In situ determination of loss and coupling loss factors by the power injection method

Journal of Sound and Vibration (1980) 70(2), 187-204

IN SITU

DETERMINATION

FACTORS

OF LOSS AND COUPLING

BY THE POWER

INJECTION

LOSS

METHOD

D. A. BIES AND S. HAMID Department of Mechanical Engineering,

University of Adelaide, Adelaide, South Australia 5000

(Received 28 July 1979, and in revised form 17 November 1979)

Vibrational energy distribution between two coupled plates is considered. Inversion of the linear power balance equations is used to determine the plate loss factors and the coupling loss factors in situ. To accomplish the determinations power was injected and measured sequentially at five points chosen at random on each plate to ensure effective statistical independence of modes. In each case the response of both plates was measured at ten randomly chosen points and mean values of response and injected power were determined. Good agreement is obtained between the predicted and measured coupling loss factors and between the in situ loss factors and loss factors determined for each plate separately also in steady state from power injection measurements. Loss factors deter-

mined by transient decay methods are consistently lower than those determined by either steady state method. It is suggested that the latter result may be true because the energy distribution among decaying modes is not the same as in steady state. During reverberant decay the more lightly damped modes predominate giving rise to an apparent loss factor which is significantly less than the steady state loss factor.

1.

INTRODUCTION

Statistical Energy Analysis (SEA) offers a very powerful means for estimating the distribution of vibrational energy in a structure built up of identifiable separate parts characterized by quantities of stored vibrational energy and mode counts within narrow frequency bands. The coupling between parts and losses of energy from the parts of the system are described by coupling loss and loss factors. In steady state linear power balance equations written in terms of the coupling loss and loss factors allow calculation of energy levels among the various parts of the system. If the various coupling loss and loss factors are known then resulting vibrational energy levels may be predicted for any arbitrary distribution of power injected into the system. If, as may be the case, however, the coupling loss and loss factors are not known or are only partly known or are to be verified for the purpose of confirmation of some prediction technique then it may be desirable to measure them in situ in a test structure. Lyon [l, see pp. 18 1 and 2171 has suggested that this may be done by measuring energy levels in the various parts of the structure for various arbitrary distributions of injected power. If the number of input power distributions is chosen sufficiently large then the resulting linear system of power balance equations may be inverted to determine in situ the coupling loss and loss factors. Lyon also observed, however, that the method has had limited success. When small values of coupling loss and loss factors are involved the quantities sometimes turn out to be negative which of course is physically impossible. The matter of measuring injected power has presented a problem. This is because the systems which seem to have been studied have been metal plates or plates and air filled cavities and only recently have successful means become available for measuring the vibrational energy injected at a single point into plates [2-S]. None is yet available for 187 0022-460X/80/100187+18

$02.00/O

@ 1980 Academic

Press Inc. (London)

Limited

188

D. A. BIES

AND S. HAMID

measuring the power injected into air filled cavities. Swift and Bies described a successful power injection measuring device for plates which is used in the experiments described here [2]. Their device measured the force and acceleration at the point of injection and the phase between these quantities. The latter measurement, which is critical, was considerably improved in precision by the simple expedient of matching the mechanical impedance of the driver to the mechanical impedance at the point of power injection into the plate. The impedance matching was accomplished in the design of the power measuring device. The latter device in turn was designed for use with a Briiel and Kjaer mini-shaker. Yet one more problem remains to be overcome if Lyon’s inversion technique is to be successfully used to measure coupling loss and loss factors in situ. Fahy has suggested that power injected at a single point will result in modes which are not statistically independent, a requirement fundamental to SEA. Under such circumstances Fahy has predicted and Swift has confirmed that energy ratios predicted on the basis of statistically independent modes in a coupled structure will not be realized in practice [3,6]. Swift found that a non-contacting lightly coupled electromagnetic drive of the order of a bending wavelength or greater in diameter produced energy ratios in good agreement with prediction but a rigidly coupled drive of the order of a quarter bending wavelength in diameter or less produced discrepancies as large as 10 dB between predicted and measured energy ratios. Thus if a single point power injection device is to be useful for the determination of coupling loss and loss factors in situ the objection raised by Fahy must be resolved. In this paper it is shown that modal statistical independence may be sufficiently well approximated by the simple expedient of injecting power at three or more points of the system chosen at random. Averages over all measurements are then used to determine mean quantities. Under such circumstances the power balance equations for a two coupled plate system have been successfully inverted to determine the coupling loss and loss factors in situ. These quantities are compared with prediction in the former case and with separate measurements in the latter case. The comparisons of predicted and measured coupling loss factors are reasonably good. It is shown that the loss factors determined by the power balance equation inversion are in good agreement with steady state determinations of the latter quantities and the steady state values are consistently higher in all frequency bands than are the same quantities determined by the usual reverberation decay technique. An explanation for the discrepancy is proposed which has implications of some importance for all determinations of loss for multimodal systems by the standard reverberation technique. It is suggested that differences between steady state and transient determinations of loss factors in general may be expected. 2. ANALYSIS The power balance equations for a system of N connected subsystems may be written as follows: Pik = l?ik + WEik f qij -0 j=l

f Ejkv,i, j=l jti

i=l,2

,...,

N.

