Linear Modal Analysis Research Articles

Development of a comprehensive flame transfer function and its application to predict combustion instabilities in a dump combustor

ABSTRACTIn combustion instability analysis, a flame transfer function (FTF) is used to describe the response of a premixed flame to incident acoustic oscillations. A comprehensive FTF is developed that relates the fluctuations in flame heat-release rate to those in density, heat of reaction, turbulent flame speed, and the flame surface area. Each of these four contributions is eventually expressed in terms of the fluctuations in acoustic pressure and velocity through the respective response functions. The flame surface-area response to acoustic oscillations is incorporated into the FTF through the -equation level-set method. In this method, we directly solve for the level-set fluctuations in terms of the velocity fluctuations and then relate the flame surface-area oscillations to . In the absence of turbulent flame-speed fluctuations, the response functions from the present -equation approach are in good agreement with those from the conventional -equation approach (where one writes ). However, when the turbulent flame-speed fluctuations are included, the two approaches differ, principally in the flame response to axial velocity fluctuations. The current FTF, generalized for any mean flame shape, is applied to analyze the response of a V-shaped mean flame located at the cross-sectional interface of a 2D dump combustor. For this flame, the effects of variation in acoustic frequency, mean Mach number, mean temperature, and mean equivalence ratio on the FTF magnitude and phase are investigated. Both axial and mixed mode acoustic perturbations are considered. For the purely axial modes, the response-function amplitudes of density and heat of reaction are seen to be independent of frequency, while those of flame speed and axial velocity exhibit harmonic-like oscillations as a function of frequency. For the mixed-mode fluctuations, the response-function amplitude of density decays monotonically with frequency, whereas the response-function amplitudes of the heat of reaction, turbulent flame speed, and velocity components show decaying oscillatory behavior in frequency. Further, it is observed that the flame-speed response function contributes the most to the overall FTF. Phase analysis of the FTF shows that the transverse velocity response is nearly in phase with the heat-release fluctuations. In addition, density and transverse velocity fluctuations always show constructive interference with pressure fluctuations. The current FTF is also incorporated into a linear modal analysis framework to predict the combustion instabilities in a 2D dump combustor, and the model predictions validated against experiments. Three mean velocity profiles—uniform, parabolic, and turbulent power law—are considered. For the three flames, the first three longitudinal and fundamental transverse unstable modes are predicted.

Collapse investigation of the Arquata del Tronto medieval fortress after the 2016 Central Italy seismic sequence

In this paper, an investigation is presented of the Arquata del Tronto medieval fortress local collapses that occurred due to the 2016 Central Italy seismic sequence. Both historical evolution and damage inspection of the fortress are discussed. Interestingly, the damage reported to the fortress after the 2016 earthquakes appears similar to what occurred more than 300 years before (due to the 1703 Apennine earthquakes), despite the restoration work of the 1920s. Linear kinematic analysis and 3D modal and seismic FE analyses are carried out to evaluate the safety of the local collapse mechanisms that occurred on the towers’ crownings. Numerical findings point out that no failure mode passes the safety check. This suggests that the reconstruction of the collapsed parts should be complemented by specific strengthening devices.

Ground effects on the stability of separated flow around a NACA 4415 airfoil at low Reynolds numbers

We perform a linear BiGlobal modal stability analysis on the separated flow around a NACA 4415 airfoil at low Reynolds numbers (Re=300–1000) and a high angle of attack (α=20°), with a focus on the effect of the airfoil's proximity to two different types of ground: a stationary ground and a moving ground. The results show that the most dominant perturbation is a Kelvin–Helmholtz mode, which gives rise to a supercritical Hopf bifurcation to a global mode, leading to large-scale vortex shedding at a periodic limit cycle. As the airfoil approaches the ground, this mode can become more unstable or less unstable, depending on the specific type of ground: introducing a stationary ground to an otherwise groundless system is destabilizing but introducing a moving ground is stabilizing, although both effects weaken with increasing Re. By performing a Floquet analysis, we find that short-wavelength secondary instabilities are damped by a moving ground but are amplified by a stationary ground. By contrast, long-wavelength secondary instabilities are relatively insensitive to ground type. This numerical–theoretical study shows that the ground can have an elaborate influence on the primary and secondary instabilities of the separated flow around an airfoil at low Re. These findings could be useful for the design of micro aerial vehicles and for improving our understanding of natural flyers such as insects and birds.

