Wishart Random Matrices in Probabilistic Structural Mechanics

Uncertainties need to be taken into account for credible predictions of the dynamic response of complex structural systems in the high and medium frequency ranges of vibration. Such uncertainties should include uncertainties in the system parameters and those arising due to the modeling of a complex system. For most practical systems, the detailed and complete information regarding these two types of uncertainties is not available. In this paper, the Wishart random matrix model is proposed to quantify the total uncertainty in the mass, stiffness, and damping matrices when such detailed information regarding uncertainty is unavailable. Using two approaches, namely, (a) the maximum entropy approach; and (b) a matrix factorization approach, it is shown that the Wishart random matrix model is the simplest possible random matrix model for uncertainty quantification in discrete linear dynamical systems. Four possible approaches for identifying the parameters of the Wishart distribution are proposed and compared. It is shown that out of the four parameter choices, the best approach is when the mean of the inverse of the random matrices is same as the inverse of the mean of the corresponding matrix. A simple simulation algorithm is developed to implement the Wishart random matrix model in conjunction with the conventional finite-element method. The method is applied vibration of a cantilever plate with two different types of uncertainties across the frequency range. Statistics of dynamic responses obtained using the suggested Wishart random matrix model agree well with the results obtained from the direct Monte Carlo simulation.     

Free Transverse and Lateral Vibration of Beams with Torsional Coupling

The free vibration of beams whose flexural motions in both principal planes are coupled with torsion is investigated by using the dynamic stiffness method. First, the governing differential equations of motion in free vibration are derived using Hamilton’s principle. The dynamic stiffness matrix is then developed from the solution of these differential equations when the oscillatory motion of the beam is harmonic. Finally, the resulting dynamic stiffness matrix is applied with particular reference to the Wittrick–Williams algorithm to carry out the free vibration analysis of a few illustrative examples. The numerical results are discussed and this is followed by some concluding remarks.     

Dynamic Field Test, System Identification, and Modal Validation of an RC Minaret: Preprocessing and Postprocessing the Wind-Induced Ambient Vibration Data

The investigation of dynamic response for civil engineering structures largely depends on a detailed understanding of their dynamic characteristics, such as the natural frequencies, mode shapes, and modal damping ratios. Dynamic characteristics of structures may be obtained numerically and experimentally. The finite-element method is widely used to model structural systems numerically. However, there are some uncertainties in numerical models. Material properties and boundary conditions may not be modeled correctly. There may be some microcracks in the structures, and these cracks may directly affect the modeling parameters. Modal testing gives correct uncertain modeling parameters that lead to better predictions of the dynamic behavior of a target structure. Therefore, dynamic behavior of special structures, such as minarets, should be determined with ambient vibration tests. The vibration test results may be used to update numerical models and to detect microcracks distributed along the structure. The operational modal analysis procedure consists of several phases. First, vibration tests are carried out, spectral functions are produced from raw measured acceleration records, dynamic characteristics are determined by analyzing processed spectral functions, and finally analytical models are calibrated or updated depending on experimental analysis results. In this study, an ambient vibration test is conducted on the minaret under natural excitations, such as wind effects and human movement. The dynamic response of the minaret is measured through an array of four trixial force-balanced accelerometers deployed along the whole length of the minaret. The raw measured data obtained from ambient vibration testing are analyzed with the SignalCAD program, which was developed in MATLAB. The employed system identification procedures are based on output-only measurements because the forcing functions are not available during ambient vibration tests. The ModalCAD program developed in MATLAB is used for dynamic characteristic identification. A three-dimensional model of the minaret is constructed, and its modal analysis is performed to obtain analytical frequencies and mode shapes by using the ANSYS finite-element program. The obtained system identification results have very good agreement, thus providing a reliable set of identified modal properties (natural frequencies, damping ratios, and mode shapes) of the structure, which can be used to calibrate finite-element models and as a baseline in health monitoring studies.     

