Higher-order Harmonic Balance Research Articles

Analysis on nonlinear stiffness isolators revealing damping thresholds

Nonlinear vibration isolators with stiffness nonlinearity show promise for a broadband isolation performance without degrading the load capacity. An analytical method is proposed to predict all the possible frequency responses of nonlinear stiffness isolators under different damping. The vibration equation is transformed into an algebraic equation through harmonic balance method. The equation is regarded as a function of the frequency instead of the amplitude, leading to a general analytical method with its complexity independent with the order of the nonlinear stiffness. A damping threshold function is proposed based on the root conditions of the frequency, and the whole damping region is divided into large damping, medium damping and small damping sub-regions accordingly. The root conditions and the frequency responses in each sub-region are analytically analyzed. All the possible responses are classified into five categories based on the damping thresholds and the existence of a jump phenomenon. Nonlinear phenomena of full-band isolation, bounded response and unbounded response are revealed according to multivalued roots of the frequency, and their sufficient and necessary conditions are presented. Simulations are performed on nonlinear isolators with cubic stiffness, high-order stiffness, and non-polynomial stiffness, respectively. The damping thresholds, frequency response categories, and the nonlinear characteristics of each category are demonstrated for the studied cases. A high-order harmonic balance method and the Runge–Kutta method are adopted to verify the effectiveness of the proposed method. This work presents a general analytical method for the prediction of frequency responses, the determination of damping thresholds, and the theoretical illustrations of unique behaviors for nonlinear stiffness isolators.

Characterizing harmonic and subharmonic solutions of the bi-stable piezoelectric harvester with a modified Harmonic Balance approach

In this study, broadband frequency-response of a bi-stable piezo-magneto-elastic harvester is extended by analytically characterizing many subharmonic-n solutions. The energy principle-based model considers the continuous analytical mode-shapes of a compound beam, rotational inertia of the tip mass, and nonlinear geometry effect. A modified higher-order Harmonic Balance (HB) approach proposed here is effective not only in characterizing the frequency-responses of harmonic-1 and subharmonic-n solutions, but also in estimating the chaotic response zones of any multi-stable harvester. Balancing between the computational cost and solution convergence, intra-well (IW) and cross-well (CW) period-n solutions of general-type upto period-4 and odd-type upto period-7 are characterized. Effectiveness and convergence of the modified HB approach are demonstrated through good qualitative and quantitative agreements observed between the numerical sweep and HB results. Dynamical behaviors of several period-n solutions from different branches are revealed through the self-evident plots of displacement and voltage time histories, frequency spectra, and phase-portraits. Novel observations regarding the role of various sub- and super-harmonics in producing only CW orbits and mixed-orbits; and subsequently, asymmetric solution branches appearing due to the nonlinear geometry effect are reported. Besides the customary voltage and power output analyses, the assessment of maximum and average strains induced in the piezoelectric material is presented. Such an assessment has proved quite useful in identifying the alternate low-strain inducing and next best power-producing solutions to harmonic-1 responses. The characterized subharmonic-n solutions produce 20μW–90 mW of RMS power and span the entire range of 1–30 Hz considered in this investigation.

A new look at the Equivalent Linearization Technique to predict LCO in aeroelastic systems with discrete nonlinearities

Control surfaces nonlinearities can lead to limit cycle oscillations (LCO). Several methods have been proposed to predict LCO, such as Harmonic Balance-based methods (HB). Describing function (DF) is the HB with a single harmonic motion assumed, and the approach can be combined with a classic eigenvalue stability analysis via the Equivalent Linearization Technique (ELT) to predict LCO. On the other hand, High-Order Harmonic Balance (HOHB) methods consider higher number of harmonics, but they lead to a more complex nonlinear algebraic system of equations. This paper introduces a new look at the ELT combining both eigenvalue analysis and describing functions. Two applications are considered. The first one is a new DF written in a matrix form for the ELT to consider both first and third harmonics in LCOs due to freeplay. The second application is an iterative procedure to combine the eigenvalue analysis with a describing function to predict LCOs in systems with both freeplay and friction nonlinearities. Numerical simulations are performed for the aeroelastic typical section airfoil. The results show that the new DF improves the classic DF by providing a more accurate prediction of LCOs that accounts for third harmonic. Also, the iterative-ELT is shown as an excellent predictor of LCO for systems with both freeplay and friction.

