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Development of an impact method to determine the vibration transfer characteristics of railway installations
Journal of Sound and Vibration (1983)87(2),
357-370
DEVELOPMENT
OF AN IMPACT
TO DETERMINE
THE VIBRATION
CHARACTERISTICS
OF RAILWAY
METHOD TRANSFER
INSTALLATIONS
E. C. BOVEY London Transport Research Laboratory, 566 Chiswick High Road, London W4 5RR, England (Received 24 May 1982)
The development of a method of vibration testing with impact excitation is described. This technique has many advantages for the investigation of railway installations and has been shown to be a reliable, controlled method for providing quantitative data. A brief outline of the theoretical basis of the method is given and also a description of two site tests where the method was developed and used to measure different aspects of vibration transmission.
1. INTRODUCTION
Vibration from railways, although not a continuous problem, is a persistent one. The consequences can range from mild disturbance to the instigation of law suits for damage, loss of earnings or capital depreciation. In surface railways the effects of vibration are generally overshadowed by the air-borne noise generated directly, but for underground systems there is no such protection and vibration or its subsequent conversion to air-borne noise is the sole manifestation of the railway operation. It is well known that there is a slightly elevated tolerance level to directly generated railway noise due to a range of social and historic factors but this level of tolerance does not seem to apply equally to noise from underground railways. London Transport has found that public complaints can be expected if noise levels caused by their underground trains exceed 40 dB(A) [l]. Particularly troublesome from the viewpoint of public complaints is the operation of newly opened underground lines. This causes a change in the local noise environment which promotes response from the community. The same effect could also result if substantial modifications were made to existing underground installations which caused them to generate increased vibrations. The desire to be able to minimize both these effects is, in part, the motivation for the project described in this paper. There is a wealth of data concerning railway vibrations comprising vibration measurements at various locations due to trains of varying configuration running on different track forms, in different types of tunnel, in different surrounding soils, etc. The difficulty arises when it is necessary to predict the vibration effects of a particular combination of circumstances and the definitive data available to assist with this task is very limited. As an example, the attenuation of ground-borne vibration with distance can be considered. Correlations of attenuations with distance have been proposed [2], and these provide useful guidelines, but in particular circumstances large variations from the norm can occur. This was found by Richards [3], who recorded such a large degree of scatter of the various measurements of attenuation and distance that it was impossible to formulate a reliable relationship. 357 @ 1983 Academic Press Inc. (London) Limited 0022-460X/83/060357+ 14 $03.00/O
E. C. BOVEY
In order to obtain more reliable information, particularly to assist in design studies, it is necessary to have not only data obtained from particular systems but also a clear understanding of the mechanism of railway vibration generation and propagation and the effects of various system parameters. There are, therefore, many areas of investigation which can be identified: for example, coupling of rail vehicle to track forms, correlation between input forces at various wheels, motion of the track, motion of the tunnel, method of vibration propagation through the surrounding soil, and attenuation of vibration with distance. The method presented in this report is designed to enable information to be obtained about many of these items. 2. METHOD
OF TESTING
The basis for the proposed method is the measurement of the mechanical transfer function between various points on a system. This function gives, as its name implies, the transfer characteristics between two points of the system and yields information in the frequency domain. In order to generate the function, simultaneous analysis must be performed on data signals representing the input force applied at one point of the system and the system response motion measured at the same or at a different point. Depending upon the type of measurement used to detect the response motion the transfer function may be calculated as one of several derivatives. In general, the measurement of response acceleration is the most convenient and the transfer function is then said to be defined in terms of inertance. Any other measurement of response motion would give a different derivative but an equally valid measure of the system transfer characteristics. There are many advantages of this method of system testing. Some of these apply generally to the method but others are particularly useful with respect to railway vibration testing. The following list presents the main benefits: (1) direct measurement of transfer function is possible; (2) effects of different components in the vibration transmission path can be more easily determined; (3) an absolute comparative method’of assessment can be attained; this means that objective comparisons can be achieved irrespective of the type of rolling stock, the condition of the rail running surface or of the train wheels or whether or not the track is jointed; (4) only short lengths of test track are required for assessment-perhaps only a few metres; (5) there is no need to run test trains so that unlimited quantities of data can be collected and analyzed; (6) the characteristics of vibration transmission can be studied; (7) the single input technique is a more rational means of testing, removing the multi-input and moving character of a train with many wheels, each having a different force input; (8) most mathematical models of track forms are based on discrete degree of freedom systems with single inputs; this method will provide information much more useful for inclusion into these studies; (9) effects of extraneous vibrations can be eliminated. The type of force used to excite the system is of great importance in this method. Three basic types of input are available: these are sine wave (or swept sine), random or impulsive. Each has its own advantages but for the particular application of railway vibration, impulsive force is by far the most appropriate. Sine wave excitation requires elaborate equipment to generate the wave form and also one has difficulty in providing sufficient force at low frequencies. Random excitation has the advantage of producing results rapidly but again requires elaborate apparatus to produce sufficient force. The use of impulsive force, however, has one major advantage in that time history analysis is possible without the use of correlation functions which would be necessary if random excitation were used and would be impossible if sine waves were used. This facility has proved to be particularly useful in some of the tests performed. The impulsive force can
AN
IMPACT
METHOD
FOR
RAILWAY
359
VIBRATIONS
also be applied relatively simply and can be easily tailored to provide the required amplitude and frequency input. Examples of a typical force input pulse and its spectrum are shown in Figure 1. The concept of impact testing has been developed mainly for structural work and is, in that application, an established and powerful method; its potential for measuring ground vibration transmission has been demonstrated by White and Mannering [4].
1
-I.
I
I
I
I
0
1
a”
1
IO-10 500
Frequency
(Hz
1
/
1
I
I
I
1
Frequency
(Hz)
I
I
0
I
, 500
Figure 1. Typical force input pulse. (a) Time history; (b) power spectrum.
It is recognized, however, that measurements in which train generated vibration is used are ultimately necessary to ensure that the impact method sees the system in exactly the same way as does a train. One area of uncertainty in this respect is whether lateral excitation of the rail by trains is significant since in the impact method as developed to date only vertical force is applied. The analysis of data when using the impact method yields the magnitude and phase of the transfer function and also the coherence function, all of which provide information about the system and also about the quality and reliability of the recorded data. The functions are described in Appendix 1. 3. APPLICATION OF THE IMPACT METHOD TO RAILWAY VIBRATION In general the application of the impact method for investigation of railway vibrations is fairly straightforward. Some points, however, do need consideration. For most structural applications the force input is measured by a load cell contained in the impacting head itself, in many cases a hammer. It was found more suitable for railway applications to locate the load cell statically on the impacted surface and to rest on top of it a resilient pad. The impact is then made on top of this sandwich with the impact head. Response acceleration has always been measured by using a piezoelectric accelerometer of appropriate sensitivity. The accelerometer mounting, however, varied according to where the accelerometer was positioned: magnetic attachment was used for rail and tunnel wall measurement, gravity mounting for sleeper or tunnel foundation mounting and a triaxial mounting block, which is described later, for measurement on soft soil. For the vibration signals measured, all three methods were considered to allow the accelerometers to faithfully duplicate the system motion without effecting the system by their attachment, however, subsequent testing showed the tri-axial block to be somewhat deficient in this respect. The investigation of the system with only vertical excitation is thought to be adequate, at least for the first stages of development of the impact method. There are, of course,
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practical difficulties in arranging for lateral forces but in any case it is considered that they are of minor significance with respect to vibration generation. Although quite high lateral dynamic wheel forces can exist it is unlikely that these will translate into equivalent forces on the tunnel structure; they are probably, to a large extent, absorbed by the lateral stiffness and mass of the rail-sleeper system. It is recognized that for rail excitation the non-linear characteristics of the track form could and probably would significantly effect the dynamic response when the rail is subjected to the superimposed dead weight of a train. For the results presented here, this has not been taken into account but work is already in hand to investigate this.
