Optimal Vibration Absorber with Nonlinear Viscous Power Law Damping and White Noise Excitation

A tuned mass damper with a nonlinear power law viscous damper excited by white noise is considered. The system is analyzed by statistical linearization and stochastic simulation with the objective of minimizing the standard deviation of the response. It is shown that the optimal parameters for the tuned mass damper are unaffected by the magnitude of the structural damping in the linear case. However, in the nonlinear case the structural damping influences the equivalent parameters obtained by statistical linearization and thereby indirectly the optimal values for the damper parameters. Results from stochastic simulation show good agreement with results from statistical linearization in terms of the standard deviation of the response. It is shown that the optimal damping, which can be obtained by the passive device, is the same for the linear and nonlinear damper. However, for the nonlinear tuned mass damper the optimal parameters will depend on both structural damping and excitation intensity (or vibration amplitude). The results are presented in such a way that they can be used directly for the design of a tuned mass damper with damping governed by a nonlinear viscous power law.     

Spectral Method for Nonlinear Stochastic Partial Differential Equations of Elliptic Type

This paper is concerned with the numerical approximations of semi-linear stochastic partial differential equations of elliptic type in multi-dimensions. Convergence analysis and error estimates are presented for the numerical solutions based on the spectral method. Numerical results demonstrate the good performance of the spectral method.     

Semiactive Neurocontrol for Seismic Response Reduction Using Smart Damping Strategy

A new semiactive control strategy that combines a neurocontrol system with a smart damper is proposed to reduce seismic responses of structures. In the proposed semiactive control system, the improved neurocontroller, which was developed by employing a training algorithm based on a cost function and a sensitivity evaluation algorithm to replace an emulator neural network, produces the desired active control force, and then a bang-bang-type controller clips the control forces that cannot be achieved by a smart damper (e.g., a variable orifice damper, controllable fluid damper, etc.). Therefore, the proposed semiactive control strategy is fail-safe in that the bounded-input, bounded-output stability of the controlled structure is guaranteed. Numerical simulation results show that the proposed semiactive control system that employs a neural network-based control algorithm is quite effective in reducing seismic responses.     

Spectral Approach to Equivalent Statistical Quadratization and Cubicization Methods for Nonlinear Oscillators

Random vibrations of nonlinear systems subjected to Gaussian input are investigated by a technique based on statistical quadratization, and cubicization. In this context, and depending on the nature of the given nonlinearity, statistics of the stationary response are obtained via an equivalent system with a polynomial nonlinearity of either quadratic or cubic order, which can be solved by the Volterra series method. The Volterra series response is expanded in a trigonometric Fourier series over an adequately long interval T, and exact expressions are derived for the Fourier coefficients of the second- and third-order response in terms of the Fourier coefficients of the first-order, Gaussian response. By using these expressions, statistics of the response are determined using the statistics of the Fourier coefficients of the first-order response, which can be readily computed since these coefficients are independent zero-mean Gaussian variables. In this manner, a significant reduction of the computational cost is achieved, as compared to alternative formulations of quadratization and cubicization methods where rather prohibitive multifold integrals in the frequency domain must be determined. Illustrative examples demonstrate the reliability of the proposed technique by comparison with data from pertinent Monte Carlo simulations.     

Earthquake Response of Elastic Single-Degree-of-Freedom Systems with Nonlinear Viscoelastic Dampers

Investigated are the steady-forced and earthquake responses of single-degree-of-freedom (SDF) systems with a nonlinear viscoelastic damper (VED), which consists of a nonlinear fluid viscous damper (FVD) connected in series to a linear elastic bracing element (chevron or inverted V-shaped braces). For a wide range of bracing stiffness, nonlinear dampers are advantageous because they achieve essentially the same reduction in system responses but with a significantly reduced force. Damper nonlinearity has little influence on the structural response in the velocity-sensitive region of the spectrum even if the bracing is fairly flexible, but differences up to 16% were observed in other spectral regions. As expected, supplemental damping reduces structural response and the response reduction depends on the bracing stiffness, with this dependence varying with the spectral regions. For practical applications, a procedure is presented to estimate the design values of structural deformation, structural force, foundation shear, and damper force directly from the earthquake design (or response) spectrum. Finally, a procedure is presented to determine the damper and bracing properties necessary to limit the structural deformation to some design value or to the structural capacity.     

