Applied Structural and Mechanical Vibrations 2009 Part 16 pptx

such a case, if allowed, it is advisable either to change the rate of the signal repetitions or force them to happen at irregular instants, or to filter the interfering frequency if it falls out of the signal bandwidth. If the signal is not repetitive, it can in most cases be made repetitive by repeatedly exciting the phenomenon that causes it, such as when several impact responses from an accelerometer are obtained by subsequent hammer blows and are waveform-averaged together. As long as the noise can be considered uncorrelated, the time of occurrence of the blows and their time separation are unifluential and only their total number, i.e. the number of averages, determines the S/N ratio at the output. As a concluding observation, it should be realized that synchronous averaging is actually a bandwidth-narrowing technique. The measurement bandwidth increasingly reduces at a rate proportional to the integration time Ti nt or, equivalently, to the number of averages m and shrinks on the signal while leaving out the noise. As opposed to filtering in the frequency domain, the signal harmonic components need not be adjacent for the method to be effective but may as well be separated. The averaging process is able to detect them and enhance their amplitude over wideband noise by selectively concentrating the energy accumulated from signal repetitions. 15.6 Analogue-to-digital conversion Analogue signals take their name from the fact that their time behaviour is analogous to, i.e. an exact replica of, that of the real-world quantity that they represent. Analogue signals, therefore, are continuous functions of time and can assume an infinite number of values within a range. Conversely, digital signals, also called numerical signals, have defined values only at discrete time instants and can assume only a finite number of stepping values within a range. For instance, if we measure the temperature of a room with an electronic thermometer and continuously plot the results on a strip-chart recorder, we obtain an example of an analogue signal or, better stated, an analogue representation of the temperature as a function of time. In contrast, if we decide to take the measurement once every hour and to round-off the readings to a resolution of say 1°C, we obtain a sequence of data pairs, i.e. the time of measurement and the corresponding reading, which represent the temperature over time in a digital form. It is important to notice that, as the above example illustrates, the digital representation of a signal implies both a time discretization, called sampling, and an amplitude value round-off, called quantization. Sampling and quantizing are fundamental steps in passing from analogue to digital signals, i.e. in performing an analogue-to-digital conversion. It is important to note that a time-discrete signal is not necessarily a digital signal unless amplitude quantization also occurs. On the other hand, it is essentially impossible to quantize a signal without acquiring its value for a time interval, however short. Therefore, practical amplitude quantization Copyright © 2003 Taylor & Francis Group LLC involves time sampling. In digital signals the time sampling most invariably occurs at regularly spaced instants; the interval between successive samples is called the sampling interval and the number of samples per unit time is called the sample rate. The quantized values of digital signals are usually coded into binary format, that is they are expressed as numbers in base 2. The binary numeration system makes use of two symbols, 0 and 1, which are called bits from the contraction of ‘BInary digiT’ The choice of the base-2 coding is motivated by the fact that it is particularly easy and convenient to obtain electronic circuits which use two voltage or current levels to represent the equivalent of binary digits 1 and 0. In contrast, it would be rather difficult and inefficient to obtain ten different voltage or current levels necessary to implement the decimal coding. The numeration in base 10 is well suited to humans, but rather problematic for adoption by machines. Binary-coded signals obey the formal rules of the Boolean logic, which is based on the two states ‘false’ and ‘true’ that can be made to correspond to binary levels 0 and 1. For this reason, binary-coded digital systems are usually called logic systems. Most often digital signals are represented by voltage levels. When the absence of signal, i.e. a voltage level low, is coded as 0 and the signal presence, i.e. a voltage level high, is coded as 1 the logic is said to be positive. When the inverse correspondence applies, the logic is called negative. Most of today’s electronic instrumentation makes an extensive use of digital circuitry and processing techniques to manipulate signals. In fact, due to the advent and