Applied Structural and Mechanical Vibrations Theory, Methods and Measuring Instrumentation Part II Measuring instrumentation Vittorio Ferrari Copyright © 2003 Taylor Francis Group LLC 13 Basic conce.
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-toone 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 Conversely, the former sources, as they are known, can be compensated for by applying a proper correction factor to the measurement results After the correction, the expected value of systematic errors is zero Although followed for a long time, the approach based on the concept of measurement error has an intrinsic inconsistency due to the impossibility of determining the value of a quantity with absolute certainty In fact, the true value of a quantity is unknown and ultimately unknowable, since it could only be determined by measurement which, in turn, is recognizably imperfect and can only provide approximate results As a consequence, the measurement error is unknowable as well, since it represents the deviation of the measurement result from the unknowable true value As such, the concept of error can not provide a quantitative and consistent mean to qualify measurement results on a theoretically sound basis As a solution to the problem, a different approach has been developed in the last few decades and is currently adopted and recommended by the international metrological and standardization institutions [2] It is based on recognizing that when performing a measurement we obtain only an estimate of the value of the measurand and we are uncertain on its correctness to some extent This degree of uncertainty is, however, quantifiable, though we not know precisely how much we are in error since we not know the true value The term measurement uncertainty can be therefore introduced and defined as the parameter that characterizes the dispersions of the values that could be attributed to the measurand In other words, the uncertainty is an estimate of the range of values within which the true value of a measurand lies according to our presently available knowledge Therefore uncertainty is a measure of the ‘possible error’ in the estimated value of a measurand as obtained by measurement It is worth noting that the result of a measurement can unknowably be very close to the value of the measurand, hence having a small error, nonetheless it may have a large uncertainty On the other hand, even when the uncertainty is small there is no absolute guarantee that the error is small, since some systematic effect may have been overlooked because it is unknown or not recognized and, as such, not corrected for in the measurement result From this standpoint, a different meaning can be attributed to the term true value in which the adjective ‘true’ loses its connotation of uniqueness and becomes formally unnecessary The true value, or simply the value, of a measurand can be conventionally considered as the value obtained when the measurement with lowest possible uncertainty according to the presently available knowledge is performed, i.e when an exemplar measuring method which minimizes and corrects for every recognized influencing effect is used In practical cases, the idea of an exemplar method should be commensurate with the accuracy needed for the particular application; for instance, when we measure the length of a table with a ruler we consciously disregard the influence of temperature on both the table and the ruler, since we consider this effect to be negligible for our present measuring needs We simply Copyright © 2003 Taylor & Francis Group LLC acknowledge that our result has an uncertainty which is higher than the best obtainable, but is suitable for our purposes However, we may be in the situation of negligible uncertainty of the instrument (the ruler in this case) compared to that caused by temperature on the measurement object (the table), for which we are therefore able to detect and measure the thermal expansion The converse situation is that of negligible uncertainty of the measurement object compared to that of the measuring instrument and procedure This is the case encountered when testing an instrument by using a reference or standard of low enough uncertainty to be ignored Thus the value of the reference or standard can be conventionally assumed as the true value, and the test thought of as a mean to determine the errors of the measuring instrument and procedure Quantifying such errors and correcting those due to systematic effects is actually no different from performing a calibration of the measuring instrument under test Summarizing, the introduction of the concept of uncertainty removes the inconsistency of the theory of errors, and directly provides an operational mean for characterizing the validity of measurement results In practice, there are many possible sources of uncertainty that, in general, are not independent, for example: incomplete definition of the measurand, effect of interfering environmental conditions and noise, inexact calibration and finite discrimination capability of measuring instruments and variations in their readings in repeated observations under apparently identical observations, unconscious personal bias in the operation of the experimenter In principle, the influence of each conceivable source of uncertainty could be evaluated by the statistics of repeated observations In the practical cases this is essentially impossible and, therefore, many source of uncertainty can be more conveniently quantified