(1)

The coupling loss factors nii are related by the consistency equations [I, see pp. 122 and 211; 71 niqij

=

ajVji7

(2)

where ni and nj are the modal densities of the ith and ith subsystems. In equations (1) the convention has been adopted that nii represents the loss factor for the ith subsystem and nij represents the coupling loss factor describing power flow from the ith to the fib subsystem.

IN SITU

DETERMINATION

189

OF LOSS FACTORS

Pik is the power injected into the ith subsystem and & is the energy stored in the ith subsystem due to the kth distribution of injected power into all of the N parts of the system. The index k is introduced to indicate that the subscripted quantities depend upon the distribution of injected power among all of the parts of the system. For example, N arbitrary distributions of Pik will produce N* linearly independent equations in the quantities Tij according to equations (1). w is the centre band radian frequency of the narrow frequency band considered. Equations (1) represent the transient state; in the steady state, on the other hand, as the energies are independent of time, ri,, = 0,

i=1,2

,...,

N.

(3)

Under steady state conditions equations (1) become a linear algebraic system in either the Eik or the vi? In the most general case where all subsystems are interconnected and in consequence no vii)s are zero there will be N(N - 1) coupling loss factors and N loss factors giving a total of N* coupling loss and loss factors [ 1, see pp. 18 1 and 2 171. As suggested, however, it is always possible to determine all of the nits by constructing N arbitrary distributions of Pik, measuring the resulting N* values of the Erk in steady state and using equations (3) to invert equations (1). Conversely, given all of the vii’s, one can determine the Eik’s by equations (1) for any arbitrary distribution of injected power Pik. For the purpose of determining the loss and coupling loss factors in situ a good strategy to adopt might be to generate as many equations as possible, by using equations (1) and a large number of distributions of injected power. If the modal densities of the subsystems can be independently estimated the consistency equations (2) may be used to reduce the number of unknown coupling loss factors. More equations than unknowns may thus be generated. A system of M linear algebraic equations in N unknown quantities of the following form will result: A, = ?! arnnq, -Pm, ??=l

m=l,2

,...,

M,

(4)

M>N.

(5)

In equation (4) the injected power Pik has been replaced with an arbitrarily ordered series P,,, ; similarly nij has been replaced with an ordered series q,, and the coefficients a,,,”are dependent upon the measured values of stored energy Eik and perhaps calculated values of modal densities. The small quantities A,,, are the result of small errors in the experimental determinations of the quantities a,,,”and P,,,. Following standard least squares procedure one can choose the values of T,, to minimize the sum of the squares of the errors A,,,. One forms the sum and sets the partial derivatives equal to zero as follows: S=

y A;, m=l

($) dS/dq, = F A,a,,,, = 0,

n=1,2

WI=1

,...,

N.

(67)

Equations (7) may be rewritten by using equations (4) as follows:

(8) The equations (8) constitute a linear algebraic system of N equations which may be inverted to determine the N unknowns qn. As the coefficients of equations (8) become less dependent upon individual measurements as M exceeds N a strategy for improvement of

190

D. A. BIES AND

S. HAMID

precision is offered but of course at the expense of a great increase in number of measurements. Equations (1) provide an alternative approach for determining the nii’s. In principle N arbitrary initial distributions of Eik could be established and all of the Pik could be set equal to zero. The resulting N2 differential equations (1) could then be solved for the vii’s* Clearly the mathematical description of the transient case is much more complicated and in consequence more difficult to use to determine the vii’s than is the steady state method. This approach, however, may have a more serious flaw than complexity as it has been assumed implicitly that all of the vii (i # j) are time independent: that is, they are characterized by a relaxation time very small compared to the times of all transient events. This assumption in general may not be true. Relaxation time appears not to have been previously considered in connection with SEA but it is a very familiar concept important to the consideration of perturbed systems in dynamic equilibrium [S]. Certainly equations (1) describe such a system and the time average quantities Pik and Eik related by equations (1) imply that a finite time is required to establish equilibrium; a relaxation time, perhaps of the order of a mean free path transit time, is implied. If r is the implied relaxation time then for steady state one requires that averaging times t be much greater than 7: t >>7.