Analysis of super-harmonic resonance and periodic motion transition of fractional nonlinear vibration isolation system

The precision instruments and equipment are often utilized in low-frequency and micro-amplitude vibration systems, in which many vibration isolators of rubber materials are widely used. Ignoring the low-frequency amplitude will result in errors in the fatigue life design of the vibration isolators and predicting the dynamic response of each frequency band accurately becomes necessary. However, integer-order models cannot describe the frequency dependence of rubber materials, while the fractional-order models can describe it instead. On the other hand, the elastic restoring force is strongly nonlinear under large deformation, and the vibration of the nonlinear system contains multiple harmonic components. In order to solve those issues, the fractional nonlinear Nishimura model is used to characterize the constitutive relation of vibration isolators such as air springs, which are made of carbon black filled natural rubber. The high-order harmonic balance method is used to obtain the steady-state response of the vibration system, while the fourth-order Runge–Kutta method is applied to simulate the dynamic response of the system in the low-frequency region, and the Lyapunov exponent is used to determine the stability of the system. Furthermore, the influence of parameters on the amplitude–frequency characteristics of different frequency bands is also studied, and a method to solve the optimal damping coefficient is proposed based on the primary resonance amplitude–frequency curves. The results show that there is a diversity of periodic motions in the process of adjacent super-harmonic resonance transition. Numerical simulations also demonstrate that multi-periodic motions coexist in the system. The motion transition law between the polymorphic coexistence region and its adjacent regions is summarized.

Iterative techniques for analyzing nonlinear vibrating dynamical systems in the frequency domain

The nonlinear response of prototypical structures experiencing harmonic excitation is studied using novel techniques called iterative harmonic analysis (IHA) and iterative modal analysis (IMA). First, a simple damped oscillator with a cubic hardening stiffness nonlinearity is studied, and IHA is used in this single-degree-of-freedom system. In this first section, a high-order harmonic balance is applied, and IHA is employed in order to find the amplitude coefficients for different harmonics and their codependence. Additionally, a set of nested sums are identified that describe the harmonic coupling explicitly. Secondly, a pinned–pinned nonlinear beam of rectangular cross-section is studied, and IMA is applied to find the amplitude coefficients for different modes and their codependence. The nonlinearity is introduced through the membrane effect, where axial strain due to transverse deflection becomes a significant contributor to the system behavior. Typical frequency-domain methods cannot be easily applied to these systems as the solutions of the differential equations lead to intricate coupling between coefficients of the solution, and no analytical expression exists for those coefficients. Based upon these examples, other nonlinear systems may also be considered in future work using either a modal-based or finite element model. Finally, the advantages of the new method (reduced computational cost) as well as the limitations (effectively the same as those of an nth-order harmonic balance) are emphasized.

Vibration Prediction of Bladed Disks Coupled by Friction Joints

The present review article addresses the vibration behavior of bladed disks encountered e.g. in aircraft engines as well as industrial gas and steam turbines. The utilization of the dissipative effects of dry friction in mechanical joints is a common means of the passive mitigation of structural vibrations caused by aeroelastic excitation mechanisms. The prediction of the vibration behavior is a scientific challenge due to (a) the strongly nonlinear contact interactions involving local sticking, sliding and liftoff, (b) the model order required to accurately describe the dynamic behavior of the assembly, and (c) the multi-disciplinary character of the problem associated with the need to account for structural mechanical as well as fluid dynamical effects. The purpose of this article is the overview and discussion the current state of the art of vibration prediction approaches. The modeling approaches in this work embrace the description of the rotating bladed disk, the contact modeling, the consideration of aeroelastic effects, appropriate model order reduction techniques and the exploitation of the rotationally periodic nature of the problem. The simulation approaches cover the direct computation of periodic, steady-state externally forced and self-excited vibrations using the high-order harmonic balance method, the formulation of the contact problem in the frequency domain, methods for the solution of the governing algebraic equations and advanced simulation approaches, including the concept of nonlinear modes.

