Applied Structural and Mechanical Vibrations 2009 Part 14 ppsx

resistive elements that are cemented on a structure of which they measure the local deformation through the variation in resistance caused by elongation or contraction. Strain gauges function on a principle based on the expression which gives the resistance of a uniform conductor of resistivity ρ , length L and cross-section area A. The fractional change in resistance is then given by (14.1) where v is Poisson’s ratio of the conductor material, and is the strain. According to eq (14.1), strain gauges do not actually measure displacement but strain, i.e. the average gauge elongation or contraction divided by the gauge length. The parameter K is called the gauge factor, which accounts for the resistance variations due to dimensional changes, represented by the term (1+2v), and for those caused by the strain-induced resistivity variations This latter effect is called the piezoresistive effect. Depending on the material of which the strain gauge is made, the gauge factor assumes different values, ranging from close to 2 for nickel-copper (constantan) and 2.1 nickel-chromium (karma) alloys, to about 3.5 for isoelastic, to above 100 for semiconductors. Metal alloy strain gauges are the most widely used and, as shown in Fig. 14.2, they typically have the form of grid foils of various dimensions and geometry supported by an insulating backing carrier which allows them to be bonded to the body under test. The backing carrier performs the fundamental function of transferring the strain from the specimen to the gauge with maximum fidelity. The nominal values of resistance are normally 120, 350, 700 or 1000 Ω, with strain-induced variations that are usually quite small, as low as few parts per million (ppm), and therefore require special care in their measurement. Moreover, the temperature appreciably influences both the gauge resistance and the gauge factor, producing the so-called thermal output, which is due to the temperature coefficient of resistance (TCR) and of gauge factor (TCGF) combined with the thermal expansion of the specimen. A typical solution is given by the use of the Wheatstone bridge configuration with voltage or current excitation of either DC or AC type (Chapter 15 ). Special arrangements including multiple active and/or dummy gauges are used to maximize linearity and compensate for the thermal effects, and proper wiring techniques allow for lead wire resistance cancellation in distant connections between the bridge and the excitation and amplification circuitry. With properly employed good-quality signal conditioning circuits, strain levels in the microstrain range can be ordinarily detected and values lower than 1 µ ε are possible. Such figures enable the use of strain gauges for stress Copyright © 2003 Taylor & Francis Group LLC metal foils bonded to the transducer structure, but may be of different materials and construction. For instance, they may be conductors deposited in thin- or thick-film technology, or integrated semiconductors, such as in silicon micromachined sensors. 14.4.3 Linear variable differential transformers The linear variable differential transformer (LVDT) is a transducer based on the magnetic induction principle. It is made by a transformer with one primary coil and two secondary coils with a movable core of ferromagnetic material which is placed coaxially in the coils without touching them, as shown in Fig. 14.3. The core terminates in a shaft or plunger which is attached to the target object either by threading or spring loading. The transducer body containing the primary and secondary coils is mounted in the reference position. As the core moves, it produces a variation of the magnetic coupling depending on its position along the coil axis. When the primary coil is excited with a sinusoidal voltage of amplitude V E an induced voltage V o is collected across the secondary coils, which is linearly related to the core position through the mutual inductance coefficient M. Since the secondary coils are connected in series opposition, M equals zero when the core is centered and it changes sign according to the sign of the core off-centre position. For the rotary variable differential transformer (RVDT) the operating principle is the same, with the difference that the rotary core movement allows the measurement of angular rather than linear displacement. A measure of the core displacement can then be obtained by rectifying the voltage V o while taking into account of its phase relative to V E . This readout operation is typically carried out by dedicated electronic circuitry, and is usually obtained by employing an oscillator and a phase-sensitive demodulation stage followed by low-pass filtering (Chapter 15). Fig. 14.3 Schematic diagram of linear variable differential transformer (LVDT). Copyright © 2003 Taylor & Francis Group LLC The oscillator signal for the primary coil excitation is preferably sinusoidal to avoid the generation of harmonics, and often a particular frequency is recommended by the transducer manufacturer for obtaining maximum compensation of residual phase shift at null core position. In the case of dynamic measurements, the oscillator frequency should be set at a value up to ten times higher than the highest motion frequency, to avoid