Statistical Energy Analysis (SEA) based approaches for the mid- and high-frequency vibro-acoustic analysis of complex systems Robin S. Langley 1 Department of Engineering, University of Cambridge Cambridge CB2 1PZ, UK

ABSTRACT Methods are presented for predicting the statistics of the frequency response functions of complex random vibro-acoustic systems. The statistical properties include the ensemble mean, the ensemble variance, level crossing rates, extreme values, and quefrencies. The approach is based on a combi- nation of statistical energy analysis (SEA), random point process theory, and random matrix theory. The blocked modes of the SEA subsystems are assumed to have natural frequency statistics that are governed by the Gaussian Orthogonal Ensemble, and this enables the statistical properties of the response to be found without any need for Monte Carlo simulations. The use of SEA leads to a relatively low number of degrees of freedom in the model, and thus the approach is numerically efficient and well suited to the design stage, where many design options may be explored. The meth- ods presented can also be combined with the finite element method by employing the diffuse field reciprocity relation, allowing response predictions to be made across the full frequency range. .

1. INTRODUCTION

Statistical Energy Analysis (SEA) is a well known technique for predicting the vibro-acoustic re- sponse of a complex and random engineering system to high frequency excitation [1]. The original version of the method yields the ensemble average of the vibrational or acoustical energy in each subsystem of the total system. The method has been extended [2] to enable the prediction of the variance of the response in addition to the mean, and this allows the two parameter log-normal dis- tribution to be used to predict confidence intervals for the subsystem energies [3]. Some years ago it was not uncommon to hear the view that SEA does not “work” unless the modal overlap of the system is high; it is now clear that the method works extremely well at low modal overlap, the issue being that the variance is high when the modal overlap is low, so that the mean prediction may not agree with a particular realization of the system built in the laboratory. It is perhaps less well known that SEA has been further extended to predict the higher order statistics of energy frequency response functions, such as level crossing rates, extreme values, and quefrencies [4,5]. The variance theory and the more recent developments are based on the fact that the natural frequencies and mode shapes of a random subsystem tend to follow the statistical distributions predicted by the Gaussian Orthog- onal Ensemble of random matrices [6]. The main aim of the present paper is to summarize these more recent developments and to indicate how they might be further extended by employing a hybrid method that combines SEA with the finite element method.

1 RSL21@eng.cam.ac.uk

sil, —_ Ain ly | inter.noise. 21-24 AUGUST SCOTTISH EVENT CAMPUS. ? O? ? GLASGOW

2. AN ENHANCED VERSION OF SEA.

This section reviews the standard SEA equations which are commonly used to predict ensemble av- erage response values in random systems. The higher order statistical properties of frequency re- sponse functions are then considered, and it is shown that information beyond that predicted by stand- ard SEA is required to compute these properties. An extension to SEA is then described which allows the higher order statistical properties to be found.

2.1. Conventional SEA Equations A built-up system with randomly uncertain properties can often be represented as an assembly of relatively simple components or “subsystems”, and there is engineering interest in predicting the en- semble statistics of the vibrational energy of each subsystem when an excitation is applied to one or more of the subsystems. This topic has been the subject of research for more than half a century, and Statistical Energy Analysis (SEA) is now an established technique for predicting the ensemble aver- age values of the subsystem energies [1]. The SEA equations express a vibrational power balance for each subsystem, and when written in matrix notation the equations have the form

in ˆ = CE P , (1)

where

ˆ 2E[ ]/ j j j E T n = , (2)

≠ = + ∑ , (3)

jj j j jk j k j C n n ωη ωη

, jk jk j C n j k ωη = − ≠ . (4)

E[ ] j T j n j η Here is the ensemble average of the kinetic energy of subsystem j , and and are respec- tively the modal density and the loss factor of the subsystem. The modal density can be found ana- lytically for most types of component [1], and so this quantity can be assumed to be known. The

jk η coefficients are referred to as coupling loss factors, and there are many ways of calculating these values, ranging from modal methods [1], wave transmission approaches [7], and a method based on a diffuse field reciprocity relation [8], and so again these quantities can be assumed to be known. The right hand side of Equation 1 is a vector of ensemble averaged power inputs arising from external excitation, and the power input to subsystem j can be written as [1]