Synthesis of nonlinear frequency responses with experimentally extracted nonlinear modes

Determining frequency response curves is a common task in the vibration analysis of nonlinear systems. Measuring nonlinear frequency responses is often challenging and time consuming due to, e.g., coexisting stable or unstable vibration responses and structure-exciter-interaction. The aim of the current paper is to develop a method for the synthesis of nonlinear frequency responses near an isolated resonance, based on data that can be easily and automatically obtained experimentally. The proposed purely experimental approach relies on (a) a standard linear modal analysis carried out at low vibration levels and (b) a phase-controlled tracking of the backbone curve of the considered forced resonance. From (b), the natural frequency and vibrational deflection shape are directly obtained as a function of the vibration level. Moreover, a damping measure can be extracted by power considerations or from the linear modal analysis. In accordance with the single nonlinear mode assumption, the near-resonant frequency response can then be synthesized using this data. The method is applied to a benchmark structure consisting of a cantilevered beam attached to a leaf spring undergoing large deflections. The results are compared with direct measurements of the frequency response. The proposed approach is fast, robust and provides a good estimate for the frequency response. It is also found that direct frequency response measurement is less robust due to bifurcations and using a sine sweep excitation with a conventional force controller leads to underestimation of maximum vibration response.

A variationally bounded scheme for delayed detached eddy simulation: Application to vortex-induced vibration of offshore riser

We present a bounded and positivity preserving variational (PPV) method for the turbulence transport equation of Spalart–Allmaras based delayed detached eddy simulation (DDES). We employ the developed solver to simulate the vortex-induced vibration of a slender flexible riser immersed in a turbulent flow. The fluid-structure interface problem is solved by recently proposed partitioned iterative scheme [1], which consists of nonlinear force correction for a stable coupling of the Navier–Stokes equations with the low-mass structure subjected to strong inertial effects from the surrounding incompressible flow. In our fluid-structure model, we consider the flexible cylindrical riser as a long tensioned beam via linear modal analysis. In the present contribution, we first validate the PPV-based DDES solver for the flow past a three-dimensional stationary cylinder at two subcritical Reynolds numbers of Re=3900 and Re=140,000 based on the diameter of the cylinder. We next simulate a three-dimensional flexible riser model under a pinned-pinned condition at Re=4000 and compare our results with that of the experimental measurements. We assess the response characteristics at various locations along the span of the riser and discuss the orbital trajectories at those locations. In particular, the numerical study emphasizes on (i) The effectiveness of the modal analysis in modeling the riser dynamics and a positivity-preserving variational method in DDES for turbulence modeling, (ii) validation of response amplitudes and motion trajectories, and (iii) new insights on the trajectories and vortical structures of the flexible riser undergoing vortex-induced vibrations (VIV). We confirm a standing wave response and complex chaotic response of riser VIV observed in the recent experiments.

Acoustic resonance in the potential core of subsonic jets

The purpose of this paper is to characterize and model waves that are observed within the potential core of subsonic jets and relate them to previously observed tones in the near-nozzle region. The waves are detected in data from a large-eddy simulation of a Mach 0.9 isothermal jet and modelled using parallel and weakly non-parallel linear modal analysis of the Euler equations linearized about the turbulent mean flow, as well as simplified models based on a cylindrical vortex sheet and the acoustic modes of a cylindrical soft duct. In addition to the Kelvin–Helmholtz instability waves, three types of waves with negative phase velocities are identified in the potential core: upstream- and downstream-propagating duct-like acoustic modes that experience the shear layer as a pressure-release surface and are therefore radially confined to the potential core, and upstream-propagating acoustic modes that represent a weak coupling between the jet core and the free stream. The slow streamwise contraction of the potential core imposes a frequency-dependent end condition on the waves that is modelled as the turning points of a weakly non-parallel approximation of the waves. These turning points provide a mechanism by which the upstream- and downstream-travelling waves can interact and exchange energy through reflection and transmission processes. Paired with a second end condition provided by the nozzle, this leads to the possibility of resonance in limited frequency bands that are bound by two saddle points in the complex wavenumber plane. The predicted frequencies closely match the observed tones detected outside of the jet. The vortex-sheet model is then used to systematically explore the Mach number and temperature ratio dependence of the phenomenon. For isothermal jets, the model suggests that resonance is likely to occur in a narrow range of Mach number,$0.82<M<1$.

