A threshold for the use of Tikhonov regularization in inverse force determination

In the analysis of structure-borne sound from installed machinery, it is important to be able to estimate the operational forces. Assuming that their location is known, indirect approaches based on matrix inversion can be used to reconstruct the operational forces from a set of measured operational responses and corresponding matrix of frequency response functions. In common with many such inverse problems, matrix ill-conditioning can affect the reliability of the results. Methods such as pseudo-inversion of over-determined matrices, singular value rejection, and Tikhonov regularization have been used previously to overcome this and it has been found that Tikhonov regularization generally performs well in reducing the errors in the reconstructed forces. However, full-rank pseudo-inversion (unregularized solution) gives better results than Tikhonov regularization in some cases, particularly with low condition numbers. Since the need for regularization is greatest when the matrix is ill-conditioned, this suggests the introduction of a threshold above which Tikhonov regularization is used and below which pseudo-inversion is used. In this study, the extent to which response errors are amplified in the force estimates is considered and plotted against the matrix condition number. This allows a threshold condition number to be identified above which Tikhonov regularization gives improved results. It is found that the threshold is related not only to the condition number but also to the matrix dimensions including the extent of over-determination. A simple empirical formula is obtained for this threshold that is usable for matrices in a wide range of matrix dimensions.     

Computation of the analysis error covariance in variational data assimilation problems with nonlinear dynamics

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function. The data contain errors (observation and background errors), hence there will be errors in the optimal solution. For mildly nonlinear dynamics, the covariance matrix of the optimal solution error can often be approximated by the inverse Hessian of the cost functional. Here we focus on highly nonlinear dynamics, in which case this approximation may not be valid. The equation relating the optimal solution error and the errors of the input data is used to construct an approximation of the optimal solution error covariance. Two new methods for computing this covariance are presented: the fully nonlinear ensemble method with sampling error compensation and the ‘effective inverse Hessian’ method. The second method relies on the efficient computation of the inverse Hessian by the quasi-Newton BFGS method with preconditioning. Numerical examples are presented for the model governed by Burgers equation with a nonlinear viscous term.     

湍流大气中哈特曼传感器的模式波前复原误差

 分别采用Zernike和Karhunen-loeve两种模式波前复原法,分析了子孔径斜率测量不受噪声影响的理想情况下,哈特曼传感器对大气湍流畸变波前的模式复原误差与大气湍流相干长度、传感器的结构尺寸、模式复原阶数等的关系。结果表明Karhunen-loeve模式法比Zernike模式法的波前复原误差更小些。     

VIBRATION SUPPRESSION OF A BEAM STRUCTURE BY INTERMEDIATE MASSES AND SPRINGS

A method based on the dynamic Green function has been proposed to determine the optimum values of masses and/or springs and their locations on a beam structure in order to confine the vibration at an arbitrary location. In the analysis, the beam is driven by a harmonic external excitation. The added masses on the beam and the springs attached are modelled as simple reactions that provide transverse forces to the beam. These forces act as secondary forces that reduce the response caused by the external force. Numerical simulation shows that the vibration of the beam can be confined in a certain region by the presence of masses and springs in best arrangement. This method is demonstrated for both a simply supported and a cantilever beam. An experimental set-up was designed in which a simply supported beam is excited by an electrodynamic shaker and the response of the beam is measured using an He-Ne laser system. This assures very accurate measurements and avoids any additional loading effects as in the case of accelerometers. Comparisons of the theoretical and the experimental results show good agreement.     

Inverse radiation problem of temperature distribution in one-dimensional isotropically scattering participating slab with variable refractive index

In this paper, an inverse analysis is performed for estimation of source term distribution from the measured exit radiation intensities at the boundary surfaces in a one-dimensional absorbing, emitting and isotropically scattering medium between two parallel plates with variable refractive index. The variation of refractive index is assumed to be linear. The radiative transfer equation is solved by the constant quadrature discrete ordinate method. The inverse problem is formulated as an optimization problem for minimizing an objective function which is expressed as the sum of square deviations between measured and estimated exit radiation intensities at boundary surfaces. The conjugate gradient method is used to solve the inverse problem through an iterative procedure. The effects of various variables on source estimation are investigated such as type of source function, errors in the measured data and system parameters, gradient of refractive index across the medium, optical thickness, single scattering albedo and boundary emissivities. The results show that in the case of noisy input data, variation of system parameters may affect the inverse solution, especially at high error values in the measured data. The error in measured data plays more important role than the error in radiative system parameters except the refractive index distribution; however the accuracy of source estimation is very sensitive toward error in refractive index distribution. Therefore, refractive index distribution and measured exit intensities should be measured accurately with a limited error bound, in order to have an accurate estimation of source term in a graded index medium.     

