Finite-Element Harmonic Analysis of Sandwich Plates with Periodically Modulated Faces and a Viscoelastic Core

1. Introduction

Sandwich plates with a thin viscoelastic core are a standard means of reducing flexural vibration. The faces constrain the core, so bending drives the core into shear and dissipates energy. That mechanism is known as constrained-layer damping (CLD). In buildings a sandwich of this kind is used as a lightweight floor or a wall panel of one bay. The faces are a stiff isotropic polymer composite and the core is a soft viscoelastic polymer. These are modelling values for a structural sandwich, not a named commercial product and not a sub-millimetre damping tape. The viscoelastic core attenuates flexural vibration of that floor or partition under base motion. Kerwin [1] and DiTaranto [2] identified it for beams. Mead and Markus [3] then gave the three-layer kinematics that still underlie most forced-vibration analyses of the stack. Yan and Dowell [4] extended the governing equations to sandwich plates and beams. Rao [5] gave the frequencies and loss factors of sandwich beams for the principal boundary conditions. Finite-element implementations follow the same template, with the core represented either by a complex modulus [6,7,8,9] or by a frequency-domain viscous or fractional operator [10,11,12,13]. Layerwise and zig-zag theories refine the through-thickness kinematics when the core is thick or compressible [14,15,16,17,19,20,21]. Recent plate-level studies still treat a uniform or piecewise-uniform stack: spectral collocation of a viscoelastic sandwich [22], embedded viscoelastic plies [23], soft-core delamination [24], auxetic cores [25], functionally graded constraining layers [26], and active or hybrid CLD and layout optimisation [27,28,29,30,31]. Reviews of viscoelastic damping and of hybrid active–passive treatments set the broader design context [32,33,34].

Those CLD models treat each layer as uniform, or at most piecewise uniform, in the plane. A second literature treats periodicity as a wave-control device. Phononic crystals, locally resonant plates, and ribbed or patched waveguides are analysed through dispersion diagrams and attenuation bands [35,36,37,38,39,40,41,42]. Honeycomb, truss and other discrete cores introduce periodicity through a microstructure [43,44,45], not through a prescribed continuum field on an otherwise homogeneous sandwich. Viscoelastic cores that are periodic by cells do produce stop bands on sandwich beams and plates [46,47,48,49]. Cupiał and Nizioł [50] analysed a viscoelastic mid-layer that remains uniform. The two streams therefore leave an intermediate case: a sandwich plate whose face thicknesses and core shear moduli vary slowly across the planform. That case has already been reduced by a Navier–Galerkin projection of the coupled face-membrane and core-shear equations, first on a sandwich beam with harmonic parameters [51] and then on simply supported sandwich plates with cosine-modulated faces, a viscoelastic core and distributed supported masses [52,53,54]. Those reductions supply frequency-response operators, but they remain tied to a Navier basis on a single rectangle.

That slow modulation is not a phononic crystal. Its wavelength is comparable to the plate, so a coarse structural mesh cannot resolve a Bragg or local-resonance gap, and none should be claimed. It is nonetheless strong enough to change the area-averaged stiffness, the local shear lever arm between the faces, and therefore the forced frequency-response function (FRF). Such fields arise in tapered faces, in a core whose thickness or storage modulus is patterned on the scale of the plate, and as the leading long-wave residual of an imperfect lay-up. A designer who must place or suppress the first few resonances under base motion then needs that FRF: which peaks move, and by how much. A Bloch band diagram does not answer that question.

The present paper addresses the intermediate case with a rectangular finite element that can be assembled on an arbitrary mesh and mass layout. The element is reproduced from the stated kinematics and interpolation: Kirchhoff faces, first-order shear in a Kelvin–Voigt core, bilinear membrane fields, Adini–Clough–Melosh bending [8,55], and cosine modulation of both faces and core. The harmonic problem is solved under base acceleration and attached point masses. Before any parametric scan, a Navier–Galerkin sine series of the same family [52,53] verifies the element on two single-mass plates that share the planform and the total attached mass of the later campaigns: one plate is cosine-modulated and one is uniform. Three archives then isolate independent controls. The first varies the core wavenumbers and the core thickness. The second varies the four face wavenumbers, the relative thickness amplitude, and the two mean face thicknesses. The third relocates a fixed total attached mass on the same plate. The first three centre-point FRF peaks are reported as grid values. That separation is the novelty of the study, and it is numerical. Uniform CLD elements do not carry an in-plane cosine field, and phononic analyses do not give the first forced peaks of a finite modulated plate. The Galerkin reductions do both, but only on one simply supported rectangle. The element is not tied to that basis: the same matrices accept any rectangular mesh and any set of nodal masses. No experiment is used to support these comparisons.

The numerical scope is a soft core with Young’s modulus neglected ($E_v=0$), omitted rotatory inertia, a Kelvin–Voigt law whose loss factor grows linearly with frequency [6,12,56], and a rectangular mesh on which the reported spectra have converged. The cosine fields are not claimed to produce phononic stop bands, and no experiment is reported.

The remainder of the paper is organised as follows. Section 2 states the kinematics, the finite-element discretisation, the harmonic solve and the Galerkin series used to verify the element. Section 3 reports the Galerkin verification and the three parametric archives. Section 4 concludes.

2. Kinematics and Finite-Element Formulation

2.1. Kinematics and Constitutive Model

The plate occupies the rectangle $\Omega =[0,L]\times [0,W]$. Figure 1 shows the simply supported planform with nine equal point masses on a $3\times 3$ lattice. Figure 2 is the through-thickness stack: a top face of thickness $h_c(X,Y)$, a core of constant thickness $h_v$, and a bottom face of thickness $h_p(X,Y)$. The core mid-plane is $z=0$, so the core occupies $z\in [-h_v/2,h_v/2]$. The face mid-planes are

$$z_p=-{\displaystyle \frac{h_v+h_p}{2}},z_c=+{\displaystyle \frac{h_v+h_c}{2}}.$$

(1)

The bottom-face/core interface is $z=-h_v/2$ and the top-face/core interface is $z=+h_v/2$. Subscripts c, v and p denote the top (constrained) face, the viscoelastic core and the bottom face.