4. DEVELOPMENT
OF THE IMPACT METHOD
At the outset there existed many uncertainties. Factors requiring investigation included the site of the impact, the frequency range of interest and the frequency range that could be practically achieved, and the reliability of the method in terms of consistency of track characteristics both in space and time. Most important of all was the relationship between characteristics measured by the test method and those relevant to the transmission of vibration from trains. On a more practical level it remained to be determined the number of data samples required to obtain reasonable accuracy, the types of equipment to use and their sensitivities and the method of data transmission to minimize interference from noise, particularly coherent noise on both channels. In initial testing a 6.35 kg (14 lbf) sledge hammer was used, fitted into a rudimentary framework that enabled it to be directed at a specific location on its downward swing. The force transducer was placed at that location on the rail and a resilient pad rested on top of it. The hammer impact was applied by hand. These tests were designed to determine the level of force required and its frequency content. The force amplitude was controlled by the arc of swing of the hammer and the frequency range was determined by the stiffness of the resilient pad which itself was a function of material properties and pad shape. Many types of pad were tested in the laboratory but the best results were obtained with a mouldable material called Devcon Flexane 80 which is a room temperature curing urethane. It became apparent that a force amplitude of approximately 200 kg peak would suffice for initial work. This level of force gave reasonable response at a distance in laboratory tests on a short section of rail and sleeper and also could be easily managed by the hammer arrangement. The force pulse was shaped by varying the resilient pad design to provide energy in the range O-200 Hz which covered the frequency range of interest and produced a satisfactory power density of the input spectrum as shown in Figure 1. It should be noted that the reduction in power content of the input force at progressively higher frequencies does not affect the measurement of vibration characteristics since the resultant transfer function is normalized to take account of this. Measurement of tunnel vibrations with the arrangement described above showed that sufficient response signal levels could be obtained on the tunnel wall in the vicinity of the excitation point, sufficient, that is, to give a reasonable signal to noise ratio over the frequency range of interest. It soon became clear, however, that the manual method of impacting the track was impractical since the weight of the hammer and the number of impacts required made the physical requirements of the operator very demanding and a reliable series of impacts could not be obtained. It was decided to develop an automatic method to provide the required force impulse. The equipment designed was based on the operating principle of an automatic drop hammer and the realization of this design
AN IMPACT
Figure
METHOD
2. General
FOR
RAILWAY
view of drop hammer
VIBRATIONS
361
impacter.
is shown in Figure 2. This has the advantage of applying not only a calibrated force pulse to the rail but also one of constant shape and frequency content. The system response has always been measured by accelerometers. These form the only practical means of measurement of absolute motion and form a self-contained, compact measurement system. Piezoelectric accelerometers of various sensitivities have been used exclusively throughout the testing and have been found to be generally satisfactory at all frequencies of interest. There is one small problem, however, inherent in measurements of acceleration, namely that the acceleration of a structural response at low frequencies is invariably small despite the fact that relatively large quantities of energy may be used for excitation and that resultant displacement amplitudes may be large. This situation is further aggravated since the measuring system noise is relatively high at low frequencies. The result is a poor signal to noise ratio in this region. This
0
500
Frequency
Figure
3. Typical
coherence
function
showing
(Hz)
effect of extraneous
noise at low frequencies.
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situation is reflected in the coherence function of which an example is shown in Figure 3. In order to overcome this problem, and to gain confidence in the computed transfer function, ensemble averaging of sample transfer functions must be made according to the definition given in Appendix 1. A transfer function produced in this manner can have its confidence limits continuously improved by increasing the number of averages made. Unfortunately the relationship is not linear but follows an inverse square root law and a compromise must be found between accuracy required and practicability of multiple averaging. For the railway vibration testing, 32 averages have been chosen as a suitable number. This number of averages gives 90% confidence limits on the measurement of the transfer function magnitude (IHI) as given in Table 1. TABLE 1 Relationship between coherence function and confidence limits
Measured value of coherence function 0.2 0.5 0.9
90% confidence limits for IHl A r 32 averages 256 averages +3*8 -7.1 +2.1 -2.8 +0.8 -0.8
+1*5 -1.8 +0.8 -0.9 +0*3 -0.3
The improvement in confidence limits obtained by averaging 256 samples is clearly evident but such a process would increase the testing time by an unacceptable factor. It has, however, been possible to substantially improve the confidence limits in a way that was not anticipated. It has been mentioned that spatial variation of track form parameters is a very likely characteristic of the track support system and, therefore, that measurement of the system with a single point of excitation would not yield representative results. In order to overcome this problem, the track parameters are measured at several different locations including over-sleeper and between-sleeper rail excitation with corresponding response measurement positions. Five different locations have been used for most of the testing. The transfer function estimates generated for these conditions are averaged to remove (or reduce) the effects of parameter variability. This averaging is performed as an extension of the 32 averages used to generate the estimate of the transfer function associated with a single point of excitation. The effect of this extension is to further average out the extraneous noise in addition to averaging out the variation in the estimates of H, Appendix 2 shows the details of this process. A further effect of this extended averaging is seen on the coherence function. The coherence function of the 5 x 32 averages now incorporates the effect of the variability in H and this, in general, causes a reduced value of coherence function relative to that obtained from one set of 32 averages, the amount of reduction being proportional to the variability of the estimates of H. This is also demonstrated in Appendix 2 and provided that the coherence from a set of 32 averages is fairly high an approximation can be obtained for the variance of H. This, of course, is a function of frequency. Figure 4 shows this effect and the frequencies at which coherence has been depressed indicate variability in track parameters for nominally the same form of track.
AN
0
IMPACT
METHOD
FOR
RAILWAY
VIBRATIONS
363
0 Frequency
(Hz)
(b)
Figure 4. Effect of track form variability on coherence measurement. Track form consisted of Pandrol clips and concrete sleepers in new, fully concreted tsack bed. (a) Transfer function showing single point measurement (fi) and averaged multipoint measurement (H); (b) coherence function.
