Nonlinear Modal Analysis Research Articles

The influence of damping on the limit cycles for a self-exciting mechanism

The objective of the present work is to investigate qualitative aspects of mode-coupling instability of self-excited friction-induced oscillations in the presence of structural damping and a cubic nonlinearity. In a previous study, Sinou and Jezequel demonstrated the effects of structural damping in determining stable and unstable zones and indicated that the damping ratio of the coupling modes may be a key factor for avoiding bad design. In this paper, we propose to complete this study by examining the influence of structural damping on limit cycle amplitudes in order to achieve a complete design including not only the evolution of stable/unstable areas but also the evolution of limit cycle amplitudes as functions of the structural damping and nonlinear system parameter. For the sake of simplicity, a two-degree-of-freedom minimal model is constructed and analysed to examine the effects of structural damping and cubic nonlinearity on the limit cycles. The nonlinear behaviour and stable limit cycle amplitudes are determined through a Complex Nonlinear Modal Analysis which makes use of the nonlinear unstable mode governing the nonlinear dynamics of structural systems in unstable areas. Based on this nonlinear modal approach, we will produce and explain qualitative and quantitative results of limit cycle evolutions in the presence of structural damping and nonlinearity.

Validation of power system non-linear modal analysis methods

The validity of two power system non-linear modal analysis methods, i.e. normal form method and modal series method, are investigated in this work. The second-order participation factors are derived for modal series method and qualitatively compared with that of normal form method. To evaluate the validity of analysis methods, a valid region that satisfies certain error index is defined. The test results of a two-area four-generator power system show that the valid regions of both modal series method and linear modal method are not affected by second-order resonance condition, while that of normal form method is affected significantly. It is found that as the system stress increases, the error index and its growing speed of modal series method are the smallest, and that of normal form method take second place. A non-linearity index for normal form method is defined based on the original index and its validity is verified by the case studies using the same test system.

Stability analysis and non-linear behaviour of structural systems using the complex non-linear modal analysis (CNLMA)

Herein, a novel non-linear procedure for producing non-linear behaviour and stable limit cycle amplitudes of non-linear systems subjected to super-critical Hopf bifurcation point is presented. This approach, called Complex Non-Linear Modal Analysis (CNLMA), makes use of the non-linear unstable mode which governs the non-linear dynamic of structural systems in unstable areas. In this study, the computational methodology of CNLMA is presented for the systematic estimation of the non-linear behavior of structural system, and it is applied to a complex system with quadratic and cubic non-linearities. Results from this non-linear method are compared with results obtained by integrating the full original non-linear system. This procedure appears very interesting in regard to computational time, formulation, and also necessitates very few computer resources. Thus the CNLMA is suited to computational implementation for various complex non-linear system with many degrees-of-freedom.

Nonlinear normal modes and their application in structural dynamics

Recent progress in the area of nonlinear modal analysis for structural systems is reported. Systematic methods are developed for generating minimally sized reduced-order models that accurately describe the vibrations of large-scale nonlinear engineering structures. The general approach makes use of nonlinear normal modes that are defined in terms of invariant manifolds in the phase space of the system model. An efficient Galerkin projection method is developed, which allows for the construction of nonlinear modes that are accurate out to large amplitudes of vibration. This approach is successfully extended to the generation of nonlinear modes for systems that are internally resonant and for systems subject to external excitation. The effectiveness of the Galerkin-based construction of the nonlinear normal modes is also demonstrated for a realistic model of a rotating beam.