Damping in Shear Beam Structures and Estimation of Drift Response

Under pulse-type ground motions modal analysis is not quite efficient for estimating the elastic response of multi-degree-of-freedom systems, in particular when the effects of higher modes are significant. This paper first shows that the assumption of nondispersive damped waves for shear beams leads to inconsistent response estimation. Subsequently, a closed form time domain dispersive damped wave solution to the partial differential equation of motion is presented and it is verified with frequency domain solutions. Finally, using the solutions to the differential equation of motion, the response of frame structures with energy dissipating devices is studied.     

Simulation of Nonstationary Stochastic Processes by Spectral Representation

This paper presents a rigorous derivation of a previously known formula for simulation of one-dimensional, univariate, nonstationary stochastic processes integrating Priestly’s evolutionary spectral representation theory. Applying this formula, sample functions can be generated with great computational efficiency. The simulated stochastic process is asymptotically Gaussian as the number of terms tends to infinity. This paper shows that (1) these sample functions accurately reflect the prescribed probabilistic characteristics of the stochastic process when the number of terms in the cosine series is large, i.e., the ensemble averaged evolutionary power spectral density function (PSDF) or autocorrelation function approaches the corresponding target function as the sample size increases, and (2) the simulation formula, under certain conditions, can be reduced to that for nonstationary white noise process or Shinozuka’s spectral representation of stationary process. In addition to derivation of simulation formula, three methods are developed in this paper to estimate the evolutionary PSDF of a given time-history data by means of the short-time Fourier transform (STFT), the wavelet transform (WT), and the Hilbert-Huang transform (HHT). A comparison of the PSDF of the well-known El Centro earthquake record estimated by these methods shows that the STFT and the WT give similar results, whereas the HHT gives more concentrated energy at certain frequencies. Effectiveness of the proposed simulation formula for nonstationary sample functions is demonstrated by simulating time histories from the estimated evolutionary PSDFs. Mean acceleration spectrum obtained by averaging the spectra of generated time histories are then presented and compared with the target spectrum to demonstrate the usefulness of this method.     

ONDA: Computer Code for Nonlinear Seismic Response Analyses of Soil Deposits

This paper describes a newly developed computer code for performing one-dimensional nonlinear dynamic analysis (ONDA) of soil deposits. The code has been developed by revisiting the 1982 work by Ohsaki with the purpose of simulating the ground response to an earthquake of moderate intensity (i.e., values of peak ground acceleration on stiff soil on the order of 0.15 to 0.25g, which are typical of many sites in Italy). In the Ohsaki model a horizontally stratified soil deposit is idealized as a discrete mechanical system composed of a finite number of lumped masses connected with a series of springs and dashpots. Nonlinearity is modeled by assuming (1) a “backbone” curve that describes the initial monotonic loading of the stress-strain curve, and (2) a “rule” that simulates the unloading-reloading paths and stiffness degradation undergone by soil as seismic excitation progresses. Typically, the backbone curve is obtained from conventional cyclic undrained loading laboratory tests. The rule generally used is the so-called Masing criterion, which assumes that the unload-reload branches of the stress-strain curve have the same shape as the initial loading curve but are affected by a scale factor (n) equal to 2. In this work, the Masing criterion has been modified by assuming a scale factor (n) not necessarily equal to 2. It turns out that a factor n greater than 2 allows the simulation of cyclic hardening, while cyclic softening can be modeled by assuming decreasing values of n even smaller than 2. Pyke proposed in 1979 to use a scale factor (n) lower than 2 to simulate cyclic degradation. According to Pyke, the n parameter is a function of the mobilization factor. The generalization of the Masing criterion allows ONDA to properly simulate the phenomena of soil hardening and soil degradation, giving it the capability to compute the permanent strains developed during a seismic event. The procedure required to evaluate the model parameters is also described in the paper. Note that the laboratory tests examined gave values of n between 2 and 6 for a strain level not greater than 0.3%. In ONDA the numerical solution of the nonlinear equations of motion is obtained using the unconditionally stable Wilson θ algorithm (with θ ≥ 1.37). The new method has been used to predict the seismic response at two sites in Italy. For these case studies, the maximum input acceleration was not greater than 0.3g and the computed shear strains were less than 0.2%. The ONDA results have been compared with those computed with SHAKE, EERA (equivalent-linear analysis), and NERA (nonlinear analysis).     

