Harmonic Balance Research Articles

A novel permanent magnet vibration isolator with wide stiffness range and high bearing capacity

Quasi-zero stiffness (QZS) vibration isolators have attracted extensive attention because of their excellent performance in low-frequency vibration isolation and high bearing capacity. However, its practical engineering application remains challenges due to the magnitude and adaptability of the negative stiffness. This paper proposes a permanent magnet type variable stiffness (PMVS) mechanism with a wide stiffness range from negative to positive. Based on the semi-analytical model of magnetic force, the QZS isolator composed of the spiral spring in parallel with the PMVS mechanism is developed, which has both stiffness adjustment and load adaptability. The nonlinear dynamic equation with 5th polynomial restoring force is established then solved by the harmonic balance method (HBM). The dynamic behaviours is investigated and validated through the experimental prototype and good isolation performance is observed. Building upon the wide stiffness adjustment and large magnitude of the PMVS mechanism. the proposed isolator achieves a stiffness range of 4.15–153.45 N/mm under a safe load up to 120 kg, which implies that the natural frequency of the isolator can reach an ultralow value of 0.93 Hz within the safe load (120 kg).

Research on flexible beam-type nonlinear vibration isolators suitable for low frequencies

Ship vibration not only causes safety issues, but also easily generates noise pollution. Therefore, this paper introduces the quasi-zero stiffness (QZS) vibration isolator to solve the problem of low-frequency vibration. In terms of static analysis, this paper proposes a discretized beam constraint model (DBCM) to study the static properties of flexible beams with large deformation, and the finite element analysis (FEA) verifies the accuracy of DBCM. Regarding dynamic analysis, this paper combines the harmonic balance method (HBM) and the pseudo-arc length method (PALM) to analyze the dynamic response and explore the influence of excitation and damping ratio on the response stability. In addition, the performance of equivalent linear isolators is compared, revealing significant advantages of QZS isolator in terms of isolation bandwidth and peak transmissibility. The experimental data demonstrates that QZS isolator’s initial isolation frequency (2.14 Hz) is 68.5% lower than that of linear isolators (6.79 Hz), and the isolation efficiency at 5.35 Hz can reach 90%, making it suitable for low-frequency isolation. Furthermore, experimental data validate the accuracy of the theoretical model and FEA. The static and dynamic theoretical models can optimize the structural parameters of flexible beams and serve as valuable inspiration and reference for designing and improving isolator.

Nonlinear Dynamics and Combination Resonance of a Flexible Turbine Blade with Contact and Friction of Shrouds

Flexible shrouded blades are commonly adopted in the last stages of steam turbines where complicated dynamical behavior can be induced by dry friction force generated on contacting interfaces between adjacent shrouds and the geometric nonlinearity due to the structural flexibility of the blades. In this paper, combination resonance caused by contact and friction forces generated on shroud interfaces is investigated, which is a concurrence of 1:3 internal resonance involving the first and second modes in the flapwise direction and the primary resonance of the first flapwise mode. The stiffness and damping at the contact interface are obtained by linearizing the contact and friction forces between shrouds through the harmonic balance method. The vibrating blade is modeled as a beam with a concentrated mass of which the responses under the combination resonance are solved through the multiple-scale method. Sensitivities of response with respect to the angle of shrouds, contact stiffness and rotation speed are illustrated, and the influences of these parameters on the periodicity and amplitudes of steady responses are demonstrated. The parametric regions where the combination resonance occurs are pointed out. Finally, parametric analyses are presented to show how the amplitude–frequency relation of the multiple-scale solutions under the combination resonance vary with detuning and design parameters. The present research provides a designing basis for improving the dynamic performance of flexible shrouded blades and suppressing vibrations of blades by adjusting structural parameters in practical engineering.

