Abstract The long-term simulation of biological systems at the cellular level presents a significant computational challenge due to pronounced multiscale behavior and severe numerical stiffness. While implicit solvers offer improved stability in solving stiff systems, they can incur prohibitively high computational costs. Alternative conventional methods, such as mass scaling or unit transformation, are often inadequate because model stiffness may arise from multiple causes beyond inertial effects. To address these limitations, this study proposes a model-reduction technique integrated with an adaptive Runge–Kutta solver, which substantially reduces computational time. The effectiveness of the proposed approach is demonstrated by successfully simulating a 14-day adipogenic differentiation process in less than 1 h and 9 min of computational time on a typical desktop computer. Numerical evaluations conducted on a linear spring–mass–damper benchmark, a nonlinear Duffing oscillator, and a detailed human mesenchymal stem cell model show that the proposed approach substantially outperforms conventional scaling techniques and stiff solvers, establishing it as a robust and efficient tool for the long-term simulation of large, multiscale, stiff systems.
Abstract The dynamic response of rocking rigid blocks subjected to sinusoidal base excitation exhibits rich nonlinear behavior, with strong sensitivity to initial conditions and external parameters that often leads to chaotic dynamics. Classical analyses have identified bands of forcing frequencies that appear favorable for sustained rocking, whereas others are considered unfavorable due to overturn. This work revisits this assumption by explicitly controlling the initial phase of a second finite-duration sinusoidal pulse. Using an autonomous embedding that enforces phase scheduling and systematic sweeps over the full range ϕ∈[0,2π), we show that many frequencies previously deemed unfavorable can sustain bounded oscillations when phase alignment is chosen appropriately. Phase acts not only as a controllable initial condition but also through the implicit time-dependence of the excitation, reshaping the immediate torque balance right after activation. We quantify these effects via long-term response maps in the (tr,ϕ) plane (reaction time tr) and a complementary dimensionless momentum-impulse metric, revealing windows of stability within the “saw-like” diagrams reported when phase is uncontrolled. The results suggest phase-based control as a simple and effective strategy for mitigating overturn in rocking blocks, refine our understanding of chaotic transitions, and motivate amplitude-phase–frequency codesign for robust stabilization.
Abstract Fractals, with their intricate structures and self-similar patterns, continue to fascinate mathematicians and scientists alike. In this article, we introduce a novel generalization for generating Mandelbrot, Julia, and Multicorn sets by extending the M-iteration process through incorporation of the s-convex combination. We first establish an escape criterion for complex polynomials of the form xk+1+c, which forms the basis of our escape-time algorithms. These algorithms enable the construction of diverse sets of fractals under the proposed iteration scheme, unveiling new structural variations. Additionally, we present graphical representations to visualize these complex sets and conduct a detailed analysis of the interplay between the iteration parameters and two key numerical measures: the average escape time and the nonescaping area index. Our findings reveal highly nonlinear dependencies, shedding new light on the intricate dynamics of fractal evolution.
Abstract A physical system exhibiting subcritical Hopf bifurcation followed by a fold bifurcation of the limit cycles presents three operational regimes: a single limit cycle around an unstable equilibrium, a pair of stable and unstable limit cycles around a stable equilibrium, and a globally stable equilibrium. In this study, we investigate whether such an oscillator can be utilized to nonresonantly suppress single-mode instability in structural systems. To this end, we analyze a system comprising a negatively damped linear primary oscillator coupled with a lightweight secondary oscillator capable of subcritical Hopf responses. The secondary oscillator is modeled using a modified van der Pol oscillator exhibiting the stated three regimes of operations. For the single limit cycle regime, prior work (Tandel et al., 2023, “Vibration Stabilization by a Nonresonant Secondary Limit Cycle Oscillator,” Nonlinear Dyn., 111(7), pp. 6043–6062) provided a criterion for conditional stabilization, defining the threshold of instability in the primary system for given secondary oscillator parameters. The present study demonstrates that the double limit cycle regime offers conditional stabilization over a significantly larger instability range, leveraging the benefits of the subcritical Hopf dynamics. Finally, the globally stable equilibrium regime offers unconditional stabilization, though its stabilization limit remains equivalent to that using the single limit cycle regime. Hence, a secondary vibration absorber with a stable equilibrium surrounded by an unstable limit cycle which is further surrounded by a larger amplitude stable limit cycle is beneficial as it offers conditional stabilization of an unstable primary system with an arbitrary degree of instability.