a priori by analysing with scientific judgment the pool of available information, such as tabulated data, previous measurement results, instrument specifications The results of the two evaluation methods are called respectively type A and type B uncertainties, which are classified as different according to their derivation but not differ in nature and, therefore, are directly comparable A detailed treatment of the methods used to evaluate uncertainty can be found in [2] and [3] 13.4 Measuring instrument functional model Irrespective of the measured variable and the operating principle involved, a measuring instrument can be represented by the block diagram of Fig 13.1 This is a simplified and general model which focuses on the very fundamental features that, with various degrees of sophistication in the implementation, are typical of every measuring instrument The measuring instrument can be seen as composed of three cascaded blocks, which provide an information transfer path from the measurand quantity to the observer The first block, named the sensing element, is the Copyright © 2003 Taylor & Francis Group LLC Fig 13.1 Functional model of (a) a measuring instrument and (b) an electronic measuring system stage being in contact with the measurand and interacting with it in order to sense its value This interaction should be perturbing as little as possible so that negligible load is produced by the instrument on the measured object, as discussed in Section 13.7 The output of the sensing element is in the form of some physical variable which is in a known relationship with the measurand If we take, for example, a mercury glass thermometer, than the sensing element is constituted by the mercury, and its output is the thermal expansion of the fluid volume in the bulb As we shall see, in electronic instruments and systems the sensing element function is performed by sensors and transducers The second block, named the variable-conversion stage, accepts the output of the sensing element and converts it into another variable and/or manipulates it with the general aim of obtaining a representation of the signal more suitable to its presentation, yet preserving the original information content In our example of the glass thermometer the variable-conversion stage is the capillary tube that transduces the volume expansion into the elongation of the fluid column The third block, named the presentation stage, undertakes the final translation of the measurement signal into a form which is perceived and understood by humans, once again preserving the original information content The role of this stage is straightforward, but its importance should not be overlooked In fact, the degree of discrimination between closely spaced values of the measurand that an instrument allows, i.e the resolution, is strongly related, among other factors, to the design and construction of its presentation stage This can be readily recognized if we think at our glass thermometer for which the presentation stage is the gridmark pattern on the capillary tube Although the mercury expansion is a continuous function of temperature, the discrete spacing of the gridmarks enables discrimination Copyright © 2003 Taylor & Francis Group LLC no better than 0.1 °C to the naked eye, which is, nevertheless, all that is needed in many applications It should be observed that the distinction between functional blocks does not necessarily reflect a physical separation of such blocks in the real instruments On the contrary, there are many cases in which several functions are somewhat distributed among different pieces of hardware so that it is difficult, besides essentially useless, to distinguish and parse them Nowadays, most of the measurement tasks in any field are performed by instruments and systems which measure physical quantities by electronic means Basically, the use of electronics in measuring instrumentation offers higher performance, improved functionality and reduced cost compared to purely mechanical systems A very general block representation of an electronic measuring instrument or system is given in Fig 13.1(b), which is fairly similar to that of Fig 13.1(a) with some important differences In this case the sensing function is performed by sensors, or transducers, which respond to the physical stimulus caused by the measurand with a corresponding electrical signal Such a signal is then amplified, possibly converted into a digital format and processed in order to extract the information of interest contained in the sensor signal, and filter out the unwanted spurious components All such processing operations are carried out in the electrical domain irrespective of the nature of the measurand, and therefore they may take advantage of the high capabilities of modern electronic elaboration circuitry The obtained results can then be presented to the observer through a display stage, and/or possibly stored into some form of memory device, most typically electronic or magnetic The memory storage capability offered by many electronic measuring instruments is of fundamental importance, as it enables analysis, processing and comparisons on measurement data to be performed offline, that is, arbitrarily later than the time when the data are captured Some instruments are optimized for extremely fast cycles of data storage-retrieval-processing so that they can perform specialized functions, such as filtering, correlation or frequency transforms, in real time, i.e with a delay inessential for the