Harmonic Balance Research Articles

Improving the Solution Procedure of Incremental Harmonic Balance Method for Multi-Degree-of-Freedom Self-Excited Vibration Systems

Improving the Solution Procedure of Incremental Harmonic Balance Method for Multi-Degree-of-Freedom Self-Excited Vibration Systems

An Efficient Analysis of Amplitude and Phase Dynamics in Networked MEMS-Colpitts Oscillators

Abstract This paper explores the interactions of both phase and amplitude in a network of N MEMS-Colpitts oscillators that are resistively coupled. The numerical simulations of the extensive networks of oscillators, required for emerging applications such as clock synchronization and neuromorphic computing, become computationally prohibitive as the number of oscillators increases. This complicates the design and evaluation of such systems, as understanding the effects of changes in coupling and design parameters can require many simulations. This study employs the method of multiple scales (MS) in combination with the harmonic balance method to convert the coupled differential equations governing the system of oscillators into a set of nonlinear evolution equations for the amplitude and phase of the oscillators. The amplitude and phase evolve on a timescale that is slow, commensurate with the damping in the system, compared with the fast timescale of the oscillation frequencies. The approach used in this study to describe the amplitude and phase dynamics offers significant computational efficiency (gains of 10× to 50× are shown) compared to direct numerical integration while maintaining an accurate representation of the response. The results of the presented simulations demonstrate the effect of coupling strength on the dynamics of the network, accounting for both phase and amplitude dynamics.

Enhancing the performance of solar-powered EV charging stations using the TOSSI-based CTF technique

In this paper, the design and analysis of a novel solar-powered EV-charging system employing a third-order sinusoidal signal integrator (TOSSI) based-CTF (character of triangular function) is proposed. The TOSSI-based CTF is used to extract fundamental active components by eliminating harmonic distortions from the load currents. This control structure has the unique capability of active current separation, employing simple mathematical operations. Moreover, the response is further refined with the help of optimized gain parameters. The designed system is capable of handling various power quality issues, including harmonic mitigation, current balancing, and power factor improvement. The EV battery is primarily charged by solar-PV power employing a bidirectional DC-DC converter. Alternatively, the EV battery may be charged by the grid supply during the unavailability of sunlight by taking the power quality issues into due consideration. The suggested control topology is used to enhance the dynamic operation of solar-powered EV charging stations experiencing solar power intermittency and variation of load. Using MATLAB, the efficacy of the proposed control topology is tested for different operating scenarios. The suggested control topology is also verified and validated using prototype hardware developed in the laboratory, where the suggested controller has proven its utility over and above existing state-of-the-art controllers in the domain.

Ultra-low frequency and broadband flexural wave attenuation using an inertant nonlinear metamaterial beam

Attenuation of elastic waves in the extremely low frequency range is a considerable challenge. While linear acoustic metamaterials can manipulate elastic waves, their effectiveness is often limited to a narrow frequency range. The present paper proposes a novel inertant nonlinear metamaterial beam to address the challenging problem of the ultra-low frequency broadband flexural wave attenuation. The amplitude-frequency response and dispersion relations of the flexural waves are obtained using the first-order harmonic balance method, where four different resonant units are investigated and the resulting band gaps are compared. Finite element (FE) simulations are also conducted to evaluate the transmission spectra of the flexural vibration through a finite metamaterial beam. The good consistency between the band gaps predicted by the analytical model and the FE simulation verifies the correctness and effectiveness of the developed analytical approach. The results of this study demonstrate that introducing the inertance and softening nonlinearity into the resonant units can significantly widen the flexural wave band gaps within the low-frequency range. This finding provides a viable solution for engineering applications that require low-frequency vibration suppression and isolation.

A composite structure device: vibration control and energy harvesting of tool system in aviation thin-walled parts milling process