(9)

If the inequality (9) is not satisfied then the qii of equations (1) are probably time dependent, if they can be defined, and the resulting Eik will not be quasi-stationary: that is, at any instant they will not have steady state values. Whether inequality (9) is satisfied or perhaps even necessary the mathematical complexity and the difficulty of making the required measurements with sufficient precision has prevented the use of equations (1) to determine the coupling loss and loss factors in situ. Rather, the alternative of measuring nii of the N individual parts prior to assembly has been used where possible, and where this is not possible values of vii have been arbitrarily assumed based on some prior experience. Similarly the vij are calculated or estimated by some procedure. Consider now a single uncoupled subsystem and suppose that the Ei are the modal energies resulting from some distribution of injected power among the modes of the subsystem in a narrow frequency band. In this case the total energy E stored in the subsystem is the sum of all modal energies in the frequency band: E = f Ei. i=l

(10)

Differentiation of equation (10) with respect to time provides an equation for the transient response of the subsystem which may be written in terms of modal loss factors as follows [9] (small damping has been assumed):

B=--W f

Eiqii.

i=l

(11)

Alternatively if one injects power in steady state the total power P is the sum of the power flows into the various modes of the narrow frequency band. In this case the steady state response may again be written in terms of the modal loss factors as follows:

P=w f Eiqii. i=l

(12)

IN SITU

DETERMINATION

OF LOSS FACTORS

191

The right-hand sides of equations (11) and (12) are formally the same which suggests that one defines an average value of loss factor as follows: z

Eivii =&iv

(13)

i=l

Equation (13) may be used to integrate equation (1 l), leading to the following familiar equation: ?Y = 2*2O/(fT60).

(14)

In equation (14) the centre band frequency isf = w/27r and the 60 dB decay reverberation time is ThO. Alternatively use of equation (13) allows equation (12) to be rewritten as follows: Ki= PItiE: (15) Equation (13) has been proposed as a definition for the average loss factor [9]; however, such a definition is unsatisfactory as it depends upon the distribution of stored energy among the various modes and there is no reason to suppose that the modal energy distribution is unique. For example, when, as will be shown, different values for average loss factor are obtained from steady state and reverberation decay measurements, one must conclude that the modal energy distributions are quite different in the two cases. For the lightly damped plates considered in this paper the best that can be done is to assume that the power flow into any mode is on average equal to the power flow into any other mode of the uncoupled plate. One must therefore assume that one can arrange the experiment so that in steady state the assumption of equal mode power flow is valid. Under this circumstance equation (13) then serves as the definition of average loss factor. The ideas expressed here are consistent with the assumption of possibly light coupling between modes of the plate and it is worth noting that only in the case of strong coupling between modes would one expect equal modal energies [7]. In the latter case equation (13) would then lead to

Apparently equation (16) can be expected to hold only when the coupling between modes is much greater than the damping of any mode [7]. It is proposed that use of equation (15) will lead to a value of internal loss factor as defined by equation (13) under the condition of equal modal power flow. Whether use of equation (14) also leads to the same value depends upon whether or not the inequality (9) is satisfied. 3. TWO COUPLED

PLATES

The power balance equations (equations (1)) lead to the following useful formula under conditions of steady state (equation (3)) for two coupled subsystems: &/EI

= &(VZ

+

(17)

7122).

The subsystems are indicated by the subscripts 1 and 2 in equation (17). Alternatively E2/&

=

(n2/d7?21/(7121+

722)).

(18)

In the latter equation the consistency relation given by equation (2) has been used. Either of equations (17) and (18) may be used to calculate the ratio of energy stored in system two to system one, given the appropriate coupling loss and loss factor for system two.

192

D. A. BIES

AND

S. HAMID

For the plates considered in this investigation the modal densities were calculated by using the following formula valid for thin plates [l, see p. 2821: n(f)=d?AJhCL,

(19)

where A is the plate surface area, h is the plate thickness, and CL is the longitudinal wave speed in the plate material. The coupling loss factors were calculated by using the following formula [lo]: ~12 =

W~Ul~1Ad712,

(20)

where L is the length of the junction, A, is the area of plate one, and k1 is the bending wave number for plate one. The bending wave number is given by kl= (w’i?/hcJ”.

(21)

The transmission coefficient T was calculated by using the approximate 712= 712(0)2*754X/(l +3*24X),

formula [3]

X = hJh2.

(22,W

In equation (22) the transmission coefficient 712for random incidence vibrational energy has been written in terms of the normal incidence transmission coefficient ~~~(0).For thin plates the latter quantity may be calculated as follows [ 111: 712(O)= 2[$1’2+ $-1’2]-2,

(24)

where

(25) The material densities of plates one and two have been represented by p1 and p2 respectively in the above equation. In the special case considered in this investigation of plates of the same material equation (25) simplifies to the ratio of plate thicknesses so that equation (24) becomes 712(O)= 2[x5’4 + x-5’4]-2.

4. MEASUREMENT

OF DAMPED

(24a)

PLATE LOSS FACTORS

For this investigation light gauge mild steel flat plates were used which were of the order of 1 mm in thickness and 0.2 m2 in area, being generally rectangular but with non-parallel edges. They were lightly damped by the addition of a felt material bonded to one side of the plates with an adhesive. The choice of damping material was based upon an experimental investigation of various alternatives [3]. The material chosen gave loss factors significantly higher than those of the undamped plate but not too high for the present purpose. Thus the added damping was chosen so that internal loss for each plate dominated all other losses. In particular as the plates would be primarily edge radiators in the frequency range investigated the complication of a change in the edge radiation due to a change in edge condition when the plates were subsequently welded together along a common edge could be ignored. For test, a plate was supported and driven as described in Appendix A. Prior to the joining of two plates to form a two plate model the loss factor of each plate was determined in two ways over the frequency range from 400 Hz to 5000 Hz inclusive in one third octave bands. The lower frequency bound was chosen so that the mode count in the lowest frequency band was approximately 4, which is the minimum number required for the purposes of SEA, and the upper frequency bound was chosen so that damping which was

