The root-mean-square (RMS) response of various points in a system comprised of two parallel plates coupled at a point undergoing high frequency, broadband transverse point excitation of one component is considered. Through this prototypical example, asymptotic modal analysis (AMA) is extended to two coupled continuous dynamical systems. It is shown that different points on the plates respond with different RMS magnitudes depending on their spatial relationship to the excitation or coupling points in the system. The ability of AMA to accurately compute the RMS response of these points (namely, the excitation point, the coupling points, and the hot lines through the excitation or coupling points) in the system is shown. The behavior of three representative prototypical configurations of the parallel plate system considered is: two similar plates (in both geometry and modal density), two plates with similar modal density but different geometry, and two plates with similar geometry but different modal density. After examining the error between reduced modal methods (such as AMA) to classical modal analysis (CMA), it is determined that these several methods are valid for each of these scenarios. The data from the various methods will also be useful in evaluating the accuracy of other methods including statistical energy analysis (SEA).

## Introduction

Current work in statistical mechanics, energy methods, and modal analysis explores the behavior of vibrating dynamical systems responding in regions beyond the practical limits of the finite element method (FEM) and classical modal analysis (CMA). Even single-component problems in these scenarios have characteristics that make FEM and CMA simulations cumbersome, costly, or inaccurate due to small wavelengths in the response that require very fine meshes or a large number of excited modes demanding large matrix solvers. Design and analysis problems involving coupled, continuous components experiencing high-frequency and/or broadband excitation require more efficient methods.

In 1960s, Lyon, Maidanik, Ungar, Fahy, Sharton, and others proposed statistical energy analysis (SEA) (e.g., see Ref. [1]), which showed that if energy is assumed to be equally distributed among the responding modes of the system in question, its mean response could be quickly determined. Initially, SEA addressed only the behavior of a single continuous component, and did not allow for spatial variation of a component's response. In the next decade, other alternatives to FEM and CMA began to arise in an attempt to build generalized approaches to even networks of coupled systems, while still capturing spatial variation on components. In 1984, the spectral energy method (SEM) was developed by Patera [2]—employing Chebyshev polynomials as basis functions to describe fluid flow in a channel expansion. SEM has been developed for many other problems in both fluid and solid mechanics, from the axisymmetric Navier–Stokes equation [3] to entry flow in a contraction channel [4] to plates reinforced by beams [5]. In 1985, Dowell and Kubota presented asymptotic modal analysis (AMA) [6], which reduces the complexity of the results of the Ritz method by identifying elements of system transfer functions that are slowly varying for high-frequency, broadband excitation. From its original investigation of the transverse displacement of a rectangular plate, AMA has been applied to plate systems [7,8], acoustic cavities [9], and systems with multiple points of excitation [10]. In 1987, yet another method of dynamical system analysis was proposed by Nefske and Sung [11]. This method models dynamical systems as control volumes and considers the energy flow through the control volume. This converts the modal formulation of the elastodynamic wave equations to a conduction partial differential equation, which may be analyzed much more easily via the finite element method. From this work, energy flow analysis (EFA) was born, and it was further applied to rods and beams by Wohlever and Bernhard [12]. Even more recently, Maxit and Guyader extended SEA to include systems with nonuniform modal energy density using a method called statistical modal energy distribution analysis (SmEdA), that is born of CMA equations, but uses SEA coupling loss factors to make its response predictions [13].

Each of these methods began by analyzing different prototypical systems, overcoming obstacles such as different coordinate systems [14], spatial parametric or geometric distributions [15], and property uncertainty [16]. Reinforcements, discontinuities [17], and even nonlinear systems have been investigated [18,19]. After a few iterations and further refinement, these techniques are being applied to product-related problems [20,21].

Each of these advances of the various methods brings us closer to an effective generalized theory for the high-frequency behavior of elastodynamic systems. The most recent steps have involved developing theories for coupled continuous systems, such as coupled plates [2224]. The present work seeks to extend AMA to coupled systems through a point-coupled parallel rectangular plate prototypical system, offering a method for determining the root-mean-square (RMS) response throughout the system quickly and accurately while capturing the spatial variation shown by Crandall [25]. The previous AMA derivations will be summarized, and a general model for coupled systems undergoing high-frequency, broadband point excitation will be presented.