Abstract The intense vibration generated by the high-speed milling cutter in the processing process not only seriously affects the surface roughness but also may damage important components of the machine tool, resulting in significant economic losses. Therefore, a composite structure of piezoelectric energy-collector (PEC) and nonlinear energy sink (NES) is innovatively constructed by combining theoretical research and numerical analysis to achieve more efficient vibration suppression and energy recovery in milling. Using the Hamilton principle, the electromechanical model of the composite structure is developed. Then, based on the Galerkin truncation to discretized displacement functions, we solve the amplitude-frequency response of the tool structure by using the harmonic balance method. The validity analysis of the proposed method can be verified by rigorous comparison with the results of the Runge-Kutta method. A novel NES-PEC composite structure presented in this work shows excellent vibration suppression capability in the milling process. More importantly, the structure also enables self-powered sensing of the tool system, significantly reducing dependence on external power sources. This innovative design substantially improves the stability and reliability of milling processes and opens up new ways to optimize the energy efficiency of tool systems. In summary, this paper’s modeling and solving methods, vibration reduction techniques, and energy harvesting strategies have laid a solid foundation for applying a high-speed milling cutter system. More relevant literature on applying vibration suppression and energy harvesting devices in milling must be reviewed. This paper fills in the research gap and provides a valuable reference for future tool systems design and optimization.

Signal growth in a pure time-modulated transmission line and the loss effect.

We present the first comprehensive study for signal growth in transmission lines (TL) with purely time-modulated characteristic impedance (infinite superluminality). This study pioneers the investigation into the effects of varying the cell's electrical length and the impact of loss on momentum bandgaps and amplification levels. It also thoroughly examines how time-modulated transmission line truncation by a static load influences the sensitivity of amplification gain to the relative phase between the incoming signal and modulation, comparing these findings with the case of parametric amplification. Varying is accomplished by loading TLs with a sinusoidally time-modulated capacitor (TMC). The study starts with a simple lumped model cell to facilitate understanding of the phenomena. Following this, transmission lines are introduced, and the effects of incorporating loss are examined. To accomplish this, three models are investigated: a lossless L-C TL lumped model loaded with a shunt lossless TMC and a TL loaded with a shunt lossless and lossy TMC. Dispersion diagrams are plotted and momentum bandgaps are identified at a modulation frequency double the signal frequency. Within the momentum bandgap, only imaginary frequencies are found and correlated to momentum bandgap width and signal growth level. Signal growth is confirmed using harmonic balance and transient simulations, and the results are consistent with the dispersion diagram outcomes.

Discussion of S. Ponsioen, S, Jain and G. Haller: “Model reduction to spectral submanifolds and forced-response calculation in high-dimensional mechanical systems”, Journal of Sound and Vibration 488, 2020, pages 1–23

The authors of the discussed article considered two examples (1) a finite element model of a linear beam with a cubic spring attachment having 10 to 10,000 degrees of freedom, and (2) a finite element model of a geometrically nonlinear Timoshenko beam having 21 degrees of freedom. The authors concluded that Harmonic Balance (HB) becomes intractable for those examples. In this discussion, we falsify this conclusion. We demonstrate that very well-tractable computation effort is obtained when properly using the HB tool NLvib that was also employed in the discussed article. In particular, for (1), the well-known exact condensation to the nonlinear part of the equation system is carried out. For (2), suitable parameters are set for the path continuation and the Newton-type solver. With this, the computation takes from split seconds to about two minutes instead of the many hours reported in the discussed article.

Study on vibration control performance of pendulum TMD with additional stoppers and its application on high-rise buildings

Although pendulum tuned mass damper (PTMD) is one of the most classic and commonly used vibration control devices, it has clear limitation due to the natural feature of linear TMD. Additional stoppers for the pendulum string (PTMD-AS) were proposed and introduced to improve the PTMD's performance by triggering its nonlinearity. A 2-DOF model was established to analyze the dynamic response of the system subjected to harmonic excitation, and the governing equations were formulated using the Lagrange equation. The extended incremental harmonic balance (EIHB) method and the Runge-Kutta (R-K) method were utilized to calculate the frequency response and time history of the system. Nonlinear dynamic characteristics of the pendulum with stiffness hardening were explored in detail. Sensitivity analyses were performed to investigate the effect of stopper position. It was found that aperiodic responses or multiple solutions could be induced when the pendulum underwent significant stiffness hardening upon passing the additional stoppers. Finally, the effectiveness and robustness of PTMD-AS are demonstrated in a numerical simulation of a high-rise building subjected to random wind excitation based on wind tunnel experiments.