Face thicknesses and the core shear moduli vary periodically. In global coordinates $(X,Y)$,

$$h_c=c_{1m}+b_{1r}f(k_1,k_2;X,Y),$$

(2)

$$h_p=c_{3m}+b_{3r}f(k_3,k_4;X,Y),$$

(3)

$$G_r=e_{a1}+b_{a1}f(k_a,k_b;X,Y),$$

(4)

$$G_i=e_{c1}+b_{c1}f(k_a,k_b;X,Y),$$

(5)

$$f(k_x,k_y;X,Y)=cos\left({\displaystyle \frac{2k_x\pi }{L}}\left(X-{\displaystyle \frac{L}{2}}\right)\right)cos\left({\displaystyle \frac{2k_y\pi }{W}}\left(Y-{\displaystyle \frac{W}{2}}\right)\right).$$

(6)

The four face wavenumbers are independent: $k_1,k_2$ act only on $h_c$, and $k_3,k_4$ act only on $h_p$. If a wavenumber vanishes, the corresponding cosine is identically 1, so that face varies in at most one direction; if both wavenumbers of a face vanish, that face has the uniform thickness $c+b$. The storage and loss coefficients of the core share the pair $(k_a,k_b)$. Thus $e_{a1}$ and $b_{a1}$ have the dimension of stress, while $e_{c1}$ and $b_{c1}$ have the dimension of stress times time. If every amplitude vanishes, the plate is uniform. Element integrals use a local frame $(x,y)$ centred on the element, with $X=x+x_0^{\left(e\right)}$ and $Y=y+y_0^{\left(e\right)}$.

Figure 3 shows f for two isotropic pairs. The area average of f on $\Omega$ is $\mu (k_x,k_y)=\sigma \left(k_x\right)\sigma \left(k_y\right)$, where $\sigma \left(0\right)=1$ and $\sigma \left(k\right)=sin\left(k\pi \right)/\left(k\pi \right)$ for $k\ne 0$. Integer wavenumbers therefore leave the mean modulus or mean thickness unchanged, while $k_x=k_y=1/2$ gives $\mu ={(2/\pi )}^2\approx 0.405$. The cubic mean $\langle h^3\rangle$ that enters bending stiffness still depends on the amplitude when $\mu =0$, because $\langle f^2\rangle >0$.

The kinematic and constitutive hypotheses are as follows. Deformations are small and the strain–displacement relations are linear. The three layers are perfectly bonded and share a z-independent deflection w. Each face is an isotropic Kirchhoff plate in plane stress; face shear is neglected. The core in-plane displacement is linear in z (first-order shear). The core shear law is Kelvin–Voigt,

$$\mathsf{\tau }=G_r\gamma +G_i\dot{\gamma },$$

(7)

with $\mathsf{\tau }={[{\tau }_{xz},{\tau }_{yz}]}^{\mathrm{T}}$ and $\gamma ={[{\gamma }_{xz},{\gamma }_{yz}]}^{\mathrm{T}}$. $E_{\alpha }$ is the Young’s modulus of layer $\alpha$, that is the in-plane direct stiffness, and ${\nu }_{\alpha }$ is Poisson’s ratio. $E_v$ is that modulus for the core. It is not a shear modulus: core shear is carried by $G_r$ and $G_i$. The formulation retains core membrane and bending stiffness when $E_v\ne 0$. In the examples that stiffness is neglected: $E_v$ changes the reported spectra only negligibly, and it is set to zero. Kinetic energy retains mid-plane translation of each layer and omits rotatory inertia. Variable thickness is treated locally: interface continuity and $h^3/12$ use the thickness at the integration point.

The independent mid-plane fields are $(u_c,v_c)$, $(u_p,v_p)$ and w. The Kirchhoff rotations are

$${\theta }_x:={\displaystyle \frac{\partial w}{\partial y}},{\theta }_y:=-{\displaystyle \frac{\partial w}{\partial x}},$$

(8)

so ${\theta }_x$ and ${\theta }_y$ are not additional independent fields. A nodal generalised displacement is

$${\mathbf{d}}_i={\left[u_{ci},v_{ci},u_{pi},v_{pi},w_i,{\theta }_{xi},{\theta }_{yi}\right]}^{\mathrm{T}}.$$

(9)

Through the thickness of each face,

$$u^{\alpha }=u_{\alpha }-(z-z_{\alpha }){\displaystyle \frac{\partial w}{\partial x}},v^{\alpha }=v_{\alpha }-(z-z_{\alpha }){\displaystyle \frac{\partial w}{\partial y}},\alpha \in \{c,p\}.$$

(10)

Interface continuity at $z=\pm h_v/2$ then gives

$$u{|}_{z=h_v/2}=u_c+{\displaystyle \frac{h_c}{2}}{\displaystyle \frac{\partial w}{\partial x}},u{|}_{z=-h_v/2}=u_p-{\displaystyle \frac{h_p}{2}}{\displaystyle \frac{\partial w}{\partial x}},$$

(11)

and the analogous pair for v. Linear interpolation through the core, together with ${\gamma }_{xz}=\partial w/\partial x+\partial u^v/\partial z$, yields the Mead–Markus core fields

$$\begin{array}{cc}\hfill u_v& ={\displaystyle \frac{1}{2}}\left[u_c+u_p+{\displaystyle \frac{h_c-h_p}{2}}{\displaystyle \frac{\partial w}{\partial x}}\right],\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill v_v& ={\displaystyle \frac{1}{2}}\left[v_c+v_p+{\displaystyle \frac{h_c-h_p}{2}}{\displaystyle \frac{\partial w}{\partial y}}\right],\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill {\gamma }_{xz}& ={\displaystyle \frac{1}{h_v}}\left[u_c-u_p+\left({\displaystyle \frac{h_c+h_p}{2}}+h_v\right){\displaystyle \frac{\partial w}{\partial x}}\right],\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill {\gamma }_{yz}& ={\displaystyle \frac{1}{h_v}}\left[v_c-v_p+\left({\displaystyle \frac{h_c+h_p}{2}}+h_v\right){\displaystyle \frac{\partial w}{\partial y}}\right].\hfill \end{array}$$

(15)

The shear strain is produced by the relative in-plane slip of the faces together with the lever arm $(h_c+h_p)/2+h_v$, which is the distance between the face mid-planes. In terms of (8), $\partial w/\partial x=-{\theta }_y$ and $\partial w/\partial y={\theta }_x$.