The quantification of structure-borne transmission paths by inverse methods. Part 1: Improved singular value rejection methods

Structure-borne sound from installed machinery is often transmitted into a receiver structure via many connection points and several co-ordinate directions at each of them. In order to quantify the contributions from the various connection points, the operational forces at the interfaces, or an equivalent set of forces at some other locations, should be determined. These forces may be combined with measured transfer functions to determine their contributions to the sound at the receiver locations. Inverse methods are becoming widely used, in which a matrix of measured accelerances is inverted at each frequency and used with operational acceleration data to find the forces. Due to poor conditioning of this matrix, however, the results can often be unreliable. In this paper, using both simulations and measurements, an assessment is made of the success and failure of various strategies for dealing with the problems of ill conditioning, in particular over-determination and singular value rejection. In each case the test structure is a rectangular plate, and a wide frequency range is covered to include regions of both low and high modal overlap. Critical for the rejection of singular values is a suitable threshold. It is established that previously used thresholds, based on estimates of error in either accelerances or operational responses, cannot be used universally. An alternative approach is developed in which the accelerance matrix is perturbed by a different amount for each sample of the operational responses. Based on this approach a more robust strategy is proposed which takes account simultaneously of the effect of errors in both the accelerances and operational responses.     

Shape identification of underwater objects using backscattered frequency signals

The inverse problems in the area of the acoustic scattering often concern the determination of the size, shape, and orientation of an object using the scattered field data. This paper presents a method to retrieve the shape information of an underwater object using illuminated lengths, which can be obtained from the ramp response signatures of the object. An ellipsoidal object submerged in water is considered. Both the low and high backscattered frequency data have been employed to calculate the illuminated lengths. The calculated results show that the illuminated lengths will be more accurate, if only the high-frequency-range data are employed. For ellipsoidal objects, any three illuminated lengths that are not of a same plane can in theory fully determine the shape of the ellipsoid. As the calculated illuminated lengths contain numerical errors, the calculated results of the three semiaxes of the ellipsoid will deteriorate and become unreliable, especially when the three incident directions of the illuminated lengths become close. The reason is that the condition number of the coefficient matrix becomes big in such situations, which leads to an increase of the relative error upper limit in the calculated results. To avoid such errors in close incident wave cases, it is found that the use of more than three incident waves works very well in the shape identification of an underwater object.     

A numerical procedure for estimation of the melt depth in laser material processing

In this study, an algorithm based on conjugate gradient method (CGM) is applied to estimate the unknown time dependent melt depth during laser material processing in liquid phase. The determination of the melt depth is treated as a one-dimensional, transient, inverse heat conduction problem (IHCP). It is assumed that no prior information is available for the functional form of the unknown melt depth, but it can be estimated by an inverse analysis with temperature measurements near the heated surface. The algorithm has been applied to aluminum, titanium and fused quartz and accurate melting depth and temperature distributions can also be returned. In addition, this methodology can also be applied to solve other problems such as calculating the cutting forces in nanomachining by atomic force microscopy (AFM), and estimating the heat sources in a X-ray lithographic process.     

Probability density functions for hyperbolic and isodiachronic locations

Animal locations are sometimes estimated with hyperbolic techniques by estimating the difference in distances of their sounds between pairs of receivers. Each pair specifies the animal's location to a hyperboloid because the speed of sound is assumed to be spatially homogeneous. Sufficient numbers of intersecting hyperboloids specify the location. A nonlinear method is developed for computing probability density functions for location. The method incorporates a priori probability density functions for the receiver locations, the speed of sound, winds, and the errors in the differences in travel time. The traditional linear approximation method overestimates bounds for probability density functions by one or two orders of magnitude compared with the more accurate nonlinear method. The nonlinear method incorporates a generalization of hyperbolic methods because the average speed of sound is allowed to vary between different receivers and the source. The resulting "isodiachronic" surface is the locus of points on which the difference in travel time is constant. Isodiachronic locations yield correct location errors in situations where hyperbolic methods yield incorrect results, particularly when the speed of propagation varies significantly between a source and different receivers.     

