Frequency domain analysis and identification of block-oriented nonlinear systems

Block-oriented nonlinear models including Wiener models, Hammerstein models and Wiener-Hammerstein models, etc. have been extensively applied in practice for system identification, signal processing and control. In this study, analytical frequency response functions including generalized frequency response functions (GFRFs) and nonlinear output spectrum of block-oriented nonlinear systems are developed, which can demonstrate clearly the relationship between frequency response functions and model parameters, and also the dependence of frequency response functions on the linear part of the model. The nonlinear part of these models can be a more general multivariate polynomial function. These fundamental results provide a significant insight into the analysis and design of block-oriented nonlinear systems. Effective algorithms are therefore proposed for the estimation of nonlinear output spectrum and for parametric or nonparametric identification of nonlinear systems. Compared with some existing frequency domain identification methods, the new estimation algorithms do not necessarily require model structure information, not need the invertibility of the nonlinearity and not restrict to harmonic inputs. Simulation examples are given to illustrate these new results.     

兰姆波非线性效应的实验观察

对兰姆波非线性效应进行了实验观察。首先,对实验所采用的Ritec系统进行了介绍,尤其对实验系统的声学部分进行了详细考虑,特别是对二次谐波的时域响应进行了简洁直观的分析。根据兰姆波具有强烈非线性效应的条件,采用斜劈换能器发射和接收兰姆波。改变发射与接收斜劈换能器的间距,对一定频率范围内的兰姆波及二次谐波在固体板表面的振幅进行了测量。在基频与二倍频兰姆波相速度相等所对应的频率附近,分析了二次谐波振幅随传播距离的定量关系,从实验上证实了二次谐波在传播过程中具有积累增长效应的结论。     

平衡衔铁受话器的非线性特性与失真仿真模型

平衡衔铁受话器(BAR)具有尺寸小、电声转换效率高和灵敏度高等特点。在大振幅振动时, BAR存在显著的非线性,并导致较严重的失真。利用磁路的集总参数模型(LPM)深入研究BAR的非线性,确立可表征其非线性特性的4个非线性参数,采用迭代计算方法研究平衡衔铁磁阻对非线性特性的重要影响。基于所提出的网格移动和旋转等效的有限元模型(FEM),考虑平衡衔铁磁阻的非线性,准确仿真计算得到非线性参数,再将它们代入到非线性LPM模型中,最终建立FEM与LPM相结合的失真仿真模型。实验结果表明,该失真仿真模型可比较准确地预测BAR在不同加载电压时的总谐波失真以及二次和三次谐波失真。     

Dynamic errors and their compensation in seismic low-tuned transducers

Seismic low-tuned transducers can give rise to considerable dynamic errors. The usual damping ratio of 0·5?0·7 extends their usable range down to the vicinity of the resonant frequency only for the case of harmonic inputs. An undistorted response for other inputs necessitates a much higher ratio of the measured to resonant frequencies. Also, a lower damping ratio can lower the requirements from this ratio for cases involving periodic inputs.Dynamic compensation can be used to decrease the dynamic distortion. Relatively simple analog and digital compensating systems can be used, and their realization and associated problems are described. The distortion reduction by a factor of 10 seems feasible for many practical cases, although a careful choice of the compensating parameters may be necessary for cases involving noisy systems.     

1:50 scale acoustic models for objective testing of auditoria

The particular problems of small-scale models, namely air absorption and transducers, are discussed and it is found that with a dehumidified atmosphere (2–3 per cent relative humidity) and currently available transducers the measurement of reverberation time, as well as impulsive type measurements such as early to late energy ratio and early decay time, is possible at a scale of 1:50. The upper frequency limit for reverberation time is the 2 kHz octave equivalent and the 1 kHz octave for impulse measurements. Comparable measurements made in an auditorium and its 1:50 scale model showed good agreement. Some results of measurements in a model theatre and concert hall are also reported. It is concluded that 1:50 scale acoustic models enable one to measure quantities which cannot be calculated from drawings and offer a valuable aid for the development and confirmation of an auditorium design.     

An analysis of the distortion effects of Coulomb damping on the vector plots of lightly damped systems

An analysis of the vector plot responses of lightly damped single degree-of-freedom systems with Coulomb damping has been made. The vector plots, as derived by using both an exact and an approximate method (the method of harmonic balance) are compared and it is shown that the distortion of the normally circular vector locus is due to the Coulomb damping. Although the vector plots of such systems are distorted it is also shown that the frequency gradient criterion is still applicable for location of a natural frequency even when the frictional force levels approach the excitation force levels. To permit estimation of the modal damping of these systems a criterion by means of which the limits of the useful frequency range can be specified is suggested. The criterion, which is based upon the quadrature input power necessary to excite the mode of vibration, is found to be equivalent to that obtained from the half power point theory when applied to linear systems.     

