Harmonic Balance Research Articles

Nonlinear dynamics of a dual-rotor system with active elastic support/dry friction dampers based on complex nonlinear modes

In this study, the nonlinear dynamics of a dual-rotor system with active elastic support/dry friction dampers (ESDFDs) are investigated based on complex nonlinear modes (CNMs). The finite element method (FEM) combined with a full-3D friction model is introduced to construct the governing equation for the system. Additionally, the Craig–Bampton technique is applied to downscale the finite element model of the system. Based on the reduced order model (ROM), the nonlinear modal damping ratio of the target mode is employed to measure the dry friction damping performance of active ESDFD. The effects of the active ESDFD position, normal force, and tangential contact stiffness on the nonlinear modal damping ratio and modal frequency are analysed. Moreover, the softening characteristics of the active ESDFD are revealed, and the critical speed intervals of the active ESDFD/dual-rotor system are determined. Furthermore, by using the harmonic balance–alternating frequency/time domain (HB–AFT) method, the steady-state response of the system under unbalanced excitation is calculated. The accuracy and effectiveness of nonlinear modal analysis are validated based on the relationships between nonlinear modes and steady-state unbalanced responses. Conversely, the vibration mitigation effects of active ESDFD are determined by the unbalanced response amplitude. Additionally, the controllable region and optimal normal force for effective vibration control in the target mode are defined. Depending on the controllable region, a control strategy for turning on/off the optimal normal force is developed. The findings demonstrate that the developed control strategy enables the active ESDFD to significantly reduce the response amplitude of the dual-rotor system across various excitation levels, showing substantial potential for engineering applications.

Intermodulation spectroscopy and the nonlinear response of two-level systems in superconducting coplanar-waveguide resonators

Two-level system (TLS) loss typically limits the coherence of superconducting quantum circuits. The loss induced by TLS defects is nonlinear, resulting in quality factors with a strong dependence on the circulating microwave power. We observe frequency mixing due to this nonlinearity by applying a two-tone drive to a coplanar waveguide resonator and measuring the intermodulation products using a multifrequency lock-in technique. This intermodulation spectroscopy method provides an efficient approach to characterizing TLS loss in superconducting circuits. Using harmonic balance reconstruction, we recover the nonlinear parameters of the device-TLS interaction, which are in good agreement with the standard tunneling model for TLSs. Published by the American Physical Society 2024

Nonlinear resonant analyses of graphene oxide powder reinforced hyperelastic cylindrical shells containing flowing-fluid

The resonant responses are investigated for the graphene oxide powder reinforced hyperelastic cylindrical (GOPRHC) shells containing flowing-fluid, and the shells are subjected to the radial harmonic excitations. Employing the improved Donnell's nonlinear shell theory, Halpin-Tsai model, hyperelastic constitution relation, velocity potential theory and Lagrange equation, the governing equation of motions are obtained for the GOPRHC shells containing flowing fluid. The amplitude frequency and force amplitude curves are presented by using the harmonic balance method and the pseudo-arc length continuation method. The effects of three distributions of the GOP, weight fraction of the GOP and fluid velocity on the natural frequency for the GOPRHC shells are discussed. The influences of different parameters on the dynamical responses for the GOPRHC shells containing flowing-fluid are conducted, including the external excitation, weight fraction of the GOP, fluid velocity and structural parameters. The results show that the GOPRHC shells with Hypere-X distribution containing flowing-fluid have the largest frequency. The super-harmonic resonance responses appear and present the synchronization effects with the primary resonant responses in the GOPRHC shells containing flowing-fluid. The increases of external excitations, fluid velocity, weight fraction of the GOP and structural parameters enrich the resonant responses for the GOPRHC shells. Compared to the fluid-filled hyperelastic cylindrical shells, the existence of the flowing-fluid makes the GOPRHC shells more prone to the chaotic vibrations. The hysteresis phenomena of chaotic vibrations occur in the GOPRHC shells containing flowing-fluid.

