Incremental Harmonic Balance Method Research Articles

Dynamic Stability and Control of Vortex Induced Oscillations of Tension Leg Platform Tethers

Dynamic stability and control of sway oscillations of Tension Leg Platform (TLP) tether under regular vortex shedding are investigated. The TLP is modelled as a single degree of freedom simply supported uniform beam under pretension, assuming transverse oscillation to take place only in its fundamental natural mode. The tether is subjected to parametric excitation due to the heave motion of the TLP. The incremental harmonic balance (IHB) method is employed to investigate the period-1 and period-2 responses of the TLP as a function of the detuning parameter. The same system is then investigated under time-delayed state feedback control. Suitable control parameters are selected from the analysis of the local stability of the system equilibrium. The stability of solutions is analyzed by the semidiscretization method with the modified Floquet theory. The stable and periodic responses obtained by the IHB method are verified by directly integrating the equation of motion using the fifth-order Runge-Kutta algorithm at discrete points. Finally, the efficacy of the control scheme has been investigated in the plane of response amplitude versus excitation amplitude. The results show that appropriate selections of control parameters can effectively reduce the response amplitude to a great extent with the improved measure of stability.

Nonlinear dynamic modelling and analysis of a rotating composite thin-walled beam considering ice coating

For rotating structures operating in severe weather conditions, ice coating phenomenon is an established issue that impairs the reliability and effectiveness of the structures by changing the dynamic behaviors. To explore the influence mechanisms of ice coating on the dynamic behaviors of rotating structures, this work presents an analytical dynamic model for nonlinear forced vibration of a composite thin-walled beam under both rotating and ice coating conditions. The ice coating is considered as C-shaped configuration which can affect the system parameters of the beam, especially aerodynamic performance. With effect of out-of-plane warping taken into account, the nonlinear dynamic equation of the beam is established using Hamilton's principle. The Galerkin discrete method together with the incremental harmonic balance method (IHB) as well as pseudo arclength continuation technique are adopted to obtain the periodic response of the nonlinear system. Then, a detailed parametric study on natural frequency, frequency-response and bifurcation behaviors is numerically conducted. Special attention is paid to the effects of ice coating, rotating angular velocity, inflow velocity and design parameters on the linear and nonlinear dynamic behaviors of the beam. Results reveal a prominent change in natural frequency, vibration amplitude and cusp bifurcation of multivalued region under the combined effects of ice coating and rotating motion. Such ice-induced changes exhibit the requirement of vibration control and optimization design consideration for rotating structures operating in icing coating conditions.

Dynamic Analysis of a Double-Cable-Stayed Shallow Arch Model Under Multi-Frequency Excitation by IHB Method

To reveal the dynamic properties of cable-stayed bridges under complex external excitations, a double-cable-stayed shallow arch model induced by multi-frequency harmonic excitations is established. The cable and shallow arch exhibit primary or subharmonic resonances by sweeping excitations simultaneously and asynchronously. Galerkin method, one/two time scales Incremental Harmonic Balance (IHB) method, and Runge–Kutta (RK) method are applied to explore the nonlinear dynamic responses. Research results demonstrate that both the cables and shallow arch present soft and hard spring properties, despite the very weak hard spring of the shallow arch; one-way dynamic interaction among the cables and the shallow arch is discovered when sweeping excitations asynchronously.

Harmonic in-plane patch loading induced nonlinear dynamic instability of unidirectional and bidirectional stiffened composite plates

A finite element framework is established to study the nonlinear dynamic instability of stiffened plates subjected to in-plane harmonic patch loading. An eight-noded isoparametric degenerated shell element and a three-noded curved beam element are used to model the skin and the stiffener, respectively. The Green–Lagrange strain displacement relationship is adopted to formulate the system matrices for the plate and the stiffener, adopting the total Lagrangian approach. The Bolotin method is adopted to trace the boundaries of linear dynamic instability region. The Incremental Harmonic Balance (IHB) method is used to investigate the nonlinear instability behavior of stiffened plates. The effect of varying loading patches along with the number of stiffeners in unidirectional ( x-directional and y-directional) and bidirectional ( x as well as y-directional) stiffened plates on the nonlinear frequency response is studied. It is observed that the cross-stiffened plates with the higher number of stiffeners are dynamically more stable than the unidirectional stiffened plates. Moreover, the stiffened plates with smaller loading patches are dynamically more stable than those with larger loading patches.