Equivalent linear modal parameter estimation for a nonlinear aeroelastic system

A nonlinear dynamic system modal analysis methodology is applied to a two degree-of-freedom, nonlinear aeroelastic system. Modal analysis is an essential tool in the flight flutter testing of new and modified aircraft structures, and is used to identify system modal parameters such as frequency and damping. Traditional spectral methods based on the fast Fourier transform, rely on the assumption of system linearity. It has been shown that even isolated nonlinearities in the aircraft structure can contribute to errors in the estimated values of frequency and damping obtained from such a modal analysis. The reverse multiple-input/single-output method, described by Bendat, is a spectral analysis technique applicable to the frequency response function obtained from a nonlinear dynamical system. The technique uses a band-limited random forcing input to obtain the system response and permits the separation of the linear and nonlinear portions of the frequency response function within the frequency domain. The linear portion of the frequency response may then be used to recover the values of frequency and damping for the equivalent linear system. Results are presented for the application of this method to the specific case of a two degree-of-freedom, structurally nonlinear, aeroelastic system. It is shown that the method may be used to successfully recover the linear frequency response function using the input and output data for the nonlinear system.

Application of a Nonlinear Modal Instability Approach to Brake Systems

A method, called the Complex NonLinear Modal Analysis (CNLMA), is proposed for the calculation of the periodic solutions of nonlinear mechanical systems with continued nonlinearities. The equivalent linearization concept and the notion of nonlinear complex modes are applied in order to analyze the steady-state responses of autonomous nonlinear systems. The CNLM-Analysis appears very interesting in regard to computational time; it also necessitates very few computer resources. This method was applied to study instability phenomena in a nonlinear model with a constant brake friction coefficient.

Nonlinear Dynamic Analysis of MEMS Switches by Nonlinear Modal Analysis

A nonlinear modal analysis approach based on the invariant manifoldmethod proposed earlier by Boivin et al. [10] is applied in this paperto perform the dynamic analysis of a micro switch. The micro switch ismodeled as a clamped-clamped microbeam subjected to a transverseelectrostatic force. Two kinds of nonlinearities are encountered in thenonlinear system: geometric nonlinearity of the microbeam associatedwith large deflection, and nonlinear coupling between two energydomains. Using Galerkin method, the nonlinear partial differentialgoverning equation is decoupled into a set of nonlinear ordinarydifferential equations. Based on the invariant manifold method, theassociated nonlinear modal shapes, and modal motion governing equationsare obtained. The equation of motion restricted to these manifolds,which provide the dynamics of the associated normal modes, are solved bythe approach of nonlinear normal forms. Nonlinearities and the pull-inphenomena are examined. The numerical results are compared with thoseobtained from the finite difference method. The estimate for the pull-involtage of the micro device is also presented.

Nonlinear Pressurization and Modal Analysis Procedure for Dynamic Modeling of Inflatable Structures

Covers advancements in spacecraft and tactical and strategic missile systems, including subsystem design and application, mission design and analysis, materials and structures, developments in space sciences, space processing and manufacturing, space operations, and applications of space technologies to other fields.

River meandering dynamics.

The Ikeda, Parker, and Sawai river meandering model is reexamined using a physical approach employing an explicit equation of motion. For periodic river shapes as seen from above, a cross-stream surface elevation gradient creates a velocity shear that is responsible for the decay of small-wavelength meander bends, whereas secondary currents in the plane perpendicular to the downstream direction are responsible for the growth of large-wavelength bends. A decay length D=H/2C(f) involving the river depth H and the friction coefficient C(f) sets the scale for meandering, giving the downstream distance required for the fluid velocity profile to recover from changes in the channel curvature. Using this length scale and a time scale T, we explicitly trace the observed length scale invariance to the equations of motion, and predict similar time and velocity scale invariances. A general time-dependent nonlinear modal analysis for periodic rivers reveals that modes higher than the third mode are needed to describe upstream migration of bend apexes just before oxbow cutoff, and are important to accurate calculations of the time and sinuosity at cutoff.

NORMAL MODES OF A NON-LINEAR CLAMPED–CLAMPED BEAM

Non-linear modal analysis approach based on invariant manifold method proposed earlier by Shaw and Pierre (Journal of Sound and Vibration164, 85–124) is utilized here to obtain the non-linear normal modes of a clamped–clamped beam for large amplitude displacements. The results obtained for the fundamental normal mode are compared with the corresponding reported experimental and theoretical studies. The effects of modal coupling are examined in greater detail. The limitation of the present method for analyzing non-linear behavior is highlighted.

Critical wavelength for river meandering.

