Vibrations Fundamentals and Practice ch09

Vibrations Fundamentals and Practice ch09 Maintaining the outstanding features and practical approach that led the bestselling first edition to become a standard textbook in engineering classrooms worldwide, Clarence de Silva''s Vibration: Fundamentals and Practice, Second Edition remains a solid instructional tool for modeling, analyzing, simulating, measuring, monitoring, testing, controlling, and designing for vibration in engineering systems. It condenses the author''s distinguished and extensive experience into an easy-to-use, highly practical text that prepares students for real problems in a variety of engineering fields.

de Silva, Clarence W “Signal Conditioning and Modification” Vibration: Fundamentals and Practice Clarence W de Silva Boca Raton: CRC Press LLC, 2000 Signal Conditioning and Modification Signal modification is an important function in many applications of vibration The tasks of signal modification can include signal conditioning (e.g., amplification, and analog and digital filtering), signal conversion (e.g., analog-to-digital conversion, digital-to-analog conversion, voltage-to-frequency conversion, and frequency-to-voltage conversion), modulation (e.g., amplitude modulation, frequency modulation, phase modulation, pulse-width modulation, pulse-frequency modulation, and pulse-code modulation), and demodulation (the reverse process of modulation) In addition, many other types of useful signal modification operations can be identified For example, sampleand-hold circuits are used in digital data acquisition systems Devices such as analog and digital multiplexers and comparators are needed in many applications of data acquisition and processing Phase shifting, curve shaping, offsetting, and linearization can also be classified as signal modification This chapter describes signal conditioning and modification operations that are useful in vibration applications Signal modification plays a crucial role in component interfacing When two devices are interfaced, it is essential to guarantee that a signal leaving one device and entering the other will so at proper signal levels (voltage, current, power), in the proper form (analog, digital), and without distortion (loading and impedance considerations) For transmission, a signal should be properly modified (by amplification, modulation, digitizing, etc.) so that the signal-tonoise ratio of the transmitted signal is sufficiently large at the receiver The significance of signal modification is clear from these observations The material covered in this chapter is intimately related to what has been discussed in chapters on signal analysis and instrumentation (see Chapters and 8) 9.1 AMPLIFIERS The level of an electrical signal can be represented by variables such as voltage, current, and power Across variables, through variables, and power variables that are analogous can be defined for other types of signals (e.g., mechanical) as well Signal levels at various interface locations of components in a vibratory system must be properly adjusted for correct performance of these components and the overall system For example, input to an actuator should possess adequate power to drive the actuator A signal should maintain its signal level above some threshold during transmission so that errors due to signal weakening are not excessive Signals applied to digital devices must remain within the specified logic levels Many types of sensors produce weak signals that must be upgraded before they can be fed into a monitoring system, data processor, controller, or data logger Signal amplification concerns proper adjustment of the signal level for performing a specific task Amplifiers are used to accomplish signal amplification An amplifier is an active device that needs an external power source to operate Although active circuits — amplifiers in particular — can be developed in the monolithic form using an original integrated-circuit (IC) layout so as to accomplish a particular amplification task, it is convenient to study their performance using the operational amplifier (op-amp) as the basic building block Of course, operational amplifiers are widely used not only for modeling and analyzing other types of amplifiers, but also as basic building blocks in building these various kinds of amplifiers For these reasons, the present ©2000 CRC Press discussion on amplifiers will focus on the operational amplifier An introduction to this topic was presented in Chapter 9.1.1 OPERATIONAL AMPLIFIER The origin of the operational amplifier dates back to the 1940s when the vacuum tube operational amplifier was introduced Operational amplifier or op-amp got its name due to the fact that it was originally used almost exclusively to perform mathematical operations; for example, in analog computers Subsequently, in the 1950s, the transistorized op-amp was developed