frequency overlapping. Since the maximum recommended excitation frequencies of typical LVDTs are in the order of 20 kHz, the useful measurement bandwidth is generally limited to several kilohertz. The mechanical input impedance of LVDTs is mainly given by the mass associated with the core inertia in threaded-core types, with an added spring effect when spring-loaded plunger-type cores are used. Friction is generally almost absent, since there is no contact during the core motion within the transducer. This fact offers an ideally unlimited resolution and virtually no hysteresis, which represent fundamental advantages of LVDTs compared, for instance, with resistive potentiometers. Practical devices can reach resolutions better than 0.01% of the range, therefore submicron displacements may be appreciated with transducers with a stroke of few millimetres. These features, joined to a typically rugged construction and good immunity to environmental factors and electromagnetic interference, makes LVDTs the first choice for transducers in many precision measurement applications. They are often considered the electrical equivalent of the mechanical dial gauge, or micrometer. As a drawback, LVDTs are not generally cheap and require dedicated signal conditioning electronics, which is comparatively costly. In this respect, some devices which include the excitation and amplification electronics within the transducer case are particularly advantageous, providing a DC voltage output signal ready to be acquired. As a precaution, nonmagnetic materials such as aluminium or plastic should be used for the mounting fixture in order not to alter the sensitivity. Similar in shape to the LVDTs is a type of variable-inductance transducer which includes only two series-connected coils in an autotransformer configuration, i.e. analogous to the secondary windings of the LVDTs with the primary absent. When the core is halfway between the coils their respective inductances are equal, while they become different according to the amount and sign of the core off-centre displacement. The transducer then essentially works as an inductive potentiometer whose cursor is represented by the core, with the advantage of no internal electrical contact. The inductance imbalance can be measured by connecting the transducer in an AC-excited bridge and reading the correspondent bridge output. 14.4.4 Inductive transducers The functioning principle is based on the variable inductance of a coil wound on a core caused by the changes in the magnetic flux reluctance when the Copyright © 2003 Taylor & Francis Group LLC distance from a ferromagnetic target varies. If the measured object is ferrous it can act as the target, otherwise a ferromagnetic target must be attached to the object. The inductance changes are usually measured in an AC bridge circuit, or by making it part of a resonant circuit and detecting the resonant frequency shift. Two geometries may be used, namely the closed-loop and open-loop magnetic system, as shown in Fig. 14.4. In both cases, provided that the magnetic permeability of both the core and the target are much greater than that of air practically equal to that of vacuum (hence the requirement for a ferromagnetic target) the coil inductance L may be approximated by (14.2) where A is the area of the core facing the target, N is the number of coil turns and d is the distance from the target. It can be observed that the relationship between L and d is nonlinear, therefore these transducers are best suited for use as proximity sensors rather than distance measuring devices. To obtain linear operation, electronic linearization circuitry is generally added, often within the sensor housing, and typical residual linearity errors are in the order of ±1% FS. This type of transducer is inherently sensitive to stray magnetic fields, and to ferromagnetic materials in proximity of the sensing coil (especially in the open-loop magnetic geometry), therefore attention should be paid to this aspect in its positioning and mounting. 14.4.5 Eddy-current transducers When a coil is driven by an alternating voltage it generates an electromagnetic field. If an electroconductive object is placed in proximity of the coil, the Fig. 14.4 Schematic diagrams of inductive displacement transducers based on variable reluctances: (a) closed magnetic loop design, (b) open magentic loop design. Copyright © 2003 Taylor & Francis Group LLC electromagnetic field induces eddy currents (the name comes from the circular nature of their flow within the object). Such eddy currents in turn produce an electromagnetic field which opposes to the original field, and has the ultimate effect of changing both the inductance and the quality factor, i.e. the losses, of the coil. Therefore, from the measurement of the coil impedance, the distance from the object can be derived. The principle is generally applied by making use of a two-coil arrangement, which includes a driving coil and a sensing coil both oriented with their axis perpendicular to the target, as shown in Fig. 14.5. Both shielded and unshielded constructions exist, which differ in that the former provides a more directional field that ensures a higher immunity from stray effects caused by metallic objects near the sides of the transducer. When two or more transducers need to