Accurate prediction of ship roll motion is challenging due to strong nonlinear damping effects that directly impact vessel safety. This study introduces two novel graph theoretic spectral algorithms, namely, the Stable Set Polynomial of the Complete Bipartite Graph Algorithm and the Clique Polynomial of the Complete Graph Algorithm for solving nonlinear ship roll motion equations with and without external excitation. The proposed methods transform the governing nonlinear differential equations into sparse algebraic systems using spectral polynomial representations over short time evolution intervals, enabling efficient and stable computation. Their performance is evaluated through systematic comparisons with the Homotopy Perturbation Method (HPM) and a standard high order explicit Runge-Kutta time integration scheme. To extend short time solutions to longer prediction horizons, a multilayer perceptron-based extrapolation strategy is employed. Numerical results, supported by root-mean-square error analyses and parameter space heatmaps, demonstrate improved accuracy and stability over existing methodologies, establishing the proposed algorithms as effective alternatives for nonlinear ship roll motion prediction.
Abstract In complex frictional systems, friction-induced vibration (FIV) and noise are ubiquitous and intricate issues. Achieving high-precision simulation of the vibration response is crucial for the diagnosis of system dynamic properties and vibration control. However, frictional surfaces with multiple contact points introduce nonsmoothness, resulting in unpredictable vibration responses and posing significant challenges for numerical methods to maintain accuracy over long-term analyses. This study proposes a new physics-informed neural network (PINN) method designed to enhance the adaptability between physical constraints and neural network training. The method introduces loss functions with state transition boundary modification (STBM) derived from the physical governing equations. Additionally, a data expansion and regression (DER) strategy for processing linear complementarity problem (LCP) is implemented in the optimizer, significantly improving simulation accuracy for complex stick–slip vibration processes in multicontact frictional systems. By combining these two innovations, the proposed method, referred to as BMDER-PINN, was validated through simulations of stick–slip vibration in a two-degree-of-freedom (2DoF) frictional system. Compared with conventional time-stepping methods, this approach ensures higher accuracy in longer simulations while also enabling large time steps, thereby offering a promising calculation method for improving nonsmooth dynamics simulations.
Accurate modeling of robotic manipulator dynamics is vital for achieving precise and robust control, particularly under nonlinear and coupled motion conditions. This study explores the capability of machine learning techniques to capture the nonlinear dynamics of an articulated manipulator and predict joint torques with high fidelity. Four function approximation schemes-artificial neural network-Bayesian optimization (ANBO), support vector machines (SVMs), Gaussian processes (GPs), and decision trees-are optimized using Bayesian hyperparameter tuning and systematically evaluated. The results show that: (1) among artificial neural network (ANN) architectures, the multiple-input single-output (MISO) configuration achieved the highest prediction accuracy; (2) ANNs consistently provided reliable torque estimation across all joints, while GPs offered comparable performance except at low torque magnitudes; and (3) decision trees (DTs) and SVMs yielded lower accuracy, reflecting limited ability to capture complex nonlinear behaviors. Overall, the findings demonstrate the effectiveness of ANBO-based models for adaptive inverse dynamics estimation and highlight their potential for future exploration of adaptive dynamic modeling, particularly in scenarios where variable payloads are involved to further optimize the performance and control of many robotic systems.