A generalized incremental harmonic balance method by combining a data-driven framework for initial value selection of strongly nonlinear dynamic systems

A generalized incremental harmonic balance method by combining a data-driven framework for initial value selection of strongly nonlinear dynamic systems

A Low Mach Preconditioned Harmonic Balance Solver for Cavity Flutter Computations

Abstract Labyrinth seal flutter is a critical phenomenon in turbomachinery, as it can lead to severe structural vibrations and potential component damage. Accurate prediction and mitigation of flutter are paramount to ensuring the reliability and performance of modern turbomachinery systems. This paper explores the numerical computation of a labyrinth seal flutter test case using a low Mach preconditioned harmonic balance (HB) solver and investigates how this approach can improve the accuracy and response time of flutter computations. HB solvers have gained prominence in turbomachinery computations for their ability to efficiently capture unsteady flow phenomena and significantly reduce computational time compared to time-domain analyses. In labyrinth seals, however, the flow is often characterized by low Mach numbers, and preconditioning for these conditions has been shown to significantly improve convergence and accuracy. The goal of this paper is to demonstrate how to implement low Mach preconditioning in a HB solver in the frequency domain. We employ iterative preconditioning to alleviate the stiffness associated with density-based solvers under low Mach conditions and analyze the effect of the preconditioning parameters on the convergence rate. Furthermore, we address inaccuracies linked to the classical Roe solver in low Mach scenarios by adapting it to the low Mach preconditioned governing equations. Through the combined utilization of iterative preconditioning and a preconditioned Roe solver, this study aims to improve convergence rates and the overall quality of flutter predictions. We demonstrate the method with an academic labyrinth seal test case originally presented by Corral et al. (“Higher Order Conceptual Model for Labyrinth Seal Flutter,” ASME J. Turbomach., 143(7), p. 071006). While previous investigations have primarily relied on linearized frequency domain solvers and reduce-order models, in this research a preconditioned HB solver is applied to this test case.

Isolation Performance of the Quasi-Zero Stiffness Isolation System Enhanced by Mixed Tuned Inerter Damper

The low-frequency isolation performance of the transverse spring-rocker quasi-zero stiffness (QZS) isolation system has been demonstrated. However, the QZS isolation system could encounter a deterioration of the low-frequency isolation performance under strong external excitation, due to the hardening phenomenon of its amplitude-frequency response (AFR). To end this problem, a mixed tuned inerter damper (MTID) which combines the advantages of the tuned inerter damper (TID) and the tuned viscous mass damper (TVMD) is proposed in this research. The analytical solution of the AFR of the MTID-QZS isolation system under harmonic excitation is solved by the harmonic balance method (HBM) and verified by numerical method. Whereafter, the corresponding stability analysis of AFR is conducted, and the force transmissibility (FT) of the MTID-QZS is defined to qualify the isolation performance. The effects of the tuning frequency of MTID, the inertance-to-mass ratio, and damping ratios on AFR and FT are conducted, and optimized parameter combinations are suggested to obtain distinguished ultra-low-frequency isolation performance. The comparison among the MTID-QZS system, TID-QZS system, TVMD-QZS system, and QZS system shows that the MTID-QZS system has obvious advantages in suppressing the low-frequency and high-frequency vibration either under strong or weak external excitation.

Nonlinear harmonic resonant behaviors and bifurcation in a Two Degree-of-Freedom Duffing oscillator coupled system of Tension Leg Platform type Floating Offshore Wind Turbine

The surge-heave coupled motion model of a Tension Leg Platform type Floating Offshore Wind Turbine (TLP FOWT) is first established. By taking the tendons stretching and platform set-down motion into consideration, the nonlinear model is a Duffing oscillator coupled system. We analyze the nonlinear feature in the surge-heave coupled motions by the semi-analytical algorithms in nonlinear dynamics. The harmonic balance method is applied to obtain the analytical solutions of the nonlinear system under wind and wave excitations. The analytical solution verifies the excellent accuracy. To further achieve the heave motion response, we organize the bifurcation analysis and discuss the effects of the dynamic parameters on the heave responses. The results reveal the multi components of heave motion, which consist of constant, primary harmonic resonate and higher order harmonic resonate. In addition, the change of dynamic parameters has various effects on the response. The increment of wave load leads to the expansion of heave response, and an instantaneous expansion appear. Both linear surge and heave damping only affect the amplitudes of heave motion. The nonlinear surge stiffness and heave stiffness can barely affect the primary harmonic resonance amplitudes, but they present different effects on the higher order harmonic resonance component.

