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Vibration Measurement & Control

by Brian McLauchlan

© B. McLauchlan TAFE SI 2006

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

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TABLE OF SYMBOLS δ .......................... ..density, kg/m3 f........................... ..frequency, Hz ff .............................forcing frequency, Hz fn ............................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/sec2 y.............................position in meters

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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

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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.

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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 =

1 k 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 )

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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:

X = A sin ( ϖ t ) where 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.
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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 = A sin(ωt )

displacement

X = Aω cos(ωt )
X = − Aω 2 sin(ωt )
••

velocity

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.

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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.

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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 m/sec )
2

UNITS USED m , mm , um

m/sec , mm/sec m/sec2 , " g " ( % of 9.8

ACCELERATION

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
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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
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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.

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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 0.1 1 Frequency Ratio 0 10

Figure 1.6 Response to Forced Vibration (1 DOF Spring-Mass)

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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 V i b r a t i o n 8 6 4 2 0 -2 -4 -6 -8 -10 time

Peak RMS

Peak to peak

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
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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
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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
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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.

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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.
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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.

Time Amplitude

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.

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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.
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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.

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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

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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.
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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 spectra

Error 95% c.i.

Length of data record for full scale frequency

8....... ........ 2.8dB 16..... ........ 2.0dB 32..... ........ 1.46dB 64..... ........ 1.06dB

100Hz 32 64 128 256

1kHz 3.2 6.4 12.8 25.6 23

10kHz 0.32 0.64 1.28 2.56
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128... ........ 0.74dB 256... ........ 0.54 dB

512 51.2 5.12 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

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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 made. be

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.
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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

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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.

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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)
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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).
Electronics Mass 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.

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Mass Connector

Cantilever Beam Strain Gages 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.

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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.

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

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

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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

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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. Calibrator Transducer

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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

θ
acceleration= 9.81 x sin θ

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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

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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 1 + e 2 2 + e 2 3 +......+ e 2 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%

Error = 12 + 15 . 2 + 32 + 0.42
= +/- 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
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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.
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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
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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

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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.

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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

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varying rigidity. This greatly increases the complexity of the analysis of the machine and its' vibration.

Response to Forced Vibration
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 0.1 1 Frequency Ratio 0 10

Figure 4.1 Frequency Response Curve One Degree Of Freedom System
For instance, many vehicle suspension systems are designed considering all the degrees of freedom possible. (eg motor cars, trains) This may require analysis of over 20 degrees of freedom ! This type of analysis can only be reasonably achieved by using computer programs called simulations that solve the mathematical models of the system continuously in time. Fortunately, for many purposes use of a single degree of freedom system as an APPROXIMATION to the real system is possible. In this course we
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will only deal with single degree of freedom systems. It is essential however to be aware of the limitation of this approximation when designing vibration isolation for machines. 4.1.3 The Response Curve And Transmissiblity The response curve shown in figure 4.1 has two axes. The horizontal axis is the ratio between the natural frequency and the applied ( "forcing ") frequency. The vertical axis is labelled TRANSMISSIBILITY. We have used this axis to determine either force ratio or displacement ratio for the vibrating systems. The term TRANSMISSIBILITY refers to the ratio of transmitted force or displacement to that applied to the system. (see figure 4.3) As an example, we may consider the reduction in force to the foundation of a machine which has isolators providing a ratio of forced to natural frequency of 4. This gives a TRANSMISSIBILITY of 0.04 (vertical axis, zero damping). This means that force caused by the vibration of the machine when operating (ie at the forcing frequency) will be reduced to 0.04 of the original at the machine foundation or base. The transmissibility is 0.04 or a (1 - 0.04) x 100 = 96% reduction in force level. The transmissibility is a description of the quality of performance of a vibration isolation system and may be applied as a design standard. It may describe either force or displacement reduction of the system. Although we have only calculated single degree of freedom examples, transmissiblity may be applied to any system. An alternative means of expressing this requirement is the ISOLATION. The isolation value is frequently used on data sheets for commercial vibration control products. Isolation is a measure of the reduction of vibration and is expressed as a percentage. The calculation above for Transmissibility = 0.04 showed that this was a 96% reduction in vibration level. The 96% is the isolation. This leads to the relationships: Isolation % = (1- Transmissibility) x100 Transmissibility = (1- Isolation)
(note: use the decimal form of percentage to make this formula work; eg 96% =0.96)

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Force on support G

Input displacement Y

m

m

Input exciting force F T = G/F

Displacement of Mass X T= X/Y

figure 4.3 Showing Transmissibility for force and displacement

4.2 Static Deflection When calculating the natural frequency of a system, the following formula is used:

ω=

k m

With any system the spring constant k must be determined to suit the above formula and the transmissiblility required.

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Consider the spring constant: F= kx where k = spring constant, N/m F = force , N x = displacement , m

If a machine is lowered onto an isolation system, the springs will deflect from their unloaded position by an amount proportional to the mass of the machine. This is called the static deflection. If the static deflection is known we can calculate the spring constant. We thus have, using the above equation;

k=
so,

F x

k=

mg x static

If this information is substituted in the equation for natural frequency we have:

ω=
or

mg mx static

ω=

g x

We thus have another way of expressing the natural frequency of a system in terms only of the static deflection. How is this useful ? For a required transmissibility, say 0.1, we can determine the required frequency ratio to be about 3.4 (figure 4.1) By knowing the forcing frequency we determine the required natural frequency.

ωnatural =ωforcing / 3.4
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We may now determine the required static deflection directly from the equation developed above. The suppliers of vibration isolators often supply information in this form (static deflection and mass) so that this simple relationship can be used for design.