2 in, E[ ] / 4 j j j P F n M π = , (5)

j M where F the generalized force applied to the subsystem, and is the mass of the subsystem. 2.2 Higher Order Statistical Properties of Frequency Response Functions An example of the energy frequency response function (FRF) of a random system is shown in Figure 1 [5]. Three curves are shown, corresponding to the results obtained when the properties of the system are slightly changed – i.e. the Figure shows three members of the ensemble of FRFs associated with the random system. The standard SEA equations described in the previous section will predict

sil, —_ Ain ly | inter.noise. 21-24 AUGUST SCOTTISH EVENT CAMPUS. ? O? ? GLASGOW

b

sil, —_ Ain ly | inter.noise. 21-24 AUGUST SCOTTISH EVENT CAMPUS. ? O? ? GLASGOW

Figure 1: Three realizations of the energy response of a plate with randomly placed masses. The horizontal line represents a level b , and the interest is in the number of times this level is crossed by the energy. the ensemble average value of the FRF, but it is clear that other statistical properties are also of engi- neering interest. For example, the variance of the FRF will provide important information on the spread of the results, and there may also be interest in the probability that the FRF will exceed some critical level b , as indicated in the Figure. The analysis of variance and the emergence of the log- normal distribution for the energy statistics has been described elsewhere [2,3], and so the present work will focus crossing rates, exceedance probabilities, and related measures. The prediction of the exceedance probability requires a consideration of the joint probability density function (jpdf) of the FRF and the rate of change of the FRF with frequency. Letting the FRF relate to the kinetic energy, the jpdf has the form

( , / ) ( , ) p T T p T s ω ∂ ∂ =

Energy (J) 2500 3000 Frequency (rad/s) 3500 4000

where s is used to represent the frequency derivative of the FRF. The mean rate at which the FRF up-crosses the critical level b can be written in terms of the jpdf in the form [5]

∞ + = ∫ (6)

0 ( ) ( , )d b sp b s s ν ω

To clarify the terminology, the word “rate” refers to up-crossings occurring along the frequency axis, so the interest is in the number of up-crossings per unit frequency interval. It is shown in reference [5] that (on the assumption that the natural frequencies and mode shapes of a subsystem conform to the universal distribution predicted by the Gaussian Orthogonal Ensemble) Equation 6 leads to

2 2 1/2 ( ) [2 ( )] ( ) b s bp b ν ω σ π σ µ + − = + , (7) where

    −   = −          

2

ln 1 1 ( ) exp 2 2 ( )

b c p b c bc π

2

, (8)

1 1

    = + =   +  

2 2 1 2 2 2 ln(1 / ), ln 1 / c c µ σ µ σ µ

, (9,10)

2 2 E[ ], Var[ ], Var[ / ] s T T T µ σ σ ω = = = ∂ ∂ . (11-13)

µ 2 σ 2 s σ Here and are the mean and variance of the kinetic energy, and is the mean squared value of s (the mean value being approximately zero); the crossing rate as given by Eq. (6) is fully deter-

µ 2 σ 2 s σ mined by the three parameters , , and . It is shown in reference [5] that if the modal overlap of the system is greater than 2, then a good approximation to the maximum crossing rate (i.e. the rate of crossing the most crossed level b ) can be written as

− +     = + + ≈      

1/2 2 2 2 2 2 max 1 2 ( ) 2 (1 / )ln(1 / ) 2 b s ν βσ πµ σ µ σ µ π ωη

. (14)

max ( ) b ν + 2 / ( ) ωη Were to have units of Hz then Equation 14 would identify the quantity as the mean “circular frequency” of the fluctuations in the energy FRF. Howeve r, the FRF is expressed as a

ω max ( ) b ν + 1 (ra d/s ) − 2 / ( ) ωη function of , meaning that has units of and hence obviously has units 2 / ( ) ωη of s/rad. Rather than being a circular frequency, represents a “quefrency” as used in cepstral analysis [9]; nonetheless it can be physically interpreted as the mean rate of fluctuation of the FRF in terms of phase rotation, and as discussed in reference [5] this result is consistent with earlier work in the field of random FRFs [10].

