## Abstract

The paper concerns the statistical energy analysis (SEA) of two conservatively coupled oscillators, sets of oscillators and continuous subsystems under broadband excitation. The oscillator properties are assumed to be random and ensemble averages found. Account is taken of the correlation between the coupling parameters and the oscillator energies. For coupled sets of oscillators or continuous subsystems, it is assumed that the coupling power between a pair of oscillators is proportional to the difference of either their actual energies or their ‘blocked’ energies, and expressions for the ensemble averages and coupling loss factors (CLFs) are found. Various observations are made, some of which differ from those that are commonly assumed within SEA. The coupling power and CLF are governed by two parameters: the ‘strength of connection’ and the ‘strength of coupling’. The CLF is proportional to damping at low damping and independent of damping in the high damping, weak coupling limit. Equipartition of energy does not occur as damping tends to zero, except for the case of two oscillators that have identical natural frequencies. While attention is focused on spring-coupled oscillators, similar results hold for more general forms of conservative coupling. The examples of two spring-coupled rods and two spring-coupled plates are considered. Conventional SEA and the coupled oscillator results are in good agreement for weak coupling but diverge for strong coupling. For strong coupling and weak connection, the coupled oscillator results agree well with an exact wave analysis and Monte Carlo simulations.

## 1. Introduction

Statistical energy analysis (SEA) has become an established method for modelling the noise and vibration behaviour of complex, built-up structures at higher frequencies (Lyon & Dejong 1995). The response is described in terms of the flow of energy through the structure. It is recognized that the properties of the structure are uncertain, and hence a statistical description is required. In principle, at least, the structure is assumed to be drawn from an ensemble of similar structures whose properties are random and estimates of the ensemble average response are required.

The earliest derivations of the SEA equations concerned two coupled oscillators and two coupled sets of oscillators whose properties are random. Lyon & Maidanik (1962) showed that for broadband excitation, the energy flow between the two conservatively coupled oscillators is proportional to the difference in their blocked energies, i.e. the energies when the other oscillator is fixed. This result was then applied to the coupling power between two specific modes of coupled, multi-modal subsystems, which can be regarded as comprising sets of oscillators. Lyon & Eichler (1964) extended the results to coupled oscillator sets, various assumptions being made to show that the coupling power is proportional to the difference in the mean blocked modal energies of the subsystems. The coupling loss factor (CLF) was introduced and estimated using wave approaches rather than from the modal approach itself. A more formal approach was developed by Newland (1968), in which a statistical description for the properties of the sets of oscillators was adopted and ensemble averages of the coupling power found. Finally, it was shown by Scharton & Lyon (1968) that the coupling power between two oscillators is also proportional to the difference in their actual energies. It is on this base that SEA is founded.

In this paper the SEA of coupled oscillators and coupled sets of oscillators is revisited and departs from these earlier works in that various assumptions are removed or relaxed. In particular, account is taken of the correlation between the coupling parameters for the oscillator or mode pairs and their energies. Consequently, some of the conclusions differ from those of these earlier works.

Section 2 concerns the SEA of two spring-coupled oscillators. The oscillator properties are assumed to be random and ensemble averages found. Various expressions for powers and energies are derived and discussed. The results are then extended to the case of two spring-coupled sets of oscillators. Ensemble averages are again taken, the ensembles being defined in terms of the statistics of the natural frequencies of the oscillators in each set. In §4, application of the coupled oscillator theory to coupled, continuous, multi-modal subsystems is discussed. Expressions for the ensemble averages of coupling power, energy response and consequent CLF are derived in terms of the interaction of oscillator pairs.

It is seen that the coupling power and CLF can be written in terms of two parameters: the ‘strength of connection’ between the sets of oscillators and the ‘strength of coupling’. The strength of coupling depends on both the damping and the strength of connection. To find the CLFs, further assumptions are required. First, it is assumed that the interaction of a pair of oscillators is independent of the existence of the other oscillators (equivalent to ‘weak connection’). Two approaches are then suggested, in which it is assumed that the coupling power between each oscillator pair is proportional to either their actual or their ‘blocked’ energies. Two similar but different expressions for the CLFs result. These are identical in the weak coupling limit or for weak connection. For two oscillators, it is seen that equipartition of energy does not occur as damping tends to zero, except in the case where the uncoupled oscillators have identical natural frequencies. Equipartition of energy does not hold for the ensemble averages. The strength of coupling is seen to depend on damping and decreases as the damping increases. The CLF also depends on damping, being proportional to damping when the coupling is strong and tends asymptotically to a constant for the high damping, weak coupling limit.