Membrane strains and the common Kirchhoff curvature are the engineering measures [4,50]

$${\epsilon }_0^{\alpha }=\left[\begin{array}{c}{\displaystyle \frac{\partial u_{\alpha }}{\partial x}}\\ {\displaystyle \frac{\partial v_{\alpha }}{\partial y}}\\ {\displaystyle \frac{\partial u_{\alpha }}{\partial y}}+{\displaystyle \frac{\partial v_{\alpha }}{\partial x}}\end{array}\right],\kappa =\left[\begin{array}{c}{\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}\\ {\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}\\ 2{\displaystyle \frac{{\partial }^{2}w}{\partial x\partial y}}\end{array}\right],$$

(16)

with $\alpha \in \{c,p,v\}$ and $(u_v,v_v)$ from (12)–(13). The curvature $\kappa$ is the engineering curvature, linear in the second derivatives of w. That linearisation is the small-displacement hypothesis stated above: the nonlinear terms of the Green strain are omitted. Each isotropic layer uses the plane-stress matrix

$${\mathbf{D}}_{\alpha }={\displaystyle \frac{E_{\alpha }}{1-{\nu }_{\alpha }^2}}\left[\begin{array}{ccc}1& {\nu }_{\alpha }& 0\\ {\nu }_{\alpha }& 1& 0\\ 0& 0& (1-{\nu }_{\alpha })/2\end{array}\right],$$

(17)

membrane stiffness $h_{\alpha }{\mathbf{D}}_{\alpha }$ and bending stiffness ${\mathbf{D}}_{b\alpha }=h_{\alpha }^3{\mathbf{D}}_{\alpha }/12$. The leading diagonal entries $E_{\alpha }/(1-{\nu }_{\alpha }^2)$ are direct stiffnesses. The last diagonal entry is the in-plane shear stiffness $E_{\alpha }/\left[2(1+{\nu }_{\alpha })\right]$. ${\mathbf{D}}_{\alpha }$ is not the transverse shear stiffness of the core. The local strain in layer $\alpha$ is ${\epsilon }^{\alpha }={\epsilon }_0^{\alpha }-{\zeta }_{\alpha }\kappa$ with ${\zeta }_{\alpha }\in [-h_{\alpha }/2,h_{\alpha }/2]$.

2.2. Finite-Element Discretisation

A uniform $m\times n$ partition of $\Omega$ into rectangles of size $\Delta x=L/m$ and $\Delta y=W/n$ is the default mesh. If an attached mass does not lie on a node, an additional grid line is inserted through that station, so some rectangles are smaller than the nominal cell; the element below is written for a general rectangle. Each element has four nodes, ordered anticlockwise from the lower-left corner, and 28 degrees of freedom ${\delta }^e={[{\mathbf{d}}_1^{\mathrm{T}},{\mathbf{d}}_2^{\mathrm{T}},{\mathbf{d}}_3^{\mathrm{T}},{\mathbf{d}}_4^{\mathrm{T}}]}^{\mathrm{T}}$.

The four membrane fields $u_c$, $v_c$, $u_p$ and $v_p$ are interpolated with the complete bilinear monomials $[1,x,y,xy]$. The deflection uses the ACM incomplete quartic [8,55]

$$\begin{array}{cc}\hfill w& =a_1+a_{2}x+a_{3}y+a_4{x}^2+a_{5}xy+a_6{y}^2\hfill \\ \hfill & +a_7{x}^3+a_8{x}^{2}y+a_{9}x{y}^2+a_{10}{y}^3+a_{11}{x}^{3}y+a_{12}x{y}^3.\hfill \end{array}$$

(18)

The seven interpolated fields are collected in a $7\times 28$ monomial matrix $\mathbf{P}(x,y)$, whose last two rows are $\partial w/\partial y$ and $-\partial w/\partial x$ acting on the coefficients of (18). Collocating $\mathbf{P}$ at the four nodes $(\pm a,\pm b)$ produces the configuration matrix $\mathbf{C}\in {\mathbb{R}}^{28\times 28}$. The shape-function matrix is $\mathbf{N}=\mathbf{P}{\mathbf{C}}^{-1}$, with rows ${\mathbf{N}}_1,\dots ,{\mathbf{N}}_7$ so that $u_c={\mathbf{N}}_1{\delta }^e$, $v_c={\mathbf{N}}_2{\delta }^e$, $u_p={\mathbf{N}}_3{\delta }^e$, $v_p={\mathbf{N}}_4{\delta }^e$, $w={\mathbf{N}}_5{\delta }^e$, ${\theta }_x={\mathbf{N}}_6{\delta }^e$ and ${\theta }_y={\mathbf{N}}_7{\delta }^e$. The core interpolants follow from (12)–(15) and (8):

$${\mathbf{N}}_8=\frac{1}{2}\left[({\mathbf{N}}_1+{\mathbf{N}}_3)+\frac{h_c-h_p}{2}(-{\mathbf{N}}_7)\right],$$

(19)

$${\mathbf{N}}_9=\frac{1}{2}\left[({\mathbf{N}}_2+{\mathbf{N}}_4)+\frac{h_c-h_p}{2}{\mathbf{N}}_6\right],$$

(20)

$${\mathbf{N}}_{10}=\frac{1}{h_v}\left[({\mathbf{N}}_1-{\mathbf{N}}_3)+\left(\frac{h_c+h_p}{2}+h_v\right)(-{\mathbf{N}}_7)\right],$$

(21)

$${\mathbf{N}}_{11}=\frac{1}{h_v}\left[({\mathbf{N}}_2-{\mathbf{N}}_4)+\left(\frac{h_c+h_p}{2}+h_v\right){\mathbf{N}}_6\right],$$

(22)

hence $u_v={\mathbf{N}}_8{\delta }^e$, $v_v={\mathbf{N}}_9{\delta }^e$, ${\gamma }_{xz}={\mathbf{N}}_{10}{\delta }^e$ and ${\gamma }_{yz}={\mathbf{N}}_{11}{\delta }^e$.

ACM interpolation enforces ${\theta }_x=\partial w/\partial y$ and ${\theta }_y=-\partial w/\partial x$ inside the element, but the normal rotation is not continuous across element edges. The element is therefore a nonconforming thin-plate rectangle. Membrane fields are $C^0$.

The strain–displacement matrices are

$$\begin{array}{cc}\hfill {\mathbf{B}}_p^c& =\left[\begin{array}{c}{\partial }_x{\mathbf{N}}_1\\ {\partial }_y{\mathbf{N}}_2\\ {\partial }_y{\mathbf{N}}_1+{\partial }_x{\mathbf{N}}_2\end{array}\right],{\mathbf{B}}_p^p=\left[\begin{array}{c}{\partial }_x{\mathbf{N}}_3\\ {\partial }_y{\mathbf{N}}_4\\ {\partial }_y{\mathbf{N}}_3+{\partial }_x{\mathbf{N}}_4\end{array}\right],\hfill \\ \hfill {\mathbf{B}}_p^v& =\left[\begin{array}{c}{\partial }_x{\mathbf{N}}_8\\ {\partial }_y{\mathbf{N}}_9\\ {\partial }_y{\mathbf{N}}_8+{\partial }_x{\mathbf{N}}_9\end{array}\right],{\mathbf{B}}_b=\left[\begin{array}{c}{\partial }_{xx}{\mathbf{N}}_5\\ {\partial }_{yy}{\mathbf{N}}_5\\ 2{\partial }_{xy}{\mathbf{N}}_5\end{array}\right],{\mathbf{B}}_s=\left[\begin{array}{c}{\mathbf{N}}_{10}\\ {\mathbf{N}}_{11}\end{array}\right].\hfill \end{array}$$

(23)

When $h_c$ and $h_p$ vary, ${\mathbf{N}}_8$ and ${\mathbf{N}}_9$ depend on thickness, so ${\mathbf{B}}_p^v$ contains $\partial h/\partial x$ and $\partial h/\partial y$ terms. The shear rows ${\mathbf{N}}_{10}$ and ${\mathbf{N}}_{11}$ use the local thicknesses but not their derivatives. That is the local-thickness approximation stated with those hypotheses.