Active Control of Quasi-Zero-Stiffness Vibration Isolator with Variable Load

Quasi-zero-stiffness (QZS) isolator has great application potential in the field of low-frequency vibration isolation due to its high-static and low-dynamic (HSLD) nonlinear stiffness characteristic, but it is precisely this characteristic that makes it very sensitive to load changes. Once the load changes, causing it to deviate from the equilibrium position, it no longer qualifies zero-stiffness characteristic, and the vibration isolation capacity will be significantly decrease. To make the QZS isolator possess variable load capacity and be more suitable for engineering practice, an active QZS vibration isolator based on electromagnetic actuator is designed in this paper, which eliminates the influence of load changes through the active force of the electromagnetic actuator. First, the dynamic equation of the isolator is established by Newton–Euler method, and the dynamic characteristic of the isolator under standard load and load variations are analyzed through improved incremental harmonic balance (IHB) method based on discrete Fourier transform (DFT). Next, the improved particle swarm optimization (PSO) algorithm are employed to optimize the Proportion Integration Differentiation (PID) controller parameters. Then, the vibration isolation performance of QZS isolator in controlled and uncontrolled and linear systems in the same control state are compared in frequency domain and time domain, respectively. Finally, the performance of active QZS isolator under load variation is discussed. The results indicate that the isolation performance of the QZS isolator under active control is significantly better than in uncontrolled conditions and the controlled linear system. When the load changes, real-time compensation through the actuator output control force can also enable the QZS isolator to achieve a better vibration isolation performance.

Bio-inspired quasi-zero stiffness vibration isolator with quasilinear negative stiffness in full stroke

Simpler structure, better adaptability to variable loads, and wider linear stable region have always been pursued by nonlinear quasi zero stiffness (QZS) isolator. Inspired by the balancing and stabilizing effect of the leg and wing in the take-off and landing of birds, this paper proposes a novel bio-inspired QZS vibration isolator (BI-QZSVI). The BI-QZSVI is composed of two primary components: the legged negative stiffness structure, which comprises a two-link mechanism and a set of oblique springs, and a pair of vertical springs that imitate wings. A static model is established based on the virtual work principle, the nonlinear dynamic model is then constructed using Lagrangian equations, and an analytical solution is derived through implementation of the harmonic balance method. An experimental platform is established to verify the effectiveness of the theoretical results. The results reveal that the proposed isolator outperforms existing typical isolators by offering a larger quasi-zero stiffness region, as evidenced by the two proposed range and linearity indexes. The unique structure effectively tackles geometric nonlinearity to establish a linear relationship between the load and stiffness. Notably, the negative stiffness of the isolator exhibits exceptional linearity throughout the entire stroke, ensuring a broad stability region even under high excitation levels. Furthermore, the isolator demonstrates remarkable efficacy in reducing vibrations with frequencies exceeding 3 Hz. Overall, the proposed isolator is an efficient low-frequency isolator with simple structure, strong adaptability, and wide linear range.

Low-frequency vibration absorption of magnetic quasi-zero-stiffness structures with lever mechanism

Dynamic vibration absorbers (DVAs) are widely employed in diverse engineering systems for their capacity to mitigate structural vibrations. However, traditional absorbers exhibit sensitivity to natural frequencies and present challenges in tuning within the low-frequency range. This study introduces a lever mechanism to enhance the low-frequency vibration absorption performance of magnetic quasi-zero-stiffness structures, resulting in a lever-type DVA (L-DVA). We elucidate the design philosophy behind the proposed L-DVA. An analytical model is derived based on the Lagrange equation and the frequency-response relationship is determined using the harmonic balance method. We conduct numerical and experimental analyses to assess the impact of the lever ratio, mass ratio, frequency ratio, nonlinear stiffness coefficient ratio, and damping ratio on vibration absorption performance. Furthermore, we fabricate a prototype of the L-DVA with an adjustable magnetic quasi-zero-stiffness structure, and the experimental results align with simulation outcomes. The result shows that this study is the ease of customization in the vibration absorption capabilities of magnetic quasi-zero-stiffness structures, achieved by adjusting the lever ratio. An increase in the lever ratio and tip mass effectively shifts the anti-resonant peak to a lower frequency. The decrease in the frequency ratio leads to a reduction in the anti-resonant frequency. This study validates the efficacy and feasibility of utilizing the L-DVA for low-frequency vibration absorption.

Vibration Response Analysis of a Time-varying Stiffness Wheel with Contrate Gear under Unbalanced Excitation

The vibration and response of multi-contact interface micro-motion time-varying stiffness wheel are analyzed, and the nonlinear vibration response characteristics caused by the unbalance force of multi-contact interface contrate gear face of a gas turbine rotor are studied. Firstly, the local sliding model of dry friction-damped contact is extended to establish a holistic and local unified sliding model. The equivalent stiffness and damping of the damping device are calculated by means of the equivalent linearization method and the first harmonic balance method. Finally, the finite element model of multi-contact contrate gear rotor wheel is established to analyze the influence of different unbalanced masses on vibration response under nonlinear frictional contact. The results show that the response caused by the unbalance mass of multi-contact interface wheel with contrate gear is mainly radial response, and the peak value of resonance response increases with the increase of unbalance force.