4.3 Center Of Gravity When machinery is being installed with vibration isolators it is often necessary to consider the position of the center of gravity of the machine. This is the position where all the mass of the machine can be considered to act through a single point giving the same static force. If the center of gravity is known, vibration isolators can be positioned at equal displacements from this point to ensure equal loading. Equal loading on each isolator is preferable as this means that all isolators are of the same type. This prevents confusion during maintenance and minimises spare parts stock. The center of gravity is calculated in the conventional method of mechanics using moments about an arbitrary point. It is preferable to mount the isolators vertically in the plane of the centre of gravity. This will prevent some possible additional rocking vibration. This arrangement is not always practicable and is usually reserved for machines that are very large or that have a particularly difficult vibration problem. It should be seriously considered also in the case of a machine that is very tall compared to its width. These situations are illustrated in figure 4.3.

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a) conventional mounting

Centre of Gravity

b) mounting in plane of center of gravity

Figure 4.3 Center Of Gravity And Mounting Of Isolators

4.3.1 Location Of Isolators We have already observed that the isolators should be mounted so as to have equal vertical static force. There are a number of other practical concerns regarding the positioning of isolators so that they operate as designed.

a)Horizontal spacing: In figure 4.4 a) the isolators are shown positioned at equal spacing from the center of gravity however, the isolators are also very close together. This could cause problems of rocking vibration. It is thus preferable to mount the isolators as far apart as is practicable.

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b)Lateral loading: If the isolators are to operate as desired it must be ensured that the required spring constant is achieved. It is possible that the arrangement of mechanical drives or other connections to the machine will add stiffening to the spring constant of the isolators. Remember that a decrease in the spring constant will increase the isolation and so an increase in the spring constant will DECREASE the isolation. Careful attention to the design will ensure that this problem does not arise. Figure 4.5 a) shows examples of designs that interfere with the operation of the isolators. Methods of improving the design are shown in figure 4.4 b).

a) close mounting (unsatisfactory)

b) far apart mounting (desirable)

Figure 4.4 Position Of Isolators

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a) external loading on isolators (unsatisfactory)

b) correct design to remove isolator side loads

Figure 4.5 Design Of Isolator Installation
c) Connection of pipes electrical etc. In a manner similar to that described in part b), the stiffness may be effected by pipework or other connections to a machine that is fitted with vibration isolators. Because this is not always a serious reduction in spring constant, these items are often ignored. The additional problem here is that the vibration may be transmitted along these connections, causing excessive noise and potenially excessive stresses that could lead to failure of parts. This is illustrated in figure 4.6 a) where a number of unsatisfactory arrangements are shown. In 4.6 b) alternative arrangements are shown. Often these alternatives also provide allowance for thermal expansion of machines and certainly will reduce the incidence of failure in connections.

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a) direct connection to machinery (unsatisfactory)

b) connections that isolate vibration

Figure 4.6 Connections To Vibrating Machines
d) Spring Constant with Multiple Isolators In all the analysis, we have assumed a spring / mass system with a specific spring constant but have only referred to one spring. Most realistic isolation designs for machines require 4 or more springs. How do we relate the spring constant from the analysis to the practical application? Springs can be connected in either series or parallel as shown in figure 4.7. The same results apply to either tension or compression springs.

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Springs in Series or Parallel

Series

Parallel

K K K K K

K

K K total = K + K + K + K K

K total = 1 / (1/K + 1/K + 1/K + 1/K)

If K = 100 N/m for all springs: K total (Series) = 25 N/m K total (Parallel) = 400 N/m

Figure 4.7 Spring Constant for Multiple Springs
Figure 4.7 shows that springs in series decrease the spring rate whereas the same springs in parallel increase the spring rate (compared to the rate of a single spring) When supporting a machine with multiple springs the spring constant determined from the analysis must be divided by the number of springs that will be used for support. The new spring constant determined will be used to select the actual springs used. For example, the required spring rate for isolation of machine vibration might be 50kN/m. If 4 springs are to be used for supporting the machine, then each spring will require a spring constant of 50/4 =12.5 kN/m.

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4.4 Types Of Isolators 4.4.1 Materials Many materials may be used for isolation of vibration. A list of the common types follows:

Material felt

Form that it is used felt mat felt composites cork blocks cork composites rubber with non - metallic reinforcement Neoprene and cork natural rubber synthetic rubbers

cork

composite materials

rubbers

wire mesh metal spring coil spring leaf springs air bellows

air

Each material has an area of specific application related to the available static deflection. This means that selection in some applications may be limited to one type of isolator. Figure 4.8 shows the typical range of application for the types of material.

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Deflection mm 250

Natural Frequency Hz 1.0

Isolator

Air

25 Metal Springs 2.5 10 Rubbers 0.25 Cork

0.025

100
Figure 4.8 Areas Of Use Of Isolation Materials

4.4.2 Using Materials FELT A matrix of textile fibres which is usually supplied in sheet form. Usually used for applications where well balanced machines are expected to produce vibrations at acoustic frequencies. It is necessary to use a low loading per unit area to prevent settling and loss of isolation. Often used between a concrete machine base and the foundation or floor. CORK A naturally occurring substance that is processed into convenient shapes for use. Cork may be used for high compressive loads at low acoustic frequencies (50 - 60 Hz) More widely used than felt as the engineering properties are more readilly available. Able to be treated against most industrial contamination. RUBBERS These materials offer a wide range of mouldable shapes and properties that provide isolators for many applications. Rubber materials have varying resistance to industrial contaminants and the following should be considered: 1 Temperature of operation 2 Ozone 3 Sunlight
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4 Liquids eg oils The manufacturers'data should be consulted to ensure adequate life in the particlular environment that is to be considered. METAL SPRINGS Used in the range 3 - 10 Hz these isolators provide the lower frequency area of isolation after the rubbers. A wide range of spring types and sizes means that a suitable design is usually able to be achieved and provide good environmental performance. These springs however usually have very low damping and generally require external damping to achieve desired performance. AIR SPRINGS Used for high isolation at low frequency and in some cases where vibration amplitude is very small. The air spring system can provide a constant hieght with varying load while maintaining the required isolation performance. Although more complex and costly than other systems, it is often the only choice for high performance isolation. 4.5 Damping The term damping has been mentioned previously and must be considered in all isolation design. Damping refers to that part of a vibrating system that provides loss of vibration energy, either intentionally or unintenionally. Damping ensures that vibration does not continue indefinitely after excitation of the system occurs. Isolation materials have a degree of internal damping. Some values are listed in table 4.1 Material steel rubbers air damping felt/cork Approximate damping ratio 0.005 0.05 0.17 0.06

Table 4.1 Damping Ratios Of Some Materials
The effect of damping appears in figure 4.1 as the damping ratio. This is the ratio of actual damping to critical damping (see sect. 1 and 2).

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It can be seen that the isolation effect reduces as damping increases so why have damping ? Damping provides a limit to the oscillations at resonance. For many machines operation at resonance will not occur but may be experienced at start up or during run down. Other machines may not be able to completely avoid resonant operation and thus damping must keep oscillations at an acceptable level. Damping also effects the natural frequency of the system. The natural frequency of the system is altered by :

f nd = f n 1 − ξ 2
where fnd= natural frequency with damping fn= natural frequency ξ = damping ratio In many cases the effect is small, for example with rubber isolators with a damping ratio of 0.05 the change in natural frequency is less 0.2 %.

4.6 Real Isolation Actual isolation may not be exactly as predicted by the one degree of freedom model. The reductions at acoustic frequencies may be less than that in the transmissiblity graph. This comes from the simplified mathematics used for this system. In a real system the mass and stiffness of the support structure are important in achieving reasonable performance. 4.6.1 Support Structure As a guide, the mass of the supporting structure should be at least 1000 times greater than the mass that is vibrating. The foundation stiffness is also of concern because it acts as another spring in the system. The natural frequency fn of the spring mass system of the isolated machine should be lower than the lowest natural frequency of the foundation structure. The natural frequency of the foundation should also not be an exact multiple of fn. To assist in this it is better to have as low stiffness isolator as possible when the foundation is suspected of some significant flexibility such as a suspended floor. An example of this is a machine operating at 1500 rpm could be isolated on a basement floor of a building with 95% isolation by a static deflection of 8.6 mm.
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For suspended floors such values as: 10 m span 12 m span 15 m span 10 mm static deflection 10.7 mm " " 11.2 mm " "

are recommended by a machine supplier. 4.6.2 Mounting Frame Any frame that is used to mount a vibrating machine should be designed with maximum feasible stiffness to prevent amplification of vibration. 4.6.3 Lateral Stability When designing any vibration isolation installations it is clear that it is desireable to have a relatively large static deflection. This provides minimum transmissibility. An isolator with low stiffness (ie large static deflection) may also have a low lateral stiffness. This may affect other modes of vibration but also may cause problems with excessive motion of the isolated machine. This type of problem is usually more significant if the machinery is in motion as in a vehicle. Acceleration or retardation of the vehicle will mean a large longitudinal or lateral force must be transmitted by the vibration isolators. The large force and /or displacement resulting could damage the isolators or even disconnect the machine from its mountings. To overcome this motion is often limited by safety stops or resilient bumpers. This problem also often limits the practical spring constants usable and hence place a limit on the realisable maximum performance of isolators. Thus, even though it is theoretically possible to achieve a reduction of 99. 99% of the vibration it may not be practical due to the very low lateral spring constant leading to instability.

4.7 Isolation Of Equipment From Base Vibrations In most of the above examples we have been discussing the isolation of a vibrating machine from the structure that is carrying it. In some cases, such as the use of delicate equipment on a vibrating machine, the isolation is to minimise the vibration recieved by this equipment. Even with low transmissiblity isolation, damage may occur to the sensitive equipment. This is because of the resonant effect that we have already studied. Low level
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vibrations that are transmitted can be amplified by a resonance in the equipment. To avoid this problem we must know what the resonances are. We have seen already that most real systems are multi - degree of freedom systems. The analysis for these systems is complex but may often be achieved using finite element methods on a computer. We can also experimentally study the response of the system to vibration by placing the delicate equipment on a "shaker" like those used for calibration but with the ability to provide a variable frequency and level. Resonances of equipment can then be observed by accelerometers or non-contact measurement using laser interferometry.

4.8 Summary of Isolator Considerations For practical design of isolators then we should consider: the required transmissibility b) even spacing from the center of gravity c) positioning in the plane of the center of gravity d) positioning of equipment to avoid loss of isolator performance e) minimising vibration transmission through pipes etc. f) the type of isolator material and its environment g) damping of the isolator and need for additional damping h) effect of the isolator at higher frequencies i) support stiffness and mass j) lateral stability of the isolators

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Vibration - Human Effects
5.1 The Human Body The human body is composed of soft tissue and a hard skeleton with connections of firmer muscle tissue. This structure has then basically a rigid frame with masses hung on resilient (springy) mountings. This means that the human body can be considered as a complex of spring mass systems that will respond to vibration. There are two major areas of concern for the human effects of vibration: a.whole - body vibration (vibration applied to the entire body) b.hand - arm vibration (vibration applied in a limited area of the hand/arm as when using vibrating tools) 5.2 Whole Body Vibration Interest in the response of the human body to vibration began with the transport industries. Concern with passenger comfort in rail vehicles, ships, aircraft and limits for military vehicles each generated research into this area. The research has shown many similarities but, as yet little correlation between vibration levels and injury has been established. The vibration criteria are thus not as clearly defined in requirement as for instance noise exposure. In noise exposure, we were concerned with damage to the human ear. What are the effects on the human body from vibration ? 5.2.1 A Body Model The human body can be modelled as a series of spring mass systems made up of muscle and tissue as spring elements, body organs as masses and connected via the skeleton. The different spring - mass systems represent the different parts of the body, for example, the internal organs are suspended by tissues forming the spring mass system. Each system will have a different natural frequency. This means that certain vibrations will make different parts oscillate.

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Figure 5.1 The Human Body as a Spring – Mass System (Coerman et al)

5.3 Testing Human Vibration Sensitivity Most of the testing for human vibration sensitivity has been done by subjecting volunteers to varying sinusoidal vibration and recording their response. This method does not completely establish the response of the body to a complex vibration, such as multiple axis vibration, but does give an indication of trends. This means that the effects of random vibration, for instance, will not always follow the same sensitivity curves Recent work (1997) has begun to address this problem to get more accurate data..

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5.3.1 Direction Of Vibration For whole body vibration, the vibration may enter the body while in a standing or seated position. The vibration may be horizontal, vertical or lateral. The human body has differing sensitivity in each of these directions due to differing stiffness in each plane. This means that muscle effort is required to restrain the body parts, leading to fatigue. The vibration may even interfere with the organs' function, such as distortion of vision by vibration of the eye. Excessive levels of vibration can lead to tissue damage or illness. For example, oscillations from earth moving machinery may cause spinal injury. 5.4 Vibration Criteria Due to the differing uses for vibration criteria they been frequently considered in separate categories of comfort limits and exposure limits.

5.4.1 Comfort Limits These limits are established to provide passenger comfort in the transport of humans. Comfort limits are usually expressed in terms of time of exposure. That is, a lesser comfort level is acceptable for short duration trips and for longer trips lower levels of vibration must be provided. This is to ensure that the passenger will not be excessively fatigued at the end of the journey. Each mode of transport has differing vibration source mechanisms. This leads to differing comfort criteria. For example, ships must consider very low frequency vibration which does not occur in most other forms of transport. Aircraft have high frequency vibrations that are more irregular than other transport. 5.4.2 Exposure Limits These limits have been set to indicate the maximium levels that humans can be exposed to in industrial situations. In using these limits, it should be noted that the fatigue level will be higher and so task performance will reduce. If these limits are to be used, they should be considered as infrequent exposures as in an emergency situation. 5.4 The ISO Standard The vibration limits for human exposure have been expressed as an ISO standard. This standard has been used as the basis for the Australian Standard (AS2670) Although this standard is based on
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pure sinusoidal vibration tests on subjects, it has become acceptable to use this in the absence of more precise information. It is a good guide for general industrial applications and as a start point for establishing special purpose vibration standards. Weighting curves are used to give a single value measure of comfort levels. The Australian standard has also included the very low frequency extension to the ISO standard that covers the motion sickness region. Modified criteria are used as guidelines for exposure limits to protect against ill health or injury.

5.5 Instrumentation Considering the frequency range of the human sensitivity to whole body vibration, we can determine the frequency range and sensitivity of the required measuring instruments. The standard shows that a maximum level of 1.0 g will be applicable for most measurements. The frequency range will be between 50 and 100 Hz, depending on the likely vibration source spectrum. For example, aircraft measurements will require up to 100 Hz, while motor vehicle measurements will be adequately covered by a 50 Hz range. If motion sickness measurements are to be made, the low frequency range, 0.1 to 1.0 Hz, must also be considered. For these types of measurement, the use of servo accelerometer or piezo resistive accelerometer types is common. The piezo electric types sometimes cannot provide the low frequency sensitivity, unless specially selected. The measurement of vibration should be made at the point of entry of the vibration to the human body. That is, if the subject is seated, the vibration limits apply at the seat surface not the floor that the seat is attached to. This is because the seat itself may amplify the vibration that comes from the floor. The low frequency of the vibration data means that only FM tape recorders can be used to record the information. A suitable conditioning amplifier must also be selected. As the measured data may be either weighted or unweighted the assessment according to the standard will be in two possible manners:

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i. weighted data compared directly to the AS2670 curves to determine the allowable exposure time ii. unweighted data is first weighted in each frequency band and then compared to the AS2670 curves. In the first case, the RMS value from the instrumentation is directly compared to the curves. The second case may utilise computer calculation to speed the weighting and assessment process. Where vibration is steady, the first method may give the most rapid result. If vibration is unsteady, for example, random vibration in a vehicle, an averaging process may be needed to ensure the correct exposure is determined. This may be achieved by measures at various levels that are typical of the environment and summing in proportion to the time exposed at each level. This is equivalent to the Leq method used for noise. Alternatively, a sample may be taken over a representative period using an integrating process to determine the total exposure. 5.6 Hand Arm Vibration Whole body vibration is frequently able to be maintained at or below fatigue levels so that significant injury is not normally of concern. With hand arm vibration however, there is a real danger of serious and permanent injury. Vibration at high levels applied to the hands while using machinery or tools can cause disease of the blood vessels, joints and blood circulation system. The result is what is called "vibration white finger" or Reynauds' disease. Circulation of the blood becomes so poor that fingers and hands may turn white and in extreme cases permanent tissue damage or gangrene may result. The type of machinery that can cause this type of injury is that producing high levels of vibration and which must be hand held for long periods of time. For example: Chainsaws, chipping hammers, rivetting guns, power grinders, and hammer drills. These types of machines may be found in many industries and are often able to generate very high levels of vibration in human hands and arms. 5.7 Hand Arm Exposure Limits

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An ISO standard has been developed which forms the basis of the AUSTRALIAN STANDARD. The Australian Standard also includes reccommendations for medical recording of the effects of vibration exposure to the hand arm. This is because of the relatively limited experience in Australia with this disease. The standard specifies the postition for measuring vibration levels as an axis system related to the human skeleton at the wrist. This is to standardise the level allowed in the measured direction. The standard provides a weighting curve from which allowable exposure can be determined.

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5.8 Instrumentation The most appropriate instrumentation for this work is an integrating vibration meter with the standard weighting curve. This allows the measurement of vibration directly in a weighted RMS value that can be compared to the allowable exposure limits. The placement of the accelerometer to measure this vibration is probably the most difficult part of this type of measurement. The requirement is to measure at the "3rd metacarpal" (ie knuckle) of the exposed hand. To achieve this a special mounting for the accelerometer is required. Many researchers have made their own mountings but now commercial items such as that by Bruel and Kjaer are available. This type of measurement usually requires an miniature accelerometer so that the instrumentation is not too bulky so that it interferes with the operation of the machine being measured. In use the three axes of vibration must be measured by successive measurements after repositioning the accelerometer. 5.9 Action To Minimise Risk The incidence of "vibration white finger" in Australia is presently less than other areas as the climate is more favourable in this country. The disease apparently has more rapid onset with cold extremities. That is, where workers are exposed to lower temperature ranges while operating the equipment. This does not mean that workers are safe in Australia, only that they may require more time before the injury appears. To protect operators then measurements must be made so that doseage will be limited. The coupling of vibration to the hand is also important so that the wearing of glove should be ecouraged wherever possible. This is especially important where low temperatures are experienced. 5.10 Building Vibration The vibration experienced in buildings is not generally regarded as dangerous to health. There is however some concern for discomfort of humans in buildings, especially the taller ones that are subject to wind sway. The ISO standard provides guidelines for acceptable vibration levels for these situations.

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Balancing Of Machinery
6.1 Reasons For Balancing Balance refers to the even distribution of mass around the axis of rotation. Any variation from this distribution results in imbalance.

Axis of Rotation & centre of gravity aligned

Axis of Rotation & centre of gravity misaligned (not balanced)

Figure 6.1 Distribution Of Mass Of A Rotating Machine
For any rotating machinery, any out of balance will cause a rotating force proportional to the amount of unbalance. This force will cause vibration that can lead to excessive wear and/or failure of the machine. Unbalance is probably the most common source of vibration in rotating machinery. For any machine except those that rotate at very low speeds, balancing must be considered part of manufacture and maintenance. 6.2 Unbalance Force The force due to unbalance can be expressed by :

F = mrω 2
where F represents the rotating force due to imbalance. The other factors are m , the mass at radius r that is unbalanced , and w is the angular velocity. Note that the unbalance force increases in proportion to m or r but increases as the square of ω. Thus as operating speeds are increased the need for balancing becomes rapidly more important.

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Rotating Force due to unbalance

Figure 6.2 Force due to unbalance
Notice that the force is rotating at the shaft speed so the frequency of vibration due to unbalance is expected to be: Frequency of unbalance (Hz) = RPM of shaft / 60 In the example in figure 6.1 the rotor is represented by a disc. While some machines may be considered to be like this, for example, fans, many machines have rotating parts that have a considerable length. (See figure 6.3) Real rotors may have unbalance in two planes that is of the same mass but exactly 180 degrees out of alignment. Such a rotor is in STATIC balance but will still give an out of balance force at each bearing when run. This will be in the form of a couple formed by the two out of balance forces. Such a rotor is not DYNAMICALLY balanced. Static balance can be achieved by ensuring the rotor does not have a "heavy spot" when rotated at low speed. That is, the the rotor does not stop at any preferred position when when allowed to turn freely.

2 Plane Rotor

Multi-Plane Rotor

Figure 6.3 Real Machine Rotors
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The other alternative is that the out of balance masses will be of differing sizes and at differing angular positions and radii. This is the more common situation and will cause out of balance forces of differing magnitudes and phase at each bearing. A single rotor such as that in figure 6.1 is a 1 PLANE balancing problem. The rotors in figure 6.2 are examples of 2 PLANE and MULTI - PLANE balancing problems. In this course we will only discuss solutions to 1 PLANE problems. It can be seen that for any of these situations, the out of balance force is always radial and rotates at the shaft running speed. This is what allows us to detect unbalance by vibration measurements at the bearings.

6.2 Severity Standards Every real machine will have some amount of imbalance. Balancing of a machine is to ensure that this imbalance is sufficiently small to be tolerable. But what is tolerable ? The international standard ISO 1940 gives recommended quality grades for balance of rotors. This standard makes use of units of unbalance that are determined by considering the mass of the rotor to be eccentric from the axis of rotation by an amount that causes an equivalent unbalance force. The unbalanced force Fu can be expressed as :

Fu = mrω 2
Where r = m = unbalanced mass in grammes

radius from axis of rotation of m (in mm) effect of displacing the rotor mass can be

The equivalent expressed by:

Fu = Meω 2

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where :

M = mass of the rotor in kg e = eccentricity of the rotor from rotation axis

From these equations :

Meω 2 = mrω 2
or

e=

mr M

g .mm / kg

This gives the units referred to in ISO 1940. There is some concern in industry however that the balance quality grades were developed some years ago and reflect the need and technology of low to medium speed machines. It has been suggested that the ISO standard be used as a guide to orders of magnitude and attempts made to balance to as high a standard as possible, for a given application. 6.3 BALANCING OF ROTORS (Single Plane Balancing) In practice, a rotor may be balanced in place with simple vibration measurements if it can be assumed to be a single plane rotor such as a fan. Measurement of at least vibration level is required for balancing. If angular shaft position can also be measured, balancing will be more rapid. The measuring system will be discussed in a later section. 6.3.1 The Four Run Method Of Balancing This method requires only the measurement of vibration at the bearing closest to the rotor to be balanced. The method is as follows: a Install the accelerometer on the bearing b Place three marks at 120 degrees to each other on the rotor and mark them 1 , 2 , 3. c Run the machine and read the vibration level, Vi mm/sec. d Attach a trial mass at point 1 e Run the machine and read the vibration level , A1 mm/sec f Move the trial mass from point 1 to point 2 g Run the machine and read the vibration level , A2 mm/sec h Move the trial mass from point 2 to point 3 i Run the machine and read the vibration level , A3 mm/sec j Plot the data as circles.
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k From the diagram measure Vf from the center of the first circle to the intersection of the A1 , A2 , A3 circles. l Calculate the magnitude of the balance mass by:

mbalance =

Vf Vi

mtrial

m Place the balance mass at the same radius as that used for the trial mass and in the direction of the vector Vf n Check run the machine to ensure adequate balance Note : the trial mass can be determined from : m = Rotor mass angular velocity x Peak vibration level Radius of fixing

or kg mm/sec m = ------ x r/sec mm 6.3.2 Balancing Using Vibration And Position Measurement This method uses both measurement of bearing vibration and shaft angular position. Instrumentation for this will be discussed seperately. The method is as follows: a b c d e Position the accelerometer on the bearing Position the shaft transducer Mark the rotor at 90 degree increments Run the machine and measure the vibration Vi, mm/sec Fix a trial balance weight at 90 degrees against rotation from the angular position of the unbalance vector (use the previous method for an estimate of the balance weight) f Run the machine and meaure the vibration At, mm/sec g Remove the trial weight. h Plot the vector Vi to scale and position i Plot the vector At to scale on this diagram
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j Determine the magnitude of the balance weight from the ratio of Vi to At, that is: Vi At

m

balance

=

x mtrial

k Determine the direction of mounting from the angle OFB taken in the direction against rotation and from the present position of the trial mass. l Position the calculated balance weight at the same radius used for the trial weight m. Run the machine and check the vibration to ensure adequate balance has been achieved.

6.4 BALANCING OF ROTORS (Two Plane Balancing) Two plane balancing requires the determination of mass, angular position and radius for the balance weight at each end of the rotor. This requires many more trials than the methods above. Usually this type of rotor is balanced in a commercial balancing machine. 6.5 BALANCING OF ROTORS (Multi - Plane Balancing) The problems of multi - plane balancing are relatively complex and usually require the assistance of someone experienced in this type of work and the use of computer software for the calculations. 6.6 Instrumentation For Balancing The instrumentation for balancing single plane rotors is no more complex than that used for other vibration measurements. In section 6. 3. 1 the only items needed are an accelerometer, a conditioning amplifier and a meter to read vibration level, usually in mm/sec. The instrumentation should be chosen with adequate frequency response to measure at the running speed of the machine to be balanced. At times an additional filter will be required to reduce the signal at frequencies other than that of the running speed. In the method described in 6.3.2 the instrumentation as above is used with the addition of some form of angular position sensor. This sensor could be a non - contact proximity probe that is triggered by a key way on the shaft, a strip of reflective material or other method. From this detector the phase angle of the unbalance vector can be determined. This can be done on an oscilloscope by
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comparing the accelerometer data with the detector pulse position and calculating the phase angle. Alternatively, commercial instruments are available that calculate the phase angle directly from the data. In either case the data is used to apply the method described above. For two plane balancing, balancing machines are available as complete units. These use transducers at both bearings simultaneously. The data is reduced electronically to a readout that provides magnitude and position data for balancing at each end.

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Vibration Specification
9.1 When Is A Specification Required ? It is clear that a vibration specification may be included in a general specification that will be used when calling for tenders. There may also be reasons for vibration specifications within an organisation for example, for quality control. We should thus consider all these possibilities. The table below lists some possible areas that a specification for vibration may be applied.

Manufacturing : Type Of Vibration Product vibration Product vibration Product vibrationh Maintenance : Type Of Vibration Machine vibration Machine vibration Machine vibration Machine vibration Equipment purchase Type Of Vibration Machine vibration Machine vibration purchase Machine vibration

Specification For: quality control human body exposure human hand exposure

Example: electric motor helicopter power tools

Specification For: repair quality condition monitoring human body human hand : Specification For: performance criterion human body exposure human hand exposure

Example: after overhaul any machine exposure exposure

Example: pump quality tractor powertool purchase

Table 9.1 Application Of Specifications
9.2 Standards For Specifications Where possible, the existing standards from SAA, ISO or other sources should be used for specifications.

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9.3 Specification Content In creating a specification it is important that the required vibration levels be specified and that the type and position of instrumentation be included in the specification test procedures. The standards above provide ranges of values for specified conditions. The standards generally do not provide a specific value for the required conditions or positioning or type of instrumentation. 9.3.1 Level Specification The level of vibration must be selected by the specification writer to suit the conditions of use of the item specified. For example, the level of allowable vibration for a military vehicle for whole body vibration will be able to be set somewhat higher than that required for an executive limousine. In choosing the allowable level some consideration should be given to wear and age of components to ensure that the item specified has adequate performance for life between overhauls. This may require additional testing of "worn in" prototypes or some other means of predicting future performance. When specifying vibration levels it is important to consider the averaging time over which the measurement is to be taken. Clearly, for a machine operating at constant speed and load, the vibration should be relatively constant. For a machine subject to varying speed or load or both however, the vibration level will vary considerably in time. Averaging of the signal with time may be acceptable for some circumstances and not others. 9.3 2 Specification Of Frequency Consideration of the frequency content of the vibration must also be made, although this may be controlled using the standard weighting curves in the standards mentioned above. If the standard weighting curves are used these should be specifically mentioned and include the particular means of determining weighted vibration level. This is because the differing ways of determing weighted value may lead to slightly different answers. The means of determining the weighted level are: a. using an electronic weighting curve b. using a frequency analyser and mathematical weighting c. evaluating limits at each frequency individually

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It is usually preferable to use a. above however this means that if the data is weighted as it is recorded then the unweighted data is not available for further analysis. For many applications then it may be better to record the unweighted data and then process either with a weighting filter or a frequency analyser. The specification of frequency should also consider the range of frequencies to be measured. This should be specified as a frequency band and should be within the specified abilities of the specified instruments. 9.3.3 Instrumentation Specification The aim here is not to specify a particular brand of instrumentation but to ensure that the ability of the instrumentation system to measure the vibration accurately is ensured. Rather than specify each part of the instrumentation system it is often preferable to simply describe the overall performance of the system. This may still require some extra details for some applications for transducer performance particularly. We should specify the following: • • • • • • • • • • • required minimum sensitivity in appropriate vibration units expected maximum vibration level frequency range to be measured in Hz overall error in determining vibration level overall error in frequency determination or weighting methods or standards of calibration data recording method data analysis method ownership of data or copies to be supplied method of reporting results special requirements

9.3.4 Position Of Instrumentation The position of instrumentation is extremely important. Position usually refers to transducer position only. Small changes in position can make a large difference in the measured data and cause disagreement over acceptable performance. Where possible, utilise standard positions and fixing of transducers and fixtures that are of adequate stiffness to ensure accurate measurement. It is preferrable to measure the position of the transducers so that repeat measurements can be made in the same location and any calculation using the vibration data can be carried out correctly.

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9.4 Operating Conditions It is important that the operating conditions when measuring be specified. Some machines may only require measurements at constant speed and constant load. Others may require a range of speeds and loads to be measured or that run up or run down in speed be measured. Each required condition should be specified and the test conditions for each clearly indicated. This may require additional instrumentation such as tachometers, wattmeters (for load measurement), dynamometers etc. The accuracy of this equipment should also be specified. Operating conditions also may require certain environmental conditions such as temperature to be considered. The specification writer should provide test conditions that are appropriate but also those that can be realistically expected to be able to be produced and controlled at the time of test. This will apply particularly to product quality control where vibration measurements are made at the end of a production line. The type of loading and means of loading may dictate the type of testing possible.

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Appendix
Relationship of Vibration Amplitude Parameters Velocity can be regarded as the time rate of change of displacement. Using calculus, the first derivative of displacement is velocity as shown. Similarly, we can consider acceleration to be the time rate of change of velocity - and thus the first derivative of velocity OR acceleration is the second derivative of displacement. This can be seen below:

X = A sin(ωt )

displacement

.
V = dx = Aω cos(ωt ) dt

velocity

A=

d2x = − Aω 2 sin(ωt ) dt 2

acceleration

Mathematical formulation of transmissibility T (or amplification ratio)

T=
ξ = damping ratio

[1 − (ω / ω ) ] + [2ξ (ω / ω )]
2 2 n n

1 + [2ξ (ω / ω n )]

2 2

ωn = natural frequency, rad/s ω = forcing frequency, rad/s T = transmissibility = Force ratio = displacement ratio Force ratio = Vibration force / force transmitted to structure Displacement ratio = Displacement of vibrating part / displacement of supporting part.

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Vibration Exercises
1. A spring mass system has mass = 50kg, k=56kN/m.What is the natural frequency of this system ? What is the natural angular frequency of this system ? (Answer: f = 5.33 Hz, ϖ = 33.5 rad/sec) 2. If a spring mass system has m = 210 kg and natural frequency = 2.2 Hz, what is the spring constant k for this system. (Answer: k = 40 kN/m ) 3. A 1000 kg machine is supported on four springs, which are identical and carry equal proportions of the machine weight. If the springs deflect 5 mm when the machine is place on them, find: a.the spring constant for this system b.the spring constant for each spring (Answer: a. 1.96 MN/m, b. 490 kN/m) 4.A machine has a normal running speed of 1500 rpm. It has a total mass of 390 kg and is supported on four springs with a spring constant of 200 kN/m each. Find a. the natural frequency b. the forcing frequency c. the ratio of forcing frequency to natural frequency. (Answer: a. 7.21 Hz, b. 25 Hz, c. 3.47 )

5. For a harmonic vibration, the equation for displacement with time of the vibrating mass is: x = A sin ( ϖ t ) If the frequency for this vibration is 1000 Hz and A = 0.01 mm, find the value of x ( the displacement) when : a. t = 0.0 sec b. t = 0.00025 sec c. t = 0.00050 sec d. t = 0.00075 sec e. t = 0.0010 sec Sketch the curve between these points.

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f. where do the maximum and minimum values occur ? g. what is the magnitude of these values ? (Answer: a. 0, b. 0.01mm, c. 0, d.- 0.01mm, e.0, f. at b and d, g. the max/min values are 0.01 and -0.01 = A ) 6.Vibration displacement, velocity and acceleration areall related by: x = A sin ( ϖ t ) x = A ϖcos (ϖ t ) x = -A ϖ2 sin (ϖ t ) If a vibration has an acceleration of 0.1 g peak at 10 Hz, find the corresponding peak values of velocity and displacement. (Answer: a. 15.6 mm/sec, b. 0.24 mm) 7. Two vibrating systems are shown in the sketches. transmissibility for both systems: Determine

100 mm peak

3.2 N peak 320 N peak

1mm peak

(no friction)

(Answer: a. 0.01, b. 0.01)

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8.

What frequency ratio ( ie f/fn) is needed to provide: a. T = 0.4 b. T = 0.06 T is transmissibility c. T = 0.03 with damping ratio = 0 (Answer: a. 1.95, b. 4.1, c. 5.7)

9. Repeat question 8 for damping ratio = 0.1. effect of the increased damping on isolation. (Answer: a. 2.2, b. 6, c. 8)

What

is

the

10. A vibrating machine is to be isolated from its base with T = 0.2. The machine has a mass of 295 kg and a forcing frequency of 200 Hz. Find: a. natural frequency b. k to provide the required isolation. (Answer: a. 83.3 Hz, b. 80 MN/m) 11. A large turbine vibrates when operating at normal speed of 48 000 rpm. If the machine mass is 621 kg find: a. natural frequency b. system spring constant c. spring constant for each spring if a total of four equally loaded springs support the machine. The machine is to be provided with 90 % isolation. (Answer: a. 250 Hz, b. 1.5 GN/m, c. 383 MN/m)

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12.SELECTION OF TRANSDUCERS We are told that a peizo - electric accelerometer has a "mounted resonance frequency" of 18 kHz. Is this accelerometer suitable for measurement of vibration of a machine that is known to have significant vibration at 3kHz? 13.ERROR DETERMINATION When measuring vibration on a machine,instruments were used with the following errors: transducer +/- 1% amplifier +/- 1.2% tape recorder +/- 3.5% FFT analyser +/- 0.5% The recorded data was analysed with the FFT analyser set on 100 Hz full scale. A recording of 5 minutes duration is available for analysis. What is the expected amplitude error? 14.DIGITAL SAMPLING Data analysis of our vibration measurements will be made with a computer fitted with an analog to digital converter. The intended analysis is of amplitude distribution and frequency.The frequency analysis will be carried out by a FFT program available for the computer. The data is recorded on an FM tape recorder at a tape speed of 38.1 mm/sec. This speed has a linear frequency response to 1kHz. We are only concerned with data up to 500 Hz so the tape recorder is adequate for the purpose. Can we digitise this data? What precautions should we take to ensure correct analysis? What sample frequency should be chosen for digitising ? 15. CALIBRATION A servo accelerometer is to be calibrated by the static method. (ie using the earths' gravitational field for a reference value) If the accelerometer can measure a maximum of 0.5g,it cannot be calibrated by tilting through 90 degrees. Find the angle,measured from the horizontal,that provides a 0.5 g calibration for this accelerometer. 16. A spring/mass system has a mass m = 10 kg and a spring constant k = 500 N/m. Determine the natural frequency of this system.

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17. A spring has a 30 kg mass suspended on it. An additional 5kg mass is added and the spring deflects a further 5mm from its previous position. Find the natural frequency of the system: a. with the 30 kg only b. with the 5 kg added 18. The system in the first exercise has a vibration of +/- 50mm at 2 Hz applied to the spring. Find the magnitude of vibration of the mass. Assume zero damping. 19. Explain the term RESONANCE. If a spring/mass system has a natural frequency of 10 Hz,what conditions will cause resonance ? 20. For the system in question 19, a vibrating force is applied to the mass. The spring mass system is to be isolated from the base that it is mounted on. Rubber springs are used with a damping ratio of 0. 2. If the force on the base is to be kept to a maximum of +/- 200 Newtons,what is the maximum force that can be applied to the mass at a frequency of 35 Hz ? 21. On a tractor,whole body vibration measured an RMS magnitude of 0. 4 m/sec at 2.5 Hz. Determine the fatigue limit for this vibration if it is applied in the Az direction. 22. What would be the level of vibration for the same exposure time and frequency as question 21, but considering instead the comfort limit. 23. For vibration at 5 Hz, what level of vibration is at the exposure limit for 1 minute. Consider the Az direction only. 24. A vibration has a level of 0.4 g at 10Hz. fatigue limit in the Az direction? What is the

25. What is the combined effect of the following vibrations in the Ax directions. Consider fatigue limits only and calculate the allowable time of exposure. Hz m/sec 1.0 0.5 2.0 0.4 10.0 0.8

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Glossary Of Vibration Terms
CALIBRATION The process of comparing the measuring system with a known standard value to check accuracy DAMPING Loss of energy in a system that prevents continuous motion DAMPING RATIO Ratio of the actual amount of damping to a standard value Fast Fourier Transform This is a mathematical method for reducing time varying data to a frequency spectrum FFT Abbreviation for Fast Fourier Transform FILTER An electronic device that allows some frequencies to pass through while restricting or eliminating others. Types of filter are :band pass,high pass and low pass FREQUENCY The rate at which a periodic wave repeats itself. Measured in Hertz (Hz) g A common way of measuring acceleration for human effects evaluations. "g" = 9.8 m/s2 (acceleration due to gravity) REYNAUDS' DISEASE Damage to human tissues and circulatory system resulting in VWF and sometimes amputation SEGMENTAL Refers to vibration that is applied to the bone structure of the hand and arm

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SIGNAL A signal is an information carrying wave. This term may describe the actual acoustic wave or its electrical analog as measured by a transducer SPECTRUM The frequency content of vibration can be expressed graphically as a spectrum TRANSDUCER A device for converting energy from the parameter being measured into convenient form for measurement. Often the energy measured is converted to electricity TRANSMISSIBILITY A ratio that expresses the amount of force or displacement transmitted in a spring mass system VWF Vibration White Finger or Reynauds disease (see above) WAVELENGTH The length between successive similar features of a wave, such as from a peak to peak. Measured in meters. WEIGHTING A band pass filter that reduces the level of signal at some frequencies while maintaining or increasing others. Used to measure with a single value, the effect of the signal on humans or machines. WHOLE BODY Vibration that is applied to the whole human body rather than just a portion is called Whole Body vibration. Typically it is found in transportation vehicles.

TAFE Mechanical Engineering

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