In §4, application of the coupled oscillator theory to coupled, continuous, multi-modal subsystems is discussed. Finally, the examples of two spring-coupled rods and two spring-coupled plates are considered. Comparisons are made with the results of conventional SEA, an exact wave analysis for one-dimensional subsystems and numerical Monte Carlo simulations (MCS).

## 2. Two spring-coupled oscillators

This section concerns two spring-coupled oscillators. Similar results hold for more general cases of conservative coupling (electronic supplementary material 1). The cases considered there are general conservative coupling through spring, mass and gyroscopic elements (Scharton & Lyon 1968) and an interface spring configuration that can arise in Craig–Bampton component mode synthesis models. The frequency average behaviour for white noise excitation is reviewed. Then ensemble averages are taken over a distribution of properties of the system. In subsequent sections these results are applied to coupled sets of oscillators.

### (a) Two coupled oscillators under broadband excitation

Consider the system shown in figure 1. Here, *x*_{1,2}, *m*_{1,2}, *k*_{1,2} and *c*_{1,2} are the displacement, mass, stiffness and damping of the oscillators, respectively, and *k* is the stiffness of the coupling spring (a list of symbols is given in electronic supplementary material 2). The forces *f*_{1}(*t*) and *f*_{2}(*t*) are assumed to be random, stationary and statistically independent, with spectral densities and that are constant over some frequency band B, which is wide enough to contain both natural frequencies of the system. This system was considered by Lyon & Maidanik (1962), see also Mace & Ji (2006) and the analysis will not be repeated here. The total input power *P*_{1}, oscillator energy *E*_{1} and coupling power *P*_{12} can be found by integrating over the frequency band B. The energy of each oscillator is taken to be twice the kinetic energy (there are no significant consequences of this). This is because the potential energy in the coupling spring cannot be unambiguously ascribed to one or other oscillator. It is also assumed that B is wide enough so that the limits of integration can be extended to (0, ∞) without introducing significant errors. Under these circumstances,(2.1)(2.2)where *Δ*_{1}=*c*_{1}/*m*_{1} is the half-power bandwidth of oscillator 1 (throughout, similar expressions for oscillator 2 follow by interchanging subscripts or superscripts). The coupling parameter(2.3)while(2.4)are the blocked natural frequencies, i.e. the natural frequencies when one oscillator is fixed. Note that the input powers are independent of the natural frequencies, but the coupling power and energies depend strongly on them, and, in particular, on the difference between them. Finally, from equations (2.1) and (2.2), it follows that(2.5)(2.6)

Equation (2.5) is the familiar statement of coupling power proportionality, applied here to two spring-coupled oscillators under broadband random excitation. The coupling term *β* depends on the coupling stiffness *k* and the properties of the oscillators but not on the oscillator energies. Note that *β* depends strongly on the difference of the subsystem uncoupled natural frequencies through the term . Similar expressions can be found for other types of coupling (Lyon & Maidanik 1962 and electronic supplementary material 1). They are somewhat complicated due to the number of parameters involved, but the underlying physics is the same, especially the dependence of the coupling term on a group of parameters related to the coupling mechanism (here the term (*k*^{2}/*m*_{1}*m*_{2})) and a term depending on the oscillator properties that involves the difference of the natural frequencies. In the above, the oscillator natural frequencies are taken as the blocked natural frequencies. The uncoupled system can equally be regarded as that in which the coupling spring *k*=0, i.e. *ω*_{1,2} are defined in terms of the free natural frequencies and . The power–energy relation holds equally for both definitions, although there are slight differences in the elements of the coupling term. In principle, therefore, it is not relevant whether the uncoupled natural frequencies *ω*_{1,2} are given by blocking (or clamping) the interface or by cutting (or freeing) it.