Equation 6 can be used to compute the mean number of times a response level b is up-crossed in

A B ω ω ω < < a frequency interval , to give

ω

ω ν ω ω + + = ∫ . (15)

B

( ) ( )d

b N b

A

Additionally, if the up-crossings of the level b are taken to represent a Poisson process then the prob- ability that the response will lie below b throughout the whole interval is given by [5]

ω ν ω ω +     = −       ∫ , (16)

ω

B

0 ( )exp ( )d

b P P b

A

0 ( ) P b A ω ω = where is the probability that the response lies below b at . If Equation 16 is employed

0 ( ) 1 P b = with then the equation has the alternative interpretation that P is the probability of neither

A ω ω = up-crossing nor down-crossing the level b , regardless of the initial conditions at . The ap- proximations involved in Equation 16 are fully discussed in reference [5].

sil, —_ Ain ly | inter.noise. 21-24 AUGUST SCOTTISH EVENT CAMPUS. ? O? ? GLASGOW

In addition to level crossing rates, reference [5] considered a number of the statistical properties of the peaks of the energy FRF. It was shown that the mean rate of occurrence of peaks can be approximated by

σ ν π σ   =    

1 2

, (17)

r P s

Finally it was shown that the mean trough-to-peak height can be approximated by

s E[ ] / 6( ) R π ωησ = . (18)

In order to apply the equations of this section to the FRFs of a built-up system, it is necessary to

µ σ s σ r σ compute the parameters , , , and for each subsystem. Only the first of these parameters is predicted by standard SEA, and so the other parameters require an enhanced version of the method, and this is described in the following subsection. 2.3 Enhanced SEA It is shown in reference [5] that SEA can be extended to predict the following quantities for each subsystem

2 2 2 2 2 2 , , ˆ , = Var( ), = Var( ), = Var( ) j j j j j j s j j j r j j j n E n E n E n E µ σ σ σ ′ ′′ = , (19-22)

where a prime represents differentiation with respect to frequency. Expressions for these quantities are available in closed form and they can be evaluated rapidly as a post-processing calculation once the mean SEA equations, Equation 1, have been solved. The derivation and the detailed expressions for these quantities are too lengthy to include here, but full details are given in reference [5]. Given these results, the statistical properties outlined in Section 2.2 can be computed.

3. NUMERICAL EXAMPLE

In order to validate the foregoing theory, an example system consisting of two plates that are coupled by a number of linear springs is considered. Both plates are rectangular and simply-supported with

0.8m 0.67m × planform dimensions . One plate is arranged vertically above the other, and five springs of stiffness k are attached between the two plates. The spring stiffness is assigned the value

4 1 10 ( /1500) N/m k ω = × 12 1 n ωη , which yields a coupling factor that is approximately constant over 1 mm h = 1.5 mm the frequency range of interest. Plate 1 has thickness and Plate 2 has thickness , 11 2 2 10 N/m E = × 3 7800 kg/m ρ = and both plates are made of steel, with Young’s modulus , density 0.3 ν = and Poisson’s ratio . In what follows the forced response over the frequency range 1500 to 2500 rad/s is considered, which covers around 20 to 30 resonant modes in each plate. The damping in the plates is assigned so that Plate 1 has a constant modal overlap factor of 2.5, and Plate 2 has a constant modal overlap factor of 4.0. Accurate benchmark results for the response of the system have been calculated by using the Lagrange-Rayleigh-Ritz method, with the modes of the uncoupled simply-supported plates used as basis functions. In order to randomize the system in the benchmark calculations 10 masses have been added to each plate in random locations, with each mass having 2% of the mass of the relevant plate. Results have been obtained for the case in which each plate is

sil, —_ Ain ly | inter.noise. 21-24 AUGUST SCOTTISH EVENT CAMPUS. ? O? ? GLASGOW

sil, —_ Ain ly | inter.noise. 21-24 AUGUST SCOTTISH EVENT CAMPUS. ? O? ? GLASGOW

1 α − excited by a point force: the force acting on Plate 1 has amplitude and that on Plate 2 has am- α plitude . The results predicted by Equations 19-21, together with the quefrency predicted by Equation 14, are shown in Figure 2, where very good agreement with the benchmark calculations can be seen.

08 06 t t t & S ‘Noha

(b)

(a)

Relative variance a eee ee eee 0.2 04 06 08 Alpha

‘Aloha

(c)

(d)

g 3 ° (8) fouayend onnooy3,

Figure 2: Statistical parameters of the energy at A ω ω = as a function of the loading parameter α . Solid curve, Plate 1 SEA prediction; dashed curve, Plate 2 SEA prediction; * benchmark results, Plate 1; + benchmark results, Plate 2. (a) mean, (b) variance, (c) variance of frequency derivative, (d) quefrency .