SYNERGIZING ACADEMIC ANIMATION AND AI INTEGRATION FOR ENRICHED LIBRARY EXPERIENCE

Libraries must maintain a harmonic balance between technology and the human aspect to ensure patrons receive the best possible service. This conceptual paper combines academic animation and artificial intelligence to create a library experience. This study proposes using animation to enhance patrons' library experience. Using artificial intelligence and machine learning in a natural language can enrich the library experience.

Harmonic Response of a Highly Flexible Thin Long Cantilever Beam: A Semi-Analytical Approach in Time-Domain With ANCF Modeling and Experimental Validation

Abstract The main objective of this work is to use the time variational method (TVM), a semi-analytical approach to evaluate steady-state responses in the time-domain for absolute nodal coordinate formulation (ANCF) modeled systems. The gradient-deficient ANCF beam element's performance is demonstrated for a highly flexible cantilever beam under gravity and impulse loading, with comparisons to experiments. The damping behavior is compared for the Rayleigh proportional and the Navier–Stokes (NS) damping model for a gradient-deficient ANCF beam element. Classical finite element method (FEM) beam formulation's shortcomings in predicting large deflections of thin, flexible cantilever beams are highlighted. Unlike the harmonic balance method (HBM), TVM reduces the computational time for harmonic response evaluation compared to numerical integration techniques and handles nonlinear forces in the time-domain. The harmonic response is evaluated by exciting the cantilever beam in the linear region for both experiments and TVM computations.

A nonlinear breathing crack model for characterization of chaotic dynamics of damaged large-amplitude vibrating plates

Large-amplitude vibrating plates (LVPs) inherently exhibit intricate chaotic behaviors stemming from geometric nonlinearity. Such chaotic dynamics could be substantially influenced by the occurrence of a breathing crack, characterized by a cyclic opening and closing process. However, current models for breathing cracks primarily focus on beam-like structures and often rely on the assumption of periodic dynamical responses, which is inadequate for depicting the chaotic vibrations experienced by cracked LVPs. To that end, this study presents a novel nonlinear breathing crack model (NBCM) to precisely characterize the breathing effect on chaotic dynamics of large-amplitude vibrating plates containing a parallel part-through breathing crack. By refining the line spring model and devising a mathematical expression for crack parameters and dynamical responses, the NBCM-based governing equation for cracked LVPs is derived with a Duffing type expression having a quadratic term. Through the utilization of the harmonic balance method and the Hilbert–Hughes–Taylor algorithm, both analytical and numerical dynamic responses of the cracked LVP based on the NBCM are obtained. Comprehensive nonlinearity analyses are then conducted, including hardening nonlinearity, breathing nonlinearity, and bifurcation nonlinearity. Comparison results demonstrate the superiority of the presented nonlinear breathing crack model in characterizing the chaotic dynamics of LVPs with breathing cracks. Accordingly, the presented NBCM shows significant potential for identifying abnormalities within practical engineering plate-like structures when considering the effects of crack breathing.

Improved energy harvesting by enhanced nonlinearities: New phenomena and experimental demonstration

This paper investigates a nonlinear piezoelectric energy harvesting system that enhances conversion of mechanical energy to electrical energy by integrating piezoelectric materials with a nonlinear system. The research simplifies a flexible beam with macro fiber composite (MFC) piezoelectric patches using a lumped-parameter approach and derives the dynamic equations of the system using the Lagrange equations. The incremental harmonic balance method (IHBM) is then employed to solve these equations, simulating the nonlinear dynamics involved in the harvested voltage. Key findings include the initial discovery of multiple voltage islands which are voltage jumps that improve energy conversion efficiency, confirmed by experimental data validating the IHBM simulations. The study reveals that these phenomena are due to the enhanced nonlinearities of the system, particularly the geometric nonlinearity caused by the spring k1 and the nonlinear magnetic force between magnets. Adjusting k1 allows the observation of multiple voltage islands, and increasing d0 results in a broad-frequency constant voltage, offering new insights. The research concludes that the nonlinear piezoelectric energy harvester can effectively convert vibration energy into electrical one, as demonstrated by successfully powering multiple light-emitting diodes (LEDs). Bifurcation diagrams illustrate the adaptability of the system, showing transitions among periodic, quasi-periodic, and chaotic states for different parameters.