The prototypical system is shown in Fig. 1. RMS responses in every region of interest on these plates will be studied.

## Summary of Analytical Methods

In this section, various analytical methods for describing the behavior of the prototypical dynamical system in question will be explored. Ultimately, the work of Dowell et al. [6,8] will be extended, presenting AMA for coupled continuous systems. In this section, we consider a nominal configuration for illustrative purposes. In Sec. 3, we consider several alternative configurations to study the sensitivity of the results to system parameters.

### Component Equation Solutions Via Eigenmode Expansion.

Asymptotic modal analysis is based upon the eigen-expansion of the system energy. The contributions to the Lagrangian for this system are
$T=∑m1Mm1a˙m12+∑m2Mm2a˙m22V=∑m1Mm1ωm12am12+∑m2Mm2ωm22am22λf=λ[∑m1am1ψm1(x1,0,y1,0)−∑m2am2ψm2(x2,0,y2,0)]$
(1)
Through Hamilton's principle, Fourier transforms, and algebraic manipulation, these will ultimately generate the transfer functions used to determine the RMS response from frequency-domain information via
$w¯q2=∫0∞|Hq|2ΦFdω$
(2)

where q is a generic response point or query point, $w¯q$ is the mean-square transverse response at point q, Hq is the transfer function to that response, and $ΦF$ is the input force power spectrum.

#### Constraint and Modal Coordinate Transfer Functions.

After modal decomposition and applying Hamilton's principle, the modal equations of motion can be found
$Mm1(a¨m1+2ζm1ωm1a˙m1+ωm12am1)=λψm1(x1,0,y1,0)+Fψm1(x1,F,y1,F)$
(3)
$Mm2(a¨m2+2ζm2ωm2a˙m2+ωm22am2)=−λψm2(x2,0,y2,0)$
(4)
$∑m1am1ψm1(x1,0,y1,0)−∑m2am2ψm2(x2,0,y2,0)=0$
(5)

where $ψm1$ and $ψm2$ are the component mode shapes associated with plate 1 and plate 2, respectively. Using classical modal analysis in the time domain (CMA-TD), these equations are all we need to observe how the system behaves in the asymptotic modal limit. As Figs. 2 and 3 show, there are regions of relatively uniform RMS response on both plates 1 and 2, but with some exceptions. The intensification zones (hot lines, hot points, and their mirror images) on plate 1 are a phenomenon resulting from a direct broadband excitation

Using the trial solutions
$F=F0eiωtλ=Λeiωtam1=Am1eiωtam2=Am2eiωt$
(6)
and the short-hand notation
$ψm10=ψm1(x1,0,y1,0)ψm20=ψm2(x2,0,y2,0)ψm1F=ψm2(x1,F,y1,F)Sm1=(ωm12−ω2)+i2ζm1ωm1ωSm2=(ωm22−ω2)+i2ζm2ωm2ω$
(7)
These equations may be rewritten as
$Mm1Am1Sm1=Λψm10+F0ψm1FMm2Am2Sm2=−Λψm20∑m1Am1ψm10=∑m2Am2ψm20$
(8)
$Am1=Λψm10+F0ψm1FMm1Sm1Am2=−Λψm20Mm2Sm2$
(9)
Substituting the isolated amplitudes into the constraint equation yields the constraint transfer function
$Hλ=ΛF0=−∑m1ψm1Fψm10Mm1Sm1∑m1ψm102Mm1Sm1+∑m2ψm202Mm2Sm2$
(10)
and the equation for the modal amplitude of plate 2 is, therefore,
$HAm2=−Am2ΛΛF0=−ψm20Mm2Sm2Hλ$
(11)
The transfer function for $Am1$ is a little more complicated as it involves a direct relationship to the input force and an indirect relationship through the constraint
$HAm1=Am1F0=Hλψm10+ψm1FMm1Sm1=ψm10Mm1Sm1Hλ+ψm1FMm1Sm1$
(12)

