On the crossing points of the Lamb modes and the maxima and minima of displacements observed at the surface

This article elaborates on the crossing points of the frequency–wavenumber branches for the symmetric and anti-symmetric Lamb modes in a homogeneous plate. It is shown both theoretically as well as experimentally that at these crossing points either the normal or the longitudinal components of modal displacement attain an extreme value, i.e. a maximum or it vanishes. This behavior is assessed herein using a method due to Mindlin, who showed that the dispersion curves for a plate with mixed boundary conditions – which are associated with uncoupled shear and dilatational modes – provide bounds to the spectral lines of the free plate. Therefore, a subset of the crossing points of the symmetric and antisymmetric Lamb modes for a free plate coincide with the crossing points for a plate with mixed boundary conditions.     

阻尼受击板的时域建模及声音合成

为满足声源辨识中对合成冲击声的迫切需求,建立了球-板撞击的时域模型,提出一种时域快速求解方法,并进行了实验验证.首先给出一种将时域有限差分法(FDTD)和模态展开法(MEM)相结合的时域混合方法,求解板的振动方程,并解决了混合方法中MEM的模态截断和初值问题,及两种方法中阻尼的一致性问题;随后,给出了简支矩形板的冲击声计算结果,通过与FDTD方法的运算量进行对比,验证了混合方法的高效性;最后,针对自由边界下的L形板进行了实验验证.结果表明,与传统FDTD方法相比,时域混合方法在保证合成冲击声精度的前提下可将计算效率提高100至1200倍。      

Mode propagation in curved waveguides and scattering by inhomogeneities: application to the elastodynamics of helical structures

This paper reports on an investigation into the propagation of guided modes in curved waveguides and their scattering by inhomogeneities. In a general framework, the existence of propagation modes traveling in curved waveguides is discussed. The concept of translational invariance, intuitively used for the analysis of straight waveguides, is highlighted for curvilinear coordinate systems. Provided that the cross-section shape and medium properties do not vary along the waveguide axis, it is shown that a sufficient condition for invariance is the independence on the axial coordinate of the metric tensor. Such a condition is indeed checked by helical coordinate systems. This study then focuses on the elastodynamics of helical waveguides. Given the difficulty in achieving analytical solutions, a purely numerical approach is chosen based on the so-called semi-analytical finite element method. This method allows the computation of eigenmodes propagating in infinite waveguides. For the investigation of modal scattering by inhomogeneities, a hybrid finite element method is developed for curved waveguides. The technique consists in applying modal expansions at cross-section boundaries of the finite element model, yielding transparent boundary conditions. The final part of this paper deals with scattering results obtained in free-end helical waveguides. Two validation tests are also performed.     

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.     

A study of modal characteristics and the control mechanism of finite periodic and irregular ribbed plates

An analytical solution is presented in this paper to investigate the control mechanism and modal characteristics of finite periodic and irregular ribbed plates. Peak responses of a finite periodic ribbed plate were examined where they were grouped into two sets of propagation zones according to the coupling mechanism at beam/plate interfaces. Details of modal characteristics in pass bands of the periodic ribbed plate were elucidated and the control mechanism was discussed. Modes in each pass band that are governed by shear force couplings were characterized by one of the beam flexural modes whose modal responses could be represented approximately by those of the corresponding orthotropic plate modes. Modes in the second set of pass bands were found to retain the resonance frequencies of the corresponding modes of the unribbed base plate. Higher order orthotropic plate modes were also identified, which could not be grouped into any pass bands defined by the classical periodic theory. The control mechanism leading to vibration confinement in disordered and irregular ribbed plates was also discussed. It was found that beam spacing irregularity attributes to localization of the group of modes associated with flexural wave couplings but not the group of modes associated with moment couplings.     

A hybrid modal analysis for enclosed sound fields

A hybrid modal expansion that combines the free field Green's function and a modal expansion will be presented in this paper based on a review and an extension of the existing modal analysis theories for the sound field in enclosures. The enclosed sound field will be separated into the direct field and reverberant field, which have been treated together in the traditional modal analysis. Studies on a point source in rectangular enclosures show that the hybrid modal expansion converges notably faster than the traditional modal expansions, especially in the region near the source, and introduces much smaller errors with a limited number of modes. The hybrid modal expansion can be easily applied to complex sound sources if the free field responses of the sources are known. Damped boundaries are also considered in this paper, and a set of modified modal functions is introduced, which is shown to be suitable for many damped boundary conditions.     