Freeplay-Induced Limit-Cycle Oscillations in a T-Tail: Numerical vs Experimental Validation

To investigate the effect of control-surface freeplay, an aeroelastic wind-tunnel model of a T-tail with freeplay in the control chain was developed. The T-tail rig presents a conventional vertical fin represented only by its main structural component: the main spar. No aerodynamic sectors have been used to reproduce the aerodynamic contribution of a horizontal tail. This was chosen because of the limited size of the wind-tunnel chamber. Two numerical models were developed. The first one describes the dynamics of the tail by a state-space system with nonlinearity represented as a lumped element in the actuator feedback loop, and the second is described by the high-order harmonic balance approach, where the response of the nonlinear system is approximated with a periodic signal. The two approaches are then validated against experimental measurements collected during wind-tunnel testing. The results of this validation process as well the numerical and experimental approaches adopted are reported in this paper.

Period Doubling Motions of a Nonlinear Rotating Beam at 1:1 Resonance

A rotating beam subjected to a torsional excitation is studied in this paper. Both quadratic and cubic geometric stiffening nonlinearities are retained in the equation of motion, and the reduced model is obtained via the Galerkin method. Saddle-node bifurcations and Hopf bifurcations of the period-1 motions of the model were obtained via the higher order harmonic balance method. The period-2 and period-4 solutions, which are emanated from the period-1 and period-2 motions, respectively, are obtained by the combined implementation of the harmonic balance method, Floquet theory, and Discrete Fourier transform (DFT). The analytical periodic solutions and their stabilities are verified through numerical simulation.

A high-order harmonic balance method for systems with distinct states

A pure frequency domain method for the computation of periodic solutions of nonlinear ordinary differential equations (ODEs) is proposed in this study. The method is particularly suitable for the analysis of systems that feature distinct states, i.e. where the ODEs involve piecewise defined functions. An event-driven scheme is used which is based on the direct calculation of the state transition time instants between these states. An analytical formulation of the governing nonlinear algebraic system of equations is developed for the case of piecewise polynomial systems. Moreover, it is shown that derivatives of the solution of up to second order can be calculated analytically, making the method especially attractive for design studies.The methodology is applied to several structural dynamical systems with conservative and dissipative nonlinearities in externally excited and autonomous configurations. Great performance and robustness of the proposed procedure was ascertained.

Dual Time Stepping Algorithms With the High Order Harmonic Balance Method for Contact Interfaces With Fretting-Wear

Contact interfaces with dry friction are frequently used in turbomachinery. Dry friction damping produced by the sliding surfaces of these interfaces reduces the amplitude of bladed-disk vibration. The relative displacements at these interfaces lead to fretting-wear which reduces the average life expectancy of the structure. Frequency response functions are calculated numerically by using the multi-harmonic balance method (mHBM). The dynamic Lagrangian frequency-time method is used to calculate contact forces in the frequency domain. A new strategy for solving nonlinear systems based on dual time stepping is applied. This method is faster than using Newton solvers. It was used successfully for solving Nonlinear CFD equations in the frequency domain. This new approach allows identifying the steady state of worn systems by integrating wear rate equations a on dual time scale. The dual time equations are integrated by an implicit scheme. Of the different orders tested, the first order scheme provided the best results.

