Free Vibrations of Taut Cable with Attached Damper. I: Linear Viscous Damper

Free vibrations of a taut cable with an attached linear viscous damper are investigated in detail using an analytical formulation of the complex eigenvalue problem. This problem is of considerable practical interest in the context of stay-cable vibration suppression in bridges. An expression for the eigenvalues is derived that is independent of the damper coefficient, giving the range of attainable modal damping ratios and corresponding oscillation frequencies in every mode for a given damper location without approximation. This formulation reveals the importance of damper-induced frequency shifts in characterizing the response of the system. New regimes of behavior are observed when these frequency shifts are large, as is the case in higher modes and for damper locations further from the end of the cable. For a damper located sufficiently near the antinode in a given mode, a regime of solutions is identified for which the damping approaches critical as the damper coefficient approaches a critical value. A regime diagram is developed to indicate the type of behavior in each mode for any given damper location.     

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.     

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.     

Dynamic Behavior of Stay Cables with Rotational Dampers

Vibration reduction in stay cables by means of viscous dampers is of great interest in cable damage prevention and serviceability of structural system supported by such cables. The paper presents a study on the effectiveness, as well as the limits, of rotational viscous dampers and springs inserted at the two ends of a bending-stiff taut cable; influence of rotational stiffness of the springs is also investigated. After a nondimensional expression of the equation of motion has been obtained, as in other cases of nonproportionally damped continuous structures, complex modal analysis is pursued, obtaining complex eigenvalues and eigenfunctions. Comparison with intermediate dampers, widely used in bridge engineering, is performed showing the range of nondimensional parameters for which the proposed approach is of interest. Finally, a numerical technique based on complex mode superposition is presented in order to evaluate time domain responses for transversal distributed excitation. As an example, the procedure is applied to a wind-exposed cable.     

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.     

Timoshenko Beam with Tuned Mass Dampers to Moving Loads

The structural analysis of a Timoshenko beam system with tuned mass dampers (TMDs) under moving-load excitation is presented. A proposed simplified two-degrees-of freedom system based on the first mode of the Timoshenko beam is employed for the design of TMDs. The dynamic characteristics of a Timoshenko beam, such as the structural-resonant and phase-resonant velocities, and the effectiveness of a TMD for vibrational control are especially emphasized. A practical example of an elevated railway subjected to the Japanese Shinkansen (SKS) high-speed bullet train is included.     

Analytical Study on Bending Effects in a Stay Cable with a Damper

The effects of bending on the modal properties of a stay cable with a transverse damper are analytically studied. Considering that the value of the flexural rigidity in the stay cable is small in practice, an explicit asymptotic formula for the modal damping of a cable with a general type of damper is derived. For a viscous damper, the asymptotic formula obtained is compact, accurate, and thus is very suitable for practical design. Furthermore, for the first few vibration modes of interest, the asymptotic solution is independent of the modal index. It is shown that flexure in the cable reduces the maximum attainable modal damping, possibly up to 20%, while it significantly increases the optimal damping coefficient of the damper.     

Joint First-Passage Probability and Reliability of Systems under Stochastic Excitation

The first-passage probability, describing the probability that a scalar process exceeds a prescribed threshold during an interval of time, is of great engineering interest. This probability is essential for estimating the reliability of a structural component whose response is a stochastic process. When considering the reliability of an engineering system composed of several interdependent components, the probability that two or more response processes exceed their respective safe thresholds during the operation time of the system is an equally essential quantity. This paper proposes simple and accurate formulas for approximating this joint first-passage probability of a vector process. The nth order joint first-passage probability is obtained from a recursive formula involving lower order joint first-passage probabilities and the out-crossing probability of the vector process over a safe domain. Interdependence between the crossings is approximately accounted for by considering the clumping of these events. The accuracy of the proposed formulas is examined by comparing analytical estimates with those obtained from Monte Carlo simulations for stationary Gaussian processes. As an example application, the reliability of a system of interconnected equipment items subjected to a stochastic earthquake excitation is estimated by linear programming bounds employing marginal and joint component fragilities obtained by the proposed formulas.     