The increasing complexity of autonomous systems and underwater acoustics technologies necessitates the development of progressive signal dispensation methods that can handle nonlinear dynamics and mitigate the effects of noise. The problem lies in improving signal dispersion methods for autonomous systems and underwater acoustics to enhance accuracy, robustness, and overall system performance in complex environments. The objectives are to develop advanced nonlinear and noise-resilient signal processing techniques, enhance system performance, improve data accuracy, and ensure robust communication and navigation in autonomous systems and underwater acoustic environments. Adaptive bilateral kernel filtering (ABKF) enhances data preprocessing by effectively reducing noise, smoothing, and preserving edge details in nonlinear signal environments. Nonlinear iterative partial least squares (NIPALS) efficiently model complex relationships in data, improving signal extraction and reducing noise in autonomous systems and underwater acoustics. Reweighted sparse signal decomposition (RSSD) enhances noise resilience by effectively separating signals from noise, improving data quality in submerged audibility and autonomous systems. Hypergraph partitioning algorithm (HGPA) improves signal processing by efficiently partitioning complex data, enhancing performance, and reducing noise in autonomous systems and underwater acoustics. These systems can process real-time data with greater precision. The findings show that energy consumption varies with node count (6-20) across methods ABKF, NIPALS, RSSD, and HGPA. Energy starts near zero at six nodes, peaking around 350 units for ABKF, implemented in python software. Future scope focuses on real-time optimization, dynamic environment adaptability, and enhanced integration for autonomous navigation and underwater communication systems.
Abstract Wear and rolling contact fatigue (RCF) are regarded as two prominent factors that deteriorate the long-term service performances and aggravate the maintenance costs of metro rails on the small-radius curves; therefore, an effective countermeasure against them is needed. In this work, a long-term rail wear and RCF evolution prediction model based on vehicle–track coupled dynamics and surface material wear and fatigue damage theory is performed, in which an enhanced wheel/rail non-Hertzian contact method is involved. On the other hand, a series of candidate rail profiles are constructed and generated by applying the Gaussian function correction (GFC) method. An improved nondominated sorting genetic algorithm-II (NSGA-II) method is applied for the optimization design of rail profiles, and then we propose two types of rail profiles which take the rail wear or RCF as the optimization objectives. Further, the long-term rail wear and RCF evolution performances subjected to the raw rail profiles and optimally designed ones are compared. The results demonstrate that the optimized rail profiles contribute profitably to alleviate the wheel flange–rail gauge corner frictional contact and, subsequently, can slow down the wear and RCF development of rails on sharp-radius curves. This study can offer a theoretical inspection for wear- and RCF-resistant design of rail grinding profiles.
Abstract This paper addresses the complex vibration issues arising from the coupling of nonlinear dynamic cutting forces with internal nonlinear factors in the bearing-rotor system during machining of high-speed motorized spindles. By introducing a dynamic cutting force model and integrating it with Timoshenko beam theory and Hertz contact theory, a coupled dynamic model of the bearings-rotor-cutting system for high-speed motorized spindles is established. Based on this model, the Newmark-β method is employed to solve the dynamic equations and conduct numerical simulations, systematically analyzing the effects of spindle speed and bearing radial clearance on the system's dynamic response. The results reveal that the introduction of dynamic cutting forces significantly alters the system's bifurcation behavior. Through an internal resonance mechanism, it advances the critical instability point and induces period-doubling bifurcation and chaotic motion. Increasing the bearing radial clearance intensifies nonlinear coupling effects, causing the vibration response to evolve from periodic motion through period-doubling and chaos before stabilizing under certain conditions. This reveals the influence patterns of coupled internal and external nonlinear sources on the system's vibration response. This research provides a theoretical basis for structural optimization and cutting parameter selection in high-speed motorized spindles.
This paper explains the stress behavior of filled and vulcanized rubbers subject to large deformations by using the fractional derivatives proposed in a paper in this series (Fukunaga et al., 2025, "A Fractional Derivative Interpretation of Viscoelastic Rubbers. I. Thermodynamically Consistent Fractional-Derivative Models for Finite Strain," ASME J. Comput. Nonlinear Dyn., 20(11), p. 111009, Paper I). The proposed rubber model consists of two fractional-derivative terms and one elastic term arranged in parallel. The orders of two fractional derivatives are alpha similar or equal to 0.5 and beta
This work presents a data-driven framework for the design and optimization of elastic metamaterials composed of a hexagonal honeycomb unit cell with embedded cantilever-type resonators. A feed-forward neural network (FFNN) is adopted as a surrogate dynamic model to explore both direct and inverse modeling approaches, with the aim of integrating them into an effective design workflow. The surrogate is then employed to optimize the geometric parameters in order to maximize the bandgap width while enforcing a prescribed central frequency. To generate the training data, two analytical models are developed based on an orthotropic plate formulation: one treating the resonator as a uniform beam with a lumped tip mass, and the other representing it as a two-segment beam. A third dataset is obtained from extensive finite element simulations. The study compares the performance of the FFNN across the three datasets, highlighting how the underlying data source affects the accuracy and generalization of the surrogate model. The optimized design is fabricated using 3D printing and experimentally validated through laser scanning vibrometry, confirming the effectiveness of the proposed framework.
