Finite-element analysis of the fundamental frequency of a sandwich plate with pyramidal lattice truss core: Comparison of the discrete and continuous core models

Abstract

This paper presents a comparative analysis of discrete and continuous models for the pyramidal lattice truss core in sandwich plates. The study focuses on determining the fundamental vibration frequency as the basis for model comparison. The finite element method, implemented in ANSYS, was employed to address the dynamic problem. The sandwich plate with a pyramidal lattice truss core was modeled using shell and beam finite elements, while the continuous core plate was represented with shell and solid volume finite elements. The effects of the number of pyramidal cells, core thickness, and lattice truss core strut diameter on the fundamental frequency were examined. Fundamental frequencies obtained from both core models were compared, revealing that the continuous core model consistently overestimates the fundamental frequency. Additionally, a design algorithm for sandwich plates with a pyramidal lattice truss core was developed. The procedures for finite element model generation, fundamental frequency calculation, and structural design were implemented using APDL.

Keywords

sandwich plate pyramidal lattice truss core vibration analysis fundamental frequency finite-element modelling sandwich plate design

Introduction

Sandwich plates have long been utilized in diverse engineering applications because of their high stiffness and strength. Traditionally, these plates have incorporated a limited range of core types to join the facings, with honeycomb cores being the most prevalent. The increasing adoption of additive manufacturing technologies has facilitated the development of novel core designs for sandwich plates. These new cores are cellular structures characterized by various spatially distributed architectures, consisting of repeating cells that completely fill the designated space. Owing to their potential applications, cellular structures have become a focus of intensive research. Comprehensive reviews of prior studies can be found in the works of Hanks et al.,¹ Hunt et al.,² Sajjad et al.,³ Tyagi and Manjaiah,⁴ Yan et al.,⁵ Liu et al.,⁶ Miao et al.,⁷ and Khan and Riccio.⁸

Many of the cellular structures reviewed in these articles are suitable as cores for sandwich plates, depending on the specific application. Among these, the pyramidal lattice truss core is of particular practical interest. Its unit cell comprises four struts that converge at a single point. The widespread adoption of this core type is largely attributed to its manufacturing simplicity. In recent years, extensive research has focused on the pyramidal lattice truss core. Notable findings are reported in the studies by Yungwirth et al.,⁹ Wu et al.,¹⁰ Li et al.,¹¹ Wu et al.,¹² Hu et al.,¹³ Ye et al.,^14,15 Zhang et al.,^16,17 Meng et al.,¹⁸ Yang et al.,¹⁹ Tao et al.,²⁰ Ma et al.,²¹ Zhao et al.,²² Chen et al.,²³ Zhang et al.,²⁴ Fan et al.,²⁵ Cadart et al.,²⁶ and Sun et al.²⁷

Yungwirth et al.⁹ evaluated the ballistic performance of edge-clamped stainless-steel sandwich panels by subjecting them to mid-span impacts with a spherical steel projectile. Each sandwich panel comprised two identical face sheets and a pyramidal truss core. Wu et al.¹⁰ conducted compression and shear experiments to investigate the mechanical properties of lattice structures composed of carbon fibre composites. Li et al.¹¹ examined the impact of vacuum thermal cycling on the compression and shear performance of composite sandwich panels with pyramidal lattice truss cores. Wu et al.¹² studied the mechanical performance of hierarchical pyramidal-pyramidal grid materials as base materials for sandwich structures. Hu et al.¹³ focused on the mechanical properties of hierarchical pyramidal-pyramidal grid materials used as core materials in sandwich structures subjected to compressive and transverse shear loading. Ye et al.¹⁴ employed axial compression tests and finite element methods to assess the mechanical properties and energy absorption of pyramidal lattice truss structures. In a subsequent study, Ye et al.¹⁵ investigated the influence of relative density on the compression performance and energy absorption characteristics of pyramidal lattice truss composite structures. Zhang et al.¹⁶ designed and analysed the compression behaviour of continuous carbon fibre reinforced composite lattice sandwich structures, developing theoretical models to predict the compressive strength of tetrahedral, pyramidal, and hexagonal structures. Zhang et al.¹⁷ proposed the fabrication of octet truss structures using continuous carbon fibre composites with additive manufacturing and provided both theoretical and experimental investigations of their mechanical properties at varying fibre volume fractions. Meng et al.¹⁸ derived Legendre-Ritz solutions for the vibration characteristics of double-layer lattice truss sandwich plates, equating the pyramidal lattice core to a homogeneous plate based on the equivalence principle. Yang et al.¹⁹ analysed the nonlinear static and dynamic responses of sandwich plates constructed with aluminium pyramidal lattice cores and composite face layers. Tao et al.²⁰ explored the effects of geometrical parameters and relative density on the deformation mechanisms and mechanical properties of sandwich structures with lattice metamaterial cores through experiments, numerical simulations, and theoretical predictions. Ma et al.²¹ investigated the nonlinear dynamics, bifurcation, and chaotic properties of composite piezoelectric lattice sandwich plates with four simply supported edges. Zhao et al.²² examined the compression behaviour of pyramidal lattice truss cores in sandwich structures using experimental, theoretical, and finite element analyses. Chen et al.²³ performed free vibration analysis and multiobjective robust optimisation of pyramidal truss core sandwich plates. Zhang et al.²⁴ studied the compressive behaviour and energy absorption capacity of additively manufactured lattice structures using uniaxial compression tests. Fan et al.²⁵ developed an analytical model to analyse the bending response of lattice core sandwich structures based on homogenisation and dehomogenisation methods. Cadart et al.²⁶ proposed an optimal penalty method for joint stiffening in beam models of additively manufactured lattice structures, concluding that beam element simulation within the finite element method is a more efficient alternative to traditional solid finite element simulations. Sun et al.²⁷ developed a variable-stiffness bio-inspired metamaterial based on carbon fibre-reinforced composites (CFRPs) by integrating a biomimetic helical design with S-shaped spring units. The authors established a theoretical model for the S-shaped spring, quantifying the relationship between linear stiffness characteristics and geometric parameters.