be mounted in close proximity, the shielded construction is preferred to minimize mutual interference. Eddy-current inductive sensors are responsive to both the magnetic permeability µ and the electrical conductivity s of the target material but, as opposed to the variable-reluctance type, they do not require that the magnetic permeability is high. Therefore, they operate properly even with non- ferromagnetic yet conductive target materials, or at temperatures higher than the Curie temperature of ferromagnetic materials. As a drawback, different target materials give different sensitivities and, therefore, require different calibrations. Moreover, the thickness of the target is also influential on the sensitivity, since the eddy currents have a finite penetration depth δ in the material which depends on its µ and s and on the field frequency f through the relationship (14.3) Fig. 14.5 Inductive displacement transducer based on eddy currents: (a) schematic diagram of the two-coil configuration; (b) shielded design; (c) unshielded design ([4, p. 279], reproduced with permission). Copyright © 2003 Taylor & Francis Group LLC On the basis of this formula it is possible to use eddy-current probes to measure the thickness of metal foils or coatings and to detect material cracks. Eddy-current sensors are almost always provided with built-in or external signal-conditioning electronic circuits which drive the coil, amplify the signal and linearize it to a typical value of ±1% FS. A resolution as high as ±0.1% FS can be obtained. Typically, with small- size short-range devices with 5 mm of probe diameter and 0–1 mm measuring distance, submicron resolutions can be achieved. The time and temperature stability can be very high making eddy-current sensors very suitable for long- term operation even in harsh and dirty environments. A frequency response typically ranging from zero to several kilohertz or a few tens of kilohertz and the absence of mechanical loading because of their noncontact operation make eddy-current sensors ideal for the measurement of vibration. For instance, they are well suited and widely used for measuring the vibrations and the eccentricity of rotating shafts, or the looseness of bearings. As far as the mounting is concerned, attention should be paid to ensuring that the lateral dimensions of the target are at least two to three times the probe diameter and the target surface is as flat as possible. Especially for the unshielded version, the side-mounting of more transducers or the use of metallic fixtures may perturb the sensitivity and, therefore, the recommendations of the manufacturer should be followed to keep distances at safe values. 14.4.6 Capacitive transducers Capacitive transducers are based on the principle that the capacitance of two electrical conductive bodies (armatures) separated by a dielectric medium varies if either the dielectric constant of the medium or the system geometry vary. The change in the dielectric properties of the separating medium is exploited, for instance, in liquid level or air humidity sensors. The change of geometry is well suited to use in dimensional measurements, such as linear and rotational displacement sensing. For instance, the principle can be applied in devices where an armature terminating in a shaft is guided to move between fixed armatures in a cylindrical geometry, giving rise to a capacitive linear potentiometer. The capacitive effect is well suited to noncontact displacement measurements according to the expression of the capacitance C of two parallel plates of area A separated by an air gap d given by (14.4) where ε is the dielectric constant of air practically equal to that of vacuum Copyright © 2003 Taylor & Francis Group LLC With reference to the above formula, two alternative methods may be used, as illustrated in Fig. 14.6, namely the variation of the distance d between the armatures and the variation of the area of overlap A. The noncontact capacitive probes are based on the former principle, i.e. they use capacitance variations to measure the air-gap d between two parallel conductive plates, of which one is the fixed reference and the second is attached to the moving object. If the object material is conductive it may act as the electrode, otherwise it may be equipped with a metal target or made conductive with the aid of conductive paint or rubbed graphite. As opposed to the eddy-current probes, the conductivity value of the target is not influential on the sensitivity. Capacitive transducers for nonconductive targets are also on the market, but their sensitivity typically depends on the target material and are mostly suited to proximity detection. Equation (14.4) shows that the capacitance between two conductor plates varies nonlinearly with the plate spacing d. The problem may be partially overcome by operating the transducer over a reduced portion of its usable range to approach linearity. A better and elegant solution is given by employing an electronic readout scheme which provides an output signal proportional to the modulus of the transducer impedance |Z|. Since the output signal is proportional to d at a fixed frequency ω . A further limitation to linearity comes from the fringing effect caused by the electric field lines diverging from parallel at the border of the plates