It used discrete elements such as bipolar junction transistors and resistors Still, it was too large in size, consumed too much power, and was too expensive for widespread use in general applications This situation changed in the late 1960s when the integrated-circuit (IC) op-amp was developed in the monolithic form, as a single IC chip Today, the IC op-amp, which consists of a large number of circuit elements on a substrate of typically a single silicon crystal (the monolithic form), is a valuable component in almost any signal modification device An op-amp can be manufactured in the discrete-element form using, say, ten bipolar junction transistors and as many discrete resistors or alternatively (and preferably) in the modern monolithic form as an IC chip that may be equivalent to over 100 discrete elements In any form, the device has an input impedance Zi, an output impedance Zo, and a gain K Hence, a schematic model for an op-amp can be given as in Figure 9.1(a) The conventional symbol of an op-amp is shown in Figure 9.1(b) Typically, there are about six terminals (lead connections) to an op-amp For example, there are two input leads (a positive lead with voltage vip and a negative load with voltage vin), an output lead (voltage vo), two bipolar power supply leads (+vs and –vs), and a ground lead Note from Figure 9.1(a) that under open-loop (no feedback) conditions, vo = Kvi (9.1) in which the input voltage vi is the differential input voltage defined as the algebraic difference between the voltages at the positive and negative lead; thus, vi = vip − vin (9.2) The open-loop voltage gain K is very high (105 to 109) for a typical op-amp Furthermore, the input impedance Zi could be as high as MΩ and the output impedance is low, on the order of 10 Ω Because vo is typically to 10 V, from equation (9.1), it follows that vi ≅ since K is very large Hence, from equation (9.2), vip ≅ vin In other words, the voltages at the two input leads are nearly equal Now, if one applies a large voltage differential vi (say, V) at the input, then according to equation (9.1), the output voltage should be extremely high This never happens in practice, however, because the device saturates quickly beyond moderate output voltages (of the order of 15 V) From equations (9.1) and (9.2), it is clear that if the negative input lead is grounded (i.e., vin = 0), then vo = Kvip (9.3) and if the positive input lead is grounded (i.e., vip = 0), then vo = − Kvin Accordingly, vip is termed noninverting input and vin is termed inverting input ©2000 CRC Press (9.4) FIGURE 9.1 Operational amplifier: (a) a schematic model, and (b) conventional symbol EXAMPLE 9.1 Consider an op-amp having an open loop gain of × 105 If the saturation voltage is 15 V, determine the output voltage in the following cases: (a) (b) (c) (d) (e) (f) µV at the positive lead and µV at the negative lead –5 µV at the positive lead and µV at the negative lead µV at the positive lead and –2 µV at the negative lead –5 µV at the positive lead and –2 µV at the negative lead V at the positive lead and negative lead grounded V at the negative lead and positive lead grounded SOLUTION This problem can be solved using equations (9.1) and (9.2) The results are given in Table 9.1 Note that in the last two cases, the output will saturate and equation (9.1) will no longer hold ©2000 CRC Press TABLE 9.1 Solution to Example 9.1 vip µV –5 µV µV –5 µV 1V vin vi vo µV µV µV µV 1V µV –7 µV µV –3 µV 1V –1 V 0.3 V –0.7 V 0.7 V –0.3 V 15 V –15 V 2 –2 –2 Ⅺ Field effect transistors (FETs) — for example, metal oxide semiconductor field effect transistors (MOSFETs) — could be used in the IC form of an op-amp The MOSFET type has advantages over many other types; for example, higher input impedance and more stable output (almost equal to the power supply voltage) at saturation, making the MOSFET op-amps preferable over bipolar junction transistor op-amps in many applications In analyzing operational amplifier circuits under unsaturated conditions, one can use the following two characteristics of an op-amp: Voltages of the two input leads should be (almost) equal Currents through each of the two input leads should be (almost) zero As explained earlier, the first property is credited to high open-loop gain, and the second property to high input impedance in an operational amplifier These two properties are repeatedly used to obtain input-output equations for amplifier circuits and systems 9.1.2 USE OF FEEDBACK IN OP-AMPS An operational amplifier is a very versatile device, primarily due to its very high input impedance, low output impedance, and very high gain However, it cannot be used without modification as an amplifier because it is not very stable in the form shown in Figure 9.1 Two factors that contribute to this problem are: Frequency response Drift Stated another way, op-amp gain K does not remain constant; it can vary with