Interpreting proper orthogonal modes of randomly excited vibration systems

Proper orthogonal modes (POMs) of displacements are interpreted for linear vibration systems under random excitation. Excitations are considered for which the Fourier transform is convergent, meaning that the input must have zero mean, and no sustained sinusoidal component. In such a case, the POMs in undamped discrete linear symmetric systems can represent linear natural modes if the mass distribution is known. POMs in one-dimensional distributed-parameter self-adjoint systems can approximately represent the linear normal modes if the mass distribution is known. Simulation examples are presented. Simulations show that these ideas are also applicable under light modal damping.     

Projection-free proper orthogonal decomposition method for a cantilever plate in supersonic flow

This paper explores the use of the proper orthogonal decomposition (POD) method for supersonic nonlinear flutter of a cantilever plate or wing. The aeroelastic equations are constructed using von Karman plate theory and first-order piston theory. The two-dimensional POD modes (POMs) in xy plane are determined from the chaotic results given by the traditional Rayleigh–Ritz (RR) approach. For a specific structure, the POMs need to be calculated once and then can be used for various parameters of interest. The derivatives of the POMs are calculated numerically to avoid the complex projection from the POMs to the Rayleigh–Ritz modes (RRMs). Numerical examples demonstrate that the POD method using 4 POMs can obtain accurate limit cycle oscillation (LCO) results with substantial computational cost savings, compared with 12 RRMs by the Rayleigh–Ritz method. The POD method is employed for the analysis of the chaotic oscillations. It is also demonstrated that the POD modes are robust over a range of flight parameters.     

System identification-guided basis selection for reduced-order nonlinear response analysis

Reduced-order nonlinear simulation is often times the only computationally efficient means of calculating the extended time response of large and complex structures under severe dynamic loading. This is because the structure may respond in a geometrically nonlinear manner, making the computational expense of direct numerical integration in physical degrees of freedom prohibitive. As for any type of modal reduction scheme, the quality of the reduced-order solution is dictated by the modal basis selection. The techniques for modal basis selection currently employed for nonlinear simulation are ad hoc and are strongly influenced by the analyst's subjective judgment. This work develops a reliable and rigorous procedure through which an efficient modal basis can be chosen. The method employs proper orthogonal decomposition to identify nonlinear system dynamics, and the modal assurance criterion to relate proper orthogonal modes to the normal modes that are eventually used as the basis functions. The method is successfully applied to the analysis of a planar beam and a shallow arch over a wide range of nonlinear dynamic response regimes. The error associated with the reduced-order simulation is quantified and related to the computational cost.     

表面声道对深海风成噪声垂直空间特性的影响规律

水下风成噪声的垂直空间特性包括噪声垂直方向性和垂直相关性,研究海洋环境对其影响规律对提升声呐性能、增加海洋环境参数反演的准确性具有重要意义.本文利用Pekeris割线下的简正波理论描述噪声的传播过程,研究了深海环境下存在表面声道时,表面声道以下噪声垂直空间特性的变化规律及其原因.研究表明,在临界深度以上,表面声道的存在导致噪声垂直方向性在水平凹槽边缘靠近海底方向上的峰值升高,噪声垂直相关性随垂直距离增加先后周期地向正相干和负相干方向偏移;在临界深度以下,表面声道的存在导致水平方向上的噪声能量增强,噪声垂直相关性整体向正相干方向偏移.当表面声道的参数变化时,表面声道的厚度变化对噪声垂直空间特性影响较大,而表面声道内的声速梯度变化对噪声垂直空间特性几乎没有影响.结合各类简正波的变化分析表明,存在表面声道时,噪声源激发的折射简正波阶数增加,强度增强,是表面声道引起噪声垂直空间特性变化的主要原因.     

Extraction of acoustic normal mode depth functions using vertical line array data

A method for extracting the normal modes of acoustic propagation in the shallow ocean from sound recorded on a vertical line array (VLA) of hydrophones as a source travels nearby is presented. The mode extraction is accomplished by performing a singular value decomposition (SVD) of individual frequency components of the signal's temporally averaged, spatial cross-spectral density matrix. The SVD produces a matrix containing a mutually orthogonal set of basis functions, which are proportional to the depth-dependent normal modes, and a diagonal matrix containing the singular values, which are proportional to the modal source excitations and mode eigenvalues. The conditions under which the method is expected to work are found to be (1) sufficient depth sampling of the propagating modes by the VLA receivers; (2) sufficient source-VLA range sampling, and (3) sufficient range interval traversed by the source. The mode extraction method is applied to data from the Area Characterization Test II, conducted in September 1993 in the Hudson Canyon Area off the New Jersey coast. Modes are successfully extracted from cw tones recorded while (1) the source traveled along a range-independent track with constant bathymetry and (2) the source traveled up-slope with gradual changes in bathymetry. In addition, modes are successfully extracted at multiple frequencies from ambient noise.     

