A hybrid approach for the modal analysis of continuous systems with discrete piecewise-linear constraints

The analysis of continuous systems with piecewise-linear constraints in their domains have previously been limited to either numerical approaches, or analytical methods that are constrained in the parameter space, boundary conditions, or order of the system. The present analysis develops a robust method for studying continuous systems with arbitrary boundary conditions and discrete piecewise-linear constraints. A superposition method is used to generate homogeneous boundary conditions, and modal analysis is used to find the displacement of the system in each state of the piecewise-linear constraint. In order to develop a mapping across each slope discontinuity in the piecewise-linear force-deflection profile, a variational calculus approach is taken that minimizes the L₂ energy norm between the previous and current states. An approach for calculating the finite-time Lyapunov exponents is presented in order to determine chaotic regimes. To illustrate this method, two examples are presented: a pinned-pinned beam with a deadband constraint, and a leaf spring coupled with a connector pin immersed in a viscous fluid. The pinned-pinned beam example illustrates the method for a non-operator based analysis. Results are used to show that the present method does not necessitate the need of a large number of basis functions to adequately map the displacement and velocity of the system across states. In the second example, the leaf spring is modeled as a clamped-free beam. The interaction between the beam and the connector pin is modeled with a preload and a penalty stiffness. Several experiments are conducted in order to validate aspects of the leaf spring model. From the results of the convergence and parameter studies, a high correlation between the finite-time Lyapunov exponents and the contact time per period of the excitation is observed. The parameter studies also indicate that when the system's parameters are changed in order to reduce the magnitude of the impact velocity between the leaf spring and the connector pin, the extent of the regions over which a chaotic response is observed increases.     

Active modal control for sound field in a room

I.IntroductionInlate198()'s,thestudiesonactivenoisecontrolhavebeengraduallyturnedfromtheductsandopenspacestothemorepracticaIenclosedsoundfields.SomespeciaIworkinaccordancewiththeactiveattenuationofcabinnoisehasstartedwiththeoreticalmechanisms,systemdesignandtransducerdevelopmellt.Intensivereflectionsandtheformedstandingwavedistributioni11al1enclosedsoundfieldarealwaysthemaindifficultiesforactivenoisecontrolinthesecases.Ontheotherhand,normalmoderesponseofthefieldalsoimplementsanewmethod,theap…     

Minimal input models for sound level prediction in fitted enclosed spaces

Highly advanced computer models for the prediction of sound fields in rooms are now available. However, these tools are complex and require a skilled acoustician to use effectively and hence there is a need for more simpler models. A simple model needs to be accurate and quick to use, but most importantly should require a minimum amount of input data to construct the model. This is only achievable if the scope of the model is reduced to one or two acoustic parameters. Three simple models were investigated two empirical based formulae and a geometric acoustic model. The models were validated in six configurations of an experimental room simulating a textile workshop and two real engineering workrooms. It was found that all the models executed near instantaneously, but the obtainable prediction accuracy and consistency was proportional to the amount of input data. The models are now available on the Web, running directly inside Netscape or Internet Explorer.     

A hybrid method of modal synthesis using vibration tests

The assembly of structures along continuous boundaries presents great difficulty in the context of modal synthesis. In order to solve such problems, a method is proposed in which a hybrid model is defined reflecting the dynamic behavior of a structure loaded along a boundary. It is based on Weinstein's method and corresponds to a generalization of the impedance matrix method. Generalized boundary co-ordinates are defined from branch modes obtained by introducing mass loading along the boundary. Thus, the hybrid model can be derived from testing as a result of two independent modal identifications. The method permits high precision prediction of the influence of strong structural modifications. Thus, in the case of rectangular plates, it has been possible to find the modes of a cantilever plate and of a plate with stiffeners from the free modes.     