On the local domain ${\omega }^e=[-a,a]\times [-b,b]$, with a and b equal to half the element sides, the element matrices are

$$\begin{array}{cc}\hfill {\mathbf{m}}^e=& {\int }_{{\omega }^e}{\rho }_c{h}_c\left({\mathbf{N}}_1^{\mathrm{T}}{\mathbf{N}}_1+{\mathbf{N}}_2^{\mathrm{T}}{\mathbf{N}}_2+{\mathbf{N}}_5^{\mathrm{T}}{\mathbf{N}}_5\right)\mathrm{d}\omega \hfill \\ & +{\int }_{{\omega }^e}{\rho }_p{h}_p\left({\mathbf{N}}_3^{\mathrm{T}}{\mathbf{N}}_3+{\mathbf{N}}_4^{\mathrm{T}}{\mathbf{N}}_4+{\mathbf{N}}_5^{\mathrm{T}}{\mathbf{N}}_5\right)\mathrm{d}\omega \hfill \\ & +{\int }_{{\omega }^e}{\rho }_v{h}_v\left({\mathbf{N}}_8^{\mathrm{T}}{\mathbf{N}}_8+{\mathbf{N}}_9^{\mathrm{T}}{\mathbf{N}}_9+{\mathbf{N}}_5^{\mathrm{T}}{\mathbf{N}}_5\right)\mathrm{d}\omega ,\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill {\mathbf{k}}^e=& \sum _{\alpha \in \{c,p,v\}}{\int }_{{\omega }^e}\left(h_{\alpha }{\left({\mathbf{B}}_p^{\alpha }\right)}^{\mathrm{T}}{\mathbf{D}}_{\alpha }{\mathbf{B}}_p^{\alpha }+{\left({\mathbf{B}}_b\right)}^{\mathrm{T}}{\mathbf{D}}_{b\alpha }{\mathbf{B}}_b\right)\mathrm{d}\omega \hfill \\ & +{\int }_{{\omega }^e}{h}_v{G}_r{\mathbf{B}}_s^{\mathrm{T}}{\mathbf{B}}_s\mathrm{d}\omega ,\hfill \end{array}$$

(25)

$${\mathbf{c}}^e={\int }_{{\omega }^e}{h}_v{G}_i{\mathbf{B}}_s^{\mathrm{T}}{\mathbf{B}}_s\mathrm{d}\omega .$$

(26)

If $E_v=0$, the core membrane and bending blocks vanish and ${\mathbf{B}}_p^v$ is not formed. The only damping in ${\mathbf{c}}^e$ is core shear viscosity. A base acceleration $S_0=1\mathrm{m}/{\mathrm{s}}^2$ produces the consistent inertial load

$${\mathbf{f}}^e={\int }_{{\omega }^e}{\mathbf{N}}_5^{\mathrm{T}}({\rho }_c{h}_c+{\rho }_p{h}_p+{\rho }_v{h}_v)\mathrm{d}\omega .$$

(27)

The integrals are evaluated by four-point Gauss–Legendre quadrature in each local direction, with $h_c$, $h_p$, $G_r$ and $G_i$ sampled at the Gauss points. The element matrices, the assembly and the harmonic solve are an in-house implementation in MATLAB. No commercial finite-element package is used. The global matrices $\mathbf{M}$, $\mathbf{C}$, $\mathbf{K}$ and $\mathbf{F}$ are assembled by the standard element-to-global map [8,9].

All four edges are simply supported. On $X=0$ and $X=L$ the essential conditions are $v_c=v_p=w={\theta }_x=0$; on $Y=0$ and $Y=W$ they are $u_c=u_p=w={\theta }_y=0$. The in-plane displacement normal to the edge and the rotation associated with the normal slope remain free (moment-free natural condition). Constrained degrees of freedom j are treated by zeroing the corresponding rows and columns, setting $K_{jj}=1$ and $M_{jj}=1/\left({\omega }_{max}^2{10}^{10}\right)$, where ${\omega }_{max}^2$ is the largest unconstrained ratio of diagonal stiffness to diagonal mass; the associated damping diagonal is left at zero.

Each attached mass $m_k$ at a mesh node ${\mathbf{x}}_k$ is added to the five translational entries of that node’s $7\times 7$ block (rotations receive no mass), and $m_k$ is added only to the transverse force for $S_0=1 \mathrm{m}/{\mathrm{s}}^2$. Those five entries are $u_c$, $v_c$, $u_p$, $v_p$ and w, and each receives the full $m_k$. The in-plane inertia is therefore assigned to both faces. That choice is intentional: each face receives the full $m_k$, rather than a share of one mass or a fixture bonded to one face only. The transverse entry receives $m_k$ once, and that is the inertia which shifts the base-driven flexural spectrum. If the two faces have the same in-plane velocity $\dot{u}$, the two in-plane entries together store kinetic energy $m_k{\dot{u}}^2$, twice the energy of a single rigid mass $m_k$ moving with one face. The tabulated spectra use this five-entry assembly and have not been recomputed with a one-face attachment. If ${\mathbf{x}}_k$ does not coincide with an existing node, an additional grid line is inserted so that every mass sits on a node.

2.3. Harmonic Analysis

After assembly the discrete system is

$$\mathbf{M}\ddot{\mathbf{U}}+\mathbf{C}\dot{\mathbf{U}}+\mathbf{K}\mathbf{U}=\mathbf{F}{S}_0{\mathrm{e}}^{\mathrm{i}\omega t}.$$

(28)

The steady state $\widehat{\mathbf{U}}\left(\omega \right){\mathrm{e}}^{\mathrm{i}\omega t}$ satisfies

$$\left(\mathbf{K}+\mathrm{i}\omega \mathbf{C}-{\omega }^2\mathbf{M}\right)\widehat{\mathbf{U}}\left(\omega \right)=\mathbf{F}{S}_0.$$

(29)

The sparse complex system is factored at each frequency. The operator $\mathbf{K}+\mathrm{i}\omega \mathbf{C}$ is identical to a complex-stiffness assembly that uses $G^{*}=G_r+\mathrm{i}\omega G_i$ [6,11]. The corresponding loss factor $\eta \left(\omega \right)=\omega G_i/G_r$ therefore grows linearly with frequency. The reported centre-point spectrum is

$$W_{\mathrm{c}}\left(\omega \right)=\left|\widehat{w}({\mathbf{x}}_{\mathrm{c}},\omega )\right|,$$

(30)

plotted as ${log}_{10}{W}_{\mathrm{c}}$ against $f=\omega /\left(2\pi \right)$.

