Nonlinear Frequency Response Analysis Research Articles

Analysing dynamic deep stall recovery using a nonlinear frequency approach

Based on bifurcation theory, nonlinear frequency response analysis is a recent development in the field of flight dynamics studies. Here, we consider how this method can be used to inform us on how to devise the control input such that the system transitions from an undesirable equilibrium solution—an aircraft deep stall solution in our case—to a desirable solution. We show that it is still possible to induce a large-amplitude oscillation via harmonic forcing of the pitch control device and escape the otherwise unrecoverable deep stall, despite very little control power available in such a high angle-of-attack flight condition. The forcing frequencies that excite these resonances are reflected as asymptotically unstable solutions using bifurcation analysis and Floquet theory. Due to the softening behaviour observed in the frequency response, these unstable (divergent) solutions have slightly lower frequencies than the value predicted using linear analysis. Subharmonic resonances are also detected, which are reflected in the time-domain unforced responses. These nonlinear phenomena show strong dependency on the forcing/perturbation amplitude and result in complex dynamics that can impede recovery if the existing procedures are followed. The proposed method is shown to be a useful tool for nonlinear flight dynamics analysis as well as to complement the rather thin literature on deep stall analysis—a topic of relevance for recent research on unconventional landing techniques in unmanned aerial vehicles. A full description of the aircraft model used, the unstable F-16 fighter jet, is provided in the appendix.

Capacity Fade Detection in Lithium-Ion Batteries Using Non-Linear Frequency Response Analysis(Nfra) Under Multiple Open-Circuit Voltages (Ocvs)

Capacity Fade Detection in Lithium-Ion Batteries Using Non-Linear Frequency Response Analysis(Nfra) Under Multiple Open-Circuit Voltages (Ocvs)

Bending-torsional-axial-pendular nonlinear dynamic modeling and frequency response analysis of a marine double-helical gear drive system considering backlash

The double-helical gear system was widely used in ship transmission. In order to study the influence of backlash on the nonlinear frequency response characteristics of marine double-helical gear system, according to the structural characteristics of double-helical gear transmission, considering the time-varying meshing stiffness, backlash, damping, comprehensive transmission error, external load excitation, and other factors, a three-dimensional bending-torsional-axial-pendular coupling nonlinear dynamic modeling and dynamic differential equation of 24-DOF double-helical gear transmission system were established. The Runge–Kutta numerical method was used to analyze the influence of backlash, time-varying meshing stiffness, damping, error and external load excitation on the amplitude frequency characteristics. The results show that the backlash can cause the runout of the double-helical gear system, and the system has first harmonic and second harmonic response. With the increase of backlash, the amplitude of the system increases and the jumping phenomenon remains unchanged. The amplitude frequency response of the system is stimulated by time-varying meshing stiffness and comprehensive transmission error, and restrained by damping and external load excitation. The vibration displacement amplitude of the system increases with the increase of vibration displacement and has little effect on the state change of the system. The vibration test of double-helical gear is carried out. The frequency response components obtained by numerical simulation are basically consistent with the experimental results, which proves the correctness of the theoretical calculation. It provides a technical basis for the study of vibration and noise reduction performance of double-helical gear.

Nonlinear frequency response analysis: a recent review and perspectives

The nonlinear frequency response analysis (NFRA) can be seen as an extension of electrochemical impedance spectroscopy. NFRA gives a full and detailed representation of the system response and can establish a connection between model parameters and the experimentally observed phenomena. In this article, different theoretical NFRA approaches and the most recent application examples are discussed. A simple electrochemical example is used to showcase the benefits and disadvantages of analyzing the system response by using different approaches. In addition, it was shown how to extract experimental harmonic values and analyze them.

