Applied Structural and Mechanical Vibrations 2009 Part 13 pot

12.7 Summary and comments In the light of the preliminary results on probability and statistics given in Chapter 11, this chapter has considered the subject of random vibrations. Random vibrations arise in a number of situations in engineering practice. More specifically, when it is not possible to give a deterministic description of the vibratory phenomenon under investigation but repeated observations show some underlying patterns and regularities, we resort to a description in terms of statistical quantities and we speak of a ‘random (or stochastic) process’. This is precisely the subject of Section 12.2, where we also note that, in practical situations, the engineer’s representation of a random process is an a so-called ‘ensemble’, i.e. a number of sufficiently long time histories (samples) which can be used, by averaging across the ensemble at specific instants of time, to calculate (or better ‘estimate’) all the quantities of interest. Luckily, a large number of natural vibratory phenomena have—or can be reasonably assumed to have—some properties that allow a noteworthy simplification of the analysis. These properties are stationarity and ergodicity (Section 12.2.1). There exist different levels of stationarity and ergodicity but, broadly speaking, the first property has to do with the fact that certain statistical descriptors of the process do not change with time, while the second property refers to the circumstance in which a sufficiently long time record can be considered as representative of the whole process. Furthermore, ergodicity implies stationarity and, in practice, when there is evidence that a given process is stationary, ergodicity is also tacitly assumed so that we can (1) record only one (sufficiently long) time history and (2) describe the process by taking time averages along this single sample rather than calculating ensemble averages across a number of different samples, the two types of averages being equal because of ergodicity. It should be noted, however, that the assumption of ergodicity is, more often than not, an educated guess rather than a proven fact. Just as deterministic vibrations can be analysed in the time domain or in the frequency domain, there is the possibility of doing the same with random vibrations. However, some complications of mathematical nature do arise and the problem is tackled by Fourier-transforming correlation functions rather than the time signal itself (Wiener-Khintchine relations). This procedure leads to the concept of spectral density, whose definition and properties are the subject of Section 12.3, and to the notions of narrow-band and wide- band random processes. Then, with all the above results at our disposal, we can consider the problem of determining the (random) response of a (deterministic) linear system to a random stationary source of excitation. Proceeding in order of increasing complexity—one input and one output, one output and more than one input, MDOF and continuous systems—we do so in Sections 12.4 and 12.5, where we establish the fundamental input-output relationships for linear systems and note that, once again, the system’s characteristics are represented Copyright © 2003 Taylor & Francis Group LLC in terms of IRFs in the time-domain FRFs in the frequency domain. Moreover, also in this case, there is the possibility of expressing the output characteristics in terms of modal IRFs and FRFs. Also, in the final part of Section 12.4, we pay due attention to the fact that the steady-state condition—in which a stationary input produces a stationary output—is not reached immediately after the onset of the input, but some time has to pass before the system, so to speak, adjusts to its new state of motion. During this time, the response is clearly nonstationary because its statistical characteristics (typically, its mean value if different from zero and its variance) vary from zero to their stationary value. Finally, in order to give the reader an idea of the richness of the subject of random vibrations, which is now a specialized field of activity and research in its own right, Section 12.6 deals with specific topics of particular interest. Sections 12.6.1 and 12.6.2 are strictly related and consider, respectively, the threshold crossing rates and peak distributions of stationary narrow-band processes, while Section 12.6.3 introduces some basic concepts of fatigue damage of engineering materials and gives a brief account of how, based on our limited knowledge of the details of material fatigue, we can attack the frequently encountered problem of fatigue damage due to random excitation. In this circumstance, when this excitation is in the form of a narrow-band random process, it is also shown how we can use the results of Sections 12.6.1 and 12.6.2 to estimate the mean time to failure. References 1. International Standard ISO 4866–1990, Mechanical Vibration and Shock— Vibration of Buildings—Guidelines for the Measurement of Vibrations and Evaluation of Their Effects on Buildings. 