Outphasing power amplifiers (PAs) enable high efficiency under output back-off through constant-envelope operation, making them attractive for wireless power transfer (WPT) systems. However, in practical wearable scenarios, overall efficiency is strongly influenced by the interaction between the power combiner and the coupling-dependent impedance of the inductive link-an effect that has not been systematically characterized. In this work, we present a system-level analytical framework that jointly models outphasing PA operation, Wilkinson and Chireix combiner behavior, and coupling-dependent WPT load variations. The proposed approach enables explicit determination of optimal operating conditions as a function of the coupling coefficient. The analysis shows that for the Wilkinson combiner, the optimal outphasing angle is always θ = 0°, with efficiency primarily limited by wireless link coupling. In contrast, for the Chireix combiner, peak efficiency consistently occurs at the compensation angle θc, while its magnitude is strongly modulated by coupling conditions. These results translate into practical design insights: Chireix-based architectures with θc = 60°-70° can outperform Wilkinson combiners by up to 8% points at moderate coupling (k ≈ 0.2), whereas Wilkinson configurations offer greater robustness under weak or dynamically varying coupling (k < 0.15). Circuit-level harmonic balance simulations for both Wilkinson and Chireix combiners further confirm the analytical trends, demonstrating that the predicted efficiency behavior is physically realizable for both topologies. The proposed framework provides actionable guidelines for early-stage design and architecture selection in wearable WPT systems without requiring device-level or full electromagnetic simulations.

Piezoelectric flutter systems exhibit rich nonlinear dynamics under aerodynamic excitation. Alongside their beneficial effects in terms of energy harvesting, nonlinearities may also induce chaotic responses, potentially compromising system performance. Unlike bistable energy harvesters in which chaotic behaviors have not been reported, tristable flutter systems exhibit experimentally observed aperiodic oscillations; yet, the underlying chaos mechanisms remain theoretically unexplored. This paper investigates the nonlinear dynamics of a tristable piezoelectric flutter system, elucidating the transition from periodic oscillations to chaos for the first time. Using a harmonic balance method, Lyapunov exponent analysis, and bifurcation mapping, the complete evolutionary pathway is revealed. Originating from the central potential well, the system undergoes Hopf bifurcations followed by a symmetry-breaking bifurcation that triggers period-doubling cascades, which are fundamentally different from classical flutter instabilities. The chaotic regime features positive Lyapunov exponents and fractal basin boundaries, representing unique dynamical behavior. Analyses reveal that chaotic responses reduce power output by up to 80%. Parametric studies show that reducing cubic pitching stiffness substantially lowers the critical wind speed for interwell limit cycles, effectively suppressing pure chaotic regions. This work provides the theoretical foundation for chaos phenomena in tristable piezoelectric flutter systems and actionable design guidelines for performance enhancement.

Speaking Patient's Language: Assessment of Readability and Fidelity of Artificial Intelligence-Optimized Consent Forms.

A harmonic balance normal form parametrisation for single mode reduction of nonlinear vibrating systems

This study presents a unified analytical–numerical framework for the El Borhamy–Rashad–Sobhy equation with Duffing-type nonlinearity, augmented by an external harmonic forcing term. The used application in the study is motivated by the electromechanical dynamics of salient-pole synchronous machines. Starting from an energy-based formulation, the harmonic balance method is used to obtain closed-form frequency-response relations. The local stability of the periodic orbit is further quantified via a Floquet-based test derived from the linear variational equation, yielding stability-classified frequency-response curves. While the method of multiple scales captures primary, superharmonic, and subharmonic resonance conditions and the associated stability boundaries are analyzed. Beyond the weakly nonlinear regime, numerical bifurcation analysis is performed to trace a period-doubling route to chaos, supported by Poincaré sections, the largest Lyapunov exponent, and time-series diagnostics. The system is further coupled with linear and nonlinear piezoelectric energy-harvesting branches, to demonstrate that, the large-amplitude responses enhance harvested power and the cubic stiffness can be tuned for bandwidth optimization. Overall, the results bridge perturbation theory with nonlinear time-domain diagnostics, providing design-relevant insights for broadband energy harvesters and electromechanical systems operating under parametric excitation and external forcing.