Minimization of sound radiation from baffled beams through optimization of partial constrained layer damping treatment

An optimization study is presented with aim to minimize the sound power radiated by a simply supported, baffled beam with constrained layer damping (CLD) treatment. The governing equation of motion for the calculation of time-harmonic response of a partially CLD covered beam is derived first on the basis of energy approach. Assumed-modes method is used to solve the equation with obtained frequency response functions at different beam locations, which are further used for the calculation of its radiated sound power into half free-space by using Rayleigh’s integral. The optimization problem is then formulated to minimize the sound power radiated by the beam over a frequency range of interest covering multiple resonant modes. A genetic algorithm-based penalty function method is employed to search for the optimum of location/length of the CLD patch and the shear modulus of viscoelastic layer. Optimal results show that for a simply supported beam with a transverse force applied at its central location, it is not necessary to fully cover the structure using CLD patch in order to achieve the largest reduction in the sound power radiated by the beam over a frequency range. With inclusion of the amount of damping material to be minimized, the optimal CLD coverage length is only one-fourth of the base beam’s. Moreover, the optima of three design variables, the CLD coverage length, location on the beam and the shear modulus of viscoelastic layer, are highly relevant to each other.     

An approach to estimating and extrapolating model error based on inverse problem methods:towards accurate numerical weather prediction

Model error is one of the key factors restricting the accuracy of numerical weather prediction(NWP). Considering the continuous evolution of the atmosphere, the observed data(ignoring the measurement error) can be viewed as a series of solutions of an accurate model governing the actual atmosphere. Model error is represented as an unknown term in the accurate model, thus NWP can be considered as an inverse problem to uncover the unknown error term. The inverse problem models can absorb long periods of observed data to generate model error correction procedures. They thus resolve the deficiency and faultiness of the NWP schemes employing only the initial-time data. In this study we construct two inverse problem models to estimate and extrapolate the time-varying and spatial-varying model errors in both the historical and forecast periods by using recent observations and analogue phenomena of the atmosphere. Numerical experiment on Burgers’ equation has illustrated the substantial forecast improvement using inverse problem algorithms. The proposed inverse problem methods of suppressing NWP errors will be useful in future high accuracy applications of NWP.     

Estimation of structural wave numbers from spatially sparse response measurements

A method is presented for estimating the complex wave numbers and amplitudes of waves that propagate in damped structures, such as beams, plates, and shells. The analytical basis of the method is a wave field that approximates response measurements in an aperture where no excitations are applied. At each frequency, the method iteratively adjusts wave numbers to best approximate response measurements, using wave numbers at neighboring frequencies as initial estimates in the search. In comparison to existing methods, the method generally requires far fewer measurement locations and does not require evenly spaced locations. The number of locations required by the method scales with the number of waves that propagate in the structure, whereas the number of locations required by existing methods scales with the minimum wavelength. In addition, the method allows convenient inclusion of the analytic relationships between wave numbers that exist for flexural vibrations of beams and plates. Advantages of the method are illustrated by an example in which a beam is excited by a transverse force at one end. Using analytic data and experimental measurements, the method produces a wave field that matches response measurements to within 1 percent. One interesting feature of the new method is that, when applied to analytic data, it supplies more robust wave number estimates using responses at unevenly spaced locations.     

Spatial resolution limits for the reconstruction of acoustic source strength by inverse methods

One method for deducing the strength of an acoustic source distribution from measurement of the radiated field involves the inversion of the matrix of frequency response functions relating the field measurement points to the strengths of a number of point sources used to represent the source distribution. In practice, the frequency response function matrix to be inverted may very often be ill-conditioned. This ill-conditioning will also often result in an ill-posed problem and thus regularization algorithms are used to produce reasonable solutions. For this purpose, Tikhonov regularization has been applied, and generalized cross-validation (GCV) has been introduced as an effective method for determining the proper amount of regularization without prior knowledge of either the source distribution or the contaminating errors. In the present work, the emphasis is placed on the relationship between the spatial resolution of the reconstructed source distribution and the small singular values of the frequency response function matrix to be inverted. However, the use of Tikhonov regularization often suppresses the effect of small singular values and these are in turn often associated with high spatial frequencies of the source distribution. Thus, the process of regularization produces a useful estimate of the acoustic source strength distribution but with a limited spatial resolution. Furthermore, in the field of Fourier acoustics, the spatial resolution of the reconstructed source distribution is usually limited by the wavelength of the radiation. This paper expresses the relationship between estimation accuracy, spatial resolution, noise-level and source/sensor geometry, when a range of inverse sound radiation problems are regularised using Tikhonov regularization with GCV. The results presented form the basis of guidelines that enable the reconstruction of acoustic source strength with a resolution that is finer than the intrinsic half-wavelength limit.     