particular application Transducers and electronic signal amplification and processing will be treated in Chapters 14 and 15 respectively A fundamental fact resulting from both block diagrams of Fig 13.1 is that the measuring instrument occupies the position at the interface between the observer and the measurand Moreover, all of them are under the global influence of the surrounding environment This influence is generally a cause of interference on the information transfer path from the measurand to the observer, producing a perturbing action which ultimately worsens the measurement uncertainty This fact may be represented by considering the output y of a measuring instrument being a function not only of the measurand x, as we ideally would like to happen, but also of a number of further quantities qi related to the effects of the boundary conditions Such quantities are named the influencing or interfering quantities Typical influencing quantities may Copyright © 2003 Taylor & Francis Group LLC be of an environmental nature, such as temperature, barometric pressure and humidity, or related to the instrument operation, such as posture, loading conditions and power supply Besides observing that y, x and the quantities qi are actually functions of time y(t), x(t) and qi(t), we may even consider time itself as an influencing quantity, since in the most general case the output of a real measuring instrument depends to some extent on the time t at which the measurement is performed This means that the same input combination of measurand and influencing quantities applied at different time instants of the instrument’s operating life may, in general, produce different output values due to instrument ageing and drift Considered as an influencing quantity, time has a peculiar nature due to the fact that, unlike what theoretically can be done for the qis, the observer cannot exert any kind of control over it Developing a formal description of measuring instruments which globally takes into account all the involved variables as functions of time with the aim of deriving the time evolution of the output is a difficult task Usually, a more practicable approach is followed which, besides, provides a better understanding of the instrument performances and a deeper insight into its operation It consists of distinguishing between static and dynamic behaviour, each of which can be analysed separately Operation under static conditions can be analysed by neglecting the time dependence of the measurand and the influencing quantities, therefore avoiding the solution of complicated partial-derivative differential equations The consequent reduction in complexity enables a detailed description of the output-to-measurand relationship and the evaluation of the impact due to influencing quantities On the other hand, the analysis of dynamic operation is essentially performed by taking into account the time evolution of the measurand only and the resultant time dependence of the instrument output, thereby requiring only ordinary differential equations The effect of the influencing quantities on dynamic behaviour is generally evaluated by a semiquantitative extension of the results obtained for the static analysis Though this approach it is not strictly rigorous, it offers a viable solution to an otherwise unmanageable problem and, as such, it is of great practical utility 13.5 Static behaviour of measuring instruments Let us assume that the measurand x and the influencing quantities qis are constant and independent of time It should be noted that this assumption is not in contradiction with regarding x and the qis as variables In fact, we consider that the x and the qis are subject to variations over a range of values, but we not take into account the time needed by such variations to take place In other words, we consider only the static combinations of constant inputs once the transients have died out Under such an assumption, the relationship between the instrument output y and the measurand x, the Copyright © 2003 Taylor & Francis Group LLC qis and the time t at which the measurement is performed is given by the following expression: (13.1) where fg is a function which defines the global conversion characteristic of the measuring instrument The differential of y is given by (13.2) The quantities and represent the sensitivities of the measuring instrument in response to the measurand x, the ith influence quantity qi and the time t The term is responsible for the time stability of the conversion characteristic or, better, of its instability Higher values of imply a more pronounced ageing effect on the instrument and require a more frequent calibration An instrument for which is called time-invariant The instrument is the more selective for x the lower the value of the terms are compared to so that their effect on the output is negligible with respect to the 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 In principle, the contribution of the significant influence quantities could be experimentally evaluated by varying each of them in turn over a given interval, 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 determination of the measurand-to-output relationship The curve obtained in this way is the static calibration or conversion characteristic of the instrument under given conditions of the influencing quantities Under varying conditions, a family of calibration characteristics is obtained, which contain information on the impact of the considered qis Assuming