Modal reduction of mathematical models of biological molecules

This paper reports a detailed study of modal reduction based on either linear normal mode (LNM) analysis or proper orthogonal decomposition (POD) for modeling a single α-d-glucopyranose monomer as well as a chain of monomers attached to a moving atomic force microscope (AFM) under harmonic excitations. Also a modal reduction method combining POD and component modal synthesis is developed. The accuracy and efficiency of these methods are reported. The focus of this study is to determine to what extent these methods can reduce the time and cost of molecular modeling and simultaneously provide the required accuracy. It has been demonstrated that a linear reduced order model is valid for small amplitude excitation and low frequency excitation. It is found that a nonlinear reduced order model based on POD modes provides a good approximation even for large excitation while the nonlinear reduced order model using linear eigenmodes as the basis vectors is less effective for modeling molecules with a strong nonlinearity. The reduced order model based on component modal synthesis using POD modes for each component also gives a good approximation. With the reduction in the dimension of the system using these methods the computational time and cost can be reduced significantly.     

Bilinear modal representations for reduced-order modeling of localized piecewise-linear oscillators

A novel reduced-order modeling method for vibration problems of elastic structures with localized piecewise-linearity is proposed. The focus is placed upon solving nonlinear forced response problems of elastic media with contact nonlinearity, such as cracked structures and delaminated plates. The modeling framework is based on observations of the proper orthogonal modes computed from nonlinear forced responses and their approximation by a truncated set of linear normal modes with special boundary conditions. First, it is shown that a set of proper orthogonal modes can form a good basis for constructing a reduced-order model that can well capture the nonlinear normal modes. Next, it is shown that the subspace spanned by the set of dominant proper orthogonal modes can be well approximated by a slightly larger set of linear normal modes with special boundary conditions. These linear modes are referred to as bi-linear modes, and are selected by an elaborate methodology which utilizes certain similarities between the bi-linear modes and approximations for the dominant proper orthogonal modes. These approximations are obtained using interpolated proper orthogonal modes of smaller dimensional models. The proposed method is compared with traditional reduced-order modeling methods such as component mode synthesis, and its advantages are discussed. Forced response analyses of cracked structures and delaminated plates are provided for demonstrating the accuracy and efficiency of the proposed methodology.     

Aircraft interior noise models: Sidewall trim,stiffened structures,and cabin acoustics with floor partition

As part of the continuing development of an aircraft interior noise prediction model, in which a discrete modal representation and power flow analysis are used, theoretical results are considered for inclusion of sidewall trim, stiffened structures, and cabin acoustics with floor partition. For validation purposes, predictions of the noise reductions for three test articles (a bare ring-stringer stiffened cylinder, an unstiffened cylinder with floor and insulation, and a ring-stringer stiffened cylinder with floor and sidewall trim) are compared with measurements.     

On the acoustic field in a Pekeris waveguide with attenuation in the bottom half-space

The acoustic field in a Pekeris channel with an attenuating basement is critically examined, based on contour integrations of the wave number integrals for the two domains. In both regions, the field consists of a finite sum of proper (square integrable) normal modes plus a branch line integral around a hyperbolic cut. For low bottom attenuation, only "trapped" modes exist but as the loss increases additional "dissipation" modes contribute to the mode sum. A Newton-Raphson iterative procedure is introduced for finding the complex eigenvalues of the modes and a new expression is derived which shows that the total number of proper (trapped plus dissipation) modes supported by the waveguide increases essentially linearly with rising bottom attenuation. Approximations for the complex shape functions of the modes in the water column and the basement are developed and compared with the exact shape functions determined from the Newton-Raphson procedure. An expression derived for the modal attenuation coefficient scales in proportion to the square of the mode number and inversely with the square of the frequency. Stationary-phase approximations for the branch line integrals for both domains are developed, which serve to illustrate the asymptotic range dependence of the lateral wave but otherwise have little utility.     