IN SITU DETERMINATION

OF LOSS FACTORS

193

frequency dependent was not too high [3]. The loss factor was determined by using the standard reverberation decay technique and equation (14). The loss factor was also determined from a steady state determination of the plate response and the input power by using equation (15). The input power was measured by using a power injection transducer [2] which is described in Appendix B. Subsequent to the joining of the plates the loss factors of the plates were determined by inversion of the power balance equations as discussed earlier. In this case also the injected power was measured by using the power injection transducer. The results for the three determinations of the loss factors for the two plates investigated are shown in Figures 1 and 2. The figures show quite clearly that the loss factors determined in steady state prior to plate assembly are consistently higher than those determined by the standard reverberation decay technique. Furthermore, the figures show equally clearly that the loss factors determined in situ by power balance equation inversion are in good agreement with the values determined prior to assembly by the steady state method. The data in Figures 1 and 2 strongly imply that the inequality (9) is not satisfied in the transient case. The form of equation (13) suggests that whatever the initial modal energy distribution those modes least attenuated will persist and eventually determine the value of the measured average loss factor; thus the value of the latter quantity can be expected to be time dependent if condition (9) is not satisfied. This suggests that if steady state energy equilibrium is initially achieved prior to cessation of power injection but subsequently lost during reverberant decay the initial decay should give a value of loss factor the same or close to that determined in steady state [9]. This indeed was found to be the case as described in Appendix A but it was only possible to determine an initial decay different from the long time 30 dB decay used to obtain the data shown in Figures 1 and 2 in the first four lowest frequency bands, and these determinations were very poor at best as they were evident only in the first few decibels of decay. Nonetheless, in spite of the uncertainty in determination, values closer to those determined in steady state, rather than from

One-thwd

octave centre bond frequency (Hz)

Figure 1. Plate one internal loss factors qll. Method of determination: 0, uncoupled plate steady state method; 0, uncoupled plate reverberant decay method: 0, in sifu power injection method.

194

D. A. BIES AND S. HAMID

I0.02 -

0.01 -

8

0.005

-

0002

-

0.001

-

a 4

OGiw5’

’ 400

’ 500

’ 630

One-

’ 600

third

’ ’ 1 ’ 1 1 1 IOCO I250 16002GW250031504CC05W0

octave centre

band frequency

1

(Hz)

Figure 2. Plate two internal loss factors qz2. Method of determination: 0, uncoupled plate steady state method; 0, uncoupled plate reverberant decay method; 0, in situ power injection method.

reverberant decay, were obtained in the two lowest bands and in all cases the values were greater than those obtained from the long time reverberant decay. Based upon our experience reported here and experience reported elsewhere we have concluded that the reverberation decay method does not offer a viable means for the determination of steady state loss factors for plates [2,3]. A detail of the test arrangement requires mention. For the reverberant decay determinations of the plate loss factors the plates were driven by a non-contacting electromagnet which was biased with a permanent magnet. During decay the weak coupling between the permanent magnet and the plate continued; no effort was made to remove it. The weak magnetic field would induce eddy currents in the plate due to the vibration of the plate and in consequence the damping of the plate motion would be increased. The latter effect was ab:,ent during the steady state determination of the plate loss factors. Thus if the induced eddy current damping were significant it would have the effect of increasing the apparent loss factors determined by reverberation decay and thus decreasing the apparent difference between these quantities and those determined in steady state. The conclusion that condition (9) is not satisfied in the transient case is further strengthened.

5. INVESTIGATION OF TWO PLATE MODELS 5. I. SIMULATION OF STATISTICAL INDEPENDENCE Subsequent to determination of plate loss factors the plates were welded together at a common edge to form a two coupled plate model. The plates were joined approximately at right angles to each other. The experimental arrangement is described in Appendix A. The power injection transducer used in these experiments allowed measurement of the power injected at a single point, which was adequate for the determination of steady state loss factors of the uncoupled plates but, as pointed out by Fahy, such excitation is inadequate for the purposes of SEA. For SEA analysis all modes must be statistically independent, whereas point excitation ensures that they are all coupled and not independent. The

IN SITU

DETERMINATION

OF LOSS

195

FACTORS

question thus to be considered was that of how many points of excitation chosen at random must be used to adequately simulate statistical independence for the purposes of SEA. For these tests plate one was first driven by using the non-contacting electromagnetic drive and the resulting energy ratios in each one third octave band were determined [3]. The ratios so determined were subsequently used as reference levels. Next, plate one was driven at five randomly selected points in sequence and in each case the resulting energy ratios in one third octave bands were determined. The latter ratios were then treated as a single set to determine an average response level. The average response level is compared with the non-contact reference level in Figure 3. For comparison, ratios taken from reference [3], determined for two coupled plate systems with plate one driven at a single point, are also shown in the figure. It is apparent from reference to the data in the figure that the simple expedient of sequentially driving the driven plate at five points chosen at random is sufficient to adequately simulate the non-contacting drive situation. We therefore feel justified in assuming that the latter case adequately satisfies the requirements of SEA [3]. Review of our data suggests that as few as three randomly chosen points of excitation may be adequate.

