1 Introduction

Metal foam, as a novel class of porous materials, has a series of advantages such as good thermal insulation, low density, excellent capacity of damping and effective capacity of energy dissipation [1]. One can disperse foams continuously and smoothly along one or more directions to adjust the local parameter of metal foams for achieving specific functions. Due to their excellent characteristics, metal foam structures have attracted much attention and their mechanical properties have been studied by some researchers. Chen et al. [2] utilized the Timoshenko beam theory to analyze forced and free vibrations of metal foam beams. Jabbari et al. [3] explored the buckling problem of metal foam beams using the classical plate theory. Rezaei and Saidi [4] studied free vibration of beams made of metal foams based on the Carrera unified formulation. By utilizing the Timoshenko and Euler–Bernoulli beam theories, Wang et al. [5] performed wave propagation analysis of a metal foam beam. Jasion et al. [6] presented global and local buckling analysis of metal foam plates utilizing the finite element method. Zheng et al. [7] used a cell-based finite element model to analyze dynamic uniaxial impact behavior of metal foams. Liu et al. [8] investigated the dynamic behavior of metal foam material via the impact Hopkinson bar test. Wang et al. [9] studied the nonlinear vibration of a cylindrical shell made of metal foam materials.

Structural mechanical behaviors have attracted much attention of researchers [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]. Sandwich structures have a wide range of applications in aerospace, transportation and civil engineering industries. Typically, sandwich structure consists of three parts: two single face layers and a core [27,28,29,30]. Metal foam as the core of sandwich structures has many advantages such as high energy absorption ability, high resistance to impact, high strength-to-weight ratio and high stiffness-to-weight ratio [31,32,33]. Thus, some researches have paid their attention to this type of structures. Zhang et al. [34] studied the low-velocity impact response of MFCS beams using numerical, experimental and analytical methods. Jing et al. [35] analyzed the failure and deformation modes of MFCS beams via experiment. By utilizing the finite element method, the mechanical impedance of MFCS beams was investigated by Strek et al. [36]. Grygorowicz et al. [37] presented buckling analysis of MFCS beams by using the ANSYS software. Under the three-point bending testing, Omar et al. [38] investigated the flexural properties of MFCS beams. Smyczynski and Magnucka-Blandzi [39] explored the stability problem of sandwich beams with metal foam core. Caliskan and Apalak [40] experimentally investigated low-velocity impact bending response of MFCS beams.

Usually, structures are embedded in elastic foundations in engineering applications. Winkler’s elastic foundation [41], a widely used one-parameter model [42,43,44,45,46,47], is consisted of countless closed-spaced linear springs. Winkler’s foundation assumes no interaction between the springs, and the continuity of the foundation soil is not considered. In order to avoid this limitation, several two-parameter models were proposed [48,49,50,51,52,53]. Among them, the Winkler–Pasternak elastic foundation considers the interaction among the points in the foundation and the effect of transverse shear deformation of the foundation. Therefore, the Winkler–Pasternak elastic foundation is a more accurate model to depict the elastic foundation effect on structures [54,55,56,57,58,59,60,61].

The Chebyshev collocation method is a kind of numerical methods based on the Chebyshev polynomials of the first kind. It is worth mentioning that this method can achieve high precision within the range allowed by the effective number of bits of computers. This method can solve ordinary differential equations and its main advantage is that it is easy to deal with the singularity problem. In particular, the Chebyshev collocation method possesses a better computational accuracy and efficiency than Galerkin’s method and finite element method [62, 63].

In the existing literature, vibration of MFCS beams embedded in Winkler–Pasternak elastic foundation has not been investigated. In this article, the CCM is applied to investigate free vibration of this advanced structure. The Timoshenko beam theory is used to model the present system. Three types of foam distribution for the core are considered. Hamilton’s principle is used to obtain boundary conditions and governing equations. Additionally, the influences of various parameters are discussed on the free vibration of MFCS beams.

2 Theory and formulation

2.1 Metal foam core sandwich beam

A sandwich beam with metal foam core embedded in the Winkler–Pasternak elastic foundation is shown in Fig. 1. The top and bottom face layers of the beam are steel, and the core is steel foam. It is assumed that the face layers are bonded perfectly to the core. The width and length of the beam are denoted by b and L, respectively. The total thickness of the sandwich beam \(h_\mathrm{t}\) is \(h_\mathrm{t}=h_\mathrm{c}+2h_\mathrm{l}\), where \(h_\mathrm{c}\) and \(h_\mathrm{l}\) are the thicknesses of the core and single face layer, respectively. The Winkler parameter and Pasternak parameter of the Winkler–Pasternak elastic foundation are \(K_\mathrm{w}\) and \(K_\mathrm{p}\), respectively. For the core, three types of foam distribution are considered, as shown in Fig. 2.