5. SITE
TESTING
AT ALDWYCH
The impact method has been used at two sites on the London Transport underground system. One of these was on the Jubilee Line at Regent’s Park and is described in the next section. The tests described in this section were designed to investigate track form characteristics and to’develop the method for this application. The tests were performed on a spur line in a section of tunnel between Holborn and Aldwych Stations which carries a three car shuttle service at peak travel periods only. This enables track possession to be obtained during the daytime and greatly facilitates work. A further advantage is that the section of track is used by the Permanent Way Department to test new forms of track and several different forms are available in short lengths for comparative testing. Initial tests with manually applied impact were carried out at Aldwych to investigate the feasibility as described in the previous section and on the basis of these it was decided to proceed with more elaborate development of the method including the construction of the automatic drop hammer. The design of this was such as to maintain the input
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force spectrum within the same frequency range but to enable the peak force to be increased to 700 kg. Subsequent tests at Aldwych were all carried out with this equipment. Much of the testing performed was designed to build up confidence in the test method. To this end examination was made of the effects of allowing the passage of service traffic between measurements of transfer function. Mixed results were obtained for this series of tests. For some track forms repeatability was good as shown in Figure 5
cl
500
Frequency
Figure
5. Change
in measured
transfer
function
(Hz)
due to passage
of rail traffic over test site.
relatively poor reproduction was obtained. There was, however, a clear distinction between the types of track form which exhibited these different characteristics. The track forms that gave good repeatability were newly installed, i.e., freshly concreted into the tunnel foundations, while the other track forms consisted of old wooden sleepers loosely fitted into the foundations and clearly subject to considerable relocation of track components due to the passage of traffic. A further characteristic of these latter types of track was the lower valued transfer function magnitude and the poor coherence functions that they generated. Each track form was examined in turn and a selection of the transfer functions obtained for these is shown in Figure 6. The functions are reproduced in full despite the fact that poor coherence indicates uncertainty in the values at certain frequencies particularly in the range 200-500 Hz where the frequency content of the hammer input was, by design, very low. As mentioned in the previous section the particular method of averaging the coherence function allowed some assessment of track form variability to be obtained and this indicated very good reproducibility for track forms 4, 5 and 6, concrete blocks or concrete sleepers in a fully concreted tunnel floor, but poorer stability for track forms with wooden sleepers. All measurements of track characteristics described in this section were taken with the rail unloaded. This admittedly causes the track form to exhibit different characteristics from those‘when it is loaded but provides a useful introduction to the method. The measurement of transfer function on loaded track will be discussed in a later section. The fact that the track is unloaded also allows irregularities in track installation, for example uneven sleepers or small clearances, to exert a strong influence over the local dynamic characteristics of the track form.
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IMPACT
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(Hz)
Figure 6. Examples of transfer functions of different track forms measured at Aldwych. (a) Track form 4-concrete sleepers in fully concreted track bed; (b) track form S-concrete blocks in fully concreted track bed; (c) track form 6-concrete blocks in concreted track bed with 4 ft drain channel; (d) track form 7-old wooden sleepers in concreted track bed with 4 ft drain channel.
6. SITE
TESTING
AT REGENTS
PARK
The second series of site tests took place at Regent’s Park. These were designed to investigate ground transmission characteristics rather than performance of track systems, although tunnel characteristics are considered to play an important role in the results obtained. The testing involved vertically impacting the concrete foundation of a bored tube tunnel and measuring ground response at various points on the ground surface above. For this application the input force was increased to 1200 kg by increasing the hammer weight. The particular site was chosen because of the relatively uniform geological structure of the ground in that area, being a deep layer of reasonably uniform London Clay resting on a gravel base about 10 m below the tunnel level. The tunnel is of jointed concrete segment construction and lies approximately 30 m below the ground surface in the test area. A further feature of the site is the fact that over the area of interest, and for some distance around, it is uncluttered by buildings or other civil engineering structures of any type. The opportunities for wave reflections are, therefore, much reduced and this represents as near perfect conditions as can be realized for any site on the London Transport System. In order to measure the transfer function, simultaneous signals of excitation and response are required. It was not possible at this site to obtain a direct cable link between the excitation location below ground and the measurement position above ground. The only means by which the data could be transmitted was to employ a radio telemetry system and this was developed with use of f.m. modulation superimposed on a VHF carrier. The force signal from the tunnel was brought to ground level by cable at the nearest underground station and fed to a radio transmitter at the top of a nearby building.
E. C. BOVEY
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Data was then transmitted to a mobile laboratory in Regent’s Park about 300 m away. Voice communication was achieved in a similar manner. Triaxial measurements of ground response acceleration were made for all tests. The accelerometers of high sensitivity were mounted on a large steel plate which was rested on the ground and coupled to it by long steel spikes. The accelerometer signals were led by cable to the mobile laboratory where they were recorded along.with the force signal. Some signal analysis was also performed at the time using a Nicolet 660A Spectral Analyser. A comprehensive series of tests was performed to examine many characteristics of the system. Data was collected to enable investigation of (a) system homogeneity, i.e., that within the test area, transfer functions measured between similar positions relative to the tunnel gave the same result, (b) the directional characteristics of the tunnel vibration, (c) the sensitivity of the system to small changes in excitation position, (d) the attenuation of transmitted vibration with distance, and (e) the mechanism of wave propagation through the surrounding ground. Much of the data still requires analysis but some interesting results have already been obtained. It is not appropriate to give details in this report but two figures are presented which demonstrate the type of information which can be obtained by the impact method. Figure 7 shows a three dimensional diagram of the variation of transfer function with
0
of
accelerometer tunnel centre
from hne Cm)
200
100 Frequency
Displacement
(Hz)
Figure 7. Variation of transfer function magnitude with distance measured at Regent’s Park.
distance. The measurement direction was horizontal and normal to the tunnel axis. This effectively gives the attenuation of vibration with distance from the tunnel. Figure 8(a) shows a typical time history of an acceleration measured 30 m from a point on the ground directly above the tunnel. The acceleration in this case is vertical. This display of the time history was achieved by signal averaging and Figure 8(b) shows the trace of a single sample time history. The signal enhancement is achieved by the averaging process and was only possible because a consistent force impact was applied in the tunnel. It is possible to obtain wave propagation speeds from this type of measurement and by examination of the complete set of such time histories an indication may be obtained of the motion of the tunnel and the mechanism of the wave propagation in the ground.
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250
Figure 8. Time history of transient response to excitation in tunnel at Regent’s Park. Measurement position was 30 m from a ooint on the around immediatelv over the tunnel. (a) Averane of 32 time history samples; (b) single sample of time history.
7. FUTURE
DEVELOPMENT
AND
STUDIES
The immediate future must, to a large extent, be used to analyze the data already obtained and to obtain as much information from these as possible particularly with respect to ground transmission. The most important outstanding question, however, is the relationship between results obtained by the impact method and the vibrations induced by trains themselves. This uncertainty falls broadly into three areas. (a) The effect of the train dead load on the transmission characteristics of track forms is the first uncertainty. This is clearly an important factor and one on which testing has already begun. Dead load can be practically applied to the track only by gravity loads since reaction against any part of the tunnel could cause serious modifications to the system under investigation. Initial tests have already been completed with a train itself as the load and with the track excited through a train wheel by excitation on the top surface of the wheel, to which access is obtained through the carbody. This method clearly involves the bogie and wheelsets in the mechanical system and a method of mass cancellation is currently being developed to remove their influence. (b) The effects of inputs from a multi-wheeled train and possible correlation of the forces involved constitute the second area of uncertainty. Tests carried out at Regent’s Park should provide some information concerning the former; the effect of the latter is more difficult to determine and the impact method alone may not be able to provide much relevant information.