An approximate treatment of reflection coefficient in the phased beam tracing method for the simulation of enclosed sound fields at medium frequencies

The phased beam tracing method (PBTM) was suggested as a medium-frequency simulation technique for the calculation of impulse response, although main assumptions of geometric acoustics still hold. The phased method needs the reflection coefficient for characterizing the acoustic property of a surface and the complex wave number for describing the propagation characteristics. In this study, two types of approximate real reflection coefficients derived from the measured absorption coefficient were tested for a practical applicability. As a test example, pressure impulse responses and energy impulse responses computed from the PBTM were compared with those from the measurement and the ordinary beam tracing method. The PBTM employing the approximate reflection coefficients greatly increased the accuracy of the prediction compared to the ordinary beam tracing method, in particular at the medium-frequency range in octave bands above the Schroeder cutoff frequency. A comparison was made between angle-dependent and angle-independent reflection coefficients in the calculation of acoustic measures. Although the angle-dependent reflection coefficient yielded best matched results with measured data, but the simple angle-independent reflection coefficient can be also used with a reasonably good precision.     

A complete method for efficient fuzzy modal analysis

The objective of this paper is to determine both the fuzzy eigenvalues and eigenvectors of a finite element model defined with fuzzy parameters. The proposed method introduces the concepts of mode shape pairing and the functional dependence of eigensolutions with respect to design parameters. High-order approximations are then introduced to limit the computational cost associated with variability management. Numerical test cases are used to highlight the abilities of this method to predict behaviour modifications due to variations in the physical parameters.     

A hybrid SEA/modal technique for modeling structural-acoustic interior noise in rotorcraft

This paper describes a hybrid technique that combines Statistical Energy Analysis (SEA) predictions for structural vibration with acoustic modal summation techniques to predict interior noise levels in rotorcraft. The method was applied for predicting the sound field inside a mock-up of the interior panel system of the Sikorsky S-92 helicopter. The vibration amplitudes of the frame and panel systems were predicted using a detailed SEA model and these were used as inputs to the model of the interior acoustic space. The spatial distribution of the vibration field on individual panels, and their coupling to the acoustic space were modeled using stochastic techniques. Leakage and nonresonant transmission components were accounted for using space-averaged values obtained from a SEA model of the complete structural-acoustic system. Since the cabin geometry was quite simple, the modeling of the interior acoustic space was performed using a standard modal summation technique. Sound pressure levels predicted by this approach at specific microphone locations were compared with measured data. Agreement within 3 dB in one-third octave bands above 40 Hz was observed. A large discrepancy in the one-third octave band in which the first acoustic mode is resonant (31.5 Hz) was observed. Reasons for such a discrepancy are discussed in the paper. The developed technique provides a method for modeling helicopter cabin interior noise in the frequency mid-range where neither FEA nor SEA is individually effective or accurate.     

A modal test method using sound pressure transducers based on vibro-acoustic reciprocity

A modal test method that uses sound pressure transducers at fixed locations and an impact hammer roving over a test structure is developed in this work. Since sound pressure transducers are used, the current method deals with a coupled structural–acoustic system. Based on the vibro-acoustic reciprocity, the method is equivalent to one, where acoustic excitations at fixed locations are given and the resulting acceleration of the test structure is measured. The current method can eliminate mass loading due to use of accelerometers, which can destroy existence of repeated or close natural frequencies of a symmetric structure. It can also avoid effects of a nodal line of a mode and an inactive area of a local mode, and measure all the out-of-plane modes within a frequency range of interest, including global and local ones. The coupling between the structure and the acoustic field in a structural–acoustic system introduces asymmetry in the model formulation. An equivalent state space formulation is used for a damped structural–acoustic system and the associated eigenvalue problem is derived. The biorthonormality relations between the left and right eigenvectors and the relations between the structural and acoustic components in the left and right eigenvectors are proved. The frequency response functions associated with the current method are derived and their physical meanings are explained. The guidelines for using the current method, including the types of structures that are suitable for the method, the positions of the sound pressure transducers, and the orientation of the test structure relative to the transducers, are provided. Modal tests were carried out on an automotive disk brake using the traditional and current methods, where multiple accelerometers and microphones were used to measure its dynamic responses induced by impacts, respectively. The differences between the measured natural frequencies using the current method and those from the finite element model of the disk brake are less than 3 percent for the first 18 elastic modes, and the modal assurance criterion values of the associated mode shapes are all above 90 percent. The current method was also used to measure the natural frequencies, damping ratios, and mode shapes of a light circuit board.     