### (b) Ensemble averages

Assume that the system properties are not known exactly, but are random and defined by some joint probability density function . How *p* is defined in practice is problematical. Here, it is assumed that all parameters are known exactly except for the oscillator natural frequencies, which are both random and distributed over some frequency band *Ω*. If this band of uncertainty is fairly small compared with the bandwidth of excitation B, then it can equally be assumed that *ω*_{1} is known and that the natural frequency spacing *δ*=(*ω*_{1}−*ω*_{2}) is random and uniformly probable in some band *Ω*. As a result, the ensemble of two, spring-coupled oscillators is defined by(2.7)If *Ω* is large compared with *Δ*_{1,2} but small compared with *ω*_{1,2}, then the ensemble average input power and subsystem energy (equation (2.1)) are given by(2.8)(2.9)where both the expectation E[.] and . represent the ensemble average. Defining , it follows that(2.10)where *ω* is the centre frequency of the band *Ω* and where(2.11)is a measure of the *strength of connection* between the oscillators. The ensemble average energy is thus(2.12)

#### (i) Coupling loss factors

Equations (2.8) and (2.12) relate the ensemble- and frequency-averaged energies and input powers by(2.13)where **A** is a matrix of energy influence coefficients (EICs). By inverting this matrix, it follows that(2.14)where **L** is a matrix of damping loss factors *η*_{1}, *η*_{2} and CLFs *η*_{12}, *η*_{21}. The CLFs are thus(2.15)where(2.16)In equation (2.16), *γ* is a parameter describing the *strength of coupling*, while *κ* describes the *strength of connection*. A similar expression for the strength of coupling parameter was found by Finnveden (1997) by considering a power series expansion of a coupling matrix for oscillators with fixed natural frequencies. Convergence of the expansion is guaranteed if *γ*<1. The CLFs in equation (2.15) thus relate the ensemble average coupling power and energies, i.e.(2.17)whereas the constant of proportionality *β* relates frequency averages for a specific pair of oscillators.

### (c) Discussion

The above analysis reveals some conclusions that are in contrast to those normally drawn in SEA. First, equipartition does not occur in the limit *Δ*→0 except for the special case of two coupled oscillators that have identical uncoupled natural frequencies. For example, if only oscillator 1 is excited, then the energy ratio as *Δ*→0 becomes(2.18)Equipartition of ensemble average energies will never occur. This is consistent with the behaviour that has been observed via wave (Mace 1993) and system-mode (Mace 2003) approaches. Second, two distinct parameters govern the coupling power and CLF: the strength of connection and the strength of coupling. Third, the CLF depends on damping. At low damping, *η*_{12} is proportional to *η*, while at high damping, it asymptotes to the constant *ωη*_{12}=*πκ*^{2}/2*Ω*. Such behaviour has been noted before in terms of wave analysis (Mace 1993), in terms of the modes of the system as a whole (Mace 2003) and from finite-element analysis (FEA; Yap & Woodhouse 1996; Mace & Rosenberg 1998), so it is not surprising that it is also evident in the ensemble average behaviour of two coupled oscillators.

The reason why these observations differ from those of previous work (Lyon & Maidanik 1962; Lyon & Eichler 1964; Newland 1968; Scharton & Lyon 1968; Lyon & Dejong 1995) is that the CLF is defined from the expected value of the coupling power, i.e. . However, in the earlier analyses, the further assumption was made that the terms in brackets are statistically independent, so that(2.19)The CLF estimated in this manner, i.e. by ensemble averaging *β*, is(2.20)However, this is clearly an approximation: *β* and (*E*_{1}−*E*_{2}) are strongly correlated, i.e. if *β* is large, then (*E*_{1}−*E*_{2}) is small, and vice versa.

## 3. Two spring-coupled sets of oscillators

Consider now the case of two spring-coupled subsystems a and b (figure 2). Each subsystem is regarded as being a set of oscillators and, when coupled, each oscillator in one set shares energy with all the oscillators in the second set. Here, oscillators *j* and *k* in sets a and b are coupled by the spring . In this section, relations between the input powers, coupling powers and subsystem energies and their ensemble averages are found. In §3*a*, the system power and energy relationships are stated. Following that, some comments are made concerning the ensemble and ensemble averages. Expressions for the input and dissipated powers are then found—this is fairly straightforward. The coupling powers are more problematical, however, and various assumptions must be made. Two different approximate approaches are developed, assuming different forms for the coupling power: first, that the coupling power between two oscillators is proportional to the difference of their actual energies and, second, that it is proportional to the blocked energy difference. The two approaches give the same results for weak coupling or, for weak or strong coupling, if the strength of connection is weak. The latter can predict negative values for oscillator energies in the strong coupling limit.