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(c) The final area of uncertainty is the effect of horizontal force input from a train on the vibration levels generated. No plans to investigate this have yet been formulated but an adaptation of the automatic hammer will be required. As has been stressed, the correlation between train vibration and impact excitation results is ultimately required and with that end in mind train vibrations have always been recorded where impact measurements have been taken. This will continue to be the case until the impact method is fully proven.
8. CONCLUSION The method described has been developed to provide objective and quantitative measurements of the characteristics of railway vibration transmission, but another and equally important motive is to gain a better insight into the mechanism of that transmission and, therefore, ultimately to be able to provide guidance on tunnel and track design. The impact method is independent of the characteristics of trains and therefore enables the parameters of the civil engineering structure to be determined absolutely. The ability to average the received data enables good rejection of background or instrumentation noise to be effected and the use of impact excitation facilitates time history analysis to determine, for example, wave propagation speeds. This is particularly useful for a system where travelling waves are of far greater importance than standing waves. Track possession is necessary for performing this type of testing but during the course of a possession a large amount of data can be collected. It is not necessary to run test trains for any form of this test method but the use of a stationary train to provide dead load for some forms of testing may be necessary. REFERENCES 1. London Transport Research Laboratory Report 1980 Complaints of railway noise received by
London Transport 1969-79. 2. J. E. MANNING, R. G. CANN and J. J. FREDBERG 1974 U.S. Department of Transportation, Report No. CJTMA-MA-06-0025-74-5.
Prediction and control of rail transit noise and vibra-
tion, a state of the art assessment. London Transport Research Laboratory Report 1969 Noise and vibration
due to the operation of trains on the Victoria line. R. G. WHITE and M. E. J. MANNERING 1975 Journal of the Society of Environmental Engineers. Techniques for measuring the vibration transmission characteristics of the ground. R. K. OTNES and L. ENOCHSON 1979 Applied Time Series Analysis. New York: John Wiley & Sons. J. S. BENDAT and A. G. PIERSOL 1971 Random Data Analysis and Measurement Procedures. New York: John Wiley & Sons.
APPENDIX 1: FUNCTIONS USED IN SPECTRAL ANALYSIS Spectral analysis is a method of analyzing data in the frequency domain. The spectral characteristics are of particular value since physical systems have many interesting and significant properties related to the spectral content of the excitation to which the system is subjected. This applies to mechanical and electrical systems alike. Although of immense value, frequency analysis is not always the type of analysis which is most appropriate for investigating a system. Once a set of data has been frequency analyzed certain information may be irretrievably lost, in particular information in the amplitude domain. For some systems this may be important, for example where material yield or fatigue is of importance. It is possible, however, provided that the correct functions are evaluated to retain in the frequency domain a complete description of the system. Transformations
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of the frequency functions generated can then yield information to enable system response in any situation to be determined. The frequency response function or transfer function is one function which satisfies this requirement. The frequency analysis of data has been considered in detail by many authors [S, 61, and therefore only a brief description will be presented here. The transfer function H(f) is defined as A(f) = $,,(f)/&(f) where S,.,(f) is the cross spectral density and S,,(f) is the auto spectral density of the input. The caret above the function symbols indicates that the functions are estimates: i.e., the function values are subject to a certain degree of uncertainty depending on the measurement procedure. Estimates of spectral densities used here are produced by ensemble smoothing of a set of raw estimates, as for example in the case of S,,(f) where this estimate is produced by the process
w h ere Xi(f) and X?(f) are raw estimates or, in shorthand form, $,,(f) =X(f)X*(f), of the Fourier transform and its conjugate of the input signal x(t). The use of smoothing greatly improves the statistical accuracy of the transfer function estimate despite the fact that the raw estimates can be of poor quality, although in ideal circumstances a raw estimate can be practically perfect. Another function which is of great use and which is frequently considered in conjunction with the transfer function is the coherence. This is defined as y* = ]S~x,(~)\2/S1~ (&,, (f), where S,,(f) is the auto spectral density of the output y(t). In the measurement of most types of system, and in particular when measuring the characteristics of railway systems, the measured output contains some signal which is not caused by the system input. This, in many cases, is the reason why raw estimates of transfer function are unreliable. A sample transfer function will be calculated by using a sample response signal which contains a certain amount of extraneous noise. The Fourier transform of the sample response yi(t) is given in this case by Y;(f) = H(f)Xi(f)+ZVi(f), where Ni(f) is the Fourier transform of the extraneous noise n;(t). The smoothing of the transfer function averages out the noise to a degree dependent on the number of sample averages and the level of extraneous noise. This latter factor is directly indicated in a quantitative manner by the value of the coherence function, It can easily be shown that the coherence function is also given by y* = ]H(f)\*Sxxx(f)/&,(f): i.e., the coherence function is a measure of the output power that is due to the input divided by the total output power. Measurement of the transfer function can be made with any form of input desired. In practice the use of impulse excitation and transient response has many advantages. The sample length for each excitation/response sequence is chosen such that the input and output signals are entirely contained within the sample time window. This means that there is no frequency smearing due to truncation effects and also that any time delay in the system transfer characteristics does not depress the coherence function as would occur if random excitation were used. APPENDIX 2: EFFECT OF VARIABILITY OF H ON THE MEASURED FUNCTIONS In this Appendix the effect of measuring nominally identical transfer functions is investigated. Averaging is performed in two stages, firstly to produce an estimate of a particular value of H and secondly to average the individual values of H. The notation of Appendix 1 is used but with the indication of the independent variable f omitted. By
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definition $*, = yx*, and so one has &,, = Hm +Nx*, where Hi is the true value of a particular transfer function estimated by the first averaging process. A =s’,,/L!Zx,,= (HiF+NX*)/xx”. By continuing the averaging for successive estimates of H one obtains --d = &,/&, = (HXX* + NX*)/XX*. The extraneous noise is, therefore, subject to both averaging process and its effect on the estimate H is reduced to a low level. In addition the use of consistent force impulses for excitation purposes causes the input spectrum to be extremely constant and consequently the variability of XX* is very small; this can then be ignored in the second averaging process. These two factors lead to the result H -H, which is a good approximation provided that the noise n(t) is not of excessive magnitude. A similar process can be applied to the coherence function: c’ = I& 1*/&& =((HiX+N)X*)
= (xY*)*/yy*
xx*
((H:X*+N*)X)
(H&+N)(H:X*+N*)XX* HiHT (xX*)” =(HJrrXX*+NN*)xx* provided that n(t) is of sufficiently low value such that terms of the type NX” are relatively small after the first averaging process; if this is not so then --(H&X*+NX*)(H:XX*+N*X) “=(HiHTXX*+H&N*+H”X*N+EF)E?* Continuing the averaging for successive values of H one obtains --(HXX* + NX*) (H*XX* + N*X) + -(HH*XX*+HXN*+H*X*N+NN*)XX*’ Since XX* is sensibly constant because of the consistency of input and the first averaging procedure and also since terms of the type X*N will be relatively small, following double averaging the above expression reduces to -~*=EG*~~*~(*~~X+NN*). If the noise spectrum NN* is relatively low then one obtains -**=HH*/HH*=@*z/((r*+p*), Y where k and (T* are the mean and variance of the magnitude of H. If the noise spectrum W NN* is of significant value then, provided that the value of Hi is, by inspection, close to that of H, (l/+*)-(~/~*)~(HH*-H~H*)/HH*z~*/~** The above derivations are only approximate and, therefore, should only be used as a guide to determine, for example, the number of estimates of H that should be made at different locations on nominally the same system.
