulated progressive-damage scenarios confirmed the method’s capacity to detect incipient deterioration under varying environmental conditions. Overall, this work establishes a robust and generalizable framework that integrates modal analysis, probabilistic modeling, and climate projections to improve the resilience, interpretability, and long-term applicability of structural monitoring systems.
This study investigates the nonlinear dynamic (ND) behavior of obliquely stiffened porous functionally graded (PFG) doubly curved shallow shells (DCSSs) exposed to external excitation, with particular emphasis on primary resonance (PR) and 1:2 internal resonance (IR). Both the shell and the oblique stiffeners (OSs) are made of PFG materials. Two PFG layouts are examined: (1) shells graded from ceramic-rich at the surfaces toward metal-rich, with uniform porosity through the thickness; and (2) the same grading is used, but the porosity varies through the thickness. In both layouts, the stiffeners are functionally graded (FG) members with uniform porosity through their thickness. A mechanical model is developed for PFG shells reinforced by two crossed families of OSs, where the stiffener angles can be identical or different. Based on first-order shear deformation theory (FSDT), von K & aacute;rm & aacute;n kinematics, and Hooke's law, the stress-strain relationships are established, and the governing partial differential equations are obtained via Hamilton's principle. These equations are then converted into a two-degree-of-freedom nonlinear ordinary differential system using the Galerkin method. After that, the method of multiple scales (MMSs) is used to derive a four-dimensional system of nonlinear averaged equations. The MMS-based analytical formulation is therefore established within a weakly nonlinear framework, in which small-amplitude responses, weak external excitation, and light damping are assumed. Finally, numerical simulations are conducted to explore key response features, such as phase portraits, Poincar & eacute; sections, and time histories, showing how material gradation patterns and stiffener orientation affect the shells' nonlinear dynamics.
The increasing levels of air pollutants produced by human activities create a demand for efficient gas detectors, among which passive acoustic sensors are one of the most promising technologies. This work contributes to the field by developing a gas sensor based on a finite-size phononic crystal with a defect. Unit cells are made of dual Helmholtz resonators attached to a rectangular waveguide. The defect is introduced as a spacer between two periodic arrays of resonators and forms a one-dimensional cavity defined by Bragg reflectors. As a result, the structure’s transmission spectrum exhibits a narrow resonant peak within the stop band, and its spectral characteristics are susceptible to the acoustic properties of the medium filling the structure. The sensing response is analyzed using the transfer matrix method and validated by the finite element method, with thermoviscous losses taken into account to assess realistic sensor performance. Several hazardous gases, including carbon dioxide and ammonia, are considered as test samples. For the considered configurations, the estimated sensitivity is 1.38 Hz.s/m. The limit of detection is associated with the minimum detectable sound-speed variation, 0.04 m/s, for the defect mode resonance, depending on the gas and the length of the spacer. This enables the detection of small concentration changes in gas mixtures.
In this study, we investigate the stochastic Kakutani-Matsuuchi model (SKMM) with multiplicative noise in the It & ocirc; sense, which describes the propagation of internal gravity waves in stratified fluids such as the Earth's atmosphere and ocean. These waves, generated by density or temperature variations, play a fundamental role in transferring energy and momentum across the system. The multiplicative noise term accounts for random fluctuations whose intensity depends on wave amplitude, thereby providing a realistic description of noise-wave interactions in geophysical environments, while the It & ocirc; framework ensures a rigorous mathematical treatment of such randomness and its cumulative effect on system evolution. By applying the Sub-ODE method, we derive a broad spectrum of exact analytical solutions, including bright soliton, periodic wave, rational-type, hyperbolic-type, and singular structures. Their geometrical characteristics are explored through 3D graphical representations obtained under different values of the random noise parameter, which reveal distinctive behaviors such as localization, periodic modulation, algebraic decay, and blow-up dynamics. These findings deepen the understanding of nonlinear wave phenomena governed by the SKMM and demonstrate the versatility of the Sub-ODE approach in capturing the impact of stochastic influences. The results are expected to provide a valuable reference for modeling wave propagation in oceanic and atmospheric systems where stochastic effects cannot be ignored.
In this article, we investigate a generalized Fornberg-Whitham (gFW) equation. Based on the Lax pair and Darboux transformation, one-soliton solution, breather solutions, lump solution, lump-1-strip, lump-periodic, manifold periodic, and rogue wave solutions for the gFW equation are obtained by choosing appropriate transformations and symbolic computation. In addition, we evaluate Ma breather, Kuznetsov-Ma breather and their corresponding rogue waves, generalized breather, and Akhmediev breathers. The obtained solutions reveal rich nonlinear wave phenomena such as localized structures, recurrence patterns, and energy concentration effects. These findings have direct applications in fluid dynamics, optical fiber communications, and plasma physics, where such nonlinear behaviors are prevalent. Moreover, the explicit construction of rogue waves and breather interactions provides insight into the prediction and control of extreme wave events in oceanography and nonlinear optics. The results further demonstrate the utility of analytical methods in exploring complex wave structures in integrable and near-integrable systems.
The nonlinear Mathieu-Duffing oscillator is a basic model of parametrically driven dynamical systems with nonlinear coefficients, and it serves as an important model for the study of such phenomena as bifurcation, chaotic transitions and resonance instabilities in engineering structures and physical systems subjected to periodic external excitations. In this work, a non-perturbative analytical approach is proposed to analyse a coupled parametric nonlinear oscillatory system and to determine its dynamical features with much more computing efficiency. The proposed methodology is based on He’s frequency formulation, which systematically linearises weakly nonlinear ordinary differential oscillators, avoiding the fundamental limitations of classical perturbation methods and being effective for large amplitude nonlinear oscillatory regimes. A prominent contribution of this work is the extension of the non-perturbative framework to the consideration of linked oscillator systems, which is a unique application domain for the method. The analytical conclusions are validated by symbolic computation in Mathematica. It is found that there is strong conformity with the original governing equations for different parametric configurations. The stability study is conducted under different operating situations, and it always shows the accuracy, analytical tractability, and numerical accuracy of the procedure. The global dynamical behavior of the system is also characterized by bifurcation diagrams, which identify critical parameter-dependent transitions, and the largest Lyapunov exponent, which quantitatively distinguishes between periodic and chaotic oscillatory regimes and defines the threshold conditions for the onset of chaos. The results agree with the conclusion that the non-perturbative technique is a powerful, and efficient for the analysis of parametrically stimulated nonlinear dynamical systems.