On the dynamic characteristics of using track nonlinear energy sinks for structural vibration control

Nonlinear energy sinks (NESs) have emerged as a promising solution to overcome the narrow effective bandwidth of linear dynamic vibration absorbers such as the tuned mass damper (TMD). In particular, track NESs are increasingly attracting attention as they are capable of generating nonlinear restoring forces by the movement of an additional mass on the curved tracks. The present study aims to investigate and explore the versatile characteristics of track NESs in structural vibration control. To this end, three different track profiles are considered, namely the profiles designed by a quartic term only, by combined positive quadratic and quartic terms, and by combined negative quadratic and positive quartic terms. Their profiles and damping coefficients are analytically derived in closed form. The frequency response functions of standalone track NESs are constructed using the Harmonic Balance Method and compared with those obtained by the 4th-order Runge-Kutta method, and the dynamic responses, including the displacements, the phase planes, the nonlinear stiffness forces, and the vibrational frequencies, of the track NESs are presented and discussed under different excitation amplitudes and frequencies. The effectiveness and robustness of using track NESs for structural vibration control are also systematically investigated and compared with TMD under harmonic and white-noise ground excitations. Results show that the closed-form solutions of optimal profile and damping coefficients are sufficiently accurate for lightly damped structures, and the frequency response functions of track NESs can be accurately estimated using the Harmonic Balance Method for low and moderate excitations, although their dynamic responses are sensitive to the excitation amplitudes and frequencies. Furthermore, in comparison to TMD, the track NESs are superior in terms of decreased stiffness and smaller working strokes. In general, the present study demonstrates the potential of using track NESs as an effective and robust alternative for structural vibration control.

The Vibration Isolation Properties of an Asymmetrical Quasi-Zero Stiffness Isolator with Geometrical Nonlinear Damping

This paper proposes a Quasi-Zero-Stiffness (QZS) vibration isolator with asymmetric stiffness and geometric nonlinear damping. The nonlinear stiffness and nonlinear damping characteristics of proposed model are studied. The motion equation of proposed system under force excitation is established, and its dynamic response is solved by the Harmonic Balance Method (HBM). In addition, the influence of nonlinear damping on force transmissibility is studied. The results indicate that the proposed isolator has a large equivalent damping ratio in the resonance region and a small equivalent damping ratio in the high-frequency region. Therefore, the proposed asymmetrical QZS isolator with nonlinear damping can supress the resonance in lowfrequency under force excitation and achieve a good highfrequency vibration isolation effect simultaneously than that with linear damping.

Research on Vibration Response of Shrouded Blades Based on Contact State

The shrouded is widely used in vibration reduction of turbine blades. In this study, the damping characteristics of the shrouded blade in gas turbine were studied. Firstly, a finite element model of the shrouded blade was established, and the full set of blades was reduced to a single sector based on the principle of cyclic symmetry. The contact state of the contact surface of the blade at different interference and speeds was calculated based on the displacement extraction method. Secondly, a microslip friction contact model was established based on the contact state of the contact surface. The nonlinear vibration response of shrouded blades was calculated based on the harmonic balance method. The results indicate that the shrouded has excellent vibration reduction performance. The interference and rotational speed of the blade will affect the damping efficiency.

EXPLORING INSECTS FREE FLIGHT: ENHANCING THE DIPTERAN FLIGHT MODEL TO INCLUDE FRACTAL EFFECTS

This paper advances fundamental knowledge of how environmental conditions and physical phenomena at different scales can be included in the differential equation that models the flight dynamics of dipteran insects. The insect’s anatomical capability of modifying their mass inertia and flapping-wing damping properties during flight are included by modeling inertia and damping forces with fractal derivatives. An expression for calculating fractal dimension linked to the temporal distribution of non-geometric quantities related to atmospheric processes such as turbulence flow is introduced using, for the first time ever, the two-scale fractal dimension definition and adopting the flow energy spectrum of eddies that occur at large and small scales. The applicability of the derived expression is illustrated with the prediction of the fractal dimension observed in turbulent flows. Then, the two-scale fractal dimension transform is used to re-write the dipteran flight equation of motion in equivalent form to derive its approximate solution using harmonic balance and homotopy perturbation methods. Numerical predictions computed from the derived approximate solutions allow to elucidate how insects and animals could adapt to flight under different environmental conditions.

