Model reduction of nonlinear systems with external periodic excitations via construction of invariant manifolds

A methodology for determining reduced order models of periodically excited nonlinear systems with constant as well as periodic coefficients is presented. The approach is based on the construction of an invariant manifold such that the projected dynamics is governed by a fewer number of ordinary differential equations. Due to the existence of external and parametric periodic excitations, however, the geometry of the manifold varies with time. As a result, the manifold is constructed in terms of temporal and dominant state variables. The governing partial differential equation (PDE) for the manifold is nonlinear and contains time-varying coefficients. An approximate technique to find solution of this PDE using a multivariable Taylor-Fourier series is suggested. It is shown that, in certain cases, it is possible to obtain various reducibility conditions in a closed form. The case of time-periodic systems is handled through the use of Lyapunov-Floquet (L-F) transformation. Application of the L-F transformation produces a dynamically equivalent system in which the linear part of the system is time-invariant; however, the nonlinear terms get multiplied by a truncated Fourier series containing multiple parametric excitation frequencies. This warrants some structural changes in the proposed manifold, but the solution procedure remains the same. Two examples; namely, a 2-dof mass-spring-damper system and an inverted pendulum with periodic loads, are used to illustrate applications of the technique for systems with constant and periodic coefficients, respectively. Results show that the dynamics of these 2-dof systems can be accurately approximated by equivalent 1-dof systems using the proposed methodology.     

A least-squares approximation of partial differential equations with high-dimensional random inputs

Uncertainty quantification schemes based on stochastic Galerkin projections, with global or local basis functions, and also stochastic collocation methods in their conventional form, suffer from the so called curse of dimensionality: the associated computational cost grows exponentially as a function of the number of random variables defining the underlying probability space of the problem. In this paper, to overcome the curse of dimensionality, a low-rank separated approximation of the solution of a stochastic partial differential (SPDE) with high-dimensional random input data is obtained using an alternating least-squares (ALS) scheme. It will be shown that, in theory, the computational cost of the proposed algorithm grows linearly with respect to the dimension of the underlying probability space of the system. For the case of an elliptic SPDE, an a priori error analysis of the algorithm is derived. Finally, different aspects of the proposed methodology are explored through its application to some numerical experiments.     

Statistical damage identification of structures with frequency changes

Model updating methods based on structural vibration data have being rapidly developed and applied to detect structural damage in civil engineering. But uncertainties existing in the structural model and measured vibration data might lead to unreliable damage detection. In this paper a statistical damage identification algorithm based on frequency changes is developed to account for the effects of random noise in both the vibration data and finite element model. The structural stiffness parameters in the intact state and damaged state are, respectively, derived with a two-stage model updating process. The statistics of the parameters are estimated by the perturbation method and verified by Monte Carlo technique. The probability of damage existence is then estimated based on the probability density functions of the parameters in the two states. A higher probability statistically implies a more likelihood of damage occurrence. The presented technique is applied to detect damages in a numerical cantilever beam and a laboratory tested steel cantilever plate. The effects of using different number of modal frequencies, noise level and damage level on damage identification results are also discussed.     

The resonant mechanism of subdivision of a gas bubble in a fluid

Small nonlinear oscillations of a bubble in a fluid at the resonance of the frequencies of the radial mode and an arbitrary deformational mode 2 : 1 are considered. The deformational mode is determined by the associated Legendre polynomial with indices n = 2, 3, ??, m = 0, 1, ??, n. The energy transfer from the radial mode into the Legendre deformational mode is described by the method of invariant normalization. An analogy is established with oscillations of the material point on the string with the frequency ratio of the vertical mode to the horizontal one of 2. During the transfer, the amplitude of the Legendre mode with indices n, m exceeds the amplitude of radial oscillations by a factor of 3n at m = 0. As index m increases, the transfer time increases considerably and the maximal amplitude of the Legendre mode increases insignificantly in this case. From here, it is concluded that the deformational Legendre mode with indices n, m = n has the greatest probability to rise. The considered effect can serve as a mechanism of subdivision of gas bubbles under varying the external pressure in the fluid.     

Nonstandard (non-σ-additive) probabilities in algebraic quantum field theory

Traditionally, physicists deduce the observational (physical) meaning of probabilistic predictions from the implicit assumption that thewell-defined events whose probabilities are 0 never occur. For example, the conclusion that in a potentially infinite sequence of identical experiments with probability 0.5 (like coin tossing) the frequency of heads tends to 0.5 follows from the theorem that sequences for which the frequencies do not tend to 0.5 occur with probability 0. Similarly, the conclusion that in quantum mechanics, measuring a quantity always results in a number from its spectrum is justified by the fact that the probability of getting a number outside the spectrum is 0. In the mid-60s, a consistent formalization of this assumption was proposed by Kolmogorov and Martin-Löf, who defined arandom element of a probability space as an element that does not belong to any definable set of probability 0 (definable in some reasonable sense). This formalization is based on the fact that traditional probability measures are σ-additive, i.e., that the union of countably many sets of probability 0 has measure 0. In quantum mechanics with infinitely many degrees of freedom (e.g., in quantum field theory) and in statistical physics one must often consider non-σ-additive measures, for which the Martin-Löf's definition does not apply. Many such measures can be defined as “limits” of standard probability distributions. In this paper, we formalize the notion of a random element for such finitely-additive probability measures, and thus explain the observational (physical) meaning of such probabilities.     

Time domain identification of standing wave parameters in gas piping systems

A method for experimentally determining the natural frequencies and modal pressures of an air or gas piping system is presented. Such information is of interest in installations where pressure pulsations caused by pumps or compressors are of importance. In the method a time domain based technique is used which was originally developed as an alternative to frequency response methods for determining the vibration parameters (natural frequencies, modes, damping factors) of structures, to avoid difficulties often encountered in interpreting complex and non-conclusive frequency response data such as arises from systems having numerous modes, some of which may be highly damped or closely spaced in frequency. In this application, a straight steel pipe with a sound source at one end and closed at the other end was used. Two microphones were used to measure the pressure at two locations in the pipe. The free pressure response following a rapidly swept sinewave input was recorded, digitized and then used in a computational procedure based on a lumped parameter representation of the system. The natural frequencies and the corresponding modal pressure ratios at the two stations, thus obtained, are compared with mention here that although in the experiment reported here an external frequency sweep excitation was used, the technique works as well with free decay response after a system shut-off, impulse response or random responses from normal system operation.     

Numerical approach for quantification of epistemic uncertainty

In the field of uncertainty quantification, uncertainty in the governing equations may assume two forms: aleatory uncertainty and epistemic uncertainty. Aleatory uncertainty can be characterised by known probability distributions whilst epistemic uncertainty arises from a lack of knowledge of probabilistic information. While extensive research efforts have been devoted to the numerical treatment of aleatory uncertainty, little attention has been given to the quantification of epistemic uncertainty. In this paper, we propose a numerical framework for quantification of epistemic uncertainty. The proposed methodology does not require any probabilistic information on uncertain input parameters. The method only necessitates an estimate of the range of the uncertain variables that encapsulates the true range of the input variables with overwhelming probability. To quantify the epistemic uncertainty, we solve an encapsulation problem, which is a solution to the original governing equations defined on the estimated range of the input variables. We discuss solution strategies for solving the encapsulation problem and the sufficient conditions under which the numerical solution can serve as a good estimator for capturing the effects of the epistemic uncertainty. In the case where probability distributions of the epistemic variables become known a posteriori, we can use the information to post-process the solution and evaluate solution statistics. Convergence results are also established for such cases, along with strategies for dealing with mixed aleatory and epistemic uncertainty. Several numerical examples are presented to demonstrate the procedure and properties of the proposed methodology.     

Geometry of the central limit theorem in the nonextensive case

We uncover geometric aspects that underlie the sum of two independent stochastic variables when both are governed by q-Gaussian probability distributions. The pertinent discussion is given in terms of random vectors uniformly distributed on a p-sphere.     

COINCIDENCE OF THERMOELASTIC AND THERMOVISCOUS ACOUSTIC WAVES IN FLUID-FILLED ELASTIC TUBES

Thermoelastic and thermoviscous acoustic wave propagation in fluid-filled steel tubes is studied using the exact three-dimensional (3-D) fluid-elastic coupled system equations for the vibration in the n=0 and 1 circumferential modes. Water- and air-filled tubes are examined. The water-filled steel tube shows a strong fluid-elastic coupling effect in the lower frequency range and the air-filled tube shows a strong thermal effect for all frequencies. An 88·9 mm outer diameter tube with 3·05 mm wall thickness is used for the study. Due to the fluid-elastic coupling introduced for air having a specific heat ratio of 1·4 (the solution uncouples when the ratio is 1·0), thermal effects are seen to be very important with the modal attenuation rate being at least 32% underestimated if the thermal effect is not included in the air-steel system. A coincidence phenomenon is accurately found directly from the coupled modes in the fluid-elastic coupled system. When coincidence occurs, the axial modal attenuation rate drops sharply, allowing the exact determination of the coincidence frequency by locating the local minimum of the modal spatial attenuation rate with increasing frequency. In the water-steel system, the coincidence frequency is seen to be 8% in error if methods are employed using the uncoupled theory for the separate fluid and elastic wall.     

INDUCED DAMPING BY A NEARLY CONTINUOUS DISTRIBUTION OF NEARLY UNDAMPED OSCILLATORS: LINEAR ANALYSIS

The induced damping in a master oscillator contributed by a set of satellite oscillators is obtained in terms of a summation over a discrete distribution of the set. (The distribution is with respect to the resonance frequencies of the satellite oscillators in the set. The distribution is cast in an ascending order and is assumed to be centered about the resonance frequency of the master oscillator in isolation). If the modal overlap parameters are less than unity, significant undulations are present in the induced damping; the less the modal overlap parameters are compared with unity, the more prominent are the undulations. The undulations are largely suppressed when the local modal overlap parameters exceed unity. Moreover, appropriately averaging the undulations yields values for the induced damping that coincide with those obtained when the modal overlap parameters exceed unity. Further, it transpires that these common values are independent of the individual modal overlap parameters. When the summation is replaced by an integration, the first order results are undulations-free and the values, so obtained, again, coincide with those pertaining to modal overlap parameters that exceed unity. Without defining the excursions in the undulations, the transition from a discrete-to-a continuous distribution, that is implied by a summation-to-an integration, must assume that the modal overlap parameters exceed unity. In particular, without careful and meaningful qualifications, it may be misleading to assume, a priori, that the modal overlap parameters are equal to zero.     

On predicting point mobility plots from measurements of other mobility parameters

Experimental modal analysis techniques are designed to exploit a property of linear vibration theory in order to construct a mathematical model of a structure from the minimum amount of measured mobility data. This property, derived from the orthogonality of normal modes, is that, in principle, all the elements of the full N × N mobility matrix can be derived from measurement of the elements in just one row or column of that matrix. The property is also used in procedural quality “checks”, in which one measures some of the derived mobilities as well as predicting them. The derivation process itself, however, gives rise to an inherent error when the frequency range covered does not include all the natural frequencies of the structure. The source of this error is discussed and some illustrations of its significance are quoted, demonstrating the need for caution when applying modal analysis methods to practical structures.     

Structural modal excitation using travelling impulses—force frequency shifting

A review of existing hardware and methods for vibration testing of large structures is given by Koss and has shown that the size of inertial vibration shakers, to achieve a specific displacement, has to increase, as a structure becomes larger. In previous papers the concept of “force frequency shifting (ffs) for structural excitation”, was introduced to develop a more compact structural vibration exciter than is presently available for low frequencies. An ffs shaker operates at a frequency much greater than the natural frequency of the structure under test but generates a modal force at the lower frequency of the structure. This effect is accomplished by moving a vibrating force back and forth across the structure while the force is applied normally to its surface. For example, the generalized force generated by an ffs shaker at the fundamental structural frequency for a simply supported beam is given by 1.65Pr/l where P is the high frequency out of balance force, r is the throw amplitude and l is the beam length. The term that reduces the efficiency of force transfer from high to low frequencies is “r/l” as, usually, the length of a structure is much greater than the throw of the force. This paper introduces another force frequency shifting approach that allows r/l to be large. This is accomplished by placing force exciters along a structure-spatial array, spaced a distance ΔX apart, and each force exciter is activated for a short period of time to simulate a travelling force traversing the structure forwards and backwards. The “force throw r “can thus be made large. Results of simulations and experiments verify that force frequency shifting can be accomplished using travelling impulses and modal identification can be achieved.     

稳态损耗因子的衰减法识别研究

根据稳态损耗因子的定义, 推导了含多阶模态的频带稳态损耗因子公式, 得到结论: 稳态损耗因子不一定介于各阶模态损耗因子之间, 而是与各阶模态对振动响应的贡献程度有关. 提出了过程损耗因子的概念, 并给出了利用频带内各模态固有频率、损耗因子和振幅计算过程损耗因子的方法. 当时间趋于无穷时, 过程损耗因子趋于只由最小模态损耗因子贡献的稳态损耗因子. 传统衰减法测试稳态损耗因子在频带内仅有单个模态或模态密集的情况下精度较高, 但对于含有多阶模态且模态不密集的中频带, 采用传统衰减法准确获取稳态损耗因子存在困难. 根据过程损耗因子的特点, 提出了利用时域衰减曲线逐步分离频带内不同衰减特性分量及其响应幅度从而获取稳态损耗因子的方法. 仿真和实验均表明: 提出的利用时域衰减数据获取稳态损耗因子的方法具有很高精度, 可以弥补传统衰减法在中频段损耗因子实验确定中的不足.     

Identification of Bayesian posteriors for coefficients of chaos expansions

This article is concerned with the identification of probabilistic characterizations of random variables and fields from experimental data. The data used for the identification consist of measurements of several realizations of the uncertain quantities that must be characterized. The random variables and fields are approximated by a polynomial chaos expansion, and the coefficients of this expansion are viewed as unknown parameters to be identified. It is shown how the Bayesian paradigm can be applied to formulate and solve the inverse problem. The estimated polynomial chaos coefficients are hereby themselves characterized as random variables whose probability density function is the Bayesian posterior. This allows to quantify the impact of missing experimental information on the accuracy of the identified coefficients, as well as on subsequent predictions. An illustration in stochastic aeroelastic stability analysis is provided to demonstrate the proposed methodology.     

Formulation and validation of a Ritz-based analytical model of high-frequency periodically layered isolators in compression

Periodically layered isolators exhibit transmissibility “stop bands” or frequency ranges in which there is very low transmissibility. A two-dimensional axisymmetric model was developed to accurately predict the location of these stop bands for isolators in compression. A Ritz approximation method was used to model the axisymmetric elastic behavior of layered cylindrical isolators. A modal analysis was performed for a single elastomer and metal layer combination or cell. A modal synthesis approach was then used to obtain a model of an n-celled isolator, from which overall isolator modal properties are determined. This model of the dynamic behavior of layered isolators was validated with experiments. Analytical and experimental transmissibilities are compared for test specimens having identical elastomer components, but different geometries and different numbers of cells. In all cases, experimental and analytical transmissibilities are in close agreement at frequencies ranging from zero to those associated with the initial roll-off of the stop bands. For three and four cell cases, minimum stop band analytical transmissibilities lie below the minimum experimental measurements, although an experimental noise floor imposed a minimum transmissibility measurement of approximately 1.4×10⁻⁴. Experiment suggests a practical isolator design could limit the minimum number of cells to three or four to ensure a pronounced stop band attenuation effect. In addition, analytical and experimental transmissibilities are compared for geometrically similar test specimens with differing elastomeric damping properties. The analytical and experimental results show that stop band effectiveness is not appreciably affected by the addition of modest damping.     

Point mobility techniques for the in situ estimation of modal densities of coupled cylindrical shells

This paper discusses the utilization of the experimentally measured value of the point mobility (transfer function of velocity on force at the excitation point) to estimate the modal density (number of structural modes per unit frequency) of cylindrical piping systems. The technique allows for a rapid in situ estimate of the modal density, has applications in statistical energy analysis and is especially suited at frequencies below the ring frequency where theoretical estimates do not account for the grouping of structural modes that occurs. Provided that feedback due to exciter-structure interaction is minimized, experimental results suggest that the conventional two-channel technique, using a dual-channel signal analyzer, gives reliable results. The three-channel technique, as discussed by Brown⁷, serves as an important test to establish the presence, or otherwise, of any feedback in a proposed experimental set-up.     

Use of modal energy transfer as a new damping device with controllable bandwidth

This paper presents a new design of nonlinear dynamic absorber (NDA) using the phenomenon of modal energy transfer between the symmetric mode and the anti-symmetric mode of a curved beam. It can reduce the resonance vibration of a primary structure with a controllable operational frequency range. The energy transfer is initiated by an autoparametric vibration and the excitation force required is lowest when the ratio of the resonance frequencies of the first symmetric mode (ω₁) and first anti-symmetric mode (ω₂) is close to 2.The resonance frequency of the first anti-symmetric mode (ω₂) can be altered to control the operational frequency range. The autoparametric vibration response can be used to create an energy-dissipative region with a controllable bandwidth. It is also possible to create a non-dissipative region in between two dissipative regions. This is useful for providing damping for a conventional dynamic absorber without adding high damping material. The damping is due to the dissipation of energy to anti-symmetric mode. Numerical calculations indicate that the resonance vibration of a primary structure can be successfully reduced using this approach. The results are verified with experimental data.     

太赫兹波发射晶体的亚波长微棱锥增透结构的设计与实验研究

光学整流方法产生太赫兹(THz)辐射常用的非线性发射晶体在THz波段都具有较高的折射率, 使得很大一部分THz波由于晶体表面的菲涅尔反射而无法有效耦合输出. 本文报道了GaP晶体THz波发射器输出表面上亚波长微棱锥增透结构的设计和实验研究. 利用有效介质模型在理论上验证了亚波长光栅结构的增透效果, 并进一步设计了适用于不同频段的增透结构的参数. 实验中, 通过微机械加工手段在GaP晶体输出端面刻划了多种亚波长微棱锥结构, 验证了其增透效果及参数对增透频带的关系. 理论与实验的符合证明该设计思想也可用于其他THz波发射晶体. 关键词： THz波 光学整流 亚波长微棱锥增透结构 微加工     

Sound radiation from point-excited structures: Comparison of plate and sphere

Acoustic radiation from a structure can be expressed in terms of “modal radiation” and “modal coefficients”. This paper investigates the contributions of these two modal properties to radiation excited by a point force. Sound radiation from two basic structures is considered: a baffled rectangular plate and a closed spherical shell. The plate behaviour is familiar, and governed by the relation between the natural frequency of a mode and its coincidence frequency. For a closed spherical shell, there are either zero or two “critical frequencies”, depending on the radius and thickness. When there are two the shell radiates well both above and below the two frequencies, and poorly in the frequency range between them.     

Generating Simple Random Graphs with Prescribed Degree Distribution

Let F be a probability distribution with support on the non-negative integers. Four methods for generating a simple undirected graph with (approximate) degree distribution F are described and compared. Two methods are based on the so called configuration model with modifications ensuring a simple graph, one method is an extension of the classical Erdös-Rényi graph where the edge probabilities are random variables, and the last method starts with a directed random graph which is then modified to a simple undirected graph. All methods are shown to give the correct distribution in the limit of large graph size, but under different assumptions on the degree distribution F and also using different order of operations.