Nonlinear frequency response analysis using MSC Nastran

A novel approach for the dynamic analysis of Lithium-ion batteries is the investigation of higher harmonic voltage responses with Nonlinear Frequency Response Analysis (NFRA). This is commonly applied in the field of acoustics [1], material science [2], for fuel cells [3] and as a state of charge estimator for Lead-Acid batteries [4]. NFRA is an adaption and advancement of Electrochemical Impedance Spectroscopy (EIS). In contrast to EIS, NFRA allows to study nonlinear processes in Lithium-ion batteries, such as electrode reactions and solid diffusion in spherical particles, without linearization. By analyzing the NFRA spectrum and correlating the dynamic responses with the corresponding EIS spectrum, a novel and deeper insight into battery processes can be provided. We will show the application of NFRA on Lithium-ion batteries with a model-based study for optimal identification and quantification of the individual processes. A state-of-the-art Single Particle Doyle-Newman model [5] is extended with a SEI and double layers at each interface and adapted to simulate NFRA of Lithium-ion batteries. In contrary to EIS, the system is excited with a high amplitude guaranteeing nonlinear behavior. The model output in the time domain is transferred via Fast-Fourier-Transformation (FFT) to the frequency domain, thereby accessing the amplitudes of the individual harmonics Y n as well as their sum in dependency of the frequency. Once a system is excited with a sinusoidal input current, it will reach for a dynamic-steady-state. We will show how state variables, e.g. solid concentration and double-layer potentials, contribute to this behavior at characteristic frequencies defined for the NFRA spectrum in Figure 1. Thereby investigating the number of oscillations needed for dynamic-steady-state analysis. With decreasing frequency, ranging from 1131 Hz to 5.3 Hz to 0.3 Hz, the influence of the anode solid concentration csurf,a increases until it becomes dominant in the low frequency region at 0.3 Hz. Further, only the double-layer potentials Δϕ a and Δϕ SEI at the anode side are influenced, indicating the anode as limiting electrode. We further focus on the investigation of the dynamic-steady-state for process identification by parameter and amplitude variations. The impact of the variation of diffusion coefficient, reaction rates and double-layer capacitances is demonstrated. Changes in SEI processes influence the overall excited frequency range, whereas changes in reaction and diffusion processes affect only certain frequency regions. Thus, frequency ranges can be assigned to processes, therefore enabling process identification. Additionally, with a variation of processes, the amplitude dependency changes. Further, individual harmonics show different characteristic progressions within the overall frequency range. Process identification on Lithium-ion batteries in general can highly benefit from these findings. Besides NFRA, we provide EIS spectra, simulated with small excitation amplitudes for linear excitation of the system, for comparison to NFRA simulation results. With NFRA, a powerful new dynamic analysis tool is created, accessing a new dimension to the electrochemical characterization of Lithium-ion batteries. Literature: [1] Everest, F. A., et al., Master Handbook of Acoustics, The McGraw-Hill Companies, 2009. [2] Wilson, J. R., et al., Electrochimica Acta, 51, p. 1389–1402, 2006. [3] Mao, Qing, et al., Electrochemistry Communications, 12, p. 1517-1519, 2010. [4] S. Okazaki, J. Electrochemical Society 132, p. 1516, 1985. [5] Doyle, Marc, et al., J. Electrochem. Soc., 143, (6), p. 1890-1903, 1996. Figure 1

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.

Determination of residual capacity, or ‘state of health’ (SOH), is important for the management of battery assets in service and when assessing batteries for re-use at the end of service. Nonlinear frequency response analysis (NFRA) has recently been demonstrated as a means to determine SOH, but the effects of temperature and state-of-charge (SOC) on battery NFRA measurements are not yet well understood. In this work, we investigate the effects of SOC on NFRA measurements performed on commercial 21700 cylindrical cells aged under different temperatures to various SOHs. Total harmonic distortion (THD%) is proposed as a capacity fade indicator. THD% in the range of 10 Hz–150 Hz are found to correlate well with SOH at all SOCs. For frequency < 1 Hz, the effect of aging on THD% is SOC dependent, with larger THDs for the new cell at low open-circuit voltages (OCVs) (2.75 V–3.2 V), and larger THDs for aged cells at higher OCVs (3.5 V–4.1 V). This work also demonstrates the feasibility of NFRA to distinguish between cells that have been aged differently based on temperature.