#### Coupled Natural Frequencies and Damping.

The expression for the coupled natural frequencies is obtained when the real part of the denominator of Eq. (10) is set to zero
$∑m1ψm102Mm1(ωm12−ωM2)+∑m2ψm202Mm2(ωm22−ωM2)=0$
(13)
Furthermore, the damped natural frequency satisfies the corresponding complex equation
$∑m1ψm102Mm1Sm1(ω̃M)+∑m2ψm202Mm2Sm2(ω̃M)=0$
(14)
where
$ω̃M=ωM(1+iζM)$
(15)
Therefore, it can be shown that the product of the coupled modal damping ratio and natural frequency is given by
$ζMωM=(ζω)m1∑m1ψm102Mm11(ωm12−ωM2)2+(ζω)m2∑m2ψm202Mm21(ωm22−ωM2)2∑m1ψm102Mm11(ωm12−ωM2)2+∑m2ψm202Mm21(ωm22−ωM2)2$
(16)

M denotes the coupled modes of the two-plate system and m1, m2 denote the component modes of each uncoupled plate. For the remainder of this investigation, we will assume a damping model such that $ζmiωmi$ is constant for all mi, but it will be seen that the following techniques still apply for different models of $ζmi$, such as constant modal damping.

#### Useful Forms of the Constraint Transfer Function.

The constraint transfer function is represented in several expressions in the discrete CMA equations for the coupled plate system. The complex characteristic equations must be carefully manipulated in order to simplify them without losing accuracy. It can be shown that, in the small-damping limit
$Smi(ω)=Ri(ω)+iGi(ω)Ri(ω)=ωmi2−ω2Gi(ω)=2(ζω)miωΨ0F=ψm1Fψm10Mm1Ψi0=ψmi02Mmi$
(17)
the following expressions are accurate [6,26]:
$|Hλ|2=(∑m1Ψ0FR1R12+G12)2+(∑m1Ψ0FG1R12+G12)2(∑m1Ψ10R1R12+G12+∑m2Ψ20R2R22+G22)2+(∑m1Ψ10G1R12+G12+∑m2Ψ20G2R22+G22)2$
(18)
$∑m1Ψ0FR1R12+G12(∑m1Ψ10R1R12+G12+∑m2Ψ20R2R22+G22)Hλ+H¯λ=−2+∑m1Ψ0FG1R12+G12(∑m1Ψ10G1R12+G12+∑m2Ψ20G2R22+G22)(∑m1Ψ10R1R12+G12+∑m2Ψ20R2R22+G22)2+(∑m1Ψ10G1R12+G12+∑m2Ψ20G2R22+G22)2$
(19)

### Classical Modal Analysis in the Frequency Domain (CMA-FD).

The mean-square response of a generic point q in a dynamical system can be described by the following integral:
$w¯q,12=∫0∞|Hq(ω)|2ΦF(ω)dω$
(20)
For a point $(xq,yq)$ on plate 1, whose modal expression is
$Wq,1=∑m1Am1ψm1(x1,q,y1,q)=∑m1Am1ψm1q$
(21)
the transverse displacement mean-square response is therefore
$w¯q,12=∫0∞|∑m1HAm1ψm1q|2ΦF(ω)dω$
(22)
where $HAm1$ is the transfer function for the modal amplitude response $Am1$
$w¯q,12=∫0∞|∑m11Mm1Sm1(ψm10Hλ+ψm1F)ψm1q|2ΦF(ω)dω$
(23)
Similarly, for plate 2
$w¯q,22=∫0∞|∑m1HAm2ψm2q|2ΦF(ω)dω$
(24)
$w¯q,22=∫0∞|∑m1ψm20ψm2qMm2Sm2Hλ|2ΦF(ω)dω$
(25)

These expressions constitute CMA-FD, and the integrals over all frequencies are evaluated by numerical quadrature.

### Discrete Classical Modal Analysis (dCMA).

To find a reduced-order model for high frequency behavior of this system, we must approximate these integrals over the frequency domain. Using an approximate representation of the coupled mode shapes and contour integration, it can be shown that [7]
$w¯q12=∑MπζMωM|∑m1HAm1(ωM)ψm1q|2ΦFw¯q22=∑MπζMωM|∑m2HAm2(ωM)ψm2q|2ΦF$
(26)

where for slowly varying $ΦF$ one may set $ΦF=ΦF(ω=ωM)$ and take it outside the integral in Eq. (20). The essence of the approximation is that the square of the mode transfer functions rapidly varies near $ω=ωM$, whereas $ΦF$ is slowly varying.