357-370
DEVELOPMENT
OF AN IMPACT
TO DETERMINE
THE VIBRATION
CHARACTERISTICS
OF RAILWAY
METHOD TRANSFER
INSTALLATIONS
E. C. BOVEY London Transport Research Laboratory, 566 Chiswick High Road, London W4 5RR, England (Received 24 May 1982)
The development of a method of vibration testing with impact excitation is described. This technique has many advantages for the investigation of railway installations and has been shown to be a reliable, controlled method for providing quantitative data. A brief outline of the theoretical basis of the method is given and also a description of two site tests where the method was developed and used to measure different aspects of vibration transmission.
1. INTRODUCTION
Vibration from railways, although not a continuous problem, is a persistent one. The consequences can range from mild disturbance to the instigation of law suits for damage, loss of earnings or capital depreciation. In surface railways the effects of vibration are generally overshadowed by the air-borne noise generated directly, but for underground systems there is no such protection and vibration or its subsequent conversion to air-borne noise is the sole manifestation of the railway operation. It is well known that there is a slightly elevated tolerance level to directly generated railway noise due to a range of social and historic factors but this level of tolerance does not seem to apply equally to noise from underground railways. London Transport has found that public complaints can be expected if noise levels caused by their underground trains exceed 40 dB(A) [l]. Particularly troublesome from the viewpoint of public complaints is the operation of newly opened underground lines. This causes a change in the local noise environment which promotes response from the community. The same effect could also result if substantial modifications were made to existing underground installations which caused them to generate increased vibrations. The desire to be able to minimize both these effects is, in part, the motivation for the project described in this paper. There is a wealth of data concerning railway vibrations comprising vibration measurements at various locations due to trains of varying configuration running on different track forms, in different types of tunnel, in different surrounding soils, etc. The difficulty arises when it is necessary to predict the vibration effects of a particular combination of circumstances and the definitive data available to assist with this task is very limited. As an example, the attenuation of ground-borne vibration with distance can be considered. Correlations of attenuations with distance have been proposed [2], and these provide useful guidelines, but in particular circumstances large variations from the norm can occur. This was found by Richards [3], who recorded such a large degree of scatter of the various measurements of attenuation and distance that it was impossible to formulate a reliable relationship. 357 @ 1983 Academic Press Inc. (London) Limited 0022-460X/83/060357+ 14 $03.00/O
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In order to obtain more reliable information, particularly to assist in design studies, it is necessary to have not only data obtained from particular systems but also a clear understanding of the mechanism of railway vibration generation and propagation and the effects of various system parameters. There are, therefore, many areas of investigation which can be identified: for example, coupling of rail vehicle to track forms, correlation between input forces at various wheels, motion of the track, motion of the tunnel, method of vibration propagation through the surrounding soil, and attenuation of vibration with distance. The method presented in this report is designed to enable information to be obtained about many of these items. 2. METHOD
OF TESTING
The basis for the proposed method is the measurement of the mechanical transfer function between various points on a system. This function gives, as its name implies, the transfer characteristics between two points of the system and yields information in the frequency domain. In order to generate the function, simultaneous analysis must be performed on data signals representing the input force applied at one point of the system and the system response motion measured at the same or at a different point. Depending upon the type of measurement used to detect the response motion the transfer function may be calculated as one of several derivatives. In general, the measurement of response acceleration is the most convenient and the transfer function is then said to be defined in terms of inertance. Any other measurement of response motion would give a different derivative but an equally valid measure of the system transfer characteristics. There are many advantages of this method of system testing. Some of these apply generally to the method but others are particularly useful with respect to railway vibration testing. The following list presents the main benefits: (1) direct measurement of transfer function is possible; (2) effects of different components in the vibration transmission path can be more easily determined; (3) an absolute comparative method’of assessment can be attained; this means that objective comparisons can be achieved irrespective of the type of rolling stock, the condition of the rail running surface or of the train wheels or whether or not the track is jointed; (4) only short lengths of test track are required for assessment-perhaps only a few metres; (5) there is no need to run test trains so that unlimited quantities of data can be collected and analyzed; (6) the characteristics of vibration transmission can be studied; (7) the single input technique is a more rational means of testing, removing the multi-input and moving character of a train with many wheels, each having a different force input; (8) most mathematical models of track forms are based on discrete degree of freedom systems with single inputs; this method will provide information much more useful for inclusion into these studies; (9) effects of extraneous vibrations can be eliminated. The type of force used to excite the system is of great importance in this method. Three basic types of input are available: these are sine wave (or swept sine), random or impulsive. Each has its own advantages but for the particular application of railway vibration, impulsive force is by far the most appropriate. Sine wave excitation requires elaborate equipment to generate the wave form and also one has difficulty in providing sufficient force at low frequencies. Random excitation has the advantage of producing results rapidly but again requires elaborate apparatus to produce sufficient force. The use of impulsive force, however, has one major advantage in that time history analysis is possible without the use of correlation functions which would be necessary if random excitation were used and would be impossible if sine waves were used. This facility has proved to be particularly useful in some of the tests performed. The impulsive force can
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also be applied relatively simply and can be easily tailored to provide the required amplitude and frequency input. Examples of a typical force input pulse and its spectrum are shown in Figure 1. The concept of impact testing has been developed mainly for structural work and is, in that application, an established and powerful method; its potential for measuring ground vibration transmission has been demonstrated by White and Mannering [4].
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-I.
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Figure 1. Typical force input pulse. (a) Time history; (b) power spectrum.
It is recognized, however, that measurements in which train generated vibration is used are ultimately necessary to ensure that the impact method sees the system in exactly the same way as does a train. One area of uncertainty in this respect is whether lateral excitation of the rail by trains is significant since in the impact method as developed to date only vertical force is applied. The analysis of data when using the impact method yields the magnitude and phase of the transfer function and also the coherence function, all of which provide information about the system and also about the quality and reliability of the recorded data. The functions are described in Appendix 1. 3. APPLICATION OF THE IMPACT METHOD TO RAILWAY VIBRATION In general the application of the impact method for investigation of railway vibrations is fairly straightforward. Some points, however, do need consideration. For most structural applications the force input is measured by a load cell contained in the impacting head itself, in many cases a hammer. It was found more suitable for railway applications to locate the load cell statically on the impacted surface and to rest on top of it a resilient pad. The impact is then made on top of this sandwich with the impact head. Response acceleration has always been measured by using a piezoelectric accelerometer of appropriate sensitivity. The accelerometer mounting, however, varied according to where the accelerometer was positioned: magnetic attachment was used for rail and tunnel wall measurement, gravity mounting for sleeper or tunnel foundation mounting and a triaxial mounting block, which is described later, for measurement on soft soil. For the vibration signals measured, all three methods were considered to allow the accelerometers to faithfully duplicate the system motion without effecting the system by their attachment, however, subsequent testing showed the tri-axial block to be somewhat deficient in this respect. The investigation of the system with only vertical excitation is thought to be adequate, at least for the first stages of development of the impact method. There are, of course,
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practical difficulties in arranging for lateral forces but in any case it is considered that they are of minor significance with respect to vibration generation. Although quite high lateral dynamic wheel forces can exist it is unlikely that these will translate into equivalent forces on the tunnel structure; they are probably, to a large extent, absorbed by the lateral stiffness and mass of the rail-sleeper system. It is recognized that for rail excitation the non-linear characteristics of the track form could and probably would significantly effect the dynamic response when the rail is subjected to the superimposed dead weight of a train. For the results presented here, this has not been taken into account but work is already in hand to investigate this.