### (a) Excitation, powers and energies: power balance relations

The subsystems are excited by white noise over some frequency band B, which contains *N*_{a} and *N*_{b} resonant modes of subsystems a and b, respectively. The expected number of modes , where *n*_{a,b} are the modal densities, such that the out-of-band, non-resonant modes are ignored.

Power balance for oscillator *j* in set a gives(3.1)where the terms represent the input, dissipated and net coupling powers, respectively. The net coupling power for oscillator *j*,(3.2)is written as the sum of powers exchanged between oscillators *j* and *k* in sets a and b, respectively, and powers exchanged between oscillators *j* and *n*, both of which are in set a. This in-set contribution is unimportant, since it has zero contribution to the net coupling power between the subsystems.

The total power input to each subsystem is assumed to be the sum of the powers input to each oscillator, i.e.(3.3)where is the power input to oscillator *j* in set a. Similarly, the subsystem total energies and dissipated powers, and the net coupling power between the subsystems, can be written as the sums of oscillator and oscillator-pair terms as(3.4)Thus, power balance for subsystem a is(3.5)

### (b) The ensemble and ensemble averages

The properties of the two sets of oscillators are random and the ensembles from which they are drawn are defined by a joint probability distribution function *p*, which depends on the oscillator masses, bandwidths and natural frequencies, the coupling spring stiffnesses and the excitation spectral densities. Clearly, the problem is extremely complicated, and the estimation of ensemble statistics requires the evaluation of an integral of very high dimension.

As in §2, assumptions will be made about the ensemble. The properties of the two sets of oscillators, the coupling stiffnesses and the excitations are assumed statistically independent (in practice, for continuous subsystems, will depend on the mode shapes of the respective modes). For simplicity, each oscillator in set a is assumed to have the same mass *m*_{a}, bandwidth *Δ*_{a} and ensemble average spectral density , i.e. , , . Similar assumptions are also made for set b. The joint probability density function that defines the ensemble then becomesStrictly, the discrete frequency problem should be solved to find the discrete frequency response, which is then frequency averaged and then ensemble averaged. This involves the evaluation of a multidimensional integral of the inverse of a large random matrix. This approach is intractable. Instead, in what follows, simplifying assumptions are made and results for the two-oscillator case introduced. Subsequently, it is assumed that a randomly selected oscillator pair is such that is random and uniformly probable within the band of excitation B and statistically independent of the coupling stiffness.

### (c) Ensemble and frequency average input and dissipated powers

The input and dissipated powers for oscillator *j* are taken to be (cf. equation (2.1)) and . Thus, every oscillator in each set is assumed to be excited equally: this is ‘rain-on-the-roof’ excitation. The total input and dissipated powers for subsystem a are thus(3.6)The power balance equation for subsystem a thus becomes(3.7)Note that the in-set coupling powers do not contribute to the net coupling power: they sum to zero. The problem thus remains of determining the coupling powers between the oscillator pairs and hence forming the SEA equations. Two possible approaches are described in §3*d*,*e*.

### (d) Coupling power proportional to the difference of actual oscillator energies

In this section it is assumed that the coupling power between two oscillators is proportional to the difference in the actual oscillator energies. For the case of two oscillators, the coupling power is given by equation (2.5), with the constant of proportionality *β* being given by equation (2.6). For the case of two sets of oscillators, these equations become(3.8)Thus, . It is now assumed that there are so many oscillators in each set that the mean value of the terms in this sum approximates the ensemble average. Under these circumstances,(3.9)The analysis in §2b(i) is then assumed to hold, i.e. the relations between ensemble average powers and energies leading to equation (2.15), and hence the CLFs are(3.10)where(3.11)Note that the conclusions made for the two-oscillator case hold here, and, in particular, that there is a measure *γ* of the strength of coupling, a measure *κ* of the strength of connection and that the CLF depends on the damping.