Fig. 1
figure 1

Metal foam core sandwich beam

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Fig. 2
figure 2

Beam cross section

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The largest size pore is located at the mid-plane of core for non-uniform foam distribution 1 (foam-I) while at the bottom and top surfaces of the core for non-uniform foam distribution 2 (foam-II). For the uniform foam distribution (foam-\(\mathrm{I}\)II), the pore size is constant. The material properties of metal foam core are expressed by using Eqs. (1)–(3) [64,65,66].

foam-I:

$$\begin{aligned} \left\{ {{\begin{array}{l} {E( z )=E_1 \left[ {1-\vartheta _0 \cos \left( {\frac{\pi z}{h_\mathrm{c} }} \right) } \right] } \\ {\rho (z)=\rho _1 \left[ {1-\vartheta _\mathrm{m} \cos \left( {\frac{\pi z}{h_\mathrm{c} }} \right) } \right] } \\ \end{array} }} \right. \end{aligned}$$
(1)

foam-II:

$$\begin{aligned} \left\{ {{\begin{array}{l} {E( z)=E_1 \left\{ {1-\vartheta _0^{*} \left[ {1-\cos \left( {\frac{\pi z}{h_\mathrm{c} }} \right) } \right] } \right\} } \\ {\rho ( z)=\rho _1 \left\{ {1-\vartheta _\mathrm{m}^{*} \left[ {1-\cos \left( {\frac{\pi z}{h_\mathrm{c} }} \right) } \right] } \right\} } \\ \end{array} }} \right. \end{aligned}$$
(2)

foam-III:

$$\begin{aligned} \left\{ {{\begin{array}{l} {E=E_1 \varphi } \\ {\rho =\rho _1 \varphi ^{{*}}} \\ \end{array} }} \right. \end{aligned}$$
(3)

where E(z) and \(\rho (z)\) are Young’s modulus and mass density of the metal foam core, respectively; \(E_{1}\) and \(\rho _{1}\) are the maximum values of Young’s modulus and mass density of the metal foam core, respectively, which are equal to those values of pure steel; \(\vartheta _0 \), \(\vartheta _0^{*} \) and \(\varphi \) are foam coefficients for foam-\(\mathrm{I}\), foam-II and foam-III, respectively; \(\vartheta _\mathrm{m} \), \(\vartheta _\mathrm{m}^{*}\) and \(\varphi ^{{*}}\) are corresponding coefficients of mass density.

For open-cell metal foams, we have [64, 67, 68]

$$\begin{aligned} \frac{E( z)}{E_1 }=\left( {\frac{\rho ( z)}{\rho _1 }} \right) ^{2}. \end{aligned}$$
(4)

The relationships between foam coefficients and mass density coefficients are expressed as

$$\begin{aligned} \left\{ \begin{array}{ll} 1-\vartheta _\mathrm{m} \cos \left( {\frac{\pi z}{h_\mathrm{c} }} \right) =\sqrt{1-\vartheta _0 \cos \left( {\frac{\pi z}{h_\mathrm{c} }} \right) }&{} \quad \hbox {foam-I} \\ 1-\vartheta _\mathrm{m}^{*} \left[ {1-\cos \left( {\frac{\pi z}{h_\mathrm{c} }} \right) } \right] =\sqrt{1-\vartheta _0^{*} \left[ {1-\cos \left( {\frac{\pi z}{h_\mathrm{c} }} \right) } \right] }&{} \quad \hbox {foam-II} \\ \varphi ^{{*}}=\sqrt{\varphi }&{} \quad \hbox {foam-III} \\ \end{array} \right. . \end{aligned}$$
(5)

Without the loss of generality, the masses of beams with different metal foam core are considered to be equal to each other, so we have

$$\begin{aligned} \int _0^{h_\mathrm{c}/2} {\sqrt{1-\vartheta _0 \cos \left( {\frac{\pi z}{h_\mathrm{c} }} \right) } \mathrm{d}z} =\int _0^{h_\mathrm{c} /2} {\sqrt{1-\vartheta _0^{*} \left[ {1-\cos \left( {\frac{\pi z}{h_\mathrm{c} }} \right) } \right] } \hbox {d}z=\int _0^{h_\mathrm{c} /2} {\sqrt{\varphi }\; \hbox {d}z} }. \end{aligned}$$
(6)

which is utilized to determine \(\vartheta _0^{*} \) and \(\varphi \) with a given value of \(\vartheta _0 \). These values are shown in Table 1. It is noted that \(\vartheta _0^{*} \) increases with \(\vartheta _0 \) and is close to the upper limit (0.9612) when \(\vartheta _0 =0.6\). Hence, in the following numerical calculations, \(\vartheta _0 \in \) [0, 0.6] is adopted.

Table 1 Foam coefficients for different foam distributions
Full size table

2.2 Governing equations

According to the Timoshenko beam theory, the displacement components of an arbitrary point in the beam can be described by [69]

$$\begin{aligned} \left\{ {\begin{array}{l} u_x (x,z,t)=u(x,t)+z\psi (x,t) \\ u_y (x,z,t)=0 \\ u_z (x,z,t)=w(x,t) \\ \end{array}} \right. \end{aligned}$$
(7)

in which \(u_{x}\), \(u_{y}\) and \(u_{z}\) are x-, y- and z- components of the displacement vector, respectively; u and w are axial and transverse displacements of a point on the mid-plane of the beam, respectively; \(\psi \) is rotation of the cross section of the beam; and t is time.

From Eq. (7), the normal strain \(\varepsilon _{xx}\) and shear strain \(\gamma _{xz}\) are obtained as

$$\begin{aligned} \varepsilon _{xx} =z\frac{\partial \psi }{\partial x}+\frac{\partial u}{\partial x}, \; \gamma _{xz} =\psi +\frac{\partial w}{\partial x}. \end{aligned}$$
(8)

The normal stress \(\sigma _{xx}\) and shear stress \(\tau _{xz}\) are expressed as [70]

$$\begin{aligned} \sigma _{xx} =E\varepsilon _{xx} , \; \tau _{xz} =\frac{E}{2( {1+\nu } )}\gamma _{xz} \end{aligned}$$
(9)

where \(\nu \) is Poisson’s ratio. It is assumed that metal foam core has the same Poisson’s ratio as steel.

