High Frequency Analysis of a Point-Coupled Parallel Plate System

Culver, Dean R.; Dowell, Earl H.

doi:10.1115/1.4036212

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.

Fig. 1

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Schematic of the point-coupled parallel plate system

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

\begin{matrix} T & = \sum_{m_{1}} M_{m_{1}} {\dot{a}}_{m_{1}}^{2} + \sum_{m_{2}} M_{m_{2}} {\dot{a}}_{m_{2}}^{2} \\ V & = \sum_{m_{1}} M_{m_{1}} ω_{m_{1}}^{2} a_{m_{1}}^{2} + \sum_{m_{2}} M_{m_{2}} ω_{m_{2}}^{2} a_{m_{2}}^{2} \\ λ f & = λ [\sum_{m_{1}} a_{m_{1}} ψ_{m_{1}} (x_{1, 0}, y_{1, 0}) - \sum_{m_{2}} a_{m_{2}} ψ_{m_{2}} (x_{2, 0}, y_{2, 0})] \end{matrix}

(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

{\bar{w}}_{q}^{2} = \int_{0}^{\infty} {| H_{q} |}^{2} Φ_{F} d ω

(2)

where q is a generic response point or query point, ${\bar{w}}_{q}$ is the mean-square transverse response at point q, H_q is the transfer function to that response, and $Φ_{F}$ 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

\begin{matrix} M_{m_{1}} ({\ddot{a}}_{m_{1}} + 2 ζ_{m_{1}} ω_{m_{1}} {\dot{a}}_{m_{1}} + ω_{m_{1}}^{2} a_{m_{1}}) \\ = λ ψ_{m_{1}} (x_{1, 0}, y_{1, 0}) + F ψ_{m_{1}} (x_{1, F}, y_{1, F}) \end{matrix}

(3)

M_{m_{2}} ({\ddot{a}}_{m_{2}} + 2 ζ_{m_{2}} ω_{m_{2}} {\dot{a}}_{m_{2}} + ω_{m_{2}}^{2} a_{m_{2}}) = - λ ψ_{m_{2}} (x_{2, 0}, y_{2, 0})

(4)

\sum_{m_{1}} a_{m_{1}} ψ_{m_{1}} (x_{1, 0}, y_{1, 0}) - \sum_{m_{2}} a_{m_{2}} ψ_{m_{2}} (x_{2, 0}, y_{2, 0}) = 0

(5)

where $ψ_{m_{1}}$ and $ψ_{m_{2}}$ 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

Fig. 2

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Response of plate 1 in the point-coupled plate system under a broadband transverse point excitation with a 600 Hz band centered at 1 kHz. One hundred and seventy-one modes are excited in plate 1. The excitation and coupling points are illustrated by vertical dashed lines. The spatial intensification zones and their mirrors are clearly visible, creating an almost tic-tac-toe-like pattern on the plate.

Fig. 3

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Response of plate 2 in the point-coupled plate system under a broadband transverse point excitation with a 600 Hz band centered at 1 kHz. Two hundred and nine modes are excited in plate 2. Spatial intensification is less clear. In this case, there appears to be a hot line and a mirror image in the x-direction, but why it exists without a corresponding hot line in the y-direction is likely a consequence of the coupling force power spectrum being neither sparse nor definitively broadband.

Using the trial solutions

\begin{matrix} F & = F_{0} e^{i ω t} \\ λ & = Λ e^{i ω t} \\ a_{m_{1}} & = A_{m_{1}} e^{i ω t} \\ a_{m_{2}} & = A_{m_{2}} e^{i ω t} \end{matrix}

(6)

and the short-hand notation

\begin{matrix} ψ_{m_{1} 0} & = ψ_{m_{1}} (x_{1, 0}, y_{1, 0}) \\ ψ_{m_{2} 0} & = ψ_{m_{2}} (x_{2, 0}, y_{2, 0}) \\ ψ_{m_{1} F} & = ψ_{m_{2}} (x_{1, F}, y_{1, F}) \\ S_{m_{1}} & = (ω_{m_{1}}^{2} - ω^{2}) + i 2 ζ_{m_{1}} ω_{m_{1}} ω \\ S_{m_{2}} & = (ω_{m_{2}}^{2} - ω^{2}) + i 2 ζ_{m_{2}} ω_{m_{2}} ω \end{matrix}

(7)

These equations may be rewritten as

\begin{matrix} M_{m_{1}} A_{m_{1}} S_{m_{1}} & = Λ ψ_{m_{1} 0} + F_{0} ψ_{m_{1} F} \\ M_{m_{2}} A_{m_{2}} S_{m_{2}} & = - Λ ψ_{m_{2} 0} \\ \sum_{m_{1}} A_{m_{1}} ψ_{m_{1} 0} & = \sum_{m_{2}} A_{m_{2}} ψ_{m_{2} 0} \end{matrix}

(8)

\begin{matrix} A_{m_{1}} & = \frac{Λ ψ_{m_{1} 0} + F_{0} ψ_{m_{1} F}}{M_{m_{1}} S_{m_{1}}} \\ A_{m_{2}} & = - \frac{Λ ψ_{m_{2} 0}}{M_{m_{2}} S_{m_{2}}} \end{matrix}

(9)

Substituting the isolated amplitudes into the constraint equation yields the constraint transfer function

H_{λ} = \frac{Λ}{F_{0}} = - \frac{\sum_{m_{1}} \frac{ψ_{m_{1} F} ψ_{m_{1} 0}}{M_{m_{1}} S_{m_{1}}}}{\sum_{m_{1}} \frac{ψ_{m_{1} 0}^{2}}{M_{m_{1}} S_{m_{1}}} + \sum_{m_{2}} \frac{ψ_{m_{2} 0}^{2}}{M_{m_{2}} S_{m_{2}}}}

(10)

and the equation for the modal amplitude of plate 2 is, therefore,

H_{A_{m_{2}}} = - \frac{A_{m_{2}}}{Λ} \frac{Λ}{F_{0}} = - \frac{ψ_{m_{2} 0}}{M_{m_{2}} S_{m_{2}}} H_{λ}

(11)

The transfer function for

A_{m_{1}}

is a little more complicated as it involves a direct relationship to the input force and an indirect relationship through the constraint

H_{A_{m_{1}}} = \frac{A_{m_{1}}}{F_{0}} = \frac{H_{λ} ψ_{m_{1} 0} + ψ_{m_{1} F}}{M_{m_{1}} S_{m_{1}}} = \frac{ψ_{m_{1} 0}}{M_{m_{1}} S_{m_{1}}} H_{λ} + \frac{ψ_{m_{1} F}}{M_{m_{1}} S_{m_{1}}}

(12)

Coupled Natural Frequencies and Damping.

The expression for the coupled natural frequencies is obtained when the real part of the denominator of Eq. is set to zero

\sum_{m_{1}} \frac{ψ_{m_{1} 0}^{2}}{M_{m_{1}} (ω_{m_{1}}^{2} - ω_{M}^{2})} + \sum_{m_{2}} \frac{ψ_{m_{2} 0}^{2}}{M_{m_{2}} (ω_{m_{2}}^{2} - ω_{M}^{2})} = 0

(13)

Furthermore, the damped natural frequency satisfies the corresponding complex equation

\sum_{m_{1}} \frac{ψ_{m_{1} 0}^{2}}{M_{m_{1}} S_{m_{1}} ({\tilde{ω}}_{M})} + \sum_{m_{2}} \frac{ψ_{m_{2} 0}^{2}}{M_{m_{2}} S_{m_{2}} ({\tilde{ω}}_{M})} = 0

(14)

where

{\tilde{ω}}_{M} = ω_{M} (1 + i ζ_{M})

(15)

Therefore, it can be shown that the product of the coupled modal damping ratio and natural frequency is given by

\begin{matrix} ζ_{M} ω_{M} = \frac{{(ζ ω)}_{m_{1}} \sum_{m_{1}} \frac{ψ_{m_{1} 0}^{2}}{M_{m_{1}}} \frac{1}{{(ω_{m_{1}}^{2} - ω_{M}^{2})}^{2}} + {(ζ ω)}_{m_{2}} \sum_{m_{2}} \frac{ψ_{m_{2} 0}^{2}}{M_{m_{2}}} \frac{1}{{(ω_{m_{2}}^{2} - ω_{M}^{2})}^{2}}}{\sum_{m_{1}} \frac{ψ_{m_{1} 0}^{2}}{M_{m_{1}}} \frac{1}{{(ω_{m_{1}}^{2} - ω_{M}^{2})}^{2}} + \sum_{m_{2}} \frac{ψ_{m_{2} 0}^{2}}{M_{m_{2}}} \frac{1}{{(ω_{m_{2}}^{2} - ω_{M}^{2})}^{2}}} \end{matrix}

(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 $ζ_{m_{i}} ω_{m_{i}}$ is constant for all m_i, but it will be seen that the following techniques still apply for different models of $ζ_{m_{i}}$ ⁠, 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

\begin{matrix} S_{m_{i}} (ω) & = R_{i} (ω) + i G_{i} (ω) \\ R_{i} (ω) & = ω_{m_{i}}^{2} - ω^{2} \\ G_{i} (ω) & = 2 {(ζ ω)}_{m_{i}} ω \\ Ψ_{0 F} & = \frac{ψ_{m_{1} F} ψ_{m_{1} 0}}{M_{m_{1}}} \\ Ψ_{i 0} & = \frac{ψ_{m_{i} 0}^{2}}{M_{m_{i}}} \end{matrix}

(17)

the following expressions are accurate [6,26]:

{| H_{λ} |}^{2} = \frac{{(\sum_{m_{1}} Ψ_{0 F} \frac{R_{1}}{R_{1}^{2} + G_{1}^{2}})}^{2} + {(\sum_{m_{1}} Ψ_{0 F} \frac{G_{1}}{R_{1}^{2} + G_{1}^{2}})}^{2}}{{(\sum_{m_{1}} Ψ_{10} \frac{R_{1}}{R_{1}^{2} + G_{1}^{2}} + \sum_{m_{2}} Ψ_{20} \frac{R_{2}}{R_{2}^{2} + G_{2}^{2}})}^{2} + {(\sum_{m_{1}} Ψ_{10} \frac{G_{1}}{R_{1}^{2} + G_{1}^{2}} + \sum_{m_{2}} Ψ_{20} \frac{G_{2}}{R_{2}^{2} + G_{2}^{2}})}^{2}}

(18)

\begin{matrix} \sum_{m_{1}} Ψ_{0 F} \frac{R_{1}}{R_{1}^{2} + G_{1}^{2}} (\sum_{m_{1}} Ψ_{10} \frac{R_{1}}{R_{1}^{2} + G_{1}^{2}} + \sum_{m_{2}} Ψ_{20} \frac{R_{2}}{R_{2}^{2} + G_{2}^{2}}) \\ H_{λ} + {\bar{H}}_{λ} = - 2 \frac{+ \sum_{m_{1}} Ψ_{0 F} \frac{G_{1}}{R_{1}^{2} + G_{1}^{2}} (\sum_{m_{1}} Ψ_{10} \frac{G_{1}}{R_{1}^{2} + G_{1}^{2}} + \sum_{m_{2}} Ψ_{20} \frac{G_{2}}{R_{2}^{2} + G_{2}^{2}})}{{(\sum_{m_{1}} Ψ_{10} \frac{R_{1}}{R_{1}^{2} + G_{1}^{2}} + \sum_{m_{2}} Ψ_{20} \frac{R_{2}}{R_{2}^{2} + G_{2}^{2}})}^{2}} \\ + {(\sum_{m_{1}} Ψ_{10} \frac{G_{1}}{R_{1}^{2} + G_{1}^{2}} + \sum_{m_{2}} Ψ_{20} \frac{G_{2}}{R_{2}^{2} + G_{2}^{2}})}^{2} \end{matrix}

(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:

{\bar{w}}_{q, 1}^{2} = \int_{0}^{\infty} {| H_{q} (ω) |}^{2} Φ_{F} (ω) d ω

(20)

For a point

(x_{q}, y_{q})

on plate 1, whose modal expression is

W_{q, 1} = \sum_{m_{1}} A_{m_{1}} ψ_{m_{1}} (x_{1, q}, y_{1, q}) = \sum_{m_{1}} A_{m_{1}} ψ_{m_{1} q}

(21)

the transverse displacement mean-square response is therefore

{\bar{w}}_{q, 1}^{2} = \int_{0}^{\infty} {| \sum_{m_{1}} H_{A_{m_{1}}} ψ_{m_{1} q} |}^{2} Φ_{F} (ω) d ω

(22)

where

H_{A_{m_{1}}}

is the transfer function for the modal amplitude response

A_{m_{1}}

{\bar{w}}_{q, 1}^{2} = \int_{0}^{\infty} {| \sum_{m_{1}} \frac{1}{M_{m_{1}} S_{m_{1}}} (ψ_{m_{1} 0} H_{λ} + ψ_{m_{1} F}) ψ_{m_{1} q} |}^{2} Φ_{F} (ω) d ω

(23)

Similarly, for plate 2

{\bar{w}}_{q, 2}^{2} = \int_{0}^{\infty} {| \sum_{m_{1}} H_{A_{m_{2}}} ψ_{m_{2} q} |}^{2} Φ_{F} (ω) d ω

(24)

{\bar{w}}_{q, 2}^{2} = \int_{0}^{\infty} {| \sum_{m_{1}} \frac{ψ_{m_{2} 0} ψ_{m_{2} q}}{M_{m_{2}} S_{m_{2}}} H_{λ} |}^{2} Φ_{F} (ω) d ω

(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]

\begin{matrix} {\bar{w}}_{q_{1}}^{2} & = \sum_{M} π ζ_{M} ω_{M} {| \sum_{m_{1}} H_{A_{m_{1}}} (ω_{M}) ψ_{m_{1} q} |}^{2} Φ_{F} \\ {\bar{w}}_{q_{2}}^{2} & = \sum_{M} π ζ_{M} ω_{M} {| \sum_{m_{2}} H_{A_{m_{2}}} (ω_{M}) ψ_{m_{2} q} |}^{2} Φ_{F} \end{matrix}

(26)

where for slowly varying $Φ_{F}$ one may set $Φ_{F} = Φ_{F} (ω = ω_{M})$ 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 $ω = ω_{M}$ ⁠, whereas $Φ_{F}$ 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:

\begin{matrix} {\bar{w}}_{q_{1}}^{2} = \sum_{M} π {(ζ ω)}_{M} Φ_{F} [{| H_{λ} |}^{2} \sum_{m_{1}} \frac{ψ_{m_{1} q}^{2} ψ_{m_{1} 0}^{2}}{M_{m_{1}}^{2} {| S_{m_{1}} |}^{2}} \\ + (H_{λ} + {\bar{H}}_{λ}) \sum_{m_{1}} \frac{ψ_{m_{1} q}^{2} ψ_{m_{1} 0} ψ_{m_{1} F}}{M_{m_{1}}^{2} {| S_{m_{1}} |}^{2}} + \sum_{m_{1}} \frac{ψ_{m_{1} q}^{2} ψ_{m_{1} F}^{2}}{M_{m_{1}}^{2} {| S_{m_{1}} |}^{2}}] \\ {\bar{w}}_{q_{2}}^{2} = \sum_{M} π {(ζ ω)}_{M} Φ_{F} {| H_{λ} |}^{2} \sum_{m_{2}} \frac{ψ_{m_{2} q}^{2} ψ_{m_{2} 0}^{2}}{M_{m_{2}}^{2} {| S_{m_{2}} |}^{2}} \end{matrix}

(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

J_{1} \equiv \frac{{(\sum_{m_{2}} Ψ_{20} \frac{R_{2}}{R_{2}^{2} + G_{2}^{2}})}^{2} + {(\sum_{m_{2}} Ψ_{20} \frac{G_{2}}{R_{2}^{2} + G_{2}^{2}})}^{2}}{{(\sum_{m_{1}} Ψ_{10} \frac{R_{1}}{R_{1}^{2} + G_{1}^{2}} + \sum_{m_{2}} Ψ_{20} \frac{R_{2}}{R_{2}^{2} + G_{2}^{2}})}^{2} + {(\sum_{m_{1}} Ψ_{10} \frac{G_{1}}{R_{1}^{2} + G_{1}^{2}} + \sum_{m_{2}} Ψ_{20} \frac{G_{2}}{R_{2}^{2} + G_{2}^{2}})}^{2}} J_{2} \equiv \frac{{(\sum_{m_{1}} Ψ_{0 F} \frac{R_{1}}{R_{1}^{2} + G_{1}^{2}})}^{2} + {(\sum_{m_{1}} Ψ_{0 F} \frac{G_{1}}{R_{1}^{2} + G_{1}^{2}})}^{2}}{{(\sum_{m_{1}} Ψ_{10} \frac{R_{1}}{R_{1}^{2} + G_{1}^{2}} + \sum_{m_{2}} Ψ_{20} \frac{R_{2}}{R_{2}^{2} + G_{2}^{2}})}^{2} + {(\sum_{m_{1}} Ψ_{10} \frac{G_{1}}{R_{1}^{2} + G_{1}^{2}} + \sum_{m_{2}} Ψ_{20} \frac{G_{2}}{R_{2}^{2} + G_{2}^{2}})}^{2}}

(28)

It follows that (see the “Nomenclature” section for a more detailed explanation)

\begin{matrix} {\bar{w}}_{q_{1}}^{2} = & \sum_{M} π {(ζ ω)}_{M} Φ_{F} J_{1} \sum_{m_{1}} \frac{ψ_{m_{1} q}^{2} ψ_{m_{1} F}^{2}}{M_{m_{1}}^{2}} \frac{1}{R_{1}^{2} + G_{1}^{2}} \\ {\bar{w}}_{q_{2}}^{2} = & \sum_{M} π {(ζ ω)}_{M} Φ_{F} J_{2} \sum_{m_{2}} \frac{ψ_{m_{2} q}^{2} ψ_{m_{2} 0}^{2}}{M_{m_{2}}^{2}} \frac{1}{R_{2}^{2} + G_{2}^{2}} \end{matrix}

(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 $(({({\bar{w}}_{q i}^{2})}_{dCMA}) / ({({\bar{w}}_{q_{i}}^{2})}_{CMA - FD}))$ 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.

Fig. 4

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Ratio of response predictions of a free point (a point not experiencing any spatial intensification) on plate 1 from dCMA to CMA-FD for various bandwidths and center frequencies ranging from 1 kHz to 100 kHz exciting prescribed numbers of modes. The bandwidths used to capture the array of modes illustrated on the x-axis are 30 Hz, 50 Hz, 100 Hz, 300 Hz, 500 Hz, and 1000 Hz.

Fig. 5

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Ratio of response predictions of a free point (a point not experiencing any spatial intensification) on plate 2 from dCMA to CMA-FD for various bandwidths and center frequencies ranging from 1 kHz to 100 kHz exciting prescribed numbers of modes. The bandwidths used to capture the array of modes illustrated on the x-axis are 30 Hz, 50 Hz, 100 Hz, 300 Hz, 500 Hz, and 1000 Hz.

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

Ψ_{10}, Ψ_{20}

⁠, and

Ψ_{0 F}

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:

\begin{matrix} \sum_{m_{1}} Ψ_{10}^{2} \frac{R_{1}^{2}}{{(R_{1}^{2} + G_{1}^{2})}^{2}} \approx \frac{1}{M_{p_{1}}^{2}} \lim_{Δ m_{1} \to \infty} \sum_{m_{1} = m_{min}}^{m_{min} + Δ m_{1}} \frac{ψ_{m_{1} 0}^{4}}{{〈 ψ_{m_{1}}^{2} 〉}^{2}} \sum_{m_{1}} \frac{R_{1}^{2}}{{(R_{1}^{2} + G_{1}^{2})}^{2}} \\ \approx \frac{9}{4} \frac{1}{M_{p_{1}}^{2}} \sum_{m_{1}} \frac{R_{1}^{2}}{{(R_{1}^{2} + G_{1}^{2})}^{2}} \\ \sum_{m_{1}} Ψ_{0 F}^{2} \frac{R_{1}^{2}}{{(R_{1}^{2} + G_{1}^{2})}^{2}} \approx \frac{1}{M_{p_{1}}^{2}} \lim_{Δ m_{1} \to \infty} \sum_{m_{1} = m_{min}}^{m_{min} + Δ m_{1}} \frac{ψ_{m_{1} 0}^{2} ψ_{m_{1} F}^{2}}{{〈 ψ_{m_{1}}^{2} 〉}^{2}} \sum_{m_{1}} \frac{R_{1}^{2}}{{(R_{1}^{2} + G_{1}^{2})}^{2}} \\ \approx \frac{1}{M_{p_{1}}^{2}} \sum_{m_{1}} \frac{R_{1}^{2}}{{(R_{1}^{2} + G_{1}^{2})}^{2}} \end{matrix}

(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:

{\tilde{J}}_{1} \equiv \frac{{(\sum_{m_{2}} \frac{R_{2}}{R_{2}^{2} + G_{2}^{2}})}^{2} + {(\sum_{m_{2}} \frac{G_{2}}{R_{2}^{2} + G_{2}^{2}})}^{2}}{{(\frac{M_{p_{2}}}{M_{p_{1}}} \sum_{m_{1}} \frac{R_{1}}{R_{1}^{2} + G_{1}^{2}} + \sum_{m_{2}} \frac{R_{2}}{R_{2}^{2} + G_{2}^{2}})}^{2} + {(\frac{M_{p_{2}}}{M_{p_{1}}} \sum_{m_{1}} \frac{G_{1}}{R_{1}^{2} + G_{1}^{2}} + \sum_{m_{2}} \frac{G_{2}}{R_{2}^{2} + G_{2}^{2}})}^{2}} {\tilde{J}}_{2} \equiv \frac{4}{9} \frac{{(\sum_{m_{1}} \frac{R_{1}}{R_{1}^{2} + G_{1}^{2}})}^{2} + {(\sum_{m_{1}} \frac{G_{1}}{R_{1}^{2} + G_{1}^{2}})}^{2}}{{(\sum_{m_{1}} \frac{R_{1}}{R_{1}^{2} + G_{1}^{2}} + \frac{M_{p_{1}}}{M_{p_{2}}} \sum_{m_{2}} \frac{R_{2}}{R_{2}^{2} + G_{2}^{2}})}^{2} + {(\sum_{m_{1}} \frac{G_{1}}{R_{1}^{2} + G_{1}^{2}} + \frac{M_{p_{1}}}{M_{p_{2}}} \sum_{m_{2}} \frac{G_{2}}{R_{2}^{2} + G_{2}^{2}})}^{2}}

(31)

and the mean-square response at a point q on plate 1 or 2 becomes

\begin{matrix} {\bar{w}}_{q_{1}}^{2} \approx & \sum_{M} π {\tilde{ζ}}_{M} ω_{M} Φ_{F} {\tilde{J}}_{1} \frac{Γ}{M_{p_{1}}^{2}} \sum_{m_{1}} \frac{1}{R_{1}^{2} + G_{1}^{2}} \\ {\bar{w}}_{q_{2}}^{2} \approx & \sum_{M} π {\tilde{ζ}}_{M} ω_{M} Φ_{F} {\tilde{J}}_{2} \frac{1}{M_{p_{2}}^{2}} \sum_{m_{2}} \frac{1}{R_{2}^{2} + G_{2}^{2}} \end{matrix}

(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

R_{1} (ω_{M})

and similarly for R₂, G₁, and G₂. The expression for the coupled damping ratio may also be rewritten

{\tilde{ζ}}_{M} ω_{M} \approx \frac{{(ζ ω)}_{m_{1}} \sum_{m_{1}} \frac{1}{R_{1}^{2}} + {(ζ ω)}_{m_{2}} \frac{M_{p_{1}}}{M_{p_{2}}} \sum_{m_{2}} \frac{1}{R_{2}^{2}}}{\sum_{m_{1}} \frac{1}{R_{1}^{2}} + \frac{M_{p_{1}}}{M_{p_{2}}} \sum_{m_{2}} \frac{1}{R_{2}^{2}}}

(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 ζ_M before modal averaging for all coupled natural frequencies excited in a response to a 600 Hz excitation band centered at 1 kHz.

Fig. 6

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The ratio of ζ_M after modal averaging to ζ_M before modal averaging. The frequency axis is normalized by the location within the excitation band. More specifically, 0 refers to ω_min and 1 refers to ω_max.

Frequency Reduction.

Equation 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:

\begin{matrix} Σ_{i} & \equiv \sum_{m_{1}} \frac{1}{R_{i}^{2} + G_{i}^{2}} \\ Σ_{R i} & \equiv \sum_{m_{1}} \frac{R_{i}}{R_{i}^{2} + G_{i}^{2}} \\ Σ_{G i} & \equiv \sum_{m_{1}} \frac{G_{i}}{R_{i}^{2} + G_{i}^{2}} \\ Σ_{ζ i} & \equiv \sum_{m_{1}} \frac{1}{R_{i}^{2}} \end{matrix}

(34)

The next steps in reducing these expressions involves modeling the relationship between the coupled and uncoupled component natural frequencies. The component natural frequencies

ω_{m_{i}}

are replaced by the following relationship, relative to a single coupled natural frequency ω_M:

ω_{m_{i}} \approx ω_{M} + \frac{γ_{i} + k}{ρ_{m_{i}}}

(35)

where the modal density of that component is $ρ_{m_{i}}$ ⁠. γ_i refers to the location of a coupled natural frequency between the two nearest component natural frequencies, as illustrated in Fig. 7.

Fig. 7

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Illustration of the definition of γ_i, where a given ω_M is illustrated by a solid vertical line and the adjacent component natural frequencies $ω_{m_{i}}$ are illustrated by the dashed vertical lines. The horizontal axis of this one-dimensional plot is frequency.

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:

Σ_{i} \approx \sum_{k = - \infty}^{\infty} \frac{1}{{[{(ω_{M} + \frac{γ_{i} + k}{ρ_{m_{i}}})}^{2} - ω_{M}^{2}]}^{2} + 4 {(ζ ω)}_{m_{i}}^{2} ω_{M}^{2}} \approx \frac{ρ_{m_{i}}^{2}}{4 ω_{M}^{2}} \sum_{k = - \infty}^{\infty} \frac{1}{{(γ_{i} + k)}^{2} + {(ζ ω)}_{m_{i}}^{2} ρ_{m_{i}}^{2}}

(36)

Using these substitutions, the sums within the overall transfer functions become sums over k rather than m_i, and have single-term expressions

Σ_{i} \approx - \frac{π ρ_{m_{i}}}{4 {(ζ ω)}_{m_{i}} ω_{M}^{2}} \frac{\sinh [2 π {(ζ ω)}_{m_{i}} ρ_{m_{i}}]}{cos (2 π γ_{i}) - \cosh [2 π {(ζ ω)}_{m_{i}} ρ_{m_{i}}]} = \frac{1}{ω_{M}^{2}} σ_{i} Σ_{R i} \approx \frac{π ρ_{m_{i}}}{2 ω_{M}} \frac{sin (2 π γ_{i})}{cos (2 π γ_{i}) - \cosh [2 π {(ζ ω)}_{m_{i}} ρ_{m_{i}}]} = \frac{1}{ω_{M}} σ_{R i} Σ_{G i} \approx \frac{π ρ_{m_{i}}}{2 ω_{M}} \frac{\sinh [2 π {(ζ ω)}_{m_{i}} ρ_{m_{i}}]}{cos (2 π γ_{i}) - \cosh [2 π {(ζ ω)}_{m_{i}} ρ_{m_{i}}]} = \frac{1}{ω_{M}} σ_{G i} Σ_{ζ i} \approx \frac{π^{2} ρ_{m_{i}}^{2}}{4 ω_{M}^{2}} \csc^{2} (π γ_{i}) = \frac{1}{ω_{M}^{2}} σ_{ζ i}

(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 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.

Fig. 8

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Relative distribution of plates 1 (solid) and 2 (dashed) natural frequencies assuming a constant modal density. The horizontal axis of this one-dimensional plot is frequency. Recall that exactly one coupled natural frequency occurs between two component natural frequencies of any component, or rather, only one coupled natural frequency exists between any two adjacent dashed lines in this plot. It can be seen that there are few cases where one might place a coupled natural frequency such that γ₁ and γ₂ are simultaneously close to 1 or simultaneously close to 0.

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 γ_i is periodic

γ_{2} = γ_{1} + \frac{1}{2}

(38)

Or, for many components, it follows that:

γ_{i} = γ_{1} + \frac{i - 1}{N_{comp}}

(39)

where

N_{comp}

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

\begin{matrix} K_{1} & \equiv \frac{σ_{R 2}^{2} + σ_{G 2}^{2}}{{(\frac{M_{p_{2}}}{M_{p_{1}}} σ_{R 1} + σ_{R 2})}^{2} + {(\frac{M_{p_{2}}}{M_{p_{1}}} σ_{G 1} + σ_{G 2})}^{2}} \\ K_{2} & \equiv \frac{σ_{R 1}^{2} + σ_{G 1}^{2}}{{(σ_{R 1} + \frac{M_{p_{1}}}{M_{p_{2}}} σ_{R 2})}^{2} + {(σ_{G 1} + \frac{M_{p_{1}}}{M_{p_{2}}} σ_{G 2})}^{2}} \end{matrix}

(40)

Then, the mean-square response of the system may be written as

\begin{matrix} {\bar{w}}_{q_{1}}^{2} \approx & \sum_{M} π ζ_{M} ω_{M} Φ_{F} K_{1} \frac{Γ}{M_{p_{1}}^{2}} \frac{1}{ω_{M}^{2}} σ_{1} \\ {\bar{w}}_{q_{2}}^{2} \approx & \sum_{M} π ζ_{M} ω_{M} Φ_{F} K_{2} \frac{1}{M_{p_{2}}^{2}} \frac{1}{ω_{M}^{2}} σ_{2} \end{matrix}

(41)

where

ζ_{M} ω_{M} \approx \frac{{(ζ ω)}_{m_{1}} σ_{ζ 1} + {(ζ ω)}_{m_{2}} σ_{ζ 2}}{σ_{ζ 1} + σ_{ζ 2}}

(42)

However, as

ζ_{M} ω_{M}

may be replaced with an expression independent of ω_M, and as the frequency is large, we may replace the remaining dependence on ω_M by its average value

\frac{1}{ω_{max} - ω_{min}} \int_{ω_{min}}^{ω_{max}} \frac{1}{ω_{M}^{2}} d ω_{M} = \frac{1}{ω_{min} ω_{max}}

(43)

Let us define a reference frequency

ω_{r} \equiv \sqrt{ω_{min} ω_{max}}

(44)

Then, the mean-square response of points throughout the system may be captured by

{\bar{w}}_{q_{1}}^{2} \approx π ζ_{M} ω_{M} {\bar{F}}^{2} \frac{Δ M}{Δ ω} \frac{Γ}{M_{p_{1}}^{2} ω_{r}^{2}} σ_{1} K_{1} {\bar{w}}_{q_{2}}^{2} \approx π ζ_{M} ω_{M} {\bar{F}}^{2} \frac{Δ M}{Δ ω} \frac{1}{M_{p_{2}}^{2} ω_{r}^{2}} σ_{2} K_{2}

(45)

where the modal density may be calculated [7] by

ρ_{m_{i}} = \frac{Δ M_{i}}{Δ ω} = \frac{A_{p_{i}}}{4 π} \sqrt{\frac{m_{p_{i}}}{D_{i}}}

(46)

the overall modal density is

\frac{Δ M}{Δ ω} = ρ_{m_{1}} + ρ_{m_{2}} or \sum_{i = 1}^{N_{comp}} ρ_{m_{i}}

(47)

and the number of points used to construct γ_i is found by

Δ M_{i} = Round [\frac{A_{p_{i}} Δ ω}{4 π} \sqrt{\frac{m_{p_{i}}}{D_{i}}}]

(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.

Table 1

Parameter set for point-coupled parallel plates excitation sweep

Parameter	Value	Unit
$l_{x_{1}}$	0.9425	m
$l_{y_{1}}$	0.7639	m
$l_{x_{2}}$	0.9549	m
$l_{y_{2}}$	1.1519	m
$x_{1, 0}$	0.1980	m
$y_{1, 0}$	0.5107	m
$x_{1, F}$	0.5850	m
$y_{1, F}$	0.2918	m
$x_{2, 0}$	0.3048	m
$y_{2, 0}$	0.3850	m
h₁	2.000	mm
h₂	1.600	mm
$ζ_{1_{1}}$	2.5	%
$ζ_{1_{2}}$	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.

Table 2

Parameter set for point-coupled parallel plates excitation sweep

Parameter	Value	Unit
$ρ_{m_{1}}$	0.0189	s
$ρ_{m_{2}}$	0.0362	s
${(ζ ω)}_{m_{1}}$	1.6959	rad/s
${(ζ ω)}_{m_{2}}$	1.3815	rad/s
$M_{p_{1}}$	11.14	kg
$M_{p_{2}}$	13.62	kg
${AR}_{p_{1}}$	1.234	—
${AR}_{p_{2}}$	0.829	—

Figures 9–12 illustrate the accuracy of AMA relative to CMA-FD. Specifically, the plots show $(({({\bar{w}}_{q i}^{2})}_{AMA}) / ({({\bar{w}}_{q_{i}}^{2})}_{CMA - FD}))$ for different cases of excitation.

Fig. 9

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Ratio of predictions of AMA to CMA-FD for the plate 1 excitation point response experiencing excitation from various center frequencies and bandwidths, exciting different numbers of modes in plate 1. Center frequencies used range from 1 kHz to 100 kHz exciting prescribed numbers of modes. The bandwidths used to capture the array of modes illustrated on the x-axis are 30 Hz, 50 Hz, 100 Hz, 300 Hz, 500 Hz, and 1000 Hz.

Fig. 10

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Ratio of predictions of AMA to CMA-FD for a sample of the plate 1 hot line response experiencing excitation from various center frequencies and bandwidths, exciting different numbers of modes in plate 1. Center frequencies used range from 1 kHz to 100 kHz exciting prescribed numbers of modes. The bandwidths used to capture the array of modes illustrated on the x-axis are 30 Hz, 50 Hz, 100 Hz, 300 Hz, 500 Hz, and 1000 Hz.

Fig. 11

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Ratio of predictions of AMA to CMA-FD for the response of a nonintensified point on plate 1 experiencing excitation from various center frequencies and bandwidths, exciting different numbers of modes in plate 1. Center frequencies used range from 1 kHz to 100 kHz exciting prescribed numbers of modes. The bandwidths used to capture the array of modes illustrated on the x-axis are 30 Hz, 50 Hz, 100 Hz, 300 Hz, 500 Hz, and 1000 Hz.

Fig. 12

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Ratio of predictions of AMA to CMA-FD for the response of a point on plate 2 experiencing excitation from various center frequencies and bandwidths, exciting different numbers of modes in plate 2. Center frequencies used range from 1 kHz to 100 kHz exciting prescribed numbers of modes. The bandwidths used to capture the array of modes illustrated on the x-axis are 30 Hz, 50 Hz, 100 Hz, 300 Hz, 500 Hz, and 1000 Hz.

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.

Fig. 13

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Ratio of computer runtimes for the two-plate system experiencing excitation from various center frequencies and bandwidths, exciting different numbers of total system modes. Center frequencies used range from 1 kHz to 100 kHz exciting prescribed numbers of modes. The bandwidths used to capture the array of modes illustrated on the x-axis are 30 Hz, 50 Hz, 100 Hz, 300 Hz, 500 Hz, and 1000 Hz.

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., $ω_{2, 0}$ comes from plate 1, $ω_{2, 1}$ from plate 2, $ω_{2, 2}$ 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.
$a_{m_{i}}$ =: plate 1 time-dependent modal amplitude
$A_{m_{i}}$ =: plate i frequency-dependent modal amplitude
$A_{p_{i}}$ =: area of plate i
D_i =: plate stiffness for plate i
F =: time-dependent input force
F₀ =: input force amplitude
G_i =: magnitude of the imaginary part of the characteristic equation for plate i
h_i =: thickness of plate i
$H_{A_{m_{i}}}$ =: modal amplitude transfer function for plate i
$H_{q_{i}}$ =: transfer function for a query point on plate i
$H_{λ}$ =: constraint transfer function
J_i =: modified transfer function expression
k =: index of mode natural frequencies
$l_{x_{i}}$ =: length of plate i in the x-direction
$l_{y_{i}}$ =: length of plate i in the y-direction
$m_{p_{i}}$ =: mass per unit area of plate i
$M_{m_{i}}$ =: modal mass of plate i
$M_{p_{i}}$ =: plate i mass
R_i =: magnitude of the real part of the characteristic equation for plate i
$S_{m_{i}}$ =: modal characteristic equation of plate i
t =: time
w_i =: plate 1 transverse displacement
$w_{q, i}$ =: time-dependent transverse displacement of a query point on plate i
x_i =: x-coordinate on plate i
$x_{i, 0}$ =: x-coordinate of the coupling on plate i
$x_{1, F}$ =: x-coordinate of the excitation on plate 1
y_i =: y-coordinate on plate i
$y_{i, 0}$ =: y-coordinate of the coupling on plate i
$y_{1, F}$ =: 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
$Δ m_{i}$ =: number of component modes in plate i
$Δ ω$ =: excitation bandwidth
ζ_M =: damping ratio of the coupled system
$ζ_{m_{i}}$ =: modal damping ratio of plate i
${(ζ ω)}_{i}$ =: damping-natural frequency constant of component i
${(ζ ω)}_{i}$ =: 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
$ρ_{m_{i}}$ =: modal density of plate i
Σ_i =: shorthand for the transfer function mode sum of plate i
$Φ_{F}$ =: excitation power spectrum
$Ψ_{0 F}$ =: mode shape-dependent coefficient for the constraint transfer function numerator
$Ψ_{i 0}$ =: mode shape-dependent coefficient for the constraint transfer function denominator relative to plate i
$ψ_{m_{i}}$ =: mode shape of plate i
$ψ_{m_{i} 0}$ =: mode shape of plate i evaluated at the coupling point
$ψ_{m_{i} F}$ =: mode shape of plate i evaluated at the excitation point
ω =: frequency
ω_M =: natural frequency of the coupled system
${\tilde{ω}}_{M}$ =: damped natural frequency of the coupled system
$ω_{m_{i}}$ =: natural frequency of plate i

Appendix

Consider the expression

{| H_{λ} |}^{2} \sum_{m_{1}} \frac{ψ_{m_{1} q}^{2} ψ_{m_{1} 0}^{2}}{M_{m_{1}}^{2} {| S_{m_{1}} |}^{2}} + (H_{λ} + {\bar{H}}_{λ}) \sum_{m_{1}} \frac{ψ_{m_{1} q}^{2} ψ_{m_{1} 0} ψ_{m_{1} F}}{M_{m_{1}}^{2} {| S_{m_{1}} |}^{2}} + \sum_{m_{1}} \frac{ψ_{m_{1} q}^{2} ψ_{m_{1} F}^{2}}{M_{m_{1}}^{2} {| S_{m_{1}} |}^{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:

\begin{matrix} Term 1 = \sum_{m_{1}} \frac{ψ_{m_{1} q}^{2} ψ_{m_{1} 0}^{2}}{M_{m_{1}}^{2} {| S_{m_{1}} |}^{2}} [{(\sum_{m_{1}} \frac{ψ_{m_{1} 0} ψ_{m_{1} F} R_{1}}{M_{m_{1}} {| S_{m_{1}} |}^{2}})}^{2} + {(\sum_{m_{1}} \frac{ψ_{m_{1} 0} ψ_{m_{1} F} G_{1}}{M_{m_{1}} {| S_{m_{1}} |}^{2}})}^{2}] Term 2 = - 2 \sum_{m_{1}} \frac{ψ_{m_{1} q}^{2} ψ_{m_{1} 0} ψ_{m_{1} F}}{M_{m_{1}}^{2} {| S_{m_{1}} |}^{2}} [\sum_{m_{1}} \frac{ψ_{m_{1} 0} ψ_{m_{1} F} R_{1}}{M_{m_{1}} {| S_{m_{1}} |}^{2}} (\sum_{m_{1}} \frac{ψ_{m_{1} 0}^{2} R_{1}}{M_{m_{1}} {| S_{m_{1}} |}^{2}} + \sum_{m_{2}} \frac{ψ_{m_{2} 0}^{2} R_{2}}{M_{m_{2}} {| S_{m_{2}} |}^{2}}) + \sum_{m_{1}} \frac{ψ_{m_{1} 0} ψ_{m_{1} F} G_{1}}{M_{m_{1}} {| S_{m_{1}} |}^{2}} (\sum_{m_{1}} \frac{ψ_{m_{1} 0}^{2} G_{1}}{M_{m_{1}} {| S_{m_{1}} |}^{2}} + \sum_{m_{2}} \frac{ψ_{m_{2} 0}^{2} G_{2}}{M_{m_{2}} {| S_{m_{2}} |}^{2}})] Term 3 = \sum_{m_{1}} \frac{ψ_{m_{1} q}^{2} ψ_{m_{1} F}^{2}}{M_{m_{1}}^{2} {| S_{m_{1}} |}^{2}} [{(\sum_{m_{1}} \frac{ψ_{m_{1} 0}^{2} R_{1}}{M_{m_{1}} {| S_{m_{1}} |}^{2}} + \sum_{m_{2}} \frac{ψ_{m_{2} 0}^{2} R_{2}}{M_{m_{2}} {| S_{m_{2}} |}^{2}})}^{2} \\ + {(\sum_{m_{1}} \frac{ψ_{m_{1} 0}^{2} G_{1}}{M_{m_{1}} {| S_{m_{1}} |}^{2}} + \sum_{m_{2}} \frac{ψ_{m_{2} 0}^{2} G_{2}}{M_{m_{2}} {| S_{m_{2}} |}^{2}})}^{2}] \end{matrix}

(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:

\begin{matrix} \sum_{m_{1}} \frac{ψ_{m_{1} q}^{2} ψ_{m_{1} 0}^{2}}{M_{m_{1}}^{2} {| S_{m_{1}} |}^{2}} {(\sum_{m_{1}} \frac{ψ_{m_{1} 0} ψ_{m_{1} F} R_{1}}{M_{m_{1}} {| S_{m_{1}} |}^{2}})}^{2} \\ - 2 \sum_{m_{1}} \frac{ψ_{m_{1} q}^{2} ψ_{m_{1} 0} ψ_{m_{1} F}}{M_{m_{1}}^{2} {| S_{m_{1}} |}^{2}} \sum_{m_{1}} \frac{ψ_{m_{1} 0} ψ_{m_{1} F} R_{1}}{M_{m_{1}} {| S_{m_{1}} |}^{2}} \sum_{m_{1}} \frac{ψ_{m_{1} 0}^{2} R_{1}}{M_{m_{1}} {| S_{m_{1}} |}^{2}} \\ + \sum_{m_{1}} \frac{ψ_{m_{1} q}^{2} ψ_{m_{1} F}^{2}}{M_{m_{1}}^{2} {| S_{m_{1}} |}^{2}} {(\sum_{m_{1}} \frac{ψ_{m_{1} 0}^{2} R_{1}}{M_{m_{1}} {| S_{m_{1}} |}^{2}})}^{2} \approx 0 \end{matrix}

(A3)

\begin{matrix} 2 \sum_{m_{1}} \frac{ψ_{m_{1} q}^{2} ψ_{m_{1} F}^{2}}{M_{m_{1}}^{2} {| S_{m_{1}} |}^{2}} \sum_{m_{1}} \frac{ψ_{m_{1} 0}^{2} R_{1}}{M_{m_{1}} {| S_{m_{1}} |}^{2}} \sum_{m_{2}} \frac{ψ_{m_{2} 0}^{2} R_{2}}{M_{m_{2}} {| S_{m_{2}} |}^{2}} \\ - 2 \sum_{m_{1}} \frac{ψ_{m_{1} q}^{2} ψ_{m_{1} 0} ψ_{m_{1} F}}{M_{m_{1}}^{2} {| S_{m_{1}} |}^{2}} \sum_{m_{1}} \frac{ψ_{m_{1} 0} ψ_{m_{1} F} R_{1}}{M_{m_{1}} {| S_{m_{1}} |}^{2}} \sum_{m_{2}} \frac{ψ_{m_{2} 0}^{2} R_{2}}{M_{m_{2}} {| S_{m_{2}} |}^{2}} \approx 0 \end{matrix}

(A4)

with similar equalities where the R terms in the numerators are replaced with the G terms. The remaining nonzero terms are

\sum_{m_{1}} \frac{ψ_{m_{1} q}^{2} ψ_{m_{1} F}^{2}}{M_{m_{1}}^{2} {| S_{m_{1}} |}^{2}} [{(\sum_{m_{2}} \frac{ψ_{m_{2} 0}^{2} R_{2}}{M_{m_{2}} {| S_{m_{2}} |}^{2}})}^{2} + {(\sum_{m_{2}} \frac{ψ_{m_{2} 0}^{2} G_{2}}{M_{m_{2}} {| S_{m_{2}} |}^{2}})}^{2}]

(A5)

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

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High Frequency Analysis of a Point-Coupled Parallel Plate System

Introduction

Summary of Analytical Methods

Component Equation Solutions Via Eigenmode Expansion.

Constraint and Modal Coordinate Transfer Functions.

Coupled Natural Frequencies and Damping.

Useful Forms of the Constraint Transfer Function.

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

Discrete Classical Modal Analysis (dCMA).

Asymptotic Modal Analysis.

Modal Averaging.

Frequency Reduction.

Results

Conclusions

Acknowledgment

Nomenclature

Appendix

References

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High Frequency Analysis of a Point-Coupled Parallel Plate System

Introduction

Summary of Analytical Methods

Component Equation Solutions Via Eigenmode Expansion.

Constraint and Modal Coordinate Transfer Functions.

Coupled Natural Frequencies and Damping.

Useful Forms of the Constraint Transfer Function.

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

Discrete Classical Modal Analysis (dCMA).

Asymptotic Modal Analysis.

Modal Averaging.

Frequency Reduction.

Results

Conclusions

Acknowledgment

Nomenclature

Appendix

References

Get Email Alerts

Cited By

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