Our analysis of the referenced publications has enabled us to identify key characteristics of studies on pyramidal lattice truss cores. Most of these investigations focus on determining the average mechanical properties of the structures examined. Typically, the subject of analysis is a segment of a sandwich plate, comprising two facing fragments and one lattice truss core fragment. Researchers determine the stress-strain state parameters of this segment using experimental tests, finite element analysis, and analytical calculations of the truss core. The findings are then used to establish the averaged mechanical properties of the lattice structure. In this process, the discrete core fragment is modeled as a continuous medium with reduced modulus and density. This modeling is achieved through homogenisation, as outlined in the monograph by Gibson and Ashby.²⁸ Consequently, the entire core of the sandwich plate is assigned the averaged mechanical properties derived from its fragments.

The analysis and design of a sandwich plate with a solid core can be conducted using various established calculation methods. To evaluate the reliability of these results, it is necessary to compare them with outcomes from calculations that consider the discrete nature of a pyramidal lattice truss core. This comparison is typically performed by determining the fundamental vibration frequency of a sandwich plate with the specified core structure. Such results are frequently utilized in composite structure design calculations. The fundamental frequency is determined by the ratio of the plate’s bending stiffness to its mass per unit area. An increase in bending stiffness is invariably accompanied by an increase in mass per unit area. Consequently, the fundamental frequency serves as an effective criterion for assessing the mass efficiency of a sandwich plate, as it reflects the combined influence of bending stiffness and mass per unit area.

This paper presents a comparative analysis of discrete and continuous models of a pyramidal lattice truss core in sandwich plates by determining the fundamental vibration frequencies for both models. The dynamic problems were solved using the finite element method in ANSYS, employing shell and beam finite elements for the discrete model and shell and solid volume finite elements for the continuous model. These solutions enabled analysis of the effects of the number of pyramidal cells, core thickness, and lattice truss core strut diameter on the fundamental vibration frequency for both models. The resulting data facilitated a comparison of model behaviors. Additionally, a practical design algorithm was developed to achieve minimal mass for a sandwich plate with a pyramidal lattice truss core, subject to a specified fundamental frequency constraint.

Geometrical parameters of the sandwich plate and pyramidal lattice truss core

A rectangular sandwich plate is composed of identical composite orthotropic facings and a pyramidal lattice truss core with struts of circular cross section (Figure 1). The lower plane of the plate is referenced to the $x y z$ coordinate system. The plate has plan sizes $a$ and $b$ .

Figure 1.

Sandwich plate with a pyramidal lattice truss core.

The thicknesses of the core and each facing are denoted by $h$ and $t$ , respectively. The core comprises $n_{a}$ pyramidal lattice unit cells along the long side $a$ and $n_{b}$ pyramidal lattice unit cells along the short side $b$ . The geometric parameters characterizing the cell structure (Figure 2) include the side lengths in plan, $s_{a}$ and $s_{b}$ , the strut diameter $d$ , and the strut slope angle $β$ .

Figure 2.

The geometric parameters of a pyramidal lattice unit cell.

The sizes $s_{a}$ and $s_{b}$ are generally defined as follows:

s_{a} = \frac{a}{n_{a}}

(1)

s_{b} = \frac{b}{n_{b}}

(2)

When arbitrary values are assigned to $n_{a}$ and $n_{b}$ , the resulting cells exhibit a rectangular shape (Figure 3).

Figure 3.

The sizes of a unit cell.

The stiffness parameters of the core along the sides of the plate $a$ and $b$ depend on the respective sizes of the unit cell in plan view. When $s_{a} \neq s_{b}$ , these stiffness parameters differ. Typically, sandwich plates are manufactured to minimize the difference between the stiffness parameters along sides $a$ and $b$ . For a pyramidal truss core, this condition can be achieved if the unit cell in plan view is nearly square. The unit cell parameters for which $s_{a} \approx s_{b}$ are defined as follows. Let the number of cells along the short side of the plate be $n_{b}$ . The value of $s_{b}$ is then calculated using equation (2). The required number of cells along the long side of the plate, $n_{a}$ , is determined as follows:

n_{a} = rounding (\frac{a}{s_{b}})

(3)

where

rounding

denotes the function that rounds a number to the nearest integer. The required size

s_{a}

is determined using equation (1). The resulting values of

s_{a}

and

s_{b}

ensure that the unit cell is nearly square in plan view. A pyramidal truss core with this unit cell configuration exhibits minimal differences between the stiffness parameters along the sides of the plate

a

and

b

Given the known values of the unit cell dimensions $s_{a}$ , $s_{b}$ , and $h$ , the strut slope angle $β$ and strut length $l$ (Figure 2) can be determined using the following formulae:

β = \arctan (\frac{2 h}{\sqrt{s_{a}^{2} + s_{b}^{2}}})

(4)

l = \sqrt{{(\frac{s_{a}}{2})}^{2} + {(\frac{s_{b}}{2})}^{2} + h^{2}}

(5)

Continuous model of the core

A continuous model of the discrete core is constructed by replacing it with a continuous orthotropic layer (Figure 4).