Damping and Viscoelastic Dynamic Response of RC Flexural Members Strengthened with Adhesively Bonded Composite Materials

A theoretical approach for the dynamic viscoelastic response of reinforced concrete (RC) beams and one-way slabs strengthened with adhesively bonded composite materials is developed. The analytical model is based on variational principles, dynamic equilibrium, and compatibility of deformations between the structural components (RC beam/slab, adhesive, composite material). The model accounts for the deformability of the adhesive layer and for its high order stress and displacement fields. The equations of motion and the boundary, continuity, and initial conditions are derived via the extended Hamilton’s principle. The Kelvin-Voigt approach is adopted for the consideration of the viscoelastic response of the adhesive material and the internal damping in the composite material and the RC member. The Rayleigh damping model is used for the external viscous damping of the RC member. The dynamic governing equations are solved using the Newmark time integration and a multiple shooting algorithm is used for the solution in space. A numerical example is presented to examine the capabilities of the model, to highlight the unique phenomena associated with the viscoelastic response of the adhesive material, and to demonstrate its influence on the local and global behavior. The results obtained using the analytical model show that the viscoelastic response of the adhesive material may significantly modify the critical shear and peeling stresses at the interfaces of the adhesive layer.     

SWAN-Mud: Engineering Model for Mud-Induced Wave Damping

This paper describes the implementation of a new dispersion relation and energy-dissipation equation obtained from a viscous two-layer model schematization in the state-of-the-art wave forecasting model SWAN to simulate wave damping in coastal areas by fluid mud deposits. This new dispersion relation is derived for a nonviscous, nonhydrostatic upper layer and a viscous, hydrostatic lower layer, covering most conditions encountered in nature. An algorithm is developed for a robust numerical solution of this new implicit dispersion relation through proper starting values in the iteration procedure. The implementation is tested against a series of analytical solutions and three schematic test cases. Next, four dispersion relations published in the literature are evaluated and compared with the new dispersion relation. The solution of the dispersion relations forms a multidimensional space. Comparison of the various model solutions through one-dimensional graphs can therefore become quite misleading, as shown in the discussion of a two-dimensional representation of the solution space, explaining for instance the variation in ambient conditions at which maximum wave damping is to be expected. The various models have been developed for a variety of conditions, such as shallow and deep water and shallow and thick mud layers; the various models agree well in their domain of applicability, but they show significant deviations when used outside their domain. Because the ambient and mud conditions may vary considerable in space and time at a particular site, the use of the new model is advocated because it covers most water depths and fluid mud thicknesses encountered in nature. The strength of the new SWAN-mud model lies in its large-scale applicability, assessing the two-dimensional evolution of wave fields in coastal areas. Therefore, the new implementation is evaluated with respect to the behavior of waves on a sloping seabed, representing real-world coasts. In all cases, the new SWAN-mud model behaves satisfactorily; a critical remaining issue, though, is the assessment of the relevant fluid mud parameters.     

Free Vibrations of Taut Cable with Attached Damper. II: Nonlinear Damper

Free vibrations of a taut cable with a nonlinear power-law damper attached near the end are considered. An approximate analytical solution for the amplitude-dependent effective damping ratios in each mode is developed by assuming the same form of solution as for the linear damper and minimizing the mean-square error in the force equilibrium at the damper. An asymptotic approximate solution for small frequency shifts reveals a nondimensional grouping of parameters allowing the development of an amplitude-dependent “universal estimation curve” for the power-law damper. The shape of the universal curve is slightly different for each value of the damper exponent, but for a given exponent the curve is nearly invariant over the same range of parameters as the universal curve for the linear damper. This formulation yields insights into the dependence of nonlinear damper performance on mode number and amplitude of oscillation, suggesting potential advantages that may be offered by a nonlinear damper over a traditional linear damper.     