High order harmonic balance formulation of free and encapsulated microbubbles

The radial responses of free and encapsulated microbubbles excited by an ultrasonic plane wave with a large wavelength in comparison with the bubble size are governed by NonLinear Ordinary Differential Equations (NL-ODEs). The nonlinear frequency response gives the harmonic content of the time response and constitutes the expected outcome of a high order harmonic analysis. In this paper, high order harmonic balance analysis of modified “RPNNP” (bubble), Hoff and Marmottant (contrast agents) models is performed with an open-source software program. For this purpose, the original NL-ODEs are recast into nonlinear systems in which the nonlinearities are at most quadratic. In the spectral domain, this recast provides close form and aliasing-free solutions of arbitrarily large numbers of harmonics. Relevant quantities such as primary and secondary resonances and the nonlinear amplitude threshold of the excitation wave are evaluated. The frequency curves drawn up characterize the bending and quantify the jump frequencies and amplitudes of each harmonic component. The results obtained with this predictive method confirm that it should provide a useful tool for nonlinear bubble detection and sizing and for contrast agent designing.

Non-resonant double Hopf bifurcations: The complex case

The analysis of the unfolding of a non-resonant double Hopf singularity is considered using the frequency domain and the normal form methodologies. A higher-order harmonic balance and the evaluation of Floquet multipliers are used to obtain accurate approximations of the limit cycles and its local bifurcations. This type of hybrid methodology using harmonic balance and normal forms gives a complementary view in the vicinity of the singularity. More specifically, some of the bifurcation curves arising in the unfolding of the double Hopf bifurcation are computed with great accuracy using the harmonic balance method compared to the classical normal form. However, this last method is able to predict the appearance of complex quasiperiodic behavior such as three-dimensional (3D) torus. Numerical simulations corroborate this observation.

Continuation of Higher-Order Harmonic Balance Solutions for Nonlinear Aeroelastic Systems

The harmonic balance method is a very useful tool for characterizing and predicting the response of nonlinear dynamic systems undergoing periodic oscillations, either self-excited or due to harmonic excitation. The method and several of its variants were applied to nonlinear aeroelastic systems over the last two decades. This paper presents a detailed description of several harmonic balance methods and a continuation framework allowing the methods to follow the response of dynamic systems from the bifurcation point to any desired parameter value, while successfully negotiating further fold bifurcations. The continuation framework is described for systems undergoing subcritical and supercritical Hopf bifurcations as well as a particular type of explosive bifurcation. The methods investigated in this work are applied to a nonlinear aeroelastic model of a generic transport aircraft featuring polynomial or free-play stiffness nonlinearity in the control surface. It is shown that high-order harmonic balance solutions will accurately capture the complete bifurcation behavior of this system for both types of nonlinearity. Low-order solutions can become inaccurate in the presence of numerous folds in the limit-cycle oscillation branch but can still yield practical engineering information at a fraction of the cost of higher-order solutions. Time-domain harmonic balance schemes are shown to be more computationally expensive than the standard harmonic balance approach.

Bifurcation analysis and limit cycle oscillation amplitude prediction methods applied to the aeroelastic galloping problem

A global stability and bifurcation analysis of the transverse galloping of a square section beam in a normal steady flow has been implemented. The model is an ordinary differential equation with polynomial damping nonlinearity. Six methods are used to predict bifurcation, the amplitudes and periods of the ensuing Limit Cycle Oscillations: (i) Cell Mapping, (ii) Harmonic Balance, (iii) Higher Order Harmonic Balance, (iv) Centre Manifold linearization, (v) Normal Form and (vi) numerical continuation. The resulting stability predictions are compared with each other and with results obtained from numerical integration. The advantages and disadvantages of each technique are discussed. It is shown that, despite the simplicity of the system, only two of the methods succeed in predicting its full response spectrum. These are Higher Order Harmonic Balance and numerical continuation.

