Allowance for shear distortion and rotatory inertia of ship hulls

The response of a ship to waves and to propeller excitation (as well as its distortion in still water) may be analysed in modal form by using a linear theory. For symmetric responses the approach has been discussed in terms of a “hull girder” treated as a simple beam. The same is true of uncoupled bending in antisymmetric motion. Simple beam theory has also been adapted for use in “coupled bending and twisting” responses of hulls with large deck openings. The theory has not, hitherto, embodied an allowance for the effects of shear distortion or rotatory inertia. It is shown in this paper how those effects may be allowed for in the analysis of symmetric response, that they do not alter the form of the more rudimentary analysis and that the response lends itself to a convenient matrix formulation.     

Nonlinear structural identification by the “Reverse Path” spectral method

When dealing with nonlinear dynamical systems, it is important to have efficient, accurate and reliable tools for estimating both the linear and nonlinear system parameters from measured data. An approach for nonlinear system identification widely studied in recent years is “Reverse Path”. This method is based on broad-band excitation and treats the nonlinear terms as feedback forces acting on an underlying linear system. Parameter estimation is performed in the frequency domain using conventional multiple-input–multiple-output or multiple-input–single-output techniques. This paper presents a generalized approach to apply the method of “Reverse Path” on continuous mechanical systems with multiple nonlinearities. The method requires few spectral calculations and is therefore suitable for use in iterative processes to locate and estimate structural nonlinearities. The proposed method is demonstrated in both simulations and experiments on continuous nonlinear mechanical structures. The results show that the method is effective on both simulated as well as experimental data.     

Vibration analysis of a continuous system subject to generic forms of actuation forces and sensing devices

This work provides a general formulation to solve vibration problems for continuous systems with damping effects, including modal, transient, harmonic and spectrum response analyses. In modal analysis, the system eigenvalues and corresponding eigenfunctions can be determined. The orthogonal relations of eigenfunctions are shown. For transient, harmonic and spectrum analyses, the generic force/actuator functions and response/sensing operators are introduced, respectively, and used to derive the system response. The time domain response is obtained for transient analysis, the frequency response function is derived for harmonic analysis and statistical quantities of response variables due to random excitation are determined in spectrum analysis. The solution for each type of analysis can be formulated and expressed in a concise format in terms of generic force/actuator and response/sensor mode shape functions. In particular, one-dimensional beam and two-dimensional plate vibration analyses are illustrated by following the developed generic formulation. This work provides the complete analytical solutions of four types of vibration analyses for continuous systems and can be applied to other engineering structures as well.     

Identification of a dynamical model for an acoustic enclosure using the wavelet transform

This paper presents results of research work in the identification of a dynamical model for an acoustic enclosure, a duct with rectangular cross-section, closed ends, and side-mounted speaker enclosures. An acoustic enclosure is excited randomly and random decrement functions are built to convert the random responses to free acoustic responses. It is shown that the estimation of resonance frequencies is possible using the wavelet transform of the system’s free response. Using a particular form of the son wavelet function, results are improved compared to those obtained with the traditionally Morlet wavelet function. An optimal value of a parameter of the son wavelet function is obtained by minimization of the wavelet entropy. The accuracy of this new technique is confirmed by applying it to a numerical example, and to an acoustic enclosure. The advantage of using the wavelet transform method over the Fourier-based modal analysis that would normally be used for the enclosure modes problem is established.     

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.     

USING THE CROSS-CORRELATION TECHNIQUE TO EXTRACT MODAL PARAMETERS ON RESPONSE-ONLY DATA

Modal parameter identification is used to identify those parameters of the model which describe the dynamic properties of a vibration system. Classical modal parameter extractions usually require measurements of both the input force and the resulting response in laboratory conditions. However, when large-scale operational structures are subjected to random and unmeasured forces such as wind, waves, or aerodynamics, modal parameters estimation must base itself on response-only data. Over the past years, many time-domain modal parameter identification techniques from output-only have been proposed. Among them, the natural excitation technique (NExT) has been a very powerful tool for the modal analysis of structures excited in operating environment. This issue reviews the theoretical development of natural excitation technique (NExT), which uses the cross-correlation functions of measured responses coupling with conventional time-domain parameter extraction under the assumption of white-noise random inputs. Then a frequency-domain poly reference modal identification scheme by coupling the cross-correlation technique with conventional frequency-domain poly reference modal parameter extraction is presented. It uses cross-power spectral density functions instead of frequency response functions and auto- and cross-correlation functions instead of impulse response functions to estimate modal parameters from response-only data. An experiment using an airplane model is performed to investigate the effectiveness of the cross-correlation technique coupled with frequency-domain poly reference modal identification scheme.     