the frequency of the input signal (i.e., frequency response function is not flat in the operating range); and also, it can vary with time (i.e., drift) Frequency response problems arise due to circuit dynamics of an operational amplifier This problem is usually not severe unless the device is operated at very high frequencies Drift problems arise due to the sensitivity of gain K to environmental factors such as temperature, light, humidity, and vibration, and as a result of the variation of K due to aging Drift in an op-amp can be significant, and steps should be taken to remove this problem It is virtually impossible to avoid gain drift and frequency-response error in an operational amplifier But an ingenious way has been found to remove the effect of these two problems at the amplifier output Because gain K is very large, by using feedback, one can virtually eliminate its effect at the amplifier output This closed-loop form of an op-amp is preferred in almost every application In particular, voltage followers and charge amplifiers are devices that use the properties ©2000 CRC Press of high Zi, low Zo, and high K of the op-amp, along with feedback through a precision resistor, to eliminate errors due to non-constant K In summary, an operational amplifier is not very useful in its open-loop form, particularly because gain K is not steady But because K is very large, the problem can be removed using feedback It is this closed-loop form that is commonly used in practical applications of an op-amp In addition to the nonsteady nature of gain, there are other sources of error that contribute to the less-than-ideal performance of an operational amplifier circuit Noteworthy are: The offset current present at the input leads due to bias currents that are needed to operate the solid-state circuitry The offset voltage that might be present at the output even when the input leads are open The unequal gains corresponding to the two input leads (i.e., the inverting gain not equal to the noninverting gain) Such problems can produce nonlinear behavior in op-amp circuits, and they can be reduced by proper circuit design and through the use of compensating circuit elements 9.1.3 VOLTAGE, CURRENT, AND POWER AMPLIFIERS Any type of amplifier can be constructed from scratch in the monolithic form as an IC chip, or in the discrete form as a circuit containing several discrete elements such as discrete bipolar junction transistors or discrete field effect transistors, discrete diodes, and discrete resistors However, almost all types of amplifiers can also be built using operational amplifier as the basic building block Because one is already familiar with op-amps, and because op-amps are extensively used in general amplifier circuitry, the latter approach — which uses discrete op-amps for the modeling of general amplifiers — is preferred If an electronic amplifier performs a voltage amplification function, it is termed a voltage amplifier These amplifiers are so common that the term amplifier is often used to denote a voltage amplifier A voltage amplifier can be modeled as vo = K v vi (9.5) where vo = output voltage vi = input voltage Kv = voltage gain Voltage amplifiers are used to achieve voltage compatibility (or level shifting) in circuits Current amplifiers are used to achieve current compatibility in electronic circuits A current amplifier can be modeled by io = Ki ii where io = output current ii = input current Ki = current gain ©2000 CRC Press (9.6) Note that the voltage follower has Kv = and, hence, it can be considered as a current amplifier Also, it provides impedance compatibility and acts as a buffer between a low-current (high-impedance) output device (the device that provides the signal) and a high-current (low-impedance) input device (device that receives the signal) that are interconnected Hence, the name buffer amplifier or impedance transformer is sometimes used for a current amplifier with unity voltage gain If the objective of signal amplification is to upgrade the associated power level, then a power amplifier should be used for that purpose A simple model for a power amplifier is po = K p pi (9.7) where po = output power pi = input power Kp = power gain It is easy to see from equations (9.5) through (9.7) that K p = K v Ki (9.8) Note that all three types of amplification can be achieved simultaneously from the same amplifier Furthermore, a current amplifier with unity voltage gain (e.g., a voltage follower) is a power amplifier as well Usually, voltage amplifiers and current amplifiers are used in the first stages of a signal path (e.g., sensing, data acquisition, and signal generation) where signal levels and power levels are relatively low Power amplifiers are typically used in the final stages (e.g., actuation, recording, display) where high signal levels and power levels are usually required Figure 9.2(a) shows an op-amp-based