High nonlinearities in Langevin transducer: a comprehensive model

The design and simulation of power transducers are difficult since piezoelectric, dielectric and elastic properties of ferroelectric materials differ from linear behavior when driven at large levels. This paper is devoted to modeling of a resonant power transducer at a high level of dynamic mechanical stress. The power transducer is subjected to a sine electrical field E of varying frequency which was considered as the excitation of the transducer.The mechanical equation of the piezoelectric element is written using electrostriction. The dielectric part is written as a nonlinear function of an equivalent electric field including stress influence (scaling relationship between electric field and mechanical stress). Using various simulations, we show then that typical resonance nonlinearities are obtained, such as jump phenomenon of transducer speed amplitude and phase, resonance peak that become asymmetric, and diminution of mechanical quality factor. As a consequence, we state that those typical nonlinearities are only due to dielectric nonlinearities, in good correlation with typical ferroelectric behavior. Moreover, this demonstrates the usefulness of scaling relationships in ferroelectrics, which explain static depoling under stress and butterfly strain hysteresis loop. The same scaling law gives here several nonlinearities for resonant transducers as well.     

A comparison of various nonlinear models of cochlear compression

The vibration response of the basilar membrane in the cochlea to sinusoidal excitation displays a compressive nonlinearity, conventionally described using an input-output level curve. This displays a slope of 1 dB/dB at low levels and a slope m < 1 dB/dB at higher levels. Two classes of nonlinear systems have been considered as models of this response, one class with static power-law nonlinearity and one class with level-dependent properties (using either an automatic gain control or a Van der Pol oscillator). By carefully choosing their parameters, it is shown that all models can produce level curves that are similar to those measured on the basilar membrane. The models differ, however, in their distortion properties, transient responses, and instantaneous input-output characteristics. The static nonlinearities have a single-valued instantaneous characteristic that is the same at all input levels. The level-dependent systems are multi-valued with an almost linear characteristic, for a given amplitude of excitation, whose slope varies with the excitation level. This observation suggests that historical attempts to use functional modeling (i.e., Wiener of Volterra series) may be ill founded, as these methods are unable to represent level-dependent nonlinear systems with multi-valued characteristics of this kind.     

Nonlinear pulsed ultrasound beams radiated by rectangular focused diagnostic transducers

A numerical model for simulating nonlinear pulsed beams radiated by rectangular focused transducers, which are typical of diagnostic ultrasound systems, is presented. The model is based on a KZK-type nonlinear evolution equation generalized to an arbitrary frequency-dependent absorption. The method of fractional steps with an operator-splitting procedure is employed in the combined frequency-time domain algorithm. The diffraction is described using the implicit backward finite-difference scheme and the alternate direction implicit method. An analytic solution in the time domain is employed for the nonlinearity operator. The absorption and dispersion of the sound speed are also described using an analytic solution but in the frequency domain. Numerical solutions are obtained for the nonlinear acoustic field in a homogeneous tissue-like medium obeying a linear frequency law of absorption and in a thermoviscous fluid with a quadratic frequency law of absorption. The model is applied to study the spatial distributions of the fundamental and second harmonics for a typical diagnostic ultrasound source. The nonlinear distortion of pulses and their spectra due to the propagation in tissues are presented. A better understanding of nonlinear propagation in tissue may lead to improvements in nonlinear imaging and in specific tissue harmonic imaging. Published in Russian in Akusticheskiĭ Zhurnal, 2006, Vol. 52, No. 4, pp. 560–570. This article was translated by the authors.     

Prediction condensed models adapted to the nonlinear structures in time domain

This paper deals with a method to study closely the stationary solution of nonlinear dynamic systems in time domain. This method is based on the exploitation of Karhunen-Loève decomposition with or without parametric modifications as well as on the characteristics of localized nonlinearities. With the application of this method at first on linear models initially condensed by Karhunen-Loève, the predictions of nonlinear responses can be obtained rapidly. This method is adapted to a condensed linear model used in the first optimization procedure of the nonlinear dynamic behaviour. This robust basis will be used as condensation basis of the modified model local per zone, which leads to a prediction of vibratory responses of complex structures modified and affected by localized nonlinearities.     

Using acoustic distortion products to measure the cochlear amplifier gain on the basilar membrane.