Figure 4.

Sandwich plate with continuous solid core.

The material of this layer is generally characterized by averaged parameters, including density $ρ_{c}$ , elastic moduli $E_{x c}$ , $E_{y c}$ , $E_{z c}$ , shear moduli $G_{x y c}$ , $G_{x z c}$ , $G_{y z c}$ , and the corresponding Poisson ratios. The pyramidal lattice truss core, however, exhibits no stiffness in the $x y$ plane. Consequently, in the continuous model of this core, $E_{x c} = E_{y c} = 0$ and $G_{x y c} = 0$ are assumed. All Poisson ratios are also taken as zero. This representation of the elastic parameters is analogous to that used for honeycomb structures.²⁹ Therefore, the continuous model of the pyramidal lattice truss core requires determination of four averaged parameters: density $ρ_{c}$ , elastic modulus $E_{z c}$ , and transversal shear moduli $G_{x z c}$ and $G_{y z c}$ . To achieve this, an elementary parallelepiped is selected within the continuous layer, with dimensions along the $x$ , $y$ , and $z$ axes denoted as $s_{a}$ and $s_{b}$ , respectively (Figure 5). A pyramidal lattice unit cell is positioned within such a parallelepiped.

Figure 5.

Unitary parallelepiped.

The averaged mechanical properties of the solid parallelepiped are determined using the corresponding mechanical properties of a strut. It is assumed that a layer of continuous core in a sandwich plate, which replaces a discrete structure, possesses the averaged mechanical characteristics of the considered elementary parallelepiped.

The averaged density of the parallelepiped is determined by first calculating the mass of the four struts that constitute a pyramidal lattice unit cell as follows:

m_{s} = π d^{2} l ρ_{s} .

(6)

The mass of the parallelepiped is expressed by the following formula:

m_{c s} = ρ_{c} s_{a} s_{b} h,

(7)

where

ρ_{c}

represents the averaged density to be determined. Equating

m_{s}

and

m_{c s}

yields

ρ_{c} = \frac{π d^{2} l ρ_{s}}{s_{a} s_{b} h}

(8)

We also aim to determine the average modulus of elasticity of the parallelepiped. To achieve this, we solve two problems involving the deformation of a pyramidal lattice unit cell and a parallelepiped, each subjected to the same force $F_{z}$ applied along the $z$ axis (Figure 6).

Figure 6.

A pyramidal lattice unit cell and an unitary parallelepiped loaded by the $F_{z}$ force.

The first problem is addressed by modeling the pyramidal lattice unit cell as a spatial truss. Each strut end is subjected to the vertical force $F_{z} / 4$ .

The elongation of the strut $δ_{s z}$ results from the longitudinal force $F_{z} / (4 \sin β)$ , as illustrated in Figure 7. The corresponding value is determined as follows:

δ_{s z} = \frac{F_{z} l}{E_{s} π d^{2} \sin β}

(9)

Figure 7.

Elongation of the strut caused by $F_{z}$ force.

The vertical displacement at the end of the strut, as shown in Figure 7, is given by

δ_{z} = \frac{δ_{s z}}{\sin β} .

(10)

By substituting equation (9) into equation (10), the following expression is obtained:

δ_{z} = \frac{F_{z} l}{E_{s} π d^{2} \sin^{2} β} .

(11)

To address the second problem, the parallelepiped as a rod with length $h$ and cross-sectional area $s_{a} \times s_{b}$ (Figure 6). The elongation of this rod under the applied force $F_{z}$ is expressed as follows:

δ_{z c} = \frac{F_{z} h}{s_{a} s_{b} E_{z c}}

(12)

where

E_{z c}

denotes the averaged modulus of elasticity of the parallelepiped. Applying the condition

δ_{z} = δ_{z c}

yields.

E_{z c} = \frac{E_{s} h π d^{2} \sin^{2} β}{s_{a} s_{b} l} .

(13)

The next step is to determine the averaged transverse shear modulus in the $x z$ plane of the parallelepiped under consideration. First, solutions are obtained for the deformation of a pyramidal lattice unit cell and a parallelepiped subjected to the same force $F_{x}$ applied along the $x$ axis (Figure 8).

Figure 8.

A pyramidal lattice unit cell and an unitary parallelepiped loaded by the $F_{x}$ force.

The solution to the first problem, considering the symmetry of the computational model, is conducted for one half of the pyramidal lattice unit cell. This structure is subjected to a shear force of $F_{x} / 2$ (Figure 9).

Figure 9.

Loading of a half pyramidal lattice unit cell by the $F_{x} / 2$ force.

The applied force induces both tensile and compressive forces in the struts. Based on the equilibrium condition for the struts forming an isosceles triangle (Figure 10), these forces are equal in value and can be determined as follows:

Figure 10.

The force diagram of the struts.

P = \frac{F_{x}}{4 \cos ϕ}

(14)

where

ϕ

denotes the angle between the strut axis and the

x

axis. The strut elongation or shortening (Figure 11) is determined by the following formula:

δ_{s x} = \frac{4 P l}{E_{s} π d^{2}}

(15)

Figure 11.

Elongation and shortening of the struts caused by $F_{x} / 2$ force.