This study investigates the primary resonance response of a composite micropipe reinforced with graphene nanoplatelets conveying fluid, with applications in remote selfpowered micro-sensors, precise drug delivery, the safety of chemical reactions, ecological assessments, and the transport of cells. A sequential pseudo-nonlinear normal mode scheme, combined with the modified Rauscher technique and harmonic balance method, is employed to derive the frequency response. The gyroscopically coupled equations are systematically reduced, at each stage, to a single degree-of-freedom equation through nonlinear normal mode framework, treating the system as a two-dimensional invariant manifold. The harmonic balance method generates nonlinear algebraic equations, which are solved using an arclength continuation method with an adaptive arc length. Critical to this analysis, Hill’s method is implemented to determine the stability of periodic solutions along the frequency response branches, enabling classification of solution branches as stable or unstable. This stability classification reveals that the bandwidth of multi-valued response coincides with the region of unstable periodic solutions, with stability transitions occurring at the saddle-node bifurcation points (turning points) of the frequency response curves. Parametric studies demonstrate that an increase in the slenderness ratio, radius ratio, flow velocity, and flow mass density and considering laminar flow instead of turbulent flow widen the multi-valued response bandwidth, while micro-scale contribution narrows it. A comprehensive sensitivity analysis reveals that the multi-valued response bandwidth and maximum frequency response amplitude are least sensitive to nanocomposite weight fraction, flow mass density, and micro-scale parameter, while exhibiting the highest sensitivity to slenderness ratio and radius ratio. The primary resonance characteristics show moderate sensitivity to the flow speed. These findings can be implemented for the design of remote self-powered micro-sensors, system identification, and structural health monitoring applications.

In coupled modular floating systems, amplitude death (AD) denotes a suppressed dynamic state characterized by the cessation of oscillatory activity. This study derives analytical solutions for AD and establishes its critical conditions under multi-frequency excitations. We demonstrate that AD is modulated by harmonic functions determined by the excitation frequencies, enabling the formulation of a fundamental solution for the AD state. Employing the harmonic balance method, this solution is obtained by resolving coefficients within a system of nonlinear algebraic equations. The derived analytical solution elucidates the distribution of AD states within the parameter design space, offering critical insights for system stability design. Furthermore, a generalized critical criterion is established, delineating the parametric boundaries governing the emergence and cessation of AD. Theoretical predictions demonstrate strong concordance with numerical simulations. As a practical application, the stability design of modular floating airports is investigated, leveraging the AD mechanism to enhance structural resilience under dynamic loading. This work provides a foundational framework for optimizing stability in complex floating systems subjected to multi-frequency environmental forces.

This paper proposes a numerical method combining a fixed-point harmonic balance finite element method (FEM) with a dynamic hysteresis model to accurately calculate the loss distribution of laminated cores under complex operating conditions. This method primarily employs the frequency-domain FEM to solve the magnetic field distribution of silicon steel laminated cores under various excitations, including power frequency multi-harmonic conditions. The resulting magnetic flux density distribution is substituted into the dynamic hysteresis model, enabling the accurate simulation of the hysteresis loop at any position of the laminated core. Hence, the loss distribution of the laminated core can be obtained. Compared to the modified Steinmetz formula, the proposed method is validated with better performance. The relative error is less than 6% between the measured and the calculated core losses of the proposed method. These results can improve the accuracy and efficiency of transformer iron loss calculation in engineering applications.