voltage amplifier Note the feedback resistor Rf that serves the purposes of stabilizing the op-amp and providing an accurate voltage gain The negative lead is grounded through an accurately known resistor R To determine the voltage gain, recall that the voltages at the two input leads of an op-amp should be virtually equal The input voltage vi is applied to the positive lead of the op-amp Then the voltage at point A should also be equal to vi Next, recall that the current through the input lead of an op-amp is virtually Hence, by writing the current balance equation for the node point A, one obtains vo − vi vi = Rf R This gives the amplifier equation Rf   vo =  + v R i  (9.9) Hence, the voltage gain is given by Kv = + Rf R (9.10) Note the Kv depends on R and Rf, and not on the op-amp gain Hence, the voltage gain can be accurately determined by selecting the two resistors R and Rf precisely Also note that the output ©2000 CRC Press FIGURE 9.2 (a) A voltage amplifier, and (b) a current amplifier voltage has the same sign as the input voltage Hence, this is a noninverting amplifier If the voltages are of the opposite sign, it will be an inverting amplifier A current amplifier is shown in Figure 9.2(b) The input current ii is applied to the negative lead of the op-amp as shown, and the positive lead is grounded There is a feedback resistor Rf connected to the negative lead through the load RL The resistor Rf provides a path for the input current because the op-amp takes in virtually zero current There is a second resistor R through which the output is grounded This resistor is needed for current amplification To analyze the amplifier, note that the voltage at point A (i.e., at the negative lead) should be because the positive lead of the op-amp is grounded (zero voltage) Furthermore, the entire input current ii passes through resistor Rf as shown Hence, the voltage at point B is Rf ii Consequently, current through resistor R is Rf ii /R, which is positive in the direction shown It follows that the output current io is given by io = ii + or ©2000 CRC Press Rf R ii Rf   io = 1 + i Ri  (9.11) The current gain of the amplifier is Ki = + Rf R (9.12) This gain can be accurately set using high-precision resistors R and Rf 9.1.4 INSTRUMENTATION AMPLIFIERS An instrumentation amplifier is typically a special-purpose voltage amplifier dedicated to a particular instrumentation application Examples include amplifiers used for producing the output from a bridge circuit (bridge amplifier) and amplifiers used with various sensors and transducers An important characteristic of an instrumentation amplifier is the adjustable-gain capability The gain value can be adjusted manually in most instrumentation amplifiers In more sophisticated instrumentation amplifiers, gain is programmable and can be set by means of digital logic Instrumentation amplifiers are normally used with low-voltage signals Differential Amplifier Usually, an instrumentation amplifier is also a differential amplifier (sometimes termed a difference amplifier) Note that in a differential amplifier, both input leads are used for signal input, whereas in a single-ended amplifier, one of the leads is grounded and only one lead is used for signal input Ground-loop noise can be a serious problem in single-ended amplifiers Ground-loop noise can be effectively eliminated using a differential amplifier because noise loops are formed with both inputs of the amplifier and, hence, these noise signals are subtracted at the amplifier output Because the noise level is almost the same for both inputs, it is canceled Note that any other noise (e.g., 60-Hz line noise) that might enter both inputs with the same intensity will also be canceled out at the output of a differential amplifier A basic differential amplifier that uses a single op-amp is shown in Figure 9.3(a) The inputoutput equation for this amplifier can be obtained in the usual manner For example, because current through the op-amp is negligible, current balance at point B gives vi − v B v B = R Rf (i) in which vB is the voltage at B Similarly, current balance at point A gives vo − v A v A − vi1 = Rf R (ii) vA = vB (iii) Now we use the property for an operational amplifier, to eliminate vA and vB from equations (i) and (ii) This gives ©2000 CRC Press FIGURE 9.3 (a) A basic differential amplifier, and (b) a basic instrumentation amplifier (v R R + v ) (1 + R R ) (1 + R R ) vi = o i1 f f f or vo = Rf R (vi − vi1 ) (9.13) Two things are clear from equation (9.13) First, the amplifier output is proportional to the “difference” and not the absolute value of the two inputs vi1 and vi2 Second, the voltage gain of the amplifier is Rf /R This is known as the differential gain Note that the differential gain can be accurately set using high-precision resistors R and Rf The basic differential amplifier, shown in Figure 9.3(a) and discussed above, is an important