Most models of the cochlea developed during the last decade have explained frequency selectivity and sensitivity of the cochlea at threshold by the use of power amplification of the acoustic wave on the basilar membrane. This power amplification has been referred to as the cochlear amplifier (CA). In this paper, a method to measure the cochlear amplifier gain as a function of position along the basilar membrane is derived from a simple model. Next, experimental evidence is presented that strongly restricts the properties of these proposed cochlear amplifier models. Specifically, it is shown that small signals generated by mechanical nonlinearities in the basilar membrane motion are not amplified during basilar membrane propagation, contrary to what would be expected from the cochlear amplifier hypotheses. This paper describes a method of measuring the cochlear power gain as a function of frequency and position, from the stapes to within 2 mm of the place corresponding to the frequency being measured. Experimental results in the cat indicate that the total gain of the cochlear amplifier, over the range of positions measured, must be less than 10 dB. The simplest interpretation of the experimental results is that there is no cochlear amplifier. The results suggest that the cochlea must achieve its frequency selectivity by some other means.     

用梅森等效电路对厚度模压电换能器暂态特性的一些分析

我们采用梅森等效电路的方法,比较大量地计算了厚度模压电换能器在阶跃式电压的激励下,当具有一些不同的结构参数时,所发射的声波波形,或在又发又收时所产生的电信号波形。近距离的测量结果进行了对比。从对比结果看来,尽管梅森等效电路是相当简化了的模型,在我们所讨论的范围内,根据这个模型所得理论结果还是近似地可用的。这包括一些理论预计的细节,如辐射声波波形对于时间轴的不对称性。但也有些问题需要澄清,如辐射声波也是平面的这个假设的适用程度。
文内简单地涉及了保护膜厚度的改变将引起波形的畸变和声波频率的改变。也简单地涉及了关于接收电信号波形的先小后大或先大后小的控制问题。     

Spatial correlation in the ambient core noise field of a turbofan engine

An acoustic transfer function relating combustion noise and turbine exit noise in the presence of enclosed ambient core noise is investigated using a dynamic system model and an acoustic system model for the particular turbofan engine studied and for a range of operating conditions. Measurements of cross-spectra magnitude and phase between the combustor and turbine exit and auto-spectra at the turbine exit and combustor are used to show the presence of indirect and direct combustion noise over the frequency range of 0-400 Hz. The procedure used evaluates the ratio of direct to indirect combustion noise. The procedure used also evaluates the post-combustion residence time in the combustor which is a factor in the formation of thermal NO(x) and soot in this region. These measurements are masked by the ambient core noise sound field in this frequency range which is observable since the transducers are situated within an acoustic wavelength of one another. An ambient core noise field model based on one and two dimensional spatial correlation functions is used to replicate the spatially correlated response of the pair of transducers. The spatial correlation function increases measured attenuation due to destructive interference and masks the true attenuation of the turbine.     

Application of a modified rational harmonic balance method for a class of strongly nonlinear oscillators

An analytical approximate technique for conservative nonlinear oscillators is proposed. This method is a modification of the rational harmonic balance method in which analytical approximate solutions have rational form. This approach gives us the frequency of the motion as a function of the amplitude of oscillation. We find that this method works very well for the whole range of parameters, and excellent agreement of the approximate frequencies with the exact one has been demonstrated and discussed. The most significant features of this method are its simplicity and its excellent accuracy for the whole range of oscillation amplitude values and the results reveal that this technique is very effective and convenient for solving conservative truly nonlinear oscillatory systems with complex nonlinearities.     

Semi-analytical computation of the acoustic field of a segment of a cylindrically concave transducer in lossless and attenuating media

Conventional ultrasound transducers used for medical diagnosis generally consist of linearly aligned rectangular apertures with elements that are focused in one plane. While traditional beamforming is easily accomplished with such transducers, the development of quantitative, physics-based imaging methods, such as tomography, requires an accurate, and computationally efficient, model of the field radiated by the transducer. The field can be expressed in terms of the Helmholtz-Kirchhoff integral; however, its direct numerical evaluation is a computationally intensive task. Here, a fast semianalytical method based on Stepanishen's spatial impulse response formulation [J. Acoust. Soc. Am. 49, 1627-1638 (1971)] is developed to compute the acoustic field of a rectangular element of cylindrically concave transducers in a homogeneous medium. The pressure field, for, lossless and attenuating media, is expressed as a superposition of Bessel functions, which can be evaluated rapidly. In particular, the coefficients of the Bessel series are frequency independent and need only be evaluated once for a given transducer. A speed up of two orders of magnitude is obtained compared to an optimized direct numerical integration. The numerical results are compared with Field II and the Fresnel approximation.     