Spreading of electron scale magnetic reconnection with a wave number dependent speed due to the propagation of dispersive waves

In this work, we study electron scale spreading of localized magnetic reconnection in the presence of a guide magnetic field, however, without the influence of ions and cross-scale coupling. These fundamental physics studies will help to understand the coupling of the electron scale spreading with the ion scales in real systems. An electron-magnetohydrodynamic (EMHD) model is employed to model the physics at electron scales. Three dimensional EMHD simulations and linear eigen mode analysis are performed for different guide field strengths. The simulations show a wave-like bi-directional spreading of magnetic reconnection at electron scales. The electron scale spreading, however, unlike the ion scale spreading by Alfvén waves, is caused by the uni- and bi-directional propagation of the dispersive flow induced and whistler wave modes, respectively. The dispersive nature of the two wave modes makes the spreading-speed dependent on the wave numbers of the unstable tearing mode, which depend on the thickness of the electron current sheet and the strength of the guide field. A model of the speed of spreading is developed, in which the spreading-speeds parallel and anti-parallel to the guide field are given by linear combinations of the group speeds of the two wave modes. The model prediction of the spreading speeds agrees well with the speeds obtained from the simulation results. For small guide fields, the spreading is asymmetric being faster in the direction of the electron flow. On increasing the guide field, the spreading becomes increasingly symmetric, with the speeds of the order of electron Alfvén speed in the guide magnetic field, due to the dominance of the whistler group speed in determining the speed of the spreading. As a consequence of the asymmetric spreading, the chain of alternate X- and O-points, formed due the growth of oblique tearing modes, extends farther in the direction of electron flow as compared to that in the direction of the guide magnetic field.

Elastic foundation effects on arch dams

Earthquake response of an arch dam should be calculated under ground motion effects. This study presents three-dimensional linear earthquake response of an arch dam. Thereby, we considered different ground motion effects and also foundation conditions in the finite element analyses. For this purpose, the Type 3 double curvature arch dam was selected for application. All numerical analyses are carried out by SAP2000 program for empty reservoir cases. In the scope of this study, linear modal time-history analyses are performed using three dimensional finite element model of the arch dam and arch dam-foundation interaction systems. According to numerical analyses, maximum horizontal displacements and maximum normal stresses are presented by dam height in the largest section. These results are evaluated for rigid and various elastic foundation conditions. Furthermore, near-fault and far-field ground motion effects on the selected arch dam are taken into account by different accelerograms obtained from the Loma Prieta earthquake at various distances.

From homotopy perturbation technique to reduced order model for multiparametric modal analysis of large finite element models

The paper focuses on the definition of a reduced order model for linear modal analysis. The aim is to supply a suitable mathematical alternative tool compatible for multiparametric analysis of large finite element model considering numerous variable parameters, numerous mode shapes and significant levels of variation. The initial full eigenvalue problem is so replaced by a reduced one considering an efficient projection basis. To build it, we propose to combine homotopy transformation and perturbation technique for each parameter direction to define a reduced order model compatible with the design space. Finally, a complete finite element application highlights the capabilities of the proposal in terms of precision and computational time.