Damping of Taut-Cable Systems: Two Dampers on a Single Stay

The mitigation of in-plane stay oscillation in cable-stayed bridges is commonly addressed by placing an external mechanical damper, linear or nonlinear, on each stay or by introducing transverse cross-ties among cables. Although the problem of a cable with a single external damper has found significant attention in the past and different techniques have been proposed for the solution of the free-vibration problem, limitations are related to the fact that the location of the damper is usually very close to the cable end (on the bridge deck side) due to geometric constraints, leading to inherently low modal damping in the fundamental modes. In this paper the installation of more than one damper on an individual stay is considered to overcome such limitations and to increase the overall performance of the system. An existing procedure, based on the linearized taut-string theory, was modified to allow for the presence of multiple external discrete viscous dampers. The case of two devices with arbitrary location has been solved, identifying advantages and disadvantages of the proposed solution. In addition, extensions of the practical “universal curve” and the interpretation thereof are presented.     

Linear Stochastic Dynamical System under Uncertain Load: Inverse Reliability Analysis

The reliability of a linear dynamical system driven by a partially known Gaussian load process, specified only through its total average energy, is studied. A simple dynamic parallel and series system reliability network is investigated for the failure analysis using the crossing theory of stochastic processes. The critical input power spectral density of the load process which maximizes the mean crossing rate of a parallel or series system network emerges to be fairly narrow banded and hence fails to represent the erratic nature of the random input realistically. Consequently, a tradeoff curve between the maximum mean crossing rate of the reliability network and the disorder in the input, quantitatively measured through its entropy rate, is generated. Mathematically, the generation of the tradeoff curve of nondominated solutions, known as the Pareto optimal set, leads to a nonlinear, nonconvex, and multicriteria optimization problem with conflicting objectives. A recently developed Pareto optimization technique, implemented through genetic algorithm, has been successfully exploited to solve the optimization problem. A suitable exploitation of stochastic process theory circumvents the task of handling an apparent robust optimization problem (i.e., optimization under uncertainty) and at the same time, reduces the dimensionality of the multiobjective optimization scheme in view of both the multiobjectivity and constraints in the aforementioned methodology. The optimally disordered inputs which simulate the worst performance of the system of a spring-supported coupled rod assembly, has been studied as a numerical illustration of the mathematical formulation.     

Dynamic Behavior of Nonlinear Cable System. IV

A suitable filter response is defined that allows studies of the effects of excitation bandwidth on the responses of geometrically nonlinear dynamic systems. For such a random excitation, as the bandwidth approaches zero, the mean square displacement response of a linear single-degree-of-freedom (SDOF) system approaches the mean square response to harmonic excitation. Studies of mean square responses of a nonlinear SDOF model of a cable system are presented. The set of nonlinear equations for the response moments are truncated using Gaussian closure and solved using continuation techniques as implemented in the program AUTO. Surfaces of turning point loci are computed in the parameter space of excitation bandwidth, excitation central frequency, and system damping. These surfaces provide values of the three parameters that separate regions with multiple mean square responses from regions with unique mean square responses. Response time histories are computed by simulation to illustrate system behavior in regions with multiple stable mean square responses.     

Semi-Implicit Integration Algorithm for Stochastic Analysis of Multi-Degree-of-Freedom Structures

This paper presents a semi-implicit integration algorithm for random vibration problems that is appropriate for analyzing large structures, nonlinear hysteretic systems, and structural control problems. This semi-implicit approach results in a recursive expression for the mean and covariance response. A state-space representation of the equations of motion is adopted for deriving the algorithm. The solution of the state-space equations is first obtained, after which the expected value of the resulting equations is taken so as to obtain the first two moments. A stability condition for the method is also derived. Three numerical examples, a linear oscillator, a Duffing oscillator, and a multi-degree-of-freedom system with hysteretic supplemental damping devices, are provided to illustrate the effectiveness of the proposed method. Results compare well with Monte Carlo simulation, indicating that the semi-implicit integration algorithm is accurate and stable.     

Stochastic Response of a Random Mass Structure

This paper proposes a polynomial approximation approach for the estimation of the stochastic response of a random mass structure subjected to an evolutionary random excitation. A bounded, monopeak, and symmetrically distributed probability density function, called λ-PDF, together with the Gegenbauer polynomial approximation, is introduced to deal with stochastic dynamic response problems of the random mass structures. The λ-PDF model is used to describe the random parameters in the engineering random structures. And then the Gegenbauer polynomial approximation method is used to reduce the random structures into its deterministic equivalent one. The numerical example shows the effectiveness of the proposed method to explore dynamic phenomena in random structures.     