A machine learning based Bayesian optimization solution to non-linear responses in dusty plasmas

Nonlinear frequency response analysis is a widely used method for determining system dynamics in the presence of nonlinearities. In dusty plasmas, the plasma–grain interaction (e.g. grain charging fluctuations) can be characterized by a single-particle non-linear response analysis, while grain–grain non-linear interactions can be determined by a multi-particle non-linear response analysis. Here a machine learning-based method to determine the equation of motion in the non-linear response analysis for dust particles in plasmas is presented. Searching the parameter space in a Bayesian manner allows an efficient optimization of the parameters needed to match simulated non-linear response curves to experimentally measured non-linear response curves.

Electrostatically actuated double walled piezoelectric nanoshell subjected to nonlinear van der Waals effect: nonclassical vibrations and stability analysis

Abstract In this paper, nonlinear vibration and frequency response analysis of double walled piezoelectric nanoshell (DWPENS) is investigated using nonclassical approach of the Gurtin–Murdoch surface/interface (GMSIT) theory. The piezoelectric nanoshell is simultaneously subjected to visco-Pasternak medium, the nonlinear van der Waals and electrostatic forces. Hamilton’s principles, the assumed mode method combined with Lagrange–Euler’s are used for the governing equations and boundary conditions. Complex averaging method combined with Arc-length continuation is used to achieve the nonlinear frequency response and stability analysis of the DWPENS. It is found that the electrostatic and piezoelectric voltages, the length to radius ratio, the nanoshell gap width, van der Waals (vdW) coefficients and other parameters can effectively change the flexural rigidity of the system which in turn affects the nonlinear frequency response. And also, increasing or decreasing of some parameters lead to increasing or decreasing the resonance amplitude, resonant frequency, the system’s instability, nonlinear behavior, and bandwidth.

Nonlinear Frequency Response Analysis of Double-helical Gear Pair Based on the Incremental Harmonic Balance Method

The torsional dynamic model of double-helical gear pair considering time-varying meshing stiffness, constant backlash, dynamic backlash, static transmission error, and external dynamic excitation was established. The frequency response characteristics of the system under constant and dynamic backlashes were solved by the incremental harmonic balance method, and the results were further verified by the numerical integration method. At the same time, the influence of time-varying meshing stiffness, damping, static transmission error, and external load excitation on the amplitude frequency characteristics of the system was analyzed. The results show that there is not only main harmonic response but also superharmonic response in the system. The time-varying meshing stiffness and static transmission error can stimulate the amplitude frequency response of the system, while the damping can restrain the amplitude frequency response of the system. Changing the external load excitation has little effect on the amplitude frequency response state change of the system. Compared with the constant backlash, increasing the dynamic backlash amplitude can further control the nonlinear vibration of the gear system.

Computer-Aided Nonlinear Frequency Response Method for Investigating the Dynamics of Chemical Engineering Systems

The Nonlinear Frequency Response (NFR) method is a useful Process Systems Engineering tool for developing experimental techniques and periodic processes that exploit the system nonlinearity. The basic and most time-consuming step of the NFR method is the derivation of frequency response functions (FRFs). The computer-aided Nonlinear Frequency Response (cNFR) method, presented in this work, uses a software application for automatic derivation of the FRFs, thus making the NFR analysis much simpler, even for systems with complex dynamics. The cNFR application uses an Excel user-friendly interface for defining the model equations and variables, and MATLAB code which performs analytical derivations. As a result, the cNFR application generates MATLAB files containing the derived FRFs in a symbolic and algebraic vector form. In this paper, the software is explained in detail and illustrated through: (1) analysis of periodic operation of an isothermal continuous stirred-tank reactor with a simple reaction mechanism, and (2) experimental identification of electrochemical oxygen reduction reaction.

