Piezoelectric Accelerometer - Assignment Example

a) Piezoelectric materials (e.g., quartz) are those which generate charge when ed to a deformation or force. Transducers based on piezoelectric effect are known as piezoelectric transducers. The charge generated in the piezoelectric crystal due to a force Fx is given by Qx = kFx = kApx (1) where k is the piezoelectric constant A is the are on which the force Fx acts px is the pressure due to Fx The output voltage of the crystal is given by E = vtpx (2) where v is the voltage sensitivity t is the thickness of the crystal A piezoelectric transducer that is used to measure vibrations, acceleration of a vibrating body in particulars, is called a piezoelecric accelerator. Of course, accelerators measure vibrations by measuring accelerations. Some of the accelerators, if required, can compute and display displacements and velocities. But, displacements and velocities are derived from accelerations. A schematic of a piezoelectric accelerator is shown in Fig. 1. Fig. 1 Piezoelectric Accelerometer Here, a small mass is spring loaded against a piezoelectric crystal. When the base vibrates, the load exerted by the mass on the crystal changes with acceleration and hence the output voltage generated by the crystal will be proportional to the acceleration. The factors affecting the range of frequencies: An ideal accelerometer should be perfectly linear. A good accelerometer should have a near linear response; a real accelerometer shows approximately linear response over a range of frequencies only. Several factors decide this range of frequencies. The response is nearly linear only if the following holds good: (3) where r = ω/ωn ω is the excitation frequency (i.e., frequency of vibration of the body whose vibration is to be measured) ωn natural frequency of the sensor ζ is the damping factor = c/ (2mωn) where m is the mass c is the damping coefficient Now, if ζ lies in between 0.65 and 0.7, from (3), the left-hand side of (3) lies between 0.96 and 1.04 for 0 ≤ r ≤0.6. Since r is small, the natural frequency of the instrument has to be large compared to the frequency of vibration to be measured. Hence, the natural frequency is the most important factor which decides the range of frequencies over which a piezoelectric accelerator can operate. As can be seen from (3), mass, damping and stiffness (stiffness comes into picture while calculating the natural frequency), and also the excitation frequency (i.e., frequency of vibration of the structure whose acceleration is to be measured) have their role in deciding the frequency range over which the measurement may reliably be made using the piezoelectric accelerometer. The other factors which have a say on the operating range are: Piezoelectric material: Thickness of piezoelectric material also has an effect, e.g., for quartz, natural frequency increases when quartz thickness decreases [1]. Mass: Natural frequency increases when the thickness of seismic mass decreases [1]. The exact variation depends on the actual material composition of seismic mass. Also, the total mass of the accelerometer should be small compared to the mass of the test structure. Sensor mounting arrangements: (i) Contact stiffness: For better results, contact surfaces of components should be as smooth as possible. (ii) Conductive plates: They should be thin so that they may be considered rigid bodies. If not, analysis becomes tough and it becomes difficult to predict how they affect the response. (iii) Direct stud mounting is better since it gives the highest natural frequency and hence a large usable frequency range. (iv) Addition of an adhesive or magnetic mounting base may shorten useful frequency range. (v) Compliant materials such as a rubber pad may badly affect the performance. Signal conditioning: This refers to amplifying the output from sensors, so that the amplified signal may be read by the display device. The type of final output required decides the type of signal conditioning to be employed. Finally, the frequency range over which the accelerometer can reliably be used depends on each of accelerometer, conditioner and the display device. It also depends on the structure whose acceleration is to be measured and also how the accelerometer is mounted on the structure. (b) Shaft speed = 2600 rpm Hence ω = {(2600 * 2π) / 60} rad/s = 272.271 rad/s f = ω / (2π) = 272.271 / (2*π) = 43.333 Hz According to [2], as per the (now obsolete) standard 2372 - 1974, for a Class II machine, maximum allowed RMS velocity for the good case is 1.12 mm/s, and for the acceptable case is 2.8 mm/s. Displacement is of the form x(t) = Asin(2πft) (4) Hence, velocity is of the form ẋ(t) = 2πfAcos(2πft) (5) And acceleration is of the form ẍ(t) = -(2πf)2Asin(2πft) (6) The RMS value for the wave form given by (4) is given by A/√2 Hence RMS value for the waveform given by x(t) = Acos(2πft) is also given by A/√2 Hence, for the good case, from (5), 1.12 = 2πfA/√2 i.e., 1.12 = 2* π * 43.333*A/√2 Hence, A = 0.00582 mm Same way, for the acceptable case, from (5), 2.8 = 2πfA/√2 i.e., 2.8 = 2* π * 43.333*A/√2 Hence, A = 0.01455 mm Now, from (6), maximum acceleration for the good case is given by (2πf)2A = (2*π*43.333)2*0.00582 = 431.440 mm/s2=0.044g (Assuming g = 9.81 m/s2) Again, from (6), maximum acceleration for the acceptable case is given by (2πf)2A = (2*π*43.333)2*0.01455 = 1078.6 mm/s2= 0.110g (Assuming g = 9.81 m/s2) Since the sensitivity is 1V/g; Voltage generated for the good case = 1*0.044 = 0.044V = 44mV Voltage generated for the acceptable case = 1*0.110 = 0.110V = 110mV Note: Since ISO 2372 refers to the whole vibrating structure, the measured (vibration) quantities at any point on the structure, in any direction, should not exceed the allowable limits. Although, it is a standard practice to measure vibrations near bearings, inner race etc., the standard (ISO 2372) does not speak anything specific to measurements at these locations. (c) Although A/D conversion is not compulsory, it is highly recommended. This is because, the digital signals may easily be processed using wide variety of devices available for the purpose. Digital computers can directly use the signals generated. In most of modern systems, the final outputs are in digital form. Referring to the Problem (b) above, shaft speed = 2600 rpm. Now, if a machine always runs at a fixed speed only, it is enough if it is tested at that particular speed. Of course, while solving (b), this was implicitly assumed. But, in reality, a machine may be subjected to different forcing frequencies, e.g, during starting and stopping, because of load variation etc. It is a standard practice in such cases to test the machine in a range of frequencies (e.g, as explained in [3]). The frequency range over which the machine has to be tested is divided into three bands of frequencies. The first band covers (0.3 - 0.8) of operating frequency, i.e., (0.3 * 2600) rpm to (0.8 * 2600) rpm, i.e., 780 rpm to 2080 rpm, i.e., 81.681 rad/s to 217.819 rad/s, i.e., 13 Hz to 34.667 Hz. The second band covers (0.8 - 1.2) of operating frequency, i.e., (0.8 * 2600) rpm to (1.2 * 2600) rpm, i.e., 2080 rpm to 3120 rpm, i.e., 217.819 rad/s to 326.726 rad/s, i.e., 34.667 Hz to 52 Hz. . The third band covers (1.2 - 3.5) of operating frequency, i.e., (1.2 * 2600) rpm to (3.5 * 2600) rpm, i.e., 3120 rpm to 9100 rpm, i.e., 326.726 rad/s to 952.952 rad/s, i.e., 52 Hz to 151.667 Hz. Hence the range of frequencies over which the accelerometer has to reliably operate is : 81.681 rad/s (or 13 Hz) to 952.952 rad/s (or 151.667 Hz). Now, the minimum voltage ever generated is when f = 13 Hz, and for the good case. The voltage generated = (1.12 *√2*(2πf)2)/( 2πf*9810 )=0.01319V The maximum voltage ever generated is when f = 151.667 Hz, and for the acceptable case. The voltage generated = (2.8 *√2*(2πf)2)/( 2πf*9810 )=0.38466V Hence, Dynamic range (dB) = 20 log10(Maximum Voltage/Minimum Voltage) = 29.297 But, for good results, the dynamic range should be 60 to 90. To get the 60 dB dynamic range, by reverse calculating, for the same minimum voltage, maximum voltage = 13.190 V. Hence f for this voltage may be calculated to be, f = 5200.689 Hz = 312041.34 rpm (It is assumed that the machine can operate at such a high speed; the measurements at these higher frequencies are not of much significance since the machine operates at a much lower rpm). A new (fourth) frequency band is created. It spans from the end of the third band upto this frequency (5200.689 Hz). Coming to the sampling frequency, according to Nyquist Sampling Theorem [4], sampling frequency should always be at least twice the highest frequency of the measured signal. For convenience, we can determine it for each band separately. Hence, sampling frequency for the first band could be 2*34.667 Hz = 69.334 Hz. Similarly, the sampling frequency for the second band = 2*52 Hz = 104 Hz, for the third band = 2*151.667=303.334 Hz, for the fourth band = 2*5200.689 Hz =10401.378 Hz Frequency resolution may also be calculated band wise. The standard practice is to set the resolution = 150 * Machine rpm, unless it falls below 800 in which case the resolution set to 800. Hence, for the first band, resolution = 150 * 2080 = 312000. For the second band, resolution = 150 * 3120 = 468000. For the third band, resolution = 150 * 9100 = 1365000. For the fourth band, resolution = 150 * 5200.689 = 780103. Number of samples that should be recorded for each spectrum = (frequency resolution * sampling frequency), for the corresponding band. Hence, the number of samples that should be recorded for the first spectrum (or band) = 21632208. For the second band = 48672000. For the third band = 414050910. For the fourth band = 8114146182. A 24 bit A/D converter is best suited for the present case. Because of phase noise, selecting a 32 bit converter above a 24 bit converter will not improve the results. It is possible to use other converters like 16 bit, and even 12 bit A/D converters, but with lower performance. Some signal conditioners like voltage amplifiers are essential. It is better if there are some more signal conditioning elements like elements which can calculate velocities and displacements from accelerations. References [1] ZHANG Zhong-cai, Yang Li-ming, CHENG Yong-sheng, "Design of a Large Measurement Range Piezoelectric Accelerometer," Proceedings of the World Congress on Engineering 2009, Vol I, WCE 2009, July 1 - 3, 2009, London, U.K. [2] "Vibration Pen Plus," SKF Condition Monitoring, SKF, 18 March 2011 < www.prospect-bearing.com.tw/cmvp405.pdf > [3] The Website, Used Vibration, 20 March 2011 [4] H Nyquist, "Certain Topics in Telegraph Transmission Theory," Trans. AIEE, vol. 47, pp. 617 - 644, Apr. 1928 [5] Singiresu S Rao, Mechanical Vibrations, Fourth Edition, Pearson Education, 2004 [6] Marks Standard Handbook for Mechanical Engineers, 8th Edition, McGraw - Hill, New York, 1978 [7] Grover G K, Mechanical Vibrations, Nem Chand & Brothers, 1996 [8] William T Thompson and Marie Dillon Dahleh, Theory of Vibration with Applications, Prentice Hall, 5th Edition, 1997 Read More