Comparing the Identification Methods for a Finite Element Mass Matrix of the Axisymmetric Shell Restricted by Rigid Ends

The paper deals with comparison of various algorithms to obtain mass-inertial characteristics of the finite element (FE) of revolution shell restricted by rigid ends. Such shells are widely used in engineering, for example, as body parts of gas turbine plants. When solving the problems of linear statics and modal analysis for complex structures containing such shells, a method of replacing a complex section of the structure by a single generalized element with previously calculated characteristics is used. In case of stiffness properties everything is obvious, since with using a numerical integration of the system of differential equations of a revolution shell, the stiffness matrix elements can be found exactly, while to obtain mass-inertial characteristics a standalone consideration is necessary. The paper considers different algorithms to build a matrix of the shell FE mass, limited by rigid ends, with an arbitrary form of the generatrix: consistent/inconsistent formulation; frequency identification algorithm; natural frequency identification and that of free element shapes. When performing numerical experiments it was found that a classical approach that uses form functions obtained from statics is inappropriate for constructing the mass matrix of the generalized element. Therefore, other algorithms were proposed to obtain mass-inertial characteristics. But those of have some restrictions too: the frequency identification algorithm requires large computational efforts and can be successfully used only in the cases of securing, which involve identification; natural frequency identification and that of free element shapes can be successfully used only in the case of free FE. Thus, when replacing the extended section by one generalized element in solving dynamics problems, it is necessary to be clearly aware of requirements imposed on the generalized element and on this basis choose one or another algorithm for obtaining mass-inertial characteristics, since the result significantly depends on the chosen identification method.

Dynamic Stability Analysis of Linear Time-varying Systems via an Extended Modal Identification Approach

The problem of linear time-varying(LTV) system modal analysis is considered based on time-dependent state space representations, as classical modal analysis of linear time-invariant systems and current LTV system modal analysis under the “frozen-time” assumption are not able to determine the dynamic stability of LTV systems. Time-dependent state space representations of LTV systems are first introduced, and the corresponding modal analysis theories are subsequently presented via a stability-preserving state transformation. The time-varying modes of LTV systems are extended in terms of uniqueness, and are further interpreted to determine the system’s stability. An extended modal identification is proposed to estimate the time-varying modes, consisting of the estimation of the state transition matrix via a subspace-based method and the extraction of the time-varying modes by the QR decomposition. The proposed approach is numerically validated by three numerical cases, and is experimentally validated by a coupled moving-mass simply supported beam experimental case. The proposed approach is capable of accurately estimating the time-varying modes, and provides a new way to determine the dynamic stability of LTV systems by using the estimated time-varying modes.

Linear modal stability analysis of bowed-strings.

Linearised models are often invoked as a starting point to study complex dynamical systems. Besides their attractive mathematical simplicity, they have a central role for determining the stability properties of static or dynamical states, and can often shed light on the influence of the control parameters on the system dynamical behaviour. While the bowed string dynamics has been thoroughly studied from a number of points of view, mainly by time-domain computer simulations, this paper proposes to explore its dynamical behaviour adopting a linear framework, linearising the friction force near an equilibrium state in steady sliding conditions, and using a modal representation of the string dynamics. Starting from the simplest idealisation of the friction force given by Coulomb's law with a velocity-dependent friction coefficient, the linearised modal equations of the bowed string are presented, and the dynamical changes of the system as a function of the bowing parameters are studied using linear stability analysis. From the computed complex eigenvalues and eigenvectors, several plots of the evolution of the modal frequencies, damping values, and modeshapes with the bowing parameters are produced, as well as stability charts for each system mode. By systematically exploring the influence of the parameters, this approach appears as a preliminary numerical characterisation of the bifurcations of the bowed string dynamics, with the advantage of being very simple compared to sophisticated numerical approaches which demand the regularisation of the nonlinear interaction force. To fix the idea about the potential of the proposed approach, the classic one-degree-of-freedom friction-excited oscillator is first considered, and then the case of the bowed string. Even if the actual stick-slip behaviour is rather far from the linear description adopted here, the results show that essential musical features of bowed string vibrations can be interpreted from this simple approach, at least qualitatively. Notably, the technique provides an instructive and original picture of bowed motions, in terms of groups of well-defined unstable modes, which is physically intuitive to discuss tonal changes observed in real bowed string.