a reference condition for which the influencing quantities are kept constant at their nominal or average values qoi, and ageing effects are neglected, it follows that the output y depends on the measurand only and eq (13.1) reduces to (13.3) Copyright © 2003 Taylor & Francis Group LLC Again the overall flow-transfer function Kf12(s) is not simply the product of Kf1(s) and Kf2(s) but it is scaled by the term which represents the loading exerted by the input port of the device on the output port of the device In this case the output of device is not working in a short-circuit condition and, accordingly, its effective transfer function is different from the short-circuit transfer function Kf1(s) The term which is in general a complex function of s, determines a flow-divider action at the interconnection between the two devices and can be called the flow loading factor Lf(s) The flow loading effect is minimized when Yo1(s) is negligible compared to Yi2(s), being ideally zero either for or In such cases Lf(s) is equal to zero and The above analysis of the loading effect for the effort and the flow devices is directly extendible to the case of effort-to-flow and flow-to-effort devices In fact, irrespective of the overall input and output quantities of the cascaded devices, i.e effort or flow, it can be realized that the loading effect of device on device will always be represented by one of the two cases already discussed Therefore the analysis necessarily reduces to taking into consideration either the effort or the flow loading factor: • • The effort loading factor Le(s) needs to be as close to zero as possible in each of the following three cascade configurations: effort/effort, effort/ effort-to-flow, flow-to-effort/effort Conversely, the flow loading factor Lf(s) needs to be as close to zero as possible in each of the remaining three cascade configurations: flow/ flow, flow/flow-to-effort, effort-to-flow/flow We therefore can conclude that, for a given pair of systems and represented by linear unilateral two-port devices, the effort and flow loading factors Le and Lf are parameters which enable to quantify the loading effect produced by system on system Furthermore, it is worthwhile observing that Le and Lf are not independent but are linked by the relationship consistent with the fact that they simply refer to different representations of the same phenomenon of interaction between the two systems The foregoing conclusions can be directly applied to the analysis of the measurement loading error by considering systems and as the measurement object and the measuring instrument respectively We will shortly show some examples of application of this concept in the measurement of mechanical dynamic quantities, such as displacement, velocity, acceleration and force To this we first need to introduce a simplification in the description of the twoport devices consisting of exploiting the so-called 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 formalism of electrical circuits can be used to describe mechanical systems [7, 8] Such a correspondence is not unique and, without defining any requirement that it should satisfy, none of the choices is preferable with respect to the others A consistent approach is firstly to agree on which quantities are taken as the effort and flow variables in the mechanical and electrical domains The choice of mating force F with voltage V as the effort variables, and velocity u with current I as the flow variables is that which renders the usual definitions of both the mechanical impedance ZM=F/u and the electrical impedance ZE=V/I consistent with that of the generalized impedance given by the effort/flow ratio Afterwards, in both domains the series connection can be defined as that in which all elements are subjected to the same value of the flow variable, while the parallel connection is that for which all elements see the same value of the effort variable Therefore, in a series electrical circuit the same current I flows through each element, while the total voltage drop V is the sum of the individual voltage drops In a parallel electrical circuit the voltage drop V is the same across each element, while the total current I is the sum of the individual currents Similarly, in a series mechanical circuit each element undergoes the same velocity u, while the total force F is the sum of the forces acting on each individual element In a parallel mechanical circuit the same force F acts on each element, while the total velocity u is the sum of the individual velocities Now, a correspondence criterion between series electrical and series mechanical circuits can be established, giving rise to the direct EM analogy This is equivalent to 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 versa) can be established, giving rise to the inverse EM analogy This is equivalent to considering the mechanical effort and flow variables analogous to the electrical flow and effort variables in a cross-linked correspondence Thus V is now analogous to u, and I is analogous to F, leading to electrical impedance V/I being analogous to mechanical mobility u/F It follows the electrical resistance R, inductance L and capacitance C are now analogous to 1/Rm, 1/K and m respectively Both EM analogies are summarized in Fig 13.13 The main drawback of the inverse EM analogy is that the mechanical and electrical impedances of the analogous components have opposite behaviours versus the frequency, as opposed to the case of the direct analogy For