The first variation of the strain energy \(U_\mathrm{s}\) is written as

$$\begin{aligned} \begin{aligned} \delta U_\mathrm{s}&=\int _0^L {\int _A {\left( {\sigma _{xx} \delta \varepsilon _{xx} +\tau _{xz} \delta \gamma _{xz} } \right) \hbox {d}A\hbox {d}x} }\\&=\int _0^L {\left[ {N_x \left( {\delta \frac{\partial u}{\partial x}} \right) +M_x \left( {\delta \frac{\partial \psi }{\partial x}} \right) +Q_x \left( {\delta \frac{\partial w}{\partial x}+\delta \psi } \right) } \right] \hbox {d}x} \\ \end{aligned} \end{aligned}$$
(10)

where A is the area of cross section of the beam; \(N_{x}\) is the axial force, \(M_{x}\) is the bending moment and \(Q_{x}\) is the transverse shear force. These stress resultants are defined by

$$\begin{aligned} N_x =A_{11} \frac{\partial u}{\partial x}+B_{11} \frac{\partial \psi }{\partial x}, \;\; M_x =B_{11} \frac{\partial u}{\partial x}+D_{11} \frac{\partial \psi }{\partial x}, \;\; Q_x =K_\mathrm{s} A_{55} \left( {\frac{\partial w}{\partial x}+\psi } \right) \end{aligned}$$
(11)

in which \(K_\mathrm{s}=5/6\) is the shear correction factor and

$$\begin{aligned} \begin{aligned} A_{11}&=\int _{h_\mathrm{c} /2}^{h_\mathrm{t} /2} {E_1 b} \hbox {d}z+\int _{-h_\mathrm{c} /2}^{h_\mathrm{c} /2} {E( z )b} \hbox {d}z +\int _{-h_\mathrm{t} /2}^{-h_\mathrm{c} /2} {E_1 b} \hbox {d}z \\ B_{11}&=\int _{h_\mathrm{c} /2}^{h_\mathrm{t} /2} {E_1 bz} \hbox {d}z+\int _{-h_\mathrm{c} /2}^{h_\mathrm{c} /2} {E( z)bz} \hbox {d}z +\int _{-h_\mathrm{t} /2}^{-h_\mathrm{c} /2} {E_1 bz} \hbox {d}z \\ D_{11}&=\int _{h_\mathrm{c} /2}^{h_\mathrm{t} /2} {E_1 bz^{2}} \hbox {d}z+\int _{-h_\mathrm{c} /2}^{h_\mathrm{c} /2} {E( z)bz^{2}} \hbox {d}z +\int _{-h_\mathrm{t} /2}^{-h_\mathrm{c} /2} {E_1 bz^{2}} \hbox {d}z \\ A_{55}&=\frac{1}{2( {1+\nu } )}\left( {\int _{h_\mathrm{c} /2}^{h_\mathrm{t} /2} {E_1 b} \hbox {d}z+\int _{-h_\mathrm{c} /2}^{h_\mathrm{c} /2} {E( z )b} \hbox {d}z+\int _{-h_\mathrm{t} /2}^{-h_\mathrm{c} /2} {E_1 b} \hbox {d}z} \right) .\\ \end{aligned} \end{aligned}$$
(12)

The first variation of additional strain energy from the elastic foundation is expressed as [55]

$$\begin{aligned} \delta U_\mathrm{ad} =\int _0^L {\left( {K_\mathrm{w} w\delta w+K_\mathrm{p} \frac{\partial w}{\partial x}\frac{\partial \delta w}{\partial x}} \right) \mathrm{d}x}. \end{aligned}$$
(13)

The kinetic energy of the beam can be written as

$$\begin{aligned} K=\int _0^L {\int _A {\frac{1}{2}\rho \left[ {\left( {\frac{\partial u_x }{\partial t}} \right) ^{2}+\left( {\frac{\partial u_y }{\partial t}} \right) ^{2}+\left( {\frac{\partial u_z }{\partial t}} \right) ^{2}} \right] } } \hbox {d}A\hbox {d}x. \end{aligned}$$
(14)

The first variation of the kinetic energy is given by

$$\begin{aligned} \delta K=\int _0^L {\left[ {I_0 \left( {\frac{\partial u}{\partial t}\frac{\partial \delta u}{\partial t}+\frac{\partial w}{\partial t}\frac{\partial \delta w}{\partial t}} \right) +I_1 \left( {\frac{\partial \psi }{\partial t}\frac{\partial \delta u}{\partial t}+\frac{\partial u}{\partial t}\frac{\partial \delta \psi }{\partial t}} \right) +I_2 \frac{\partial \psi }{\partial t}\frac{\partial \delta \psi }{\partial t}} \right] } \hbox {d}x \end{aligned}$$
(15)

where

$$\begin{aligned} I_0= & {} \int _{h_\mathrm{c} /2}^{h_\mathrm{t} /2} {\rho _1 b} \hbox {d}z+\int _{-h_\mathrm{c} /2}^{h_\mathrm{c} /2} {\rho ( z )b} \hbox {d}z+\int _{-h_\mathrm{t} /2}^{-h_\mathrm{c} /2} {\rho _1 b} \hbox {d}z \\ I_1= & {} \int _{h_\mathrm{c} /2}^{h_\mathrm{t} /2} {\rho _1 bz} \hbox {d}z+\int _{-h_\mathrm{c} /2}^{h_\mathrm{c} /2} {\rho ( z )bz} \hbox {d}z+\int _{-h_\mathrm{t} /2}^{-h_\mathrm{c} /2} {\rho _1 bz} \hbox {d}z \\ I_2= & {} \int _{h_\mathrm{c} /2}^{h_\mathrm{t} /2} {\rho _1 bz^{2}} \hbox {d}z+\int _{-h_\mathrm{c} /2}^{h_\mathrm{c} /2} {\rho ( z )bz^{2}} \hbox {d}z+\int _{-h_\mathrm{t} /2}^{-h_\mathrm{c} /2} {\rho _1 bz^{2}} \hbox {d}z. \\ \end{aligned}$$