Nonlinear Static and Dynamic Responses of a Floating Rod Pendulum

Abstract A novel nonlinear dynamics model is developed in this paper to describe the static and dynamic nonlinear behaviors of a rod pendulum partially immersed in still water. The pendulum is hinged above the water level and subject to nonlinear gravity, hydrostatic, and hydrodynamic loads, all of which are incorporated into the system dynamics. The nonlinear static behavior and stability of the pendulum have been characterized by analyzing the fixed points. It is found that Pitchfork bifurcation governs the relationship between the rod density (the control parameter) and the static equilibrium angle. The pendulum's nonlinear response to external harmonic torque is obtained using Harmonic Balance Method (HBM). The influence of system parameters, including hinge height, rod diameter, and rod density, on the nonlinear frequency response is examined. Upon altering the system parameters, particularly the rod density, it is found that the system exhibits either a softening or a hardening effect.

A bidirectional-controllable magnetorheological elastomer-based quasi-zero-stiffness isolator

To enhance the low-frequency vibration isolation bandwidth of quasi-zero stiffness (QZS) isolators under variable amplitude excitations, this paper proposes an adaptive bidirectional-controllable QZS isolator (Bi-QZS) based on magnetic-controlled smart material named magnetorheological elastomer (MRE) as positive element and tilting spring as negative element. Firstly, the positive element of isolator is composed of laminated MRE and permanent magnet to realize bidirectional stiffness adjustment. Then the magnetic circuit of vibration isolator is designed and the specific parameters are obtained by taking the magnetic field and energy consumption as the optimization objectives. Secondly, a static theoretical model is established to calculate the achievable range of stiffness variation, and match the parameters of the negative element. Static analysis shows that bidirectional stiffness control is beneficial to achieve stiffness matching. Additionally, the adjustment of bidirectional stiffness variation on the dynamic response of the system is deduced by harmonic balance method. Numerical simulation results indicate that adjusting stiffness in reverse can increase the vibration isolation bandwidth when the excitation amplitude increases, and adjusting in forward can reduce the peak value of jump region. Also, increasing the damping ratio under reverse conditions has a certain contribution to reducing the response peak. Finally, static and dynamic tests are carried out, results reveal a bidirectional stiffness adjustment capability, with a 36.4% adjustment in reverse stiffness and a 70% adjustment in forward stiffness. The resonance frequency can be reduced from 14.4 Hz to 5.8 Hz by stiffness bidirectional-controllable.

Random Vibration Analysis of Double Deck Rocking Self-Centering Pier Under Seismic Excitation

This paper investigates nonlinear random vibration of double deck rocking self-centering pier structure subjected to seismic excitation. First of all, a novel type of double deck rocking self-centering pier that offers better resilience is designed. The stochastic dynamical model of the system is then established, in which the seismic excitation is viewed as Kanai–Tajimi filtered white noise and the self-centering restoring force is characterized by classical flag-shaped hysteretic model. Subsequently, the self-centering restoring force is decomposed into equivalent quasi-linear elastic and damping forces by adopting the harmonic balance (HB) scheme. Following this, the stochastic averaging (SA) technique is implemented to derive the averaged Itô equation. The steady-state response probability density with respect to the amplitudes is solved from the averaged Fokker–Planck–Kolmogorov (FPK) equation. Finally, the effects of system parameters on the stochastic response of the double deck rocking self-centering pier structure are performed, and validated by the Monte Carlo simulation (MCS). In addition, with the help of the Laplace transform, the transition function as well as conditional power spectral density are achieved, which are combined with the steady-state probability density to obtain the power spectral density response. This research will contribute to the optimal seismic design of double-deck rocking self-centering piers.