#### (i) Conventional SEA

As discussed in §2*c*, in conventional SEA, the average of the product is assumed to equal the product of the averages, i.e. . The ensemble average . The CLFs, putting *n*_{a}=*N*_{a}/*Ω*, are(3.12)Note that equation (3.10) can thus be written as(3.13)Both analyses (leading to equations (3.10) and (3.12)) imply that the ensemble average oscillator energies are equal. They differ in the fact that, in the first analysis, the correlation between the constants and the oscillator energy differences is included, as least as an approximation. Thus, these constants are not identical over the ensemble. Note also that the analysis leading to equation (3.10) is approximate, in that strictly the energies themselves depend on the coupling powers so that the equations should be solved-then-averaged. Finally, the expressions for the CLFs are equal in the weak coupling limit, *γ*→0.

### (e) Coupling power proportional to the difference of blocked oscillator energies

An alternative approach is to note from equation (2.2) that, for two oscillators, the coupling power is also proportional to the difference of the blocked oscillator energies (*P*_{1}/*Δ*_{1}−*P*_{2}/*Δ*_{2}). If this result is assumed to hold for two oscillators of the coupled sets, then the coupling power between two such oscillators is(3.14)where from equation (2.3),(3.15)The net coupling power *P*_{ab} is again found by summing the coupling powers between each oscillator pair. Again, if there are assumed to be so many modes that the sum can be approximated by the ensemble average, then(3.16)Replacing the ensemble averages with the expression from equation (2.10), noting that and putting *n*_{a}=*N*_{a}/*Ω*, leads to(3.17)where *α*, *γ* and *κ* are as defined in equation (3.11). Substituting the coupling power into the power balance equation gives(3.18)Hence, the matrix of EICs, which relates the ensemble averages of the input powers and energies, is(3.19)By inverting this matrix, the CLFs are found to be(3.20)where *μ*_{a,b}=*n*_{a,b}*Δ*_{a,b} is the modal overlap based on the half-power bandwidth.

### (f) Comments

To summarize, in the above, the power balance equations were written for each oscillator and oscillator set. Mild assumptions were made concerning the input and dissipated powers. Solution to the power balance equations is intractable, hence assumptions were made concerning the coupling powers. Two approaches were suggested. These lead to equations (3.10) and (3.20) for the CLFs and , respectively. In both the cases, the system response is governed by two parameters: the strength of connection *κ* and the strength of coupling *γ*, given in equation (3.11). Finally, conventional SEA leads to equation (3.12) for the CLF.

The three expressions for the CLF are identical in the weak coupling limit, *γ*→0. The conventional expression ignores all correlation between the oscillator energies and the constant in the coupling power proportionality relation, overestimating the coupling powers (especially between oscillators with natural frequencies that are close) and hence overestimating the CLFs when the coupling is strong. The two other expressions become proportional to damping as the damping is reduced. The blocked expressions, equation (3.20), can diverge, with the denominator becoming zero or negative, implying (non-physical) negative subsystem energies. This happens if *κ* is large enough and the modal densities are large enough, such that if *Δ*_{a}≈*Δ*_{b}, then (*n*_{a}+*n*_{b})*πκ*/2≥1. In effect, this means that each oscillator interacts sufficiently strongly with so many other oscillators that the blocked energies are reduced substantially by the energies lost to the other oscillators, and the assumption in equation (3.14) breaks down.

The response of a specific system excited over a finite frequency band will differ from the ensemble averages given above. This arises partly from the fact that only a finite sampling of oscillator pairs (and hence terms involving ) are taken and partly from the fact that the number of modes in the band is a random variable. Furthermore, natural frequency spacing statistics will affect the variance.

## 4. Two spring-coupled continuous subsystems

This section concerns two spring-coupled continuous subsystems. Each subsystem mode is regarded as an oscillator and the spring couples these modes. The motion in subsystem a can be written in terms of a modal sum as(4.1)where and are the *j*th modal amplitude and mode shape, respectively (*x*^{(a)} may be a vector if the subsystem is two- or three-dimensional). The modal mass of the *j*th mode is(4.2)where *ρ*_{a}(*x*) is the mass density.