6~ s

0 ,Q $

o_

,

, *

4-

,

-10

,

, X

X X

2-

-6

,

X

,

,

,

,

,

X

x x

_ I1 400

500

11 630

800

One-third

I 1000 octow

I 1250 ccntrc

I I 1600 2000

,

11 2500

bond frequency

X

1 3150 4000

Y

-

X

-

I 5000

(Hz)

Figure 3. Comparison of measured energy ratios for two coupled plates. Subscripts refer to plates i and j where plate j is driven either in point contact indicated by subscript P or in distributed non-contact indicated by subscript b. Response to single point excitation is indicated by x and the mean response to excitation at five points chosen at random is indicated by 0.

A very large number of measurements are implied by the data in the figure. It is to be noted that each data point shown is the result of approximately 40 measurements in the case of the single point excitation and 120 measurements in the case of the five point excitation. In turn each measurement was the result of approximately 90 separate measurement samples: that is, in each case the plate energy level was determined from ten randomly placed acceleration measurements on the surface of the plate and approximately 90 separate samples were used to determine each mean value as described in Appendix A. In the same way approximately 20 000 measurement samples were acquired in each one-third octave band to produce the data discussed in the next section. Clearly if SEA is to be a useful tool for the investigation of large structures the implied enormous number of measurements must be reduced to manageable proportions. The use of sophisticated measurement equipment yet to be developed is implied.

196

D. A. BIES AND S. HAMID

1

I

400

I

500

I 630

I 600

One-third

I 1000

octave

I 1250

I 1600

centre

I

I

2000

2500

band frequency

I 3150

I 4000

I 5000

I

(Hz)

Figure 4. Coupling loss factor r)ir. 0, Theoretical values calculated according to equations (20), (21), (22) and (23a); 0, measured values determined by the in situ power injection method. 5.2.

POWER BALANCE

EQUATION

INVERSION

The measurements of injected power and plate responses of the previous section provide the opportunity to investigate the inversion of the power balance equations under conditions of steady state. The determination of the plate loss factors in situ has already been discussed. The results of the inversion for the determination of the coupling loss factors by the same method are shown in Figures 4 and 5. In each case the predicted coupling loss factor calculated according to equations (20) to (25) inclusive is shown for comparison. The predicted values appear to be a bit higher on average than the measured values which though somewhat scattered are reasonably self consistent.

0.02

.

0.01

g t -0 $ 0 a v

0.005

oGQ2

-:;:i::!.;:,;r;--

1

I

I

I

I

400

500

630

600

One- third

I 1000

octave

I

I

1250

1600

centre

band

I

I

I

I

I

2000

2500

3150

4000

5000

frequency

I

(Hz)

Figure 5. Coupling loss factor qsi. 0, Theoretical values calculated according to equations (30), (21), (22) and (23a); 0, measured values determined by the in situ power injection method.

5.3. COMPARISON OF PREDICTED AND MEASURED ENERGY RATIOS The power balance equations (equations (1)) may be used to predict the energy balance between two coupled plates when one plate is driven. The energy ratios so predicted may then be compared with the measured energy ratios. For the purpose of prediction the measured loss factors based upon the steady state determinations and the coupling loss factors calculated according to equations (20) to (25) inclusive are used. The comparisons

IN SITU DETERMINATION

III

I 400

I

I

I

197

OF LOSS FACTORS

I

I

11

1

I

J

500 630 800 1000 1250 16002000 25CXl 31504000 5000 One-third

octave

centre

band

frequency

(Hz)

Figure 6. Predicted and measured energy ratios for two coupled plates. Plate two is driven by using non-contacting excitation. 0, Theoretical values calculated according to equation (17) with calculated values of coupling loss factors; 0, measured values determined by the in situ power injection method.

between measurement and prediction are shown in Figure 6, the non-contacting electromagnetic drive having been used for the measurements. The predicted and experimental values agree quite well over most of the frequency range, with the largest discrepancies occurring at low frequencies. Similar results were obtained by Gibbs and Gilford [12] who attributed this behaviour to the gradual breakdown of simple power flow concepts at low frequencies. 6. CONCLUSIONS

