This paper introduces a method for the evaluation of the seismic risk at the site of an engineering project. The results are in terms of a ground motion parameter (such as peak acceleration) versus average return period. The method incorporates the influence of all potential sources of earthquakes and the average activity rates assigned to them. Arbitrary geographical relationships between the site and potential point, line, or areal sources can be modeled with computational ease. In the range of interest, the derived distributions of maximum annual ground motions are in the form of Type I or Type II extreme value distributions, if the more commonly assumed magnitude distribution and attenuation laws are used.

https://courses.washington.edu/ess502/Cornell_1968.pdf

Engineering seismic risk analysis

1 Concepts from Vibrations 2 Response of Single-Degree-of-Freedom Systems to Initial Excitations 3 Response of Single-Degree-of-Freedom Systems to Harmonic and Periodic Excitations 4 Response of Single-Degree-of-Freedom Systems to Nonperiodic Excitations 5 Two-Degree-of-Freedom Systems 6 Elements of Analytical Dynamics 7 Multi-Degree-of-Freedom Systems 8 Distributed-Parameter Systems: Exact Solutions 9 Distributed-Parameter Systems: Approximate Mathods 10 The Finite Element Method 11 Nonlinear Oscilations 12 Random Vibrations Appendix A. Fourier Series Appendix B. Laplace Transformation Appendix C. Linear Algebra

Fundamentals of Vibrations

In recent papers the author has shown that when determining optimum parameters for an absorber which minimizes the vibration response of a complex system, the latter may be treated as an equivalent single degree-of-freedom system if its natural frequencies are well separated. Emphasis was on minimizing the displacement response when the excitation was a harmonic force. In the present paper simple expressions for optimum absorber parameters are derived for undamped one degree-of-freedom main systems for harmonic and white noise random excitations with force and frame acceleration as input and minimization of various response parameters. These expressions can be used to obtain optimum parameters for absorbers attached to complex systems provided that optimization is with respect to an absolute, rather than a relative, quantity. The requirement that the natural frequencies should be well separated is investigated numerically for the different cases. The effect of damping in the main system on optimum absorber parameters is investigated also.

Optimum absorber parameters for various combinations of response and excitation parameters

1 Formulation of the equations of motion 2 Element energy functions 3 Introduction to the finite element displacement method 4 In-plane vibration of plates 5 Vibration of solids 6 Flexural vibration of plates 7 Vibration of stiffened plates and folded plate structures 8 Vibration of shells 9 Vibration of laminated plates and shells 10 Hierarchical finite element method 11 Analysis of free vibration 12 Forced response 13 Forced response II 14 Computer analysis technique

Introduction to finite element vibration analysis

The detailed morphology of impact craters is now believed to be mainly caused by the collapse of a geometrically simple, bowl-shaped “transient crater.” The transient crater forms immediately after the impact. In small craters, those less than approximately 15 km diameter on the Moon, the steepest part of the rim collapses into the crater bowl to produce a lens of broken rock in an otherwise unmodified transient crater. Such craters are called “simple” and have a depthto-diameter ratio near 1:5. Large craters collapse more spectacularly, giving rise to central peaks, wall terraces, and internal rings in still larger craters. These are called “complex” craters. The transition between simple and complex craters depends on 1/g, suggesting that the collapse occurs when a strength threshold is exceeded. The apparent strength, however, is very low: only a few bars, and with little or no internal friction. This behavior requires a mechanism for temporary strength degradation in the rocks surrounding the impact site. Several models for this process, including acoustic fluidization and shock weakening, have been considered by recent investigations. Acoustic fluidization, in particular, appears to produce results in good agreement with observations, although better understanding is still needed.

http://www.planetary.brown.edu/planetary/geo2810/papers/MeloshIvanov99.pdf

Impact crater collapse

Random Vibration in Mechanical Systems

Certain statistical properties of the response of a vibratory system depend on the excitation process. The response process can be that of displacement, of acceleration, of stress, or of some other dynamic quantity. It would be highly desirable if the statistics on such dynamic response quantities could be simply converted into statistics on the reliability of the structure or equipment in the random vibration environment. This would then provide information directly useful to the designer or test engineer. Unfortunately, this is one of the least well-developed aspects of random vibration theory. Knowledge concerning the precise nature of failure mechanisms is far from complete and even in some cases where a clear-cut failure model exists, the purely mathematical problems of translating the dynamic response statistics into useful reliability statistics appear to be beyond the present powers. This chapter outlines the present state of the art of fatigue failure. For the particular mechanism of fatigue failure, the theory has been fairly well developed. Within certain limitations, it is possible to use the dynamic response statistics to predict both the mean time to failure and the variance about this mean. With the help of certain rational hypotheses, a reasonably valid model for the fatigue failure of metals under random vibration has been assembled. The chapter discusses the background of these hypotheses and their application to a failure model for narrow-band random stress histories.

Failure Due to Random Vibration

A mechanical system is said to be vibrating when its parts undergo motions that fluctuate in time. The engineering problems that arise in random vibration usually fall within the following headings: measurement of an existing environment, design of a structure or piece of equipment that is to work in such an environment, and the specification and execution of tests to verify that equipment already built will work in such an environment.. Useful analytical tools for performing these tasks are provided by the theory of random processes, which is an outgrowth of probability theory. The chapter primarily presents the pertinent results of this theory. It discusses how in principle it is possible to give complete probabilistic information about a random process. When for theoretical reasons such information is available, it is a simple task to calculate statistical averages for the process. In dealing experimentally with a random process, the statistical averages are much simpler to measure directly than are the probability distributions which underlie them.

Characterization of Random Vibration

A vibratory system is said to be linear and time-invariant if its equations of motions take the form of linear differential equations with constant coefficients. A system with a single degree of freedom can be described by a single second-order ordinary differential equation. A two-degree freedom system can be described by a pair of coupled second-order ordinary differential equations. The excitation–response or inputoutput problem for linear time-invariant systems in terms of particular responses to particular excitations is extended to the case where the excitation is no longer an individual time history but is an ensemble of possible time histories, that is, a random process. When the excitation is a stationary random process, then the response is also a stationary random process.

William D. Mark

Papers

Random Vibration in Mechanical Systems

Failure Due to Random Vibration

Characterization of Random Vibration

Transmission of Random Vibration

Early detection of gear-tooth bending-fatigue damage by the Average-Log-Ratio ALR algorithm