widespread diffusion of microprocessors, microcontrollers, and digital signal processors (DSP) this is accomplished in an efficient an convenient way. Moreover, thanks to the availability of memory circuits and devices, digital signals are more easily stored and retrieved without degradation. On the other hand, the real world variables are typically analogue in nature. Therefore, the need is always present for devices capable of converting signals from the analogue to the digital domain and vice versa. They are respectively called analogue-to-digital converters (ADC), and digital-to- analogue converters (DAC). In the present section we will mainly concentrate on ADCs, even if many presented concepts apply equally well to DACs. 15.6.1 Quantization: resolution, number of bits, conversion time We will consider the input of an ADC as an analogue voltage signal v i (t) which, for sake of simplicity, is supposed to be always greater than zero, i.e. unipolar and positive. Each ADC has an input range represented by the fullscale value V FS which specifies the maximum input level acceptable for conversion. When the ADC output code is the maximum possible. The number of intervals into which V FS is divided is 2 n , where n is the number of bits used to represent the output in digital format. The AD conversion is performed by assigning the value of the input signal amplitude to the Copyright © 2003 Taylor & Francis Group LLC corresponding interval identified by a digital code. The resolution of an ADC is the smallest increment in the input v i which causes the output code to change by a unitary step. For example, an 8-bit ADC divides V FS into 2 8 =256 intervals numbered as 00000000, 00000001,…, 11111111. Thus the resolution of an 8-bit ADC is one part in 256, equivalent to 0.39% of V FS , which for V FS =10 V corresponds to 39 mV. A set of eight bits grouped to represent a single number is called a byte. The rightmost and leftmost bits are respectively called the least-significant bit (LSB) and the most-significant bit (MSB). Therefore, resolution and number of bits are equivalent terms defining the same concept, i.e. the width of the discretization interval referred to the full scale. Related to the resolution is the dynamic range, usually expressed in dB as 20 log 10 2n where n is the number of bits. This leads to approximately 6 dB per bit, hence an 8-bit ADC has a dynamic range of 48 dB. Depending on the conversion method and the technology, ADCs with markedly different resolutions are available. For general purpose applications 12–14 bits are typical, and for slowly varying signals 16 bits are achievable with 20 bits and beyond encountered in top-end instrumentation. For illustration purposes, the ideal conversion characteristic of a 3-bit ADC is shown in Fig. 15.29. The staircase output resulting from amplitude discretization is responsible for the quantization error, representing the intrinsically unavoidable difference between the converted output and the corresponding input. The quantization error has a typical sawtooth shape with maximum amplitude of ±0.5 LSB. This can be treated as a random noise, called quantization noise, superimposed on the input with a resulting rms value equal to If a sinusoidal signal of peak amplitude V FS /2 is taken as a reference, the S/N ratio can be calculated as the ratio between the rms signal and rms quantization noise, yielding (15.29) An ideal ADC would be limited only by the associated quantization error, that is by the resolution, which is, however, more a design parameter rather than a performance specification. Real ADCs are affected by additional non- idealities, such as offset and scale errors, nonlinearity errors, possible missing codes and temperature-induced errors, which overall combine in worsening the actual conversion accuracy and decrease the S/N ratio below the ideal limit set by the quantization error given by eq (15.29). The quantization process is not instantaneous but takes some time to be carried out. This time is called quantization or conversion time and usually depends on the type of the ADC and sometimes also on the signal amplitude. The reciprocal of the conversion time is called conversion rate. For the AD conversion to be carried out accurately it is important that the input signal be constant within the conversion time. Some ADCs are Copyright © 2003 Taylor & Francis Group LLC in a real quantization process, although it should be realized that it does not necessarily imply that quantization occurs. In this section we are primarily concerned with sampling itself for its effect on the processing of time-varying signals. The fact that the sampled signal is subsequently quantized to perform an AD conversion is not important to the following considerations. Taking again as an example the measurement of the temperature in a room, imagine that we have monitored the temperature