Crumpling sound synthesis

Crumpling a thin sheet produces a characteristic sound, comprised of distinct clicking sounds corresponding to buckling events. We propose a physically based algorithm that automatically synthesizes crumpling sounds for a given thin shell animation. The resulting sound is a superposition of individually synthesized clicking sounds corresponding to visually significant and insignificant buckling events. We identify visually significant buckling events on the dynamically evolving thin surface mesh, and instantiate visually insignificant buckling events via a stochastic model that seeks to mimic the power-law distribution of buckling energies observed in many materials. In either case, the synthesis of a buckling sound employs linear modal analysis of the deformed thin shell. Because different buckling events in general occur at different deformed configurations, the question arises whether the calculation of linear modes can be reused. We amortize the cost of the linear modal analysis by dynamically partitioning the mesh into nearly rigid pieces: the modal analysis of a rigidly moving piece is retained over time, and the modal analysis of the assembly is obtained via Component Mode Synthesis (CMS). We illustrate our approach through a series of examples and a perceptual user study, demonstrating the utility of the sound synthesis method in producing realistic sounds at practical computation times.

Incremental Deformation Subspace Reconstruction

AbstractRecalculating the subspace basis of a deformable body is a mandatory procedure for subspace simulation, after the body gets modified by interactive applications. However, using linear modal analysis to calculate the basis from scratch is known to be computationally expensive. In the paper, we show that the subspace of a modified body can be efficiently obtained from the subspace of its original version, if mesh changes are small. Our basic idea is to approximate the stiffness matrix by its low‐frequency component, so we can calculate new linear deformation modes by solving an incremental eigenvalue decomposition problem. To further handle nonlinear deformations in the subspace, we present a hybrid approach to calculate modal derivatives from both new and original linear modes. Finally, we demonstrate that the cubature samples trained for the original mesh can be reused in fast reduced force and stiffness matrix evaluation, and we explore the use of our techniques in various simulation problems. Our experiment shows that the updated subspace basis still allows a simulator to generate visual plausible deformation effects. The whole system is efficient and it is compatible with other subspace construction approaches.

Linear and nonlinear modal analysis of the axially moving continua based on the invariant manifold method

By studying the transverse dynamics of the axially moving beam, the linear and nonlinear complex modes of gyroscopic continua are investigated based on the invariant manifold method. The nonlinear partial differential equations are truncated into a set of ordinary differential equations by the Galerkin method. The invariant manifold method is then used to obtain the linear and nonlinear complex modes. The gyroscopic effect due to the axially moving speed is discussed by comparing the proportions of the trial functions in the complex mode during the modal motions. The ‘traveling wave’ phenomena are found for such a gyroscopic continuum. The power of the invariant manifold method in the application to gyroscopic systems is demonstrated and discussed by taking the axially moving continuum as an example.

Linear modal instabilities of hypersonic flow over an elliptic cone

Steady laminar flow over a rounded-tip $2\,:\,1$ elliptic cone of 0.86 m length at zero angle of attack and yaw has been computed at Mach number $7.45$ and unit Reynolds number $Re^{\prime }=1.015\times 10^{7}~\text{m}^{-1}$. The flow conditions are selected to match the planned flight of the Hypersonic Flight Research Experimentation HIFiRE-5 test geometry at an altitude of 21.8 km. Spatial linear BiGlobal modal instability analysis of this flow has been performed at selected streamwise locations on planes normal to the cone symmetry axis, resolving the entire flow domain in a coupled manner while exploiting flow symmetries. Four amplified classes of linear eigenmodes have been unravelled. The shear layer formed near the cone minor-axis centreline gives rise to amplified symmetric and antisymmetric centreline instability modes, classified as shear-layer instabilities. At the attachment line formed along the major axis of the cone, both symmetric and antisymmetric instabilities are also discovered and identified as boundary-layer second Mack modes. In both cases of centreline and attachment-line modes, symmetric instabilities are found to be more unstable than their antisymmetric counterparts. Furthermore, spatial BiGlobal analysis is used for the first time to resolve oblique second modes and cross-flow instabilities in the boundary layer between the major- and minor-axis meridians. Contrary to predictions for the incompressible regime for swept infinite wing flow, the cross-flow instabilities are not found to be linked to the attachment-line instabilities. In fact, cross-flow modes peak along most of the surface of the cone, but vanish towards the attachment line. On the other hand, the leading oblique second modes peak near the leading edge and their associated frequencies are in the range of the attachment-line instability frequencies. Consequently, the attachment-line instabilities are observed to be related to oblique second modes at the major-axis meridian. The linear amplification of centreline and attachment-line instability modes is found to be strong enough to lead to laminar–turbulent flow transition within the length of the test object. The predictions of global linear theory are compared with those of local instability analysis, also performed here under the assumption of locally parallel flow, where use of this assumption is permissible. Fair agreement is obtained for symmetric centreline and symmetric attachment-line modes, while for all other classes of linear disturbances use of the proposed global analysis methodology is warranted for accurate linear instability predictions.