Helmholtz水声换能器弹性壁液腔谐振频率研究

针对传统Helmholtz水声换能器设计中刚性壁假设的局限性,将Helmholtz腔体的弹性计入到液腔谐振频率计算中,实现低频弹性Helmholtz水声换能器液腔谐振频率精确设计.基于细长圆柱壳腔体的低频集中参数模型,导出了腔体弹性引入的附加声阻抗表达式,得到了弹性壁条件下Helmholtz水声换能器等效电路图,给出了考虑了末端修正的弹性壁Helmholtz共振腔液腔谐振频率计算公式.利用ANSYS软件建立了算例模型,仿真分析了不同材质、半径、长度时的Helmholtz共振腔液腔谐振频率.结果对比表明弹性理论值与仿真值符合得很好,相比起传统的刚性壁理论计算结果,本文的弹性壁理论得出的液腔谐振频率值有所降低,与真实情况更加接近.本文的结论可以为精确设计低频弹性Helmholtz水声换能器提供理论支持.     

Detection limits for single small flaws and groups of flaws when using focussed ultrasonic transducers

The performance of ultrasonic imaging systems, using line-focussed transducers, was evaluated theoretically by employing a simple technique based on convolution. The analysis considered the response of single reflectors, which are mainly voids, and also multiple reflectors located both on a single plane and on multiple planes, which represent a porous layer or bond-line. The limits of detection for each of the above cases, in the presence of different levels of acoustic noise in the object under test were also calculated. The results presented are expressed as a function of a dimensionless flaw size, incorporating the ultrasonic frequency and beam F-number. The results can be used to identify the best match between ultrasonic equipment and a specific imaging problem. They can also serve to guide decisions regarding the most suitable detection method for use with the chosen equipment. If the inspection is properly designed, surprisingly small flaws could be reliably detected — even in objects where significant levels of incoherent acoustic noise are found. The method and results presented in this paper for line-focussed transducers fit qualitatively the case of spherically focussed transducers. The method is highly versatile and it can be extended to cover a range of problems, outside the scope of this paper, including, quantitatively, that of the spherically focussed transducer by employing a formulation which uses more computer time and memory.     

外接LC电路的可调压电换能器的频率特性

压电换能器往往因受到温度与负载的影响,导致其谐振频率、反谐振频率和机械品质因数等特征参数发生变化。通过在压电换能器中设计用于调节的压电陶瓷片,并在调节压电陶瓷片两端外接电负载,可以实现压电换能器的频率调节,修正由于温度和负载导致的频率漂移。基于压电换能器的Mason等效电路,建立外接LC调节电路的压电换能器的谐振频率、反谐振频率模型,分析LC调节电路中调节电感对频率调节特性的影响;通过实验研究外接LC调节电路对可调压电换能器机械品质因数的影响,并验证调节电感对频率调节特性的影响。理论分析和实验研究的结果表明:随着调节电感的增大,调节电容对频率的调节灵敏度提高,可调压电换能器的频率调节宽度拓宽,但也会使压电换能器的机械品质因数降低。合理选择调节电感和调节电容能兼顾频率调节宽度和机械品质因数的要求。此研究可为后续自适应压电换能器的频率调节系统设计提供指导。     

Maximum a posteriori deconvolution of ultrasonic signals using multiple transducers

A new method for deconvolution of ultrasonic pulse-echo measurements employing multiple-transducer setup is proposed in the paper. An optimal way of estimating the material reflection sequence for a linear signal generation model using maximum a posteriori estimation is proposed. The method combines the measurements from a number of transducers covering different frequency bands yielding an optimal estimate of the reflection sequence. The main idea of this approach is to complement the information unavailable from one transducer in some frequency bands with the information from the other transducers. The method is based on the assumption that the measurements are performed using transducers with identical apertures and apodization, which are located exactly at the same position relative to the test object during the measurement. An error analysis presented in the paper proves that when the above assumptions are fulfilled, the proposed method, by utilizing more data for estimation, consistently yields more accurate reflection sequence estimates than the classical Wiener filter. Experimental evidence is presented using both simulated and real ultrasonic data as a verification of the correctness of the multiple-transducer model and the estimation scheme. An illustration of the advantages of the method is also given using real ultrasonic data.     

FREQUENCY DOMAIN ARX MODEL AND MULTI-HARMONIC FRF ESTIMATORS FOR NON-LINEAR DYNAMIC SYSTEMS

Non-linear dynamic systems respond at frequencies other than the excitation frequency; however, standard frequency response function estimators for linear systems do not accommodate this harmonic distortion. A new multi-harmonic frequency response function estimator that utilizes discrete frequency models for non-linear systems is introduced here. The multi-harmonic estimator relates the frequency response at each frequency to the input and output spectra within a given frequency band in the same way that autoregressive exogenous input models relate inputs and outputs at particular samples in the time domain. Overdetermined, least-mean-squares calculations are used to minimize model error throughout a frequency band rather than at a single frequency as in the corresponding linear estimators. The resulting multi-harmonic frequency response function models are non-parametric (e.g., vary with amplitude) when linear functions are used and parametric when non-linear functions are used. A new sensitive indicator for experimentally characterizing non-linearity is introduced.