during a period of 24 h. Then we are unable to ascertain if there have been temperature variations between daytime and nighttime if we have not taken at least two readings at a 12 h distance. Similarly, we cannot determine possible temperature fluctuation during a single 12 h daylight period if we do not take at least two readings at a 6 h interval. That is, to catch the presence of a periodicity in a continuous signal we need to sample it at a rate which is at least twice such a periodicity. The principle intuitively suggested by this example is formalized in the sampling theorem by Shannon (previously implicitly formulated by Nyquist), which states that to reconstruct a continuous signal having its highest frequency component at f M from its sampled version, the sampling frequency f S must be at least two times f M , that is it must be ensured that The Fig. 15.30 The aliasing phenomenon seen in the time domain: (a) absence of aliasing; (b) presence of aliasing. Copyright © 2003 Taylor & Francis Group LLC frequency f M is sometimes called the Nyquist frequency of the signal. Thus, the minimum allowed sampling rate, called the Nyquist rate, is twice the Nyquist frequency. The concept is illustrated in Fig. 15.30 showing the sampling of a sinusoidal signal. If f S satisfies the Nyquist condition the sampled signal is a faithful representation of the continuous signal with no information lost in the sampling, since the original waveform can be readily recovered by interpolating the sampled values. Of course, the higher f S the more the sampled waveform resembles the continuous signal, but practical limits necessarily impede an arbitrary increase of f S . On the other hand, if the sampled values are no longer uniquely representative of the original signal. In particular, it can be observed how they may as well be attributed to the dashed waveform, which is completely different from the original signal and actually nonexistent at the input. Such a spurious waveform resulting from undersampling (i.e. insufficient sampling rate) is called an ‘alias’ and the phenomenon is named ‘aliasing’. Aliasing can be better understood if the sampling process is analysed in the frequency domain. The sampling operation is actually the multiplication of the continuous time signal by a series of pulses equally spaced by 1/f S , where f S is the sampling frequency [12]. This, seen in the frequency domain, corresponds to the fact that the spectrum of the sampled signal is a periodic repetition of that of the underlying continuous waveform at a regularly spaced distance given by f S , as shown in Fig. 15.31. This follows from the fact that sampling is basically equivalent to amplitude modulation. If as in Fig. 15.31(b) the frequency bands of adjacent spectrum repetitions are separated and the original signal can be reconstructed by low-pass filtering the sampled signal. Conversely, if as in Fig. 15.31(c) the frequency bands of adjacent repetitions overlap, since each component at a frequency is folded back at a frequency f–f S superimposing on the spectrum of the original signal. This is the aliasing condition and no linear filtering can recover the original signal from the sampled version. The aliasing phenomenon finds practical applications for instance in the stroboscope, where a pulsed light illuminates a rotating or vibrating object. If the frequency f S of the light pulses is made equal to that of the moving target f M , the latter appears still. Furthermore, if f S is slightly greater than f M a negative frequency alias is produced which manifests as an apparent inversion of the target motion. Stroboscopes can then be used to determine the unknown frequency f M in a noncontact way by tuning f S until the motion apparently stops. Aliasing can only be avoided by sampling fast enough. In practical cases, the bandwidth of the input signal is not always known in advance to properly choose the sampling frequency. In addition, high-frequency interference and wide bandwidth noise can unpredictably enter the system and appear at the sampler input. All these circumstances may harmfully cause aliasing, which is generally very difficult to detect when the actual input signal is unknown. To Copyright © 2003 Taylor & Francis Group LLC The antialiasing filter, if present, is a separate circuit. Its performance specification can be somewhat relaxed compared to the ideal requirements by taking advantage of the finite quantization resolution. In fact, all the residual aliasing components falling below the rms quantization noise level are of no concern, since they are not converted. 15.6.3 Main ADC types The functioning principles of the principal ADCs types are briefly illustrated and their main characteristics are collected in Table 15.1. Parallel or flash The analogue input is applied simultaneously to a set of voltage comparators with equally spaced thresholds derived by a voltage reference at V FS and a multiple resistive divider. The output levels from all the comparators are then processed by an encoding block which yields a quantized representation of the input in binary format. This technique is the fastest available, since all the bits are determined in parallel at the same time instant. For this reason, flash ADCs may reach conversion rates of several hundred megahertz and find typical application in transient digitizers and digital oscilloscopes. On the other hand, the method leads to rather complex and expensive hardware, since for n bits of resolution (2 n –1) comparators are required. This is why flash ADCs are typically available with a maximum of 8 bits, corresponding to 255 comparators. Successive approximation The analogue input signal is applied to a single comparator which confronts it with the output from an internal digital-to-analogue converter (DAC). At the start of conversion the DAC begins a strategy of binomial search by operating subsequent bisections of its output from the initial values V FS /2 guided by the comparator output levels. At the end of the search, which lasts n clock pulses, Table 15.1 Typical values of speed range and resolution for most common ADC types * For line frequency rejection. Copyright © 2003 Taylor & Francis Group LLC where n is the number of bits of the ADC, the input of the DAC represents the analogue input in digital form and is then taken as the output of the ADC. Successive approximation ADCs are relatively fast, since they only need n comparisons to produce a n-bit output, enabling conversion rates up to 1 MHz. This fact, coupled with moderate cost, makes them general purpose devices extensively used in most data acquisition (DA) boards. As a drawback, they tend to be very sensitive to input sudden changes or spikes and then typically require a sample-and-hold stage to freeze the input during the n clock cycles needed for the conversion. Integrating The input voltage is converted into a current which is used to charge an internal capacitor at a reference voltage. The time interval necessary to complete the charging is measured by a digital counter which provides a quantized representation of the input averaged over the integration time. The most popular version is the dual-slope ADC which actually charges the capacitor with the input signal for a fixed amount of time, and then measures the variable time required to discharge the capacitor at a constant reference current. Dual-slope ADCs are able to provide resolutions as high as 20 bits and more; however, they are slow due to their inherent integrating nature. Most often the integration time is set equal to, or to a multiple of, the power-line period (20 ms at 50 Hz, and 16.66 ms at 60 Hz) in order to average out possible interference and increase to overall noise immunity. As a consequence, the highest conversion rate is 50 or 60 Hz, and even less if multiple cycle integration is adopted. They tend to be more expensive than successive approximation ADCs, and their typical use is in digital voltmeters, or in DA boards dedicated to the measurement of slowly-varying signals such as temperature, static pressure or weight. Voltage/frequency conversion The analogue input signal is converted into a pulse train with frequency proportional to the input voltage. The frequency is then measured by a digital counter, which counts the number of pulses within a fixed time interval. Such a pulse number is then taken as the ADC output. ADCs based on V/f conversion can reach resolutions as high as 24 bits, and are very immune to noise since the input is actually integrated over the counting time. On the other hand, they are slow since, as in dual slope ADCs, the quantization scheme inherently requires the input signal to be acquired for a significant time duration. As a consequence, V/f ADCs are not suitable for dynamic signals and especially find application in remote sensing of slowly-varying quantities. In such cases, the V/f conversion can be done at the remote sensor location and the frequency signal transmitted to the counter, Copyright © 2003 Taylor & Francis Group LLC in this way offering a markedly higher noise immunity than is achievable when sending analogue amplitude signals over long distances. In this regard, it is often affirmed that the frequency conversion provides a digital representation of a signal. This is incorrect, since the frequency of a signal is a continuous function of time and no quantization actually takes place until such a frequency is converted into a number by counting. The fact that a frequency signal often has the form of squarewave should not be misleadingly regarded as indicating a digital nature. It simply means that information is carried analogically in the time scale rather than in the signal amplitude, which is the exact reason why frequency signals are particularly insensitive to amplitude fluctuations due to noise. 15.7 Data acquisition systems and analysis instruments 15.7.1 Vibration meters Vibration meters are portable instruments which connect to accelerometers or handheld probes and provide the measurement and display of one or more vibration parameters. Some units are pocket-sized for on-the-spot tests. Often they measure velocity, but most frequently they measure acceleration and extract velocity and displacement by integration. The result is usually displayed on an analogue needle indicator or on a digital liquid crystal display (LCD), or frequently on both. In general, vibration meters measure the amplitude of the vibration parameter of interest over a range of frequencies, therefore giving an integral result related to the measurement bandwidth, which is generally user- selectable. By inserting a tunable narrow band-pass filter (also called a resonant filter) at the input, a selective frequency analysis can be performed by sweeping the filter frequency and taking the corresponding readings. Some units have the tunable filter internally. Typically the displayed reading is related to the rms value of the measured quantity, but almost always the instrument may also indicate the peak value or the crest factor, i.e. the ratio of peak-to-rms value. Depending on the model, some additional features may be present, such as input charge-mode or constant-current-mode amplifiers for piezoelectric accelerometers, an interface to a personal computer or printer, relay contacts to activate external controls or alarms on occurrence of threshold trespassing. Vibration meters are suitable for the measurement of continuous vibration levels, but not for transients. They are most typically used for machinery inspection and maintenance, often coupled to handheld probes. In particular, they find wide application in tests on rotating machines in a frequency range which is generally between 10 Hz and 10 kHz. Several models can be directly used to perform vibration severity and exposure measurements in accordance to ISO 2954, 2631 and 8041. Some manufacturers offer special versions usable as human hand-arm vibration meters in compliance with ISO 5349. Copyright © 2003 Taylor & Francis Group LLC 15.7.2 Tape recorders Tape recorders enable the acquisition, storage and playback of electrical signals coming from transducers by converting them into the magnetization of a ferromagnetic tape. There are two types of tape recorder, differing in the format in which the signals are transferred and stored, namely the analogue and the digital recorders. Analogue The input signal is recorded by modulating the tape magnetization as a continuous function of time. To this purpose, two alternative methods are adopted, which are the direct recording (DR) and the frequency modulation (FM) methods. In the DR mode the input signal amplitude as a function of time directly modulates the degree of magnetization of the tape along its length. In the FM mode, the input amplitude is converted into a frequency signal which is used to magnetize the tape at the saturation levels, resulting in the information being contained in the number of magnetization inversions for unit tape length. The two methods have similarities and differences. They are similar in the fact that they can use the same tape, standard audio or VHS cassettes, and, as such, several recorder models use both DR and FM and provide the option to choose between the two techniques. For both methods, the frequency response increases with the tape speed, which can also be different between recording and playback. As a consequence, for a given tape length, higher frequency response implies shorter available recording times. As for the differences, the DR mode cannot record and reproduce DC signals while, on the other hand, its upper frequency limit can be considerably high. The typical frequency response obtainable with VHS cassette recorders is from 20 Hz to well above 100 kHz. Since the degree of magnetization of the tape can change over time due to tape deterioration and ambient conditions, the DR mode provides poor preservation of the recorded information. The FM mode has the advantage that it can record DC signals, as they correspond to a magnetization at a constant frequency, but it has an upper frequency limit generally around 50 kHz which is typically lower than achievable with DR. Moreover, for a given upper frequency limit it requires a faster tape speed than DR, hence the available recording time is consequently less. The preservation of information on tapes recorded with FM is good. An important characteristic of tape recorders is the dynamic range or, equivalently, the S/N ratio. In this regard, the FM method tends to be superior to the DR on a wide-frequency-range basis. However, since the former method has typically a smaller bandwidth than the latter, the actual comparison on the same narrow band can provide somewhat different results. Anyway, the average S/N ratio achievable is around 50–60 dB. In Copyright © 2003 Taylor & Francis Group LLC [...]