PHYSICAL INTERPRETATION OF THE PROPER ORTHOGONAL MODES USING THE SINGULAR VALUE DECOMPOSITION

Proper orthogonal decomposition is a statistical pattern analysis technique for finding the dominant structures, called the proper orthogonal modes, in an ensemble of spatially distributed data. While the proper orthogonal modes are obtained through a statistical formulation, they can be physically interpreted in the field of structural dynamics. The purpose of this paper is thus to provide some insights into the physical interpretation of the proper orthogonal modes using the singular value decomposition     

A deterministic model of impact noise transmission through structural connections based on modal analysis

Vibration transmission through structural connections is modelled in a deterministic way by means of modal analysis. This model is used first to study the effect of elastic joints across the floor in the transmission of impact noise. They are an effective means of reducing impact noise propagation, and can almost eliminate it for small values of the joint stiffness. The method is also used to study the acoustic relevance of studs in lightweight floor transmission. Different ways of modelling the studs are presented and compared. For the examples developed, the best option is to use springs for modelling the studs rather than more complex models involving springs and beams. Also the different behaviour of point and line connections is verified, as well as the influence of the position of the studs.     

Virtual sensors for active noise control in acoustic-structural coupled enclosures using structural sensing: part II--Optimization of structural sensor placement

The work proposed an optimization approach for structural sensor placement to improve the performance of vibro-acoustic virtual sensor for active noise control applications. The vibro-acoustic virtual sensor was designed to estimate the interior sound pressure of an acoustic-structural coupled enclosure using structural sensors. A spectral-spatial performance metric was proposed, which was used to quantify the averaged structural sensor output energy of a vibro-acoustic system excited by a spatially varying point source. It was shown that (i) the overall virtual sensing error energy was contributed additively by the modal virtual sensing error and the measurement noise energy; (ii) each of the modal virtual sensing error system was contributed by both the modal observability levels for the structural sensing and the target acoustic virtual sensing; and further (iii) the strength of each modal observability level was influenced by the modal coupling and resonance frequencies of the associated uncoupled structural/cavity modes. An optimal design of structural sensor placement was proposed to achieve sufficiently high modal observability levels for certain important panel- and cavity-controlled modes. Numerical analysis on a panel-cavity system demonstrated the importance of structural sensor placement on virtual sensing and active noise control performance, particularly for cavity-controlled modes.     

Design of a single-polarization single-mode photonic crystal fiber double-core coupler

A simple single-polarization single-mode (SPSM) photonic crystal fiber (PCF) coupler with two cores is introduced. The full-vector finite-element method (FEM) is applied to analyze the modal interference phenomenon of the even and odd modes of two orthogonal polarizations and the power propagation within the two cores. Meanwhile, the SPSM coupling wavelength range and its corresponding coupling length for different structure parameters are numerically analyzed. The numerical results show that SPSM coupling can be realized with a broad range of wavelength, and the coupling length can be of millimeter magnitude. Moreover, the SPSM coupling wavelength range and the coupling length can be optimized by designing proper mirco-structure parameters of the coupler.     

Modal coupling in acoustic waveguides: planar discontinuities

A new derivation for a scattering matrix for reflection and transmission of higher order modes at the planar junction of two waveguides is presented. The derivation is extended to include finite junction wall impedance and offset waveguides. The resulting matrix equations are analyzed and the physical significance of the matrices is explained. As an example of the theory, analytical expressions for the coupling coefficients at a size change in a rectangular duct are developed and the resulting reflection and transmission coefficients are computed. The results should be of interest to the HVAC noise control community. The paper also shows the effects of modal truncation on the accuracy and convergence of the solution. It is shown that the proper selection of the ratio of the number of modes on either side of the discontinuity is related to the ratio of the characteristic sizes of the waveguides. Finally it is shown that at least one higher mode should be included for reasonable accuracy in the computation of plane wave reflection and transmission coefficients except at the very lowest frequencies.     

Electromagnetic cavity with arbitrary Q and small modal volume without a complete photonic bandgap

We show how an electromagnetic cavity with arbitrarily high Q and small (bounded) modal volume can be formed in two or three dimensions with a proper choice of dielectric constants. Unlike in previous work, neither a complete photonic bandgap nor a trade-off in mode localization for Q is required. Rather, scattering and radiation are prohibited by a perfect mode match of the TE-polarized modes in each subsection of a Bragg resonator. Q values in excess of 10(5) are demonstrated through finite-difference time-domain simulations of two- and three-dimensional structures with modal areas or volumes of the order of the wavelength squared or cubed.     