A fully nonlinear modal analysis identifies a critical centerline wave number q(c) for river meandering that separates long-wavelength bends, which grow to cutoff, from short-wavelength bends, which decay. Exact, numerical, and approximate analytical results for q(c) rely on the Ikeda, Parker, and Sawai [J. Fluid Mech. 112, 363 (1981)] model, supplemented by dynamical equations that govern the river migration and length. Predictions also include upvalley bend migration at long times and a peak in lateral migration rates at intermediate times. Experimental tests are suggested.

Development and Application of a Nonlinear Modal Analysis Technique for MDOF Systems

This work presents a frequency-domain technique that extends the concept of linear modal super position to nonlinear systems by using the normal nonlinear mode approach so that a generalized parameter identification method can be formulated for MDOF nonlinear systems. The methodology is compatible with existing established vibration analysis techniques such as finite element (FE) modeling and experimental modal analysis. Furthermore, once the nonlinear modal parameters are identified at some reference force level, the nonlinear response can be predicted at any arbitrary excitation level using standard modal sum mation techniques. The numerical study is focused on a 4-DOF system with friction damping nonlinearity, for which both the macro- and microslip representations are considered. Simulated nonlinear frequency re sponse functions, obtained for a given excitation level using a harmonic balance method, were subjected to a nonlinear modal analysis procedure, and the modal parameters were extracted as a function of the vibration amplitude. Micro- and macroslip representations yielded significantly different nonlinear modal parameters, though the findings were consistent with those of other researchers. The nonlinear modal analysis results and the response predictions at arbitrary forcing levels were compared against reference harmonic balance sim ulations, and very good agreement was observed for all cases investigated. It was verified that the friction damper produced highest damping for the vibration amplitude of maximum energy dissipation.

COUPLING OF NON-LINEAR SUBSTRUCTURES USING VARIABLE MODAL PARAMETERS

This paper presents a general methodology for the coupling analysis of systems with relatively weak non-linearities by assuming that the response remains harmonic under harmonic excitation. Standard coupling methods and their current shortcomings were discussed first. Two ways of obtaining non-linear modal parameter variations, namely profile constructing and parameter extracting, were presented next. The profile constructing method uses the system's spatial data directly, while the parameter extracting method is based on a non-linear modal analysis of measured response data. Through numerical test cases, it was shown that both methods yielded virtually identical results. An iterative algorithm for the coupling of non-linear subsystems was presented in a form compatible with profile building. A six-degree-of-freedom system with cubic stiffness non-linearity was chosen for a detailed numerical study. Two subsystems, one linear and the other non-linear, were coupled to obtain the modal parameter variations of the coupled system. Using the non-linear modal parameters, the response of the coupled system was predicted at various force levels and the findings were checked via direct simulations using the harmonic balance method. Finally, the methodology was validated by coupling experimentally derived non-linear modal models of two substructures. As for the numerical study, the response of the coupled non-linear structure was predicted at various force levels and the findings were checked against direct measurements. Very good agreement was obtained in all cases studied.

Variable Modal Parameter Identification for Non-Linear Mdof Systems. Part II: Experimental Validation and Advanced Case Study

The purpose of Part II is to provide an experimental validation of the methodology presented in Part I and to consider a representative engineering case, the study of which requires a relatively large numerical model. A beam system with cubic stiffness type non-linearity was used in the experimental study. The non-linear response was measured at three locations and the underlying linear system was obtained via linear modal analysis of low-excitation response data. The non-linear parameter variations were obtained as a function of the modal amplitude and the response of the system was generated for other force levels. The results were found to agree very well with the corresponding measurements, indicating the success of the non-linear modal analysis methodology, even in the presence of true experimental noise. An advanced numerical case study that included both inherent structural damping and non-linear friction damping, was considered next. The linear finite element model of a high-pressure turbine blade was used in conjunction with three local non-linear friction damper elements. It was shown that the response of the system could be predicted at any force level, provided that that non-linear modal parameters were available at some reference force level. The predicted response levels were compared against those obtained from reference simulations and very good agreement was achieved in all cases.