example, a spring element has ZM=K/s but its inverse analogous inductor Copyright © 2003 Taylor & Francis Group LLC Fig 13.13 Summary of the electromechanical analogies has ZE=sL, while its direct analogous capacitor has ZE=1/(sC) For this reason, the direct EM analogy is more often preferred and will be adopted hereafter 13.7.4 Examples Example 13.1 Consider a series-connected mass-spring-damper system excited by a force F, as shown in Fig 13.14 Attention should be paid to the Copyright © 2003 Taylor & Francis Group LLC fact that, though the elements are in a ‘side-by-side’ configuration suggesting the parallel connection, they are actually connected in series, since they undergo the same velocity Suppose first that we want to measure the acceleration a, which is the same for all the elements To 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 Copyright © 2003 Taylor & Francis Group LLC By applying the direct EM analogy the system can be converted into the electrical circuit of Fig 13.15(b) The accelerometer is represented as a twoport device with a mechanical velocity input ui and an electrical voltage output Vo, as described by the following equations: (13.25) where is the acceleration sensitivity in is the electrical output impedance, and is the mechanical input impedance The system under measurement behaves as a one-port with the flow variable, i.e the velocity, being the integral of the measurand, the acceleration It is therefore more convenient to pass to the Norton representation shown in Fig 13.15(c), where the velocity is the internal source variable, and is the internal impedance At this point it is very simple to derive the measurand-to-output transfer function in the hypothesis of no electrical loading of the accelerometer, i.e I o=0: (13.26) The term represents the flow loading factor Lf, which in this case is a velocity loading factor Lu In the ideal case of ma=0 it results that Lu=0, no loading error occurs and the output provides the measurand acceleration a=suint The presence of the accelerometer mass ma causes and brings about two effects The first effect is that at each fixed frequency so, the measured acceleration differs from the real one by a factor Lu(so), and the resulting error is of the order of ma/m The second effect is that the resonant frequency of the system is diminished, changing from in the unperturbed case to in the loaded case Both these loading effects are of critical importance in vibration measurements in general and modal analysis in particular, and should be carefully minimized by choosing an accelerometer with a mass ma being as small as possible compared to the system mass m In other words, the measuring instrument, due to its connection in series with the measuring system, should have a negligible impedance compared to Zint of the measured structure Copyright © 2003 Taylor & Francis Group LLC Fig 13.16 Measurement of displacement on the mass-spring-damper system of Fig 13.14 by an attached displacement transducer of negligible mass and damping, and stiffness Kx: (a) mechanical representation; (b) analogous electrical circuit; (c) Norton representation Example 13.2 Suppose now that we want to measure the displacement x of the same system For this purpose, we can use a spring-loaded transducer modelled by a stiffness element Kx with negligible mass and damping, as schematized in Fig 13.16(a) The resulting analogue electrical circuit is shown in Fig 13.16(b) The system under measurement is the same one-port as before, except that now the measurand, i.e the displacement, is the integral of the output flow variable, i.e the velocity The transducer is modelled as a two-port governed by the following equations: (13.27) where is the displacement sensitivity in V/m, Zo is the electrical output impedance, and is the mechanical input impedance Again, we can pass to the Norton equivalent of the one-port as shown in Fig 13.16(c) and obtain the measurand-to-output transfer function under Copyright © 2003 Taylor & Francis Group LLC the hypothesis of no electrical loading of the displacement transducer, i.e Io=0: (13.28) The term represents the flow loading factor Lf, which is now determined by the stiffness of the measuring instrument In case of static displacement, i.e for s=0, it reduces to which shows that the loading error increases the higher is the transducer stiffness Kx The dynamic behaviour is also affected In particular, the resonant frequency of the system is augmented by the measuring instrument stiffening action, changing from in the unperturbed case to in the loaded case To reduce both these errors it is necessary to use a transducer as compliant as possible, i.e with a very low Kx, in order to ensure a negligible impedance compared to Zint Often, especially in the cases of small-sized and lightweight systems, the only way to obtain such a condition is by adopting a noncontact measuring method which ensures a Kx equal to zero Example 13.3 As a third example, consider the mass-spring-damper system of Fig 13.17 held in motion at a velocity u with respect to the reference frame It should be noted that, though the system elements are in a ‘stacked’ arrangement suggesting the series connection, they are connected in parallel, since they are all subject to the same force F Fig 13.17 Parallel mass-spring-damper system held in motion at a velocity u treated in Example 13.3 Copyright © 2003 Taylor & Francis Group LLC Suppose that we are interested in measuring such a force and, to this purpose, an