A Nonlinear stiffness Inerter-Based Absorber (NIBA) is introduced to attenuate the dynamic responses of monopile-supported offshore wind turbines under combined wind and wave loading. A reduced-order dynamic model of a monopile wind turbine incorporating the proposed absorber mounted in the nacelle is developed, where the fore-aft bending behavior of the tower is idealized as a single rotational degree of freedom. The steady-state response of the nonlinear coupled system subjected to harmonic excitation is obtained using the harmonic balance technique, and the stability characteristics of the resulting periodic solutions are evaluated via Lyapunovs first method. The absorber parameters are determined through an optimization procedure that minimizes the root-mean-square displacement of the tower top. The vibration mitigation capability of the NIBA is then systematically benchmarked against that of a conventional Tuned Mass Damper (TMD) and an optimized linear Inerter-Based Absorber (IBA). The results indicate that the proposed NIBA provides enhanced vibration reduction across a wider frequency bandwidth, achieving reductions of 10.5% and 4.14% in the selected performance metric relative to the TMD and IBA, respectively. Additional time-domain and frequency-domain simulations under realistic stochastic wind and wave excitations further demonstrate that the NIBA consistently yields the largest decreases in both root-mean-square and peak tower-top displacements, while requiring a smaller maximum absorber stroke, thereby confirming its practical applicability for monopile offshore wind turbine systems.

This study investigates the forced vibrations of double simply supported Euler-Bernoulli beams with initial deflections connected through a distributed linear and nonlinear elastic layer. It is assumed that the upper beam is subjected to a concentrated transverse harmonic force located at its midspan. The integro-partial differential equations of motion are introduced. The Galerkin approach is utilized, and the mode shapes of uniform simply supported beams are used to establish the nonlinear governing equations of motion that incorporate quadratic and cubic nonlinearities. The harmonic balance method (HB) in conjunction with the pseudo arc-length continuation scheme are applied to obtain the amplitude-frequency curves. To assess the accuracy and validity of the proposed method and solution, some findings obtained by the suggested approach have been compared with those found through numerical integration, where a strong concordance was demonstrated. The influences of several factors such as the linear and nonlinear stiffness parameters, and the initial deflections of the beams on the steady state amplitudes have been examined. It is observed that these factors have considerable influences on the dynamic responses of the double beams. For the sake of generality and convenience, the results are displayed in dimensionless forms.

Abstract The bistable structures are widely utilized in energy harvesting, vibration isolation, and morphing applications due to the beneficial effects of negative stiffness. However, the jump phenomena and complex nonlinearity of bistable restoring forces make parameter identification challenging, particularly in high-dimensional systems with localized bistability. To overcome this, this paper introduces a novel integrated framework that synergistically combines harmonic balance identification with deep neural networks, capable of identifying bistable nonlinear restoring forces without prior assumption of their analytical form. The proposed approach requires only a limited set of harmonic excitation responses to achieve high-fidelity identification. The linear mass, damping, and stiffness matrices are directly estimated via harmonic coefficient balancing in the frequency domain. For high-dimensional bistable structures, fully connected deep neural networks are constructed to represent each local nonlinear restoring force, enabling scalable and flexible identification beyond conventional parametric methods. Numerical simulations conducted on three typical bistable cases demonstrate that the method accurately identifies system parameters and nonlinear forces under noise levels up to 40 dB. Experimental validations are performed on a bistable nonlinear energy sink with different potential well depths, confirming that the identified model closely matches the measured frequency response, restoring force, and reconstructed random response under band-limited noise. Compared to the pure deep neural networks-based approach, the identification efficiency is improved by a factor of nine. The results verify that the proposed integrated approach offers a generalizable and experimentally feasible solution for identifying complex bistable nonlinear systems.

Abstract This work presents a function interpolation method based on a Runge–Kutta (RK) scheme to achieve smooth tracking of periodic solutions and their bifurcation behaviors in nonlinear dynamical systems. Unlike conventional harmonic balance (HB) methods that rely on the Newton–Raphson (NR) iteration, the proposed method converts nonlinear equations into linear equations with periodic coefficients, and advances solutions along tangent directions determined by linear equations using RK-based interpolation. This framework enables robust tracking without requiring iterative correction or carefully tuned initial guesses,especially near bifurcation points. A systematic bifurcation analysis is performed by Hill’s method based on the triangular collocation method, allowing explicit identification of tangent directions associated with saddle-node (SN), symmetry-breaking, period-doubling, and Neimark–Sacker (NS) bifurcations. As a result, the method achieves smooth transitions between main branches and bifurcation branches within a unified numerical framework. The effectiveness and generality of the proposed approach are demonstrated through three representative nonlinear systems: the coupled van der Pol equation, the Mathieu–Duffing equation, and the van der Pol–Mathieu equation with external excitation. Numerical results show that the FIM can provide accurate bifurcation tracking with improved robustness near critical points while improves tracking performance and reduces computational cost relative to the incremental HB method by arc-length increments or the arc-length continuation method.