The computation of the steady-stateresponse of large finite element discretized systems subject to periodic excitations is unfeasible because of excessive run time and memory requirements. One could in principle resort to reduced order models stemming from the high fidelity counterparts, which typically require a solution time orders of magnitude smaller. However, when many simulations are required, as in the case of parametric studies, the overall effort could be still significant and the analysis process could be severely hindered. In this work, we propose a sensitivity approach to assess the influence of model parameters on the nonlinear dynamic response. As opposed to the costly evaluation of reduced order solutions over a range of excitation frequencies and model parameters, the sensitivities of a nominal response allow one to approximate the dynamic response by a simple evaluation of an expansion in the directions spanning the parameter space. Special care must be taken on the closure equation that needs to be appended to the system of equations stemming from the harmonic balance method. We discuss the limitations of the current constant frequency approach and propose an improvement. We demonstrate the merits of the proposed approach on a micro-electro-mechanical system affected by parameterized manufacturing defects. Leveraging from a previous contribution, the nonlinear response and the sensitivities are obtained from a reduced order model which is analytical in the defect parameters. Our procedure is able to deliver accurate probability density functions of quantities of interest (e.g. nonlinear resonance peaks, triple solution bandwidth, etc) against statistical distributions of manufacturing defects at negligible computational cost.

Various impedance-based and nonlinear frequency response-based methods for determining the state-of-health (SOH) of commercial lithium-ion cells are evaluated. Frequency response-based measurements provide a spectral representation of dynamics of underlying physicochemical processes in the cell, giving evidence about its internal physical state. The investigated methods can be carried out more rapidly than controlled full discharge and thus constitute prospectively more efficient measurement procedures to determine the SOH of aged lithium-ion cells. We systematically investigate direct use of electrochemical impedance spectroscopy (EIS) data, equivalent circuit fits to EIS, distribution of relaxation times analysis on EIS, and nonlinear frequency response analysis. SOH prediction models are developed by correlating key parameters of each method with conventional capacity measurement (i.e., current integration). The practical feasibility, reliability and uncertainty of each of the established SOH models are considered: all models show average RMS error in the range 0.75%–1.5% SOH units, attributable principally to cell-to-cell variation. Methods based on processed data (equivalent circuit, distribution of relaxation times) are more experimentally and numerically demanding but show lower average uncertainties and may offer more flexibility for future application.

Nonlinear frequency response functions of a chromatographic column—A critical evaluation of their potential for estimation of single solute adsorption isotherms

Non-linear vibrations and frequency response analysis of piezoelectrically driven microcantilevers

The nonlinear frequency response of a chromatographic column for the adsorption of two dissolved components is analyzed using the concept of higher order frequency response functions (FRFs) which is based on the Volterra series and generalized Fourier transform. By applying this concept a nonlinear model of a system is replaced by an infinite series of the FRFs of the first, second, etc. order. The FRFs up to the third order are derived theoretically starting from the equilibrium-dispersive model, which is used for description of a chromatographic column, and applying the harmonic probing method. The functions that relate outlet concentration changes of each component to the corresponding inlet concentration changes are derived. At the inlet of a chromatographic column, it is considered: (a) the concentration change of one of the components keeping the concentration of the other component constant and (b) the concentration change of both components keeping their ratio constant. The FRFs are calculated numerically for different steady-state concentrations and relative mixture compositions. It has been found that, despite certain differences in initial conditions, the FRFs exhibit similar behavior. For higher frequencies, the amplitudes of the FRFs tend to zero and phases to −∞. In the low frequency range, which is of interest for investigation of equilibrium parameters, these functions have similar behavior, but tend to different asymptotic values. Correlations between coefficients of competitive adsorption isotherms, i.e. partial isotherm derivatives, and the derived FRFs are established. This theoretical result offers the potential to use the analysis of the nonlinear frequency response of a chromatographic column for estimation of competitive adsorption isotherms.