4. DEVELOPMENT
OF THE IMPACT METHOD
At the outset there existed many uncertainties. Factors requiring investigation included the site of the impact, the frequency range of interest and the frequency range that could be practically achieved, and the reliability of the method in terms of consistency of track characteristics both in space and time. Most important of all was the relationship between characteristics measured by the test method and those relevant to the transmission of vibration from trains. On a more practical level it remained to be determined the number of data samples required to obtain reasonable accuracy, the types of equipment to use and their sensitivities and the method of data transmission to minimize interference from noise, particularly coherent noise on both channels. In initial testing a 6.35 kg (14 lbf) sledge hammer was used, fitted into a rudimentary framework that enabled it to be directed at a specific location on its downward swing. The force transducer was placed at that location on the rail and a resilient pad rested on top of it. The hammer impact was applied by hand. These tests were designed to determine the level of force required and its frequency content. The force amplitude was controlled by the arc of swing of the hammer and the frequency range was determined by the stiffness of the resilient pad which itself was a function of material properties and pad shape. Many types of pad were tested in the laboratory but the best results were obtained with a mouldable material called Devcon Flexane 80 which is a room temperature curing urethane. It became apparent that a force amplitude of approximately 200 kg peak would suffice for initial work. This level of force gave reasonable response at a distance in laboratory tests on a short section of rail and sleeper and also could be easily managed by the hammer arrangement. The force pulse was shaped by varying the resilient pad design to provide energy in the range O-200 Hz which covered the frequency range of interest and produced a satisfactory power density of the input spectrum as shown in Figure 1. It should be noted that the reduction in power content of the input force at progressively higher frequencies does not affect the measurement of vibration characteristics since the resultant transfer function is normalized to take account of this. Measurement of tunnel vibrations with the arrangement described above showed that sufficient response signal levels could be obtained on the tunnel wall in the vicinity of the excitation point, sufficient, that is, to give a reasonable signal to noise ratio over the frequency range of interest. It soon became clear, however, that the manual method of impacting the track was impractical since the weight of the hammer and the number of impacts required made the physical requirements of the operator very demanding and a reliable series of impacts could not be obtained. It was decided to develop an automatic method to provide the required force impulse. The equipment designed was based on the operating principle of an automatic drop hammer and the realization of this design
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Figure
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impacter.
is shown in Figure 2. This has the advantage of applying not only a calibrated force pulse to the rail but also one of constant shape and frequency content. The system response has always been measured by accelerometers. These form the only practical means of measurement of absolute motion and form a self-contained, compact measurement system. Piezoelectric accelerometers of various sensitivities have been used exclusively throughout the testing and have been found to be generally satisfactory at all frequencies of interest. There is one small problem, however, inherent in measurements of acceleration, namely that the acceleration of a structural response at low frequencies is invariably small despite the fact that relatively large quantities of energy may be used for excitation and that resultant displacement amplitudes may be large. This situation is further aggravated since the measuring system noise is relatively high at low frequencies. The result is a poor signal to noise ratio in this region. This
0
500
Frequency
Figure
3. Typical
coherence
function
showing
(Hz)
effect of extraneous
noise at low frequencies.
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situation is reflected in the coherence function of which an example is shown in Figure 3. In order to overcome this problem, and to gain confidence in the computed transfer function, ensemble averaging of sample transfer functions must be made according to the definition given in Appendix 1. A transfer function produced in this manner can have its confidence limits continuously improved by increasing the number of averages made. Unfortunately the relationship is not linear but follows an inverse square root law and a compromise must be found between accuracy required and practicability of multiple averaging. For the railway vibration testing, 32 averages have been chosen as a suitable number. This number of averages gives 90% confidence limits on the measurement of the transfer function magnitude (IHI) as given in Table 1. TABLE 1 Relationship between coherence function and confidence limits
Measured value of coherence function 0.2 0.5 0.9
90% confidence limits for IHl A r 32 averages 256 averages +3*8 -7.1 +2.1 -2.8 +0.8 -0.8
+1*5 -1.8 +0.8 -0.9 +0*3 -0.3
The improvement in confidence limits obtained by averaging 256 samples is clearly evident but such a process would increase the testing time by an unacceptable factor. It has, however, been possible to substantially improve the confidence limits in a way that was not anticipated. It has been mentioned that spatial variation of track form parameters is a very likely characteristic of the track support system and, therefore, that measurement of the system with a single point of excitation would not yield representative results. In order to overcome this problem, the track parameters are measured at several different locations including over-sleeper and between-sleeper rail excitation with corresponding response measurement positions. Five different locations have been used for most of the testing. The transfer function estimates generated for these conditions are averaged to remove (or reduce) the effects of parameter variability. This averaging is performed as an extension of the 32 averages used to generate the estimate of the transfer function associated with a single point of excitation. The effect of this extension is to further average out the extraneous noise in addition to averaging out the variation in the estimates of H, Appendix 2 shows the details of this process. A further effect of this extended averaging is seen on the coherence function. The coherence function of the 5 x 32 averages now incorporates the effect of the variability in H and this, in general, causes a reduced value of coherence function relative to that obtained from one set of 32 averages, the amount of reduction being proportional to the variability of the estimates of H. This is also demonstrated in Appendix 2 and provided that the coherence from a set of 32 averages is fairly high an approximation can be obtained for the variance of H. This, of course, is a function of frequency. Figure 4 shows this effect and the frequencies at which coherence has been depressed indicate variability in track parameters for nominally the same form of track.
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0 Frequency
(Hz)
(b)
Figure 4. Effect of track form variability on coherence measurement. Track form consisted of Pandrol clips and concrete sleepers in new, fully concreted tsack bed. (a) Transfer function showing single point measurement (fi) and averaged multipoint measurement (H); (b) coherence function.