The mean-square operators acting on the sums of modes in the excitation band generate nested double-sums. However, the diagonals of the resulting matrix of terms dominates, so the following is a good approximation:
$w¯q12=∑Mπ(ζω)MΦF[|Hλ|2∑m1ψm1q2ψm102Mm12|Sm1|2 +(Hλ+H¯λ)∑m1ψm1q2ψm10ψm1FMm12|Sm1|2+∑m1ψm1q2ψm1F2Mm12|Sm1|2]w¯q22=∑Mπ(ζω)MΦF|Hλ|2∑m2ψm2q2ψm202Mm22|Sm2|2$
(27)
We find that the modal cross-coupling is negligible, i.e., the summation products are dominated by their diagonal terms. Therefore, we may reduce the expressions for the mean-square response by defining
$J1≡(∑m2Ψ20R2R22+G22)2+(∑m2Ψ20G2R22+G22)2(∑m1Ψ10R1R12+G12+∑m2Ψ20R2R22+G22)2+(∑m1Ψ10G1R12+G12+∑m2Ψ20G2R22+G22)2J2≡(∑m1Ψ0FR1R12+G12)2+(∑m1Ψ0FG1R12+G12)2(∑m1Ψ10R1R12+G12+∑m2Ψ20R2R22+G22)2+(∑m1Ψ10G1R12+G12+∑m2Ψ20G2R22+G22)2$
(28)
It follows that (see the “Nomenclature” section for a more detailed explanation)
$w¯q12=∑Mπ(ζω)MΦFJ1∑m1ψm1q2ψm1F2Mm121R12+G12w¯q22=∑Mπ(ζω)MΦFJ2∑m2ψm2q2ψm202Mm221R22+G22$
(29)

Some precision is lost in this reduction, but it is relatively modest. Figures 4 and 5 show box plots (whose boxes and whiskers illustrate quartiles and the maxima/minima of the data set) illustrating the ratio of predictions from dCMA to CMA-FD for various center frequencies against the number of modes excited in various excitation bandwidths. Specifically, the plots show $(((w¯qi2)dCMA)/((w¯qi2)CMA-FD))$ for different cases of excitation. For example, the box plot associated with six excited modes considers the response of the system to an excitation band that captures six modes for many different center frequencies. Ideally, this ratio is 1 for all bandwidths and center frequencies, and these plots show acceptable estimates from dCMA.

For the further reduction to AMA, Eq. (29) is the starting point.

### Asymptotic Modal Analysis.

The next step in simplifying these equations involves modal averaging. Following the work of Crandall [25], the constraint transfer function expressions and mean-square response equations will lose their dependence on the mode shapes when the number of modes in the frequency bandwidth becomes large enough (approximately 30 or more). This is an effective approximation at high frequencies when the modal characteristics are slowly varying in a frequency bandwidth.