Harmonic balance applied to a 2D nonlinear finite‐element magnetic model with motion and circuit coupling

AbstractIn this work, the harmonic balance approach is applied to a 2D nonlinear finite‐element magnetic model with motion, coupled to a nonlinear circuit. The case study comprises a six‐pole three‐phase surface‐mounted permanent magnet generator connected to a six‐pulse full‐wave diode bridge rectifier. Simulations are performed at fixed generator speed in two operating cases: with an open‐circuit DC bus and supplying a load resistance. Both time stepping and harmonic balance approaches are deeply discussed focusing on the model under study, along with relevant implementation details. Harmonic balance results are compared with benchmark time stepping simulations in terms of voltage and current waveforms, progressively expanding the harmonic spectrum. The computational cost of the two approaches is reported as well. Simulation accuracy is satisfying with regard to time stepping benchmark results: relative errors on total harmonic distortion and global root‐mean‐square values are lower than 3% and 1%, respectively. However, the time stepping approach outperforms the harmonic balance one, due to the relatively short initial transient of the chosen case study. Further improvements on practical implementation are needed to exploit the potentialities of harmonic balance technique.

Phase and amplitude responses for delay equationsusing harmonic balance.

Robust delay induced oscillations, common in nature, are often modeled by delay-differential equations(DDEs). Motivated by the success of phase-amplitude reductions for ordinary differential equationswith limit cycle oscillations, there is now a growing interest in the development of analogous approaches for DDEs to understand their response to external forcing. When combined with Floquet theory, the fundamental quantities for this reduction are phase and amplitude response functions. Here, we develop a framework for their construction that utilizes the method of harmonic balance.

Bifurcations and mixed mode oscillations in a bi-stable plasma model with slow parametric excitation.

In this study, considering a bi-stable plasma model with slow parametric excitation, the bifurcation of periodic and chaotic responses as well as the resulting fast-slow motions is discussed analytically and numerically. For a nonautonomous fast sub-system, the generalized harmonic balancing method is utilized to obtain an averaged system. Bifurcation analysis about the averaged system shows that the critical manifolds form a S-shape structure. Meanwhile, supercritical and subcritical period doubling (PD) occurs on the upper branch simultaneously. As the frequency of the external excitation changes, bifurcation points on the limit cycle manifolds can present different relative locations. Moreover, an additional bi-stable structure induced by Cusp bifurcation emanates from the upper branch. On the other hand, the existence of a chaotic attractor and the corresponding boundary crisis phenomenon are verified using the Melnikov method and the basin of attraction. The structures of the numerical bifurcation diagram show good agreements with the analytical results. Considering two cases of low-frequency excitation, the corresponding fast-slow dynamics are discussed. It is found that, when the fast-slow flow passing the subcritical PD point, a low frequency with different magnitudes will lead to two patterns of bifurcation delay, i.e., the typical one and the excessive delay, which suppress the PD. As for the boundary crisis point, the slow passage effects show no distinct influence. Thus, three transition mechanisms based on two cases of the bifurcation structure are explained, including "fold of cycle-fold of cycle" type, "fold of cycle-delayed subcritical PD" type, and "fold of cycle-boundary crisis" type.

Coupled nonlinear vibration characteristics of quasi-zero-stiffness Gough-Stewart isolation platform

The quasi-zero-stiffness (QZS) Gough-Stewart vibration isolation platform is designed for six degrees-of-freedom (DOFs) low-frequency vibration isolation. Lots of studies regarding the amplitude-frequency relationship of the platform are based on the assumption that interference with the platform in one direction only provokes a response in that direction. This assumption artificially decouples a six DOFs coupling model into six single DOF equations, ignoring the coupling relationship between each DOFs, making the analysis results inaccurate. Therefore, without simplifying the nonlinear terms and taking the coupling effect into consideration, the improved Incremental Harmonic Balance Method (IHBM) is employed to discuss the coupling relationship between different DOFs of the QZS Gough-Stewart vibration isolation platform. The results show that the platform has multiple equilibrium positions. When the platform is excited from different directions, it will vibrate at different equilibrium positions, and the closer these positions are to the zero-stiffness position, the more likely the system is to exhibit jump phenomenon and sudden increases in response. The platform exhibits a single-directional response only when excited from the z-direction. Through comparative study, it is found that the cubic configuration can weaken the coupling effect between each DOFs compared with the general configuration.