Symplectic integrators offer vastly superior performance over traditional numerical techniques for conservative dynamical systems, but their application to dissipative systems is inherently difficult due to dissipative systems' lack of symplectic structure. Leveraging the intrinsic variational structure of higher-order dynamics, this paper presents a general technique for adapting existing symplectic integration schemes to arbitrary dynamical systems (conservative or dissipative). Utilizing the present method, any existing symplectic integrator can be generalized and made to incorporate any number of tuning parameters, which can be tailored for optimal performance. Indeed, for the linear test problem considered here, the tuning parameters can be chosen to achieve zero numerical error at every time-step for any given step size. This "tunability" emerges organically out of the intrinsic symplectic structure of the higher-order formulation. Another interesting result is that the process of introducing tuning parameters automatically circumvents the need to supply additional initial conditions for the higher-order formulation. That is, doubling the order of the equation does not actually introduce any additional complexity from a numerical perspective. For illustration, a simple scheme involving two tuning parameters is proposed, and the optimal parameter values are computed in general for two limit cases. In both cases, the resulting scheme is unconditionally stable and outperforms the implicit Euler method. These results open the door to an entirely new class of symplectic integrators that can be tailored to suit specific problems. Future work will focus on tailoring such schemes to viscous flows modeled by the Navier-Stokes equations.
For the past decade, vibration energy harvesting has gained increasing interest. Numerous approaches have been developed to capture unused energy from physical vibration sources, such as ocean waves, buildings and bridges. This paper introduces and evaluates a dual piecewise-linear (PWL) piezoelectric energy harvester design. A prior study involving a single piezoelectric cantilever beam with a mechanical stopper demonstrated that optimizing the gap size could enhance energy generation. This work replaces the mechanical stopper with a second piezoelectric cantilever beam to further improve the performance of PWL piezoelectric harvesters. A new two-degree-of-freedom model is proposed and analyzed both computationally and experimentally. Moreover, an enhanced methodology is employed to identify system dynamics from experimental measurements. A simple controller is created and implemented to experimentally validate the computational model. The results demonstrate improved overall performance and a broader operating frequency range for the proposed energy harvester.
Beam bending and torsional behavior are foundational to structural mechanics, forming the cornerstone of engineering analysis for slender structures across educational and research domains. Traditional instruction relies on small-deflection theory, which, while analytically convenient, can obscure critical aspects of real-world structural behavior. This study presents an integrated research and education initiative: the development of an interactive Graphical User Interface (GUI) application built on a nonlinear rod model to simulate cantilever beam mechanics-a canonical case in structural analysis education, whose framework also supports a broad range of research applications involving large deformations. The GUI enables real-time visualization and comparison of linear and nonlinear responses in both 2D and 3D, allowing users to examine the limitations of small-deflection assumptions and appreciate the broader applicability of nonlinear models. The tool builds on a previously established two-step initial value problem (IVP) formulation for solving the nonlinear cantilever problem, ensuring efficient simulation. Its modular design allows customization, ranging from simple interfaces for basic education to advanced features for in-depth analysis, making it suitable for diverse learners and researchers. When deployed in an undergraduate aerospace structures course, the tool enabled students to explore when nonlinear modeling becomes essential and to examine small-deflection assumptions that might otherwise seem abstract. This interactive platform bridges the gap between analytical theory and real-world structural behavior, offering an intuitive educational experience and serving as a stepping stone for undergraduate engagement in large-deformation research, lowering the barrier to nonlinear computational modeling and advanced structural analysis.