The experimental results reviewed above lead to several conclusions. Perhaps the most important conclusion is that the measurements of loss factors in steady state and by the reverberant decay method lead to different results. Furthermore the measured values for loss factors in situ during steady state excitation are in agreement with the measured values for these quantities determined in steady state when the subsystems are isolated by physical separation from each other. It is suggested that the reason for the discrepancy between steady state and transient determined values of loss fartors is that the energy distribution among modes of the system (in this case a lightly damped plate) during reverberant decay is not in steady state equilibrium. If the latter conclusion is true for lightly damped plates it may generally be true. In particular the standard procedure for the calibration of reverberant rooms may be questioned [ 131. The discrepancy between steady state and transient determined values of loss factor demonstrated here for lightly damped plates strongly suggests that SEA may be limited in its usefulness to the study of steady state response. To extend its use to consideration of quasi-transient response as has been suggested [14] may require considerable complication of the formalism. Certainly it will require careful experimental investigation. Loss and coupling loss factors may be determined by inversion of the power balance equations and, provided that the equations are well conditioned for inversion, good results may be expected. From a practical point of view “well conditioned” must mean that the proposed SEA model adequately represents the structure to be investigated and that the measured quantities of power injection and stored energy can be determined with adequate precision. However, the adequacy of the model and the required precision of measurement can only be determined from the results obtained from the model itself. Furthermore as adequacy of precision of measurement must imply that no term in the

D. A. BIES AND S. HAMID

198

power balance equations be less than the uncertainty in any other term the requirement for measurement precision must be linked to the adequacy of the model. Clearly an iterative procedure is implied in which the results of each previous trial are used to refine the model of the following trial until a satisfactory model and satisfactory results are obtained. A least squares fit procedure in which advantage is taken of the ability of the model to generate more equations than unknowns may be used to improve the effective precision of measurement but in general a very large number of measurements is implied. The proposed method requires that the power injected into the system be measured with reasonable precision. This can be done by using a power injection transducer but then only at a single point. To simulate statistical independence of the driven modes power must be injected at several randomly chosen points. Five points sequentially driven have been found adequate but a very large number of measurements is then required. For the investigation of other than the simplest of structures enormous numbers of measurements are implied. This in turn implies the need for the development of automatic equipment capable of rapidly and accurately making the needed measurements and reducing them to manageable form. The latter consideration suggests the use of a computer while the former strongly suggests the need for a new generation of measurement equipment. REFERENCES 1. R. H. LYON 1975 Statistical Energy Analysis of Dynamical Systems: Theory and Applications, M.I.T. Press. 2. P. B. SWIFT and D. A. BIES 1975 90th Meeting of the Acoustical Society of America, San Francisco, November 1975. Steady-state measurement of loss factors. 3. P. B. SWIPI 1977 Ph.D. Thesis, University of Adelaide. The vibrational energy transmission through connected structures. 4. F. J. FAHY and R. PIERRI 1977 Journal of the Acoustical Society of America 62, 1297. Application of cross-spectral density to a measurement of vibration power flow between connected plates. 5. G. E. OTTESEN and T. E. VIGRAN 1979 Applied Acoustics 12, 243-252. Measurement of mechanical input power with application to wall structures. 6. F. J. FAHY 1970 Journal of Sound and Vibration 11,481-483. Energy flow between oscillators: special case of point excitation. 7. P. W. SMITH, JR. 1979 Journal of the Acoustical Society of America 65, 695-698. Statistical

models of coupled dynamical systems and the transition from weak to strong coupling. 8. K. F. HERZFELD and F. 0. RICE 1928 Physical Review 31, 691-695. Dispersion and absorption of high frequency sound waves. 9. F. J. FAHY 1969 Journal of Sound and Vibration 9,506-508. Reply to the Letter to the Editor “Damping in plates” by M. J. Cracker and A. J. Price. 10. R. H. LYON and E. EICHLER 1964 Journal of the Acoustical Society of America 36, 1344-1354. Random vibration of connected structures. 11. L. CREMER,M. HECKL and E. E. UNGAR 1973 Structure Borne Sound. New York: SpringerVerlag. See chapter V, section 2, p. 320. 12. B. M. GIBBS and C. L. S. GILFORD 1976 Journal of Sound and Vibration 49,267-286. The use of power flow methods for the assessment of sound transmission in building structures. 13. BRUEL and KJAER 1978 Technical Review No. 3. Discrepancies between sound power measurements in an anechoic chamber and a reverberation room. 14. G. MAIDANIK1976 Journal of Sound and Vibration 52,171-191. Some elements in statistical energy analysis. 15. L. L. BERANEK 1971 Noise and Vibration Control. New York: McGraw-Hill Book Company, Inc. See chapter 3, Sound and vibration transducers. APPENDIX

A: INSTRUMENTATION, EXPERIMENTAL APPARATUS AND PROCEDURES

A. 1. SUPPORT STRUCI-URE

To avoid the unnecessary complications of possibly variable losses at the points of support of the test plates a support fixture was constructed to minimize such losses. The

IN SITU DETERMINATION

OF LOSS FACTORS

199

support structure consisted of a relatively massive steel block, 250 mm x 100 mm x 50 mm suspended by means of four thin steel wires from an overhead support. The edges of the steel plates were cut to provide three tabs about 25 mm wide and 25 mm long each separated by approximately 100 mm. The plates were supported vertically from above by means of these tabs inserted into a clamping slot on the support block. Due to the extreme mismatching of impedance thus achieved between the plate edge and the massive support block, negligible energy is lost to the latter. A.2. PLATE ACCELERATION MEASUREMENTS The stored vibrational energy E in a thin plate of mass M in a narrow frequency band of centre frequency w is related to the space time average acceleration over the plate (a*) by the equation E = M(a*)/w*.