Active Regulation of Elastic Waves in a Type of Two-Dimensional Periodic Structures With Piezoelectric Springs

Abstract Wave propagations exhibit direction and frequency selectivity in two-dimensional (2D) periodic structures, which provides possibilities to regulate wave dispersion and bandgap properties. Most of current researches focus on regulations of 1D waves, and there are few works about active regulations of 2D waves, especially in the structures with strong nonlinearities that have remarkable influences on dispersions. In this work, two types of 2D periodic nonlinear lattice structures with piezoelectric springs, which include a monatomic and a diatomic structure, are designed to implement controllable dispersion and propagation direction of 2D waves. Considering the strong nonlinearities caused by the cubic spring, dynamic models of the wave propagations in the two kinds of periodic structures are established, and an improved incremental harmonic balance (IHB) method is developed to implement efficient and accurate calculations of the 2D wave propagation. Influences of active and structural parameters on dispersion and bandgap properties are comprehensively studied, and the regulation ability of the piezoelectric springs is demonstrated where the proportional voltage constant is the active control parameter with particle displacements as the feedback. Results also show that a piezoelectric modulated bandgap and a critical wave vector region are created by positive and negative proportional constants, respectively, which indicate that the structures can be used to filter a wide range of low-frequency long-wavelength noises and waves at particular directions. The properties predicted by the improved IHB method are verified by numerical experiments.

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.

Outer isolated detached resonance curve and its implications of a two-stage nonlinear dynamic system

This paper presents an innovative study of a two-stage nonlinear dynamic system, providing insights into the occurrence of Large-Amplitude Long-Duration Vibration State (LALDVS) and outer isolated detached resonance curves (OIDRCs). The Lagrange equations and the incremental harmonic balance method (IHBM) are used to uncover the fundamental mechanism of OIDRCs and to discover two and three OIDRCs that have not been previously reported. Multiple OIDRCs could have significant practical implications in engineering, as they indicate the possibility of unpredictable and potentially hazardous vibrations under certain operating conditions. The existence of OIDRCs depends on the jump frequency point being a Neimark-Sacker bifurcation, and the resulting phase diagram provides vital insights into the stability and bifurcation behaviors of the system. This work also explores the effect of stiffness on the nonlinear response behavior of the system and identifies the range of stiffness values that lead to LALDVS. The correctness of the theoretical calculation results is validated by comparing them with vibration experiment and simulation calculations. These results have practical implications for engineering and energy harvesting research, including identifying potential hazards associated with vibration and designing systems that maximize energy harvesting efficiency. Overall, these findings are innovative and have significant implications for predicting potentially hazardous vibrations in engineering and the design of energy harvesting systems.

Periodic solutions and bifurcations of a tristable flutter-based energy harvester

This paper designs a tristable flutter-based energy harvester that can harvest large amplitude interwell wind-induced vibration energy. To explore its periodic responses and bifurcation characteristics, the harmonic balance method and the incremental harmonic balance method are employed to solve its approximate solutions by combining the two-point tracing algorithm. By analyzing the incremental harmonic balance solutions, three different response regions and five vibration types are found. The wind tunnel experiment verifies the existence of these vibration types and the correctness of incremental harmonic balance solutions. Five bifurcation points are analyzed using the Floquet theory, and two evolution paths of its periodic solutions are studied. Finally, the influences of the potential well depths on the stable periodic solutions are studied. The shallow potential energy well is beneficial to improving the response stability. This study aims to fill the gap in the semi-analytical and analytical solution of nonlinear flutter-based energy harvesters, contribute to the theoretical analysis of wind-induced vibration energy harvesting, and provide more possibilities for the design of high-performance wind-induced vibration energy harvesters.

Nonlinear dynamic instability and dynamic response of stiffened laminated composite plates subjected to in-plane pulsating patch loading

In this article, the nonlinear dynamic instability of stiffened laminated composite plates is studied in the finite element (FE) framework subjected to uniform in-plane harmonic patch loading. The harmonic load is applied to the two opposite sides of the stiffened plate. The linear and nonlinear time-history response analysis is also studied. The skin and the stiffener are modeled using an eight-node isoparametric degenerated shell element and a three-node curved beam element, respectively. A system of matrices is developed by considering the Green–Langrange strain–displacement relationship. In the linear case, the Bolotin method is used to analyze the dynamic instability region (DIR). The nonlinear instability behavior of the laminated composite stiffened plate is studied by applying the Incremental Harmonic Balance Method (IHB). The Newmark-β method is used to solve the linear and nonlinear time-history response equations to understand the instability behavior of the stiffened plates. The effect of the parameters such as the length of the in-plane loading patch, varying number of stiffeners in -direction and the position of the patch on the nonlinear vibrations and nonlinear dynamic response is examined.