The same operator (29) is used to verify the element against an independent Navier–Galerkin reduction [51,52,53] on two simply supported rectangles with identical faces and one central mass: a cosine-modulated plate and the corresponding uniform plate. The deflection is

$$w(X,Y)={\displaystyle \sum _{p=1}^{N_1}}{\displaystyle \sum _{q=1}^{N_2}}{W}_{pq}sin\left({\displaystyle \frac{p\pi X}{L}}\right)sin\left({\displaystyle \frac{q\pi Y}{W}}\right),$$

(31)

with $N_1=N_2=12$. The membrane fields use the compatible Navier pairs (cosine–sine for u and sine–cosine for v) and are condensed by the in-plane weak form, so the reduced unknown is the coefficient vector $\left\{W_{pq}\right\}$. The reduced mass, damping and stiffness come from the same face-membrane, face-bending and core-shear energies as (24)–(26), restricted to $h_c=h_p$ and to a common face-wavenumber pair; they are not fitted to the FEM spectrum. The centre-point amplitude reconstructed from (31) is compared with (30) on the same frequency grid.

3. Results and Discussion

Table 1 lists the locked data of the primary geometry. The frequency samples are $\omega =0.1,4.1,\dots ,748.1$ (188 frequencies). The frequency step is $\Delta f=2/\pi \approx 0.637\mathrm{Hz}$; every quoted peak is a grid point, not an interpolated maximum. Nine equal point masses occupy a $3\times 3$ lattice. The nine stations $(X_k,Y_k)$ and the common mass $m_k=270LW/9=234 \mathrm{k}\mathrm{g}$ are listed in Table 5. The plate is partitioned into a mesh of rectangular elements, and every attached mass lies on a node. The spectra on this mesh have converged.

Two campaigns lock complementary groups so that core periodicity is not confounded with face periodicity. The core campaign varies $(k_a,k_b,h_v)$ and holds the faces at $k_1=k_2=0.5$, $k_3=k_4=1.5$, $c_{1m}=0.05$, $c_{3m}=0.06$, $b_{1r}=0.3c_{1m}$ and $b_{3r}=0.4c_{3m}$. The face campaign varies the face fields and holds the core at $k_a=k_b=0.5$ and $h_v=0.20 \mathrm{m}$; wavenumber cases are labelled K and amplitude or thickness cases are labelled F. Twenty wavenumber plates (K01–K20) isolate five geometric contrasts, with $\alpha =0.30$ and $c_{1m}=c_{3m}=0.05$ locked: equal wavenumbers $k_1=k_2=k_3=k_4$; aligned one-dimensional stripes; orthogonal top and bottom stripes; top–bottom mismatch, including one-sided periodicity; and rectangular cells with $k_x\ne k_y$. Mean-thickness cases hold $\alpha =0.30$ and $k_1=k_2=k_3=k_4=0.5$, and include both $c_{1m}=c_{3m}$ and $c_{1m}\ne c_{3m}$. Face-campaign F01 and wavenumber K01 are the same plate and serve as the face-campaign reference. Core C01 is a different plate and is used only as the core-campaign reference. A third archive holds that same core-C01 plate and varies only the attached-mass layout at fixed $\sum m_k=270LW=2106 \mathrm{k}\mathrm{g}$; the cases are labelled D. Stations and masses are listed in Table 6.

A peak in this section is a local maximum of $|W_{\mathrm{c}}|$ on the frequency grid. The first three maxima are denoted $(f_1,A_1)$, $(f_2,A_2)$ and $(f_3,A_3)$, where $A={log}_{10}|W_{\mathrm{c}}|$.

Figure 4 verifies the finite-element FRF against the Galerkin series (31) on two single-mass plates with $L=3 \mathrm{m}$, $W=2.6 \mathrm{m}$, $m_0=270LW=2106 \mathrm{k}\mathrm{g}$ and $c_{1m}=c_{3m}=0.05 \mathrm{m}$. The periodic plate has $k_1=k_2=k_3=k_4=k_a=k_b=0.5$ and relative face amplitude $0.30$; the uniform plate sets every amplitude to zero. The two methods use the same geometry, the same frequency grid and the same operator (29); both plates are drawn on Figure 4, with FEM as solid lines and the Galerkin series as markers. These spectra are not overlaid on the $3\times 3$ campaign.

On the periodic plate the first peak differs by one grid step: Galerkin gives $f_1=8.93\mathrm{Hz}$, FEM gives $f_1=9.57\mathrm{Hz}$. The next two peaks coincide at $f_2=35.03\mathrm{Hz}$ and $f_3=48.40\mathrm{Hz}$. The first-peak heights are $A_1=-2.272$ (Galerkin) and $A_1=-2.264$ (FEM), a gap of $0.008$.

On the uniform plate both methods place the first two peaks at $f_1=7.66\mathrm{Hz}$ and $f_2=29.30\mathrm{Hz}$, and $A_1$ differs by $0.066$ ($-2.212$ versus $-2.146$). The third peak differs by one grid step: Galerkin $40.12\mathrm{Hz}$, FEM $39.49\mathrm{Hz}$. The element therefore reproduces the series frequencies to within one grid step on both a modulated and an unmodulated stack, and it is accepted for the parametric campaigns. The values $f_1=8.93\mathrm{Hz}$, $9.57\mathrm{Hz}$ and $7.66\mathrm{Hz}$ are the first peaks of these two verification plates. They move if the materials or the planform change, and they are not properties of the element. The two locked campaign references are different plates and are never overlaid: face-campaign F01 has $(f_1,A_1)=(12.11,-2.452)$, whereas core C01 has $(12.11,-2.491)$. All later comparisons stay inside a single campaign.

3.1. Core Shear-Modulus Modulation and Thickness

Table 2 and Figure 5 collect Series I (isotropic $k_a=k_b$), Series II (thickness at $k_a=k_b=0.5$) and Series III (uniform core and one-dimensional stripes). Raising the core wavenumber from $0.5$ to 1 drops $\mu$ from $0.405$ to 0 and therefore drops the mean storage modulus from $1.203e_{a1}$ to $e_{a1}$. The first-peak height rises from $-2.491$ to $-2.302$ (C02), which is the largest first-peak change in Series I, and $f_1$ moves down by one grid step. Further integer or half-integer wavenumbers (C03, C04) stay within $0.15$ of that height. The spatial pattern of the core, once the mean modulus is fixed, is a weak perturbation of $W_{\mathrm{c}}$ on this mesh.

Thickness acts on both mass (${\rho }_v{h}_v$) and shear strain ($\gamma \sim 1/h_v$). From C05 to C08, $A_1$ falls from $-2.358$ to $-2.521$ and $f_1$ rises by one step. The third peak moves down in frequency and down in height, so a thicker core is not a pure mass addition.