Rotorcraft Flight Control Design for Rotor Noise Abatement

This article presents the development and implementation of active noise abatement flight control laws utilizing redundant control allocation in rotorcraft. The study starts by finding a periodic equilibrium of coupled rotorcraft flight dynamics and acoustics using a modified harmonic balance trim solution method. Next, the rotorcraft nonlinear time-periodic dynamics are linearized about its periodic equilibrium and transformed into an equivalent higher-order linear time-invariant system in harmonic decomposition form. Composite aeroacoustic measures are formulated as outputs of this system. To enable faster runtime and tractable control design, a reduced 8‐state model representative of the coupled rigid‐body dynamics and acoustics is obtained through residualization. An explicit model following control law with pseudoinverse allocation to redundant control surfaces–an active horizontal stabilizer in this study–is synthesized using the 8‐state model. The article further assesses the impact of noise alleviation control on handling qualities of conventional and compound rotorcraft in forward flight. The results demonstrate that redundant control allocation is an effective method for active reduction of unsteady rotor noise, with increasing effectiveness during aggressive maneuvers. Future-generation rotorcraft, particularly future vertical lift configurations featuring high control redundancy and aggressive maneuvering capabilities, can significantly benefit from noise abatement through redundant control allocation. Furthermore, the article highlights that while implementing noise abatement control in conventional helicopters can degrade handling qualities, the use of redundant control allocation in compound configurations offers noise reduction without compromising handling qualities.

Improved harmonic balance method for analyzing asymmetric restoring force functions in nonlinear vibration of mechanical systems

The non-linear differential equation governing some mechanical systems, such as the non-linear vibrations of FG beams resting on a nonlinear foundation, behaves similarly to the oscillations of a non-linear oscillator with an asymmetric restoring force function that includes both odd and non-odd non-linear terms. The objective of this research is to obtain higher-order approximate analytical solutions for such problems by introducing an enhanced harmonic balance method. By building upon the original problem, two new symmetric systems with odd non-linear terms are introduced, and their higher-order approximate analytical solutions are derived using a novel approach. In proposed method, the restoring force function is represented by its Fourier series expansion. Unlike previous papers, linearizing the equation or taking the first derivative of the Fourier series in the subsequent iteration is unnecessary. However, to enhance the solution’s accuracy, the remaining error from each iteration is utilized in the next one. Finally, by combining the results from the two introduced systems, the analytical period and corresponding periodic solution of the original problem can be obtained. This method is applied to the governing differential equation of FG beams resting on non-linear foundations, a physical non-natural oscillator, and conservative Toda oscillator. The key advantages of this approach are its simplicity and its ability to provide highly precise solutions for both small and large amplitudes in a single iteration.

Improving quasi-zero-stiffness isolator performance: two hybrid isolators employing inerter elements

The pursuit of enhanced low-frequency vibration isolation motivates the development of quasi-zero-stiffness vibration isolators (QZS-VIs), yet with inherent limitations. The nonlinear hardening stiffness may distort the resonance peak rightward, narrowing the isolation bandwidth. Additionally, the high sensitivity to excitation variations can exacerbate the rightward distortion and jumping phenomenon. In this study, two types of hybrid isolators (QZS-IE isolators) are developed by integrating inerter elements (IEs) with traditional QZS-VI to enhance isolation performance. Their distinct mass block configurations yield varied enhancement behaviors. The unified dynamic equation for the two QZS-IE isolators is derived using harmonic balance and arc-length methods, validated by Runge-Kutta. The analyses of stability and periodicity clarify their response characteristics. Subsequently, the study investigates the effects of geometry and excitation parameters, comparing the enhancements of the two QZS-IE isolators. The results manifest that the introduction of inerter elements improves low-frequency vibration isolation, suppressing peak transmissibility and reducing isolation beginning frequency. The type I QZS-IE isolator outperforms the type II QZS-IE and traditional QZS isolators in low-frequency isolation performance and excitation sensitivity. However, the high-frequency isolation performance of the type I isolator is hindered by the ascending transmissibility curve post anti-resonant frequency. The type II QZS-IE isolator rectifies resonance peak distortion while avoiding adverse effects. The different configurations of the inerter elements and the related attenuating strategies offer a guideline for diversified vibration isolation objectives.