Harmonic Balance Approach for an Airfoil with a Freeplay Control Surface

The nonlinear aeroelastic response of a two-dimensional airfoil, including a control surface with freeplay placed in an incompressible flow, is studied. The model equations are formulated as a set of first-order ordinary differential equations. First, the dynamic response is investigated by a time integration method, and the time integration results are used for the verification of the harmonic balance results. The interesting hysteresis phenomenon and the effect of initial conditions of the subcritical bifurcation are presented. A higher-order harmonic balance method is then derived to investigate the high harmonics of the airfoil motions. The harmonic balance prediction is verified by comparison to the results from a numerical time marching integration and also by comparison to results from a previous experiment. Nomenclature a = nondimensional distance of the elastic axis from the midchord, with respect to the semichord B = damping submatrix b = semichord Ch = stiffness (per unit span) of wing in deflection C(k) = generalized Theodorsen function Cα, Cβ = torsional stiffness (per unit span) of wing around a and of aileron around c c = nondimensional distance of the control surface (aileron) hinge line from the midchord, with respect to the semichord

Double Hopf Bifurcation Analysis Using Frequency Domain Methods

The dynamic behavior close to a non-resonant double Hopf bifurcation is analyzed via a frequency-domain technique. Approximate expressions of the periodic solutions are computed using the higher order harmonic balance method while their accuracy and stability have been evaluated through the calculation of the multipliers of the monodromy matrix. Furthermore, the detection of secondary Hopf or torus bifurcations (Neimark–Sacker bifurcation for maps) close to the analyzed singularity has been obtained for a coupled electrical oscillatory circuit. Then, quasi-periodic solutions are likely to exist in certain regions of the parameter space. Extending this analysis to the unfolding of the 1:1 resonant double Hopf bifurcation, cyclic fold and torus bifurcations have also been detected in a controlled oscillatory coupled electrical circuit. The comparison of the results obtained with the suggested technique, and with continuation software packages, has been included.

The Secondary Bifurcation of an Aeroelastic Airfoil Motion: Effect of High Harmonics

The nonlinear dynamical response of a two-degree-of-freedom aeroelastic airfoil motion with cubic restoring forces is investigated. A secondary bifurcation after the primary Hopf (flutter) bifurcation is detected for a cubic hard spring in the pitch degree-of-freedom. Furthermore, there is a hysteresis in the secondary bifurcation: starting from different initial conditions the motion may jump from one limit cycle to another at different fluid flow velocities. A high-order harmonic balance method is employed to investigate the possible bifurcation branches. Furthermore, a numerical time simulation procedure is used to confirm the stable and unstable bifurcation branches.

Computation of limit cycle via higher order harmonic balance approximation and its application to a 3-bus power system

This work reviews and elaborates a frequency domain approximate method of estimating amplitude, frequency, and stability of the limit cycle near Hopf bifurcations. A new iterative method and new criterion for ascertaining stability are presented. The method and criterion are applied to a 3-bus dynamic power system model to give the amplitude, frequency and stability at its Hopf bifurcations. New analytic formulas for bifurcation curves are derived by using analytic method, some new saddle-node bifurcation at low voltage levels are found on the formulas. The influence of rotor inertia on the Hopf bifurcation is also studied.

Detecting period-doubling bifurcation: an approximate monodromy matrix approach

A quasi-analytical approach is developed for detecting period-doubling bifurcation emerging near a Hopf bifurcation point. The new algorithm employs higher-order Harmonic Balance Approximations (HBAs) to compute the monodromy matrix, useful for the study of limit cycle bifurcations. Prediction of the period-doubling bifurcation is accomplished very accurately by using this algorithm, along with a detailed approximation error analysis, without using numerical integration of the dynamical system. An example is given to illustrate the results.

Feedback Control of Limit Cycle Amplitudes from A Frequency Domain Approach

In this paper, a problem in feedback control of limit cycle amplitudes is studied. A graphical approach for this bifurcation control problem is developed by means of higher-order harmonic balance approximations for both amplitude and frequency of the system oscillatory outputs. The approach starts with the familiar Hopf bifurcation mechanism, and employs the second, fourth, and sixth-order harmonic balance approximations to generate a sequence of graphical tests for convergence analysis of the system oscillatory outputs. This sequential graphical testing leads to accurate approximations of limit cycles of small amplitudes in the system outputs. Degenerate Hopf bifurcation theory is used to formulate an appropriate control objective of capturing small-amplitude limit cycles, which can avoid reaching unstable equilibria or other undesirable limit sets. A rich cubic planar model is presented for illustration of the proposed control method.