Suppose that the subsystems are coupled by a spring of stiffness *K* attached between the points and . The spring exerts a force on each subsystem. From equation (4.1) and the corresponding equation for subsystem b, it follows that the modal force on mode *j* of subsystem a is(4.3)Note that the spring couples mode *j* in subsystem a to the modes in subsystem b (and also to the other modes in subsystem a). The intermodal coupling stiffness is thus(4.4)Hence, the strength of connection between the mode pair is(4.5)and thus depends on the mode shapes at the coupling location. This is the appropriate value of *κ*^{2} if some *a priori* information is available concerning the connection points (e.g. for end-coupled rods, see §5*b*).

Now consider the case where the coupling points are randomly located over the subsystems and assume that the probability density function of is proportional to the mass density *ρ*_{a}(*x*). The strength of connection, averaged over all interface points in each subsystem, is then(4.6)where is the total mass of subsystem a. The case where there is more than one coupling spring can be treated in a similar manner.

## 5. Examples

In this section, two examples are considered, namely two spring-coupled rods in axial vibration and plates in bending (figure 3). Estimates of the CLFs and (equations (3.10) and (3.20)) found using the present coupled-oscillator theory are compared with the results of conventional SEA (Lyon & Dejong 1995), an exact wave analysis for two coupled rods (Mace 1993) and MCS for two coupled plates. In conventional SEA, the CLF for two point-connected subsystems is given by (Lyon & Dejong 1995)(5.1)where *τ*_{ab,∞} is the infinite system power transmission coefficient given by(5.2)where *Z*_{a} and *Z*_{b} are the input impedances of the two subsystems, assuming that they extend uniformly to infinity, and *R*_{a} and *R*_{b} are the real parts of these impedances.

### (a) Two rods coupled at randomly selected points

The modal density of a rod of length *L* is , where *ρ*, *E* and are the mass density, elastic modulus and wave speed of the rod material, respectively. If the rods are connected at randomly selected points, then the strength of connection is given by equation (4.6), where *M*^{(a)}=*ρ*_{a}*A*_{a}*L*_{a} is the mass of rod a and *A*_{a} is its cross-sectional area. The CLFs, based on the assumption that the coupling power is proportional to the *actual* energy difference, then become(5.3)For an infinite rod . By defining subsystem a as the combination of rod a and the spring, then(5.4)and hence the conventional SEA estimate of CLF is(5.5)In the limit of weak coupling (i.e. *γ*→0), equation (5.3) (and the comparable expression for ) reduces to equation (5.5).

### (b) Two end-coupled rods

#### (i) Coupled oscillator theory and conventional SEA

If the two rods are end-coupled, then, at the ends of the rods, the squared mode shapes in equation (4.5) are equal to . The appropriate value for the strength of connection is now(5.6)and the CLF becomes(5.7)The input impedance of a semi-infinite rod excited at its end is *Z*_{∞}=*R*_{∞}=*ρcA*. Both the power transmission coefficient and the estimate of CLF given by conventional SEA are then four times the values given in equations (5.4) and (5.5). Once again, the coupled oscillator CLFs reduce to in the limit of weak coupling (i.e. *γ*→0).

#### (ii) Exact wave solution for one-dimensional subsystems

Mace (1993) developed a wave analysis for the response of two one-dimensional subsystems coupled at their ends. Ensemble averages are taken by assuming that the total phase change a wave experiences as it propagates through the system, when taken mod(2*π*), is random and uniformly probable. The CLF predicted by the wave approach of Mace (1993) is(5.8)(note that in Mace (1993), the modal overlaps were defined in terms of the noise bandwidth rather than the half-power bandwidth) where(5.9)where the transmission coefficient *τ*_{ab,∞} for end-coupled rods is four times the value given in equation (5.4) and is the conventional SEA expression, i.e. four times the value given in equation (5.5) for rods coupled at an arbitrary point. The ‘blocked energy difference’ expression for the CLF (equation (3.20)) is(5.10)Both the exact wave analysis (equation (5.8)) and the coupled oscillator theory (equations (5.7) and (5.10)) give the same expression for the CLF for the case of weak coupling (i.e. the conventional SEA estimate ). Otherwise, they differ in some respects.