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View Video Presentation: https://doi.org/10.2514/6.2021-0759.vid Current methods of assessing flying qualities are based on linear system theory, which cannot capture the nonlinear phenomena commonly found at high angles of attack. Although bifurcation analysis has been proven to be a powerful tool for numerically analyzing nonlinear flight dynamics, in its standard (unforced) form, the method is unsuitable for flying qualities assessment as no information on the transient response is provided. To address this shortcoming, we propose the use of an extension of bifurcation analysis that treats the aircraft as a harmonically forced system. This facilitates the evaluation of the frequency-domain dynamics where the response is unsteady by definition. It is shown that the method can identify regions with strong nonlinearities that lead to degraded flying qualities. Time simulations of the closed-loop step responses in these undesirable regions show that despite the predictions of desirable behaviors using linear analysis, the closed-loop aircraft can still be attracted to an isola, experience control reversal, or have higher overshoot than anticipated.

Nonlinear Frequency Response Analysis and Optimization of Vibration Systems with Air Springs and Auxiliary Chambers

Tuning of linear frequency and nonlinear frequency response of microelectromechanical systems is important in order to obtain high operating bandwidth. Linear frequency tuning can be achieved through various mechanisms such as heating and softening due to DC voltage. Nonlinear frequency response is influenced by nonlinear stiffness, quality factor and forcing. In this paper, we present the influence of nonlinear coupling in tuning the nonlinear frequency response of two transverse modes of a fixed–fixed microbeam under the influence of direct and parametric forces near and below the coupling regions. To do the analysis, we use nonlinear equation governing the motion along in-plane and out-of-plane directions. For a given DC and AC forcing, we obtain static and dynamic equations using the Galerkin’s method based on first-mode approximation under the two different resonant conditions. First, we consider one-to-one internal resonance condition in which the linear frequencies of two transverse modes show coupling. Second, we consider the case in which the linear frequencies of two transverse modes are uncoupled. To obtain the nonlinear frequency response under both the conditions, we solve the dynamic equation with the method of multiple scale (MMS). After validating the results obtained using MMS with the numerical simulation of modal equation, we discuss the influence of linear and nonlinear coupling on the frequency response of the in-plane and out-of-plane motion of fixed–fixed beam. We also analyzed the influence of quality factor on the frequency response of the beams near the coupling region. We found that the nonlinear response shows single curve near the coupling region with wider width for low value of quality factor, and it shows two different curves when the quality factor is high. Consequently, we can effectively tune the quality factor and forcing to obtain different types of coupled response of two modes of a fixed–fixed microbeam.

The current paper is concerned with the investigation of the relationship between the harmonic balance method (HBM) and the non-linear output frequency response function (NOFRF) approach in the analysis of non-linear systems. Both are applied to the Duffing's oscillator to demonstrate their relationships. The results reveal that, if the Volterra series representation of a non-linear system is convergent, the harmonic components calculated by the NOFRFs are a solution of the HBM. Moreover, the simulation studies show that, in the convergent cases, the NOFRF method can give more accurate results for the higher-harmonic components than the HBM. The relationship investigated in the current study between the two methods should help researchers and engineers to understand the HBM and the NOFRF methods.

This paper presents various types of windings in a power transformer, their FEM methods, frequency response analysis etc. Frequency response analysis (FRA) standards are used in all places. In frequency response, one reference spectrum is first done and after fault is done, spectrum is compared with reference one. A 3-D model of a transformer is considered to show the field distribution, inductance and capacitance matrices, force analysis etc. The forces developed in a transformer winding has also been explained and transient case has been studied.

Analogue and microwave design requires accurate and reliable simulation tools and methods to meet the design specifications. System properties are often measured in the steady state. Well‐suited algorithms for calculating the steady state can be classified into shooting methods, finite difference methods and the harmonic balance (HB) technique. Harmonic balance is a frequency domain method which approaches the problem of finding the steady state by a trigonometric polynomial. Depending on the size of the circuit and the number of Fourier coefficients of the polynomial, the resulting system of non‐linear equations can become very large. These non‐linear equations are solved by using Newton's method.The sparse linear system arising from Newton's method can be solved by direct, stationary or non‐stationary iterative solvers. Iterative methods are normally easy to parallelize or vectorize.In this paper a tool for the simulation of the steady state of electronic circuits is presented. the steady state is calculated using the harmonic balance technique. Non‐linear equations are solved by Newton's method and linear equations by preconditioned non‐stationary iterative solvers (CGS, Bi‐CGSTAB, BiCGSTAB(2), TFQMR). the run time is reduced dramatically, by up to an order of magnitude.