Vibration Measurements Enotes1 2k1i2y

Vibration Measuring Instruments

By:

Dr. B. K. Sridhara, NIE, Mysore

eNotes for the session on EDUSAT Programme

Dr. B. K. Sridhara, Asst. Professor, Mechanical Engineering, NIE, Mysore-8

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Chapter 5 Vibration Measuring Instruments 5.1 Introduction In practice the measurement of vibrations becomes necessary due to following reasons. 1. To determine natural frequencies, modal shapes and damping ratios. The measurement of frequencies of vibration and forces developed is necessary to design active vibration isolation systems. 2. The theoretically computed vibration characteristics of a machine or structure may be different from the actual values due to the assumptions made in the analysis. (To the analytical models). 3. Periodic measurement of vibration characteristics of machines and structures becomes essential to ensure adequate safety margins. (Preventive maintenance). 4. Measurement of input and resulting output vibration characteristics of a system helps in identifying the system in of its mass, stiffness and damping.

5.2 Vibration Measurement Scheme Figure 5.1 shows the basic features of a vibration measurement scheme. 1. Vibrating machine or structure. 2. Vibration transducer or pick up. 3. Signal conversion instrument. 4. Display / recording. 5. Data analysis. 2

3

1

4

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Figure 5.1 Vibration Measurement Scheme The motion of a vibrating body is converted in to an electrical signal by the vibration transducer or pick up. The transducer transforms changes in mechanical quantities such as displacement velocity, acceleration in to changes in electrical quantities such as voltage or current. (Electrodynamic pick up, electromagnetic pick up, piezo electric pick up, inductive displacement pick up, LVDT pick up, capacitive pick up). Since the output signal of a transducer is too small to be recorded directly, a signal conversion instrument is used to amplify the signal to the required value (Amplifier). The output from the signal conversion instrument can be displayed on a display unit or stored in a computer for later use (Oscilloscope, A to D converters, milli voltmeters, computers etc., the data can then be

Dr. B. K. Sridhara, Asst. Professor, Mechanical Engineering, NIE, Mysore-8

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analyzed to determine the desired vibration characteristics of the machine. Depending upon the quantity measured the vibration measuring instrument is called a vibrometer, a velocity meter, an accelerometer, a phase meter or a frequency meter. To summarise, following are the guidelines. 1. Displacement measurements may be useful for studying low frequency vibrations, where corresponding velocity and acceleration measurements are too small for practical purposes. 2. Velocity measurements may be useful at intermediate frequencies where displacement measurements are likely to be small to measure conveniently. 3. Acceleration measurements may be useful at high frequencies. Instead of the above, vibration analyzers can also be used. Several commercial vibration analyzers are available today. They consist of a vibration pick up and an FFT (Fast Fourier Transformation) analyser, a balancing kit for phase measurement and an inbuilt computer. The pick up essentially a piezo electric type with a natural frequency of 25 ks. (KHz). Built in double integration is also available for displacement plots. FFT converts time domain signal to a signal in frequency domain to identify the frequencies of concern.

5.3 Vibration pick ups: Seismic Instruments The commonly used vibration pick ups are called seismic instruments. The basic element of many vibration measuring instrument is a seismic unit which is basically a spring massdamper system mounted on a vibrating body on which measurements are to be made as shown in Figure 5.2.

Casing

x C

y

Figure 5.2Seismic Unit Depending on the frequency range utilized displacement, velocity or acceleration is indicated, by the relative motion of the suspended mass with respect to the case. Behaviour of Seismic unit Consider the equation of motion of spring-mass-damper system, subjected to base excitation, as shown in Figure 5.3.

Dr. B. K. Sridhara, Asst. Professor, Mechanical Engineering, NIE, Mysore-8

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x = X Sin (ω ωt-φ φ)

M

y = Y Sin ωt

Base Figure 5.3

..

. . – K (x-y) mx = - C (x-y) if.. Z = x-y; relative displacement the equation of motion becomes . . mZ + CZ + KZ = mω2Y Sin ωt from this Z = Y

r2 [1- r ] + [2 r]2 2 2

φ = tan-1 [2 r/1-r2]

Dr. B. K. Sridhara, Asst. Professor, Mechanical Engineering, NIE, Mysore-8

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The parameters that influence Z/Y and φ are: (1) frequency ratio r = ω/ωn. (2) Damping factor , as shown in the Figure 5.4. Range for Accelerometer

Range for Vibrometer

=0

= 0.1 Z/Y

= 0.15 = 0.25

= 0.5

= 1.0 r = ω/ω ωn Figure 5.4. Frequency response curves Figure 5.4 also shows the range of frequencies corresponding to which a seismic instrument act as a vibrometer or an accelerometer. Type of instrument is determined by the useful range of frequencies with respect to the natural frequency (ωn) of the instrument. The relative displacement Z, may represent the displacement or acceleration depending upon ωn of the seismic unit and frequency of vibrating body, ω.

5.4 Vibrometer or Seismometer It is an instrument with low natural frequency. Therefore, ω >>>>> ωn r >>>> 1, r is very large. Z/Y ≈ 1, in particular when r > 3 Z/Y ≈ 1, (independent of ) ∴Z=Y Relative displacement of the seismic mass = displacement of base.

Dr. B. K. Sridhara, Asst. Professor, Mechanical Engineering, NIE, Mysore-8

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∴Z = X-Y, X = 0, ∴Z=Y Hence the seismic mass remains stationary. It remains undisturbed in space. The ing casing moves the vibrating body. Thus the relative displacement between the casing and the mass is the true displacement of the casing. Like wise, the relative velocity between the casing and the mass is the true velocity of casing. Usually, the relative motion Z is converted into electric voltage. The seismic mass is a magnet moving relative to the coils fixed to the case, as shown in Figure 5.6.

x

0 0 0

0 0 0

Seismic mass

y Figure 5.6 The voltage generated is proportional to the rate of cutting of magnetic field. Therefore the output of the instrument is proportional to the velocity of the vibrating body. Such instruements are called velometers. A typical instrument of this kind may have a natural frequency of 1 Hz to 5 Hz and a useful range of 10 Hz to 2000 Hz. The sensitivity of such instruments may be in the range of 20 mV/cm to 350 mV/cm. Both the displacement and acceleration are available from the velocity type transducer by means of the integrator or the differeniator provided in most signal conditioner units. Limitation of Vibrometers In order to have r >>>1, ωn should be very small. This means that, the mass must be very large and the spring must have a very low stiffness. Therefore, a vibrometer is a springmass-damper system with a very large mass and a flexible spring. This results in bulky instrument, which is not desirable in many applications. In practice, a vibrometer may not have a large value of r, and hence the value of Z, may not be exactly equal to Y. In such cases the true value of Y, can be computed from: Z = Y

r2 [1- r2]2 + [2 r]2

5.5 Accelerometer It is an instrument with high natural frequency. When the natural frequency of the instrument is high compared to that of the vibrations to be measured, the instrument indicates acceleration. Dr. B. K. Sridhara, Asst. Professor, Mechanical Engineering, NIE, Mysore-8

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Then ω <<<< ωn, r <<<<< 1, the factor [r – (ω/ωn)2]2 + (2 r)2 approaches unity. ∴Z

(ω/ωn)2.Y (1/ωn2). ω2Y

Hence, Z α ω2 Y, which implies that Z is proportional to the acceleration of the vibrating body. Thus in order to make r <<<< 1, ωn should be very large. Hence K should be very large and m should be small. This means that, the instrument needs a small mass and spring of large stiffness. Therefore, the instrument will be very small in size and compact. Due to their small size and high sensitivity accelerometers are preferred in vibration measurements. The acceleration measured can be integrated once or twice with the help of modern electrical circuits to obtain velocity and displacement of the system. Thus the difference between a vibrometer and an accelerometer is in its natural frequency. In vibrometer it is very small where as in accelerometer it is very high. The principle of construction remains same.

5.6 Useful Frequency Range The useful range of accelerometer can be seen from the following graph for different amounts of damping . Useful frequency range is that range of r between which the maximum error is less than 0.01 %. The useful frequency range for un damped accelerometer is very much limited. However, with = 0.7 the useful frequency range is quite large, that is, between 0 ≤ ω/ωn ≤ 0.20, as shown in Figure 5.7.

Dr. B. K. Sridhara, Asst. Professor, Mechanical Engineering, NIE, Mysore-8

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Figure 5.7. Useful frequency range Thus an instrument with a natural frequency of 100 Hz has a useful frequency range of 0 to 20 Hz with negligible error. (Up to 20 Hz the error is less than 0.01%). Figure 5.8 shows accelerometers.

Figure 5.8. Accelerometers

Dr. B. K. Sridhara, Asst. Professor, Mechanical Engineering, NIE, Mysore-8

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Numerical Examples on Vibration Measuring Instruments 1. A vibrometer having a natural frequency of 4 rad/sec and = 0.2 is attached to a structure that executes harmonic motion. If the difference between the maximum and minimum recorded value is 8 mm, find the amplitude of vibration of structure when its frequency is 40 rad/sec. ωn = 4 rad/sec, = 0.2 Z = Relative amplitude = 8/2 = 4 mm ω = 40 rad/sec r = ω/ωn = 40/4 = 10 Z = Y

r2 [1- r2]2 + [2 r]2

Z Mean

4 mm 8 mm

= 1.0093

Z/Y = 1.0093 ∴ Y = 3.9631 mm 2. A vibrometer has a natural frequency of 10 s and has a damping ratio of 0.7. It is used, by mistake, to measure vibrations of a fan base at an exciting frequency of 180 rpm. The measured vibration velocity of the fan base is 3 mm/s. What is the actual velocity of the fan base? For a vibrometer, r2 Z = [1- r2]2 + [2 r]2 Y In the present case, ωn = 10 s = 62.8 rad/sec. Exciting frequency ω = 180 rpm = 18.84 rad/sec. Hence r = ω/ωn = 0.3 (Z/Y) = 0.09/(0.8281 + 0.1764) = 0.089 Hence Y = Z/0.089 = 33.6 mm/s. It may be noted that the actual velocity is beyond permissible limits, whereas what is read is well below the permissible limit. Hence one should be very careful in selecting the proper instrument. 3. A seismic instrument is fitted to measure the vibration characteristics of a machine running at 120 rpm. If the natural frequency of the instrument is 5 Hz and if it shows 0.004 cm determine the displacement, velocity and acceleration assuming no damping. ƒn = 5 Hz ; ωn = 2πƒn = 10π rad/sec = 31.4 rad/sec N = 120 rpm ω = 2πN/60 = 12.56 rad/sec r = ω/ωn = 12.56/31.4 = 0.4 Z = 0.004 cm = 0.0004 mm

Dr. B. K. Sridhara, Asst. Professor, Mechanical Engineering, NIE, Mysore-8

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For seismic instruments Z = Y

r2 , [1- r2]2 + [2 r]2

Z Y =

r2 [1- r2]2

=

=0

r2 [1- r2]

∴Displacement Y = Z(1-r2) / r2 = 0.021 cm Velocity V = ω Y = 2πN/60* 0.021 = 0.26 cm/sec Acceleration a = ω2Y = ω(ωY) = 3.265 cm/sec2 4. A vibrometer indicates 2 percent error in measurement and its natural frequency is 5 Hz. If the lowest frequency that can be measured is 40 Hz, find the value of damping factor. Solution: Data: ωn = 5Hz, ω = 40 Hz, error = 2% r = ω/ωn = 40/5 = 8 Z/Y = 1.02 (since the error is 2%) Z = Y

r2 [1- r2]2 + [2 r]2

(1.02)2 = 82/(1-64)2 + (16 )2 = 0.35

Dr. B. K. Sridhara, Asst. Professor, Mechanical Engineering, NIE, Mysore-8

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