due to their finite extension. As shown in Fig. 14.7, this problem may be solved with the help of the so-called guard electrode which encircles the moving armature and, by an electronic active driving circuitry, is kept at its same potential without, however, establishing any physical short-circuit between the two. In this way, the fringing effect is moved to the external border of the guard electrode, while the inner field lines in the region facing the sensitive electrode are steered to be perfectly parallel. Fig. 14.6 Variable-distance and variable-area methods for measuring displacement along the direction x making use of a parallel-plate capacitive transducer. Copyright © 2003 Taylor & Francis Group LLC electromagnetic interference. On the other hand, they are typically sensitive to the optical properties of the target object, such as colour, reflectivity, surface roughness, presence of dirt or dust, and of the optical path between the sensor and the object. Therefore, their use is generally confined to clean environments. A very simple principle makes use of a light source, such as a light-emitting diode (LED), coupled to a light detector in a side-by-side arrangement contained in a single unit which is positioned in front of the target object. The amount of reflected light collected by the photosensor depends on the target distance. The method is simple and cheap but gives a limited range of linearity, and suffers from a significant dependence on the optical properties of the target. Transducers based on the triangulation method employ the configuration depicted in Fig. 14.8. The light beam emitted by a visible or infrared light source, such as an LED or a semiconductor laser diode, is reflected by the target object and reaches a linear position-sensitive detector (PSD) at a particular point x of its length. Such a point is related to the target distance by trigonometric relationships, therefore the properly processed signal from the PSD gives a measure of the distance. Triangulation transducers typically have a working range of few millimetres around a stand-off distance that can be as high as several centimetres. The resolution is in the micron range with a frequency response no wider than few hundred hertz which is generally inversely dependent on the resolution. The method with the highest performance and cost is that based on the laser interferometer. A Michelson configuration is generally adopted in which the laser light beam is split into two beams which travel along different paths. One path has a fixed length and works as the reference, while the other one comprises the distance from the light source to the measurand object usually equipped with a mirrored reflecting target. The two beams recombine in a photodetector and, due to the high coherence of the laser light, produce a neat interference pattern whose number of fringes can be counted and related to the target distance. The achievable resolution can be as high as 1 nm, and the frequency response extends from DC to tens of kilohertz or more. Such performance makes the laser interferometer the Fig. 14.8 The optical triangular method for noncontact displacement measurement. Copyright © 2003 Taylor & Francis Group LLC preferred instrument where high-quality displacement and vibration measurements are needed, such as for laboratory calibration purposes or for highly demanding applications. 14.5 Relative-velocity measurement 14.5.1 Differentiation of displacement In principle, the output signal from any displacement sensor can be differentiated with respect to time to obtain a velocity signal. This may be done either electronically, by making use of differentiation circuits cascaded to the transducer output, or as a postprocessing step on the recorded data. This indirect approach, however, has possible problems with the fact that the process of differentiation inherently enhances the high-frequency components in a signal, since the amplitude of each sinusoidal component at a frequency ω results multiplied by a factor ω . Therefore, any spurious high-frequency component in the displacement signal is amplified to a level which may impair the detectability of the true velocity signal. The situation may be critical, for instance, with wire wound potentiometers, due to their staircase characteristic which causes stepping output signals under rapid cursor movement, as well as with AC excited transducers, such as the LVDTs, due to possible residual ripple in the output signal. Although the differentiating method may prove satisfying in several noncritical situations, the choice which provides a more general applicability and is therefore often preferred is to make use of velocity measuring transducers. 