Based on Hamilton’s principle, one obtains [71]

$$\begin{aligned} \int _{t_1 }^{t_2 } {\delta \left( {U_\mathrm{s} +U_\mathrm{ad} -K} \right) } \hbox {d}t=0. \end{aligned}$$
(16)

Applying Eqs. (10), (13) and (15) in Eq. (16), the following governing equations are obtained:

$$\begin{aligned} \left\{ {{\begin{array}{l} {\frac{\partial N_x }{\partial x}=I_0 \frac{\partial ^{2}u}{\partial t^{2}}+I_1 \frac{\partial ^{2}\psi }{\partial t^{2}}} \\ {\frac{\partial Q_x }{\partial x}-K_\mathrm{w} w+K_\mathrm{p} \frac{\partial ^{2}w}{\partial x^{2}}=I_0 \frac{\partial ^{2}w}{\partial t^{2}}} \\ {\frac{\partial M_x }{\partial x}-Q_x =I_2 \frac{\partial ^{2}\psi }{\partial t^{2}}+I_1 \frac{\partial ^{2}u}{\partial t^{2}}} \\ \end{array} }} \right. . \end{aligned}$$
(17)

Let us introduce the following dimensionless quantities:

$$\begin{aligned}&X=\frac{x}{L}, \;\; \eta =\frac{L}{h_\mathrm{t} }, \;\; \left\{ {\bar{{U}},\bar{{W}}} \right\} =\frac{\left\{ {u,w} \right\} }{h_\mathrm{t} }, A_{00} =E_1 A, \;\; I_{00} =\rho _1 A\nonumber \\&\left\{ {a_{11} ,a_{55} ,b_{11} ,d_{11} } \right\} =\left\{ {\frac{A_{11} }{A_{00} },\frac{A_{55} }{A_{00} },\frac{B_{11} }{A_{00} h_\mathrm{t} },\frac{D_{11} }{A_{00} h_\mathrm{t} ^{2}}} \right\} , \left\{ {\bar{{I}}_0 ,\bar{{I}}_1 ,\bar{{I}}_2 } \right\} =\left\{ {\frac{I_0 }{I_{00} },\frac{I_1 }{I_{00} h_\mathrm{t} },\frac{I_2 }{I_{00} h_\mathrm{t} ^{2}}} \right\} \nonumber \\&\bar{{K}}_\mathrm{w} =\frac{K_\mathrm{w} L^{4}}{A_{00} h_\mathrm{t} ^{2}}, \bar{{K}}_\mathrm{p} =\frac{K_\mathrm{p} L^{2}}{A_{00} h_\mathrm{t} ^{2}}, \;\; \tau =\frac{t}{L}\sqrt{\frac{A_{00} }{I_{00} }}, \;\; \omega _n =\varOmega _n L\sqrt{\frac{I_{00} }{A_{00} }} \end{aligned}$$
(18)

in which \(\varOmega _n \) and \(\omega _{n}\) (\(n=1,2,3\ldots \)) are dimensional and non-dimensional natural frequencies, respectively.

By using Eq. (18) in Eq. (17), the dimensionless governing equations are stated as

$$\begin{aligned} \left\{ {{\begin{array}{l} {a_{11} \frac{\partial ^{2}\bar{{U}}}{\partial X^{2}}+b_{11} \frac{\partial ^{2}\psi }{\partial X^{2}}=\bar{{I}}_0 \frac{\partial ^{2}\bar{{U}}}{\partial \tau ^{2}}+\bar{{I}}_1 \frac{\partial ^{2}\psi }{\partial \tau ^{2}}} \\ {\eta ^{2}K_\mathrm{s} a_{55} \left( {\frac{\partial ^{2}\bar{{W}}}{\partial X^{2}}+\eta \frac{\partial \psi }{\partial X}} \right) -\bar{{K}}_\mathrm{w} \bar{{W}}+\bar{{K}}_\mathrm{p} \frac{\partial ^{2}\bar{{W}}}{\partial X^{2}}=\eta ^{2}\bar{{I}}_0 \frac{\partial ^{2}\bar{{W}}}{\partial \tau ^{2}}} \\ {b_{11} \frac{\partial ^{2}\bar{{U}}}{\partial X^{2}}+d_{11} \frac{\partial ^{2}\psi }{\partial X^{2}}-K_\mathrm{s} a_{55} \eta \left( {\frac{\partial \bar{{W}}}{\partial X}+\eta \psi } \right) =\left( {\bar{{I}}_1 \frac{\partial ^{2}\bar{{U}}}{\partial \tau ^{2}}+\bar{{I}}_2 \frac{\partial ^{2}\psi }{\partial \tau ^{2}}} \right) } \\ \end{array} }} \right. \end{aligned}$$
(19)

and the dimensionless boundary conditions are written as

Clamped (C):

$$\begin{aligned} \bar{{W}}=\bar{{U}}=\psi =0. \end{aligned}$$
(20)