2. Adhikari, R. and Yamaguchi, H., A study on the nonstationarity in wind and wind-induced response of tall buildings for adaptive active control, Wind Engineering, Proceedings of the 9th Wind Engineering Conference, Vol. 3, pp. 1455–1466, Wiley Eastern Ltd., New Delhi, 1995. 3. Papoulis, A., Signal Analysis, McGraw-Hill, New York, 1981. 4. Bendat, J.S. and Piersol, A.G., Random Data—Analysis and Measurement Procedures, 2nd edn, John Wiley, New York, 1986. 5. Lutes, L.D. and Sarkani, S., Stochastic Analysis of Structural and Mechanical Vibrations, Prentice Hall, Englewood Cliffs, NJ, 1997. 6. Vanmarcke, E.H., Properties of spectral moments with applications to random vibration, Journal of the Engineering Mechanics Division, ASCE, 98(EM2), 425–446, 1972. 7. Köylüoglu, H.U., Stochastic Response and Reliability Analyses of Structures with Random Properties Subject to Stationary Random Excitation, Ph.D. Dissertation, Princeton University, Jan. 1995. 8. Newland, D.E., An Introduction to Random Vibrations, Spectral and Wavelet Analysis, 3rd edn, Longman Scientific and Technical, 1993. 9. Rice, S.O., Mathematical analysis of random noise, Bell System Technical Journal, Copyright © 2003 Taylor & Francis Group LLC 23 (1944) and 24 (1945); reprinted in Wax, N. (ed.) Selected Papers on Noise and Stochastic Processes, Dover, New York, 1954. 10. Sólnes, J., Stochastic Processes and Random Vibrations: Theory and Practice, John Wiley, New York, 1997. 11. ASTM Standard E468, American Society for Testing and Materials, Annual Book of ASTM Standards, E468–2, Section 3, Vol. 03.01, ASTM, Philadelphia, 1983, pp. 577–587. 12. Dowling, N.E., Fatigue failure predictions for complicated stress-strain histories, Journal of Materials, 7(1), 71–87, 1972. 13. Fuchs, H.O. and Stephens, R.I., Metal Fatigue in Engineering, John Wiley, New York, 1980. 14. Downing, S.D. and Socie, D.F., Simple rainflow counting algorithms, International Journal of Fatigue, 4(1), 31–40, 1982. Copyright © 2003 Taylor & Francis Group LLC Further reading to Part I Ainsworth, M., Levesley, J., Light, W.A. and Marletta, M. (eds) Wavelets, Multilevel Methods and Elliptic PDEs, Oxford Science Publications, Clarendon Press, Oxford, 1997. Barton, G. Elements of Green’s Functions and Propagation: Potentials, Diffusion and Waves, Clarendon Press, Oxford, 1989. Boas, M.L. Mathematical Methods in the Physical Sciences, 2nd edn, John Wiley, New York, 1983. Boswell, L.F. and D’Mello, C. The Dynamics of Structural Systems, Blackwell Scientific, Oxford, 1993. Broman, A. Introduction to Partial Differential Equations: from Fourier Series to Boundary-Value Problems, Dover, New York, 1970. Cercignani, C. Spazio, Tempo, Movimento. Introduzione alla Meccanica Rationale, (in Italian), Zanichelli, Bologna, 1976. Champeney, D.C. Fourier Transforms in Physics, Adam Hilger, Bristol, 1985. Chisnell, R.F. Vibrating Systems, Routledge & Kegan Paul, London, 1960. Clough, R.W. and Penzien, J. Dynamics of Structures, McGraw-Hill, New York, 1975. Diana, G. and Cheli F. Dinamica e Vibrazioni dei Sistemi Meccanici, Vols 1 and 2, Utet, Torino, 1993. Genta, G. Vibrazioni delle Strutture e delle Macchine, Levrotto & Bella, Torino, 1996. (Also available in English, Vibration of Structures and Machines, Springer- Verlag, New York, 1993–5.) Griffin, M.J. Handbook of Human Vibration, Academic Press, London, 1990. Hartog, J.P.D. Mechanical Vibrations, Dover, New York, 1984. Hewlett-Packard Application Note 243–3. The Fundamentals of Modal Testing, Hewlett-Packard Co., 1986. IOtech, Inc. Signal Conditioning and PC-based Data Acquisition Handbook, 1997. Ivchenko, G. and Medvedev, Yu.I. Mathematical Statistics, Mir Publishers, Moscow, 1990. Kolmogorov, A.N. and Fomin, S.V. Introductory Real Analysis, Dover, New York, 1970. Köylüoglu, H.U. Theory and Applications of Structural Vibrations, CIV 362—Lecture Notes, Princeton University, 1995. Landau, L.D. and Lifshitz, E.M. Meccanica, Editori Riuniti, Rome, 1982. (Also available in English: Landau and Lifshitz—Course of Theoretical Physics, Vol. 1, Mechanics, Pergamon Press.) Copyright © 2003 Taylor & Francis Group LLC Lebedev, N.N., Skalskaya, I.P. and Uflyand, Y.S. Worked Problems in Applied Mathematics, Dover, New York, 1965. Lembgrets, F. Parameter estimation in modal analysis, L.M.S. Seminar on Modal Analysis, Milan, 25–26 May, 1992. Milton, J.S. and Arnold J.C. Introduction to Probability and Statistics, 2nd edn, McGraw-Hill, New York, 1990. Mitchell, L.D. Modal test methods—quality, quantity and unobtainable, Sound and Vibration, Nov., 10–17, 1994. Norton, M.P. Fundamentals of Noise and Vibration Analysis for Engineers, Cambridge University Press, Cambridge, 1989. Ohayon, R. and Seize, C. Structural Acoustics and Vibration, Academic Press, London, 1998. Pettofrezzo, A.J. Matrices and Transformations, Dover, New York, 1966. Petyt, M. Introduction to Finite Element Vibration Analysis, Cambridge University Press, Cambridge, 1990. Piersol, A.G. Optimum resolution bandwidth for spectral analysis of stationary random vibration data, Shock and Vibration, 1(1), 33–43, 1993–4. Przemieniecki, J.S. Theory of Matrix Structural Analysis, Dover, New York, 1968. Reddy, B.D. Introductory Functional Analysis with Applications to Boundary Value Problems and Finite Elements, Springer-Verlag, New York, 1998. Richardson, M.H. Is it a mode shape, or an operating deflection shape? Sound and Vibration, Jan., 54–61, 1997. Rudin, W. Real and Complex Analysis, McGraw-Hill, New York, 1966. Rudin W. Functional Analysis, McGraw-Hill, New York, 1973. Scavuzzo R.J. and Pusey H.C. Principles and Techniques of Shock Data Analysis, edited and produced by the Shock and Vibration Information Analysis Center (SAVIAC, Arlington, Virginia), SVM-16, 2nd edn, 1996. Shephard, G.C. Spazi Vettoriali di Dimensioni Finite, Cremonese, Rome, 1969. (Also available in English: Vector Spaces of Finite Dimension, Oliver & Boyd Ltd.) Smith, J.D. Vibration Measurement and Analysis, Butterworths, London, 1989. Suhir, E. Applied Probability for Engineers and Scientists, McGraw-Hill, New York, 1997. Thomsen, J.J. Vibrations and Stability. Order and Chaos, McGraw-Hill, New York, 1997. Thomson, W.T. Theory of Vibration with Applications, 4th edn, Chapman & Hall, London, 1993. Timoshenko, S., Young, D.H. and Weaver, W. Jr. Vibration Problems in Engineering, 4th edn, John Wiley, New York, 1974. Towne, D.H. Wave Phenomena, Dover, New York, 1967. Ventsel, E.S. Teoria delle probabilitá, Edizioni MIR, 1983. Vu, H.V. and Esfandiari, R.S. Dynamic Systems. Modeling and Analysis, McGraw- Hill, New York, 1998. Zaveri, K. Modal Analysis of Large Structures—Multiple Exciter Systems, Bruel & Kjaer, 1985. Copyright © 2003 Taylor & Francis Group LLC Part II Measuring instrumentation Vittorio Ferrari Copyright © 2003 Taylor & Francis Group LLC 13 Basic concepts of measurement and measuring instruments 13.1 Introduction The importance of making good measurements is readily understood when considering that the effectiveness of any analysis is strongly determined by the quality of the input data, which are typically obtained by measurement. Since analysis and processing methods cannot add information to the measurement data but can only help in extracting it, no final result can be any better than such data originally are. With the intention of highlighting correct measurement practice, this chapter presents the fundamental concepts involved with measurement and measuring instruments. The first two sections on the measurement process and uncertainty form a general introduction. Then three sections follow which describe the functional model of measuring instruments and their static and dynamic behaviour. Afterwards, a comprehensive treatment of the loading effect caused by the measuring instrument on the measured system is presented, which makes use of the two-port models and of the electromechanical analogy. Worked out examples are included. Finally, a survey of the terminology used for specifying the characteristics of measuring instruments is given. This chapter is intended to be propaedeutic and not essential to the next two chapters; the reader more interested in the technical aspects can skip to Chapters 14 and 15 regarding transducers and the electronic instrumentation. 13.2 The measurement process and the measuring instrument Measurement is the experimental procedure by which we can obtain quantitative knowledge on a component, system or process in order to describe, analyse and/or exert control over it. This requires that one or more quantities or properties which are descriptive of the measurement object, called the measurands, are individuated. The measurement process then basically consists of assigning numerical values to such quantities or, more Copyright © 2003 Taylor & Francis Group LLC formally stated, of yielding measures of the measurands. This should be accomplished in both an empirical and objective way, i.e. based on experimental procedures and following rules which are independent of the observer. As a relevant consequence of the numerical nature of the measure of a quantity, measures can be used to express facts and relationships involving quantities through the formal language of mathematics. The practical execution of measurements requires the availability and proper use of measuring instruments. A measuring instrument has the ultimate and essential role of extending the capability of the human senses by performing a comparison of the measurand against a reference and providing the result expressed in a suitable measuring unit. The output of a measuring instrument represents the measurement signal, which in today’s instruments is most frequently presented in electrical form. The process of comparison against a reference may be direct