Complexity and response of bio-inspired energy harvesters based on wing-beat pattern

This paper aims to investigate the dynamical mechanism of bio-inspired energy harvesters based on wing-beat pattern under harmonic excitation. Due to the existence of the gravity force in the established model, the harmonic balance method is utilized to calculate the theoretical results, which has the advantage to keep the influence of gravity force. Multiple solutions are found in the high frequency region, and they are very close in the amplitude of displacement and voltage due to the special structure of the bio-inspired energy harvester. Direct time-domain analysis verifies the effectiveness of theoretical results. The influence mechanism of the equivalent stiffness is also explored, which leads to the appearance of different states. Then, the root mean square (RMS) voltage and average power are analyzed. It is observed that a smaller damping coefficient and equivalent capacitance enhance the average electrical output and achieve greater output power. Subsequently, the bifurcation and complexity properties of the harvester are discussed. Complex phenomena are observed under different external excitations and equivalent damping, including double periodic bifurcation, multiple periodic bifurcation, and chaos phenomena. The complexity analyses confirm the effectiveness of the bifurcation results. The distribution of complexity also exhibits significant fluctuations, closely correlated with the trend of the bifurcation diagram.

A Static/Dynamic Coupled Harmonic Balance Method for Dry Friction Systems Containing Rigid Body Modes

Abstract A harmonic balance method for under-constrained dry friction systems containing rigid body modes (HBM-RBM) is proposed. This method aims to overcome the encountered obstacle when applying the harmonic balance method to turbine blades damped by underplatform dampers. The inspiration for HBM-RBM comes from the free interface modal synthesis method. The key innovation involves deriving the elastic inversion of the singular stiffness matrix through the elimination of rigid body modes. In this way, the general HBM framework can be adopted, and the frequency response of under-constrained dry friction systems can be solved in a static/dynamic coupled manner. The accuracy and efficiency are both verified on a lumped parameter model and a finite element model of turbines with UPDs from a real gas turbine. A comparative study between the HBM-RBM and the commonly adopted way of imposing artificial grounding springs (HBM-AGS) is conducted. Results demonstrate that the HBM-RBM holds a significant advantage over HBM-AGS, as it eliminates the need for artificial grounding springs (AGS) and avoids the necessity for numerous trial cases to determine AGS stiffness.

Multi-frequency harmonic balance method for nonlinear vibration of pipe conveying fluid under arbitrary dual-frequency excitation

Multi-frequency harmonic balance method for nonlinear vibration of pipe conveying fluid under arbitrary dual-frequency excitation

Computing the dynamic response of a periodic structure coupled with a nonlinear junction using the harmonic balance method and Floquet-Bloch modelling

Structures constituted of assembled sub-components are often subject to nonlinear effects due to the presence of joints or contacts, which can induce higher-order harmonics of the source excitation. In the context of periodic structures, these nonlinearities cannot be tackled using the Floquet-Bloch theory in its usual harmonic formulation. This study hence presents an adaptation of the classical Floquet-Bloch theory to address multi-harmonic systems arising from a finite number of localized nonlinearities in periodic waveguides. We adopt the Harmonic Balance Method to recast the governing equations into a nonlinear algebraic system, which can then be solved using numerical continuation algorithms. To demonstrate the validity and benefit of this approach, the predictions derived from the proposed methodology are compared to results obtained through conventional Finite Element analysis. Excellent agreement and a notable reduction in computational cost are observed. This efficient procedure for analysing the nonlinear dynamic response of periodic waveguides opens up new possibilities in the study of damaged composite structures.

15 years of experience with the Triple Plane Pressure Mode Matching technique

The so-called Triple-Plane-Pressure-Mode-Matching technique (TPP for short) is a method used to isolate and quantify the acoustic part in numerical simulations of turbomachinery. Nowadays, simulations such as Harmonic Balance are commonly applied during the design phase of new turbomachinery components for both aeroacoustic and aeroelastic analyses. The TPP method was first published 20 years ago by Ovenden and Rienstra. The principle is to match the periodic pressure data from three or more neighbouring analysis planes with a subset of acoustic eigenmodes. The original method assumes that the flow in the channel section is potential so that the eigenmodes can be determined analytically. This approach can also be applied in slowly varying ducts. The TPP method is a powerful technique that is used at DLR and MTU Aero Engines. For nearly 15 years, constant joint efforts have been made to improve its applicability in industrial environment. Targeted applications are turbine stages, where the mean flow is complex and vortical wakes contaminate the results. In this paper, the latest developments of the method are presented, including the extension to curved extraction surfaces to enable its application to small blade-row spacing. The utility of the technique is illustrated by an industrial application.