Hinged (H):

$$\begin{aligned} \bar{{W}}=\bar{{U}}=b_{11} \frac{\partial \bar{{U}}}{\partial X}+d_{11} \frac{\partial \psi }{\partial X}=0. \end{aligned}$$
(21)

3 Chebyshev collocation method

In the Chebyshev collocation method (CCM), the Chebyshev interpolation grid points in the interval [\(-1\), 1] are constructed by using Gauss–Chebyshev–Lobatto collocation grid points, which are written as [72]

$$\begin{aligned} x_j =\cos \left( {\frac{j\pi }{N}} \right) , j=0,1,2,3,\dots ,N. \end{aligned}$$
(22)

Then, the \((N+1)\times (N+1)\) Chebyshev differentiation matrix \(\mathbf{D}_{1}\) can be constructed by using the Chebyshev collocation points. Herein, Lagrange polynomials of degree N are interpolated at each Chebyshev point. Differentiating the polynomials, and then evaluating the result at the collocation points, all elements in the matrix are obtained as

$$\begin{aligned} \begin{array}{l} \left( {D_1 } \right) _{00} =\frac{2N^{2}+1}{6}, \quad \left( {D_1 } \right) _{NN} =-\frac{2N^{2}+1}{6} \\ \left( {D_1 } \right) _{jj} =-\frac{x_j }{2\left( {1-x_j^2 } \right) }, \quad j=1,2,3,\dots ,N-1 \\ \left( {D_1 } \right) _{ij} =\frac{c_i \left( {-1} \right) ^{i+j}}{c_j \left( {x_i -x_j } \right) }, \quad i\ne j, i,j=0,1,2,\dots ,N \\ \end{array} \end{aligned}$$
(23)

where

$$\begin{aligned} c_i =\left\{ {{\begin{array}{ll} 2, &{}\quad i=0 \hbox { or } N \\ 1, &{}\quad \hbox {otherwise} \\ \end{array} }} \right. . \end{aligned}$$
(24)

The second-order differentiation matrix is defined as \(\mathbf{D}_{2}= (\mathbf{D}_{1})^{2}\) and so on for the higher-order differentiation matrices.

4 Application of CCM

In order to apply the CCM, firstly, we change the range of the independent variables in Eqs. (19)–(21) from \(X\in \) [0, 1] to \(\xi = (2X-1)\in [-1, 1]\).

The displacement functions of the sandwich beam for harmonic vibration are written as:

$$\begin{aligned} \left\{ {{\begin{array}{l} {\bar{{U}}(X,\tau )=U( X )\mathrm{e}^{i\omega _n \tau }} \\ {\bar{{W}}(X,\tau )=W( X )\mathrm{e}^{i\omega _n \tau }} \\ {\psi (X,\tau )=\phi ( X )\mathrm{e}^{i\omega _n \tau }} \\ \end{array} }} \right. \end{aligned}$$
(25)

in which \(i=\sqrt{-1}\). By applying Eq. (25) in Eqs. (19)–(21) and using the range of the independent variables \(\xi \), the governing equations are written as

$$\begin{aligned} \left\{ {{\begin{array}{l} {4a_{11} \frac{\partial ^{2}U}{\partial \xi ^{2}}+4b_{11} \frac{\partial ^{2}\phi }{\partial \xi ^{2}}=-\left( {\bar{{I}}_0 \omega _n ^{2}U+\bar{{I}}_1 \omega _n ^{2}\phi } \right) } \\ {\eta ^{2}K_\mathrm{s} a_{55} \left( {4\frac{\partial ^{2}W}{\partial \xi ^{2}}+2\eta \frac{\partial \phi }{\partial \xi }} \right) -\bar{{K}}_\mathrm{w} W+4\bar{{K}}_\mathrm{p} \frac{\partial ^{2}W}{\partial \xi ^{2}}=-\bar{{I}}_0 \eta ^{2}\omega _n ^{2}W} \\ {4b_{11} \frac{\partial ^{2}{U}}{\partial \xi ^{2}}+4d_{11} \frac{\partial ^{2}\phi }{\partial \xi ^{2}}-K_s a_{55} \eta \left( {2\frac{\partial W}{\partial \xi }+\eta \phi } \right) =-\left( {\bar{{I}}_1 \omega _n ^{2}U+\bar{{I}}_2 \omega _n ^{2}\phi } \right) } \\ \end{array} }} \right. \end{aligned}$$
(26)

and the boundary conditions are stated as

Clamped (C):

$$\begin{aligned} W=U=\phi =0. \end{aligned}$$
(27)

Hinged (H):

$$\begin{aligned} W=U=2b_{11} \frac{\partial U}{\partial \xi }+2d_{11} \frac{\partial \phi }{\partial \xi }=0. \end{aligned}$$
(28)