Response of plates with unconstrained layer damping treatment to random acoustic excitation,part I: Damping and frequency evaluations

For theoretical evaluation of the response of a structure under random acoustic excitation a complete understanding is required of the various modes of vibration and the modal damping associated with each mode. In order to evaluate these parameters for plates with unconstrained layer damping treatment, some of the theoretical approaches applicable are used. Experimentally observed modal frequencies and associated loss factors are compared with those estimated by different theories for all edges simply supported and all edges clamped boundaries, after accounting for the damping at supports. The modes of vibration used in the theoretical analysis for these boundaries are compared with those observed in the experiments. These results are made use of in Part II for the evaluation of response under random acoustic excitation.     

Experimental identification of viscous damping in linear vibration

This paper is concerned with the experimental evaluation of the performance of viscous damping identification methods in linear vibration theory. Both existing and some new methods proposed by the present authors [A.S. Phani, J. Woodhouse, Viscous damping identification in linear vibration, Journal of Sound and Vibration 303 (3–5) (2007) 475–500] are applied to experimental data measured on two test structures: a coupled three cantilever beam with moderate modal overlap and a free–free beam with low modal overlap. The performance of each method is quantified and compared based on three norms and the best methods are identified. The role of complex modes in damping identification from vibration measurements is critically assessed.     

Estimation of interfacial forces in time domain for linear systems

A structural path rank ordering process under transient excitations requires a good knowledge of the interfacial path forces, which are difficult to directly measure. Four time domain methods to estimate the interfacial forces are proposed and comparatively evaluated with application to linear time-invariant, proportionally damped discrete systems. First, the transient response is derived by modal analysis and a direct time domain technique to calculate the interfacial forces is outlined. Next, the frequency domain estimation methods, based on the sub-system concept are reviewed, and an inverse Fourier transform scheme is introduced. An indirect method of estimating interfacial force in transient state is then developed through an inverse procedure of modal analysis. The sub-system approach is employed to obtain the interfacial forces based on the forced vibration response of the original system and modal data of the sub-system. Finally, an approximate time domain scheme is suggested that could be used only if the system properties are known or precisely estimated. Although the proposed indirect methods are designed for eventual experimental applications, this article provides numerical feasibility studies via a simple source-path-receiver system (with parallel vibration paths) that has five translational degrees of freedom. The proposed methods are compared under ideal impulse force excitation input and a periodic sawtooth load (without and with Gaussian noise) to observe the starting transients as well as subsequent motions and interfacial forces. Preliminary comparisons with a laboratory experiment are very promising.     

Modal analysis by free vibration response only for discrete and continuous systems

This work aims to develop the algorithm for modal analysis by free vibration response only (MAFVRO), in particular for the general or non-proportional viscous damping system model. If the structural displacement or acceleration response due to free vibration can be measured, the system response matrices, including the displacement, velocity and acceleration, can be obtained through numerical differential or integration methods. These response matrices can then be applied to the developed MAFVRO method to determine the structural modal parameters. The numerical differential and integration methods are introduced and adopted to establish the modal parameter prediction program for the non-proportional damping model of MAFVRO. This work also shows the applications of MAFVRO to the multiple degree-of-freedom (mdof) systems and the cantilever beam, respectively. Both the discrete and continuous systems are demonstrated for the feasibility of the MAFVRO algorithm. The developed method uses the free vibration output response only and can obtain the structural modal parameters successfully.     

Frequency response testing in a noisy environment or with a limited power supply

The current trend in dynamic testing is towards short duration, transient testing in the field followed by detailed computer analysis of the data at some later time. For many applications, tests must be carried out in a noisy environment, such as in frequency response studies of operating machinery and vibration tests on buildings or ships where the excitation power source is small and produces a low level response compared with ambient vibration levels. The use of time domain averaging is demonstrated and it is shown to be a suitable signal enhancement procedure for testing in poor signal to noise ratio conditions, when using a transient excitation technique. The limitations and accuracy of the technique are discussed and examples of practical application are given.     

Stationary response of combined linear dynamic systems to stationary excitation

The stationary response of a broad class of combined linear systems to stationary random excitation is determined by the normal mode method. The systems are characterized by a viscously damped simple beam (or string, membrane, thin plate or shell, etc.) connected at discrete points to a multiplicity of viscously damped linear oscillators and/or masses. The solution of the free vibration problem by way of Green functions and the deterministic forced vibration problem by modal analysis for both proportional and non-proportional damping is reviewed. The orthogonality relation for the natural modes of vibration is used to derive a unique relationship between the cross-spectral density functions of the applied forces and the cross-spectral density functions of the generalized forces. Finally, the response spectral density functions and the mean square responses of the beam and oscillators are derived in closed form, exact for the proportionally damped system and approximate for the non-proportionally damped system.     