component of an instrumentation amplifier In addition, an instrumentation amplifier should possess the adjustable gain capability Furthermore, it is desirable to have a very high input impedance and very low output impedance at each input lead An instrumentation amplifier that possesses these ©2000 CRC Press FIGURE 9.35 A peak-holding circuit to the op-amp is positive under these conditions, the op-amp output will be positive The output will charge the capacitor until the capacitor voltage v equals the input voltage vi This voltage (call it vo) is in turn supplied to the second voltage follower, which presents the same value to its output (gain = for a voltage follower), but at a very low impedance level Note that the op-amp output remains at the saturated value for only a very short time (the time taken by the capacitor to charge) Now suppose that vi is smaller than v Then, the differential input of the op-amp will be negative, and the op-amp output will be saturated at the negative saturation level This will reverse-bias the diode Hence, the output of the first op-amp will be in open-circuit and, as a result, the voltage supplied to the output voltage follower will still be the capacitor voltage and not the output of the first op-amp It follows that the voltage level of the capacitor (and hence the output of the second voltage follower) will always be the peak value of the input signal The circuit can be reset by discharging the capacitor through a solid-state switch that is activated by an external pulse 9.8 SIGNAL ANALYZERS AND DISPLAY DEVICES Vibration signal analysis can employ both analog and digital procedures (Chapter is devoted entirely to this topic.) Since signal analysis results in extracting various useful information from the signal, it is appropriate to consider the topic within the present context of signal modification as well This section introduces digital signal analyzers that essentially make use of the same techniques that were described previously (in Chapter and Appendix D) Signal display devices also make use of at least some signal processing This may involve filtering and change of signal level and format More sophisticated signal display devices, particularly digital oscilloscopes, can carry out more complex signal analysis functions such as those normally available with digital signal analyzers Oscilloscopes as well are introduced in the present section, although they can also be treated under vibration instrumentation (Chapter 8) Signal-recording equipment commonly employed in vibration practice includes digital storage devices such as hard drives, floppy disks, and CD-ROMs, analog devices like tape recorders, stripchart recorders, and X-Y plotters, and digital printers Tape recorders are used to record vibration data (transducer outputs) that are subsequently reproduced for processing or examination Often, tape-recorded waveforms are also used to generate (by replay) signals that drive vibration test exciters (shakers) Tape recorders use tapes made of a plastic material that has a thin coating of a specially treated ferromagnetic substance During the recording process, magnetic flux proportional to the recorded signal is produced by the recording head (essentially an electromagnet), which ©2000 CRC Press magnetizes the tape surface in proportion to the signal variation Reproduction is the reverse process, whereby an electrical signal is generated at the reproduction head by electromagnetic induction in accordance with the magnetic flux of the magnetized (recorded) tape Several signal-conditioning circuitries are involved in the recording and reproducing stages Recording by FM is very common in vibration testing Strip-chart recorders are usually employed to plot time histories (i.e., quantities that vary with time), although they can also be used to plot such data as frequency-response functions and response spectra In these recorders, a paper roll unwinds at a constant linear speed, and the writing head moves across the paper (perpendicular to the paper motion) proportionally to the signal level There are many kinds of strip-chart recorders, which are grouped according to the type of writing head employed Graphic-level recorders, which use ordinary paper, employ such heads as ink pens or brushes, fiber pens, and sapphire styli Visicoders are simple oscilloscopes capable of producing permanent records; they employ light-sensitive paper for this Several channels of input data can be incorporated with a visicoder Obviously, graphic-level recorders are generally limited by the number of writing heads available (typically, one or two), but visicoders can have many more input channels (typically 24) Performance specifications of these devices include paper speed, frequency range of operation, dynamic range, and power requirements In vibration experimentation, X-Y plotters are generally employed to plot frequency