Then, based on the Chebyshev differentiation matrix, the left-hand sides of Eq. (26) are expressed as

$$\begin{aligned} \begin{aligned} \mathbf{EM1}&=4b_{11}([0\,\, 0\,\, 1]\otimes \mathbf{D}_2 )+4a_{11} ([1\,\, 0\,\, 0]\otimes \mathbf{D}_2 ) \\ \mathbf{EM2}&=\eta ^{2}K_\mathrm{s} a_{55} [4([0\,\,1\,\, 0]\otimes \mathbf{D}_2 )+2\eta ([0\,\,0\,\, 1]\otimes \mathbf{D}_1 )] \\&\quad -\bar{{K}}_\mathrm{w} ([0\,\,1\, \,0]\otimes \mathbf{I}_1 )+4\bar{{K}}_\mathrm{p} ([0\,\, 1\,\, 0]\otimes \mathbf{D}_2 ) \\ \mathbf{EM3}&=4d_{11} ([0\,\, 0\,\, 1]\otimes \mathbf{D}_2 )+4b_{11} ([1\,\, 0\,\, 0]\otimes \mathbf{D}_2 ) \\&\quad -K_\mathrm{s} a_{55} \eta [\eta ([0\,\, 0\,\, 1]\otimes \mathbf{I}_1 )+2([0\,\,1\,\, 0]\otimes \mathbf{D}_1 )] \\ \end{aligned} \end{aligned}$$
(29)

in which \(\otimes \) is the Kronecker product; \(\mathbf{I}_{1}\) is identity matrix of size \((N+1)\times (N+1)\); and \(\mathbf{D}_{2}=(\mathbf{D}_{1})^{2}\) is the second-order differentiation matrix. The size of EM1, EM2 and EM3 is \((N+1)\times 3(N+1)\), where N is the number of Chebyshev points. By stacking these matrices together, the global matrix EM of size \(3(N+1)\times 3(N+1)\) is achieved:

$$\begin{aligned} \mathbf{EM}\cdot {\varvec{\updelta }} =\left[ {{\begin{array}{l} \mathbf{EM1} \\ \mathbf{EM2} \\ \mathbf{EM3} \\ \end{array} }} \right] \cdot {\varvec{\updelta }} \end{aligned}$$
(30)

where \({\varvec{\updelta }}\) represents the transpose displacement vector in the form of

$$\begin{aligned} {\varvec{\updelta }} =\left[ {U_1\;U_2 \dots U_{N+1} \;W_1 \;W_2 \dots W_{N+1} \;\phi _1 \;\phi _2 \dots \phi _{N+1} } \right] ^\mathrm{T}. \end{aligned}$$
(31)

Therefore, Eq. (26) can be expressed as

$$\begin{aligned} \left[ {{\begin{array}{l} \mathbf{EM1} \\ \mathbf{EM2} \\ \mathbf{EM3} \\ \end{array} }} \right] \cdot {\varvec{\updelta }} =-\omega _n ^{2}\left[ {{\begin{array}{ccc} {\bar{{I}}_0 \mathbf{I}_1 }&{}\quad \mathbf{0}&{} \quad {\bar{{I}}_1 \mathbf{I}_1 } \\ \mathbf{0}&{} \quad {\bar{{I}}_0 \eta ^{2}{} \mathbf{I}_1 }&{} \quad \mathbf{0} \\ {\bar{{I}}_1 \mathbf{I}_1 }&{} \quad \mathbf{0}&{} \quad {\bar{{I}}_2 \mathbf{I}_1 } \\ \end{array} }} \right] \cdot {\varvec{\updelta }}. \end{aligned}$$
(32)

Based on the CCM, the boundary conditions of the beam at both ends are listed in Table 2.

Table 2 Boundary condition (B.C.) matrices
Full size table

The displacements of the beam at the left end are defined as \(U_{1}\), \(W_{1}\) and \(\phi _{1}\), and those at the right end are \(U_{N+1}\), \(W_{N+1}\) and \(\phi _{N+1}\). Hence, the displacement vector in Eq. (31) is reworded as

$$\begin{aligned} \bar{\varvec{\updelta }}=\left[ {U_1 \; W_1 \; \phi _1 \; U_{N+1} \; W_{N+1} \; \phi _{N+1} \; U_2 \; U_3 \dots U_N \; W_2 \; W_3 \dots W_N \; \phi _2 \; \phi _3 \dots \phi _N } \right] ^\mathrm{T}. \end{aligned}$$
(33)

By introducing Eq. (33) in Eq. (32), the algebraic system takes the form of

$$\begin{aligned} \left[ {{\begin{array}{cc} {\mathbf{S}_\mathrm{bb} }&{}\quad {\mathbf{S}_\mathrm{bg} } \\ {\mathbf{S}_\mathrm{gb} }&{}\quad {\mathbf{S}_\mathrm{gg} } \\ \end{array} }} \right] \left\{ {{\begin{array}{c} {\bar{{{\varvec{\updelta }} }}_\mathrm{b} } \\ {\bar{{{\varvec{\updelta }} }}_\mathrm{g} } \\ \end{array} }} \right\} =\omega _n ^{2}\left[ {{\begin{array}{cc} \mathbf{0}&{}\quad \mathbf{0} \\ \mathbf{0}&{}\quad {\mathbf{M}_\mathrm{gg} } \\ \end{array} }} \right] \left\{ {{\begin{array}{cc} {\bar{{{\varvec{\updelta }} }}_\mathrm{b} } \\ {\bar{{{\varvec{\updelta }} }}_\mathrm{g} } \\ \end{array} }} \right\} \end{aligned}$$
(34)

where the subscripts “g” and “b” refer to the points used for writing the collocation analog of governing equations and boundary conditions, respectively. \(\mathbf{S}_\mathrm{bb}\) is a matrix of size \(6\times 6\); \(\mathbf{S}_\mathrm{bg}\) is a matrix of size \(6\times [3(N+1)-6]\); \(\mathbf{S}_\mathrm{gb}\) is a matrix of size \([3(N+1)-6]\times 6\); and \(\mathbf{S}_\mathrm{gg}\) is a matrix of size \([3(N+1)-6]\times [3(N+1)-6]\). \(\mathbf{M}_\mathrm{gg}\) is the dimensionless inertia matrix related to the right-hand side of Eq. (32), and has the same size as \(\mathbf{S}_\mathrm{gg}\). This matrix is written as