A mode-based approach for the mid-frequency vibration analysis of coupled long- and short-wavelength structures

A mode-based approach is described for the mid-frequency vibration analysis of a complex structure built-up from a long-wavelength source and a short-wavelength receiver. The source and the receiver respectively have low and high modal densities and modal overlaps. Each substructure is described in terms of its uncoupled, free-interface natural modes. The interface forces and displacements are decomposed in terms of a set of interface basis functions. Enforcing equilibrium and continuity conditions along the interface hence yields an analytical solution for the vibration response of the built-up structure. Expressions for the frequency response of the source and the power transmitted to the receiver are found. The correlations between the modal properties of the source and the receiver along the interface are derived. These modify the dynamic stiffness matrix of the structure. The flexible receiver is seen to add effective mass and damping to the source. The modes of the short-wavelength receiver are then described statistically in terms of a simple standing wave model. This approximation avoids the need for a modal analysis of the receiver. The results are compared with those of other methods including fuzzy structure theory. Numerical and experimental examples for beam-stiffened plate models are presented.     

A combined finite element and modal decomposition method to study the interaction of Lamb modes with micro-defects

This paper presents a combined finite element and modal decomposition method to study the interaction of Lamb waves with damaged area. The finite element mesh is used to describe the region around the defects. On the contrary to other hybrid models already developed, the interaction between Lamb waves and defects is computed in the temporal domain. Then, the modal decomposition method permits to determine the wave reflected and transmitted by the damaged area. Modal analysis allows also identifying the mode conversions induced by the defects. These numerical results agree with previous finite element results concerning the interaction of Lamb modes with a notch. Experiments, carried out with gauged defects on an aluminum plate, are also compared to numerical predictions to validate the simulation. Compared to classical techniques of simulation, this new method allows us to investigate the interaction of Lamb modes generated at high frequency-thickness product with micro-defects as corrosion pitting.     

Numerical predictions and experiments for optimizing hidden corrosion detection in aircraft structures using Lamb modes

To increase the sensitivity of Lamb waves to hidden corrosion in aircraft structures, a preliminary step is to understand the phenomena governing this interaction. A hybrid model combining a finite element approach and a modal decomposition method is used to investigate the interaction of Lamb modes with corrosion pits. The finite element mesh is used to describe the region surrounding the corrosion pits while the modal decomposition method permits to determine the waves reflected and transmitted by the damaged area. Simulations make easier the interpretation of some parts of the measured waveform corresponding to superposition of waves diffracted by the corroded area. Numerical results permit to extract significant information from the transmitted waveform and thus to optimize the signal processing for the detection of corrosion at an early stage. Now, we are able to detect corrosion pits down to 80-μm depth distributed randomly on a square centimeter of an aluminum plate. Moreover, thickness variations present on aircraft structures can be discriminated from a slightly corroded area. Finally, using this experimental setup, aircraft structures have been tested.     

Vibration analysis of annular-like plates

The existence of eccentricity of the central hole for an annular plate results in a significant change in the natural frequencies and mode shapes of the structure. In this paper, the vibration analysis of annular-like plates is presented based on numerical and experimental approaches. Using the finite element analysis code Nastran, the effects of the eccentricity, hole size and boundary condition on vibration modes are investigated systematically through both global and local analyses. The results show that analyses for perfect symmetric conditions can still roughly predict the mode shapes of “recessive” modes of the plate with a slightly eccentric hole. They will, however, lead to erroneous results for “dominant” modes. In addition, the residual displacement mode shape is verified as an effective parameter for identifying damage occurring in plate-like structures. Experimental modal analysis on a clamped-free annular-like plate is performed, and the results obtained reveal good agreement with those obtained by numerical analysis. This study provides guidance on modal analysis, vibration measurement and damage detection of plate-like structures.     

Experimental modal substructuring to estimate fixed-base modes from tests on a flexible fixture