Fuzzy Inexact Mixed-Integer Semiinfinite Programming for Municipal Solid Waste Management Planning

Based on the concept of functional intervals, fuzzy inexact mixed-integer semiinfinite programming (FIMISIP) method is developed for municipal solid waste management planning. The method allows the uncertainties in parameters expressed as fuzzy, interval, and functional interval numbers to be directly communicated into the programming problem. The FIMISIP problem is solved by dividing it into two interactive semiinfinite programming (SIP) subproblems. Solutions reflecting the inherent uncertainties can then be generated by combining the SIP solutions into a set of decision intervals. The method is applied to a municipal solid waste management planning system for demonstrating its effectiveness in dealing with uncertain and dynamic complexities. Compared to the previous inexact programming methods, FIMISIP has the advantages as follows: (1) the dynamic complexity can be addressed by introducing the functional-interval parameters associated with time into the programming problem; (2) the FIMISIP solutions provide a set of flexible waste-management schemes to the decision makers; and (3) the FIMISIP solutions are more reliable than those from the previous ILP ones since they can be “really” optimal regardless of how the parameters vary with time within the time period. While this study is a first attempt to solve solid waste management issues under complex uncertainties, the method can be extended to other environmental management planning problems.     

逆式动力吸振器的机电模拟实验

对传统的动力吸振器加以发展，研制出新型逆式动力吸振器．研究了阻尼对逆式动力吸振器幅频特性的影响，并应用机电模拟方法进行了电模拟实验，所得结果与理论分析结果吻合较好．依据理论分析和机电模拟结果试制了逆式动力吸振器样机模型．     

轧机主传动系统的扭振分析与建模

利用拉格朗日方程建立了传动系统扭转振动分析的通用数学模型,进行了系统固有模态分析;依据动力学理论知识,求解不同的加载方式下系统的动态响应,获得了扭转振动的扭矩放大倍数,并对系统进行了灵敏度分析。在此基础上,以某轧机的主传动系统为例,求解出系统的扭振特性的关键参数,并与试验结果作对比分析。计算表明,应用该方法分析系统的扭振特性是行之有效的。     

Random Uncertainties Model in Dynamic Substructuring Using a Nonparametric Probabilistic Model

This paper presents a new approach, called a nonparametric approach, for constructing a model of random uncertainties in dynamic substructuring in order to predict the matrix-valued frequency response functions of complex structures. Such an approach allows nonhomogeneous uncertainties to be modeled with the nonparametric approach. The Craig–Bampton dynamic substructuring method is used. For each substructure, a nonparametric model of random uncertainties is introduced. This nonparametric model does not require identifying uncertain parameters in the reduced matrix model of each substructure as is usually done for the parametric approach. This nonparametric model of random uncertainties is based on the use of a probability model for symmetric positive-definite real random matrices using the entropy optimization principle. The theory and a numerical example are presented in the context of the finite-element method. The numerical results obtained show the efficiency of the model proposed.     

Acceleration Feedback-Based Active and Passive Vibration Control of Landing Gear Components

In this paper, a novel methodology is developed to absorb the vibrations of relatively large-scale aircraft structures such as landing gear components. This is accomplished using a combination of active and passive controls. A system equivalent to a Boeing 747 landing gear break rod is selected as a test bed. The expected goal of this study is to dissipate the fundamental vibration mode of the tube. A beam-type dynamic absorber and a constrained layer damping treatment are used for passive vibration control. Simulations and experimental results are provided for the dynamic absorber case. In addition, full-state feedback along with state estimation based on the “reciprocal state space” method is presented. The plant responses and estimates for both the open loop and closed loop systems are shown in simulations. An optimal controller based on acceleration measurements using piezoelectric actuators is implemented using a hardware in the loop protocol for the active vibration control of the system. The integrated controller with passive and active components absorbs the fundamental mode of the system, according to the experimental results.     