A Survey of Nonlinear Analysis Tools for Structural Systems

After a brief introduction to nonlinear behavior in structural dynamics, a number of strategies are listed for dealing with such situations. The methods employed for the detection, identification, and quantification of structural nonlinearities are surveyed by outlining several analytical and experimental procedures. These procedures are discussed in terms of their range of usefulness and ease of application. Nonlinear modal analysis techniques that use measured nonlinear frequency response data are studied in some detail, and their potential suitability to deal with practical cases is illustrated by means of an example. Throughout the article, special emphasis is placed on compatibility with existing linear analysis techniques and finite element methodology. Finally, current requirements and future trends are also discussed.

Normalized quasi-static mass — A new definition for multi-support motion problems

An efficient algorithm called the modal reaction method for calculating the modal participation factors in support motion problems has been proposed by Chen et al. [1]. In this paper, we extend this method to determine the number of modes needed to satisfy 90% of the sum of the base-shear modal mass as UBC(uniform building code) suggests. The sum of all the modes for each support in multi-support motions is found to be equal to the normalized quasi-static mass which is defined in this paper. The normalized quasi-static mass is equivalent to the total structure mass in the case of single supported structure. By extracting the reaction from the SPC force in data recovery using SOL 3 (linear modal analysis) or SOL 106 (nonlinear modal analysis) in MSC/NASTRAN, the modal participation factor and the base-shear modal mass ratio can be directly determined free from calculation of the influence vector, or the so-called quasi-static solution. To demonstrate this new concept of the normalized quasi-static mass, several examples including rod, beam, tower structures are given to check the validity of the proposed method using MSC/NASTRAN program. Finally, the minimum number of modes needed to reach 90% of the normalized quasi-static mass for each support is proposed as a reference for analysis and design engineers.

Non-linear modal analysis of structural systems featuring internal resonances

Non-linear modal analysis of structural systems featuring internal resonances

A Consistent Higher-Order Theory of Laminated Plates with Nonlinear Impact Modal Analysis

A consistent higher-order theory is developed for cross-ply laminated thick plates under transverse normal impact via an energy variational approach, in which the 3-D surface/edge boundary conditions and interlaminar displacement/stress continuities are satisfied, in an attempt to find the dynamic deformation and all six stress components throughout the plate during the impact process. The dynamic displacement field is expressed in a mixed form of in-plane double Fourier series and cubic polynomials through thickness as 12 variables for each layer. A system of modified Lagrange’s equations is derived with all surface and interface constraints included. The nonlinear impact modal analysis is performed using the Hertz contact law in a patch loading simulation and Green’s function for small time-steps linearization. The 3-D displacements are found with thickness shrinking and stresses generally unsymmetric with respect to the mid-surface. Tensile cracks are predicted at the unimpacted side.

Nonlinear modal analysis of frames by the incremental harmonic-balance method

A modal incremental harmonic-balance method for the steady-state vibration of geometrically nonhnear frames by finite-element analysis is introduced. The nonlinearity is in the coupling of the axial and flexural deformations through the induced axial force along the element. Free and forced vibrations for damped or undamped systems are studied. It is shown that the method allows accurate response curves to be constructed with minimum effort

Linear and nonlinear modal analysis of aeroelastic structural systems

This paper deals with the investigation of linear modal analysis and the autoparametric interaction of aeroelastic systems. The normal-mode frequencies and the associated mode shapes are obtained in terms of the system parameters. The main objective of the linear analysis is to explore the critical regions of autoparametric (or internal) resonance conditions, Σk i ω i = 0 (where k i are integers and ω i are the normal-mode frequencies.) The results show that for certain system parameters the condition of internal resonance is satisfied. The dynamic behavior of the structure in the neighborhood of internal-resonance conditions is obtained by considering the nonlinear coupling of the normal modes. Under certain conditions the steady-state response is derived analytically for the sum internal-resonance condition. It is concluded that the nonlinear modal analysis reveals certain types of response characteristics which cannot be interpreted within the framework of the linear theory of small oscillations.