5. SITE
TESTING
AT ALDWYCH
The impact method has been used at two sites on the London Transport underground system. One of these was on the Jubilee Line at Regent’s Park and is described in the next section. The tests described in this section were designed to investigate track form characteristics and to’develop the method for this application. The tests were performed on a spur line in a section of tunnel between Holborn and Aldwych Stations which carries a three car shuttle service at peak travel periods only. This enables track possession to be obtained during the daytime and greatly facilitates work. A further advantage is that the section of track is used by the Permanent Way Department to test new forms of track and several different forms are available in short lengths for comparative testing. Initial tests with manually applied impact were carried out at Aldwych to investigate the feasibility as described in the previous section and on the basis of these it was decided to proceed with more elaborate development of the method including the construction of the automatic drop hammer. The design of this was such as to maintain the input
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force spectrum within the same frequency range but to enable the peak force to be increased to 700 kg. Subsequent tests at Aldwych were all carried out with this equipment. Much of the testing performed was designed to build up confidence in the test method. To this end examination was made of the effects of allowing the passage of service traffic between measurements of transfer function. Mixed results were obtained for this series of tests. For some track forms repeatability was good as shown in Figure 5
cl
500
Frequency
Figure
5. Change
in measured
transfer
function
(Hz)
due to passage
of rail traffic over test site.
relatively poor reproduction was obtained. There was, however, a clear distinction between the types of track form which exhibited these different characteristics. The track forms that gave good repeatability were newly installed, i.e., freshly concreted into the tunnel foundations, while the other track forms consisted of old wooden sleepers loosely fitted into the foundations and clearly subject to considerable relocation of track components due to the passage of traffic. A further characteristic of these latter types of track was the lower valued transfer function magnitude and the poor coherence functions that they generated. Each track form was examined in turn and a selection of the transfer functions obtained for these is shown in Figure 6. The functions are reproduced in full despite the fact that poor coherence indicates uncertainty in the values at certain frequencies particularly in the range 200-500 Hz where the frequency content of the hammer input was, by design, very low. As mentioned in the previous section the particular method of averaging the coherence function allowed some assessment of track form variability to be obtained and this indicated very good reproducibility for track forms 4, 5 and 6, concrete blocks or concrete sleepers in a fully concreted tunnel floor, but poorer stability for track forms with wooden sleepers. All measurements of track characteristics described in this section were taken with the rail unloaded. This admittedly causes the track form to exhibit different characteristics from those‘when it is loaded but provides a useful introduction to the method. The measurement of transfer function on loaded track will be discussed in a later section. The fact that the track is unloaded also allows irregularities in track installation, for example uneven sleepers or small clearances, to exert a strong influence over the local dynamic characteristics of the track form.
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(Hz)
Figure 6. Examples of transfer functions of different track forms measured at Aldwych. (a) Track form 4-concrete sleepers in fully concreted track bed; (b) track form S-concrete blocks in fully concreted track bed; (c) track form 6-concrete blocks in concreted track bed with 4 ft drain channel; (d) track form 7-old wooden sleepers in concreted track bed with 4 ft drain channel.
6. SITE
TESTING
AT REGENTS
PARK
The second series of site tests took place at Regent’s Park. These were designed to investigate ground transmission characteristics rather than performance of track systems, although tunnel characteristics are considered to play an important role in the results obtained. The testing involved vertically impacting the concrete foundation of a bored tube tunnel and measuring ground response at various points on the ground surface above. For this application the input force was increased to 1200 kg by increasing the hammer weight. The particular site was chosen because of the relatively uniform geological structure of the ground in that area, being a deep layer of reasonably uniform London Clay resting on a gravel base about 10 m below the tunnel level. The tunnel is of jointed concrete segment construction and lies approximately 30 m below the ground surface in the test area. A further feature of the site is the fact that over the area of interest, and for some distance around, it is uncluttered by buildings or other civil engineering structures of any type. The opportunities for wave reflections are, therefore, much reduced and this represents as near perfect conditions as can be realized for any site on the London Transport System. In order to measure the transfer function, simultaneous signals of excitation and response are required. It was not possible at this site to obtain a direct cable link between the excitation location below ground and the measurement position above ground. The only means by which the data could be transmitted was to employ a radio telemetry system and this was developed with use of f.m. modulation superimposed on a VHF carrier. The force signal from the tunnel was brought to ground level by cable at the nearest underground station and fed to a radio transmitter at the top of a nearby building.
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Data was then transmitted to a mobile laboratory in Regent’s Park about 300 m away. Voice communication was achieved in a similar manner. Triaxial measurements of ground response acceleration were made for all tests. The accelerometers of high sensitivity were mounted on a large steel plate which was rested on the ground and coupled to it by long steel spikes. The accelerometer signals were led by cable to the mobile laboratory where they were recorded along.with the force signal. Some signal analysis was also performed at the time using a Nicolet 660A Spectral Analyser. A comprehensive series of tests was performed to examine many characteristics of the system. Data was collected to enable investigation of (a) system homogeneity, i.e., that within the test area, transfer functions measured between similar positions relative to the tunnel gave the same result, (b) the directional characteristics of the tunnel vibration, (c) the sensitivity of the system to small changes in excitation position, (d) the attenuation of transmitted vibration with distance, and (e) the mechanism of wave propagation through the surrounding ground. Much of the data still requires analysis but some interesting results have already been obtained. It is not appropriate to give details in this report but two figures are presented which demonstrate the type of information which can be obtained by the impact method. Figure 7 shows a three dimensional diagram of the variation of transfer function with
0
of
accelerometer tunnel centre
from hne Cm)
200
100 Frequency
Displacement
(Hz)
Figure 7. Variation of transfer function magnitude with distance measured at Regent’s Park.
distance. The measurement direction was horizontal and normal to the tunnel axis. This effectively gives the attenuation of vibration with distance from the tunnel. Figure 8(a) shows a typical time history of an acceleration measured 30 m from a point on the ground directly above the tunnel. The acceleration in this case is vertical. This display of the time history was achieved by signal averaging and Figure 8(b) shows the trace of a single sample time history. The signal enhancement is achieved by the averaging process and was only possible because a consistent force impact was applied in the tunnel. It is possible to obtain wave propagation speeds from this type of measurement and by examination of the complete set of such time histories an indication may be obtained of the motion of the tunnel and the mechanism of the wave propagation in the ground.
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Figure 8. Time history of transient response to excitation in tunnel at Regent’s Park. Measurement position was 30 m from a ooint on the around immediatelv over the tunnel. (a) Averane of 32 time history samples; (b) single sample of time history.
7. FUTURE
DEVELOPMENT
AND
STUDIES
The immediate future must, to a large extent, be used to analyze the data already obtained and to obtain as much information from these as possible particularly with respect to ground transmission. The most important outstanding question, however, is the relationship between results obtained by the impact method and the vibrations induced by trains themselves. This uncertainty falls broadly into three areas. (a) The effect of the train dead load on the transmission characteristics of track forms is the first uncertainty. This is clearly an important factor and one on which testing has already begun. Dead load can be practically applied to the track only by gravity loads since reaction against any part of the tunnel could cause serious modifications to the system under investigation. Initial tests have already been completed with a train itself as the load and with the track excited through a train wheel by excitation on the top surface of the wheel, to which access is obtained through the carbody. This method clearly involves the bogie and wheelsets in the mechanical system and a method of mass cancellation is currently being developed to remove their influence. (b) The effects of inputs from a multi-wheeled train and possible correlation of the forces involved constitute the second area of uncertainty. Tests carried out at Regent’s Park should provide some information concerning the former; the effect of the latter is more difficult to determine and the impact method alone may not be able to provide much relevant information.
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(c) The final area of uncertainty is the effect of horizontal force input from a train on the vibration levels generated. No plans to investigate this have yet been formulated but an adaptation of the automatic hammer will be required. As has been stressed, the correlation between train vibration and impact excitation results is ultimately required and with that end in mind train vibrations have always been recorded where impact measurements have been taken. This will continue to be the case until the impact method is fully proven.