Modal Perturbation Method and its Applications in Structural Systems

A new perturbation method is developed to solve any eigenvalue equation of the form (A0+ΔA)X? = (B0+ΔB)X?Λ? based on the solution of an original system described by A0X = B0XΛ. The eigenvectors of the modified system are expanded in a subspace spanned with a small number of vibration modes of the original system. In doing so, the former eigenvalue equation of the modified system is transformed into a set of algebraic equations, which require a significantly less computational effort to solve for the eigensolutions of complex structural systems. Four numerical examples show that the developed technique gives rise to the eigensolution of high accuracy and it is an effective approach for dynamic reanalysis of the structures with numerous degrees of freedom. In comparison with the conventional small parameter perturbation, the developed technique is applicable to a wider range of problems, and only m mode shapes are used based on the Ritz expansion so that the final solution can be derived efficiently. The technique also extends laboratory model tests for complex structures with the concept of dynamic hybrid tests numerically and experimentally.     

Simplified Model for Vertical Vibrations of Surface Foundations

A simplified model for simulating unbounded soil in the vertical vibration problems of surface foundations is presented. The model comprises a mass, a spring, and a dashpot without any internal degree of freedom. By considering three equivalent criteria, a group of equivalent models is established for a foundation-soil system. An optimal equivalent model is then determined to represent the best simplified model. The parameters of the optimal equivalent model may be obtained by a much easier and more direct method than the optimization technique used by existing models. The dynamic responses of the foundation-soil system using the optimal equivalent models are also compared with those obtained by the half-space theory and by the existing lumped-parameter models. The optimal equivalent model is found to have more accurate results than existing discrete models especially for responses at resonant frequencies. This proposed method may be easily applied to practical problems involving soil-structure interaction such as machine foundation vibration and seismic structural analysis.     

基于键合图的轧机机座系统垂向振动研究

应用键合图理论建立了轧机机座系统垂直振动键合图模型,获得了系统的固有频率和主振型。对系统在各阶模态下的能量分布进行了计算和比较,确定了系统的危险模态。分析了轧制力随时间变化的6种工况下的系统响应。此外通过有限元法对系统的固有特性进行了研究,计算发现系统的键合图模型与有限元模型具有很好的一致性。研究结果对于实际生产和系统垂振问题的识别与解决具有指导作用。     

球磨机制粉系统的线性自抗扰控制

球磨机制粉系统具有大惯性、大时滞和强耦合等特点,很难建立精确的数学模型.本文分析了球磨机制粉系统的动态特性,并为其设计分散线性自抗扰控制方案.该方案综合分散控制和线性自抗扰控制器的优点,结构简单,不依赖于对象精确模型,可以对被控对象中存在的耦合、干扰和不确定性等进行估计并补偿.根据实际现场要求,对球磨机制粉系统进行设定值跟踪实验、输入扰动实验和性能鲁棒性实验,并比较所设计方案与PID方案的控制性能.结果表明,分散线性自抗扰控制具有更强的解耦能力和抗干扰能力,且性能鲁棒性更优.      

Wind-Induced Response Characteristics of Monolayer Cable Net

Monolayer cable net system supporting glass facades is structurally sensitive to wind excitations. At present, there are limited researches on its wind-induced vibration performance, therefore it appears imperative to understand the wind-resistant behavior of this type of cable net. The wind-induced response of the monolayer cable net subjected to fluctuating wind loads is investigated with frequency-domain method in this paper, when the cable net deforms to the balance position under the mean wind loads. Some critical factors to wind-induced response are highlighted, including participation property of the modes in the dynamic vibration, and coupling effect among modes. The response spectrum of the cable net is also intensively investigated. It is shown that the first mode dominates wind-induced response significantly in all the modes, and the modes contributing to the wind-induced responses prominently are distributed in a narrow band of low order modes. When some lower modes and coupling effects among these modes are considered, the results in frequency domain agree well with the corresponding results obtained from time domain method, which are adequate for engineering practice. The characteristics of response spectrum of the nodal displacements are similar to those of the cable forces. When the wind loads and structural parameters vary in practical ranges in engineering, the resonant component in the total response sometimes occupies larger part in the total fluctuating wind response of the cable net, while the background component dominates in the wind response more commonly. Nevertheless, the first mode makes the largest contributions, no matter the background or the resonant component dominates.     