By substituting equation (14), this formula becomes:

δ_{s x} = \frac{F_{x} l}{E_{s} π d^{2} \cos ϕ}

(16)

We now determine the displacement of the force application point $F_{x} / 2$ along the $x$ axis. Neglecting the vertical displacement of this point (Figure 11), we have

δ_{x} = \frac{δ_{s x}}{\cos ϕ}

(17)

By substituting equation (16) into equation (17), we obtain

δ_{x} = \frac{F_{x} l}{E_{s} π d^{2} \cos^{2} ϕ}

(18)

To address the second problem, consider half of the parallelepiped (Figure 6) with height $h$ and area $s_{a} \times s_{b} / 2$ , subjected to a force $F_{x} / 2$ , in a pure shear state. The displacement of the upper plane of this parallelepiped along the $x$ axis is given by

δ_{x c} = \frac{F_{x} h}{s_{a} s_{b} G_{x z c}}

(19)

Here, $G_{x z c}$ denotes the required averaged shear modulus in the transverse direction. Equating the displacement $δ_{x}$ from equation (18) and $δ_{x c}$ from equation (19) yields

G_{x z c} = \frac{E_{s} π d^{2} h \cos^{2} ϕ}{s_{a} s_{b} l}

(20)

Given that $\cos ϕ = s_{a} / (2 l)$ , the expression for the modulus $G_{x z c}$ can be written as

G_{x z c} = \frac{E_{s} π d^{2} h}{4 l^{3}} \frac{s_{a}}{s_{b}}

(21)

The averaged shear modulus $G_{y z c}$ is determined analogously. The corresponding formula is

G_{y z c} = \frac{E_{s} π d^{2} h}{4 l^{3}} \frac{s_{b}}{s_{a}}

(22)

The averaged density $ρ_{c}$ , modulus of elasticity $E_{z c}$ , and shear moduli $G_{x z c}$ and $G_{y z c}$ are determined for the continuous core model. These averaged properties are calculated for a core in which the pyramidal lattice unit cell is rectangular in the $x y$ plane. Therefore, the formulae in Equation (8), (13), (21), and (22) generalize the corresponding expressions from papers,^18,19,21,23 which consider a core with a square-shaped unit cell.

Finite element models of the sandwich plate

Finite element models were developed for the sandwich plate with both a pyramidal lattice truss core and a continuous solid core. Both models were constructed using ANSYS.³⁰ The finite element model of the sandwich plate with a pyramidal lattice truss core was based on a typical unit cell (Figure 2) defined by the geometric parameters $s_{a}$ , $s_{b}$ , $h$ , $t$ , and $d$ . Two types of finite elements were used to construct the unit cell model. The facings were represented by four-node SHELL181 elements, each with six degrees of freedom per node. The circular struts were modeled using three-node BEAM189 elements, also with six degrees of freedom at each node. The construction stages of the finite element model of the considered sandwich plate are illustrated in Figure 12. The initial step involves constructing a finite element model of a typical unit cell, as shown in Figure 12(a). At this stage, the geometric parameters of the structure, material properties, and finite element sizes are specified. The model of the typical unit cell is subsequently replicated $n_{a}$ times along the plate side $a$ , resulting in a band of unit cells discretised into finite elements (Figure 12(b)). This band is then replicated $n_{b}$ times along the plate side $b$ (Figure 12(c)). The matching nodes of the finite element model are merged and renumbered. Model construction concludes with the application of boundary conditions at the edges of the facings (Figure 12(d)). These edges are considered fully clamped, so all degrees of freedom at the finite element nodes located on the edges are set to zero.

Figure 12.

Generation of the finite element model of the sandwich plate with pyramidal lattice truss core.

A finite element model of a sandwich solid core plate was developed using two types of finite elements. The facings were represented by the SHELL181 finite element (Figure 13(a)), while the solid core was modeled with the SOLID186 volume finite element (Figure 13(b)). The SOLID186 element contains 20 nodes, each possessing three degrees of freedom. The matching nodes of the facings and solid core elements were merged, followed by renumbering of the model nodes. The modeling process concluded with the application of boundary conditions at the clamped edges of the facings (Figure 13(c)).

Figure 13.

Generation of the finite element model of the sandwich plate with a continuous solid core.

Numerical results

Finite element models of a sandwich plate with a pyramidal lattice truss core and a continuous solid core are employed to analyze the fundamental vibration frequency in bending. This frequency is determined by the geometrical parameters of both the plate and the core, as well as the mechanical properties of the constituent materials. Any variation in these parameters leads to a corresponding change in the fundamental vibration frequency. To ensure the dynamic analysis remains tractable, the number of influencing parameters is limited. The calculations fix the plate’s plan dimensions, the thickness of the facings, the elastic properties, and the material densities. The fundamental vibration frequency is determined for a plate with side lengths $a$ and $b$ of 0.75 m and 0.5 m, respectively, and a facing thickness $t = 0.0009 m$ . The composite orthotropic facings are assumed to be made of carbon fibre reinforced plastic³¹ with elastic properties $E_{x} = E_{y} = 65.5 GPa$ , $G_{x y} = 4 GPa$ , and $ν_{x y} = ν_{y x} = 0.28$ . The density of this material is $ρ = 1560 {kg / m}^{3}$ . The core struts are composed of short carbon fibre reinforced thermoplastic, which has a longitudinal elastic modulus $E_{s} = 11.62 GPa$ and density $ρ_{s} = 1356 {kg / m}^{3}$ .²² The parameters varied in this study are the core height, strut diameter, and the number of unit cells along the plate’s short side. The core height $h$ ranges from 0.02 m to 0.06 m in increments of 0.005 m. The strut diameter $d$ is set to 0.002 m, 0.003 m, or 0.004 m. The number of unit cells $n_{b}$ varies from $10$ to $50$ in increments of $5$ .