(AlI

In the case of thin plates the mass loading effect of an accelerometer at the measurement point must be taken into account, especially at high frequencies. In this case the unloaded plate acceleration a is related to the measured plate acceleration a, by (a2>=L(&J.

t-42)

In equation (A2) [15], L = 1 + (mf/14+5h*)*. where m is the mass of the accelerometer in kilograms, f is the centre band frequency in Hz and h is the plate thickness in millimeters. Equations (Al) and (A2) combined give the expression used in determining plate energies from acceleration measurements in this investigation: E = ML(aL)/w*. A.3. STEADY-STATE

(A3)

MEASUREMENTS

The experimental arrangement used for steady state response measurements is illustrated in Figure 7. The same acceleration detection system was used to measure plate

Plate occclerometcr

_

I

.

Amplifier

-

$“e~pass

b-Octave

-

bond

Random noise generator

Figure 7. Schematic diagram of experimental set-up for the measurement of steady state internal loss factors.

200

D. A. BIES AND S. HAMID

response when the plate was driven in point contact, as shown in the figure, and also when the plate was driven with the non-contacting electromagnetic drive shown in Figure 8.

Electromapnctic

l-r

driver

Accelerometer

1

Voltage follower

-

‘5 -Octaveband

spectrometer

-

Rms Ior19 time averager and lin-log convertor

-

Storage CR0

Test plate

Amplifier

-

_ ?KYau

Random noise generotor

Figure 8. Schematic diagram of experimental set up for the measurement of internal loss factors by the energy decay method.

The input power measurements (see Figure 7) were made as follows. The power flow transducer provides simultaneous measurements of the applied force and resulting acceleration at the point of contact with the plate. Integration of the acceleration signal provides the required instantaneous plate velocity at the contact point. As low frequency noise, for example building vibrations, could not be avoided it was necessary to filter the velocity and force signals. As the one-third octave band filters could not be completely phase matched a phase compensation circuit was introduced which was adjusted for each one-third octave band measurement. The required compensation adjustment was previously determined during the power transducer calibration procedure described in Appendix B. The digital meter of Figure 7 deserves mention. This device consisted of a digital voltmeter interfaced with a hand calculator. The cycling time of the calculator provided the clock to sample the digital voltmeter at regular intervals of about 1.1 seconds. Each sample voltage was introduced directly into the summing circuit of the calculator. When a number of measurements had been made as indicated by the calculator display the sampling was stopped and the mean value determined by the calculator. The standard deviation was also available on demand by pressing the appropriate key on the calculator. For the measurements described in the text 90 readings were sampled each time one measurement was made. In Figure 8 the arrangement used for determination of the loss factors by the reverberation decay method is shown. In this case as previously mentioned the plate was driven with a non-contacting electromagnet. Alternatively, a Briiel and Kjaer tone burst reverberation processor was used to determine the loss factors. The two methods gave comparable results but the tone burst generator method did allow some determinations of early decay times in the first few one-third octave bands. It is of note that these early decay determinations of the loss factor in the first few one-third octave bands were all higher than those determined by the long time decay and thus they were consistently closer to the values determined by the steady state method.

IN SITU DETERMINATION APPENDIX B.

B: POWER

OF LOSS FACTORS

201

FLOW TRANSDUCER

1. CONSTRUCTION

The power flow transducer was constructed to inject power into the plates and to provide facilities for measuring the power injected. The power injected is determined as the time average of the product of the vector force and velocity at the point of attachment of the transducer to the plate. Thus besides measurement of the velocity and force amplitudes the phase between them must be carefully preserved in order that the power factor may be accurately determined. Ordinarily the force-producing driver internal impedance will be poorly matched to the input impedance of the plate. For example, a Briiel and Kjaer Mini-Shaker was used to drive the plates in the experiments described in the text. The internal impedance of this device in the frequency range of interest is a large mass inertance whereas the input impedance of the plate, on average, is a small resistance. In consequence of the large impedance mismatch the power factor, which is dependent upon the phase between the force and resulting velocity, will be very small; relatively large errors in its determination cannot be avoided and in consequence the determination of injected power generally will be subject to large error. However, the problem can easily be avoided by providing impedance matching at the point of attachment of the transducer to the plate. Thus, in the transducer illustrated in Figure 9, the permanent magnet serves the dual purpose of providing a ready means for attachment to a steel plate and a mass inertance at the point of attachment to match the input impedance to the internal impedance of the driver.

Stainless Permonent

Insulotinp material

steel magnetic

magnet

head of the compession

bolt

(force

transducers

1

Figure 9. Power flow transducer construction details.