#### Modal Averaging.

In his work “random vibration of one- and two-dimensional structures” [25], Crandall outlines the modal averages for various combinations of sinusoidal mode shapes in a continuous component experiencing a single broadband input. In the asymptotic modal limit, when many modes are excited, his work allows $Ψ10, Ψ20$, and $Ψ0F$ to be reduced. After squaring the terms in the parentheses, individual terms may be reduced via modal averaging as in the following two examples, noting that the mode shape contribution varies slowly compared to the frequency terms:
$∑m1Ψ102R12(R12+G12)2≈1Mp12limΔm1→∞∑m1=mminmmin+Δm1ψm104〈ψm12〉2∑m1R12(R12+G12)2≈941Mp12∑m1R12(R12+G12)2∑m1Ψ0F2R12(R12+G12)2≈1Mp12limΔm1→∞∑m1=mminmmin+Δm1ψm102ψm1F2〈ψm12〉2∑m1R12(R12+G12)2≈1Mp12∑m1R12(R12+G12)2$
(30)
As such, the contribution from the modal averaging of the terms in the denominator is approximated as a bulk factor of (9/4). Similarly, the contribution from the mode shapes in the numerator of J1 is also (9/4), but that of J1 is 1. Therefore, define the constraint transfer function expressions (including the effects of modal averaging) as follows:
$J̃1≡(∑m2R2R22+G22)2+(∑m2G2R22+G22)2(Mp2Mp1∑m1R1R12+G12+∑m2R2R22+G22)2+(Mp2Mp1∑m1G1R12+G12+∑m2G2R22+G22)2J̃2≡49(∑m1R1R12+G12)2+(∑m1G1R12+G12)2(∑m1R1R12+G12+Mp1Mp2∑m2R2R22+G22)2+(∑m1G1R12+G12+Mp1Mp2∑m2G2R22+G22)2$
(31)
and the mean-square response at a point q on plate 1 or 2 becomes
$w¯q12≈∑Mπζ̃MωMΦFJ̃1ΓMp12∑m11R12+G12w¯q22≈∑Mπζ̃MωMΦFJ̃21Mp22∑m21R22+G22$
(32)
where Γ is a spatial intensification coefficient that applies to plate 1 as the direct excitation is broadband. Plate 2 does not always experience spatial intensification in its response, however, because the coupling force exciting plate 2 has been filtered by the modal dynamics of plate 1. Γ takes a value of 1 if the query point and excitation point do not coincide, (3/2) if those points coincide in one dimension (i.e., coexist on a hot line), or (9/4) if the points coincide completely [25]. In these and all following equations, R1 refers to $R1(ωM)$ and similarly for R2, G1, and G2. The expression for the coupled damping ratio may also be rewritten
$ζ̃MωM≈(ζω)m1∑m11R12+(ζω)m2Mp1Mp2∑m21R22∑m11R12+Mp1Mp2∑m21R22$
(33)

The mode shapes do not significantly affect the damping ratio when the number of modes in the frequency bandwidth is sufficiently large. Figure 6 shows the ratio of the damping ratio after modal averaging to ζM before modal averaging for all coupled natural frequencies excited in a response to a 600 Hz excitation band centered at 1 kHz.

#### Frequency Reduction.

Equation (32) leaves the transfer functions in a form that no longer depends on mode shapes. Now the mode that sums within the transfer functions must be reduced. For convenience, the following notation is used for several summations:
$Σi≡∑m11Ri2+Gi2ΣRi≡∑m1RiRi2+Gi2ΣGi≡∑m1GiRi2+Gi2Σζi≡∑m11Ri2$
(34)
The next steps in reducing these expressions involves modeling the relationship between the coupled and uncoupled component natural frequencies. The component natural frequencies $ωmi$ are replaced by the following relationship, relative to a single coupled natural frequency ωM:
$ωmi≈ωM+γi+kρmi$
(35)

where the modal density of that component is $ρmi$. γi refers to the location of a coupled natural frequency between the two nearest component natural frequencies, as illustrated in Fig. 7.

k is an integer that will become the index over which the new sum operates. For example, consider the following equation illustrating an example how the new sums are formed:
$Σi≈∑k=−∞∞1[(ωM+γi+kρmi)2−ωM2]2+4(ζω)mi2ωM2≈ρmi24ωM2∑k=−∞∞1(γi+k)2+(ζω)mi2ρmi2$
(36)
Using these substitutions, the sums within the overall transfer functions become sums over k rather than mi, and have single-term expressions
$Σi≈−πρmi4(ζω)miωM2sinh[2π(ζω)miρmi] cos(2πγi)−cosh[2π(ζω)miρmi]=1ωM2σiΣRi≈πρmi2ωM sin(2πγi) cos(2πγi)−cosh[2π(ζω)miρmi]=1ωMσRiΣGi≈πρmi2ωMsinh[2π(ζω)miρmi] cos(2πγi)−cosh[2π(ζω)miρmi]=1ωMσGiΣζi≈π2ρmi24ωM2csc2(πγi)=1ωM2σζi$
(37)

One detail remains in the reduction of these transfer function sums: the values and relative values of γ1 and γ2. Their absolute values range uniformly from 0 to 1, and the length of the γi vector is equal to the length of the coupled natural frequency vector containing all ωM.