Nonlinear Dynamic Response of Piles under Horizontal Excitation

This paper presents test results from cast-in situ reinforced concrete single and group piles subjected to strong horizontal excitation. The tests were conducted for different eccentric moments simulating different excitation levels to obtain the frequency-amplitude response of the pile. Moderate nonlinear behavior is observed in both horizontal and rocking components of vibration. The experimental results were compared with dynamic interaction factor approach using nonlinear solutions. The accuracy of the nonlinear analysis in predicting the dynamic response depends on the choice of parameters that best characterize the response of boundary zone around the pile and the realistic length of pile separation. It is shown in this study that by allowing for boundary zone and separation between pile and soil, close agreement between theoretical predictions and measured response curves can be achieved.     

Combined Damping Effect of Two Dampers on a Stay Cable

The combined effect of two dampers, either on the same end or opposite ends of a stay cable, is analytically studied in this paper. By considering small distances of the dampers from the anchorages, an asymptotic formula for the modal damping ratio of the cable is derived from which the total damping effect is presented in an explicit form. It is shown that when two dampers are installed at opposite ends of the cable, the total damping effect is asymptotically the sum of the contributions from single dampers. On the contrast, if two dampers are at the same end, there is no advantage of increasing the maximum modal damping in the cable over the use of a single damper.     

Galerkin Solution for Linear Stochastic Algebraic Equations

Algebraic equations with random coefficients, referred to as stochastic algebraic equations, are used extensively to solve approximately differential equations describing mechanics problems with uncertain material properties and applied loads. This paper (1) constructs optimal and suboptimal Galerkin solutions for linear stochastic algebraic equations, (2) reviews current procedures for deriving stochastic algebraic equations from stochastic differential equations and proposes alternative methods, (3) demonstrates the implementation of the proposed Galerkin method by numerical examples, and (4) calculates statistics of the displacement field for a plate on random elastic foundation. The optimal Galerkin solution coincides with the conditional expectation of the exact solution with respect to a σ-field coarser than the σ-field relative to which the exact solution is measurable, and is unbiased. Generally, suboptimal Galerkin solutions are biased but may provide approximations for the tails of the distribution of the exact solution that are superior to those by the optimal Galerkin solution.     

Stochastic Response and Stability of a Nonintegrable System

For determining the stochastic response and stability of a strongly nonlinear single-degree-of-freedom system using the stochastic averaging technique, the size of excitations should be small such that the response of the system converges weakly to a Markov process. This condition is not often met with practical problems, and therefore, application of this method for obtaining their responses becomes difficult. Further, for systems with nonlinearities that cannot be integrated in closed form, stability analysis by examining the conditions of the two boundaries of the problem is not possible. A semianalytical method along with a weighted residual technique is presented here to circumvent these difficulties and to determine the response and stability of a strongly nonlinear system subjected to sizable stochastic excitation. The weighted residual technique is employed to correct the errors in averaged drift and diffusion coefficients resulting due to the size of the stochastic excitation. Two example problems are solved as illustrations of the method.     

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.     

Nonlinear Response of Double-Wall Cylindrical Shell Vibrations under Random Excitation

An analytical model is presented to predict the nonlinear response of a double-wall sandwich cylindrical shell system subjected to random excitation. Nonlinear spring-dashpot models are integrated into the system to characterize the behavior of the soft core. Donnell’s thin shell theory is used to develop the governing nonlinear equations of motion. A Monte Carlo simulation of stationary random processes, multimode Galerkin-type approach, and numerical integration procedures are used to develop linear and nonlinear response solutions of simply supported cylindrical shells. Numerical results include time domain response histories, root-mean-square values and response spectral densities. Parametric studies are performed to investigate the effects of nonlinearity, shell thickness, core stiffness, and thickness.     

Viscous Heating of Fluid Dampers under Small and Large Amplitude Motions: Experimental Studies and Parametric Modeling

This paper summarizes the results from a comprehensive experimental program in an effort to better understand the phenomenon of viscous heating of fluid dampers under small-stroke (wind loading) and large-stroke (earthquake loading) motions. Two dampers, one with 15-kip and one with 250-kip force output at peak design velocity, have been instrumented and tested under various amplitudes and frequencies. Temperature histories at different locations along the damper casing and within the silicon fluid that undergoes the shearing action have been recorded. Experimental data under small-stroke motions of the 250-kip damper showed that a single closed-form expression derived from first principles is capable of predicting the temperature rise at different locations of the damper with fidelity. The recorded data under long-stroke motions suggest a two-parameter law of cooling that allows the estimation of the internal temperature of the silicon oil once the external temperature on the damper casing is known. The presented cooling law is an extension of Newton’s law of cooling. The study concludes that for both dampers, the same values of the model parameters provide a good approximation of the cooling behavior. The study presents a valuable formula that can be used in practice to estimate the internal fluid temperature of the damper given the external shell temperature.