The decoupling of damped linear systems in free or forced vibration

The purpose of this paper is to extend classical modal analysis to decouple any viscously damped linear system in non-oscillatory free vibration or in forced vibration. Based upon an exposition of how exponential decay in a system can be regarded as imaginary oscillations, the concept of damped modes of imaginary vibration is introduced. By phase synchronization of these real and physically excitable modes, a time-varying transformation is constructed to decouple non-oscillatory free vibration. When time drifts caused by viscous damping and by external excitation are both accounted for, a time-varying decoupling transformation for forced vibration is derived. The decoupling procedure devised herein reduces to classical modal analysis for systems that are undamped or classically damped. This paper constitutes the second and final part of a solution to the “classical decoupling problem.” Together with an earlier paper, a general methodology that requires only the solution of a quadratic eigenvalue problem is developed to decouple any damped linear system.     

On the application of blind source separation for damping estimation of bridges under traffic loading

The accurate and reliable estimation of modal damping from output-only vibration measurements of structural systems is a continuing challenge in the fields of operational modal analysis (OMA) and system identification. In this paper a modified version of the blind source separation (BSS)-based Second-Order Blind Identification (SOBI) method was used to perform modal damping identification on a model bridge structure under varying loading conditions. The bridge model was created with finite elements and consisted of a series of stringer beams supported by a larger girder. The excitation was separated into two categories: ambient noise and traffic loads with noise modeled with random forcing vectors and traffic simulated with moving loads for cars and partially distributed moving masses for trains. The acceleration responses were treated as the mixed output signals for the BSS algorithm. The modified SOBI method used a windowing technique to maximize the amount of information used for blind identification from the responses. The modified SOBI method successfully found the mode shapes for both types of excitation with strong accuracy, but power spectral densities (PSDs) of the recovered modal responses showed signs of distortion for the traffic simulations. The distortion had an adverse affect on the damping ratio estimates for some of the modes but no correlation could be found between the accuracy of the damping estimates and the accuracy of the recovered mode shapes. The responses and their PSDs were compared to real-world collected data and patterns similar to distortion were observed implying that this issue likely affects real-world estimates.     

A wavelet approach to fault diagnosis of a gearbox under varying load conditions

Varying load can cause changes in a measured gearbox vibration signal. However, conventional techniques for fault diagnosis are based on the assumption that changes in vibration signal are only caused by deterioration of the gearbox. There is a need to develop a technique to provide accurate state indicator of gearbox under fluctuating load conditions. This paper presents an approach to gear fault diagnosis based on complex Morlet continuous wavelet transform under this condition. Gear motion residual signal, which represents the departure of time synchronously averaged signal from the average tooth-meshing vibration, is analyzed as source data due to its lower sensitiveness to the alternating load condition. A fault growth parameter based on the amplitude of wavelet transform is proposed to evaluate gear fault advancement quantitatively. We found that this parameter is insensitive to varying load and can correctly indicate early gear fault. For a comparison, the advantages and disadvantages of other measures such as kurtosis, mean, variance, form factor and crest factor, both of residual signal and mean amplitude of continuous wavelet transform waveform, are also discussed. The effectiveness of the proposed fault indicator is demonstrated using a full lifetime vibration data history obtained under sinusoidal varying load.     

Modeling and simulating of V-shaped piezoelectric micro-cantilevers using MCS theory considering the various surface geometries

Atomic force microscopy (AFM) is widely used as a tool in studying surfaces and mechanical properties of materials at nanoscale. This paper deals with mechanical and vibration analysis of AFM vibration in the non-contact and tapping modes for V-shaped piezoelectric micro-cantilever (MC) with geometric discontinuities and cross section variation in the air ambient. In the vibration analysis, Euler-Bernoulli beam theory based on modified couple stress (MCS) theory has been used. The governing equation of motion has been derived by using Hamilton's principle. By adopting finite element method (FEM), the MC differential equation has been solved. Damping matrix was considered in the modal space. Frequency response was obtained by using Laplace transform, and it has been compared with experimental results. Newmark algorithm has been used based on constant average acceleration to analyze time response of MC, and then time response results in the vibration mode, far from the sample surface have been compared with experimental data. In vicinity of sample surface, MC is influenced by various nonlinear forces between the probe tip and sample surface, including van der Waals, contact, and capillary forces. Time response was examined at different distances between MC base and sample surface, and the best distance was selected for topography. Topography results of different types of roughness showed that piezoelectric MC has been improved in the air ambient. Topography showed more accurate forms of roughness, when MC passes through sample surface at higher frequencies. The surface topography investigation for tapping and non-contact modes showed that using of these two modes are suitable for topography.     