Nonlinear Primary Frequency Response Analysis of Self-Sustaining Nanobeam Considering Surface Elasticity

This paper is focused to investigate the effects of nonlinear sources, including viscoelastic foundation and geometrical nonlinearity along with the surface elasticity and residual surface stress effects on the primary frequency response of a harmonically excited nanoscale Bernoulli-Euler beam. Due to large surface-area-to-volume ratio, the theory of surface elasticity as well as residual surface stress effects are taken into account within the beam models. The Galerkin approach accompanied by trigonometric shape functions is utilized to reduce the governing PDEs of the system to ODEs. The multiple scales perturbation method theory is applied to compute the nonlinear frequency response of nanobeam. The effects of linear and nonlinear viscoelastic damping coefficient of the medium, crystallographic directions of [100] and [111] of anodic alumina, geometrical nonlinear term and geometrical property on the nonlinear primary frequency response of nanoscale beam are investigated. The results show that theses parameters have a significant effect on the nonlinear frequency response of nanobeams in the case of primary resonance.

Nonlinear frequency response analysis of forced periodic operations with simultaneous modulation of two general waveform inputs with applications on adiabatic CSTR with square-wave modulations

Nonlinear frequency response (NFR) method was used to evaluate the forced periodic operations and their potential for improving the system performance compared to steady-state operations. The NFR method was used to evaluate the time-average performances of forced periodically operated systems subjected to simultaneous periodic modulation of two-inputs of general waveforms. As an example, a forced periodically operated adiabatic CSTR with simple nth order reaction A → νPP and square-wave modulation of two pairs of inputs were analyzed in detail. The theory is illustrated analyzing the hydrolysis of acetic acid anhydride performed in a laboratory scale adiabatic CSTR. The results showed that significant improvements could be achieved by simultaneous modulation of both pairs of inputs. Comparison with the results of numerical integration showed that the approximation using only the asymmetrical second order FRFs and the first three harmonics of the inputs provide very reliable predictions of the DC component.

Nonlinear vibration and stability analysis of piezo-harmo-electrostatic nanoresonator based on surface/interface and nonlocal strain gradient effects

In the current study, nonlinear vibration and stability analysis of piezo-harmo-electrostatic nanoresonator (PHENR) based on cylindrical nanoshell is investigated to compare nonlocal (NLT), nonlocal strain gradient (NSGT) and Gurtin–Murdoch surface/interface (GMSIT) theories with classical theory (CT). PHENR is simultaneously subjected to direct electrostatic voltage DC and harmonic excitations, structural damping and two piezoelectric layers. For this analysis, Hamilton’s principle and Galerkin technique, respectively, are used to obtain the governing equations and boundary conditions and also to solve the equation of motion. Complex averaging method combined with arc-length continuation is used to investigate dimensionless natural frequency (DNF), pull-in voltage, nonlinear frequency response and stability analysis of the piezo-harmo-electrostatic nanoresonator considering the nonlocal, material length scale, surface/interface effects, electrostatic and harmonic excitation. For this work, three nonclassical theories of NLT, NSGT and GMSIT are compared with classical theory (CT). It is concluded that ignoring surface/interface and small-scale effects leads to inaccurate results in vibrational response of the PHENR. It is found that in all boundary conditions, nonlocal scale parameter $$\bar{\mu }$$ and material length scale parameter $$\bar{\eta }$$, respectively, lead to increase and decrease in PHENR stiffness and in the nonclassical theories of NLT and NSGT, the oscillation amplitude, the range of instability and all nonlinear behavior is greater than that of the classical one. Also an increase in surface/interface parameters leads to increase or decrease in the dimensionless natural frequency, resonance amplitude, resonant frequency, the system’s instability and nonlinear behavior of PHENR.