Analysis of a fractal modification of attachment oscillator

In this paper, we consider a combined technique for a fractal modification of the attachment oscillator arising from nanotechnology. This technique is called as TSFT-GRHBM by coupling the two-scale fractal transformation and the global residue harmonic balance method. The approximations and frequencies of this fractal attachment oscillator are given without linearization. Numerical results are provided to confirm its efficiency.

Fractional Mathieu Differential Equations in Dynamic Stability of Piles

Piles are commonly used in Civil and Mechanical Engineering to provide support to superstructures and to transmit loads to deeper ground layers. During earthquakes, pile instability can result in the collapse of the entire structure. In this paper, the pile is modeled as a column surrounded by Winkler soil foundations with fractional damping. Investigation of the axially loaded pile leads to a fractional Mathieu differential equation of motion. The Bolotin method, employing harmonic balance, is proposed to obtain the approximate instability boundaries of the pile in the stability diagrams. A practical example is presented to conduct a parametric study on pile instability concerning the fractional order. The study observes parametric resonance in the first order, representing the principal instability region, which requires special attention to maintain stability. Furthermore, increasing the fractional damping order leads to a higher critical dynamic load and a slight reduction in the critical frequency ratio in each instability region. Higher-order instability regions exhibit greater sensitivity to changes in the fractional order. These results provided insights into the stability of piles under earthquake conditions.

喷杆喷雾机双摆悬架动力学特性研究

During the operation of a boom sprayer, various excitations can cause the boom to roll, yaw, or jolt, which can negatively impact the spray effect. In order to maintain the stability of the boom, it is crucial to incorporate a suspension between the vehicle body and the boom. This study established a dynamic model that describes the roll of a boom mounted on a double-pendulum suspension. The transfer function and frequency response function of the passive suspension were obtained, with the ground slope angle as input and the boom inclination angle as output. The analysis results showed that the passive suspension can partially isolate high-frequency disturbances but cannot follow low-frequency signals. For the active suspension, the transient and frequency responses were analyzed by numerical integration algorithm, the harmonic balance method and numerical solution. The results indicated that the active suspension can enable the boom to follow the ground slope, track low-frequency signals to a certain extent, and isolate high-frequency signals. The proposed mathematical model can be utilized to analyze the dynamic characteristics and provide guidance for the double-pendulum suspension design.

AI Algorithm Based Green Hydrogen Static Compensator for Harmonic Mitigation, Power Balancing, and Power Factor Correction in Nano-Grids

Nano-grids are emerging as a vital solution for sustainable energy distribution, particularly in integrating intermittent renewable energy sources. In this context, this research paper aims to address the challenges faced in Nano-grids through the innovative use of a Green Hydrogen Static Compensator (GHSC). The purpose of this study is to overcome issues related to harmonic pollution, power factor correction, and real power balancing within the Nano-grid system. The method employed involves a two-fold approach. Firstly, a modified instantaneous p-q algorithm is proposed to enhance the effectiveness of GHSC. Secondly, artificial intelligence (AI) algorithms, including deep neural networks, random forest, and XGBoost, are utilized to further refine the system's accuracy. These AI algorithms were trained using the modified p-q method, and the simulation of GHSC was performed using SimScape Toolbox in MATLAB-Simulink. The results of the study clearly demonstrate that the proposed GHSC approach, combined with the AI algorithms, outperforms traditional methods in terms of effectiveness in mitigating harmonic pollution and correcting power factors. However, it is noted that the current implementation lacks flexibility for real-time self-learning. In conclusion, the paper underscores the novelty and potential of the GHSC in conjunction with AI algorithms, while also pointing out areas for future research. Specifically, future efforts may focus on developing an automatic weight updating deep neural network that responds to real-time feedback of THD, power factor, and frequency, enhancing the system's adaptability and efficiency.

Active Vibration Control of a Quarter-Car Model via a Time-Delayed Vibration Absorber

A time-delayed vibration absorber (TDVA) is proposed to suppress the vertical vibration of a quarter-car model. First, the mechanical model of the combined system with a linear TDVA is studied. Second, the vibration suppression effect of the linear TDVA at a fixed excitation frequency is experimentally verified when the passive absorber totally loses efficacy. Third, based on the system acceleration response amplitude derived by the harmonic balance method, the control parameters optimization of the nonlinear TDVA is carried out in a wide frequency band; the numerical simulation of original vibration equations is conducted to determine the stability of the system. It is concluded that the broadband vibration suppression effect of the nonlinear TDVA is obtained with optimal control parameters. The numerical results validate the correctness of the theoretical ones.