Within the theoretical and applied research on nonlinear dynamics of periodic microstructured systems, the amplitude-dependent dispersion properties of mechanical metamaterials are attracting increasing interest. The paper investigates the nonlinear free and forced oscillations of a minimal two degrees-of-freedom (2DoF) model simulating the local interplay between plain woven yarns in pretensioned textile metamaterials. Numerical solutions are obtained by time-integrating the equations of motion, which are characterized by nonlinear forces. Nonlinearities arise from the piecewise constitutive law determined by the superposition of geometric stiffness caused by pretension and unilateral elastic stiffness due to interyarn contact. Frequency-response curves of the harmonically forced system are numerically obtained, and the nonlinear behavior arising from the piecewise and amplitude-dependent stiffness characteristics is discussed. Particular attention is given to the effective possibility of consistently describing the unilateral elastic stiffness governed by smooth Hertzian laws with linear and cubic nonsmooth approximations. It is shown that numerical solutions exhibit time periodicity disturbed by significant distortions of the linear solutions as soon as the cubic nonlinearities are activated or the oscillation amplitude exceeds a certain detachment threshold. Finally, parametric analyses disclose a marked softening trend in the frequency response functions, as well as the birth of superharmonic components in the time-histories.
Exoskeletons are wearable devices that use mechanical structures to provide additional strength, activate and/or reinforce movements, and better distribute mechanical loads across the human musculoskeletal structure when people handle mechanical loads, treating motor disabilities or mitigating physical health issues due to fatigue or musculoskeletal injuries. This work presents the modeling and simulation of a human operator utilizing a passive upper limb exoskeleton. By simulating routine tasks, such as object lifting and overhead movements, this model facilitates a detailed study of human-machine interaction. Various spring mechanisms were evaluated to determine their impact on mechanical assistance. Furthermore, since attachment pads are vital for operator comfort and device stability, the loads on these pads were analyzed across all configurations. The results demonstrate that understanding spring behavior, particularly identifying regions of constant torque, can optimize control during lifting tasks. The simulation also underscores the importance of waist and midback pads, which distribute the majority of the spring-generated loads across the operator's body.
In this paper, we focus on investigating the fast-slow dynamics and complex behaviors of multiwing butterfly bursting attractors in an alternating current (AC)-driven L & uuml; circuit controlled by a pulse function. We treat the periodic AC perturbation as a slow-varying control parameter to tune the system's multiscale dynamics, rather than merely using it as a critical bifurcation value for characterizing dynamical behaviors. By introducing a single or multiple pulse functions into the third equation of the L & uuml; circuit system, additional equilibrium points are generated, laying the foundation for the evolution of multiwing bursting. With the variation of the periodic perturbation, the system exhibits fast-slow dynamics characterized by transitions among different strange attractors, including periodic or chaotic multiwing attractors. Notably, when the periodic perturbation induces complex oscillations containing multiscroll chaotic components, a distinct delayed supercritical pitchfork bifurcation behavior emerges; this delayed bifurcation dynamic terminates in different parameter regions, leading to diverse spiking modes around various stable attractors, including equilibria, limit cycles, and chaotic trajectories. Our results enrich the research on bursting dynamics in AC-driven nonlinear systems, deepen the understanding of multiwing bursting phenomena, and provide a theoretical basis for the control and application of multiwing attractors.
Swashplate engines offer compelling advantages over traditional piston engines for low-altitude applications, particularly in power density, compact design, and fuel versatility. However, traditional design processes for swashplate engines still rely heavily on empirical paradigms, resulting in relatively outdated optimization schemes. To address this issue, this study focuses on a coaxial swashplate engine and proposes an optimization strategy based on the Nondominated Sorting Genetic Algorithm II (NSGA-II). First, a coupled engine power model and a connecting rod inertia moment model are developed, incorporating key structural parameters. Then, a dynamic penalty function-based elimination mechanism is introduced to ensure that the optimization outcomes surpass expected performance thresholds. Finally, the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method is employed to rank and select the Pareto-optimal solutions. The results demonstrate that the dynamic penalty function mechanism effectively overcomes the premature convergence drawback associated with fixed penalty methods, significantly enhancing the algorithm's stability. Through TOPSIS-based evaluation, a synergistic optimization outcome is achieved, with a relative power increase of 19%.