Finite element models of the sandwich plate were developed using macros for each combination of the parameters $d$ , $h$ , and $n_{b}$ . For sandwich plates with a pyramidal lattice truss core, the number of finite elements is determined by the core thickness $h$ , the number of cells $n_{b}$ , and the element size. The model with the fewest finite elements corresponds to $n_{b} = 10$ and $h = 0.02 m$ , comprising 10,800 shell-type elements and 3000 beam-type elements, for a total of 13,800 elements. The model with $n_{b} = 50$ and $h = 0.06 m$ contains the greatest number of finite elements, totaling 135,000, which includes 30,000 shell-type elements and 105,000 beam-type elements. For sandwich plates with a continuous solid core, the number of finite elements is governed by the core thickness. The model with $h = 0.02 m$ consists of 7,500 shell-type elements and 15,000 solid-type elements, for a total of 22,500 elements. The model with a core thickness of $h = 0.06 m$ contains the largest number of finite elements for this configuration, totaling 52,500, including 7500 shell-type elements and 45,000 solid-type elements.

The calculated fundamental frequencies of bending vibrations for the plate, obtained using two core models, are presented in Table 1 (

d = 0.002 m

), Table 2 (

d = 0.003 m

), and Table 3 (

d = 0.004 m

). Each table cell reports two fundamental frequency values: the upper value,

f_{p} (H z)

, corresponds to the pyramidal lattice truss core, while the lower value,

f_{c} (H z)

, pertains to the continuous solid core. Graphical representations of the functions

f_{p} (h, n_{b})

and

f_{c} (h, n_{b})

are provided in Figures 14–16. In constructing these functions, the parameter

n_{b}

was treated as a conditionally continuous variable. The characteristic vibration shapes of the sandwich plate for the considered core types are illustrated in Figure 17.

Table 1.

Fundamental frequencies of the sandwich plate $f_{p} (H z)$ and $f_{c} (H z), d = 0.002 m$ .

$n_{b}$	$h (m)$
$n_{b}$	0.02	0.025	0.03	0.035	0.04	0.045	0.05	0.055	0.06
10	281.3	321.7	351.6	372.4	385.9	393.6	397	397.1	394.9
10	316.7	357.5	385.9	404.3	414.8	419.5	420	417.5	413
15	381.3	422.6	446.7	457.7	459.5	455.3	447.3	437	425.5
15	439	472.7	487.5	489.8	484.3	474.3	461.8	448.1	434
20	437	470.7	483.7	483	474	460.5	444.8	428.3	411.7
20	494.7	516.5	517.9	507.7	491.5	472.7	453.1	433.9	415.4
25	427.8	459.7	470.1	466.4	454.3	438.1	420.1	401.7	383.8
25	520.4	530.9	521.2	501.7	478.2	453.9	430.3	408	387.4
30	439.4	463.2	465.2	454.3	436.8	416.6	395.8	375.6	356.5
30	521.7	523.5	506.4	481.3	453.9	427	401.7	378.4	357.2
35	442.6	458.4	453.3	436.8	415.5	392.7	370.3	349.2	329.6
35	513.6	508.1	485.5	456.6	426.9	398.6	372.6	349.1	328
40	438.6	447.5	436.9	416.7	393	369	346.1	324.9	305.5
40	498.1	487.4	461.3	430.5	399.9	371.4	345.6	322.5	302
45	430.4	433.4	418.5	395.7	370.6	346.1	323.1	302.2	283.4
45	479.9	465	436.5	404.6	373.8	345.5	320.3	298	278.2
50	419.4	417.7	399.9	375.6	350	325.4	302.9	282.5	264.3
50	460.2	442.4	412.7	380.6	350.1	322.5	298.1	276.6	257.8

Table 2.

Fundamental frequencies of the sandwich plate $f_{p} (H z)$ and $f_{c} (H z), d = 0.003 m$ .

$n_{b}$	$h (m)$
$n_{b}$	0.02	0.025	0.03	0.035	0.04	0.045	0.05	0.055	0.06
10	343.2	397.1	438.9	469.7	491	504.5	511.9	514.5	513.5
10	420.5	477.8	518.5	545.2	560.7	567.6	568.3	564.6	557.7
15	429.2	486.7	525.2	547.5	557.1	557.5	551.3	540.9	527.8
15	538.4	589	614.5	622.1	617.6	605.7	589.3	570.7	551.2
20	470	520.1	547.1	556.2	552.9	541.7	525.8	507.5	488.2
20	569.3	607.9	619.1	612.7	596	574.1	549.9	525.2	501.2
25	424.6	469.7	493.7	501.1	496.7	484.9	468.8	450.5	431.6
25	566.2	593.4	593.6	577.8	554	526.9	499.1	472.3	447
30	423.5	460.3	475.6	475.3	464.9	448.6	429.5	409.5	389.6
30	544	562.3	555.3	534.6	507.6	478.8	450.5	423.7	399
35	414.9	443.8	452	446	431.5	412.7	392.2	371.6	351.7
35	515.3	526.6	514.8	491.3	463	434	406.2	380.4	356.8
40	402.5	424.9	427.6	417.8	400.9	380.9	360	339.7	320.4
40	485.2	491.4	476.8	452.1	423.9	395.7	369	344.5	322.3
45	386.9	403.9	402.5	390.1	371.9	351.5	330.9	311.2	292.8
45	454.9	457.4	441	416.1	388.5	361.4	336.1	313	292.3
50	371.2	383.8	379.6	365.5	346.8	326.5	306.5	287.5	270
50	427.3	427	409.8	385.1	358.5	332.7	308.8	287.1	267.7

Table 3.