Their relative values may be constrained by observing that if a coupled natural frequency is near a natural frequency from plate 1, it is likely distant from a natural frequency of plate 2. Consider Fig. 8. The solid vertical lines represent locations of natural frequencies from plate 1. The dashed vertical lines represent locations of natural frequencies from plate 2. As coupled natural frequencies occur between adjacent component natural frequencies from either subsystem, it is likely that if a coupled natural frequency is close to a plate 1 natural frequency, it is distant from that of plate 2.

As there are so many responding modes, and there are so few scenarios where this assumption breaks down, this “out of phase” interpretation of the two γ vectors is sufficiently accurate for further reduction. So, as the functional dependence of all the reduced mode sum terms on γi is periodic
$γ2=γ1+12$
(38)
Or, for many components, it follows that:
$γi=γ1+i−1Ncomp$
(39)
where $Ncomp$ is the number of components in the system. Now, we may write the mean-square response of the system without any dependence on the component natural frequencies of the system. Define
$K1≡σR22+σG22(Mp2Mp1σR1+σR2)2+(Mp2Mp1σG1+σG2)2K2≡σR12+σG12(σR1+Mp1Mp2σR2)2+(σG1+Mp1Mp2σG2)2$
(40)
Then, the mean-square response of the system may be written as
$w¯q12≈∑MπζMωMΦFK1ΓMp121ωM2σ1w¯q22≈∑MπζMωMΦFK21Mp221ωM2σ2$
(41)
where
$ζMωM≈(ζω)m1σζ1+(ζω)m2σζ2σζ1+σζ2$
(42)
However, as $ζMωM$ may be replaced with an expression independent of ωM, and as the frequency is large, we may replace the remaining dependence on ωM by its average value
$1ωmax−ωmin∫ωminωmax1ωM2dωM=1ωminωmax$
(43)
Let us define a reference frequency
$ωr≡ωminωmax$
(44)
Then, the mean-square response of points throughout the system may be captured by
$w¯q12≈πζMωMF¯2ΔMΔωΓMp12ωr2σ1K1w¯q22≈πζMωMF¯2ΔMΔω1Mp22ωr2σ2K2$
(45)
where the modal density may be calculated [7] by
$ρmi=ΔMiΔω=Api4πmpiDi$
(46)
the overall modal density is
$ΔMΔω=ρm1+ρm2or∑i=1Ncompρmi$
(47)
and the number of points used to construct γi is found by
$ΔMi=Round[ApiΔω4πmpiDi]$
(48)

These algebraic expressions for the mean-square response at a point in the system are the AMA solutions to the point-coupled parallel plate problem.

## Results

These reductions or simplifications from continuous CMA-FD to AMA were evaluated for several specific configurations of the system. Different system configurations were studied yielding similar results, such as plates with different aspect ratios and different modal densities. Table 1 shows raw inputs to the system, such as geometry and material information.

The excitation scenarios tested involved an array of center frequencies and bandwidths. The center frequencies were varied from 1 Hz to 100 kHz, and six bandwidths were chosen between 30 Hz and 1 kHz (specifically 30, 50 100, 300, 500, and 1000 Hz). Several calculated system parameters are recorded in Table 2.

Figures 912 illustrate the accuracy of AMA relative to CMA-FD. Specifically, the plots show $(((w¯qi2)AMA)/((w¯qi2)CMA-FD))$ for different cases of excitation.

The advantage of using AMA is the greatly reduced computational cost. Figure 13 shows the increasing advantage of using AMA to analyze systems when the number of excited modes becomes large.

## Conclusions

These results illustrate that asymptotic modal analysis is valid for coupled continuous systems. The techniques used in previous works that involve leveraging the simplicity of the discrete system's influence on the constraint transfer function to eliminate the dependence on the mode sums from the continuous component could no longer be used. Instead, after ignoring the insignificant off-diagonal terms in the dCMA expressions, a model for the distribution of the component natural frequencies was used to find approximate values for these mode sums as the number of component modes becomes large. Finally, the relative proximity of coupled natural frequencies to component natural frequencies in plates 1 and 2 were studied and modeled. This approach is more generally applicable, and can in principle determine RMS responses for any combination of coupled, linear, continuous, and/or discrete elements. Moreover, this paves the way for an analysis of the damping and modal density conditions under which the coupling force is also broadband—when spatial intensification may occur on indirectly excited plates.