Fixed boundary conditions are often difficult if not impossible to simulate experimentally, but they are important to consider in many applications. In principle, modal substructuring or impedance coupling approaches can be used to predict the fixed base modes of a system from tests where the system has some other boundary condition if the motion at the connection point can be measured, but this approach can be highly sensitive to imperfections in the experimental measurements. This work presents two alternatives that reduce the sensitivity to experimental errors, capitalizing on recent works where additional degrees of freedom are used to improve the robustness of substructure uncoupling. The system of interest is tested while mounted on a stiff fixture, where some modes of the fixture inevitably interact with those of the system of interest. The modes of the system–fixture assembly are extracted using a modal test and then a modal substructuring approach is used to apply constraints to eliminate the motion of the fixture. Two types of constraints are proposed, one based on the modes of the fixture and the other on a singular value decomposition of the fixture motion that was observed during the test. Neither approach requires an estimate of the displacements or rotations at the points where the system of interest is connected to the fixture. The methods are validated by applying them to experimental measurements from a simple test system meant to mimic a flexible satellite on a stiff shaker table. A finite element model of the subcomponents was also created and the method is applied to its modes in order to separate the effects of measurement errors and modal truncation. The proposed method produces excellent predictions of the first several modes of the fixed-base structure, so long as modal truncation is minimized. The proposed approach is also applied to experimental measurements from a wind turbine blade mounted in a stiff frame and found to produce reasonable results.     

Horizontal refraction of propagating sound due to seafloor scours over a range-dependent layered bottom on the New Jersey shelf

Three-dimensional propagation effects of low frequency sound from 100 to 400 Hz caused by seafloor topography and range-dependent bottom structure over a 20 km range along the New Jersey shelf are investigated using a hybrid modeling approach. Normal modes are used in the vertical dimension, and a parabolic-equation approximate model is applied to solve the horizontal refraction equation. Examination of modal amplitudes demonstrates the effect of environmental range dependence on modes trapped in the water column, modes interacting with the bottom, and modes trapped in the bottom. Using normal mode ray tracing, topographic features responsible for three-dimensional effects of horizontal refraction and focusing are identified. These effects are observed in the measurements from the Shallow Water 2006 experiment. Specifically, signals from a pair of fixed sources recorded on a horizontal line array sitting on the seafloor show an intensification caused by horizontal focusing due to the seabed topography of 4 dB along the array.     

Free vibration analysis of the completely free rectangular plate by the method of superposition

While the subject of free vibration analysis of the completely free rectangular plate has a history which goes back nearly two centuries it remains a fact that most theoretical solutions to this classical problem are considered to be at best approximate in nature. This is because of the difficulties which have been encountered in trying to obtain solutions which satisfy the free edge conditions as well as the governing differential equation. In a new approach to this problem, by using the method of superposition, it is shown that solutions which satisfy identically the differential equation and which satisfy the boundary conditions with any desired degree of accuracy are obtained. Eigenvalues of four digit accuracy are provided for a wide range of plate aspect ratios and modal shapes. Exact delineation is made between the three families of modes which are characteristic of this plate vibration problem. Accurate modal shapes are provided for the response of completely free square plates.     

Active modal control simulation of vibro-acoustic response of a fluid-loaded plate

Active modal control simulation of vibro-acoustic response of a fluid-loaded plate is presented. The active modal control of the vibro-acoustic response is implemented using piezoelectric actuators/sensors. The active modal damping is added to the coupled system via negative velocity feedback. The feedback gain between the piezoelectric actuators/sensors for the modal control is obtained using the in-vacuo modal matrix and the incompressible fluid-loaded modal matrix. The modal control performance of structural vibration and acoustic radiation of a baffled plate is numerically studied. It is shown that the proposed method increases the modal damping ratio and achieves reduction in the mean square velocity and the sound power for given modes of the fluid-loaded plate.     

Measurement of relevant elastic and damping material properties in sandwich thick plates

An easy-to-implement method to measure relevant elastic and damping properties of the constituents of a sandwich structure, possibly with a heterogeneous core, is proposed. The method makes use of a one-point dynamical measurement on a thick-plate. The hysteretic model for each (possibly orthotropic) constituent is written generically as “E(1+jη)” for all mechanical parameters. The estimation method of the parameters relies on a mixed experimental/numerical procedure. The frequencies and dampings of the natural modes of the plate are obtained from experimental impulse responses by means of a high-resolution modal analysis technique. This allows for considerably more experimental data to be used. Numerical modes (frequencies, dampings, and modal shapes) are computed by means of an extended Rayleigh-Ritz procedure under the “light damping” hypothesis, for given values of the mechanical parameters. Minimising the differences between the modal characteristics yields an estimation of the values of the mechanical parameters describing the hysteretic behaviour. A sensitivity analysis assesses the reliability of the method for each parameter. Validations of the method are proposed by (a) applying it to virtual plates on which a finite-element model replaces the experimental modal analysis, (b) some comparisons with results obtained by static mechanical measurements, and (c) by comparing the results on different plates made of the same sandwich material.     

