Bilinear Frequency Approximation Research Articles

Analyzing the backbone curve of piecewise-linear non-smooth systems using a generalized bilinear frequency approximation method

Estimating the resonant frequency of non-smooth systems with contacting elements is critical for the design, control, and health monitoring of various mechanical and aerospace structures. The nonlinearity caused by the intermittent contact excludes the use of efficient linear methods; hence, predicting vibration characteristics of these systems becomes computationally challenging. Furthermore, several numerical and analytical approaches that have been developed for analyzing smooth nonlinear systems cannot be applied to analyze non-smooth oscillators. To understand the dynamic properties of non-smooth dynamical systems, the bilinear frequency approximation (BFA) method, a classical analytical tool, was developed to estimate the nonlinear natural frequency of bilinear oscillators by using the linear characteristics of underlying linear subsystems to construct a closed-form analytical solution. However, the traditional BFA method is only applicable to systems in which the gap size between the contacting element is zero, which limits the capability of this efficient analytical technique. In this paper, a generalized BFA method is proposed to extend the capability of the method to capture the nonlinear natural frequency of high-dimensional non-smooth systems where either gaps or prestress exist. The generalized BFA method employs the natural frequencies, mode shapes, and modal amplitudes of the linear subsystems as well as the gap size at the contacting degrees of freedom (DOF) to estimate the nonlinear natural frequencies at different amplitude levels. The backbone curve of non-smooth oscillators can be efficiently constructed using the proposed analytical method. The generalized BFA method is demonstrated on a single-DOF system, a three-DOF system, and a finite element model (FEM) of a cantilever plate with contacting elements. The results show that the proposed method provides an accurate approximation of the backbone curve for the first resonance, which has the most dominant vibration response, with very low computational effort, but a rather poor approximation for higher resonances. The proposed technique can be integrated with high-dimensional FEMs to analyze the nonlinear resonance of complex structures. The performance of the method is validated both numerically and experimentally in this work.

A generalized bilinear amplitude and frequency approximation for piecewise-linear nonlinear systems with gaps or prestress

Analysis of piecewise-linear nonlinear dynamical systems is critical for a variety of civil, mechanical, and aerospace structures that contain gaps or prestress that are caused by cracks, delamination, joints or interfaces among components. Recently, a technique referred to as bilinear amplitude approximation (BAA) was developed to estimate the response of bilinear systems that have no gap or prestress. The method is based on an idea that the dynamics of a bilinear system can be treated as a combination of linear responses in two time intervals both of which the system behaves as a distinct linear system: (1) the open state and (2) the closed or sliding state. Both geometric and momentum constraints are then applied as compatibility conditions between the states to couple the linear vibrational response for each time interval. In order to estimate the response for more general cases where there are either gaps or prestress in the system, a generalized BAA method is proposed in this paper. The new method requires inclusion of contact stiffness and damping to model contact behavior in the sliding state, and new equilibrium positions for each state to establish proper coordinates. The new method also finds the bilinear frequency of the system, which cannot be computed using the bilinear frequency approximation method previously developed since that method is only accurate for the zero gap and no prestress case. The generalized BAA method is demonstrated on a single degree of freedom system, a three degree of freedom system, and a cracked cantilever beam model for various gap sizes and prestress levels.

Reduced-order modeling for mistuned centrifugal impellers with crack damages

An efficient method for nonlinear vibration analysis of mistuned centrifugal impellers with crack damages is presented. The main objective is to investigate the effects of mistuning and cracks on the vibration features of centrifugal impellers and to explore effective techniques for crack detection. Firstly, in order to reduce the input information needed for component mode synthesis (CMS), the whole model of an impeller is obtained by rotation transformation based on the finite element model of a sector model. Then, a hybrid-interface method of CMS is employed to generate a reduced-order model (ROM) for the cracked impeller. The degrees of freedom on the crack surfaces are retained in the ROM to simulate the crack breathing effects. A novel approach for computing the inversion of large sparse matrix is proposed to save memory space during model order reduction by partitioning the matrix into many smaller blocks. Moreover, to investigate the effects of mistuning and cracks on the resonant frequencies, the bilinear frequency approximation is used to estimate the resonant frequencies of the mistuned impeller with a crack. Additionally, statistical analysis is performed using the Monte Carlo simulation to study the statistical characteristics of the resonant frequencies versus crack length at different mistuning levels. The results show that the most significant effect of mistuning and cracks on the vibration response is the shift and split of the two resonant frequencies with the same nodal diameters. Finally, potential quantitative indicators for detection of crack of centrifugal impellers are discussed.