Modeling of induced currents in a type-II superconducting cylinder of finite length in a transverse magnetic field

Induced currents in the bulk superconducting cylinder of finite length placed in a transverse magnetic field are calculated by the partial and complete separation of variables. The Fourier transform over the angular and axial coordinates reduces the problem to the solution of one-dimensional integral equations. The harmonic balance method is applied to account for the highly nonlinear constitutive relation between the voltage and current. The current and field distribution is compared with the results of FEA. The approximation of the current distribution by a set of cylindrical and disc layers is presented. The magnetic moment of induced currents is analyzed in the increasing applied field when the cylinder is magnetized up to the state of saturation.

Friction Saturated Limit Cycle Oscillations—Test Rig Design and Validation of Numerical Prediction Methods

Abstract In this paper, an experimental test rig for friction saturated limit cycle oscillations is proposed to provide a validation basis for corresponding numerical methods. Having in mind the application of turbine blades, an instrumented beam-like structure equipped with an adjustable velocity feedback loop and dry frictional contacts is designed and investigated. After dimensioning the test rig by means of a simplified one-dimensional beam model and time domain simulations, the specific requirements of limit cycle oscillations for the design of the frictional contact, the velocity feedback loop and the excitation system are discussed and possible solutions are presented. Also appropriate measuring principles and evaluation techniques are assessed. After commissioning of the test rig, the influence of the negative damping and the normal contact force on the limit cycle oscillations is measured and the practical stability is investigated. The test rig shows linear dynamics for sticking contact and highly repeatable limit cycles. The measured results are discussed regarding the consistency with theory and compared to the predictions of a three dimensional reduced order model solved in frequency domain by the harmonic balance solver OrAgL. It is demonstrated that the numerical modeling strategy is able to accurately reproduce the measured limit cycle oscillations, which stabilized for different contact normal forces and self-excitation levels.

Numerical analysis and experimental validation of nonlinear broadband monostable and bistable energy harvesters

Vibration based piezoelectric energy harvesting has been receiving more and more attention for supplying usable electrical energy to low-power electronic devices. This type of energy can produce the desired voltage to power any low electronic device or wireless sensor. But most of them provide lower voltage and insufficient power. In this paper, the nonlinear magnetic field is introduced to broaden the effective resonant bandwidth for overcoming this issue. The dynamic linear and nonlinear model of magnetic coupling piezoelectric vibration energy harvester is established based on the Hamilton principle, Euler-Bernoulli beam theory, piezoelectric theory, and Kirchhoff's law and law of mechanics. Numerical solutions of nonlinear energy harvesting systems are studied by using the Runge-Kutta method. The bifurcation diagram, Poincare map, power spectrum, and phase trajectory of voltage output are investigated with different excitation amplitudes and frequencies. Simulation of linear and nonlinear model were illustrated at given excitation levels. Simulation results show the harvesting performance can be improved by proper external magnetic coupling and potential energy function. The theoretical harmonic balance analysis results verify the numerical solution and influence the mechanism of excitation levels and interface resistances. Experimental verification is carried out to measure the nonlinear restoring force and examine the performance of nonlinear bistable and monostable energy harvesters. The experiment results show the nonlinear bistable and monostable energy harvesters can provide larger voltage and more power.

Vibration Suppression Study of Parallel Combined Damping Nonlinear Energy Sink on Piecewise Linear Systems

The Nonlinear Energy Sink (NES) is a passive control device capable of achieving targeted energy transfer and effectively suppressing system vibrations. This paper conducts dynamic modeling and vibration suppression analysis of parallel combined damping nonlinear energy sinks in piecewise linear systems. Initially, the system's amplitude-frequency response curve is obtained using the harmonic balance method, validated by the Runge-Kutta method for accuracy. Subsequently, an analysis is performed on the vibration reduction capabilities of the parallel combined damping nonlinear energy sink and the influence of parameters on its vibration suppression effectiveness. Finally, the study investigates changes in the main system's maximum response amplitude with varying damping under different stiffness conditions and determines optimal parameter values based on trend analysis. Research findings indicate that the vibration reduction performance of the parallel combined damping nonlinear energy sink surpasses that of a single connected damping nonlinear energy sink. Furthermore, after parameter optimization, the main system achieves superior vibration reduction effects.