$$\begin{aligned} \mathbf{M}_\mathrm{gg} =-\left[ {{\begin{array}{ccc} {\bar{{I}}_0 \mathbf{I}}&{} \quad \mathbf{0}&{} \quad {\bar{{I}}_1 \mathbf{I}} \\ \mathbf{0}&{} \quad {\bar{{I}}_0 \eta ^{2}{} \mathbf{I}}&{} \quad \mathbf{0} \\ {\bar{{I}}_1 \mathbf{I}}&{} \quad \mathbf{0}&{} \quad {\bar{{I}}_2 \mathbf{I}} \\ \end{array} }} \right] \end{aligned}$$
(35)

where I is identity matrix of size \((N-1)\times (N-1)\). The first component in Eq. (34) is

$$\begin{aligned} \bar{{{\varvec{\updelta }} }}_\mathrm{b} =-\mathbf{S}_\mathrm{bb} ^{-1}{} \mathbf{S}_\mathrm{bg} \bar{{{\varvec{\updelta }} }}_\mathrm{g} \end{aligned}$$
(36)

and the second component is

$$\begin{aligned} \mathbf{S}_\mathrm{gb} \bar{{{\varvec{\updelta }} }}_\mathrm{b} +\mathbf{S}_\mathrm{gg} \bar{{{\varvec{\updelta }} }}_\mathrm{g} =\omega _n ^{2}{} \mathbf{M}_\mathrm{gg} \bar{{{\varvec{\updelta }} }}_\mathrm{g} \end{aligned}$$
(37)

From the above formulations, the algebraic eigenvalue equations for free vibration of the MFCS beam are expressed as

$$\begin{aligned} \left( {\mathbf{S}_\mathrm{gb} \mathbf{S}_\mathrm{bb} ^{-1}{} \mathbf{S}_\mathrm{bg} -\mathbf{S}_\mathrm{gg} +\omega _n ^{2}{} \mathbf{M}_\mathrm{gg} } \right) \bar{{{\varvec{\updelta }} }}_\mathrm{g} =\mathbf{0}. \end{aligned}$$
(38)
Table 3 Comparison of dimensionless fundamental natural frequencies of FGM beam under different boundary conditions (\(L/h_\mathrm{t}=10\))
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Table 4 Comparison of dimensionless fundamental natural frequencies of metal foam beam under different boundary conditions (\(\vartheta _{0}=0.5\))
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Table 5 Comparison of dimensionless fundamental frequencies of metal foam beam (C–C, \(L/h_\mathrm{t}=20\))
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Table 6 Convergence of dimensionless fundamental natural frequency \(\omega _{1}\) of MFCS beam (\(\vartheta _{0}=0.5\), \(L=10h_\mathrm{t}\), \(K_\mathrm{w}=10^{8}\), \(K_\mathrm{p}=10^{6}\), \(h_\mathrm{c}/h_\mathrm{l}=6\))
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Table 7 The first three dimensionless natural frequencies of MFCS beam (\(\vartheta _{0}=0.5\), \(L=10h_\mathrm{t}\), \(K_\mathrm{w}=10^{8}\), \(K_\mathrm{p}=10^{6}\), \(h_\mathrm{c}/h_\mathrm{l}=6\))
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5 Results and discussion

Due to the inexistence of vibration research of MFCS beams, we consider an FGM beam to validate the present analysis. The material parameters used are as follows: (Aluminum: \(E_\mathrm{m}=70\) GPa, \(\rho _\mathrm{m}=2702\,\hbox {kg/m}^{3}\), \(\nu _\mathrm{m} =0.3\); Alumina: \(E_\mathrm{c}=380\) GPa, \(\rho _\mathrm{c}=3960\,\hbox {kg/m}^{3}\), \(\nu _\mathrm{c} =0.3\)). The dimensionless fundamental natural frequencies \(\left( {\omega _1 =\varOmega _1 L\sqrt{{\rho _\mathrm{m} }/{E_\mathrm{m} }}} \right) \) from the present analysis are compared with those given by Wattanasakulpong and Chaikittiratana [73], as shown in Table 3. The comparison result depicts very good agreement between them.

Table 8 The dimensionless fundamental natural frequency \(\omega _{1}\) of MFCS beam for different foundation parameters (\(\vartheta _{0}=0.5\), \(L=10h_\mathrm{t}\), \(h_\mathrm{c}/h_\mathrm{l}=6\))
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Fig. 3
figure 3

Dimensionless fundamental natural frequency versus slenderness ratio (\(k_\mathrm{w}=0\), \(k_\mathrm{p}=0\), \(\vartheta _{0}=0.5\), \(h_\mathrm{c}/h_\mathrm{l}=6\)): a C–C, b C–H, c H–H

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To further examine the validity of the present analysis, another two comparison studies are conducted on metal foam beams with different boundary conditions, as shown in Tables 4 and 5. In Table 4, the following material parameters are used: Young’s modulus \(E_{1}=200\) GPa, Poisson’s ratio \(\nu =1/3\) and mass density \(\rho _1 =7850\,\hbox {kg/m}^{3}\). In Table 5, the following material parameters are used: Young’s modulus \( E_{1}=130\) GPa, Poisson’s ratio \(\nu =0.34\) and mass density \(\rho _1 =8960\,\hbox {kg/m}^{3}\). The dimensionless natural fundamental frequencies \(\omega _1 =\varOmega _1 L\sqrt{{\rho _1 \left( {1-\nu ^{2}} \right) }/{E_1 }}\) are calculated and compared with the available results. It can be observed that the present results agree well with those given by Chen et al. [2] and Kitipornchai et al. [74], which bespeaks the validity of the present analysis.