Nonlinear Oscillations of Vehicle Tires with Hertzian Contact

The purpose of this work is to study the nonlinear vibrations induced due to wheel contact. The contact force is nonlinear and is considered to be caused due to a nonpolynomial 3/2 type of spring (Hertzian) as is commonly assumed to model wheel contact. Approximate periodic free vibration solutions are derived using an analytical technique based on the averaging method and is compared numerically. The case of the nonlinear wheel with nonlinear damping (van der Pol) is also studied and the possibility of the existence of a limit cycle is explored. The existence of initial condition dependent free periodic vibrations or a limit cycle under certain circumstances may be important for passenger comfort and noise generation or for vehicle safety as it may cause wear and tear of the mechanical components of the vehicle.     

Combination Resonance Instability of Curved Panels with Cutout Subjected to Nonuniform Loading with Damping

The dynamic instability of doubly curved panels with a centrally located circular cutout, subjected to nonuniform compressive in-plane harmonic edge loading is investigated. The present work deals with the problem of the occurrence of combination resonances in contrast to single (mode) resonances in parametrically excited doubly curved panels with a central circular cutout. The method of multiple scales is used to obtain analytical expressions for the single (mode) and combination resonance instability regions. It is shown that other cases of the combination resonance can be of major importance and yield a significantly enlarged instability region in comparison to the principal instability region. The effects of nonuniform edge loading, centrally located circular cutout, damping, the static, and dynamic load factors on dynamic instability behavior of simply supported doubly curved panels are studied. The results show that under localized edge loading, combination resonance zones are as important as single (mode) resonance zones. The effects of damping show that there is a finite critical value of the dynamic load factor for each instability region below which doubly curved panels cannot become dynamically unstable. A central circular cutout has the destabilizing effect on the dynamic stability behavior of doubly curved panels subjected to nonuniform edge loading. This example of simultaneous excitation of two modes, each oscillating steadily at its own natural frequency, may be of considerable interest in vibration testing of actual structures.     

Kalman滤波在Yule-Walker谱估计中的应用

在现代谱估计中,由于Yule-Walker沃克谱估计算法中平稳随机序列的长度n在某些情况下偏小,使计算出的ARMA随机过程的功率谱密度不能精确逼近真实值,所以我们在用自相关法估计AR模型参数时加入了kalman滤波器。将估计的AR模型系数及高斯白噪声作为滤波器的输入及部分参数,对最终估计的功率谱进行修正。实验结果表明,在Yule-Walker谱估计中加入kalman滤波,其计算精度及结果稳定性都有了一定的提高,可以作为解决相关问题的方法之一。     

Experimental Considerations for Determining the Damping Coefficients of Hard Coatings

This paper attempts to characterize the various methods that may be used in defining the vibration features of a plate dimensionally proportioned to represent a turbine compressor blade. The comparison is specifically devoted towards certain damping properties of a titanium plate coated with magnesium aluminate spinel (mag spinel). Two different coating thicknesses were tested: 0.13?mm?(.005?in.) per side and 0.25?mm?(.010?in.) per side [total thicknesses of 0.25?mm?(.010?in.) and 0.51?mm?(.020?in.)]. In order for the reader to appreciate the experimental variations possible in this undertaking, the writers define various tests performed. Dynamic ping tests were conducted on all specimens to determine their resonance frequencies. Laser vibrometry was used to determine the mode at each resonance frequency. Damping ratios were determined through the use of sine sweeps. It is important that damping characteristics be measured very precisely. Thus, in the approach using a sine sweep, one must make sure that the rate of sweep is sufficiently slow to evaluate the proper frequency response function. A discussion of the proper rate is provided. This project demonstrated that, in order to represent the basic damping properties of a coating, one must consider the rate of the dynamic load application. The usual method associated with the damping coefficient evaluation is referred to as the half-power method. The true value of the damping coefficient is directly associated with the rate of experimental power versus frequency.     

Calibrating the Water-Hammer Response of a Field Pipe Network by Using a Mechanical Damping Model

Hydraulic transient field tests have been conducted in a water distribution network. Existing transient models are applied to model the measured responses, but poor matches are obtained apart from the estimation of the initial rise of pressure. Possible reasons for these discrepancies include the effects of demands, entrained air, unsteady friction, friction losses associated with small lateral pipes, and mechanical damping caused by the interaction of pipes and joints with surrounding soils (including the effects of vibration and different degrees of restraint). These effects are systematically investigated by inclusion of the previously mentioned phenomena in conceptual transient models and calibration to the measured field responses. A mechanical damping-based conceptual transient model is shown to be the only model that can be accurately calibrated to the measured field responses.     