The effects of nonlocal and surface/interface parameters on nonlinear vibrations of piezoelectric nanoresonator

In current work, Gurtin–Murdoch surface/interface and nonlocal theories are presented to investigate nonlinear vibration and stability analysis of piezoelectric nanoresonator simultaneously subjected to direct and alternating voltages. Complex averaging method combined with arc-length continuation is used to achieve the nonlinear frequency response and stability analysis of the piezoelectric nanoresonator. It is concluded that ignoring surface and small-scale effects lead to inaccurate results in vibration response of the piezoelectric nanoresonator. Also, with increasing dimensionless nonlocal parameter and decreasing nanoshell stiffness, dimensionless natural frequency, the resonance amplitude, the range of instability with saddle-node bifurcations and nonlinear hardening behavior, and also resonance frequency decrease. In addition, it is found that when the surface/interface effects corresponding to higher surface/interface density are considered, natural frequency is increased. Also, regardless the Lamé’s constants lead to decreasing of piezoelectric nanoresonator stiffness, and as a result, the dimensionless natural frequency is lower than all cases. The rest of the surface/interface parameters do not have much effect on the dimensionless natural frequency. Also, with increasing electrostatic and piezoelectric voltages, the oscillation amplitude and range of the piezoelectric nanoresonator system’s instability and all nonlinear behavior of piezoelectric nanoresonator are increased.

Oxygen reduction reaction on silver electrodes under strong alkaline conditions

Oxygen reduction reaction (ORR) was studied on a silver polycrystalline electrode in different NaOH concentrations with a help of rotating disc and rotating ring disc electrodes. Soluble reaction intermediate was detected at all alkaline concentrations, but its concentration increased with an increase of the level of impurities. ORR is not NaOH concentration dependent at low concentrations (0.1 and 1 M). In 11 M NaOH ORR onsets at more positive potentials in the region where underpotential silver oxide formation shows less reversible behaviour. The nonlinear frequency response analysis spectra show significant qualitative difference with NaOH concentration indicating a high capability of this method for kinetic mechanism investigations.

Nonlinear dynamics and stability analysis of piezo-visco medium nanoshell resonator with electrostatic and harmonic actuation

• Nonlinear dynamics, vibration and stability analysis of cylindrical nanoshell resonator is studied. • Electro-elastic Gurtin–Murdoch surface/interface theory with von-Karman–Donnell's shell model are used. • The effects of different parameters on nonlinear frequency response and stability analysis are investigated. • Effects of electrostatic and harmonic excitation on nonlinear frequency response and stability analysis are studied. In this paper, nonlinear dynamics, vibration and stability analysis of piezo-visco medium nanoshell resonator (PVM-NSR) based on functionally graded (FG) cylindrical nanoshell integrated with two piezoelectric layers subjected to visco-pasternak medium, electrostatic and harmonic excitations is investigated. Nonclassical method of the electro-elastic Gurtin–Murdoch surface/interface theory with von-Karman–Donnell's shell model as well as Hamilton's principle, the assumed mode method combined with Lagrange–Euler's are considered. Complex averaging method combined with arc-length continuation is used to achieve a numerical solution for the steady state vibrations of the system. The stability analysis of the steady state response is performed. The parametric studies such as the effects of different boundary conditions, different geometric ratios, structural parameters, electrostatic and harmonic excitation on the nonlinear frequency response and stability analysis are studied. The results indicate that near the natural frequency of the nanoshell, it will lead to resonance and will have large motion amplitude and near the resonant frequency, the nanoshell shows a softening type of nonlinear behavior, and the nanoshell bandwidth increases due to nonlinear factors. In this range, nanoshell has three different ranges of motion, of which two are stable and the other unstable, and so the jump phenomenon and saddle-node bifurcation are visible in the behavior of the system. Also piezoelectric voltage influences on static deformation and resonant frequency but has no significant effect on nonlinear behavior and bandwidth and also system very sensitive to the damping coefficient and due to decrease of nano shell stiffness, natural frequency decreases. And also, increasing or decreasing of some parameters lead to increasing or decreasing the resonance amplitude, resonant frequency, the system's instability, nonlinear behavior and bandwidth.