Nonlinear coupled multi-mode vibrations of simply-supported cylindrical shells: Comparison studies

In spite of extensive studies on the nonlinear vibrations of cylindrical shells, the significant influences of some effective modes and the in-plane nonlinearity on the nonlinear coupled multi-mode vibrations are not clear. In this work, which is based on Donnell’s nonlinear shell theory and Amabili–Reddy’s third-order shear deformation theory, the nonlinear differential equations of motion of both the thin-walled and moderately thick cylindrical shells under the simply-supported boundary condition are established via Lagrange equations. These high-dimensional differential equations with the quadratic and cubic nonlinearities are solved by using an iteration procedure that is a combination of incremental harmonic balance method, pseudo-arclength method and extrapolation techniques. Turning and bifurcation points of the system are determined with the help of the direct method, the stability of solution of the frequency–response is examined by using the multi-variable Floquet theory. The numerical responses obtained by adopting two present shell theories are compared to investigate the influence of ignoring the in-plane nonlinearity on nonlinear vibrations. In the coupled multi-mode vibrations with respect to the fundamental mode m,n, apart from the regular modes i×m,j×ni=1,3,5,…,andj=0,1,2,3,… that include the axisymmetric and asymmetric modes, some irregular modes are taken into account to study the frequency–responses. Results show that present iteration procedure is efficient and successful to get the frequency–responses of the coupled multi-mode vibrations, and the influence of the irregular mode on the coupled multi-mode vibration is dependent on the relationship between the natural frequency of the irregular mode and that of the fundamental mode.

Quasi-periodic oscillation characteristics of a nonlinear energy sink system under harmonic excitation

Quasi-periodic oscillation characteristics of a nonlinear energy sink (NES) system under harmonic excitation are studied in this work. It is found that there exist uniformly distributed sidebands in frequency spectra of quasi-periodic oscillations. The traditional incremental harmonic balance (IHB) method and the IHB method with two time-scales are used to obtain accurate frequency components and amplitudes of periodic responses and two different quasi-periodic oscillations of the system, respectively, with variations of the external excitation frequency and amplitude. The two different quasi-periodic oscillation characteristics are analyzed; energy absorption of the NES system under different excitation frequencies and amplitudes is also analyzed. Stability of quasi-periodic solutions obtained from the IHB method with two time-scales is analyzed by the Floquet theory. Analytical results obtained from the traditional IHB method and the IHB method with two time-scales agree very well with those from numerical integration using the fourth-order Runge–Kutta method. Results show that the efficiency of energy absorption of a strongly quasi-periodic oscillation is greater than those of a periodic response and a weakly quasi-periodic oscillation.

Accurate nonlinear dynamic characteristics analysis of quasi-zero-stiffness vibration isolator via a modified incremental harmonic balance method

Accurate nonlinear dynamic characteristics analysis of quasi-zero-stiffness vibration isolator via a modified incremental harmonic balance method

Dynamical analysis of a fractional-order nonlinear two-degree-of-freedom vehicle system by incremental harmonic balance method

At present, there are few studies considering both nonlinear and fractional characteristics of suspension in vehicle systems. In this paper, a fractional nonlinear model of a quarter vehicle with two-degree-of-freedom (2-DOF) is innovatively proposed to describe the suspension system containing the viscoelastic material metal rubber. Given the lack of a general calculation scheme for the multi-degree-of-freedom fractional-order incremental harmonic balance method (IHBM), a general calculation scheme for the 2-DOF incremental harmonic balance method for nonlinear systems with fractional order is derived. The nonlinear dynamical properties of the presented model are acquired using this method. The accuracy of the proposed method is verified through a comparison with the power series expansion method. Afterward, the effects of the various parameters on the dynamic performance are analyzed. The vibration peak value of the fractional-order model is significantly higher than that of the integer-order model (IOM). Therefore, the suspension parameters should be designed with a margin when using IOM.