Uncertain Dynamical Systems in the Medium-Frequency Range

This paper presents a novel probabilistic model of random uncertainties for dynamical system in the medium-frequency (MF) range. This approach combines a nonparametric probabilistic model of random uncertainties for the reduced matrix models in structural dynamics with a reduced matrix model adapted to the MF range. The theory is presented and the random energy matrix relating to a given MF band, its random trace, and its random eigenvalues are studied. A numerical example is presented allowing convergence properties and stability of random responses with respect to the bandwidth to be analyzed.     

Vibration Control of Railway Bridges under High-Speed Trains Using Multiple Tuned Mass Dampers

In this paper, multiple tuned mass dampers (MTMDs) are considered for suppressing the vibration of railway bridges under high-speed trains. The interaction equations of motion between the vehicle and the bridge with MTMDs have been developed. The effectiveness of MTMDs on suppressing resonant vibration of railway bridges is examined and the optimum parameters of MTMDs for suppressing the resonant vibration are proposed. The results indicate that the use of the MTMD with the optimum parameters reduces the displacement and acceleration responses of railway bridges significantly.     

基于SIMULINK的结晶器液压振动系统可变参数研究

根据连铸结晶器液压振动系统数学模型,针对实际液压振动装置的运行状况,利用MATLAB/Simulink对结晶器液压振动系统进行动态特性分析.通过对不同控制参数和液压元件特性参数下液压缸输出位移的仿真研究,结果表明:在合理范围内减小负载刚度、负载粘性系数,增大油源压力、油液体积弹性模量可以改善系统动态特性.     

Cable Modal Parameter Identification. I: Theory

Cable modal parameters (natural frequencies and damping ratios) that represent the cable inherent dynamic characteristics play an important role in the construction, vibration control, condition assessment, and long-term health monitoring of cable-supported structures. The existing options to identify cable modal parameters through vibration measurements are somewhat limited. For this purpose, a cable dynamic stiffness based method is presented to effectively identify the cable modal parameters. In the first part of this two-part paper, the cable dynamic stiffness is analytically discussed for a viscously damped, uniform, inclined sagging cable supported at the lower end and subjected to a harmonically varying arbitrary angle displacement excitation in an arbitrary angle at the upper end when the cable is assumed to have a parabolic profile at its position of static equilibrium. Special attention is paid to the physical meaning and significance of every part of the frequency-dependent closed-form cable dynamic stiffness. Comprehensive numerical analyses have been carried out and a simplified cable dynamic stiffness is proposed for the purpose of identifying the cable modal parameters with a good accuracy over a wide range of frequencies.     

Study on Sinusoidal Reference Strategy-Based Adaptive Feedforward Control Applied to Benchmark Wind-Excited Building

The sinusoidal reference strategy (SRS) is a new strategy developed by the present authors for adaptive feedforward vibration control. The recursive-least-squares algorithm is used and a higher frequency sinusoidal signal is adopted as the reference signal. Some shortcomings of the conventional adaptive feedforward control are overcome. Numerical simulations and experimental studies have been carried out. The results show that this strategy can reduce vibration remarkably, and can adapt in real time to dynamic uncertainties and modeling errors. In this paper, the principle of SRS-based adaptive feedforward control is briefly introduced first, then simulation studies are conducted on reducing wind-induced response of the benchmark wind-excited building. The results are also compared to that obtained from linear quadratic Gaussian control.     

Penalty-Based Solution for the Interval Finite-Element Methods

A new approach for achieving guaranteed reliable results within the context of finite-element approximation of mechanical systems is developed. A reliable analysis requires that all the sources of uncertainty and errors be accommodated. The appropriateness of a partial differential equation to a given physical problem is beyond the scope of this work. Parameter uncertainty is treated as intervals in this work and guaranteed bounds on the “unknown” true solutions are obtained. In this paper an element-by-element penalty-based interval finite-element analysis of linear elastic structural mechanics and solid mechanics problem is introduced. Material and load uncertainties are handled simultaneously. Presented numerical examples illustrate the ability of the method to maintain very sharp solution enclosures even when the number of the interval parameters or the size of the problems is increased.