Equivalent source method for identification of panel acoustic contribution in irregular enclosed sound field

For the interior sound field formed by the complex vibrating structure,an identification approach of panel acoustic contribution based on equivalent source method(ESM)was presented.The normal velocity on the surface of vibrating structure was first reconstructed by using interior nearfield acoustic holography based on ESM and the prediction of whole interior enclosed sound field was realized.Then the sound pressure produced by each panel at the interested field point was respectively replaced by the radiated pressure of the enclosed interior sound field which is formed by the equivalent virtual sources located near the surface of the cavity.Combining with the reconstructed normal surface velocity,the acoustic contribution of each panel to any position in the cavity was obtained by transforming the complex enclosed non-free field into the simple interior free field.Numerical simulations and experiments are conducted,and the influences of the number of the equivalent sources and the distance between them and the reconstructed surface have been investigated.The results show that the proposed method is easier to be implemented with the same accuracy than the traditional analysis method.     

Dynamic holographic modal analysis for NDT

This paper describes crack and defect detection in structures through modification of the vibrational modal patterns and surface responses to stress. Features are made visible with dynamic holographic interferometery combined with parameter estimation. The procedure involves an unconventional, optimized, laser-illumination method. The methods are especially applicable to large structures and could prove pivotal to improved designs, monitoring and maintenance. Components and structures could be designed to better withstand operating stresses, and existing structures could be analysed to predict their response to stress. Since the modal characterization of a structure can act as a type of fingerprint, holographic interferometry can also be used to monitor structural degradation due to operating and aging. Modal characterization includes identification of resonant frequencies and also the corresponding mode shapes. Holographic interferometry provides for direct modal characterization of a structure as well as measuring its small loading dynamic response. The project demonstrated that a wide variety of defects can be located in structural components, vessels and pipes. An analytical exercise also demonstrates the ability to use global modal characteristics to determine the presence of local corrosion and erosion.     

A new quantity for comparative measurements concerning the diffusion of stationary sound fields

Both from experimental and theoretical observations it is known that the sound fields in normal empty reverberation rooms are not sufficiently diffuse at low frequencies. In order to increase the degree of diffusion, additional elements (e.g., hanging panels) are usually brought into the standard measurement procedures. Several methods for evaluating the degree of diffusion have been proposed. However, most of the associated measures of diffusion have not been any great success.In this article a new test method based on a new way of measuring diffusion is proposed. This measure is basically an estimate of the cross-correlation coefficient for the sound pressures measured with two microphones a distance r apart from each other. The specific quantity is presented, and its basic properties are investigated in some normal situations. The results obtained are compared with existing sound field theories. These comparisons indicate that the measure behaves logically and is sensitive enough.Only $\text{1}\text{3}$-octave band filtering is considered in this article. However, with some slight modifications the procedure can be designed for investigations concerning other source filters.     

Use of a hybrid adaptive finite element/modal approach to assess the sound absorption of porous materials with meso-heterogeneities

The paper discusses the sound absorptive performance of a porous material with meso-perforations inserted in a rectangular waveguide using a numerical hybrid adaptive finite element-modal method. Two specific applications are investigated: (i) the improvement of porous materials noise reduction coefficient using meso-perforations (ii) the effects of lateral air gaps on the normal incidence sound absorption of mono-layer and two-layer porous materials. For the first application, a numerical design of experiments is used to optimize the sound performance of a porous material with meso-perforations with a reduced number of numerical simulation. An example in which the optimization process is carried out on the thickness and size of the perforation is presented to illustrate the relevance of the approach. For the second application, a set of twenty fibrous materials spanning a large flow resistivity range is used. Practical charts are proposed to evaluate the influence of air gaps on the average sound absorption performance of porous materials. This is helpful to both the experimenter regarding characterization of porous material based on Standing Wave Tube measurements and for the engineer to quantifying the impact of air gaps and for designing efficient absorbers.     