First, in the exact analysis, there are two coupling parameters (*γ*_{ab}, *δ*_{ab}) rather than one (*γ*) in the coupled oscillator results. Second, the functional dependence is different for large values of *μ*_{a,b} (due to the hyperbolic functions in equations (5.9)). The form of the exact wave result is more similar to the blocked energy difference expression of equation (5.10) than to the ‘actual energy difference’ expression of equation (5.7). If the difference in the modal overlaps is small, then . If, furthermore, the modal overlaps are small, then(5.11)Noting that for this case and that *κ*^{2} is given by equation (5.6), it follows that(5.12)where *γ* is the strength of coupling arising from the coupled oscillator theory.

In summary, for weak coupling, both coupled oscillator expressions are identical to the result from the exact wave analysis and to the conventional SEA expression. For strong coupling, if the modal overlaps are small enough, then the coupling strength parameter becomes identical to that of the exact wave analysis. The blocked energy difference expression of equation (5.10) then asymptotes to that of the wave analysis. Otherwise, there are differences in the coupling parameters, although there is the same general dependency of CLF on the modal overlap.

#### (iii) Numerical example

The properties of the rods in figure 3*a* are taken to be such that , , and . Consequently, the characteristic impedances and modal densities of the two rods are *Z*_{a,∞}=*Z*_{b,∞}=1 Nm s^{−1}, *n*_{a}=0.3626 rad^{−1} and *n*_{b}=1−*n*_{a}=0.6374 rad^{−1}. Figure 4 shows various estimates of CLF as functions of damping for various values of the coupling spring stiffness, giving a range of different connection strengths. The centre frequency *ω*=100 rad s^{−1} is centred on the 100th natural frequency of the system. The conventional estimate is, of course, independent of damping, the other estimates increasing with increasing damping when the coupling is strong (i.e. for low *η*), and asymptotic to the conventional estimate for weak coupling (i.e. as *η*→∞). For this case, is very nearly equal to the exact wave estimate, while is significantly smaller than if the strength of connection is large (i.e. large *K*) and the coupling is not sufficiently weak.

### (c) Two spring-coupled plates

Consider the system comprising two spring-coupled plates shown in figure 3*b*. The modal density of a plate is , where *A*, *m* and *B* are the area, mass per unit area and bending stiffness of the plate, respectively. From equation (4.6), it follows that(5.13)For an infinite plate, , leading to(5.14)Note again that the conventional SEA estimates based on wave transmission are identical to the estimates of the coupled oscillator theory in the weak coupling limit.

#### (i) Numerical example

The material properties of both plates are taken to be *ρ*_{a,b}=10^{3} kg m^{−3}, *E*_{a,b}=10^{8} N m^{−2} and Poisson's ratio *ν*=0.38, while the plate dimensions are 0.560×0.405 m and 0.665×0.475 m, both with a thickness of 0.001 m and both plates having the same loss factor. These parameters correspond to plate modal densities *n*_{a}=0.18 rad^{−1} and *n*_{b}=0.25 rad^{−1} and characteristic impedances *Z*_{a,∞}=*Z*_{b,∞}=0.79 N ms^{−1}. Figure 5 shows the CLFs estimated from the coupled oscillator theory (both blocked and actual energy difference expressions being shown), together with the conventional SEA estimate and the estimate found from MCS (Mace & Ji 2006). The centre frequency is *ω*=1000 rad s^{−1}. Various values for the coupling spring stiffness are chosen, giving a range of different connection strengths (*κ*^{2}≈0.035, 0.14, 0.56).

In the MCS, the edges of the plates are assumed to be simply supported and the modes are found from well-known analytical expressions. Two hundred and forty different numerical estimates of powers and energies are found, each with different interface locations (12 points on plate 1 and 20 points on plate 2 were randomly chosen, avoiding the regions close to the plate edges). Frequency averages are taken over a band of width 100 rad s^{−1}, which contains approximately 18 modes of plate a and 25 modes of plate b. There is a significant variability between the individual estimates, which reduces as the damping increases. The CLF is finally estimated from the average powers and energies. The computational cost of the analytical estimates is, of course, trivial, while that of the MCS results is, of course, relatively very high.