Sensitivity of a continuum vocal fold model to geometric parameters, constraints, and boundary conditions

The influence of key dimensional parameters, motion constraints, and boundary conditions on the modal properties of an idealized, continuum model of the vocal folds was investigated. The Ritz method and the finite element method were used for the analysis. The model's vibratory modes were determined to be most sensitive to changes in the anterior-posterior length of the vocal fold model, due to the influence of three-dimensional stress components acting in the transverse plane. Anterior/ posterior boundary conditions were found to have a significant influence on the vibratory response. Overestimation of modal frequencies resulted when vibration of the structure was restricted to the transverse plane. The overestimation of each modal frequency was proportional to the ratio of longitudinal to transverse Young's modulus, and was significant for ratio values less than 20.     

Analytical modeling and vibration analysis of internally cracked rectangular plates

This study proposes an analytical model for nonlinear vibrations in a cracked rectangular isotropic plate containing a single and two perpendicular internal cracks located at the center of the plate. The two cracks are in the form of continuous line with each parallel to one of the edges of the plate. The equation of motion for isotropic cracked plate, based on classical plate theory is modified to accommodate the effect of internal cracks using the Line Spring Model. Berger?s formulation for in-plane forces makes the model nonlinear. Galerkin?s method used with three different boundary conditions transforms the equation into time dependent modal functions. The natural frequencies of the cracked plate are calculated for various crack lengths in case of a single crack and for various crack length ratio for the two cracks. The effect of the location of the part through crack(s) along the thickness of the plate on natural frequencies is studied considering appropriate crack compliance coefficients. It is thus deduced that the natural frequencies are maximally affected when the crack(s) are internal crack(s) symmetric about the mid-plane of the plate and are minimally affected when the crack(s) are surface crack(s), for all the three boundary conditions considered. It is also shown that crack parallel to the longer side of the plate affect the vibration characteristics more as compared to crack parallel to the shorter side. Further the application of method of multiple scales gives the nonlinear amplitudes for different aspect ratios of the cracked plate. The analytical results obtained for surface crack(s) are also assessed with FEM results. The FEM formulation is carried out in ANSYS.     

中频声振耦合的波函数统计能量法

为了提高中频声振耦合的计算效率,提出了波函数-统计能量法的结构-声学耦合方法,该方法从波动理论的角度出发,将波函数法(WBM)和统计能量法(SEA)结合,通过在耦合面分别施加声压激励和速度边界条件,推导了耦合面参数理论计算公式。将该方法用到长方体声腔和钢板耦合的模型中,并对100~1000 Hz的计算结果进行了实验验证。WBM-SEA模型与参考FEM-SEA模型以及实验模型的频响曲线对比结果表明,WBM-SEA与FEM-SEA以及实验结果吻合很好,验证了混合WBM-SEA的有效性。通过收敛性分析发现混合WBM-SEA方法计算时间比混合FEM-SEA方法更少。从而可以得出结论:混合波函数-统计能量法方法对中频声振耦合预测是有效的,且比FEM-SEA更加高效。      

The effects of distributed masses on acoustic radiation behavior of plates

The acoustic radiation behavior of a plate with a distributed mass loading is studied. A set of in vacuo normal modes or fluid-loaded undamped normal modes are used for modal analysis of the acoustic radiation from a plate in air or in water. Modal radiation efficiency, modal volume displacement, modal input energy and sound power level are computed to show the effects of size and location of the mass loading on the acoustic radiation of the plate. It is observed that the acoustic radiation behavior of a mode in both cases will have relatively larger changes if the mass loading is placed on an antinode of the mode shape or the mass loading is more concentrated. The acoustic radiation behavior of a mode and the radiated power of the plate in water have less change than those in air with the same mass loading due to the added mass of the water, especially for the first few modes.     

APPLICATION OF THE SPECTRAL FINITE ELEMENT METHOD TO TURBULENT BOUNDARY LAYER INDUCED VIBRATION OF PLATES

The spectral finite element method and equally the dynamic stiffness method use exponential functions as basis functions. Thus it is possible to find exact solutions to the homogeneous equations of motion for simple rod, beam, plate and shell structures. Normally, this restricts the analysis to elements where the excitation is at the element ends. This study removes the restriction for distributed excitation, that in particular has an exponential spatial dependence, by the inclusion of the particular solution in the set of basis functions. These elementary solutions, in turn, build up the solution for an arbitrary homogeneous random excitation. A numerical implementation for the vibration of a plate, excited by a turbulent boundary layer flow, is presented. The results compare favourably with results from conventional modal analysis.