双声源位置和强度识别的三维声强方法

目前在远场识别声源空间位置和强度缺乏行之有效的方法。针对此问题,提出采用四传声器进行三维声强测量,从而构建出声强、声源坐标和声功率的非车线性方程组,求解方程得出声源空间坐标和强度的方法。以3个三维声强探头对两个同频率单极子声源的识别为例,分别利用数值仿真和半消声室内的实验进行方法验证,并对声源的识别空间分辨率做了测试,得出角度识别最大误差为3.83°,为真实值的8.5%,距离识别最大误差0.1 m,为真实距离的10%。结果表明采用该方法空间坐标和声功率识别均具有很高的准确度,双声源的空间位置分辨力也优于远场声全息方法。      

Moment excitation and the measurement of moment mobilities

Frequency response functions (FRF), such as mobilities, are widely used in the analysis of vibration and structure-borne sound and it is important that this FRF data can be measured accurately for all important degrees of freedom. In some cases three translational and three rotational components of both excitation and response may be of importance; i.e. three forces and moments, and three velocities and angular velocities. Of these, the measurement of angular velocity due to moment excitation is one of the most challenging. This paper describes a known approach, sometimes referred to as the central difference method, which can be used for this purpose. The central difference method is thought to be one of the most practicable methods for measuring moment mobilities because it avoids the need for a moment exciter; instead finite differences are used to approximate the moment mobility which is a spatial derivative of the more easily measured velocity to force mobility ratio. There does however remain some doubt regarding the accuracy of the central difference method because of the finite difference approximation made and the method's possible susceptibility to random and bias errors. To better understand the finite difference error, an error analysis using a Taylor series expansion and simulated experiments for plate and beam structures are provided. It is then argued that random and bias errors associated with the measurement chain should now, with modern instrumentation, be less of a problem. An experimental validation of the method using two approaches is used to test this hypothesis. It is concluded that the central difference method provides a good balance between measurement effort and data quality making it widely applicable.     

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.     

Design of an optimal wave-vector filter for enhancing the resolution of reconstructed source field by near-field acoustical holography (NAH)

In near-field acoustical holography using the boundary element method, the reconstructed field often diverges due to the presence of small measurement errors. In order to handle this instability in the inverse problem, the reconstruction process should include some form of regularization for enhancing the resolution of source images. The usual method of regularization has been the truncation of wave vectors associated with small singular values, although the determination of an optimal truncation order is difficult. In this article, an iterative inverse solution technique is suggested in which the mean-square error prediction is used. A statistical estimation of the minimum mean-square error between measured pressures and the model solution is required for yielding the optimal number of iterations. The continuous curve of an optimal wave-vector filter is designed, for suppressing the high-order modes that can produce large reconstruction errors. Experimental results from a baffled radiator reveal that the reconstruction errors can be reduced by this form of regularization, by at least 48% compared to those without any regularization. In comparison to results using the optimal truncation method of regularization, the new scheme is shown to give further reductions of truncation error of between 7% and 39%, for the example in this article.     

Debonding detection of honeycomb sandwich structures using frequency response functions

A vibration-based non-destructive evaluation (NDE) method is proposed to determine the location and size of debonding in honeycomb sandwich beams. Although most of the existing vibration-based NDE methods need many measurement points, the method proposed here only utilizes the frequency response function (FRF) measured at one point. A parameterized damaged Timoshenko beam model is developed with the method of reverberation-ray matrix (MRRM) for the first time, and combined with the genetic algorithm (GA) to inverse the damage parameters from the measured FRF. The detection of a honeycomb sandwich beam can be divided into two steps: (1) identifying the equivalent elastic moduli and other parameters of the intact sandwich beam. (2) Identifying the debonding location and size of the damaged sandwich beam with the predetermined parameters. It is demonstrated experimentally that the method can inverse damage parameters with acceptable precision.     

Measuring short impulse responses with inverse filtered maximum-length sequences

Most loudspeakers have a non-flat frequency response which produces a long oscillating impulse response. An inverse filtering approach may be used to calculate the driving waveform necessary to equalize the response of the loudspeaker in order to radiate shorter acoustic pulses. When combined with the MLS technique, inverse filtering may be used to pre-emphasize the driving signal so that a shorter impulse response, with a prescribed waveform, is measured. This technique is described and illustrated by applying it to a distributed mode loudspeaker. Originally, this loudspeaker has a rather irregular response in a wide band. When the MLS signal is pre-emphasized with the proper inverse filter, a shorter impulse response is measured with a zero-phase cosine-magnitude spectrum.     

Forces between conducting surfaces due to spatial variations of surface potential

We describe analytical and numerical methods for calculating forces between conductors due to variations of electrostatic surface potential across their surfaces. In the simple case where the spatial variation of surface potential gives rise to uniform power spectra, we show that the electrostatic force can be large in comparison with, and scale in approximately the same way with distance of closest approach as, the Casimir force. Patch potentials that are consistent with existing experimental data could give rise to forces with a magnitude of 4% of the Casimir force at separations of 0.1 microm.