... provided including sine, random, user-defined and playback of acquired data 15.7.4 Frequency and dynamic signal analysers The analysis of signals in the frequency domain is an extremely powerful tool to investigate the nature of dynamic phenomena and mechanical vibrations in particular The evaluation of the frequency content of a complex signal may often reveal signal features and details otherwise undetectable... LVDTs and capacitive elements The important concepts of amplitude modulation and phase-sensitive detection are then presented as fundamental methods to extract the signal from AC bridges and achieve a high S/N ratio Section 15.4 is dedicated to the amplifier options for piezoelectric transducers The voltage and charge amplifiers are presented both as standalone units and in their built-in versions, and. .. Devices and Systems, Ellis Horwood, 1983 Dally, J.W., Riley, W.F and McConnell, K.G., Instrumentation for Engineering Measurements, 2nd edn, John Wiley, New York, 1993 Harris, C.M (ed.), Shock and Vibration Handbook, 3rd edn, McGraw-Hill, New York, 1988 Herceg, E.E., Handbook of Measurement and Control, Schaevitz Engineering, Pennsauken, NJ, 1986 Jones, B.K., Electronics for Experimentation and Research,... universally used instruments for measuring dynamic signals and vibrations in particular, their principle of operation and capabilities are illustrated in some detail and some hints are given for their practical usage References 1 Van der Ziel, A., Noise: Sources, Characterization, Measurement, Prentice-Hall, Englewood Cliffs, NJ, 1970 2 Motchenbacher, C.D and Fitchen, F.C., Low Noise Electronic Design, John... where fC is the centre frequency at which the bandwidth will be translated and the integer n spans the record length This is shown in the dotted part 2 of Fig 15.34 The combination of bandwidth narrowing by sample decimation and centre frequency translation by heterodyning is usually referred as a zoom operation, since the displayed frequency window can be expanded around the region of interest The strength... Processing of Signals, Macmillan, London, 1973 Mandel, J., The Statistical Analysis of Experimental Data, Dover Publications, New York, 1984 Marven, C and Ewers, G., A Simple Approach to Digital Signal Processing, Texas Instruments, 1993 Methods for the Calibration of Vibration and Shock Transducers Part I: Basic Concepts, ISO 160 63–1, Geneva, 1998 Oppenheim, A.V and Schafer, R.W., Digital Signal Processing,... Ovaska, S.J and Väliviita, S., Angular acceleration measurement: a review, Proc IEEE Instrumentation and Measurement Techn Conf., St Paul, USA, 875–880, May 18–21, 1998 Copyright © 2003 Taylor & Francis Group LLC Scavuzzo, R.J and Pusey, H.C., Principles and Techniques of Shock Data Analysis, 2nd edn, SAVIAC, Arlington, VA, 1996 Serridge, M and Licht, T.R., Piezoelectric Accelerometers and Vibration... Accelerometers and Vibration Preamplifiers, Bruel and Kjaer, 1987 Smith, J.D., Vibration Measurement and Analysis, Butterworths, London, 1989 Standard for a Smart Transducer Interface for Sensors and Actuators, IEEE 1451.2, 1997 Sydenham, P.H (ed.), Handbook of Measurement Science—Theoretical Fundamentals, Vol 1, John Wiley, New York, 1982 Tompkins, W.J and Webster, J.G., Interfacing Sensors to the IBM... reduce the noise Indeed, rms averaging can be used to obtain a good estimation of the random noise floor present in the measurement bandwidth Usually, for both time- and frequency-domain averaging, we can choose between three modes of calculating the average Let us indicate with Xi and Yi respectively the ith input and ith output of the averaging process performed on n repetitions In the linear or additive... applications Limiting the bandwidth from DC to 5 kHz on 16 channels, the available recording time can be several hours DATs invariantly come with an interface for connection to digital computers for data analysis and automatic operation control, and in most cases are portable units which can be used conveniently in the field 15.7.3 Computer-based data acquisition boards and systems In a large number . similarities and differences. They are similar in the fact that they can use the same tape, standard audio or VHS cassettes, and, as such, several recorder models use both DR and FM and provide. nature of dynamic phenomena and mechanical vibrations in particular. The evaluation of the frequency content of a complex signal may often reveal signal features and details otherwise undetectable. the bandwidth will be translated and the integer n spans the record length. This is shown in the dotted part 2 of Fig. 15.34. The combination of bandwidth narrowing by sample decimation and centre