Results for the mean rate of crossing a critical level b , together with the probability of crossing this level, are shown in Figure 3, where again good agreement with the benchmark simulations can be seen. Three curves are shown in Figures 3 (c) and (d), representing: (i) the benchmark results, (ii) the present results using Equation 16, (iii) the use of Equation 16 with the benchmark result for the crossing rate. This indicates that the main reason for the disagreement between (i) and (ii) is due to the Poisson crossing assumption that lies behind Equation 16. The fact that (i) and (iii) disagree at low levels of b but agree at higher levels of b is an indication that the Poisson assumption becomes more valid at higher values of the critical level. In random vibration applications Equation 16 tends to overestimate the probability of a crossing, due to a phenomena known as “clumping” of the cross- ings. The converse is seen in Figure 3, where Equation 16 underestimates the probability of crossing. It is argued in reference [4] that this is caused by veering between the system natural frequencies, which causes an “anti-clumping” effect.

sil, —_ Ain ly | inter.noise. 21-24 AUGUST SCOTTISH EVENT CAMPUS. ? O? ? GLASGOW

(a)

(b)

Number of crossings. 0s + ne 15 x10

Number of crossings os eeereveeeeesepee °O 1 2 a 4 Crossing level (J) 107

(c)

(d)

Probability of crossing 0s 1 Crossing level (J) 15 x10

Figure 3: Plots (a) and (b): mean number of crossings of the energy of Plate 1and Plate 2 respectively as a function of the energy level. Solid curve, present theory; symbols, benchmark calculation. Plots (c) and (d): probability that the energy of Plate 1and Plate 2 respectively will cross a specified level at least once. Dashed curve, present theory; solid curve, benchmark calculation; symbols, result ob- tained when the benchmark result for the crossing rate is used in Equation 16. 4. CONCLUSIONS

‘Burss0i3 0 Aaigegora

An enhanced version of SEA has been presented which allows the higher order statistics of the energy FRFs of a complex random system to be calculated. Another enhancement of SEA is to combine the method with the Finite Element and Boundary Element methods by employing the diffuse field reci- procity principle; this results in a hybrid approach [8] covering the full frequency spectrum. This method is now employed extensively in commercial software. The methodology described here can be applied to the SEA partition of this approach to further enhance the method. 5. REFERENCES

1. Lyon R.H. & DeJong R.G. Theory and Application of Statistical Energy Analysis . Boston: But- terworth-Heinemann, 1990. 2. Langley R.S. and Cotoni V. Response variance prediction in the statistical energy analysis of built-up systems, J. Acoust. Soc. Am. 115 , 706-718 (2004). 3. Langley RS, Legault J, Woodhouse J, et al. On the applicability of the lognormal distribution in random dynamical systems . J. Sound Vib . 332 , 3289-3302 (2013).

4. Langley R.S. The level crossing rates and associated statistical properties of a random frequency response function. J. Sound Vib . 417 , 19-37 (2018). 5. Langley, R.S. The crossing rates, exceedance probabilities, and related statistical properties of the energy frequency response functions of a random built-up system. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science , 233 , 6409-6424 (2019). 6. Weaver R.L. The unreasonable effectiveness of random matrix theory for the vibrations and acoustics of complex structures. In: Wright MCM and Weaver R (eds), New directions in linear acoustics and vibration: quantum chaos, random matrix theory, and complexity . Cambridge: Cambridge University Press, 2010. 7. Langley R.S. and Heron K.H. Elastic wave transmission through plate/beam junctions. J. Sound. Vib . 143 , 241-253 (1990). 8. Shorter P.J. and Langley R.S. Vibro-acoustic analysis of complex systems. J. Sound Vib . 288 , 669-699 (2005). 9. Randall R.B. Frequency analysis . Naerum: Bruel and Kjaer, 1987. 10. Schroeder M.R. Frequency-correlation functions of frequency responses in rooms. J. Acoust. Soc. Am . 34 , 1819-1823 (1962).

sil, —_ Ain ly | inter.noise. 21-24 AUGUST SCOTTISH EVENT CAMPUS. ? O? ? GLASGOW