The uniform high-modulus core C09 ($k_a=k_b=0$, $\mu =1$, ${\overline{G}}_r=1.5e_{a1}$) gives the smallest first peak in Table 2, $A_1=-2.572$ at $13.38\mathrm{Hz}$. One-dimensional stripes (Figure 6) lie between C01 and C09 when $\mu$ is large (C10) and near the integer-wavenumber plates when $\mu =0$ (C11). That ordering supports a mean-modulus reading: periodicity of $G_r$ and $G_i$ changes $W_{\mathrm{c}}$ mainly by changing ${\overline{G}}_r$ and ${\overline{G}}_i$, not by opening a gap.

On the present mesh, core periodicity therefore acts as a mean-property perturbation. The factor $\mu (k_a,k_b)$ ranks $A_1$ among C01, C02, C10, C11 and C09. Cases with the same $\mu$ differ by at most $0.08$ in $A_1$, which is small beside the $0.27$ gap between C02 and C09. A design that only “adds periodicity” to $G_r$ without changing ${\overline{G}}_r$ therefore moves the fundamental FRF little. This ranking is the continuum counterpart of a long-wave homogenisation argument: when the modulation wavelength is comparable to the plate, the first global mode samples an area average more than a unit-cell band structure [36,41].

Core thickness adds mass, reduces $\gamma$, and changes the lever arm in (14). $A_1$ decreases as $h_v$ increases from $0.10$ to $0.25$, then changes by less than $0.02$ from C07 to C08. The return therefore diminishes once the core is several times the face thickness, which is consistent with classical CLD optimisation: an excessively thick soft core no longer increases the constrained shear in proportion to the added mass [1].

3.2. Face-Thickness Modulation

Table 3 and Table 4 and Figure 7, Figure 8 and Figure 9 summarise the face campaign. Here $k_1$ and $k_2$ are the top-face wavenumbers in X and Y, while $k_3$ and $k_4$ are the bottom-face pair; they are independent in (2)–(3).

K01–K04 keep the four equal and raise that common value from $0.5$ to 2 (Table 3, Series I; Figure 7a). Those cases leave $f_1$ at $12.11$ or $11.48\mathrm{Hz}$. The second peak falls from $36.94\mathrm{Hz}$ (K01) to $29.94\mathrm{Hz}$ (K04), a shift of about $7\mathrm{Hz}$, while $A_2$ stays near $-4.07$. Raising all four wavenumbers together therefore rearranges the second and third resonances more than it changes the fundamental amplitude.

The remaining sixteen plates let the four wavenumbers differ (Table 3, Series II–V; Figure 8). On these symmetric faces, exchanging the top pair $(k_1,k_2)$ with the bottom pair $(k_3,k_4)$ leaves $W_{\mathrm{c}}$ unchanged to the reported precision: K09 coincides with K10, K13 with K14, and K15 with K16. Aligned x-stripes (K05) and y-stripes (K07) at half-integer wavenumber differ from each other by only $0.016$ in $A_1$ and share $f_2=38.85\mathrm{Hz}$, about $2\mathrm{Hz}$ above the two-dimensional reference K01. The orthogonal arrangement K09 matches K05 in $f_1$, $f_2$ and $f_3$, and differs by $0.012$ in $A_2$. Thus, at the centre point, the relative orientation of the two faces is a weaker control than whether the modulation is one- or two-dimensional, and the latter acts partly through $\mu$: a one-dimensional half-integer stripe has $\mu =2/\pi$, whereas K01 has $\mu ={(2/\pi )}^2$.

Direction does appear once a wavenumber is an integer. Y-aligned integer stripes (K08, K20) place $f_1$ at $10.84\mathrm{Hz}$, one grid step below the corresponding x-aligned plates (K06, K19). The planform is not square ($L=3 \mathrm{m}$, $W=2.6 \mathrm{m}$), so the two directions are not equivalent; the centre-point FRF feels that anisotropy, but still only at the level of one frequency step.

One-sided periodicity (K15, K16) gives the largest first-peak change in Table 3: $A_1=-2.517$ at $12.75\mathrm{Hz}$. The inactive face has both wavenumbers zero, so its thickness is the uniform value $c+b=0.065$ rather than a zero-mean ripple about $0.05$. That is again a mean-thickness shift, not a Bragg effect. Rectangular cells with $k_x\ne k_y$ put $f_2$ between the bounding equal-wavenumber cases: K17 and K18 ($33.12\mathrm{Hz}$) sit between K01 ($36.94\mathrm{Hz}$) and K02 ($31.85\mathrm{Hz}$); K19 and K20 ($31.21\mathrm{Hz}$) sit between K01 and K04 ($29.94\mathrm{Hz}$).

Relative amplitude $\alpha =b_{1r}/c_{1m}=b_{3r}/c_{3m}$ at fixed $c=0.05$ and $k_1=k_2=k_3=k_4=0.5$ moves $f_2$ from $31.85\mathrm{Hz}$ ($\alpha =0$) to $42.67\mathrm{Hz}$ ($\alpha =0.60$). The first-peak height stays inside $[-2.472,-2.401]$. Because $\langle h^3\rangle$ grows with $\alpha$ even when $\mu$ is held fixed, the upward drift of $f_2$ and $f_3$ is the expected stiffening, not evidence of a Bragg mechanism.

Mean face thickness is the strongest control in the archive. Symmetric faces with $c_{1m}=c_{3m}=c$ and $\alpha =0.30$ give $f_1=9.57\mathrm{Hz}$ at $c=0.03$ and $f_1=16.57\mathrm{Hz}$ at $c=0.08$; $A_1$ falls from $-2.365$ to $-2.611$. The second peak more than doubles in frequency over that interval. Those shifts dwarf the core-wavenumber effect in Table 2.

The two means need not be equal (Table 4, last block; Figure 9b). Exchanging $c_{1m}$ with $c_{3m}$ leaves $W_{\mathrm{c}}$ unchanged to the reported precision: F10 coincides with F11, and the same inversion holds for an untabulated pair at $k_1=k_2=k_3=k_4=1$. The faces have identical moduli and densities, so turning the stack over does not change the centre-point spectrum. What matters is the unordered pair $\{c_{1m},c_{3m}\}$.

At fixed sum $c_{1m}+c_{3m}=0.10$, the equal split F01 has $f_2=36.94\mathrm{Hz}$, the mild split F12 has $f_2=38.85\mathrm{Hz}$, and the $0.03/0.07$ split F10 has $f_2=43.31\mathrm{Hz}$ with $f_1$ one grid step higher. The same ordering appears at sum $0.12$: F13 ($0.08/0.04$) lies above F08 ($0.06/0.06$) in $f_1$, $f_2$ and $f_3$. The lever arm in (14) depends on $h_c+h_p$, hence only on the sum, but the bending stiffness of each face scales as $h^3$. For a fixed sum the convexity of $x^3$ on $[0,\infty )$ gives $c_{1m}^3+c_{3m}^3\ge 2{\left((c_{1m}+c_{3m})/2\right)}^3$, with equality only when $c_{1m}=c_{3m}$, so an unequal pair is the stiffer plate. The difference $h_c-h_p$ that enters (12) does not break the inversion symmetry of $W_{\mathrm{c}}$ on this mesh.

Face periodicity therefore enters $W_{\mathrm{c}}$ mainly through $\langle h^3\rangle$ and the higher resonances. To move the fundamental under base excitation, change the pair $(c_{1\mathrm{m}},c_{3\mathrm{m}})$. To rearrange $f_2$ and $f_3$ at nearly fixed $f_1$, change the common wavenumber, the thickness amplitude, or the split of a fixed sum. Assigning distinct $k_1,k_2,k_3,k_4$ is a finer correction on this mesh.

3.3. Attached-Mass Layout

The eight layouts in Table 6 keep the core-C01 plate and the total attached mass $270LW$. Each row of that table gives the occupied stations $(X,Y)$ and $m_k$. D01 occupies all nine nodes of Table 5 with equal mass and is identical to core C01.

Table 5. Locations of the nine equal point masses on the campaign plate ($L=3 \mathrm{m}$, $W=2.6 \mathrm{m}$). Each mass is $m_k=270LW/9=234 \mathrm{k}\mathrm{g}$. The ordering matches the MATLAB driver (rows of increasing Y, then X).

k	X ($\mathbf{m}$)	Y ($\mathbf{m}$)	${\mathit{m}}_{\mathit{k}}$ ($\mathbf{k}\mathbf{g}$)
1	$0.75$	$0.65$	234
2	$1.50$	$0.65$	234
3	$2.25$	$0.65$	234
4	$0.75$	$1.30$	234
5	$1.50$	$1.30$	234
6	$2.25$	$1.30$	234
7	$0.75$	$1.95$	234
8	$1.50$	$1.95$	234
9	$2.25$	$1.95$	234

Table 5. Locations of the nine equal point masses on the campaign plate ($L=3 \mathrm{m}$, $W=2.6 \mathrm{m}$). Each mass is $m_k=270LW/9=234 \mathrm{k}\mathrm{g}$. The ordering matches the MATLAB driver (rows of increasing Y, then X).

k	X ($\mathbf{m}$)	Y ($\mathbf{m}$)	${\mathit{m}}_{\mathit{k}}$ ($\mathbf{k}\mathbf{g}$)
1	$0.75$	$0.65$	234
2	$1.50$	$0.65$	234
3	$2.25$	$0.65$	234
4	$0.75$	$1.30$	234
5	$1.50$	$1.30$	234
6	$2.25$	$1.30$	234
7	$0.75$	$1.95$	234
8	$1.50$	$1.95$	234
9	$2.25$	$1.95$	234

Table 6. Mass-layout peaks on the core-C01 plate at fixed $\sum m_k=270LW=2106 \mathrm{k}\mathrm{g}$. Stations lie on the lattice of Table 5 (coordinates in $\mathrm{m}$, masses in $\mathrm{k}\mathrm{g}$). D01 is identical to core C01. Frequencies are grid values in $\mathrm{Hz}$; $A_i={log}_{10}|W_{\mathrm{c}}|$.

Case	Stations $(\mathit{X},\mathit{Y})$ and ${\mathit{m}}_{\mathit{k}}$	${\mathit{f}}_1$	${\mathit{A}}_1$	${\mathit{f}}_2$	${\mathit{A}}_2$	${\mathit{f}}_3$	${\mathit{A}}_3$
D01	nine nodes of Table 5; $m_k=234$	$12.11$	$-2.491$	$37.58$	$-4.187$	$47.76$	$-4.222$
D02	same; $m_k=156,234,312$ on $Y=0.65,1.30,1.95$	$12.11$	$-2.482$	$36.94$	$-4.183$	$47.76$	$-4.279$
D03	same; $m_k=156,234,312$ on $X=0.75,1.50,2.25$	$12.11$	$-2.478$	$37.58$	$-4.199$	$46.49$	$-4.235$
D04	$(0.75,0.65)$, $(2.25,0.65)$, $(1.50,1.30)$, $(0.75,1.95)$, $(2.25,1.95)$; $m_k=421.2$	$12.75$	$-2.455$	$36.94$	$-3.853$	$56.68$	$-5.392$
D05	$(1.50,0.65)$, $(0.75,1.30)$, $(1.50,1.30)$, $(2.25,1.30)$, $(1.50,1.95)$; $m_k=421.2$	$11.48$	$-2.365$	$45.85$	$-4.113$	$69.41$	$-5.051$
D06	$(1.50,0.65)$, $(1.50,1.30)$, $(1.50,1.95)$; $m_k=702$	$10.84$	$-2.386$	$36.30$	$-4.273$	$43.31$	$-4.221$
D07	$(0.75,0.65)$, $(1.50,1.30)$, $(2.25,1.95)$; $m_k=702$	$12.11$	$-2.404$	$28.66$	$-3.844$	$44.58$	$-4.420$
D08	$(0.75,1.30)$, $(1.50,1.30)$, $(2.25,1.30)$; $m_k=702$	$10.84$	$-2.436$	$31.85$	$-4.068$	$50.95$	$-4.531$

Table 6. Mass-layout peaks on the core-C01 plate at fixed $\sum m_k=270LW=2106 \mathrm{k}\mathrm{g}$. Stations lie on the lattice of Table 5 (coordinates in $\mathrm{m}$, masses in $\mathrm{k}\mathrm{g}$). D01 is identical to core C01. Frequencies are grid values in $\mathrm{Hz}$; $A_i={log}_{10}|W_{\mathrm{c}}|$.