Abstract The function interpolation method (FIM) is a continuous solution-tracking technique that combines the triangular collocation method with the Runge–Kutta method. Previously, the incremental harmonic balance (IHB) method was utilized to generate initial solutions for the FIM but the IHB method may encounter convergence limitations. In this work, three strategies are introduced to provide initial solutions of the FIM. Strategy 1 eliminates secular terms to derive solutions for linear differential systems, thereby establishing a connection between the Lindstedt-Poincaré (LP) method and the IHB method. Strategy 2 controls a certain system parameter so that one equation in the nonlinear system degenerates into a solvable linear differential equation, from which solutions for the entire system can be constructed. Strategy 3 transforms nonlinear differential systems into algebraic equations by making the excitation frequency equal to zero, making initial solutions straightforward to obtain. After obtaining initial solutions, the FIM can extend them to any place. Four examples are presented to verify the effectiveness of the strategies: the coupled van der Pol oscillator, wave propagation in the nonlinear diatomic chain, the coupled van der Pol oscillator with external excitation, and the van der Pol-Mathieu oscillator with external excitation. The results show that the FIM provides more accurate initial values for semi-numerical and semi-analytical methods compared to the LP method. Generally, solutions from the FIM can satisfy computational requirements. Moreover, solutions from the FIM and the IHB method agree well with numerical integration, which further verifies the reliability of initial solution strategies for the FIM.

Exploring the potential of simultaneous resonance in multi-frequency atomic force microscopy.

This study develops a weighted cost function-based optimization framework for tuning the time delay parameter in a nonlinear vehicle suspension with time-delayed acceleration feedback control. A quarter-vehicle model is first established to characterize suspension dynamics. The steady-state response is derived analytically using the harmonic balance method, while a stability analysis of the linearized system delineates the feasible ranges of time delay and feedback gain coefficient. A composite cost function is then formulated by integrating weighted contributions from the sprung mass acceleration, suspension dynamic deflection, and tire dynamic load. Optimization of the time delay parameter is carried out for both positive and negative feedback control configurations. A comparative performance evaluation between constant and frequency-dependent time delay strategies is conducted across the human-sensitive frequency band (4-8 Hz). Numerical simulations validate the theoretical predictions, demonstrating average reductions of 45.97% in the cost function, 35.54% in the sprung mass acceleration, and 30.65% in the tire dynamic load. Although a slight increase in the suspension dynamic deflection is observed, the results confirm that optimized time delay can simultaneously improve ride comfort and driving safety.

Multistable systems generate multiple stable equilibrium points and energy barriers through multilevel potential well configurations, enabling directional vibration energy transfer and efficient dissipation via inter-well oscillations and barrier-crossing motions, and have become an important research direction in vibration control. To this end, this paper proposes a magnetic nonlinear energy sink with designable stability for vibration mitigation. The device exploits the coupling between the cantilever-beam elasticity and dipole–dipole magnetic interactions to enable designable regulation of the multistable potential energy. Based on magnetic dipole theory and the Euler–Bernoulli beam model, an analytical nonlinear mechanical model of the NES is established, and equilibrium bifurcation analysis is performed to reveal the effects of geometric parameters on the number of stable equilibria and the evolution of the multistable potential energy structure. A nonlinear dynamic system model of the linear oscillator coupled with the NES is developed, and numerical simulations are conducted to analyze the dynamic responses under different stability configurations, revealing the regulatory effects of multiple parameters on inter-well transition thresholds, attractor evolution, and energy dissipation. The harmonic balance method is also employed for cross-validation of the steady-state predictions. On this basis, a vibration reduction efficiency index is defined to quantitatively evaluate the device performance. The results demonstrate that the designable regulation of stability characteristics can significantly reshape the system’s dynamic response, enabling efficient vibration suppression over a wide excitation range, with a reduction efficiency of up to 86%. The present findings provide new insights into the regulation and utilization of multistable characteristics and their application in broadband vibration control.