Nonlinear Frequency Response Analysis as a Tool for Identification of Adsorption Kinetics: Case Study—Pore-Surface Diffusion Control

In the present paper, the Nonlinear Frequency Response (NFR) analysis is applied for theoretical study of kinetics of adsorption governed by pore-surface diffusion. The concept of higher-order frequency response functions (FRFs) is used. Based on a nonlinear mathematical model for adsorption of pure gas and spherical adsorbent particles, the theoretical first- and second-order FRFs, which relate the adsorbate concentration in the particle to the surrounding pressure (F1(ω) and F2(ω,ω)), have been derived. The obtained FRFs have been simulated for different steady-state pressures and different ratios (between zero and one) of surface to pore diffusion coefficients. The results show that, unlike F1(ω), F2(ω,ω) exhibits features which unambiguously distinguish the pore-surface diffusion model from pure pore diffusion and micropore diffusion. Based on the characteristic features of F1(ω) and F2(ω,ω), a new methodology for direct estimation of the separate values of the pore and surface diffusion coefficients has been established.

Non-linear Frequency Response and Stability Analysis of Piezoelectric Nanoresonator Subjected to Electrostatic Excitation

Abstract The effects of surface energy on the non-linear frequency response and stability analysis of piezoelectric cylindrical nano-shell as piezoelectric nanoresonator are investigated in the current paper using Gurtin–Murdoch surface elasticity and von Karman–Donnell’s theory. The nanoresonator is embedded in visco-Pasternak medium and electrostatic excitation. The governing equations and boundary conditions are derived using Hamilton’s principle and also the assumed mode method is used for changing the partial differential equations into ordinary differential equations. Complex averaging method combined with arc-length continuation is used to achieve an approximate solution for the steady-state vibrations of the system. The validation of the mentioned system is achieved with excellent agreements by comparison with numerical results. The parametric studies such as the effects of geometrical and material properties, different boundary conditions, the ratio of length to radius L / R $L/R$ for different values of the voltages V D C ${V_{{\rm{DC}}}}$ and V A C ${V_{{\rm{AC}}}}$ , the gap width of the nanoresonator b / L $b/L$ , the effect of the voltages V D C ${V_{{\rm{DC}}}}$ and V A C ${V_{{\rm{AC}}}}$ and also the effect of piezoelectric voltage V p ${V_p}$ are conducted on the non-linear frequency response and stability analysis of the piezoelectric nanoresonator.

Understanding nonlinearity in electrochemical systems

Commonly applied electrochemical methods for the analysis and diagnosis of the processes in and state of electrochemical cells, such as Electrochemical Impedance Spectroscopy and Current Interrupt analysis, are either limited to linear analysis or the signal is quite unspecific which hampers to extract significant and precise nonlinear information. We present a systematic insight into how significant information can be extracted from a promising alternative nonlinear dynamic electrochemical analysis technique, Nonlinear Frequency Response Analysis. Further, we present a fundamental and in-depth study of impact of processes at electrodes on nonlinear behavior. Model based analysis of a reaction process with Butler-Volmer kinetics and of a diffusion process are thereby used to understand and interpret the excitation of higher harmonics. A reaction with a symmetric current potential relation thereby causes an excitation of the third harmonic, whereas for the diffusion process the second and third harmonic are excited. Nonlinearities caused by diffusion are limited to low frequencies. Further, parameter variations of exchange current density, double layer capacitance and diffusion coefficient as well as variation of the input signal show that the symmetry of the nonlinear behavior between current and potential is responsible for the excitation of the second and third harmonic. The tangent method is presented as a suitable method to quantitatively evaluate and compare influences of each process and parameter on spectrum features and the related characteristic frequency range. The work thus serves as a guideline for using and interpreting nonlinear frequency response spectra.