The Singularity of Expanded Jacobian Matrix in Incremental Harmonic Balance Method Directly Locates Bifurcation Points of Steady-State Responses

Abstract As a semi-analytical approach, the incremental harmonic balance (IHB) method is widely implemented for solving steady-state (including both periodic and quasi-periodic) responses through an iteration process. The iteration is carried out through a Jacobian matrix (JM) and a residual vector, both updated in each iteration. Though the JM is known to be singular at certain bifurcation points, the singularity is still an open question and could play a pivotal role in real applications. In this study, we define and calculate an expanded JM (EJM) by applying an expanded solution expression in the IHB iteration. The singularity of the EJM at several different bifurcation points is proved in a general manner, according to the bifurcation theory for equilibria in nonlinear dynamical systems. Given the possible bifurcation type, furthermore, the singularity is applied to locate the corresponding bifurcation point directly and precisely. Considered are the cases of the period-doubling, symmetry breaking, and Neimark-Sacker bifurcations of periodic and/or quasi-periodic responses.

Bifurcation and chaos of Li-doped 6,6,12-graphyne impacted by hydrogen atoms: A new way to explore hydrogen storage

Hydrogen energy, as a green new energy source, has great significance for environmental protection due to its widespread application. This article uses nonlinear dynamics theory to study the problem of hydrogen energy storage firstly, which is different from traditional methods by using simulation and experimental methods. In this paper, the bifurcation and chaos of Li-doped 6,6,12-graphyne impacted by hydrogen atoms are studied. The perturbation of 6,6,12-graphene by Li atoms is treated as the Gauss white noise. The incremental harmonic balance method (IHB) is used for drawing the amplitude-frequency response curve of the system under simple harmonic excitation. Theoretical analysis shows that the inhomogeneity of 6,6,12-graphyne stiffness can affect the system stability, which decides the actual hydrogen storage capacity of graphyne. And then simulation and experiment results confirm the results of theoretical analysis. The maximum increase hydrogen storage efficiency can be more than 10% by selecting appropriate distribution of Li atoms.The research results of this article have a certain promoting effect on the industrial application of hydrogen energy.

Nonlinear dynamics of ionic liquid enhanced soft composite membrane under electro-mechanical loading

The incorporation of ionic liquids (ILs) into dielectric elastomer composites has attracted significant interest due to their potential applications in soft actuators and optical devices. Their soft and hyperelastic nature make them a distinct dynamic. In this paper, nonlinear dynamics of IL enhanced soft composite (ILESC) dielectric membrane subjected to electro-mechanical loading is investigated. A mixed micromechanical model with the incorporation of an electric double layer (EDL) and Maxwell-Wagner-Sillars (MWS) polarization is developed to estimate mechanical and electrical properties of the ILESC. Governing equations are derived via an energy method with the framework of the hyperelastic membrane theory, Neo-Hookean constitutive model and the couple dielectric theory. These equations are discretized by Taylor series expansion (TSE) and differential quadrature (DQ) methods and are solved by incremental harmonic balance (IHB) method combined with arc-length technique. The developed model and results are verified by comparing to experimental data and previous results. The influence of volume fraction, shape and size of IL fillers, static and dynamic applied electric field and excitation amplitude on forced vibration and resonant response of the ILESC membrane are comprehensively analyzed. It is found that ILESC membrane with slender ellipsoidal ILs displays larger amplitude, lower peak amplitude frequency and more obvious softening behavior.

Periodic solutions and frequency lock-in of vortex-induced vibration energy harvesters with nonlinear stiffness

The phenomenon of amplitude jump usually means the presence of multi-solution ranges and unstable periodic solutions, which has not been fully explored in the field of vortex-induced vibration energy harvesting. In this study, semi-analytical periodic solutions of vortex-induced vibration energy harvesters (VIVEHs) are obtained using the incremental harmonic balance method, and their stabilities are determined via the Floquet theory. It is found that the VIVEH with linear stiffness and those with three-order nonlinear stiffness have four saddle-node bifurcation points and two multi-solution ranges, which leads to the occurrence of amplitude jumps. The hardening effect can shift frequency lock-in regions to higher wind speeds, thus increasing the aerodynamic forces, and realizing wide wind speed bandwidths and high voltage outputs. The wind tunnel experiments verify the accuracy of theoretical calculations, the occurrence of amplitude jump phenomena, and the existence of multi-solution ranges. The VIVEH with nonlinear stiffness experimentally reached a maximum voltage of 9.87 V, and a frequency lock-in region of 2.27–5.36 m/s.