Efficient modal control strategies for active control of vibrations

Some efficient strategies for the active control of vibrations of a beam structure using piezoelectric materials are described. The control algorithms have been implemented for a cantilever beam model developed using finite element formulation. The vibration response of the beam to an impulse excitation has been calculated numerically for the uncontrolled and the controlled cases. The essence of the method proposed is that a feedback force in different modes be applied according to the vibration amplitude in the respective modes i.e., modes having lesser vibration may receive lesser feedback. This weighting may be done on the basis of either displacement or energy present in different modes. This method is compared with existing methods of modal space control, namely the independent modal space control (IMSC), and modified independent modal space control (MIMSC). The method is in fact an extension of the modified independent space control with the addition that it proposes to use the sum of weighted multiple modal forces for control. The proposed method results in a simpler feedback, which is easy to implement on a controller. The procedure is illustrated for vibration control of a cantilever beam. The analytical results show that the maximum feedback control voltage required in the proposed method is further reduced as compared to existing methods of IMSC and MIMSC for similar vibration control. The limitations of the proposed method are discussed.     

Modal parameter identification through Gabor expansion of response signals

A new method of identifying modal parameters by decomposing response signals with Gabor transform is presented in this paper to estimate natural frequencies, damping ratios and mode shapes of linear time invariant systems. According to Gabor expansion theory, responses of a multi-degree-of-freedom system can be decomposed into uncoupled signal components, each vibrating at a single natural frequency. From these uncoupled signals, modal parameters are subsequently extracted with common methods. The proposed method can process stationary and non-stationary responses and requires no input signal except for the response signals generated by unknown excitation acting on a system. In the sense of less restriction on the in-out signals, the approach based on time-frequency decomposition is very general. A simulation study on a simply supported beam under non-stationary excitation has demonstrated that the proposed method is effective in parameter estimation.     

倒伏胁迫下的玉米冠层结构特征变化与光谱响应解析

倒伏胁迫是玉米生产中的主要灾害之一,严重影响玉米的产量、品质和机械收获能力。解析不同倒伏胁迫强度下玉米冠层结构变化规律及其光谱响应机理,是玉米倒伏灾情大范围遥感监测的基础。分别在玉米抽雄期、灌浆中期设置茎倒、茎折、根倒3种强度的倒伏处理,基于田间多频次持续观测实验,分析生育期、倒伏类型对玉米冠层结构动态变化及其自我恢复能力的影响;采用传统光谱变换与连续小波变换方法对倒伏玉米冠层高光谱进行处理,选取叶面积密度（LAD）为玉米倒伏冠层结构特征指标,筛选叶面积密度最佳敏感波段和小波系数,基于随机森林法构建叶面积密度高光谱响应模型,利用未参与建模的实测样本验证模型精度,重点探讨小波分解尺度和光谱分辨率对LAD光谱响应能力的影响规律。研究结果表明：叶面积密度作为单位体积内叶面积总量的冠层结构表征指标,与倒伏胁迫强度具有较好的响应关系,灌浆期的倒伏玉米LAD普遍高于抽雄期,抽雄期LAD整体表现为茎折＞根倒＞茎倒＞未倒伏,灌浆期LAD整体表现为根倒＞茎折＞茎倒＞未倒伏;经连续小波变换后,玉米倒伏冠层光谱对玉米倒伏LAD的响应能力普遍优于传统光谱变换,随着小波分解尺度的增加,LAD与敏感波段的相关性越强,其中10尺度相关系数最高,达0.74;连续小波变换所构建的模型精度普遍优于传统光谱变换,其中由原始光谱小波变换后构建的LAD响应模型精度最高,检验样本的R2为0.811,RMSE为1.763,表明连续小波变换技术可凸显和利用冠层光谱中的细微信息。因此,叶面积密度可有效定量表征不同倒伏胁迫程度的玉米冠层结构变化特征,连续小波变换能有效提升冠层光谱对倒伏玉米结构变化的响应能力,基于随机森林法构建的倒伏玉米叶面积密度诊断模型具有较高的精度和稳定性,可为区域尺度的夏玉米倒伏灾情遥感监测提供先验知识。     

The effect of trailing end geometry on the vibration of a circular cantilevered rod in nominally axial flow

The effect of “trailing end” geometry on the vibration of a circular cantilevered rod in nominally axial water flow is reported. Eleven different trailing end geometries were tested. For each end geometry, rms displacement response and damping were measured and are presented as functions of mean axial flow velocity. The various geometric end shapes are ordered according to their effectiveness in attenuating rod vibration. The results, which are in qualitative agreement with similar experimental results from tests on flat plates, provide the designer with empirical information that allows selecting a trailing end geometry that minimizes response to parallel flow excitation. The results have application in the design of nuclear reactor fuel pins and certain instrumentation probes.