In what follows, the MFCS beam shown in Fig. 1 will be dealt with. If not specified, the following geometry and material parameters are used: \(b=0.2\) m, \(h_\mathrm{t}=0.1\) m, \(E_{1}=200\) GPa, \(\rho _1 =7850\) kg/m\(^{3}\), \(\nu =0.33\). The dimensionless natural frequency is introduced as:

$$\begin{aligned} \omega _n =\varOmega _n L\sqrt{\frac{I_{00} }{A_{00} }}\left( {n=1,2,3\ldots } \right) . \end{aligned}$$

Table 6 shows the convergence of dimensionless fundamental natural frequencies of MFCS beams with different boundary conditions and different foam distributions. It is seen that the method has very high rate of convergence; when \(N=9\), the natural frequencies for different boundary conditions and different foam distributions converge. Therefore, the number of Chebyshev points, namely, \(N=9\), is adopted in the following calculations.

Table 7 illuminates the first three dimensionless natural frequencies of MFCS beams with various foam distributions and boundary conditions. It is noted that the foam-I beam has the highest natural frequency, while the foam-II beam has the lowest natural frequency among different types of foam distribution. This indicates that the foam-I provides the largest structural stiffness, while the foam-II leads to the smallest structural stiffness. In addition, the natural frequency of C–C sandwich beam is the highest while that of H–H sandwich beam is the lowest. This can be expected because C–C edges provide higher level of constraint, while H–H edges provide relatively moderate level of constraint.

Table 8 gives the dimensionless fundamental natural frequencies of the MFCS beam with different foundation parameters. When \(K_\mathrm{w}=0\) and \(K_\mathrm{p}=0\), it means that the Winkler–Pasternak foundation vanishes. The table shows that when the Winkler foundation exists, the natural frequency of the MFCS beam increases. Moreover, the addition of the Pasternak foundation results in the higher natural frequency of the MFCS beam. This phenomenon shows that the Winkler–Pasternak foundation makes the MFCS beam stiffer.

Fig. 4
figure 4

Dimensionless fundamental natural frequency versus foam coefficient \(\vartheta _{0}\) (\(L/h_\mathrm{t}=10\), \(k_\mathrm{w}=0\), \(k_\mathrm{p}=0\), \(h_\mathrm{c}/h_\mathrm{l}=6\)): a C–C, b C–H, c H–H

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Figure 3 shows the dimensionless fundamental natural frequency versus slenderness ratio for different boundary conditions and foam distributions. It can be seen that the dimensionless fundamental natural frequency decreases with increasing slenderness ratio. Additionally, when the slenderness ratio is small, the descent rate of natural frequency is rapid. With the further rise in slenderness ratio, the natural frequency changes less and less significantly.

Figure 4 gives the dimensionless fundamental natural frequency versus foam coefficient for different boundary conditions and foam distributions. It is seen that different foam distributions show different variation tendency with foam coefficient. With the increase in foam coefficient, the natural frequency of the foam-I beam keeps rising while that of the foam-II beam keeps falling. As for the foam-III beam, the natural frequency changes slightly with the foam coefficient.

Fig. 5
figure 5

Dimensionless fundamental natural frequency versus core-to-face ratio \(h_\mathrm{c}/h_\mathrm{l}\) (\(\vartheta _{0}=0.5\), \(L=10h_\mathrm{t}\), \(K_\mathrm{w}=10^{8}\), \(K_\mathrm{p}=10^{6})\): a C–C, b C–H, c H–H

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Figure 5 shows the dimensionless fundamental natural frequency versus core-to-face ratio \(h_\mathrm{c}/h_\mathrm{l}\) for various foam distributions and boundary conditions. It can be seen that the natural frequency increases initially and then decreases with the core-to-face ratio for various boundary conditions. In addition, the frequency difference of MFCS beams with different foam distributions becomes more and more obvious with the increase in core-to-face ratio. This indicates that foam distribution will play more important effect on vibration characteristics of MFCS beams when the core-to-face ratio is high.

6 Conclusions

In this paper, the Chebyshev collocation method is applied to study free vibration of sandwich beams with metal foam core. The model is proposed in the framework of the Timoshenko beam theory. Three types of foam distribution are taken into account. The governing equations and boundary conditions are obtained using Hamilton’s principle. The study shows:

  1. (i)

    The Chebyshev collocation method has very high rate of convergence and achieves high precision within the range allowed by the effective number of bits of computers for free vibration of MFCS beams.

  2. (ii)

    Among different foam distributions, foam-I provides the largest stiffness, while foam-II leads to the smallest stiffness of MFCS beams.

  3. (iii)

    With the rise in core-to-face ratio, natural frequencies of MFCS beams increase initially and then decrease, and foam distribution plays more and more important effect on the vibration characteristics.

  4. (iv)

    For the foam-I sandwich beam, the natural frequency increases with the foam coefficient. For the foam-II beam, however, the natural frequency decreases with the foam coefficient. As to the foam-III beam, the natural frequency is not sensitive to the foam coefficient.

  5. (v)

    The Winkler–Pasternak elastic foundation makes the MFCS beam stiffer and leads to the higher natural frequency.