14.5.2 Electrodynamic transducers The operating principle is based on the Faraday-Lenz law of magnetic induction for which the electromotive force (EMF), i.e. the voltage E, generated in a closed circuit is equal to the time derivative of the magnetic flux Φ linked with such a circuit. That is, with the minus sign representing the fact that the magnetic flux generated by the induced current caused by E opposes to the original flux Φ. On this principle, it is possible to develop self-generating sensors where velocity is converted into variations of the magnetic flux concatenated with a coil, and therefore produces a proportional output voltage signal. A simple and effective method is that of making use of a permanent bar magnet positioned inside a coil and free to move relative to it. There are two alternatives, called the moving-coil and the moving-magnet designs, differing in that the former has the element fixed to the reference point, while the latter has it attached to the moving object. The moving-magnet geometry is widely used for linear velocity transducers which generally base on the two-coil configuration shown in Fig. 14.9. The Copyright © 2003 Taylor & Francis Group LLC [...]... coefficient and geometry, and in the region of interest can be considered independent of frequency When charge output is considered, the force sensitivity SQf is generally expressed in picocoulombs per newton (pC/N) and in the Laplace domain is given by (14. 20) where the mechanical transfer function has the expressions given in eqs (14. 17), (14. 18) and (14. 19) respectively for the rigidly backed, hammer and. .. the values of the motion displacement and frequency This method offers very high accuracy and significant flexibility in varying the level and frequency of the test vibration On the other hand, the cost is high and the typical use is confined to the laboratory calibration of primary standard transducers, or of transfer standard transducers of the highest quality and stability Though sensitivity is normally... integrated circuits [9] Such devices are factory calibrated and temperature compensated, and due to their low cost and size can be used for instance in permanent monitoring systems for machinery and structures [10] 14. 9 Accelerometer choice, calibration and mounting 14. 9.1 Choice As can be expected, none of the above illustrated principles and technologies for accelerometers is generally the best choice... thermally and over time, and it offers good repeatability It represents, therefore, the best solution for transducers used for continuous monitoring over prolonged durations at temperatures between –190 and 240°C and for calibration reference standards It has a piezoelectric coefficient d of 2.3 pC/N, which is a rather low value and, as such, the charge sensitivity SQa is not very high On the other hand,... most widely used transducer for the measurement of vibrations, and they are well suited to the measurement of both continuous vibrations and transients, or shocks Their functioning principle requires a high resonant frequency to obtain a wide measuring bandwidth, therefore they tend to have high stiffness and small mass This makes them typically rugged and small size devices, especially in the highfrequency... providing displacement and velocity signals by means of time integration, which is a technique intrinsically insensitive to noise and high-frequency disturbances due to its averaging nature 14. 8 Accelerometer types and technologies 14. 8.1 Piezoelectric Piezoelectric elements are insulators which become electrically polarized when subject to mechanical stress (direct piezoelectric effect), and conversely Copyright... manufacturer and called the rigidly mounted resonant frequency For the hammer mounting configuration, m1 is the mass of the hammer impact head and tip, while m2 is that of the hammer body including optional extenders At side 2 it is present only the inertia of m2 and no external force is applied, hence F2 is zero Therefore, the equivalent circuit is that shown in Fig 14. 17(b), and it follows that (14. 18)... resulting in a narrow region of resonance and in a steep phase change Usually, the flat-band region is individuated by the upper and lower limiting frequencies where the voltage sensitivity is within 5% of its midband value given by When proper mounting is adopted, operation up to f0/3 and f0/5 typically ensures a deviation from the midband sensitivity of 12% (1 dB) and 6% (0.5 dB) respectively Piezoelectric... accelerometers compliant with the IEEE 145 1 standard, which have onboard memory to store calibration and identification parameters in a transducer electronic data sheet (TEDS) Silicon micromachined sensors offer cost reduction as one of the major benefits, and for this reason they are increasingly installed permanently on machinery for routine monitoring and online fault diagnosis 14. 9.2 Calibration The calibration... electronic circuitry as well as sensing and actuating capabilities on the same chip By making use of photomasking and chemical etching, silicon micromachining allows for the fabrication of three-dimensional suspended structures, such as bridges and diaphragms, with typical dimensions of few micrometres Silicon has excellent elastic behaviour and mechanical properties and, by adding dopant impurities, it . length L and cross-section area A. The fractional change in resistance is then given by (14. 1) where v is Poisson’s ratio of the conductor material, and is the strain. According to eq (14. 1),. making it part of a resonant circuit and detecting the resonant frequency shift. Two geometries may be used, namely the closed-loop and open-loop magnetic system, as shown in Fig. 14. 4. In both. finite penetration depth δ in the material which depends on its µ and s and on the field frequency f through the relationship (14. 3) Fig. 14. 5 Inductive displacement transducer based on eddy currents: