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 [22–24]. 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.

where *q* is a generic response point or query point, $w\xafq$ is the mean-square transverse response at point *q*, *H _{q}* is the transfer function to that response, and $\Phi F$ is the input force power spectrum.

#### Constraint and Modal Coordinate Transfer Functions.

where $\psi m1$ and $\psi m2$ 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

#### Coupled Natural Frequencies and Damping.

*M* denotes the coupled modes of the two-plate system and *m*_{1}, *m*_{2} denote the component modes of each uncoupled plate. For the remainder of this investigation, we will assume a damping model such that $\zeta mi\omega mi$ is constant for all *m _{i}*, but it will be seen that the following techniques still apply for different models of $\zeta mi$, such as constant modal damping.

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

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

*q*in a dynamical system can be described by the following integral:

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

### Discrete Classical Modal Analysis (dCMA).

where for slowly varying $\Phi F$ one may set $\Phi F=\Phi F(\omega =\omega M)$ 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 $\omega =\omega M$, whereas $\Phi F$ is slowly varying.

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 $(((w\xafqi2)dCMA)/((w\xafqi2)CMA-FD))$ 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.

*J*

_{1}is also (9/4), but that of

*J*

_{1}is 1. Therefore, define the constraint transfer function expressions (including the effects of modal averaging) as follows:

*q*on plate 1 or 2 becomes

*R*

_{1}refers to $R1(\omega M)$ and similarly for

*R*

_{2},

*G*

_{1}, and

*G*

_{2}. The expression for the coupled damping ratio may also be rewritten

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 *ζ _{M}* before modal averaging for all coupled natural frequencies excited in a response to a 600 Hz excitation band centered at 1 kHz.

#### Frequency Reduction.

*ω*:

_{M}where the modal density of that component is $\rho mi$. *γ _{i}* 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:

*k*rather than

*m*, and have single-term expressions

_{i}One detail remains in the reduction of these transfer function sums: the values and relative values of *γ*_{1} and *γ*_{2}. Their absolute values range uniformly from 0 to 1, and the length of the *γ _{i}* vector is equal to the length of the coupled natural frequency vector containing all

*ω*.

_{M}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.

*γ*vectors is sufficiently accurate for further reduction. So, as the functional dependence of all the reduced mode sum terms on

*γ*is periodic

_{i}*ω*, and as the frequency is large, we may replace the remaining dependence on

_{M}*ω*by its average value

_{M}*γ*is found by

_{i}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.

Parameter | Value | Unit |
---|---|---|

$lx1$ | 0.9425 | m |

$ly1$ | 0.7639 | m |

$lx2$ | 0.9549 | m |

$ly2$ | 1.1519 | m |

$x1,0$ | 0.1980 | m |

$y1,0$ | 0.5107 | m |

$x1,F$ | 0.5850 | m |

$y1,F$ | 0.2918 | m |

$x2,0$ | 0.3048 | m |

$y2,0$ | 0.3850 | m |

h_{1} | 2.000 | mm |

h_{2} | 1.600 | mm |

$\zeta 11$ | 2.5 | % |

$\zeta 12$ | 2.5 | % |

Material | Carbon steel | — |

F | 10 | N |

Parameter | Value | Unit |
---|---|---|

$lx1$ | 0.9425 | m |

$ly1$ | 0.7639 | m |

$lx2$ | 0.9549 | m |

$ly2$ | 1.1519 | m |

$x1,0$ | 0.1980 | m |

$y1,0$ | 0.5107 | m |

$x1,F$ | 0.5850 | m |

$y1,F$ | 0.2918 | m |

$x2,0$ | 0.3048 | m |

$y2,0$ | 0.3850 | m |

h_{1} | 2.000 | mm |

h_{2} | 1.600 | mm |

$\zeta 11$ | 2.5 | % |

$\zeta 12$ | 2.5 | % |

Material | Carbon steel | — |

F | 10 | N |

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.

Parameter | Value | Unit |
---|---|---|

$\rho m1$ | 0.0189 | s |

$\rho m2$ | 0.0362 | s |

$(\zeta \omega )m1$ | 1.6959 | rad/s |

$(\zeta \omega )m2$ | 1.3815 | rad/s |

$Mp1$ | 11.14 | kg |

$Mp2$ | 13.62 | kg |

$ARp1$ | 1.234 | — |

$ARp2$ | 0.829 | — |

Parameter | Value | Unit |
---|---|---|

$\rho m1$ | 0.0189 | s |

$\rho m2$ | 0.0362 | s |

$(\zeta \omega )m1$ | 1.6959 | rad/s |

$(\zeta \omega )m2$ | 1.3815 | rad/s |

$Mp1$ | 11.14 | kg |

$Mp2$ | 13.62 | kg |

$ARp1$ | 1.234 | — |

$ARp2$ | 0.829 | — |

Figures 9–12 illustrate the accuracy of AMA relative to CMA-FD. Specifically, the plots show $(((w\xafqi2)AMA)/((w\xafqi2)CMA-FD))$ 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., $\omega 2,0$ comes from plate 1, $\omega 2,1$ from plate 2, $\omega 2,2$ 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**D*=_{i}plate stiffness for plate

*i**F*=time-dependent input force

*F*_{0}=input force amplitude

*G*=_{i}magnitude of the imaginary part of the characteristic equation for plate

*i**h*=_{i}thickness of plate

*i*- $HAmi$ =
modal amplitude transfer function for plate

*i*- $Hqi$ =
transfer function for a query point on plate

*i*- $H\lambda $ =
constraint transfer function

*J*=_{i}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*R*=_{i}magnitude of the real part of the characteristic equation for plate

*i*- $Smi$ =
modal characteristic equation of plate

*i**t*=time

*w*=_{i}plate 1 transverse displacement

- $wq,i$ =
time-dependent transverse displacement of a query point on plate

*i**x*=_{i}*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*y*=_{i}*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.

- Γ
=_{i} modal average coefficient for plate

*i*- $\Delta mi$ =
number of component modes in plate

*i*- $\Delta \omega $ =
excitation bandwidth

*ζ*=_{M}damping ratio of the coupled system

- $\zeta mi$ =
modal damping ratio of plate

*i*- $(\zeta \omega )i$ =
damping-natural frequency constant of component

*i*- $(\zeta \omega )i$ =
damping-natural frequency of the coupled system

*λ*=time-dependent Lagrange multiplier (constraint force)

- Λ =
frequency-dependent Lagrange multiplier (constraint force)

*ρ*=_{i}mass density of plate 1

- $\rho mi$ =
modal density of plate

*i*- Σ
=_{i} shorthand for the transfer function mode sum of plate

*i*- $\Phi F$ =
excitation power spectrum

- $\Psi 0F$ =
mode shape-dependent coefficient for the constraint transfer function numerator

- $\Psi i0$ =
mode shape-dependent coefficient for the constraint transfer function denominator relative to plate

*i*- $\psi mi$ =
mode shape of plate

*i*- $\psi mi0$ =
mode shape of plate

*i*evaluated at the coupling point- $\psi miF$ =
mode shape of plate

*i*evaluated at the excitation point*ω*=frequency

*ω*=_{M}natural frequency of the coupled system

- $\omega \u0303M$ =
damped natural frequency of the coupled system

- $\omega mi$ =
natural frequency of plate

*i*

### Appendix

*R*terms in the numerators are replaced with the

*G*terms. The remaining nonzero terms are

which contributes, in part, to the definition of *J*_{1}.