A modified IHB method for nonlinear dynamic and thermal coupling analysis of rotor-bearing systems

This paper presents a modified incremental harmonic balance (IHB) method to analyze the nonlinear dynamic and thermal coupling characteristics of rotor-bearing systems after thermal balance. The IHB method is modified by tensor contraction and fast Fourier transform (FFT) to efficiently calculate the residual vector and Jacobian matrix. Nonlinear dynamic and thermal full coupling is implemented through the nonlinear part of the Jacobian matrix. The presented method achieves the simultaneous and efficient calculation of the motion and heat balance equations in the frequency domain according to the improvement measures. The effectiveness of the presented method is tested using the vibration response and temperature variation analysis for the coupled thermo-mechanical model of a rotor-ball bearing system. Furthermore, the performance comparisons show the good consistency of results between the presented method and the step-by-step (S-S) method, which is a partitioned iterative solution method for dealing with fully coupled problems in the time domain. The presented method has greater computational efficiency, more accurate temperature prediction, and better recognition of nonlinear phenomena in the system rather than the S-S method. Consequently, the presented method has a broad application prospect in solving fully coupled thermo-mechanical problems of more complex and higher dimensional rotor-bearing systems.

Analysis on nonlinear characteristic of coupling heave and pitch motions of sandglass-type floating body

For its special shape of floating body with oblique side and external expansion, the heave and pitch motions of the sandglass-type Floating Production Storage and Offloading unit (FPSO) have coupling influence and instability phenomenon under harmonic wave excitation force. In order to study the nonlinear characteristic of the sandglass-type floating body, this paper presents the improved incremental harmonic balance (IHB) method in conjunction with the incremental arc-length method, and adopts the Floquet theory to determine the stability of the periodic solutions. Therein, the discrete Fourier transformation is introduced in the traditional IHB method to solve the problem of no analytical expressions for nonlinear FK and hydrostatic forces of the sandglass-type floating body. Considering the dependence of the heave and pitch hydrodynamic coefficients on wave frequency, the coefficients of the Fourier series in the IHB method are modified to cope with the memory effect. Taking a classic Spar platform as the test case, the accuracy of the method in this paper is validated and can be used to analyze the nonlinear coupling stability of the large-scale floating body. Then, the nonlinear responses corresponding to single-DOF heave and single-DOF pitch respectively are studied, and the nonlinear characteristics of hardening-type stiffness are verified by considering the different effects of damping and wave amplitude. Next, the interactive effects between pitch and heave motions of the sandglass-type model are further discussed, which are mainly reflected in two frequency bands representing the hardening-type nonlinearity of pitch and heave motions. Finally, the influence of different upper inclination angle of the floating body on heave and pitch coupling motions is analyzed. The results show that the smaller the upper angle, the larger the nonlinear motion responses. Consequently, the nonlinear characteristics of the heave and pitch motions should be fully noticed and considered in the shape design of the sandglass-type FPSO.

Nonlinear vibration analysis and stability analysis of rotor systems supported on SFD by combining DQFEM, CMS and IHB methods

In this paper, the differential quadrature finite element method (DQFEM), component mode synthesis (CMS) method and incremental harmonic balance (IHB) method are combined to propose a methodology to perform vibration analysis and stability analysis of rotor systems supported on fluid-film bearings that exhibit localized nonlinearities such as squeeze-film damper (SFD). In multi-degree of freedom systems with SFD, a method for dynamic analysis of the system by reducing the degrees of freedom is first suggested in this paper. DQFEM are used to create the mass matrix, stiffness matrix, and gyroscopic matrix of the system. Then, the system is divided into linear system and nonlinear system, and the motion equation of the linear system reduces the degrees of freedom using the free interface CMS method, and the motion equation of the nonlinear system is written without reducing the degrees of freedom. By synthesizing the equations of motion of each component, the motion equation of the entire system is created. That is, the motion equation of the whole system uses hybrid coordinates. The motion equation of the system is solved using the modified IHB method. To evaluate the effectiveness of the suggested method, the computed solutions are compared with the results presented in the previous literature, and the results are very consistent. Also, stability analysis of the obtained solutions is performed by the Floquet theory. The method suggested in the paper can be effectively used to analyze the dynamic behavior of rotor systems exhibiting localized nonlinearities such as SFD.