elastic-element force transducer is adopted and attached to the system as shown in Fig 13.18(a) By applying the direct EM analogy the system can be converted into the electric circuit of Fig 13.18(b) The force transducer is represented as a two-port device with a mechanical force input Fi and an electrical voltage output Vo , as described by the following equations: (13.29) where is the force sensitivity in V/N, Zo is the electrical output impedance, and is the mechanical input impedance The system under measurement again behaves as a one-port but now the effort variable, i.e the force, is the measurand quantity In this circumstance it is advisable to pass to the Thevenin representation as shown in Fig 13.18(c), Fig 13.18 Measurement of force on the mass-spring-damper system of Fig 13.17 by a force transducer of stiffness KF: (a) mechanical representation; (b) analogous electrical circuit; (c) Thevenin representation Copyright © 2003 Taylor & Francis Group LLC where the force is the internal source variable, and is the internal impedance Again assuming no electrical load for the transducer, i.e Io=0, the measurand-to-output transfer function can be determined: (13.30) This time, the stiffness KF of the measuring instrument determines the effort loading factor Le represented by the term The loading error is now smaller the more rigid the transducer is, consistent with the fact that it is connected in parallel with the system and, therefore, its mechanical impedance should be much greater than Zint As a concluding comment, it should now be clear what we qualitatively anticipated at the beginning of this section by saying that it is theoretically possible to compute the loading error and compensate for it In fact, it appears to be sufficient to analyse the system with the method outlined above to determine, depending on the case at hand, either Le or Lf However, this would require the generalized impedances of all the involved elements to be exactly known, which is a situation that rarely occurs in practice The preferred and most used approach is that of properly choosing the instrument in order to minimize its impact by maintaining Le (or Lf) as small as possible 13.8 Performance specifications of measuring instruments The quality of measurements taken and their suitability for our purposes depend on the measuring instrument characteristics which, therefore, need to be properly specified Measuring instruments are complex systems, and specifying their performance in a clear and consistent way is not a trivial task It is of fundamental importance that a suitable terminology is both understood and adopted [9] The reason is not merely a matter of formal rigorousness, but rather lies in the substantial need to express concepts in a concise and objective way, avoiding incompleteness and misinterpretations The following is a glossary of recommended terms which apply to measuring instruments in general, irrespective of their operating principle and construction Though it is not exhaustive, it is intended to help the reader in understanding the basic characteristics of an instrument to properly chose and operate it Copyright © 2003 Taylor & Francis Group LLC Range (input/output) The input range identifies the limits between which the measurand can be applied with the instrument operating properly and in compliance with its specifications Such limits have the dimensions of the measurand The output range identifies the limits between which the instrument output lies when the measurand is within the input range Such limits have the dimensions of the instrument output Span The span is the difference between the limits which define the range (either input or output) Full-scale value The full-scale value identifies the upper limit of the range (either input or output) When the lower limit is zero, the full-scale value equals the span When this is not the case, full-scale value and span differ and, strictly speaking, they should not be confused However, it is common practice to pay little attention to this subtle matter and in many occasions, such as when specifying linearity or other parameters, full-scale value and span are used as synonyms Sensitivity The sensitivity S is the rate of change of the instrument output y with respect to input measurand x, i.e S=dy/dx In the general case, the sensitivity is a function of the measurand value x, i.e S=S(x) For linear instruments the sensitivity is constant throughout the range and takes the name of scale factor, calibration factor or conversion coefficient When the measurand and the instrument output are functions of time and linearity holds, they can be decomposed into a superposition of sinusoids and the sensitivity becomes in general a complex function of frequency representing the instrument frequency response function Linearity Linearity is the ability of the instrument to follow a prescribed linear relationship between input and output, either in the case they are static quantities or functions of time The nonlinearity, or nonlinearity error, is the maximum deviation of the instrument output, at any measurand value, from that calculated assuming such a linear relationship To properly quantify the nonlinearity it is essential to specify which line is assumed as the reference The following two alternatives are generally adopted: • • End-point linearity: the reference line is the one passing through the limiting points, or end points, of the instrument calibration characteristic Least-squares linearity: the reference