Optimal Tuning and Placement of Power System Stabilizers Based Eigenvalue

<p>In this paper, an eigenvalue assignment based Particle Swarm Optimization and Participation Factor for Optimal tuning and placement of power system stabilizers is proposed. The proposed approach presents a two-step methodology to find optimal location and parameters of PSS. The Participation Factor method is computed using the modal analysis toolbox from DIgSILENT, and used to determine the power system stabilizers optimal location. A Particle Swarm Optimization algorithm is written in MATLAB to search the power system stabilizers optimal parameters. Two eigenvalue-based objective functions to ensure a maximum damping of the inter-area modes as well as of the local modes by assigning them in a robust stability area are considered. The performance of the proposed approach is tested and examined on the four-machine two-area power system. Linear modal analysis and non-linear time domain simulations show the robustness of the proposed approach.</p>

Effect of Different Turbine-generator Shaft Models on the Sub-synchronous Resonance Phenomenon in the Double Cage Induction Generator Based Wind Farm

This paper focuses on the turbine-generator shaft models of double-cage Induction Generator (IG) based wind farm and its influence on the sub-synchronous resonance phenomenon. For this purpose, six different shaft models as 6-mass, 4-mass, 3-mass I, 3-mass II, 2-mass and 1-mass models are considered for double-cage IG wind-turbine. By using the linear modal analysis method, the effects of the different multi mass model of double-cage IG wind-turbine on the SSR phenomenon are studied. The results obtained by eigenvalue analysis show that the model of double-cage IG wind-turbine has an important effect on the detecting of Subsynchronous Resonance (SSR) phenomenon. The analytical results are validated by detailed electromagnetic transient simulation using PSCAD/EMTDC software.

Second-order oscillation mode study of hydropower system based on linear elastic model and modal series method

Summary Low-frequency oscillation occurs easily in a hydropower system with weak connected grid. Some oscillation behaviors cannot be explained by linear analysis method. According to unexplained oscillation frequency of hydropower stations, second-order oscillation mode of complex hydropower stations with weak grid is studied based on linear elastic model and modal series analysis method in this paper. Firstly, detailed hydropower station models with hydro-mechanical-electrical coupling are introduced. Especially, a new linear elastic model is used to model hydraulic conduit system. Secondly, hydro-mechanical-electrical coupling characteristics of hydropower stations are studied based on the whole model. Finally, the second-order oscillation mode and associated index of hydropower stations are calculated and studied by modal series method. Through numerical simulation of a single unit finite bus system, some useful results are achieved. Copyright © 2016 John Wiley & Sons, Ltd.

A machine learning approach to nonlinear modal analysis

Although linear modal analysis has proved itself to be the method of choice for the analysis of linear dynamic structures, its extension to nonlinear structures has proved to be a problem. A number of competing viewpoints on nonlinear modal analysis have emerged, each of which preserves a subset of the properties of the original linear theory. From the geometrical point of view, one can argue that the invariant manifold approach of Shaw and Pierre is the most natural generalisation. However, the Shaw–Pierre approach is rather demanding technically, depending as it does on the analytical construction of a mapping between spaces, which maps physical coordinates into invariant manifolds spanned by independent subsets of variables. The objective of the current paper is to demonstrate a data-based approach motivated by Shaw–Pierre method which exploits the idea of statistical independence to optimise a parametric form of the mapping. The approach can also be regarded as a generalisation of the Principal Orthogonal Decomposition (POD). A machine learning approach to inversion of the modal transformation is presented, based on the use of Gaussian processes, and this is equivalent to a nonlinear form of modal superposition. However, it is shown that issues can arise if the forward transformation is a polynomial and can thus have a multi-valued inverse. The overall approach is demonstrated using a number of case studies based on both simulated and experimental data.