State-of-Health Diagnosis of Lithium-Ion Batteries Using Nonlinear Frequency Response Analysis

Estimation of the State-of-Health (SOH) of Lithium-ion Batteries (LIBs) is commonly conducted using in-situ measurement methods, such as Incremental Capacity Analysis (ICA) and Differential Voltage Analysis (DVA) as well as impedance based techniques. In this study, we present an alternative method for SOH estimation: The nonlinear dynamic measurement method Nonlinear Frequency Response Analysis (NFRA) is shown to be able to estimate capacity fade of LIBs due to loss of active material. Capacity loss correlates with the quotient of the root mean square of the second and the third harmonic response for different excitation amplitudes in the frequency range sensitive to electrochemical reactions at approximately 1 Hz. The results of the experimental cycle-aging study are validated and further analyzed by using a reaction model containing Butler-Volmer kinetics with a dynamic charge balance of the electrode. Simulations show that the NFR quotient and capacity fade due to loss of specific surface area correlate exactly. We identify the NFR quotient as a reliable, easily measurable parameter for the diagnosis of the SOH of LIBs. Therefore, this study reveals a novel approach for SOH estimation of LIBs based on dynamic analysis with NFRA.

Nonlinear frequency response analysis on lithium-ion batteries: Process identification and differences between transient and steady-state behavior

Electrochemical Impedance Spectroscopy (EIS) is the most commonly applied method for electrochemical characterization of processes in batteries. Nevertheless, one of its limitations is the linear excitation of the system, denoting loss of information about the natural nonlinearities of the processes in batteries. Here, the potential of a nonlinear dynamic analysis method such as Nonlinear Frequency Response Analysis (NFRA) can be revealed. A reliable interpretation of the response signal of a LiB requires to understand its sensitivities to key parameters and processes. This work thus uses a Single-particle battery model to systematically analyze the sensitivity of the response to input and cell parameters and to the evaluation time. Thereby, additional experimental evaluation reveals that an initially excited system shows higher nonlinearities than the same system in its steady-state excitation state; yet, the steady-state signal should provide for a more reliable interpretation, especially regarding experimental effects. Simulation thereby indicates similar time constants for the initial excitation and the EIS. NFRA is also shown to be an excellent method to study the nature of the Lithium-ion transport process across the SEI/electrolyte interface. In contrast to EIS, NFRA allows to discriminate whether the process is nonlinear or linear, as in most transport processes. Further, ageing effects caused by changes due to the SEI are shown to influence the spectrum in the low-, mid- and high-frequency domain and changes in the reaction and diffusion influence the mid to low and low frequency range, respectively. Individual higher harmonics show different sensitivities to the single nonlinear processes, which can be used for more in-depth studies of processes at its characteristic frequency range. The presented methodology to use NFRA promises to significantly advance electrochemical characterization and state estimation of Lithium-ion batteries. Combination of NFRA and EIS is further feasible to gain an even deeper characterization of processes than with the conventional EIS alone.

Identification of Lithium Plating in Lithium-Ion Batteries using Nonlinear Frequency Response Analysis (NFRA)

Nonlinear Frequency Response Analysis (NFRA) is a novel dynamic analysis method for Lithium-ion batteries. In contrast to the most commonly applied Electrochemical Impedance Spectroscopy (EIS), NFRA is not limited to the linear response of the system, as it uses higher sinusoidal excitation currents IAC of approximately 1.5 C. Higher harmonic response signals Yn with n2 are analysed. Thereby, additional dynamic information about the system is accessible. Here, the authors evaluate the potential of NFRA to detect the safety critical ageing process of Lithium plating. NFRA results are analysed and compared to EIS for cells aged at −10∘C with plating occurence and for cells without plating aged at 25∘C. A proof of concept for detection of Lithium plating with NFRA is presented. The third harmonic, Y3, is strongly sensitive to plated Lithium at time constants characteristic for electrochemical reactions for the analysed cells. Here, Lithium plating causes an increase of charge transfer resistance between the anode and the electrolyte, which affects the nonlinear behaviour. Ex-situ analysis validated that plating occurs only for the cells aged at −10∘C, but not for the cells aged at 25∘C. Further, it is presented that Y1, analysed with EIS, is not sufficient for the diagnosis of this ageing process. Therefore, NFRA is suggested as a new, reliable method for the identification of Lithium plating in Lithium-ion batteries.