The accuracy of AMA's predictions is consistent for different configurations of the prototypical point-coupled parallel plate system. In all configurations that were considered in this investigation (including varying modal density, geometry, and other physical characteristics of the system), AMA converged to estimating system responses within 35% of the true (CMA-FD) value. This is especially encouraging as Figs. 2 and 3 show variation in the nonintensified regions that spans up to 50% of the mean response value, so 35% error is an acceptable figure for many applications where FEM or CMA are computationally prohibitive. The authors welcome the opportunity to collaborate with SEA experts on computations that compare AMA predictions to those of SEA concerning the prototypical system in question.

Through the work of Crandall [25] and subsequent authors, AMA captures spatial variation caused by intensification from excitation points in both the hot lines and hot points. However, in coupled systems, the modal averaging techniques become more nuanced, especially with transfer functions that involve mode sums in the denominator from different components. This element of AMA is one of the potential sources for remaining error between AMA and CMA. Terms that are neglected in the system transfer functions from the small-damping approximation could also contribute to these differences.

The relative separation of coupled natural frequencies to those of plates 1 and 2 is deserving further study. If the entries in the ordered component natural frequencies from both plates alternate (i.e., $ω2,0$ comes from plate 1, $ω2,1$ from plate 2, $ω2,2$ from plate 1, etc.), then, the assumption that if γ is close to 0 or 1 in the plate 1 mode sum, it is close to 0.5 in the plate 2 mode sum and vice versa can be clearly illustrated. However, if those are not ordered, then, the relative distribution of γ in plates 1 and 2 mode sums is less obvious.

The present work opens the door for AMA to be applied to other prototypical dynamical systems, such as two plates coupled by two or more discrete points, three or more stacked plates where adjacent plates are coupled at a point and ultimately any number of plates coupled at any number of points. Combinations of continuous systems (beams, plates, and shells) connected by elastic or dissipative elements are in view as well.

In addition there are opportunities for improving the general methodology. First, the nature of modal averaging in these intricate transfer function expressions may be more closely studied. Second, scenarios in which the coupling force is effectively broadband (and the consequences there of) may be explored. Finally, nonlinear continuous components and more intricate system connections may be studied.

## Acknowledgment

The authors acknowledge with appreciation the support of the Army Research Office and the guidance of Dr. Ralph Anthenien and Dr. Sam Stanton.

## Nomenclature

• Note that i may be 1 or 2 to refer to the driven or auxiliary plate, respectively.

•
• $ami$ =

plate 1 time-dependent modal amplitude

•
• $Ami$ =

plate i frequency-dependent modal amplitude

•
• $Api$ =

area of plate i

•
• Di =

plate stiffness for plate i

•
• F =

time-dependent input force

•
• F0 =

input force amplitude

•
• Gi =

magnitude of the imaginary part of the characteristic equation for plate i

•
• hi =

thickness of plate i

•
• $HAmi$ =

modal amplitude transfer function for plate i

•
• $Hqi$ =

transfer function for a query point on plate i

•
• $Hλ$ =

constraint transfer function

•
• Ji =

modified transfer function expression

•
• k =

index of mode natural frequencies

•
• $lxi$ =

length of plate i in the x-direction

•
• $lyi$ =

length of plate i in the y-direction

•
• $mpi$ =

mass per unit area of plate i

•
• $Mmi$ =

modal mass of plate i

•
• $Mpi$ =

plate i mass

•
• Ri =

magnitude of the real part of the characteristic equation for plate i

•
• $Smi$ =

modal characteristic equation of plate i

•
• t =

time

•
• wi =

plate 1 transverse displacement

•
• $wq,i$ =

time-dependent transverse displacement of a query point on plate i

•
• xi =

x-coordinate on plate i

•
• $xi,0$ =

x-coordinate of the coupling on plate i

•
• $x1,F$ =

x-coordinate of the excitation on plate 1

•
• yi =

y-coordinate on plate i

•
• $yi,0$ =

y-coordinate of the coupling on plate i

•
• $y1,F$ =

y-coordinate of the excitation on plate 1

•
• γ =

percent of a component modal spacing between a coupled natural frequency and its nearest component natural frequency.