To address the detuning sensitivity of conventional linear vibration absorbers under wide-frequency harmonic excitation and the limited effectiveness of nonlinear energy sinks (NESs) in low-energy regimes, this study investigates a bistable nonlinear energy sink (BNES) based on a negative-stiffness support. A coupled model of the primary system and the BNES is established, and the analytical steady-state amplitude-frequency relationship of the system is derived using the harmonic balance method. The accuracy of the analytical solutions is verified through numerical integration. Based on the first Lyapunov method, the instability regions of the system are identified, and the effects of system parameters on the amplitude-frequency response of the primary structure are analyzed. On this basis, a comprehensive performance index that accounts for both peak suppression and average vibration reduction over the frequency band is constructed, and an improved particle swarm optimization algorithm is employed for parameter optimization. The results demonstrate that the optimized BNES can effectively suppress isolated high-amplitude response branches and significantly reduce the response of the primary system within the resonance frequency band, exhibiting superior broadband vibration mitigation performance and enhanced stability.

Abstract Galloping-induced vibrations pose a significant threat to the stability and durability of slender bluff-body structures exposed to wind flows. Inspired by the successful application of quasi-zero stiffness (QZS) isolator in base excitation scenarios, this study aims to investigate the effectiveness of this novel passive nonlinear method for suppressing galloping-induced vibrations. First, the mathematical model of the coupled system is established and parametric studies are performed to determine the optimal system parameter. Next, to evaluate its overall performance compared to traditional passive control methods, specifically linear and cubic absorbers, a comprehensive comparative study is conducted on these three methods. The results show that up to the wind speed of 20 m/s, which corresponds to the No. 8 Beaufort wind scale (fresh gale), linear absorber outperforms both nonlinear absorbers, with a cut-in wind speed improvement of 188.9%, compared to 26.8% for both nonlinear absorbers. Besides, QZS absorber outperforms pure cubic absorber in terms of the jump wind speed. This suggests that the effectiveness of QZS absorber in base excitation scenarios is not applied to flow-induced excitation scenarios. Furthermore, the nonlinear dynamic behaviour of the QZS absorber is explored based on the numerical continuation technique via MATCONT, and bifurcations are observed. Finally, the system is nondimensionalised, and an approximate-analytical solution is derived using the harmonic balance method. This analytical approach further illustrates the complex bifurcation characteristics of the system.

To address the insufficient convergence robustness, strong dependence on the initial guess, and computational-efficiency bottlenecks of the incremental harmonic balance (IHB) method for strongly nonlinear systems, this paper develops an adaptive framework that integrates a trust-region strategy with a hybrid Hessian matrix. The framework reformulates the harmonic-balance iteration as a constrained nonlinear least-squares optimization problem, constructs a symmetric hybrid Hessian matrix by blending the Gauss–Newton and Newton directions, and uses a trust-region algorithm to adaptively regulate the step size, thereby jointly optimizing the iterative path and convergence behavior. Numerical results show that the proposed method significantly enhances convergence robustness, reduces sensitivity to initial guesses, and improves computational efficiency while maintaining high accuracy. It also captures both stable and unstable periodic solutions in strongly nonlinear systems.

Periodic unsteady flow problems often exhibit computational inefficiency when conventional time-domain methods are employed. This phenomenon can be attributed to the lengthy transition phase before reaching the ultimate periodic state. To address this problem, the harmonic balance method (HBM) is employed for aeroacoustic applications through the implementation of convective monopole sources, frequency-domain gust excitation, and nonreflective boundary conditions. A set of test cases is considered to evaluate this method, including monopole sources with uniform background flows, the pitching oscillations of airfoils, leading-edge noise of gust–airfoil interactions, and the response of gust–cascade interactions. The results show good agreement with analytical and experimental data, with normalized root-mean-square errors typically below 10% across most considered cases. The findings from this research demonstrate a comprehensive range of potential applications for HBM in the fields of fluid dynamics and aeroacoustics.