The coherence of reverberant sound fields

A new method of measuring spatial correlation functions in reverberant sound fields is presented. It is shown that coherence functions determined with appropriate spectral resolution contain the same information as the corresponding correlation functions, and that measuring such coherence functions is a far more efficient way of obtaining this information. The technique is then used to verify theoretical predictions of the spatial correlation between various components of the particle velocity in a diffuse sound field. Other possible applications of the technique are discussed and illustrated with experimental results obtained in an ordinary room.     

Prediction of the sound reduction index: a modal approach

Different methods for the calculation of the sound reduction index using modal analysis are described. The calculations use two steps: a vibratory study to determine the transverse displacement of the plate and a study of radiation. Orthotropic plates are studied in particular and the resulting calculation algorithm has been programmed. Initial hypotheses are indicated, as well as results obtained for various plates or partitions. Modal analysis calculation results are then compared to results of the Cremer-Sewell approach.     

A probabilistic method for analysis of sound localization performance

We propose a novel probabilistic method for quantitative analysis of the sound localization performance. The analysis is based on the two kinds of probability distributions estimated from a single dataset containing listening test results for sound localization. The quantitative parameters of the von Mises probability distributions provide meaningful interpretations on the localization performance. The mean direction represents the listener’s perceptual bias, and the shape parameters and the circular variance provide information on how much the responses are concentrated about the mean direction. The front-back confusion can be determined more systematically by the proposed method than the conventional one, especially for the responses near the boundary of front-back confusion region based on the conventional criterion. The proposed method can be easily extended to analyze the up-down and left-right confusions. To investigate the feasibility of the proposed method, the already published dataset originally obtained by Iida et al. was analyzed using the proposed probabilistic method. The results showed that the proposed method can provide meaningful and reasonable interpretations on the localization performance.     

Fast convergence modal analysis for continuous systems

A continuous system has an infinite number of degrees of freedom (n.d.o.f.) in a dynamic analysis. The dynamic stiffness method is able to produce an infinite number of natural modes with use of only a finite number of co-ordinates. The associated modal analysis is the only widely applicable approximate method for computing the response without discretizing the continuous system by methods such as the finite element method, in which the infinite n.d.o.f. is not retained. However, this modal analysis converges very slowly as the number of modes is increased if the loading distribution does not follow the patterns of the first few modes. A method is suggested in this paper to accelerate the convergence. A mixed mass matrix is introduced according to the reciprocal theorem to evaluate the initial transient while retaining the infinite n.d.o.f. with a fininte number of co-ordinates. Explicit formulae are given for space frames.     

The evaluation of binaural playback systems for virtual sound fields

By means of virtual acoustics technologies the impulse response in a location of a real enclosure can be used to create a virtual room. Also in this virtual space, typically implemented in a dead room, an impulse response can be measured. From both “real” and respective “virtual” impulse response a group of room-acoustical parameters can be calculated. The match of the two groups of indicators mostly depends on the layout of the playback system and on the implemented processing. In this paper a procedure of virtual measurement in a dead room is developed and some typical layouts of playback systems employing two loudspeakers are compared. Based on the consistency of the room-acoustical indicators derived from “real” and “virtual” impulse responses can be investigated. It is shown how the performance depends on the angular separation of loudspeakers and the efficiency of the cross-talk is also tested and discussed. Finally, a set-up which is suitable for psychoacoustics tests is indicated.