Design of shaped piezoelectric modal sensor for beam with arbitrary boundary conditions by using Adomian decomposition method

The theory for designing distributed piezoelectric modal sensors is well established for beam structures. However, the current modal sensor theory is limited in scope in that it can only be applied in the case of classical boundary conditions (i.e., either clamped, free, simply supported or sliding). In this paper a solution to the problem of finding the shape of piezoelectric modal sensors for a beam with arbitrary boundary conditions is proposed, using the Adomian decomposition method (ADM). A general expression for designing the shape of a piezoelectric modal sensor is presented, in which the output signal of the designed sensor is proportional to the response of the target mode. Other modes are filtered out. The modal sensor shape is expressed as a function of the second spatial derivative of the structural mode shape function. Based on the ADM and employing some simple mathematical operations, the closed-form series solution of the second spatial derivative of the mode shapes can be determined. Then the shapes of the designed modal sensors are obtained. Finally, some numerical examples are given to demonstrate the feasibility of the proposed modal sensors. It is shown that, for classical boundary conditions, the shapes of the modal sensors based on the ADM agree well with analytical and numerical results given in the literature. For general boundary conditions it is found that the shape of the modal sensors is influenced by the number of modes of interest because the second spatial derivatives of the mode shapes are not orthogonal to one another. The modal sensors for general boundary conditions can be considered as modal filters within a limited frequency band.     

Estimation of efficiency of vertical antenna arrays in underwater sound channels

The gain of the vertical antenna array in a randomly inhomogeneous ocean waveguide is numerically simulated on the assumption that the useful signal is generated by a finite set of mutually uncorrelated discrete spectrum modes and is received against the ocean noise background. It is shown that the choice of themost effective weight distribution of the array and the thus obtained gain strongly depend on both the modal spectra of the signal and noise and the number and position of the receiving elements in the channel. The critical factor is the mutual orthogonality of the waveguide modes at the array input, which indicates the possibility of specially arranging the elements for the known (expected) mode spectrum of the signal.     

IDENTIFICATION OF MODAL PARAMETERS FROM MEASURED OUTPUT DATA USING VECTOR BACKWARD AUTOREGRESSIVE MODEL

In this paper, a modal identification system that is based on the vector backward autoregressive (VBAR) model has been developed for the identification of natural frequencies, damping ratios and mode shapes of structures from measured output data. The modal identification using forward autoregressive approach has some problems in discriminating the structure modes from spurious modes. On the contrary, the VBAR approach provides a determinate boundary for the separation of system modes from spurious modes, and an eigenvalue filter for the selection of physical modes is existent in the proposed method. For convenience of application, the backward state equation established from VBAR model is transformed into a forward state equation, which is termed as transformed VFAR model in this paper. In addition, the extraction of equivalent system matrix of state equation of motion for structures from the transformed VFAR model has been developed, and then the normal modes can be calculated from the identified equivalent system matrix. Two examples of modal identification are carried out to demonstrate the availability and effectiveness of the proposed backward approach: (1) Numerical modal identification for a three-degree-of-freedom dynamic system with noise level in 20% of r.m.s of measured output data; (2) experimental modal identification of a cantilever beam. Finally, to show the advantage of the proposed VBAR approach on the selection of physical modes, the modal identification by stochastic subspace method was performed. The results from both methods are compared.     

基于模型选择的模式波前重构算法研究

基于Zernike模式的波前重构算法通常忽略实际波前像差构成的差异, 而用一定数量的低阶Zernike模式进行波前重构, 导致模式混淆或耦合等问题, 进而影响波前重构的精度. 根据信息论中的最小描述长度准则对重构模型的阶数进行了选择, 在此基础上应用非线性优化算法计算重构系数, 并最终实现波前重构; 对不同信噪比条件下振幅均匀分布入射光束的波前进行了重构. 结果表明: 该算法不但能够实现相对于现有算法相对较高的波前重构精度, 并且具有优良的噪声适应性, 体现了模型选择在模式法波前重构算法中的意义.     

All-fiber few-mode erbium-doped fiber amplifier supporting six spatial modes

Using the few-mode erbium-doped fiber(FM-EDF) with a simple two-layer erbium-doped structure, we demonstrate an all-fiber FM-EDF amplifier. The gain equalization among the six spatial modes supported by the FM-EDF is achieved when only the pump in the fundamental mode(LP_(01)) is applied. When the signals in six spatial modes are simultaneously amplified, the average modal gain is about 15 dB, and differential modal gain is about 2.5 dB for the signal at 1550 nm.