line is the one which best fits the experimental points of the instrument calibration characteristic by the method of the least squares Copyright © 2003 Taylor & Francis Group LLC Usually, the nonlinearity is expressed as a percentage of the full-scale output (%FSO or %FS) and it is, therefore, a dimensionless number Resolution Resolution identifies the minimum variation in the measurand which produces a detectable variation at the instrument output Such a minimum variation is called resolution threshold or discrimination threshold Therefore, the resolution describes the ability of the instrument of discriminating small changes in the measurand An ideal instrument with unlimited discrimination capability is said to have continuous or infinite resolution The resolution is specified by quoting the resolution threshold and may be referred to the input or to the output In the former case it is usually expressed as an absolute value having the dimensions of the measurand quantity; in the latter case it is mostly expressed as a percentage of full-scale output (%FSO or %FS) Notice that the improper use of the term sensitivity in place of resolution is sometimes encountered Since sensitivity has a markedly different significance, this misuse should be avoided Dynamic range Dynamic range is defined as the range of measurand values contained between the minimum detectable level, given by the resolution, and the full-scale value The dynamic range DR is often expressed in decibel, that is DR (dB)= 20 log10DR For a given full-scale value, the instrument with the highest resolution (i.e the highest discrimination capability) has the widest dynamic range Repeatability Repeatability is the ability of an instrument of providing the same output reading when the same value of the measurand is repeatedly applied in the same conditions It expresses the amount of random fluctuations that affect the instrument output in the ideal condition of a perfectly defined and stable measurand value Reproducibility Reproducibility is the ability of an instrument of providing the same output reading when the same value of the measurand is subsequently applied in different conditions, such as different time, observer, location Hysteresis Hysteresis is the effect by which an instrument provides, for each given value of the measurand, two different outputs when such a value has been approached by increasing or by decreasing the measurand, resulting in an upward and a downward calibration characteristic The hysteresis error is the maximum difference between such upward and downward calibration characteristics Copyright © 2003 Taylor & Francis Group LLC The hysteresis error is usually expressed as a percentage of the full-scale output (%FSO or %FS) and it is, therefore, a dimensionless number Stability Stability is the ability of an instrument of providing the same output reading over time when a value of the measurand is applied and maintained constant To properly express the stability it is advisable to specify the duration of the time interval to which it refers To provide a qualitative reference, the terms short-, medium- or long-time stability are often used Drift The drift of an instrument is its tendency to provide an output reading which gradually changes over time without any relationship with the measurand Drift can be caused by lack of stability, by instrument ageing or by the action of influencing quantities, e.g the thermal drift related to temperature fluctuations Temperature influence Temperature influence is the effect in which the instrument output for a constant applied value of the measurand changes when the environmental temperature changes It is usually expressed as a temperature coefficient defined as the output variation, absolute or fractional, for a unitary variation of the temperature Bandwidth The bandwidth of a measuring instrument is the range of frequencies of the input measurand for which the magnitude of the instrument frequency response function is no lower than dB of its peak value A decrease of dB is equivalent to an attenuation of Response time The response time is the length of time required for the instrument output to reach a specified percentage of its final value (typically 95% or 98%) when subject to a step change in the measurand Time constant The time constant is the length of time required for the instrument output to reach the 63.2% of the final value when subject to a step change in the measurand For a first-order instrument the time constant coincides with the reciprocal of the transfer function –3 dB cutoff angular frequency ω c Accuracy The accuracy of an instrument is the extent to which its output may differ from the true value of the measurand, with the significance given to the term ‘true value’ discussed in Section 13.2 Therefore, high accuracy means Copyright © 2003 Taylor & Francis Group LLC low measurement uncertainty Accuracy may be distinguished as static and dynamic The static accuracy, or better the inaccuracy, is determined by such effects as nonlinearity, nonrepeatability, hysteresis, drift, instability and environmental influences The dynamic accuracy is influenced also by the effects of bandwidth limitation and finite response time Accuracy may be expressed by quoting the associated uncertainty (for a given confidence level) as an absolute interval, as a percentage of the reading (%rdg) or as a percentage of the full-scale output (%FSO or %FS) In the first case it has the dimensions of the measurand, while in the