Case	Stations $(\mathit{X},\mathit{Y})$ and ${\mathit{m}}_{\mathit{k}}$	${\mathit{f}}_1$	${\mathit{A}}_1$	${\mathit{f}}_2$	${\mathit{A}}_2$	${\mathit{f}}_3$	${\mathit{A}}_3$
D01	nine nodes of Table 5; $m_k=234$	$12.11$	$-2.491$	$37.58$	$-4.187$	$47.76$	$-4.222$
D02	same; $m_k=156,234,312$ on $Y=0.65,1.30,1.95$	$12.11$	$-2.482$	$36.94$	$-4.183$	$47.76$	$-4.279$
D03	same; $m_k=156,234,312$ on $X=0.75,1.50,2.25$	$12.11$	$-2.478$	$37.58$	$-4.199$	$46.49$	$-4.235$
D04	$(0.75,0.65)$, $(2.25,0.65)$, $(1.50,1.30)$, $(0.75,1.95)$, $(2.25,1.95)$; $m_k=421.2$	$12.75$	$-2.455$	$36.94$	$-3.853$	$56.68$	$-5.392$
D05	$(1.50,0.65)$, $(0.75,1.30)$, $(1.50,1.30)$, $(2.25,1.30)$, $(1.50,1.95)$; $m_k=421.2$	$11.48$	$-2.365$	$45.85$	$-4.113$	$69.41$	$-5.051$
D06	$(1.50,0.65)$, $(1.50,1.30)$, $(1.50,1.95)$; $m_k=702$	$10.84$	$-2.386$	$36.30$	$-4.273$	$43.31$	$-4.221$
D07	$(0.75,0.65)$, $(1.50,1.30)$, $(2.25,1.95)$; $m_k=702$	$12.11$	$-2.404$	$28.66$	$-3.844$	$44.58$	$-4.420$
D08	$(0.75,1.30)$, $(1.50,1.30)$, $(2.25,1.30)$; $m_k=702$	$10.84$	$-2.436$	$31.85$	$-4.068$	$50.95$	$-4.531$

Reweighting the same nine stations is a weak perturbation (Figure 10a). A $20/30/40$ split along Y (D02) or along X (D03) leaves $f_1=12.11\mathrm{Hz}$ and moves $A_1$ by at most $0.013$. The first centre-point peak therefore feels the total mass and the occupied lattice, not a mild row-to-row imbalance.

Relocating the mass onto fewer stations is stronger (Figure 10b). The five-point cross (D05) and the three-point midlines (D06, D08) put more mass on the antinode of the fundamental, so $f_1$ falls to $11.48\mathrm{Hz}$ or $10.84\mathrm{Hz}$ and $A_1$ rises. The five-point corner-and-centre set (D04) does the opposite: $f_1$ rises one grid step to $12.75\mathrm{Hz}$, because four of the five masses sit nearer the simply supported edges. The three-point diagonal (D07) keeps $f_1=12.11\mathrm{Hz}$ but drops $f_2$ from $37.58\mathrm{Hz}$ to $28.66\mathrm{Hz}$, the largest second-peak shift in Table 6. Thus, at fixed total mass, the useful control is which nodes carry the mass, not a rearrangement of weights on the full $3\times 3$ lattice. These shifts remain smaller than a change of both mean face thicknesses, and they must not be compared with Table 3 and Table 4: those tables use a different locked face pair.

None of the FRFs exhibits a stop band. The modulation spans half a wave to two waves across the plate, so its wavelength is comparable to the structural modes, and the mesh resolves only those long waves [36,41,47,48]. The spectra should not be read as a phononic-crystal calculation of the kind reported for piecewise-periodic viscoelastic cores [46,47,49].

The Kelvin–Voigt operator over-damps high frequency relative to a constant-loss or fractional model [11,12,56]. Peak heights above about 40 $\mathrm{Hz}$ are therefore more model-dependent than $f_1$. The spectra are computed on a mesh of rectangular elements, with every attached mass on a node. Those results have converged. This mesh is adequate for the first few global modes of a simply supported rectangle [8,9]; it cannot represent steep thickness gradients if the wavenumbers were raised much beyond 2. The frequency step is fixed: $\Delta f\approx 0.64\mathrm{Hz}$. A one-step change in a quoted peak is at that resolution. It is not a smaller shift that the sampling has resolved. Soft-core kinematics omit core compression, which matters for thick or soft cores [14,15,19,24]. On the face-thickness archive the fundamental lies between about $9\mathrm{Hz}$ and about $17\mathrm{Hz}$. ISO 10137 and AISC Design Guide 11 judge floor vibration by frequency-weighted acceleration under walking, not by a strength limit [57,58]. Design Guide 11 treats a floor near or below about $9\mathrm{Hz}$ as one that can respond resonantly to a walking harmonic, and a stiffer floor as one whose response is essentially transient [58]. The curves in this paper are base-acceleration spectra of a simply supported plate. They are not that walking-load check. They do show where a bay-sized sandwich floor or partition of this construction sits relative to the division those documents use. A lightweight aluminum-foam sandwich slab has been compared with a conventional floor frame under dynamic loading [59]. Crowd walking-load models for vibration serviceability have been compared [60]. The tables describe this element, this grid and this constitutive law.

4. Conclusions

A Mead–Markus rectangular element is formulated for a sandwich plate whose face thicknesses and core shear moduli vary slowly across the planform, and the centre-point frequency-response function is computed under base acceleration and attached point masses. The main conclusions are as follows.

• A Navier–Galerkin sine series verifies the element on two single-mass plates that share the later planform and the later total attached mass, one cosine-modulated and one uniform. First-peak frequencies differ by one grid step on the modulated plate and coincide on the uniform plate; the next peaks agree to the same tolerance. The three later archives lock different layer pairs and are not interchangeable.
• Core periodicity acts mainly through the area-averaged storage modulus. A uniform high-modulus core gives the smallest first peak; an integer-wavenumber core, whose modulation averages to zero, gives the largest. Once that mean is fixed, the in-plane pattern of the core is a weak perturbation. Adding periodicity without changing the area average therefore moves the fundamental little.
• The two mean face thicknesses are the strongest control. Raising both means together moves the fundamental over a far wider interval than any core-wavenumber change in the archive, and it more than doubles the second-peak frequency. At a fixed sum an unequal split is stiffer than two equal faces; exchanging the two means leaves the centre-point spectrum unchanged. Distinct face wavenumbers and the relative thickness amplitude rearrange the higher peaks at a nearly fixed fundamental.
• At fixed total attached mass, occupying a midline, a cross or a diagonal rearranges the centre-point spectrum; reweighting the full $3\times 3$ lattice does not.
• None of the spectra contains a phononic stop band. The cosine fields span at most two waves across the plate, so the calculation is a long-wave forced-response study, not a dispersion analysis of a phononic crystal. The Kelvin–Voigt core over-damps high frequency relative to a constant-loss model, rotatory inertia is omitted, and the reported spectra have converged on the rectangular mesh used here. The tabulated peaks describe this element, this grid and this constitutive law. Differences of one frequency step remain at the resolution $\Delta f\approx 0.64\mathrm{Hz}$. A comparison with a laboratory frequency-response test, or with an existing solid or shell formulation, was not available for the modulated plates and is left as later work. Two routes are open. A base-excitation test on a simply supported rectangle of the same planform can be set against the first three centre-point peaks. Separately, the uniform plate can be rebuilt with shell faces and a solid core whose shear modulus is $G^{*}=G_r+\mathrm{i}\omega G_i$, and compared with the Galerkin series of this paper, before the cosine fields are put back.