Parametric reduced-order models for predicting the vibration response of complex structures with component damage and uncertainties

Modeling and fast reanalysis techniques are proposed for predicting the dynamic response of complex structures with uncertainty represented by parameter variability (in geometric and material properties) at component-level. The novel models allow for accurate reanalyses and are useful in many applications where the model of the pristine structure may not capture the changes in the system-level response due to component-level parameter variations. Herein, such models are obtained by using a novel approach based on a modified concept of component mode synthesis. The novel models, referred to as parametric reduced-order models, are developed for the general case of multiple substructures with parameter variabilities. Three types of parameteric variabilities are considered: (a) geometric (thickness) variability, (b) structural deformations (dents), and (c) cracks. For the first case, a novel parametrization of component-level mass and stiffness matrices is employed to predict the system-level response. For the second case, a novel approximate method based on static mode compensation is implemented. For the third case (cracks), a generalized formulation for the bi-linear frequency approximation is used. The predicted vibration responses of complex structures are shown to agree very well with results obtained using a much more computationally expensive commercial tool.

Addendum to: “A FDTD analysis on magnetized plasma of Epstein distribution and reflection calculation” [Computer Physics Communications 180 (1) (2009) 55–60]: A short comment on the equivalence of the shift-operator FDTD method and the bilinear frequency approximation technique for modeling dispersive electromagnetic applications

Recently, the shift operator finite difference time domain (SO-FDTD) method has been introduced by Yang [Computer Physics Communications 180 (1) (2009) 55–60] for modeling dispersive electromagnetic applications. This letter discuss the equivalence of this scheme and the well-known bilinear frequency approximation technique used in the field of digital signal processing. Although the SO-FDTD scheme looks different at first glance, it has been observed that it is fully equivalent to the bilinear frequency approximation technique. Consequently, the SO-FDTD scheme is indeed a bilinear transform operation which is unfortunately not cited by the author.

Estimation and veering analysis of nonlinear resonant frequencies of cracked plates

In this paper, veering phenomena in the nonlinear vibration frequencies of a cantilevered cracked plate are investigated, and an efficient method for estimating these frequencies is proposed. Of particular interest is the vibration response in parameter regions where the natural frequency loci show veerings. For a representative finite element model, it is shown that the veerings due to crack length variation involve the switching of mode shapes and modal interactions. The nonlinearity caused by the crack closing effect is then introduced, and its effect on the vibration response near the veerings is discussed. The nonlinear forced response analysis is carried out using a hybrid frequency/time domain method, which is based on the method of harmonic balance. The nonlinear vibration response near loci veerings and crossings due to the variation of crack length is investigated in detail. Finally, a novel method for estimating the nonlinear resonant frequency is introduced by generalizing the concept of bilinear frequency approximation, and the method is validated with the results of nonlinear forced response analysis for several veering regions.

On the Equivalence of Several FDTD Formulations for Modeling Electromagnetic Wave Propagation in Double-Negative Metamaterials

Recently, several approaches have been proposed to incorporate double-negative media into finite-difference time-domain simulators. In this letter, a rigorous comparative study of the formulation introduced by Karkkainen and Maslovski and the bilinear frequency approximation technique applied by Feise (IEEE Trans. Antennas Propag., vol. 52, pp. 2955-2962, Nov. 2004) is presented. Although these two formulations are based on different approximation principles, we show that both lead to equivalent difference schemes. Moreover, these two methods are equivalent to a third one, previously published for dispersive dielectrics, which is based on a different approach. The stability of these formulations is also analyzed by means of the von Neumann method.

Unconditionally stable Crank-Nicolson nearly PML algorithm for truncating linear Lorentz dispersive FDTD domains

In this paper, unconditionally stable formulations of the nearly perfectly matched layer are presented for truncating linear dispersive finite-difference time-domain (FDTD) grids. In the proposed formulations, the Crank-Nicolson and bilinear frequency-approximation techniques are used to obtain the update equations for the field components in linear dispersive media. A numerical example carried out in a one-dimensional Lorentz dispersive FDTD domain is included and it has been observed that the proposed formulations not only give accurate results, but also completely remove the stability limit of the conventional FDTD algorithm

Finite-Difference and Pseudospectral Time-Domain Methods Applied to Backward-Wave Metamaterials

Backwards-wave (BW) materials that have simultaneously negative real parts of their electric permittivity and magnetic permeability can support waves where phase and power propagation occur in opposite directions. These materials were predicted to have many unusual electromagnetic properties, among them amplification of the near-field of a point source, which could lead to the perfect reconstruction of the source field in an image [J. Pendry, Phys. Rev. Lett. \textbf{85}, 3966 (2000)]. Often systems containing BW materials are simulated using the finite-difference time-domain technique. We show that this technique suffers from a numerical artifact due to its staggered grid that makes its use in simulations involving BW materials problematic. The pseudospectral time-domain technique, on the other hand, uses a collocated grid and is free of this artifact. It is also shown that when modeling the dispersive BW material, the linear frequency approximation method introduces error that affects the frequency of vanishing reflection, while the auxiliary differential equation, the Z transform, and the bilinear frequency approximation method produce vanishing reflection at the correct frequency. The case of vanishing reflection is of particular interest for field reconstruction in imaging applications.