data (e.g., psd, frequency-response functions, response spectra, and transmissibility curves, as defined in Chapters and 4), although they can also be used to plot time-history data Many types of X-Y plotters are available, most of them using ink pens on ordinary paper There are also hard-copy units that use laser printing or heat-sensitive paper in conjunction with a heating element as the writing head The writing head in an X-Y plotter is moved in the X and Y directions on the paper by two input signals that form the coordinates for the plot In this manner, a trace is made on stationary plotting paper Performance specifications of X-Y plotters are governed by such factors as paper size; writing speed (inches per second, centimeters per second); deadband (expressed as a percentage of the full scale), which measures the resolution of the plotter head; linearity (expressed as a percentage of full scale), which measures the accuracy of the plot; minimum trace separation (inches, centimeters) for multiple plots on the same axes; dynamic range; input impedance; and maximum input (millivolts per inch, millivolts per centimeter) Today, the most widespread signal recording device is in fact the digital computer (memory, storage) and printer combination This, and also the other (analog) devices used in signal recording and display, make use of some signal modification to accomplish their functions These devices will not be discussed in the present section, however 9.8.1 SIGNAL ANALYZERS Modern signal analyzers employ digital techniques of signal analysis, as described in Chapter 4, to extract useful information that is carried by the signal Digital Fourier analysis using fast Fourier transform (FFT) is perhaps the single common procedure used in the vast majority of signal analyzers (see Appendix D) As noted before, Fourier analysis will produce the frequency spectrum of a time signal It should be clear, therefore, why the terms digital signal analyzer, FFT analyzer, frequency analyzer, spectrum analyzer, and digital Fourier analyzer are synonymous to some extent, as used in commercial instrumentation literature A signal analyzer typically has two (dual) or more (multiple) input signal channels To generate results such as frequency response (transfer) functions, cross-spectra, coherence functions, and cross-correlation functions, one needs at least two data signals and, hence, a dual-channel analyzer In hardware analyzers, digital circuitry rather than software is used to carry out the mathematical operations Clearly, they are very fast but less flexible (in terms of programmability and functional capability) for this reason Digital signal analyzers, regardless of whether they use the hardware ©2000 CRC Press or the software approach, employ some basic operations These operations, carried out in sequence, are: Anti-alias filtering (analog) Analog-to-digital conversion (i.e., signal sampling) Truncation of a block of data and multiplication by a window function FFT analysis of the block of data These operations were explained in Chapter Also noted are the following facts: If the sampling period of the analog-to-digital convertor (ADC) is ∆T (i.e., the sampling frequency is 1/∆T), then the Nyquist frequency fc = This Nyquist frequency is the upper limit of the useful frequency 2∆T content of the sampled signal The cutoff frequency of the anti-aliasing filter should be set at fc or less If there are N data samples in the block of data used in the FFT analysis, the corresponding record length is T = N∆T Then, the spectral lines in the FFT results are separated at a frequency spacing of ∆F = 1/ T In view of the Nyquist frequency limit, there will be only N/2 useful spectral lines in the FFT result Strictly speaking, a real-time signal analyzer should analyze a signal instantaneously and continuously as the signal is received by the analyzer This is usually the case with an analog signal analyzer But, in digital signal analyzers, which are usually based on digital Fourier analysis, a block of data (i.e., N samples of record length T) is analyzed together to produce N/2 useful spectral lines (at frequency spacing 1/T) This is, then, not a truly real-time analysis But for practical purposes, if the speed of analysis is sufficiently fast, the analyzer can be considered real-time, which is usually the case with hardware analyzers and also modern, high-speed, software analyzers The bandwidth B of a digital signal analyzer is a measure of its speed of signal processing Specifically, for an analyzer that uses N data samples in each block of signal analysis, the associated processing time can be given by Tc = N B (9.121) Note that the larger the B, the smaller the Tc Then, the analyzer is considered a real-time one if the analysis time (Tc) of the data record is less than the generation time (T = N∆T) of the data record Hence, one requires that Tc < T or N