Vibration Measurement & Control

Vibration Measurement & Control by Brian McLauchlan

Authors Note: These notes are provided for student use in National Module EA 7766L The notes are not to be reproduced in any form without the author s written permission. While every attempt has been made to ensure accuracy of the materials in these notes, the author accepts no responsiblity for any liability or loss in respect to the application of the information presented. Brian S. McLauchlan 1990-2007 1

TABLE OF SYMBOLS δ.....density, kg/m 3 f.....frequency, Hz f f...forcing frequency, Hz f n...natural frequency, Hz k...spring constant, N/m m...mass, kg ϖ...angular frequency, rad/sec x...displacement in meters X...displacement in meters at time t v...velocity in m/sec a...acceleration in m/sec 2 y...position in meters 2

Table of Contents Vibration - Concepts... 4 Vibration - Measurement... 13 Instrumentation For Vibration Measurement... 27 Vibration Isolation... 43 Vibration - Human Effects... 60 Balancing Of Machinery... 67 Balancing Of Machinery... 67 Vibration Specification... 74 Appendix... 78 Vibration Exercises... 79 Glossary Of Vibration Terms... 84 3

Vibration - Concepts 1.1 Introduction Many machines and processes in engineering generate vibration. In a few cases this vibration is intentional as in vibrating sorting screens, ultrasonic cleaners and earth compaction machines. Mostly though, the vibration is an undesireable effect. The vibration generated can cause a number of effects that are troublesome. The most serious are related to fatigue and injury to humans exposed either to the vibration directly or the effects of noise caused by the vibration. Vibration may also contribute to excessive wear, fatigue failure and other premature failure of machine components. Many vibration problems are due to inadequate engineering design of a product, or the use of a machine in a manner that has not considered the possible effect of vibration. In this course we will investigate the basics of vibration with the intent of being able to measure and assess problem areas. There are many very complex vibration problems that require sophisticated computer modelling to solve. It is often the case however that awareness at the design stage will eliminate or reduce the vibration to a level that is acceptable, using relatively simple methods. 1.2 Simple Harmonic Oscillation The simple model of a spring and a mass may be used to investigate the basic ideas of vibration. In this model a spring of spring constant k, suspends a mass m. Spring constant is measured in Newtons / meter (N/m). Mass is of course, in kilograms. Figure one shows this model with a spring constant of 400 N/m and a mass of 4 kg. If the mass is displaced down, then released, the mass will oscillate at a frequency that is independant of the amount of initial displacement. This frequency is called the NATURAL FREQUENCY. The equation that determines the natural frequency of this system involves both the spring constant and the mass. As the spring constant is increased, so the natural frequency increases. 4

k m Figure 1.1 - Vibrating Spring/Mass System The effect of the increased mass is to decrease the natural frequency so that the form of the relationship is: fn k = 1 2π m Our system in figure one thus has a natural frequency of 1.6 Hz. Note that the angular frequency, ϖ, is related to the frequency f by the factor 2 π, so that this equation can be written as: ω = 2πf so ω = k m If our simple system is set in motion we may measure the displacement over a period of time. We will find that the displacement repeats after a time called the PERIOD, which is the inverse of frequency. In the case of our example in figure one, the frequency is 1.6 Hz and so the period is 0.625 seconds. The fact that the vibration repeats is described by the term PERIODIC and the motion of one period is called a CYCLE. If we plot the cycle of displacement over the time of one period we will find that the result is a curve like that in figure 2. This is a curve that is able to be described by the familiar sine function. x = A sin ( ϖ t ) 5

where the magnitude x goes from a maximum value of A to a minimum of - A over a cycle related to an angular function ω and the time t. Where the object moves in this manner it is known as SIMPLE HARMONIC MOTION. V i b r a t i o n time Figure 1.2 Displacement / Time For A Periodic Vibration For displacement the sine function can be written as: where X = A sin ( ϖ t ) X - displacement from rest position, m at time t. A - peak displacement, m. ϖ - angular frequency, rad/sec t - time, seconds. 1.3 Displacement - Velocity - Acceleration In most considerations of vibration problems we will deal with one of three possible parameters for vibration measurement. The first we have described above. The other two are velocity and acceleration. To understand the relationship of the three parameters, displacement, velocity and acceleration is important to an understanding of vibration. 6

Displacement - A measure of the distance a vibrating body moves. Velocity - A measure of the speed of motion of a vibrating body. Acceleration - A measure of the rate of change of speed (velocity) of a vibrating body. The equations for each can be written as shown below. ( Note: see appendix A for details of the derivation of these equations.) X = Asin( ωt) displacement X = Aω cos( ωt) velocity X 2 = Aω sin( ωt) acceleration For a particular vibration, the parameters in these equations, A and ϖ, are constant and common. This means that these three measures of vibrations are always related in a predictable way. This is fine but does all this mathematics mean much in a real problem? Well, let's consider the physical significance of these equations. 7

V i b r a t i o n 10 8 6 4 2 0-2 -4-6 -8-10 time Displacement Velocity Acceleration Figure 1.3 Relationship Of Displacement, Velocity & Acceleration In figure 1.3, each of the above equations has been plotted for one cycle, with the displacement as the reference for time zero. For our vibrating spring - mass system this diagram shows that : i.at time zero, the velocity is maximum with displacement and acceleration zero. The mass is moving past its rest point. ii.at 1/4 cycle later the velocity has reduced to zero with displacement maximum and acceleration maximum in the other direction. The mass has stopped at the peak of a cycle. iii.at 1/2 cycle the displacement and acceleration have again become zero while the velocity is a maximum. The mass is again passing its rest point. iv.at 3/4 cycle the displacement is at a negative maximum with acceleration a maximum in the opposite direction. The velocity is zero. v.the cycle is complete with displacement, velocity and acceleration at their original values. We see from this that the mathematics describes what is happening to the mass at any time in the cycle. We will use the various measures of vibration - displacement, velocity and acceleration to assess problems of machine vibration. 8

1.4 Units Before any assessment can be made however, we must be aware of the units used in vibration. These are summarised in the table below. PARAMETER DISPLACEMENT VELOCITY UNITS USED m, mm, um m/sec, mm/sec m/sec 2 ) ACCELERATION m/sec 2, " g " ( % of 9.8 Table 1.1 Units For Vibration Measurement We will make use of these units but must first consider further the problem of our vibrating spring mass system. 1.5 Forced Vibrations So far we have caused the spring mass system to vibrate only at its natural frequency. The mass is displaced and then released causing a series of oscillations. What happens if the system is pushed by a force that also oscillates? Figure 1.4 shows the system acted on by an external force causing displacement of the base, that has a periodic nature. x(t) x'(t) m Figure 1.4 Forced Oscillations Of The Spring Mass System We might expect some oscillation and that it will depend on the frequency of the "EXCITING FORCE" and the natural frequency of our spring mass system. If the differential equation for the system 9

is solved and the frequency response for the system is plotted we have a diagram like that shown in figure 1.5. Amplification 10 1 0.1 0.01 0.1 1 10 Frequency Ratio Figure 1.5 Response Of 1 Degree Of Freedom System Figure 1.5 calls our spring - mass system a ONE DEGREE OF FREEDOM SYSTEM because the motion of the mass is described by only one displacement measurement. (ie displacement is along one axis only) The response diagram shows some important information that shall now be considered. First, the mass has a response which theoretically goes to infinity when the exciting force coincides in frequency with the natural frequency. This response is known as RESONANCE. In practice the response at resonance will not be infinite due to losses in other parts of the system. However the resonant behaviour is significant because the system responds with a greater displacement than that applied! This is clearly undesirable. It can also be seen that below the resonance frequency, the response climbs steadily and is always more than the applied displacement. Above resonance however, the response drops 10

rapidly, showing less and less displacement with increases in frequency. How can this knowledge help us? If the frequency of the forced vibration is always above that of the NATURAL FREQUENCY of the system, then the vibration of the mass is less than the applied vibration. In fact, if the applied vibration is more than 3 times the natural frequency, the vibration of the spring/mass is less than 10% of the input vibration. This means that we have ISOLATED the mass from the vibration to the extent that only 10% of the vibration gets to the mass. A similar situation applies if the mass in our system has the forcing frequency applied to it directly. Consider a small diesel engine. When operating, the rotating and reciprocating parts of the engine will cause a vibration at the running speed of the engine. In our simple model the engine is the mass and provides also the forced vibration. We wish to isolate the vibration of the engine from the mounting base of the engine. If we use a spring mounting with a stiffness that ensures a NATURAL FREQUENCY of 3 times less than that of the engine running frequency, we will ISOLATE the mounting base of the engine from 90% of the vibration produced by the engine. 1.6 Damping In the section above, it was observed that the vibration at resonance is limited by the system losses. The loss can be controlled to provide a more suitable frequency reponse for the system. The provision of suitable energy losses in a system is termed DAMPING. Damping will have the effect shown in figure 1.6. In the figure the term DAMPING RATIO is used to express the amount of damping used. The value where damping ratio is equal to 1.0 is called CRITICAL DAMPING. 11

Critical damping is defined by: ccritical = 2 km Damping greater than critical will have a ratio greater than 1.0. Damping less than critical, will have a ratio less than 1.0. Damping causes the response at natural frequency to be reduced but causes the shape of the response curve to alter at other points. This will be discussed in detail in section Amplification Ratio 10.00 0.05 0 0.1 0.2 0.5 1.00 1.0 Damping Ratio 1.0 0.10 0.5 0.2 0.1 0.05 0.01 0 0.1 1 10 Frequency Ratio Figure 1.6 Response to Forced Vibration (1 DOF Spring-Mass) 12

Vibration - Measurement 2.1 Introduction The vibration of machines can be considered as an oscillatory motion of part or all of the machine. So far we have seen that a simple vibrating system, the spring/mass system, has a harmonic motion. This motion can also be termed PERIODIC because it repeats itself exactly over fixed time period. In this section we will consider what other vibratory motions are possible and the basic ideas for analysing these motions. 2.2 What Are We Measuring? The motion of the vibrating system is measured with the units described in a previous section (1.4). What was not specified was what amplitude was to be specified with these units. Figure 2.1 shows a sinusoidal waveform with the possible ways of measuring amplitude. 10 8 V i b r a t i o n 6 4 2 0-2 -4-6 RMS Peak Peak to peak -8-10 time Figure 2.1 Measuring Vibration Amplitude The equations of motion specify the PEAK amplitude (see 1.3) but when measuring our measuring device could be constructed to measure any of PEAK, PEAK to PEAK or RMS amplitude values. Many measuring systems measure RMS (Root Mean Square) values because this value is proportional to the power in the vibrations of a system. This means that care should be taken to establish what is being measured by an instrument. In particular, when the procedure of 13

converting between displacement, velocity and acceleration is used, the correct amplitude must be known for correct integration. 2.3 Harmonic Periodic Vibration The vibration that has been described so far, that has a single frequency and is sinusoidal can also be described as harmonic periodic vibration. Harmonic is an alternative term for sinusoidal, and periodic means repeating regularly. A sinusoid is able to be described precisely by knowing its frequency and amplitude. 2.4 Vibration That Is Not Harmonic Our simple spring mass system gives rise to harmonic periodic vibrations. This is not the only possible type of vibration that we may encounter. In fact it is probably the least likely to be found in most engineering systems. We should first consider the possiblity of a vibration that is periodic but not harmonic. That is, its motion is not described by a simple sinusoidal signal, but the motion may repeat itself continuously in time. Such a vibration can be termed periodic and an example is shown in figure 2.3. Amplitude 10 5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21-5 -10-15 time Figure 2.3 Periodic Non Harmonic Vibration If we wish to determine the frequency content of this signal, how could it be done? A mathematician called Fourier determined that for any complex signal, its frequency content could be found by considering the complex signal as the sum of a series of sine and cosine functions. In the example above, which is the acceleration of the piston in an engine, the signal can be analysed into two sine 14

signals of differing amplitude and frequency. In this example the Fourier analysis can be seen to give the wave form in figure 2.3 using two harmonically related sinewaves. This is illustrated in figure 2.4. 10 5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21-5 Amplitude -10-15 time Figure 2.4 Fourier Components Of A Signal So far we have dealt with relatively simple types of vibration. Much of the time, however, we will be dealing with a vibration that is much more irregular than those we have seen previously. This irregular type of vibration is termed RANDOM VIBRATION. Random vibration is characterised by irregular motion cycles that never repeat themselves exactly. This means that the analysis will be somewhat more complicated. In view of this added complexity, you may be tempted to ask, how common is this type of vibration?. Consider the motion of any form of vehicle used for transport, such as cars, trains or aircraft. It is rare that any of these vehicles will experience purely periodic vibrations. In fact much design effort is expended to avoid certain periodic vibration that may result in resonance and consequent damage due to excessive vibration amplitude. Vehicle Car Aircraft Ship Rail wagon Common periodic vibration Suspension resonance from corrugated road surface Body vibration due to turbulence Roll due to sea swells Yaw instability above design speed 15

Much machinery vibration can occur as random vibration with added periodic components. It is thus a common and important type of vibration to consider in engineering. Random vibration may have a signal that is like that of figure 2.6. This shows the complex nature of the motion of a component experiencing this type of vibration. Figure 2.6 Typical Random Vibration Signal 2.5 Analysis of Vibration Signals Recall that a periodic harmonic signal can be completely specified by an amplitude and a frequency. For other signals we are also interested in these parameters. To help express these we use a special graph called a frequency spectrum that plots amplitude on the vertical axis and frequency on the horizontal axis. 16

2.5.1 Spectrum Of A Periodic, Harmonic Vibration What does the frequency spectrum of the vibration of our spring mass system look like? We have only one frequency and a single amplitude. The spectrum thus appears as a single line at the natural frequency of the spring mass system with an amplitude depending on the size of the motion of the mass. Figure 2.2 shows a typical spectrum for our simple spring mass system. Amplitude Frequency Figure 2.2 Frequency Spectrum Of Harmonic Vibration If a complex periodic wave is broken up into its' Fourier components, a frequency spectrum can also be constructed. If each Fourier component is a sinusoidal signal of a certain amplitude the spectrum will be a series of peaks on the spectrum. The example in figure 2.4 is represented as a frequency spectrum in figure 2.5. Amplitude f1 f2 Frequency Figure 2.5 Frequency Spectrum For The Signal Of Fig.2.3 In the frequency spectra shown above we have indicated an amplitude. This could be the peak amplitude of the signal or it could be the RMS amplitude. 17

Because the Fourier spectral analysis breaks a complex wave into sine or cosine components the RMS or peak amplitude can be easily converted from one to the other. Amplitude Time f1 f2 Frequency Figure 2.7 Creating a Frequency Spectrum Figure 2.7 shows how the signal, which is varying in time, is broken up into components which can be shown on the frequency sprectrum graph. The frequency spectrum is like a cross section at a point in time of all the components that make up the signal being studied. How can we analyse a random vibration signal? What do we use for amplitude? What do we use for frequency? It is clear that when we have a periodic signal, we can predict from its appearance over one cycle, the future cycles. With a random signal this is not possible. Theoretically we have a signal that must be infinite in length and the whole signal should be studied. 18

Fortunately statistics can be used to analyse the properties of this infinite signal by using a sample from the signal. Like most statistical procedures this means that the sample we take must be a representative sample, otherwise our analysis will be inaccurate. For example, if I intended to use a statistical method to find the most popular food in Australia and I asked only adult males over 70 in Alaska for their opinion my results would not be particularly valuable! Similarly, when analysing vibration signals I must use a good sample, usually determined by having a long enough sample where conditions are stabilised. This means that the following will be approximately constant: Speed of a machine (eg velocity of a car; rpm of a motor) Loading on a machine (eg power output of an engine) Forced vibration (eg quality of a road surface that a car travels on) Just what is a long enough sample under these conditions is dependant on the frequency content of the signal and will be further discussed in the data analysis section, but basically requires that lower frequencies require longer recording times for the same accuracy as higher frequencies. (see page 23) 2.5.2 Analysis Of Random Vibration Signals A random vibration signal may be analysed using an amplitude analysis and/or a frequency analysis method. The simplest means of assessing random vibration is to measure the RMS signal level over a period of time. This will give and idea of the average energy content of the signal and is useful for many applications. An analysis of the PROBABILITY of occurence of a particular amplitude value will give an asessment of what sort of vibration levels can be expected and how frequently a particular level will occur. For example, a motor car travelling along a road will have some vibration felt by the passengers continuously. Large bumps will give larger vibration levels, but less frequently. This type of analysis commonly results in a normal curve that is met frequently in naturally occurring processes. Figure 2.8 shows a Normal ( or Gaussian) curve generated by this type of analysis. 19

Frequency Mean or average Spread Vibration Amplitude Figure 2.8 Normal Curve From Vibration Amplitude Analysis Like the other signals we have dealt with, we are commonly also interested in the frequency content of the random vibration signal.fortunately, Fourier analysis is applicable to random signals in the same way as other data. The Fourier analysis of a random signal results in an infinite number of sinusoidal components of different amplitudes and frequencies. The resulting spectrum is theoretically a continuous curve rather than single line values. Figure 2.9 shows the type of curve that could result from this type of analysis. 20

Amplitude Frequency Figure 2.9 Frequency Spectrum For Random Vibration Because we cannot deal with infinite numbers of amplitudes and frequencies, this curve is usually approximated by a series of lines representing frequency values. A frequency spectrum for a random vibration is sometimes called a spectral density or power spectrum. These are all measures of the frequency content of a random vibration signal. 2.5.3 Filtering We have seen how to classify the time behaviour of a signal and that it can be broken into frequency components and expressed as a frequency spectrum. The question then arises as to how are we going to achieve the frequency analysis to get a frequency spectrum. The basic idea of frequency analysis rests on an understanding of filters. We have heard of filters in mechanical systems. These are used to limit the particle size passed through a fuel or lubricating system, for example. In a similar way electronic filters can be made which restrict the frequencies that are allowed to pass through them. A filter may be either a LOW PASS, HIGH PASS or BAND PASS FILTER. The FREQUENCY RESPONSE CURVES for these types of filter are illustrated in figure 2.10. The low pass filter will allow only frequencies 21

up to a certain value to pass through. Any higher will be blocked. In a similar manner the high pass filter will allow only those frequencies above a certain level to pass through. The point where the frequencies will become blocked can be changed by design. A Low Pass High Pass Band Pass Figure 2.10 Low Pass, High Pass And Band Pass Filters The band pass filter can be considered to be a combination of a low pass filter and a high pass filter. This gives a filter that will pass frequencies over only a narrow range. We can make such a narrow band filter with either a fixed value of range or may make it a fixed width and variable frequency so we can tune it to the band that we want. It is this band pass filter that makes frequency analysis possible. If a complex signal is measured by a transducer, the electronic signal representing the transducer signal may be passed through a band pass filter and the level measured. This level will be only the amplitude of the frequencies that are passed by the filter. If a range of filters are used then the whole range of frequency of interest can be covered in small frequency increments. The most common set of band pass filters used are in octave or 1/3 octave bands. An octave covers a frequency range such that the lowest frequency in the range is half the value of the highest frequency in the range. Filters for octave bands are generally labelled by the middle or center frequency of the whole band. One third octave band filters break each octave band range into three. Octave and 1/3 octave filters are more commonly used for noise measurement, however 1/3 octave bands are used in a number of standards, especially those concerned with human effects of vibration. 22

2.5.4 Narrow Band Analysis To achieve the discrimination needed for vibration analysis, filters with narrow bands are needed. This can be achieved by constructing more filters OR by using a technique known as Fourier Analysis. Fourier Analysis is a mathematical technique that can determine frequency content in as narrow bands as desired. It is usually achieved using electronic systems such as an FFT analyser or a computer. (note : FFT is Fast Fourier Transform, a mathematical simplification to make the calculation quicker) 2.5.5 Limitations Of Frequency Analysis The use of any filtering technique for frequency analysis must consider the limitations of the equipment. Any filter does not cut off completely those frequencies outside its bandwidth. There is also a limitation on the minimum time required for a filter to accurately determine the magnitude of the filtered data. This is called the averaging time. Limitations Of FFT Analysis Due to the popularity of FFT analysis it is considered essential to mention some important limitations in use of this technique. a. averaging time - the averaging time must be carefully selected to ensure that the amplitude is accurately measured. For many acoustic measurements this will not be significant but the same averaging for lower frequency analysis, often the case for vibration work, may cause significant errors. The FFT process must average (sometimes called "ensemble averaging") over a number of spectra to give accurate amplitude results. The following should be considered: Number of Error Length of data record spectra 95% c.i. for full scale frequency 100Hz 1kHz 10kHz 8...... 2.8dB 32 3.2 0.32 16...... 2.0dB 64 6.4 0.64 32...... 1.46dB 128 12.8 1.28 64...... 1.06dB 256 25.6 2.56 23

128...... 0.74dB 512 51.2 5.12 256...... 0.54 db 1024 102.4 10.24 time in seconds Notes: c.i. - confidence interval,this means that all data are within the error limit specified with 95% confidence. (this means that there is a 95% probability that the error will be no greater than that listed) Length of data record - means how long the recording of the data must be, with operating conditions constant, to give enough time for the analysis. Table 2.2 Averaging Times For FFT Analysers 24

Example: This table shows that for 95% confidence that if I require the accuracy of the data to be no worse than +/- 0.54 db, and I set a maximum frequency of 1000 Hz on the analyser, I will require 256 averages. These 256 averages will take 102.4 seconds of data or 102.4/60 = 1.7 minutes. The table has some important consequences. The time limits often cause problems for data recording at low frequency. For instance,if the analyser is now used at 100 Hz full scale setting,a recording of 17 MINUTES is required for the highest accuracy! ( 1024/60 = 17) Often then, a trade off of accuracy and time recording must be made. b. windows - The FFT process is a mathematical simplification. It relies on the correct data being input to give accurate answers. In the FFT process a block of data is converted to digital information and then frequency analysis is performed. The next block of data is treated similarly and then averaged with the first block. If the two blocks of data do not "fit together" like acontinuous wave, the FFT analysis process will calculate non- existing frequencies! To overcome the problem of data blocks with "non - fitting" ends a process called WINDOWING is used. This process ensures that data always fits together. A "window" here is a mathematical weighting curve that is used to ensure that the sampling that occurs with the FFT averaging process still has acceptable accuracy. It uses windows called Hanning, Hamming or rectangular. The Hanning and Hamming windows are for use with continuous signals. The rectangular window is usually only used for analysis of impulsive signals. c. Anti - aliasing There is always a problem with limiting the maximum frequency of a signal when converting from analogue to digital, as required by FFT analysers and computers. When a signal is to be digitised it is converted to a series of numbers. Each number represents a sample point of the continuous wave that is the acoustic or vibration data. If the sample points are taken too far apart the frequency data is not able to be accurately specified. 25

This is because the sample data may represent the signal frequency measured or multiples of that signal at higher frequency. To prevent aliasing,the higher frequency components must be removed by a LOW PASS FILTER. This filter is usually provided by the manufacturers of FFT analysers but must be provided also when using a computer for FFT analysis. This aspect is often overlooked when using computer based systems and can lead to serious errors. The frequency of sampling should then be set to be at least twice the limiting frequency of the low pass filter. It is preferable in fact to set the sample frequency higher for greater confidence in the frequency data. The problem of aliasing should not be overlooked as incorrect data analysis will result if aliased data is used. Like most instrumentation, it is possible to get out values from frequency analysers that seem reasonable but may not be accurate. Ensure that the correct procedure is used for any frequency analysis. Sample Period Sample Period time time Signal constructed from samples. Signal constructed from samples. Figure 2.10 Sampling to Eliminate Aliasing When Digitising 26

Instrumentation For Vibration Measurement 3.1 Transducers A transducer is a device that converts a small amount of the energy of the quantity to be measured into another form of energy, usually electrical. This conversion is to allow for ease of measurement as typically it is easier to record and analyse electrical signals. In vibration measurement, we may wish to measure the vibration displacement, velocity or acceleration.the choice of measurement will depend on how the data obtained from the measurement wil be used. Because of this choice we have transducers suitable for measuring each quantity. The transducers may make use of the following techniques for energy conversion: a.piezo - electric effect b.piezo - resistive effect (silicon strain gauge) c.inductance d.capacitance e.resistance f.optical 3.1.1 Displacement Transducers These may be either non - contact inductive or capacitance types, resistance types or may use a DIFFERENTIAL TRANSFORMER to measure displacement. Alternately, double integration of an accelerometer signal may be used to provide displacement. The non-contact types are often used to measure shaft postion in rotating machinery. They operate by sensing the change in a magnetic or electrical field between the sensor and the shaft as the shaft varies its motion relative to its bearings. The differential transformer uses a set of three coils of wire wound on a cylinder common to all. (fig 3.1) An alternating voltage is fed to the center winding at a fixed frequency and level. A slug moves inside the cylinder and its position determines the proportion of signal induced in each of the other two coils. 27

Output Slug Input Figure 3.1 Differential Transformer Displacement Transducer The slugs' displacement is thus able to be determined. In use, the coils are attached to a component and the slug to another so that the relative displacement between the components is measured. Resistance types use an electrical POTENTIOMETER (variable resistor ) that moves in response to the displacement applied. The potentiometer is supplied with a steady voltage and the varying resistance provides a varying voltage signal proportional to displacement. The differential transformer and potentiometer types are usually used for relatively low frequency measurement up to about 20 Hz. Displacements up to about 300 mm can be measured. Higher frequencies are possible with the other types (up to 200kHz) but often this is possible for only very small displacements. 3.1.2 Velocity Transducers Velocity measurement may make use of all the types of transducers above, excepting the potentiometer type. In addition, the signal from an accelerometer may be integrated to give velocity. 3.1.3 Acceleration Transducers Probably the most common vibration measuring transducer is the accelerometer. The measurement of acceleration can be made by many types of transducers and the signal is able to be integrated to give either velocity or displacement signals. (Note: although theoretically possible, the differentiation of signals from displacement to velocity etc. are usually avoided due to stability problems with electronic differentiation) 28

Piezo Electric Accelerometers These are a very common type of accelerometer with a wide range of frequency small size and rugged construction. They use a mass that is attached to a crystal of material that varies its charge in response to mechanical stress. (fig 3.2) If the mass is accelerated, the crystal provides a signal proportional to the acceleration. Piezo accelerometers cover a wide range of sensitivity from about.00005 m/sec.sec (5 x 10 g) to 200, 000 m/sec.sec (20, 400g). Mass Electronics Piezocrystal Base Connector Figure 3.2 Basic Arrangement of a Piezoelectric Accelerometer Piezo Resistive Accelerometers These types of accelerometer use a silicon strain gauge. This type of strain gauge is not the metal foil type commonly used for engineering strain measurements. The metal foil type is more linear and much easier to handle than the silicon type.for permanent measurement situations however, the silicon type has the advantage of giving higher sensitivity. These accelerometers use a mass supported on a beam which has the strain gauges attached.(fig.3.3) The strain in the beam is proportional to the applied force and hence the acceleration. These types can give good sensitivity and a response at zero Hertz, with good high frequency response. 29

Mass Connector Strain Gages Cantilever Beam Base Figure 3.3 Piezoresistive Accelerometer They are usually larger than an equivalent piezo electric type with the exception of some special micro types designed for very high level acceleration measurements. Servo Accelerometers These types of accelerometer use a technique of measuring the force required to restore a mass to its rest postion when accelerated.this force is of course, proportional to the acceleration applied to the accelerometer.this type of accelerometer can be made very sensitive to low acceleration levels, with a frequency response from 0 to up to 500 Hz. 3.1.4 Optical Transducers Optical transducers have not been used extensively in general purpose vibration measurement. Recently instruments have been produced that allow relatively easy use of laser interferometer techniques. Although very expensive, these methods are extremely powerful for studying vibration of large surfaces or where a noncontact measurement is needed, such as on fast moving machine components. 3.2 Selection Of A Transducer With any vibration measurement, we must have some idea of the range of vibration amplitude and frequency that is to be measured so that an appropriate transducer can be selected. We should not simply select a transducer with very high amplitude and frequency range performance, as the sensitivity small vibration levels may not be adequate. Frequency response data is needed to select the transducer and may be quite different for different transducer designs. 30

The useable part of the frequency response is on the flat or LINEAR part of the curve and if used outside this range, the results given will be incorrect due to change of sensitivity. Care should also be taken not to expose very sensitive transducers to shock as they may be damaged. This includes transport in vehicles to the measurement site. The mass of the transducer is also important.recall that the natural frequency of a vibrating system is related to spring constant and mass in the system. Adding a significant mass to the system will change the systems' vibrational character. The transducer mass must thus be very small compared to the system. The type of environment should also be considered when choosing the transducer.the temperature range, moisture level, dust, possibility of impact etc. should all be taken into account. For very difficult environments or permanent mounting in industrial situations, special ruggedised types of transducer are available. 3.3 Transducer Mounting A vibration transducer will measure ALL the vibration that occurs at the measuring point. This means that the mounting of the transducer must not provide additional vibration to that being measured. Mostly this means that the transducer should be connected well to the item being measured and any brackets used for mounting should be very stiff. 31

Piezo-electric transducer attachment (Pt 1 courtesy Bruel & Kjaer) 32

Piezo-electric transducer attachment (Pt 2 courtesy Bruel & Kjaer) 33

3.4 Use Of Conditioning Amplifiers Many of the transducers for vibration measuremnt require the use of a CONDITIONING AMPLIFIER.This is an electronic device that : amplifies the small signal from the transducer to a more useful level and range may provide power supply to the transducer allows the measured signal to be recorded in other instruments. ensure that the signal from the transducer does not overload the recording devices A conditioning amplifier is not essential for some transducers, but is recommended for all vibration measurement to ensure consistent results. The conditioning amplifier is essential for piezo electric transducers, due to the very small signal level generated by these devices. Some conditioning amplifiers also contain the integration circuits needed to convert acceleration signals to either velocity or displacement. Whatever conditioning is applied it must always be remembered to isolate, electrically, the transducer from the machine being measured when mains electrical supply is used. This is to prevent electrical noise pickup by the earth connections. (called ground loops) Transducer Conditioning Amplifier Meter or Other Measuring Device Figure 3.4 Conditioning Amplifier and Transducer 34

3.5 Calibration In any measuring system the dimensions measured must be assured to be correct. This is especially true where the system is measuring complex data such as random vibration, as the data is not easily determined to be correct by inspection. Calibration is the term given to the process of comparing the measuring system to a reference standard measurement and determining the systems'response. In vibration measuring systems, calibration is able to be carried out in the field, before measuring data, for most types of transducer. In addition, regular recalibration checks of transducers should be made by the equipment supplier or the national standards authority. In Australia this authority is the National Measurement Laboratory. 3.5.1 Methods Of Calibration Calibration must use a reference source for vibration and a means of determining the value of the reference. a. Vibration exciters - These devices range from hand held size to very large structural testing devices. Some devices have a fixed, stable level of vibration that is suitable as a reference for calibration without further equipment. Other devices require the additional use of a reference accelerometer or other transducer for determining the vibration level. The smaller devices are suitable for smaller piezo electric transducers. The larger devices are only limited by the mass of the transducer and the available force from the exciter. Transducer Calibrator 35

b. Displacement calibration - For displacement transducers the use of some length reference may be adequate for calibration. (static only) For example, differential transformers may be calibrated by gauge blocks or precisely machined reference gauges. DVDT Calibration reference length (guage blocks) c. Static calibration with gravity - The use of servo accelerometers, piezo resistive accelerometers or other types of transducer with a response at zero Hz may allow the calibration to be done by tilting in the earths' gravitational field. Accelerometer Angle to provide desired static acceleration due to gravity acceleration= 9.81 x sin θ θ 36

d. Low frequency pendulum - A pendulum may be used to oscillate an accelerometer in the earths' gravity to give low frequency calibration. Pendulum Accelerometer e. Centrifuge - An accelerometer may be placed in a centrifuge to allow very high accelerations to be developed. Centrifuge Accelerometer 37

Many of the calibration systems described above are not suitable for field work. Thus reliance must be placed on the sensitivity of the transducer ( eg volts/g, mv/mm, mv/m/sec) and if this is the case a calibration of the voltages in the system must be made to give an accurate reference. This can be done using the internal calibration on some conditioning amplifiers or a precision voltage reference source. Whenever possible, the whole vibration system should be calibrated. That is, when the transducer is recieving the calibration level, the amplifier and recording devices should also be operated to record the level right through the system. This tends to prevent errors in calculation of system performance as the data recorded can be compared to the recorded reference signal. 3.6 System Errors Any measurement system has an error associated with that measurement. It is essential that some consideration of the level of error of measurement be made. The frequently used estimate is that of the method of expected error. This is a value that can be expected for the ordinary circumstances. It is expressed as: Error = e 2 + e 2 + e 2 +... + e 2 1 2 3 n the en are the errors for each part of the measurement (ie each instrument ) For example, consider the following system: a. transducer error +/- 1% b. conditioning amplifier error +/- 1.5% c. tape recorder error +/- 3% d. frequency analyser error +/- 0.4% 2 2 2 Error = 1 + 15. + 3 + 04. 2 = +/- 3.5 % While this may appear to be a large error, it is typical of most field measurements which are of the order of +/- 5% accuracy. Note that this is not the worst case error which can be as much as 1 + 1.5 +3 + 0.4 = +/- 5.9 % for the example above. This is the error that would occur if the worst error occurred in each instrument at the same time. This process is also only considering system 38

errors, not errors of use of the equipment. Incorrect use, poor calibration or malfunctioning equipment can give unpredictable errors. 3.7 Measurement Of Shock Comments about shock measurement have been left to this section as this area is particularly sensitive to the concern of errors and calibration. Transducers that are to measure shock must be carefully constructed to give a sensitivity that is in the direction of the measurement axis and no other. For example, some types ofgeneral purpose accelerometers are adequate for general measurement but under shock conditions they may exhibit sensitivity in other directions leading to false results. It is preferable that transducers for shock measurement be calibrated at the levels of shock to be measured. This will ensure a reasonable confidence in the transducer performance. At the same time the conditioning amplifier should be considered as this must give a rapid rise time to follow the impulse or shock data. The tape recorder or data analyser also must be "fast enough" to record the data reliably. Test them all at calibration if possible. Calibration can be done using a falling pendulum which collides with a barrier. This is a specialised area and assistance should be sought to ensure accurate results. 3.8 Data Recording Vibration data may make use of the following means of data recording: chart recorders tape recorders continuous analysis digital recorders computers 3.8.1 Recording On Paper Chart a.pen Recorders these may be used to record the amplitude variations with time so that these can be studied manuallly. They can't be used for signals that change much more rapidly than 200 Hz. They are useful to study variation in level with time, peak levels and decay rates in buildings. b.printers - some printers can make a chart like the pen recorders. More frequently they are used to give a permanent record of vibration levels at various times during a measurement. Useful for long term studies to give hard copy that can be plotted later. 39

c.lcd Chart Recorders - able to take high speed recordings up to about 10 khz, these chart recorders are often built as data loggers with extensive hard drive or CD memory. 3.8.2 Continuous Analysis Some instruments take in and analyse data continuously to provide an overall assessment of vibration over a period of time. Data is often processed or weighted by frequency to allow evaluation over a specific range of frequencies and excluding others. This type of analysis is often chosen to evaluate the effects of vibration on humans. 3.8.3 Tape Recording The tape recorder is a very valuable instrument for vibration measurements. There are a number of ways of making tape recordings. The performance of the different methods is affected always by tape speed. For higher tape speeds, a higher frequency response is possible. a. Direct Recording - Direct recording (DR) the most common type of acoustic recording method. A frequency range of 20 to 15 khz is typical, but higher maximum frequency is possible depending on tape speed. Data must be replayed at the recorded speed to give correct frequency information. b. FM Recording - instrumentation tape recorders allow FM (frequency modulation ) recording. The frequency range begins at 0 Hz and extends to a limit controlled by tape speed. For the same tape speed the upper limit of FM is less than DR. FM is generally more accurate than DR and is able to replay data at any tape speed without loss of relative frequency data. This allows for example, detailed analysis of fast changing data by slowing down the tape speed. FM requires about 4 times more tape than DR to accomodate the same data. (because of the frequency range limits) c.digital Recording - Digital recording uses conversion of data to a digital code which is then stored on tape. The coding of this method makes it extremely accurate. New technology is making digital recording available in compact devices that makes it very useful for vibration recording. Digital recording requires more tape than FM for the same data in a ratio about 2:3. Being used more extensively as new equipment enters the market, especially in the DAT format with special input electronics. This type of recorder 40

allows direct input to computer via a digital output port. This saves a digitising process which is otherwise needed for computer data processing. 3.8.4 Digital Memory By coding the data into digits, it may be stored in an electronic memory of the type used in computers. Devices such as event recorders and some data loggers use this method. This technique is usually only used for relatively short bursts of data such as explosions due to the limits of storage of the memory. 3.8.5 Computers Computers have been used for many years for data analysis. Special input electronics modules have been available for multichannel measurement and analysis. In recent years, the development of high quality sound cards for PC and laptops have provided a relatively low cost data entry system. 3.8.6 Calibration Recording devices must be calibrated so that the recorded values can be interpreted later. The most direct and simple way to calibrate is to record the transducer calibration. This gives a reference level on the recording device. CALIBRATION is essential for confidence in measurement 41

3.9 Data Analysis Instrumentation The use of correct instrumentation for data analysis will allow thorough and meaningful results from vibration measurement. The types of possible data analysis has been discussed. The type of equipment needed to carry out these analyses will now be described. 3.9.1 Vibration Meter A very common and simple means of measuring vibration is to use a vibration meter. This type of device gives an overall reading of the vibration level, usually an RMS value. The vibration meter may include signal conditioning and integrating sections so that an accelerometer can be used for measuring acceleration, velocity or displacement. This type of instrument may also include weighting curves that may be used for the assessment of particular types of vibration such as machinery or human comfort. 3.9.2 Weighting Curves As with acoustic measurement, the use of weighting curves is also common in vibration analysis. The weighting curve can be a filter or it can be produced by numerically adjusting frequency data that is produced by an FFT analyser. Weighting curves are used primarily in vibration analysis for the assessment of human effects of vibration. The numerical adjustment of data referred to above is usually performed on a computer. 3.9.3 Frequency Analysers Much use is made in vibration analysis of frequency analysers. The most commonly used analyser is a constant bandwidth FFT (Fast Fourier Transform) instrument. While it is possible to utilise constant percentage bandwidth instruments such as one-third octave analysers, as used in acoustic analysis, these instruments generally cannot provide the detail needed at lower frequencies for machine vibration analysis. It is essential in most measurements of vibration that the frequency can be determined with an accuracy that can discriminate between the different parts that are causing vibration and this is easily achieved with the FFT analyser. FFT analysers are often implemented as hardware or firmware units but may also be implemented as software applications in PCs or laptops. 42

Vibration Isolation 4.1 Machine Vibration All machinery has some vibration. This fact has already been used to advantage in machine condition monitoring. In many cases however we wish to either isolate the machine vibration from other equipment or from people or isolate the equipment from vibration. As we have seen already, the vibration of a spring mass system can be expressed in the form of a frequency response curve. (see figure 4.1). 4.1.2 Degrees Of Freedom The curve in figure 4.1 is the response to forced vibration of a "single degree of freedom " system. Such a system has motion that is restricted to one direction of displacement. If the possible axes of displacement are considered for three dimensions, (figure 4.2) Figure 4.2 Position In Three Dimensions it can be seen that for a general three dimensional position in space of a single rigid body, that six degrees of freedom are possible. ( ie six possible directions of displacement) The single degree of freedom system then has only one possible direction of displacement which may be either a translational or rotational displacement. Very few real machines are single degree of freedom systems. Most are composed of many parts with connections of 43