remaining two it is a dimensionless number Precision The precision of an instrument has been long identified with its ability to provide the same output reading when the same measurand value is applied in the same conditions A precise instrument therefore ensures a low scattering of the readings The term repeatability is currently preferred to define this property, and the use of the term precision is discouraged by the international metrological organizations The term precision is definitely not synonymous with accuracy, and should not be confused with it 13.9 Summary Measurement is the experimental procedure to assign numerical values to real-world quantities called measurands, in order to describe them quantitatively To this purpose, measuring instruments are used To allow comparison between measurement results of the same quantity obtained by different instruments they need to be calibrated against a reference which is traceable to a conventionally agreed standard The extent to which the result of a measurement approximates the value of the measurand has traditionally been evaluated with reference to the concepts of random and systematic errors This approach has the problem that errors are defined with respect to the true value of the measurand which, however, is an unknown and unknowable quantity The inconsistency is removed by introducing the concept of measurement uncertainty, which represents a quantitative estimate of the range of values within which the value of the measurand lies with a given confidence level when using a particular instrument in specified conditions The lower the measurement uncertainty, the higher the accuracy Many traditional measuring instruments are nonelectronic, but most of the measurement tasks are today performed by electronic measuring instruments and systems Compared to purely mechanical systems, they offer higher performance, improved functionality and reduced cost Irrespective of its construction, the measuring instrument occupies the position at the interface between the observer and the measurand and is Copyright © 2003 Taylor & Francis Group LLC under the influence of the environment This influence generally produces a perturbing action due to several influencing quantities which need to be recognized and kept under control to avoid a detrimental effect on the measurement accuracy The behaviour of a measuring instrument can be analysed both statically and dynamically, depending on considering the measurand as a constant or as a function of time For linear instruments, the superposition principle holds and the dynamic behaviour can be analysed making use of the Fourierand Laplace-transform methods During the measurement process, the measuring instrument interacts with the measured system and unavoidably perturbs it by altering its energetic equilibrium As a consequence, the measurement result necessarily differs from the unperturbed measurand value This is called the loading effect caused by the measuring instrument on the measured system To analyse the loading effect, linear instruments can be represented as two-port devices with a proper choice of the effort, or across, and flow, or through, variables When mechanical systems are involved which lend themselves to a lumped-element representation, it may be convenient to make use of the electromechanical analogy to analyse the interaction between instrument and system with the formalism of electrical circuits This method allows us to determine which parameters are responsible for the loading effect and what can be done to reduce it to a minimum To properly choose and operate measuring instruments, and to effectively exchange information with other people on the subject of measurement, it is important to know and make use of the correct terminology Striving to so as a habit helps to avoid incompleteness and misinterpretations, and provides a deeper insight into the fundamental concepts related to measurement and measuring instruments References Finkelstein, L, and Watts, R.D., Mathematical models of instruments— fundamental principles, in B.E.Jones (ed.), Instrument Science and Technology, Adam Hilger, Bristol, 1982, Ch 2 Guide to the Expression of Uncertainty in Measurement, International Organization for Standardization (ISO), Geneva, 1993 Calibration: Philosophy in Practice, 2nd edn, Fluke Corporation, Everett, WA, 1994 Dettmann, J.W., Mathematical Methods in Physics and Engineering, Dover Publications, Mineola, NY, 1988 Oppenheim, A.V., Willsky, A.S and Young, I.T., Signals and Systems, Prentice Hall, Englewood Cliffs, NJ, 1983 Dobelin, E.O., Measurement Systems Application and Design, McGraw-Hill, New York, 1975 Raven, F.H., Automatic Control Engineering, McGraw-Hill, New York, 1968 Copyright © 2003 Taylor & Francis Group LLC D’Azzo, J.J and Houpis, C.H., Linear Control System Analysis and Design, McGraw-Hill, New York, 1975 International Vocabulary of Basic and General Terms in Metrology (VIM), International Standardization Organization (ISO), Geneva, 1993 Copyright © 2003 Taylor & Francis Group LLC ... (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... They are the zeroth-, first- and second-order instrument models For a zeroth-order instrument the input-output relationship in the time domain is given by (13.11) which in the s-domain becomes (13.12)... that the two-port is linear and time-invariant, and indicate with and Fo(s) the -transforms of and fo In place of the -transforms, the transforms could be used as well, but since the -transform