•
• Γi =

modal average coefficient for plate i

•
• $Δmi$ =

number of component modes in plate i

•
• $Δω$ =

excitation bandwidth

•
• ζM =

damping ratio of the coupled system

•
• $ζmi$ =

modal damping ratio of plate i

•
• $(ζω)i$ =

damping-natural frequency constant of component i

•
• $(ζω)i$ =

damping-natural frequency of the coupled system

•
• λ =

time-dependent Lagrange multiplier (constraint force)

•
• Λ =

frequency-dependent Lagrange multiplier (constraint force)

•
• ρi =

mass density of plate 1

•
• $ρmi$ =

modal density of plate i

•
• Σi =

shorthand for the transfer function mode sum of plate i

•
• $ΦF$ =

excitation power spectrum

•
• $Ψ0F$ =

mode shape-dependent coefficient for the constraint transfer function numerator

•
• $Ψi0$ =

mode shape-dependent coefficient for the constraint transfer function denominator relative to plate i

•
• $ψmi$ =

mode shape of plate i

•
• $ψmi0$ =

mode shape of plate i evaluated at the coupling point

•
• $ψmiF$ =

mode shape of plate i evaluated at the excitation point

•
• ω =

frequency

•
• ωM =

natural frequency of the coupled system

•
• $ω̃M$ =

damped natural frequency of the coupled system

•
• $ωmi$ =

natural frequency of plate i

### Appendix

Consider the expression
$|Hλ|2∑m1ψm1q2ψm102Mm12|Sm1|2+(Hλ+H¯λ)∑m1ψm1q2ψm10ψm1FMm12|Sm1|2+∑m1ψm1q2ψm1F2Mm12|Sm1|2$
(A1)
Now, let us expand the numerators of the three terms individually while ensuring that each term has the same denominator. Denote Term 1, Term 2, and Term 3 as these numerators, described in detail below:
$Term 1=∑m1ψm1q2ψm102Mm12|Sm1|2[(∑m1ψm10ψm1FR1Mm1|Sm1|2)2+(∑m1ψm10ψm1FG1Mm1|Sm1|2)2]Term 2=−2∑m1ψm1q2ψm10ψm1FMm12|Sm1|2[∑m1ψm10ψm1FR1Mm1|Sm1|2(∑m1ψm102R1Mm1|Sm1|2+∑m2ψm202R2Mm2|Sm2|2) +∑m1ψm10ψm1FG1Mm1|Sm1|2(∑m1ψm102G1Mm1|Sm1|2+∑m2ψm202G2Mm2|Sm2|2)]Term 3=∑m1ψm1q2ψm1F2Mm12|Sm1|2[(∑m1ψm102R1Mm1|Sm1|2+∑m2ψm202R2Mm2|Sm2|2)2+(∑m1ψm102G1Mm1|Sm1|2+∑m2ψm202G2Mm2|Sm2|2)2]$
(A2)
Expanding these terms and adding them together creates patterns of summations of similar form, differing only by mode shape. Most of these approach zero in the asymptotic modal limit, and are summarized in the following:
$∑m1ψm1q2ψm102Mm12|Sm1|2(∑m1ψm10ψm1FR1Mm1|Sm1|2)2 −2∑m1ψm1q2ψm10ψm1FMm12|Sm1|2∑m1ψm10ψm1FR1Mm1|Sm1|2∑m1ψm102R1Mm1|Sm1|2 +∑m1ψm1q2ψm1F2Mm12|Sm1|2(∑m1ψm102R1Mm1|Sm1|2)2≈0$
(A3)
$2∑m1ψm1q2ψm1F2Mm12|Sm1|2∑m1ψm102R1Mm1|Sm1|2∑m2ψm202R2Mm2|Sm2|2 −2∑m1ψm1q2ψm10ψm1FMm12|Sm1|2∑m1ψm10ψm1FR1Mm1|Sm1|2∑m2ψm202R2Mm2|Sm2|2≈0$
(A4)
with similar equalities where the R terms in the numerators are replaced with the G terms. The remaining nonzero terms are
$∑m1ψm1q2ψm1F2Mm12|Sm1|2[(∑m2ψm202R2Mm2|Sm2|2)2+(∑m2ψm202G2Mm2|Sm2|2)2]$
(A5)

which contributes, in part, to the definition of J1.

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