8. CONCLUSION The method described has been developed to provide objective and quantitative measurements of the characteristics of railway vibration transmission, but another and equally important motive is to gain a better insight into the mechanism of that transmission and, therefore, ultimately to be able to provide guidance on tunnel and track design. The impact method is independent of the characteristics of trains and therefore enables the parameters of the civil engineering structure to be determined absolutely. The ability to average the received data enables good rejection of background or instrumentation noise to be effected and the use of impact excitation facilitates time history analysis to determine, for example, wave propagation speeds. This is particularly useful for a system where travelling waves are of far greater importance than standing waves. Track possession is necessary for performing this type of testing but during the course of a possession a large amount of data can be collected. It is not necessary to run test trains for any form of this test method but the use of a stationary train to provide dead load for some forms of testing may be necessary. REFERENCES 1. London Transport Research Laboratory Report 1980 Complaints of railway noise received by
London Transport 1969-79. 2. J. E. MANNING, R. G. CANN and J. J. FREDBERG 1974 U.S. Department of Transportation, Report No. CJTMA-MA-06-0025-74-5.
Prediction and control of rail transit noise and vibra-
tion, a state of the art assessment. London Transport Research Laboratory Report 1969 Noise and vibration
due to the operation of trains on the Victoria line. R. G. WHITE and M. E. J. MANNERING 1975 Journal of the Society of Environmental Engineers. Techniques for measuring the vibration transmission characteristics of the ground. R. K. OTNES and L. ENOCHSON 1979 Applied Time Series Analysis. New York: John Wiley & Sons. J. S. BENDAT and A. G. PIERSOL 1971 Random Data Analysis and Measurement Procedures. New York: John Wiley & Sons.
APPENDIX 1: FUNCTIONS USED IN SPECTRAL ANALYSIS Spectral analysis is a method of analyzing data in the frequency domain. The spectral characteristics are of particular value since physical systems have many interesting and significant properties related to the spectral content of the excitation to which the system is subjected. This applies to mechanical and electrical systems alike. Although of immense value, frequency analysis is not always the type of analysis which is most appropriate for investigating a system. Once a set of data has been frequency analyzed certain information may be irretrievably lost, in particular information in the amplitude domain. For some systems this may be important, for example where material yield or fatigue is of importance. It is possible, however, provided that the correct functions are evaluated to retain in the frequency domain a complete description of the system. Transformations
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of the frequency functions generated can then yield information to enable system response in any situation to be determined. The frequency response function or transfer function is one function which satisfies this requirement. The frequency analysis of data has been considered in detail by many authors [S, 61, and therefore only a brief description will be presented here. The transfer function H(f) is defined as A(f) = $,,(f)/&(f) where S,.,(f) is the cross spectral density and S,,(f) is the auto spectral density of the input. The caret above the function symbols indicates that the functions are estimates: i.e., the function values are subject to a certain degree of uncertainty depending on the measurement procedure. Estimates of spectral densities used here are produced by ensemble smoothing of a set of raw estimates, as for example in the case of S,,(f) where this estimate is produced by the process
w h ere Xi(f) and X?(f) are raw estimates or, in shorthand form, $,,(f) =X(f)X*(f), of the Fourier transform and its conjugate of the input signal x(t). The use of smoothing greatly improves the statistical accuracy of the transfer function estimate despite the fact that the raw estimates can be of poor quality, although in ideal circumstances a raw estimate can be practically perfect. Another function which is of great use and which is frequently considered in conjunction with the transfer function is the coherence. This is defined as y* = ]S~x,(~)\2/S1~ (&,, (f), where S,,(f) is the auto spectral density of the output y(t). In the measurement of most types of system, and in particular when measuring the characteristics of railway systems, the measured output contains some signal which is not caused by the system input. This, in many cases, is the reason why raw estimates of transfer function are unreliable. A sample transfer function will be calculated by using a sample response signal which contains a certain amount of extraneous noise. The Fourier transform of the sample response yi(t) is given in this case by Y;(f) = H(f)Xi(f)+ZVi(f), where Ni(f) is the Fourier transform of the extraneous noise n;(t). The smoothing of the transfer function averages out the noise to a degree dependent on the number of sample averages and the level of extraneous noise. This latter factor is directly indicated in a quantitative manner by the value of the coherence function, It can easily be shown that the coherence function is also given by y* = ]H(f)\*Sxxx(f)/&,(f): i.e., the coherence function is a measure of the output power that is due to the input divided by the total output power. Measurement of the transfer function can be made with any form of input desired. In practice the use of impulse excitation and transient response has many advantages. The sample length for each excitation/response sequence is chosen such that the input and output signals are entirely contained within the sample time window. This means that there is no frequency smearing due to truncation effects and also that any time delay in the system transfer characteristics does not depress the coherence function as would occur if random excitation were used. APPENDIX 2: EFFECT OF VARIABILITY OF H ON THE MEASURED FUNCTIONS In this Appendix the effect of measuring nominally identical transfer functions is investigated. Averaging is performed in two stages, firstly to produce an estimate of a particular value of H and secondly to average the individual values of H. The notation of Appendix 1 is used but with the indication of the independent variable f omitted. By
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definition $*, = yx*, and so one has &,, = Hm +Nx*, where Hi is the true value of a particular transfer function estimated by the first averaging process. A =s’,,/L!Zx,,= (HiF+NX*)/xx”. By continuing the averaging for successive estimates of H one obtains --d = &,/&, = (HXX* + NX*)/XX*. The extraneous noise is, therefore, subject to both averaging process and its effect on the estimate H is reduced to a low level. In addition the use of consistent force impulses for excitation purposes causes the input spectrum to be extremely constant and consequently the variability of XX* is very small; this can then be ignored in the second averaging process. These two factors lead to the result H -H, which is a good approximation provided that the noise n(t) is not of excessive magnitude. A similar process can be applied to the coherence function: c’ = I& 1*/&& =((HiX+N)X*)
= (xY*)*/yy*
xx*
((H:X*+N*)X)
(H&+N)(H:X*+N*)XX* HiHT (xX*)” =(HJrrXX*+NN*)xx* provided that n(t) is of sufficiently low value such that terms of the type NX” are relatively small after the first averaging process; if this is not so then --(H&X*+NX*)(H:XX*+N*X) “=(HiHTXX*+H&N*+H”X*N+EF)E?* Continuing the averaging for successive values of H one obtains --(HXX* + NX*) (H*XX* + N*X) + -(HH*XX*+HXN*+H*X*N+NN*)XX*’ Since XX* is sensibly constant because of the consistency of input and the first averaging procedure and also since terms of the type X*N will be relatively small, following double averaging the above expression reduces to -~*=EG*~~*~(*~~X+NN*). If the noise spectrum NN* is relatively low then one obtains -**=HH*/HH*=@*z/((r*+p*), Y where k and (T* are the mean and variance of the magnitude of H. If the noise spectrum W NN* is of significant value then, provided that the value of Hi is, by inspection, close to that of H, (l/+*)-(~/~*)~(HH*-H~H*)/HH*z~*/~** The above derivations are only approximate and, therefore, should only be used as a guide to determine, for example, the number of estimates of H that should be made at different locations on nominally the same system.