Fundamental frequencies of the sandwich plate $f_{p} (H z)$ and $f_{c} (H z), d = 0.004 m$ .

$n_{b}$	$h (m)$
$n_{b}$	0.02	0.025	0.03	0.035	0.04	0.045	0.05	0.055	0.06
10	369.8	431.7	481.6	519.9	548	567.2	579	585	586.2
10	486.7	556.6	607.4	641.4	661.6	671.1	672.5	668.1	659.6
15	430.4	494.7	541.6	572.3	589.5	595.8	593.9	586.2	574.5
15	575.1	637.8	673.2	687.6	686.8	676.1	659.3	638.9	616.8
20	453.7	510	545.4	563	567.1	561.4	549.3	533.2	515
20	575.3	625.4	646.7	647.6	635.3	615.2	591.2	565.6	540
25	387.7	434.9	464.8	480.1	483.7	478.8	468.3	454.1	438
25	547.2	585.7	596.8	589.4	571.1	547.1	520.8	494.2	468.4
30	378.6	417.9	440.1	448.4	446.2	436.9	423.2	407.1	390
30	510	539.4	543.9	532.2	511.6	486.8	460.7	435.1	410.6
35	364.1	396.5	412.5	415.5	409.5	397.6	382.4	365.6	348.5
35	470.7	493.3	493.5	479.7	458.5	434.2	409.2	385.1	362.4
40	348.7	375.6	386.8	386.2	377.7	364.3	348.6	331.9	315.2
40	435.3	452.8	450.3	435.6	414.7	391.4	367.9	345.4	324.4
45	331.4	353.6	361.2	358.2	348.2	334.2	318.5	302.2	286.3
45	401.7	415.5	411.4	396.5	376.4	354.3	332.4	311.5	292.2
50	315.3	333.9	338.9	334.1	323.2	309.1	293.6	278	262.8
50	373.1	384.1	378.9	364.2	344.9	324.1	303.6	284.2	266.3

Figure 14.

Functions $f_{p} (h, n_{b})$ and $f_{c} (h, n_{b})$ of the sandwich plate with the pyramidal lattice truss core and continuous solid core, $d = 0.002 m$ .

Figure 15.

Functions $f_{p} (h, n_{b})$ and $f_{c} (h, n_{b})$ of the sandwich plate with the pyramidal lattice truss core and continuous solid core, $d = 0.003 m$ .

Figure 16.

Functions $f_{p} (h, n_{b})$ and $f_{c} (h, n_{b})$ of the sandwich plate with the pyramidal lattice truss core and continuous solid core, $d = 0.004 m$ .

Figure 17.

Fundamental vibration mode shapes of the sandwich plate with (a) a pyramidal lattice truss core (b) continuous solid core.

The analysis of the obtained results enables several conclusions regarding the behavior of vibration frequencies in the investigated structures:

1. For each strut diameter $d$ , specific combinations of the number of cells $n_{b}$ and core thickness $h$ result in maximal fundamental frequencies of plate bending vibrations for both core models. The corresponding values of $d$ , $h$ , and $n_{b}$ at which these maxima, $f_{p \max} (H z)$ and $f_{c \max} (H z)$ , are achieved are presented in Table 4.

2. For each $d$ , there is a range of $h$ and $n_{b}$ values in which increasing these parameters results in a decrease in the fundamental frequency of bending vibrations for sandwich plates with pyramidal lattice truss and continuous solid cores. In contrast, for sandwich honeycomb plates, increasing the honeycomb core thickness leads to an increase in the fundamental bending vibration frequency of the structure.^32,33

Table 4.

Maximum values of the fundamental frequency of the sandwich plate with pyramidal lattice truss core and continuous solid core.

$d (m)$	$h (m)$	$n_{b}$	$f_{p \max} (Hz)$	$d (m)$	$h (m)$	$n_{b}$	$f_{c \max} (Hz)$
0.002	0.03	20	483.7	0.002	0.025	25	530.9
0.003	0.045	15	557.5	0.003	0.035	15	622.1
0.004	0.045	15	595.8	0.004	0.035	15	687.6

We compare the vibration frequencies of the sandwich plate obtained using the two core models. The relative difference between the frequencies $f_{p}$ and $f_{c}$ is determined using the following formula:

η = \frac{f_{p} - f_{c}}{f_{p}} 100 %

(23)

The $η$ values were calculated using three tables that present the vibration frequencies: Table 1 ( $d = 0.002 m$ ), Table 2 ( $d = 0.003 m$ ), and Table 3 ( $d = 0.004 m$ ). Analysis of the data indicates that for most variations in the parameters $d$ , $h$ , and $n_{b}$ , the value of $η$ is negative, meaning $f_{c}$ exceeds $f_{p}$ . The largest difference between $f_{c}$ and $f_{p}$ is observed for the thinnest core $(h = 0.02 m)$ . The corresponding maximum relative differences $η$ are $- 21.6 %$ ( $d = 0.002 m$ ), $- 33.3 %$ ( $d = 0.003 m$ ), and $- 41.1 %$ ( $d = 0.004 m$ ).

The analysis indicates that as the core thickness $h$ increases, the value of $η$ decreases. This reduction in the difference between the frequencies $f_{p}$ and $f_{c}$ can be explained as follows. An increase in $h$ results in a longer strut length $l$ (see Equation (5)), which in turn causes a substantial reduction in the transverse shear moduli of the core, $G_{x z c}$ and $G_{y z c}$ (see Equations (21) and (22)). Consequently, for plates with large core thickness, the facings primarily provide the bending stiffness. As previously noted, a comparable influence of thickness $h$ on the fundamental frequency has been observed for sandwich plates with pyramid lattice cores. Therefore, the difference between the frequencies $f_{p}$ and $f_{c}$ diminishes with increasing core thickness as a result of reduced transverse shear stiffness.