Analysis of a brake squeal functional model using a linear parameter varying perspective

Brake squeal is a limit cycle vibration induced by mode coupling instability that depends on operating conditions such as applied pressure, temperature, and disc velocity. This work proposes a simplified functional model of brake squeal that reproduces the main characteristics observed in a full-scale industrial test campaign: vibration growth, limit cycle saturation, vibration decay and parametric dependence. The proposed functional model differs from the well-known Hoffmann model by the introduction of a nonlinear contact law and a quasi-static pressure loading. First, using a harmonic balance perspective, non-linear forces are shown to lead to a pressure and amplitude dependent contact stiffness. This Linear Parameter Varying perspective allows complex mode computations in the pressure/amplitude domain which are then correlated with a series of transient responses of the nonlinear modes for three different pressure profiles. The chosen profiles represent usual experiments: drag where a constant pressure is applied, pressure ramps and pressure oscillations mimicking the effect of wheel spin on the contact surfaces.

Inertial amplification as a performance enhancement method for snap-through vibration energy harvester

In recent years, there has been a lot of interest in exploiting the bistable behavior of snap-through systems to harvest energy from vibration sources. The efficient operation of any bistable VEH depends on its ability to exhibit large-amplitude interwell motion. Under weak ambient excitation, bistable VEH performs marginally because of the confinement of motion to a single well. Frequency up-conversion, multi-stability, and adaptive techniques are some of the performance enhancement strategies suggested for bistable-VEH. Considering the VEH's space constraints, the above designs are hard to implement in practical cases. This study introduces an inertial amplification mechanism (IAM) as a simple passive strategy to enhance the performance of a snap-through VEH, a concept not explored in previous studies. The addition of IAM increases the effective mass without increasing the physical mass and thereby enhances energy harvesting, especially from weak ambient excitation sources. The dynamics and performance of the enhanced snap-through VEH are investigated analytically and numerically under harmonic and random excitations. The harmonic balance method (HBM) derives the frequency-amplitude relationship, which shows a hardening behavior and an increase in bandwidth. The effective potential method provides a closed-form expression for the joint probability density function (Joint PDF), which is governed by the Fokker-Planck equation. The joint PDF shows a transition from bimodal to unimodal with an increase in the value of the geometrical parameter. The stochastic averaging method is employed to obtain the stationary probability density function, which defines the long-term dynamics of the VEH. The effects of noise intensity, mass ratio, and inertial amplifier angle on the dynamics are investigated. Finally, the performance of the proposed VEH is compared with a conventional snap-through VEH, an equivalent linear VEH, and a multistable VEH under harmonic and random excitation conditions. The findings suggest that the snap-through VEH with the IAM has advantages over the linear and multistable nonlinear VEH in terms of extracting energy from low-intensity harmonic and random excitation sources. This simple augmentation strategy preserves the original system's bistability, eliminating the need for the complex design of a multistable VEH.

دراسة علمية حول العزم الزاوي وعلاقته بالخصائص الميكانيكية لشفرات التوربينات

The general trend now in all countries of the world and all sectors is to achieve sustainability. One of how this sustainability can be achieved is to develop and improve the properties of raw materials, and in the field of turbine-powered power plants, sustainability must be achieved in light of the global crisis facing the energy sector, in this study, which aims to study the effect of angular torque on Mechanical properties of turbine blades, so that we can develop and improve the properties of those blades Angular torque is very important in turbine blades because it drives the turbine to rotate. Angular torque results from a force acting on a rotating body. In the case of turbine blades, the force of the gas flow produces angular torque. Turbine blades typically consist of thin, curved metal sheets. When gas passes through the blades, it exerts a force on the blades. This force forces the blades to rotate around its axis the greater the angular torque it produces. The greater the angular torque, the faster the turbine rotates. This study provides a mathematical model to calculate the angular torque resulting from turbine blades. The model was used to study the effect of various factors on the angular torque, including the gas flow speed, the shape of the blades, and their mechanical properties. Then, a simulation was run using the ANSYS program to determine the relationship between the angular torque and the mechanical properties of the turbine blades. The results indicated a large deviation of the flexible blades and unstable rotation of the rotating blade systems, due to the coupling effect of rotation and vibration, where the mechanical properties of mass/inertia distributions, damping, and stiffness are related to changes in rotational speed and local deformation in rotor blade systems, thus, the harmonic balance must occur and the mechanical properties of the blades be improved. Keywords: Angular Torque, Mechanical Properties, Raw Materials, Turbine blades, mathematical model, flow speed, ANSYS, rotation and vibration, harmonic balance.