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Über dieses Buch

A. GENERAL REMARKS During the last century, probabilistic methods for design and analysis of engineering systems have assumed a prominent place as an engineering tool. No longer do engineers naively believe that all problems can be analyzed with deterministic methods; but rather, it has been recognized that, due to unc- tainties in the model and the excitation, it may only be possible to describe the state of a system in terms of some random measure. Thus, with the need to address safety and design issues adequately and simultaneously to minimize the cost of a system, much attention has been given to the development of probabilistic criteria which can be applied in a systematic manner [l]t. These techniques allow for uncertainties in the parameters of the model as well as for uncertainties in both the static and dynamic loadings to be considered and therefore give a better measure of the reliability of a system. Widespread application of probabilistic methods can be found in disciplines ranging from civil, mechanical and electrical engineering to biology, economics and political science.

Inhaltsverzeichnis

Frontmatter

Chapter I. Introduction

Abstract
During the last century, probabilistic methods for design and analysis of engineering systems have assumed a prominent place as an engineering tool. No longer do engineers naively believe that all problems can be analyzed with deterministic methods; but rather, it has been recognized that, due to uncertainties in the model and the excitation, it may only be possible to describe the state of a system in terms of some random measure. Thus, with the need to address safety and design issues adequately and simultaneously to minimize the cost of a system, much attention has been given to the development of probabilistic criteria which can be applied in a systematic manner [1]†. These techniques allow for uncertainties in the parameters of the model as well as for uncertainties in both the static and dynamic loadings to be considered and therefore give a better measure of the reliability of a system. Widespread application of probabilistic methods can be found in disciplines ranging from civil, mechanical and electrical engineering to biology, economics and political science.
B. F. Spencer

Chapter II. Problem Definition and Formulation

Abstract
Of the models utilized to date to describe the hysteretic constitutive relation for a simple system, one of Che most versatile and tractable for dynamic applications is the modified Bouc model [19]. The original model was modified by Wen [127] in order that the smoothness of the transition between the pre-yield and post-yield regions of the force-deflection curve could be controlled. The constitutive relation for the modified Bouc hysteresis model can be stated for the SDOF oscillator in Figure 1.1 as
$$Q(x(\tau ),\,\dot x(\tau ),\,0 < \,\tau \, < \,t;\,t) = M(2\zeta \omega \dot x\, + \,a{\omega ^2}x\, + \,(1 - a){\omega ^2}z)$$
(2.1) where z is an evolutionary variable described by the first order differential equation
$$\dot z\, = \, - \,\gamma |\dot x|\,|z{|^{(n - 1)}}z\, - \,\beta \dot x|z{|^{n\,}} + \,A\dot x$$
(2.2) and ω is the undamped natural frequency of the oscillator when α = 1. The parameters α, β, γ, n and A are shape parameters of the hysteresis loops which can also be functions of time. The quantities
$$Ma{\omega ^2}x\,and\,M(1 - a){\omega ^2}z$$
in Equation 2.1 are the linear and hysteretic portions of the total restoring force, respectively, and M(2ζωx) is the force associated with viscous dissipation. Baber and Wen [6] have shown that the hysteretic restoring force for the SDOF oscillator can be effectively modeled by Equations 2.1 and 2.2. Their report provided insight into the effect of each of the shape parameters on the hysteretic constitutive relation. These effects shall be discussed here for completeness.
B. F. Spencer

Chapter III. Numerical Solution of the First Passage Problem

Abstract
Consider the general form of the three-dimensional convection-diffusion equation given by
$${\partial \over {\partial x}}{k_x}{{\partial \psi } \over {\partial x}} + {\partial \over {\partial y}}{k_y}{{\partial \psi } \over {\partial y}} + {\partial \over {\partial z}} - {u_x}{{\partial \psi } \over {\partial z}} - {u_y}{{\partial \psi } \over {\partial y}} - {u_z}{{\partial \psi } \over {\partial z}} + c{{\partial \psi } \over {\partial t}} + Q = 0$$
(3.1) where kx/C, ky/C and kz/C are the diffusion coefficients; ux/C, uy/C and uz/C are the velocities; Q/C is the source term; and the boundary conditions are appropriately defined. The nonlinear partial differential equations developed in Chapter II which govern the first passage behavior of the simple hysteretic oscillator are degenerate forms of Equation 3.1. The nature of Equation 3.1 for many engineering problems gives little hope for analytical solution; thus numerical methods must be adopted to solve the problem.
B. F. Spencer

Chapter IV. Validation of Results

Abstract
Given the cumulative distribution function of oscillator reliability R, the computation of the ordinary moments of time to first passage failure can be easily calculated by Equation 2.30. The inverse problem, calculation of the cumulative probability distribution function given the ordinary moments of the distribution, is a much more formidable task and will be addressed in Chapter V. In this section, however, the comparison of the ordinary moments of first passage time found in Chapter III by directly solving the steady state Pontriagin-Vitt equation will be compared with those found by using Equation 2.29 in conjunction with the transient solution of the first passage problem. Equation 2.29 can be approximated by
$${T^{(r)}}({_o}) \simeq \,\sum\limits_{i = 0}^\infty {{{[({\tau _{i\,}} + \,{\tau _{i + l}})/2]}^r}\,[R({\tau _i}|{{}_o}) - R({\tau _{i + l}}|} {_o})]$$
(4.1) where R is the cummulative distribution function of oscillator reliability.
B. F. Spencer

Chapter V. Estimating Oscillator Reliabilty Using Ordinary Moments

Abstract
Often it is of interest to obtain estimates of the probability density function and cumulative distribution function when only the ordinary or central moments of a distribution are available. This can be a difficult task, however. The inverse problem of finding the moments of a distribution wherein the distribution is prescribed is relatively simple and has been presented in Chapter IV. In this chapter, a technique reported by Dowson and Wragg [35, 130] which consists of maximizing the entropy of the system while satisfying moment constraints is employed to determine an approximate distribution.
B. F. Spencer

Chapter VI. Conclusions and Recommendations

Abstract
The first passage problem for the single degree-of-freedom oscillator incorporating the modified Bouc hysteresis model has been solved by a Petrov-Galerkin finite element method. Solutions have been obtained for the first six ordinary moments of first passage time and for the cumulative distribution and probability density functions. Results have been presented for several hysteretic oscillators, and the accuracy of these results has been demonstrated by extensive Monte Carlo simulation. Finally, an effective method to estimate the reliability of a system having several prescribed moments has been examined. In the absence of an analytical solution, these results are the most accurate reported to date.
B. F. Spencer

Backmatter

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