or, more often, indirect. In the former case, the instrument provides the capability of comparing the unknown measurand against reference samples of variable magnitude and detecting the occurrence of the equality condition (e.g. the arm-scale with sample masses, or the graduated length ruler). In the latter case, the instrument’s functioning is based on one or more physical laws and phenomena embodied in its construction, which produce an observable effect that is related to the measurand in a quantitatively known fashion (e.g. the spring dynamometer). The indirect comparison method is often the more convenient and practicable one; think, for instance to the case of measurement of an intensive quantity such as temperature. Motion and vibration measuring instruments most frequently rely on an indirect measuring method. Regardless of whether the measuring method is direct or indirect, it is fundamental for achieving objective and universally valid measures that the adopted references are in an accurately known relationship with some conventionally agreed standard. Given a measuring instrument and a standard, the process of determination and maintenance of this relationship is called calibration. A calibrated and properly used instrument ensures that the measures are traceable to the adopted standard, and they are therefore assumed to be comparable to the measures obtained by different instruments and operators, provided that calibration and proper use is in turn guaranteed. If we refer back to the definition of measurement, it can be recognized that measurement is intrinsically connected with the concept of information. In fact, measuring instruments can be thought of as information-acquiring machines which are required to provide and maintain a prescribed functional relationship between the measurand and their output [1]. However, measurement should not be considered merely as the collection of information from the real world, but rather as the extraction of information which requires understanding, skill and attention from the experimenter. In particular, it should be noted that even the most powerful signal postprocessing techniques and data treatment methods can only help in retrieving the information Copyright © 2003 Taylor & Francis Group LLC embedded in the raw measurement data, but have no capability of increasing the information content. As such, they should not be misleadingly regarded as substitutive to good measurements, nor a fix for poor measurement data. Therefore, carrying out good measurements is of primary importance and should be considered as an unavoidable need and prerequisite to any further analysis. A fundamental limit to the achievable knowledge on the measurement object is posed at this stage, and there is no way to overcome such a limit in subsequent steps other than by performing better measurements. 13.3 Measurement errors and uncertainty After realizing the importance of making good measurements as a necessary first step, we may want to be able to determine when measurements are good or, at least, satisfying to our needs. In other words, we become concerned with the problem of qualifying measurement results on the basis of some quantifiable parameter which characterizes them and allows us to assess their reliability. We are essentially interested in knowing how well the result of the measurement represents the value of the quantity being measured. Traditionally, this issue has been addressed by making reference to the concept of measuring error, and error analysis has long been considered an essential part of measurement science. The concept of error is based on the reasonable assumption that a measurement result only approximates the value of the measurand but is unavoidably different from it, i.e. it is in error, due to imperfections inherent to the operation in nonideal conditions. Blunders coming from gross defects or malfunctioning in the instrumentation, or improper actions by the operator are not considered as measuring errors and of course should be carefully avoided. In general, errors are viewed to have two components, namely, a random and a systematic component. Random errors are considered to arise from unpredictable variations of influence effects and factors which affect the measurement process, producing fluctuations in the results of repeated observation of the measurand. These fluctuations cancel the ideal one-to- one relationship between the measurand and its measured value. Random errors cannot be compensated for but only treated statistically. By increasing the number of repetitions, the average effect of random errors approaches zero or, more formally stated, their expectation or expected value is zero. Systematic errors are considered to arise from effects which influence the measurement results in a systematic way, i.e. always in the same direction and amount. They can originate from known imperfections in the instrumentation or in the procedure, as well as from unknown or overlooked effects. The latter sources in principle always exist due to the incompleteness of our knowledge and can only be hopefully reduced to a negligible level. Copyright © 2003 Taylor & Francis Group LLC [...]