Element Level System Identification with Unknown Input with Rayleigh Damping

A novel system identification procedure is proposed for nondestructive damage evaluation of structures. It is a finite element-based time-domain linear system identification technique capable of identifying structures at the element level. The unique features of the algorithm are that it can identify a structure without using any input excitation information and it can consider both viscous and Rayleigh-type proportional damping in the dynamic models. The consideration of proportional damping introduces a source of nonlinearity in the otherwise linear dynamic algorithm. However, it will also reduce the total number of damping coefficients to be identified, reducing the size of the problem. The Taylor series approximation is used to transform a nonlinear set of equations to a linear set of equations. The proposed algorithm, denoted as the modified iterative least square with unknown input algorithm, is verified with several examples considering various types of structures including shear-type building, truss, and beams. The algorithm accurately identified the stiffness of structures at the element level for both viscous (linear) and proportional (nonlinear) damping cases. It is capable of identifying a structure even with noise-contaminated response information. An example shows how the algorithm could be used in detecting the exact location of a defect in a defective element. The algorithm is being developed further and is expected to provide an economical, simple, efficient, and robust system identification technique that can be used as a nondestructive defect detection procedure in the near future.     

Dynamic Stability of Ferromagnetic Beam-Plates with Magnetoelastic Interaction and Magnetic Damping in Transverse Magnetic Fields

The dynamic stability of a soft ferromagnetic beam-plate in a transverse magnetic field is investigated. The fundamental equations of the ferromagnetic beam-plate including magnetoelastic interaction and magnetic damping are derived. Based on a magnetoelastic linearized method and perturbation technique, an expression for magnetoelastic stability of the ferromagnetic beam-plate system including magnetization and magnetic damping is explicitly obtained. In the analysis, the actual magnetic field coupled with the deformed ferromagnetic beam-plate is divided into a rigid magnetic field for the undeformed plate and a disturbed field for the deformed plate due to small displacement. It is shown that there exists two stable states involving magnetic damping stable oscillation and overdamping asymptotically stable motion, before the beam-plate loses its stability by static divergence. The effect of magnetic and geometrical parameters of the ferromagnetic beam-plate on critical magnetic field is discussed in detail.     

New Method for the Evaluation of Material Damping Using the Wavelet Transform

Material damping is a fundamental parameter required for dynamic analysis of geotechnical and civil infrastructure. The material damping ratio is very difficult to measure in situ. A new methodology for in situ measuring of material damping using surface waves is presented in this work. This methodology is successfully evaluated on laboratory scale models and numerical simulations. Ultrasonic waves are used in this work because of the size of the laboratory models. The output force of an ultrasonic piezoelectric transmitter is modeled by using a Morlet function. The wave attenuation and phase variation of propagating surface waves with distance are analyzed using the wavelet transform. Numerical results show that the material damping ratio calculated using the wavelet transform gives a global value that represents an average damping ratio for the frequency bandwidth imposed by the seismic or ultrasonic source. Experimental results, from tests on a cemented sand and a concrete plate, show good agreement with published damping values.     

Object-Oriented Modeling of Structural Analysis and Design with Application to Damping Device Configuration

This paper presents an object-oriented (OO) software framework for computer-aided structural analysis and design research, where different structural analysis methods and design procedures need to be implemented and investigated. The framework is designed with four basic modules: structure, load, analysis, and design. Each module includes a set of cooperating interfaces and classes. Through the predefined interfaces, the framework provides architecture for many structural design applications. A variety of similar entities, such as different design applications, design procedures, and analysis methods, can be built on this architecture by implementing the necessary interfaces. The clearly defined interactions between the modules accommodate the future extensions within the modules. The final OO design of the framework can be communicated by many well-known design patterns, and it is described by unified modeling language. The framework is then customized to the application for optimizing the configuration of energy dissipation devices in a given structure. By implementing a few interfaces, this paper illustrates how this OO framework accommodates changes, and how reusability and extensibility can be achieved.