While the conventional SEA estimate is, of course, independent of damping, both the numerical and analytical expressions increase with damping for low damping, then asymptote to the conventional estimate (or close to it) as the damping increases sufficiently so that the coupling becomes weak. The CLF increases as *K* increases (and hence, of course, as *τ*_{ab,∞} increases). The agreement between the coupled oscillator estimates and the MCS results is better for weaker connection. For this case, agrees better with the results of the MCS than does .

## 6. Concluding remarks

This paper concerned the SEA of coupled oscillators, coupled sets of oscillators and coupled continuous subsystems. The properties of the oscillators are uncertain and a statistical description is required, the system being drawn from an ensemble of similar systems whose properties are random. Consequently, estimates of the ensemble average behaviour are required. While attention was focused on spring-coupled sets, similar results apply for more general forms of conservative coupling (electronic supplementary material 1), including general coupling through spring, mass and gyroscopic elements as considered by Scharton & Lyon (1968).

While the earliest studies in SEA concerned coupled oscillators, this work departs somewhat from these studies, in that various assumptions are removed or relaxed. Consequently, some of the results, conclusions and observations differ from those of conventional SEA. In particular, account was taken of the correlation between the coupling parameter for an oscillator pair and their energies. For sets of oscillators, it was assumed that the interaction of each oscillator pair is not affected by the presence of a third oscillator. Regarding the coupling power between each oscillator pair, two approaches were suggested: the coupling power being assumed to be proportional to the difference of either their actual energies or their blocked energies. Very similar, but different, expressions for the ensemble averages of coupling power, energy response and the CLFs were found. These become identical in the weak coupling limit. Differences between the two expressions might give some indication of the accuracy of the approximations made.

It was seen that the behaviour depends on two parameters: the strength of connection between the oscillators and the strength of coupling. In terms of the modes of the structure as a whole, a large strength of connection corresponds to mode shapes of the whole structure which are ‘global’ rather than ‘local’ (i.e. the energy of the mode is spread throughout the structure rather than being confined primarily to one subsystem). In terms of waves, it corresponds to a large wave transmission coefficient at the boundary between two subsystems. The strength of coupling depends on the strength of connection and the damping, and thus relates transmission and dissipation effects. Equipartition of energy does not occur as damping tends to zero, except in the case of two oscillators with identical natural frequencies. Equipartition does not hold for ensemble average energies. The CLF depends on damping: it is proportional to damping at low damping and is independent of damping in the high damping, weak coupling limit, where it asymptotes to the value predicted by conventional SEA. These observations are fully consistent with previous analyses based on wave, modal and finite-element approaches (Mace 1993, 2003; Yap & Woodhouse 1996; Mace & Rosenberg 1998). Finally, the examples of two spring-coupled rods and two spring-coupled plates were considered. In summary, the coupled oscillator theory gives accurate estimates of the CLFs for either weak coupling or weak connection.

The first motivation for this work was to re-examine the underlying physics of the interaction of oscillator sets, since the conventional SEA approaches lead to conclusions and observations that are contradictory to results found from wave, mode and numerical studies, especially for stronger coupling. The resulting conclusions relate to the strength of coupling and, a second factor, the strength of connection. The second motivation was to lay the basis of a technique for estimating SEA parameters from discrete, FEA subsystem models, without the need to solve the global eigenvalue problem. This work is ongoing.

Strong coupling effects are important in developing an understanding of the underlying physical phenomena that govern the interaction of uncertain dynamic systems. In practical engineering applications, however, strong coupling is perhaps of less importance—if the coupling is strong, the difference between the actual energy, which is the quantity of interest in practice, and the energy predicted under an (erroneous) assumption of weak coupling is not likely to be large. Indeed, assuming (erroneously) that the coupling is weak may well serve to make the model of the system better conditioned and more robust. Thus, the second motivation, that of providing systematic, FE-based numerical approaches, is perhaps more valuable in a practical sense.

## Acknowledgments

The authors gratefully acknowledge the financial support provided by The Leverhulme Trust.

## Footnotes

Electronic supplementary material is available at http://dx.doi.org/10.1098/rspa.2007.1824 or via http://www.journals.royalsoc.ac.uk.

- Received October 21, 2006.
- Accepted January 22, 2007.

- © 2007 The Royal Society