A sandwich plate with a homogeneous continuous core does not exhibit the local dynamic effects characteristic of a discrete core since the core is modelled as a solid block, and it is attached to the whole layer’s surfaces. However, small waves may develop in the facings of a sandwich plate with a pyramidal lattice truss core, with wavelengths comparable to the distance between the vertices of a pyramidal lattice unit cell. The analysis of such local dynamic effects may warrant a separate investigation focused on higher vibration modes. The present manuscript aims to determine the fundamental frequency of a sandwich plate using two models of the pyramidal lattice truss core. Future research will address the local dynamic effects that arise in the structure under consideration.

Sandwich plate design

The computational time required to generate the plate model and solve the dynamic problem depends on the number of finite elements. An analysis was conducted to determine the minimum number of finite elements in a typical unit cell necessary to accurately calculate the fundamental bending frequency of a sandwich plate. Employing a model with the minimal required number of finite elements is advantageous during the design process, as it enables efficient exploration of design parameters. Minimizing computational resources during this exhaustive search is desirable and depends on the size of the finite element model. In this analysis, the sizes of finite elements in a typical unit cell were incrementally increased. For each configuration, the fundamental bending vibration frequency of the plate was calculated and compared to the frequency obtained using the original finite element sizes. The element sizes were increased until the relative difference between the frequencies exceeded 2%.

Based on the analysis, a model of a typical unit cell with the minimum required number of finite elements was established. This model, illustrated in Figure 18, comprises eight shell-type finite elements and eight beam-type finite elements. We will now demonstrate the convergence of results obtained using a model of a typical unit cell with the minimum required number of finite elements. Specifically, we consider a sandwich plate with a pyramidal lattice truss core, where $n_{b} = 20$ , $h = 0.04 m$ , and $d = 0.003 m$ . The initial finite element model of a pyramidal lattice unit cell comprises 32 SHELL181 elements and 16 BEAM189 elements, yielding a fundamental vibration frequency of $f = 552.9 Hz$ . For structural design, the finite element model consists of 8 SHELL181 elements and 8 BEAM189 elements, resulting in a fundamental frequency of $f = 541.8 Hz$ . The relative difference between these frequencies is 2%. A similar convergence analysis was conducted for the variable design parameters.

Figure 18.

The unit cell model with the minimum required number of finite elements.

Utilizing this model of a typical unit cell significantly reduces the total number of finite elements required to determine the fundamental frequency of bending vibrations of a sandwich plate with a pyramidal lattice truss core. For the dynamic problem considered, the total number of finite elements in the plate model is reduced by an average of 77.5%. Simultaneously, the relative difference between the frequencies determined by the original and reduced finite element models of the plate does not exceed 3% on average.

A practical design algorithm for a sandwich plate with a pyramidal lattice truss core is proposed. In this design, the objective function is the mass of the structure, while the specified fundamental bending vibration frequency serves as a constraint. The selection of appropriate parameters for the sandwich plate is based on the previously developed finite element model of the structure with the minimum required number of finite elements.

In solving the design problem, parameters of the sandwich plate such as $a$ , $b$ , $t$ , $E_{x}$ , $E_{y}$ , $G_{x y}$ , $v_{x y}$ , $v_{y x}$ , $ρ$ , $E_{s}$ , and $ρ_{s}$ are assumed to remain constant. These values correspond to those previously used in the calculation of the vibration frequency of the structure. The number of varying parameters for the discrete core of the sandwich plate is limited by setting $h = 0.03 m$ . The design parameters in this problem are the diameter of the struts $d$ and the number of unit cells $n_{b}$ , which can be varied within the following ranges: $0.002 m \leq d \leq 0.004 m$ and $10 \leq n_{b} \leq 50$ . The selected ranges for $d$ and $n_{b}$ define the design space.

The desired design problem can be formulated as follows. The objective is to identify combinations of parameters $d$ and $n_{b}$ within the design space such that the fundamental frequency of the plate vibrations, ${\bar{f}}_{p}$ , equals $400 Hz$ while minimizing the mass of the core. To facilitate this search, the function $ξ (d, n_{b})$ is defined as follows:

ξ (d, n_{b}) = f_{p} (d, n_{b}) - {\bar{f}}_{p},

(24)

where

f_{p} (d, n_{b})

denotes the vibration frequency of the plate as calculated by the finite element model.

The design algorithm for the specified structure comprises the following sequential steps:

1. The variation range of parameter $d$ is partitioned into $k_{d}$ increments, each of size $s_{d}$ . The value of $d_{i}$ within this range is determined as follows:

d_{i} = d_{1} + s_{d} (i - 1), i = 1, 2, \dots, k_{d} + 1,

(25)

where

d_{1} = 0.002 m

2. The variation range of the parameter $n_{b}$ is partitioned into $k_{n}$ increments, each of size $s_{n}$ . The value of $n_{b j}$ within this range is defined as follows:

n_{b j} = n_{b 1} + s_{n} (j - 1), j = 1, 2, \dots, k_{n} + 1,

(26)

where

n_{b 1} = 10

3. For each value of $n_{b j}$ , $k_{d} + 1$ plate vibration frequencies $f_{p i j} (d_{i}, n_{b j})$ are determined, and $k_{d} + 1$ corresponding values of the function $ξ_{i j} (d_{i}, n_{b j})$ are calculated.

4. Given two consecutive values of the functions $ξ_{i j}$ and $ξ_{i + 1, j}$ , the following inequality is established:

ξ_{i j} ξ_{i + 1, j} < 0

(27)

5. If equation (27) is not satisfied, the index $j$ is incremented by one, and the calculations for $n_{b j + 1}$ are performed according to equation (3).

6. If equation (27) is satisfied, the strut diameter is determined using the bisection method. The corresponding value of this diameter is denoted as $d_{n j}$ . The calculation accuracy for $d_{n j}$ should not exceed the fabrication accuracy achievable with the employed technology.

7. Given the parameters $d_{n j}$ and $n_{b j}$ , the mass of the core $m_{n j}$ is calculated.

8. The index $j$ is increased by one, and calculations for $n_{b j + 1}$ are performed according to (3).

9. The calculation continues until $j$ equals $k_{n} + 1$ .

In some cases, the desired diameter $d_{n j}$ may not be identified for every value of $n_{b j}$ , where $j = 1, \dots, k_{n} + 1$ . The total number of successful diameter determinations is denoted as $k_{n d}$ . This counting procedure should be incorporated into the design algorithm by initializing $k_{n d}$ to zero before step 3 and incrementing it by one at step 8.

Once the calculations are complete, the values of $d$ and $n_{b}$ that give the minimum core mass $m_{c}$ must be selected from the $k_{n d}$ combinations of $d_{n k}$ and $n_{b k}$ , $k = 1, 2, \dots, k_{n d}$ .

The design algorithm was implemented in APDL and structured as a macro. This macro facilitated the selection of parameters $d$ and $n_{b}$ , which define the solution to the design problem. The selected values were $d = 0.00212 m$ and $n_{b} = 12$ . For these parameters, the core mass $m_{c}$ was 0.2 kg. The sandwich plate incorporating the optimized pyramidal lattice core exhibited a fundamental vibration frequency of 400.3 Hz. At the conclusion of the design process, the value of $k_{n d}$ was $18$ . Calculations were performed with $k_{d} = 10$ , $s_{d} = 0.0002 m$ , $k_{n} = 40$ , and $s_{n} = 1$ . The strut diameter was determined with a precision of 0.00001 m.

The vibration frequency determined during the design of the plate was verified. Calculations were performed using the finite element model applied in the parametric analysis described in previous section. The resulting bending vibration frequency of the structure is 408.9 Hz. The relative difference between this value and the design frequency (400.3 Hz) is $2.1 %$ . These results confirm that the sandwich plate with a minimal core mass also achieves minimal total mass.

Conclusions

The paper presents a comparative analysis of discrete and continuous models for the pyramidal lattice truss core of a sandwich plate. The fundamental vibration frequency of the sandwich plate was selected as the basis for model comparison. A continuous model of the pyramidal lattice truss core was developed, and its average mechanical properties were determined. The finite element method, implemented in ANSYS, was employed to address the dynamic problem. The finite element model of the sandwich plate with a pyramidal lattice truss core utilized shell and beam elements, while the model of the sandwich plate with a continuous core incorporated shell and solid volume elements.

The effects of the number of pyramidal cells, core thickness, and lattice truss core strut diameter on the fundamental vibration frequency of the sandwich plate were examined. For each strut diameter, a specific combination of cell number and filler thickness was identified that maximizes the fundamental bending vibration frequency. The study determined the ranges of core thickness and cell number in which increasing these parameters results in a decrease in the fundamental bending vibration frequency for sandwich plates with a pyramidal lattice core. This frequency behavior differs from that of sandwich plates with a honeycomb core, where increasing core thickness leads to an increase in the fundamental bending vibration frequency.

A comparison of the fundamental vibration frequency values obtained from discrete and continuous core models was conducted. The results indicate that the computational model of the sandwich plate with a solid core consistently overestimates the fundamental vibration frequency. Consequently, this model does not provide reliable results across a broad range of design parameters and should be approached with caution.

A practical design algorithm was developed for sandwich plates with a pyramidal lattice truss core, where the objective function is the structural mass and the fundamental frequency serves as a constraint. The design utilized a model with the minimum number of finite elements necessary to achieve the required frequency calculation accuracy. The selection of two key parameters of the pyramidal lattice truss core constitutes the solution to the design problem.

The procedure for generating finite element models of the sandwich plate was implemented using APDL and presented as macros. These macros enable rapid modification of model parameters, calculation of the plate’s fundamental frequency, and efficient execution of the design process.

The proposed approach for addressing the dynamic problem of sandwich plates with a pyramidal truss core is applicable to the research and design of sandwich structures featuring strut-based unit cell cores.

The discrete model developed for the pyramidal lattice truss core enables the investigation of higher vibration modes in sandwich plates. This model facilitates analysis of the dynamic behaviour of sandwich plates under various boundary conditions for the facings.

Supplemental material

Suppplemental Material - Finite-element analysis of the fundamental frequency of a sandwich plate with pyramidal lattice truss core: Comparison of the discrete and continuous core models

Suppplemental Material for Finite-element analysis of the fundamental frequency of a sandwich plate with pyramidal lattice truss core: Comparison of the discrete and continuous core models by Alexander V. Lopatin, Sergey A. Pikulin, and Elijah A. Lopatin in Journal of Sandwich Structures & Materials

Footnotes

ORCID iD

Alexander V. Lopatin

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors state that there are no known competing financial interests or personal relationships that could have influenced the work reported in this paper.

Supplemental material

The ANSYS finite element model used in this study is available on GitHub at: .

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