... while keeping the measurand and the other qis constant and monitoring the instrument output In practice, this is hardly possible and usually the contribution is estimated partly from experimental data and partly from theoretical predictions Of course, it is expected that the instrument is mostly responsive to the measurand x, and, therefore, the above procedure is primarily applied to the experimental... considering the mechanical effort and flow variables analogous to the electrical effort and flow variables Thus V is analogous to F, and I is analogous to u It follows that the electrical resistance R, inductance L and capacitance C are analogous to mechanical resistance Rm, mass m and compliance 1/K respectively Conversely, a correspondence between series electrical and parallel mechanical circuits (and vice... electromechanical analogy 13. 7.3 The electromechanical analogy The electromechanical (EM) analogy is based on the fact that the linear Copyright © 2003 Taylor & Francis Group LLC differential equations describing mechanical systems and electrical circuits are formally identical Therefore, a correspondence can be established between lumped mechanical components and lumped electrical elements, and the... twoport devices: (13. 19a) (13. 19b) (13. 19c) (13. 19d) The terms and Kfe(s) are the open-circuit effort-, shortcircuit flow-, short-circuit effort-to-flow- and open-circuit flow-to-efforttransfer functions respectively For a given device they are not mutually independent, consistent with the fact that eqs (13. 19a) to (13. 19d) are just different representations of the same unique system, and the following... at hand, it may be more convenient to choose one representation or another among the six combinations available One possible choice which is convenient for analysing the loading effect is that of considering ei and fo as independent variables, and eo and fi as dependent variables We will hereafter assume that the two-port is linear and time-invariant, and indicate with and Fo(s) the -transforms of and. .. measurand If all the terms were ideally zero, the instrument would respond to the measurand only and would be called specific for x In the real cases, given the desired level of accuracy and estimated the ranges of variability of x and the qis, the comparison between and the allows us to determine the influence quantities which actually play a role and need to be taken into account in the case at hand... and Laplace transform respectively Copyright © 2003 Taylor & Francis Group LLC In the Fourier transform method, solving eq (13. 5) in the time domain becomes equivalent to solving the following complex algebraical equation in the frequency domain (13. 7) where X(ω) and Y(ω) are complex functions of the angular frequency ω called the Fourier (or ) transform of x(t) and y(t), given by (13. 8a) (13. 8b) and. .. as the linear ramp and exponential functions Again, solving eq (13. 5) in the time domain becomes equivalent to solving the following complex algebraical equation in the domain of the complex angular frequency (13. 9) Copyright © 2003 Taylor & Francis Group LLC where X(s) and Y(s) are complex functions being the Laplace (or transforms of x(t) and y(t) given by -) (13. 10a) (13. 10b) and the complex function... this purpose, an accelerometer of mass ma is attached to the mass m as shown in Fig 13. 15(a) Fig 13. 14 Series mass-spring-damper system excited by a force F treated in Examples 13. 1 and 13. 2 Fig 13. 15 Measurement of acceleration on the mass-spring-damper system of Fig 13. 14 by an attached accelerometer of mass m a (a) mechanical representation; (b) analogous electrical circuit; (c) Norton repesentation... sensitivity In particular, the step response of a zeroth-order instrument is a step function itself as shown in Fig 13. 4, and the sinusoidal frequency response T(ω ) is flat throughout the frequency axis (Fig 13. 5) An example of a zeroth-order instrument is a resistive potentiometer displacement transducer For a first-order instrument the input-output relationship in the time domain is given by (13. 13) which . J.S. and Piersol, A.G., Random Data—Analysis and Measurement Procedures, 2nd edn, John Wiley, New York, 1986. 5. Lutes, L.D. and Sarkani, S., Stochastic Analysis of Structural and Mechanical Vibrations, . Stochastic Processes and Random Vibrations: Theory and Practice, John Wiley, New York, 1997. 11. ASTM Standard E468, American Society for Testing and Materials, Annual Book of ASTM Standards, E468–2,. measurand and the other q i s constant and monitoring the instrument output. In practice, this is hardly possible and usually the contribution is estimated partly from experimental data and partly