The power flow transducer illustrated in Figure 9 consists of a stainless steel casing enclosing two annular piezoelectric crystals mounted on a stainless steel bolt contained under compression between the bolt head and a tensioning nut. The bolt head carries a Briiel and Kjaer accelerometer type 4344. A small magnet is attached to the driving end with a brass bolt. The other end of the compression bolt is threaded for mounting the transducer on a Briiel and Kjaer Mini-Shaker. In use the transducer and Mini-Shaker assembly were mounted on a cross slide fixed to a rigid frame. The arrangement enabled the transducer driving head to be moved to various positions on the driven plate. The power flow transducer can be represented by an equivalent mobility circuit as shown in Figure 10. In this case, the identifications are as follows. Current i is analogous to

202

D. A. BIES

AND S. HAMID

4

-1

I rli I

ZP

I

L-rJ

i

I

I I -

_l

Figure 10. Power flow transducer mobility circuit.

crystal force, F,. Capacitances M1 and M2 represent masses on the shaker side of the transducer and plate side of the transducer respectively. Inductances KB and Kc represent the spring rates of the compression bolt and crystal respectively. Conductance CL represents losses due to any movement of the accelerometer lead. The voltage V across the plate impedance 2, is analogous to the velocity of the plate at the driven point. For time average quantities, (i,. V)=((i3-i2-iL).

V)=(i3.

V)-(i2.

V)-(i,.

V),

but M2 is purely reactive and so i2 and V are in quadrature: i.e., (iz . V) = 0. Now i3= is + ib+ i7+ ie. However, the piezoelectric crystal and compression bolt have very low mechanical resistances and so C, and C, are both very large. Thus, i5 and is are negligibly small. Hence is = i6+ i7. The forces on the crystal and bolt depend on their respective moduli of elasticity and cross-sectional areas. Therefore, for a particular crystal and bolt combination force on bolt = constant = k/i,. force on crystal Thus, ~‘3= C& where COis a constant. Therefore, (i,. V)=C&.

V)-(i,.

V).

When the transducer is not driving the plate, (ip. V) = 0. In this case power loss in the transducer lead and other losses on the plate side of the transducer are given by the equation (iL . V) = Co&. V).

During calibration of the transducer, this power loss was set to zero by using the phase compensation network. Thus, after calibration, (i, . V) = C& . V) = power input

to

plate = pi*

This is true on the assumption that (it. V) is constant and independent

of the load.

IN SITU

B.2.

EXPERIMENTAL

DETERMINATION

DETERMINATION

OF LOSS

OF TRANSDUCER

203

FACTORS PLATE

SIDE

MASS,

b&

The mass MZ in the mobility circuit represents the plate side mass of the transducer. This was determined experimentally by driving the transducer with a band-limited signal from a Briiel & Kjaer Random Noise Generator. Various size masses were added to the transducer head and for each added mass, the force and acceleration voltages from the transducer were recorded. The procedure was repeated for different frequency bands of excitation and also with sine-wave excitation from a signal oscillator. A graph of the ratio (free voltage)/(acceleration voltage) versus added mass was plotted as shown in Figure 11. The magnitude of M2 was read where the value of the ratio became zero. This was found to

Added mass (g )

Figure 11. Power flow transducer response

as a function

of driven added mass.

When the transducer is driving the plates, the magnet is attached to the transducer head by means of a brass bolt. Therefore, the mass of the magnet and brass bolt have to be added to the plate side mass MZ in order to obtain the effective driving mass Me of the transducer (also known as the calibration mass). Hence, Me = M2 + (mass of magnet and brass bolt) = 29 + 13.77 = 42.77 g. B.3.

TRANSDUCER

CALIBRATION

The Briiel & Kjaer Spectrometers used were calibrated by using their internal reference signals. Thenceforth all power flow measurements were taken with particular amplifier settings of the spectrometers. The phase compensator settings were determined for each l/3 octave frequency band of excitation under no-load conditions of the transducer. In order to calibrate the transducer accelerometer, a reference accelerometer with known sensitivity was used. A Briiel & Kjaer accelerometer type 4344 with a sensitivity of 2.98 mV/g was mounted back-to-back with the transducer accelerometer. For each l/3 octave frequency band excitation, the transducer and reference accelerometer signals were passed through a digital averaging meter to yield the corresponding time-averaged voltages, (V,, > and ( V,r) respectively. The transducer accelerometer calibration constant, CT, was given by Cr = C&( V&/( V,,)) m/s’/mV, where C’,t is the known reference accelerometer calibration constant.

204

D. A. BIES

AND

S. HAMID

The multiplier had built-in facilities such that for two input signals X and Y, it was possible to obtain the three outputs, X2, Y* and XY. Therefore, when the transducer was driving the plates, force signal VF and velocity signal Vv were phase compensated from previously determined phase compensator settings and passed through the multiplier and then time-averaged through the digital averaging meter. The three outputs were (V’,), (V’,) and (V,). These were recorded for each l/3-octave frequency band. From the above, the force calibration constant, CF was obtained by using the equations CF

=

Mecd(

vtrans

>/ Jm)

newtons/mV

where Me is the effective driving mass of the transducer velocity calibration constant, Cv, is given by CV

= (G/u)((V~~~~~WG%~

(as determined

earlier). The

m/s/mV.

Hence, the power flow calibration constant, Cw is given by CW = CFCv watts/mV.