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

Piezoelectric transduction has received great attention for vibration-to-electric energy conversion over the last five years. A typical piezoelectric energy harvester is a unimorph or a bimorph cantilever located on a vibrating host structure, to generate electrical energy from base excitations. Several authors have investigated modeling of cantilevered piezoelectric energy harvesters under base excitation. The existing mathematical modeling approaches range from elementary single-degree-of-freedom models to approximate distributed parameter solutions in the sense of Rayleigh–Ritz discretization as well as analytical solution attempts with certain simplifications. Recently, the authors have presented the closed-form analytical solution for a unimorph cantilever under base excitation based on the Euler–Bernoulli beam assumptions. In this paper, the analytical solution is applied to bimorph cantilever configurations with series and parallel connections of piezoceramic layers. The base excitation is assumed to be translation in the transverse direction with a superimposed small rotation. The closed-form steady state response expressions are obtained for harmonic excitations at arbitrary frequencies, which are then reduced to simple but accurate single-mode expressions for modal excitations. The electromechanical frequency response functions (FRFs) that relate the voltage output and vibration response to translational and rotational base accelerations are identified from the multi-mode and single-mode solutions. Experimental validation of the single-mode coupled voltage output and vibration response expressions is presented for a bimorph cantilever with a tip mass. It is observed that the closed-form single-mode FRFs obtained from the analytical solution can successfully predict the coupled system dynamics for a wide range of electrical load resistance. The performance of the bimorph device is analyzed extensively for the short circuit and open circuit resonance frequency excitations and the accuracy of the model is shown in all cases.

/pdf/an-experimentally-validated-bimorph-cantilever-model-for-qzwpq105mk.pdf

An experimentally validated bimorph cantilever model for piezoelectric energy harvesting from base excitations

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

The dependence of muscle force on muscle length gives rise to a "spring-like" behavior which has been shown to play a role in the execution of single-joint posture and movement. This paper extends this concept and considers the influence of the apparent mechanical behavior of the neural, muscular and skeletal system on the control and coordination of multiple degree of freedom posture and movement.

A rigorous definition of "spring-like" behavior is presented. From it a numerically quantifiable, experimental test of spring-like behavior is formulated. It is shown that if the steady-state force-displacement behavior of a limb is not spring-like, this can only be due to the action of inter-muscular feedback, and can not be due to intrinsic muscle properties. The directional character of the spring-like behavior of a multiple degree of freedom system is described. The unique way in which synergistic coactivation of polyarticular muscles may modulate the directional properties of the spring-like behavior of a multiple degree of freedom system is explained.

Dynamic aspects of postural behavior are also considered. The concept of mechanical impedance is presented as a rigorous dynamic generalisation of the postural stiffness of the limb. The inertial behavior of the system is characterised by its mobility. As with the stiffness or impedance, in the multiple degree of freedom case it has a directional property. The way in which the apparent kinematic redundancy of the musculo-skeletal system may be used to modify its dynamic behavior is explained. Whereas the inertial behavior of a single limb segment is not modifiable, it is shown that the apparent inertial behavior of a multiple degree of freedom system may be modulated by repositioning the joints.

A unified description of the posture and movement of a multi-joint system is presented by defining a "virtual trajectory" of equilibrium positions for the limb which may be specified by the neuro-muscular system. The way in which this approach may lead to a simplification of some the apparent computational difficulties associated with the control of multi-joint motion is discussed.

The mechanics of multi-joint posture and movement control

Random Vibration in Mechanical Systems

Dynamics of Mechanical and Electromechanical Systems

Engineering Analysis: A Survey of Numerical Procedures

Publisher Summary This chapter focuses on statistical relations between excitation and response processes for a broad range of structural elements including beams, cables, arches, plates, membranes, and shells. The analysis of the dynamic response of continuous structures under random excitation was initiated in connection with Brownian motion (Van Lear and Uhlenbeck, 1931). More recent developments arose out of work on aerospace problems connected with jet-noise excitation. The chapter presents general formulations and a number of solution procedures. It focuses on approximate procedures. The chapter highlights experimental procedures and presents a few comparisons between measurements and analytical predictions. Special consideration is given to stationary wide-band excitation of uniform structures. Under certain circumstances, the random response distributions exhibit patterns that become asymptotically very simple as the number of responding modes increases.

Random Vibration of One- and Two-Dimensional Structures

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.

Stephen H. Crandall

Papers

Random Vibration in Mechanical Systems

Dynamics of Mechanical and Electromechanical Systems

Engineering Analysis: A Survey of Numerical Procedures

Random Vibration of One- and Two-Dimensional Structures

Failure Due to Random Vibration