Jellyfish-inspired bistable piezoelectric-triboelectric hybrid generator for low-frequency vibration energy harvesting

In pursuit of enhancing the output power of energy harvesters, this paper introduces a novel jellyfish-inspired bistable piezoelectric-triboelectric hybrid generator (JIB-PTHG) with low potential energy barrier characteristics and low-frequency, broadband energy harvesting capabilities. The JIB-PTHG comprises two flexible beams, and two rigid links, connected via a spring. Utilizing the harmonic balance method and Runge-Kutta fourth-order algorithm, a comprehensive analysis of the mechanical properties of the low-barrier bistable structure is conducted. Static equilibrium bifurcation and dynamic response analysis are performed. Subsequently, an electromechanical coupling model incorporating piezoelectricity and triboelectricity is developed to theoretically predict the output voltage of the two modules, revealing a positive correlation between displacement and output voltage. Experiment validation demonstrated that adjusting the length of the rigid links can effectively increase the deformation of the piezoelectric beam and the relative displacement of the triboelectric layers, thus improving the generator's output performance. Specifically, the TENG module exhibited exceptional energy harvesting performance within the frequency range of 3.5–6.5 Hz, achieving a voltage of 1050 V, a power of 1.84 mW, and the ability to illuminate 70 LED lights under an excitation amplitude of 17.5 mm and a frequency of 5.5 Hz. The PEG module, displayed high output performance within the frequency range of 4-7 Hz, reaching a voltage of 6.2 Vs and a power of 12.8 μW at an excitation frequency of 7 Hz and an amplitude of 17.5 mm. Both theoretical and experimental findings indicate that the JIB-PTHG has achieved improved output power within the frequency range of 3.5–6.5 Hz. Furthermore, the JIB-PTHG has the potential to harness bridge vibration energy and convert it into useful electrical energy for powering wireless sensor nodes in bridge health monitoring systems.

Aeroacoustics computation for propellers based on harmonic balance solution

In this paper, a novel open-source framework is presented for the evaluation of noise emissions produced by aerodynamic bodies exhibiting dominant motion at specific frequencies. This behavior is frequently encountered in propellers used for urban air mobility applications. The reduced order model called harmonic balance is employed to compute the unsteady flow solution, reducing the computational cost. The approach tackles K different frequencies by capturing the flow solution at N discrete time instances within a single period, where N is defined as 2K+1. The time history of the conservative variables on the solid surfaces is reconstructed through a Fourier interpolation. The Kirchhoff Ffowcs Williams Hawkings integral formulation, integrated into SU2, is used to compute the sound pressure level perceived by farfield observers. The integral formulation propagates the acoustic solution with a computational cost independent of observer distance. The noise emission of a pitching wing and a propeller is computed with the proposed framework. Regarding a small, isolated propeller operating at low Reynolds numbers in forward flight, we conducted a comparison of the aerodynamic and aeroacoustic results between a fully time-accurate solution and a steady-state Reynolds Averaged Navier-Stokes solution within a rotating reference frame. The harmonic balance results demonstrated improved agreement with the fully unsteady solution across the first three blade-pass frequencies and exhibited this consistency over a broader range of propagation angles.

An investigation of a self-powered low-frequency nonlinear vibration isolation system

High-static-low-dynamic stiffness vibration isolation are of utmost significance in low-frequency vibration isolation. However, their intricate mechanisms are always challenging for environmental vibration changes. In this study, a self-powered nonlinear vibration isolation system with two stages is studied, with the aiming of obtaining the broadband vibration isolation. Magnetic springs of negative stiffness combined with the coil could transform mechanical vibration into electrical energy in lower stage. The feedback electrical energy could power the piezoelectric actuator in the upper stage. Based on the governing equations measured in stable equilibrium, the harmonic balance analysis is used to derive the frequency response functions of the transmissibility and the output power in the mechatronic resonance. The frequency band of vibration isolation extends to the lower frequencies and two resonance peaks reduce to the lower levels. The numerical simulations results validate the analytical solutions. Then the shock isolation performance is illustrated under the round step displacement and rounded pulse displacement. The parameter study demonstrates numerically that introduction of the high-static-low-dynamic stiffness improves the shock isolation performance. Moreover, the lower stage mechanical energy could drive the piezoelectric stack, resulting in high-efficiency shock isolation. Finally, an experiment is employed to prove the high-efficiency vibration isolation.