Experimental and Numerical Investigation on

Exp Tech https://doi.org/10.1007/s40799-017-0220-3

Experimental and Numerical Investigation on Vibration of Sandwich Plates with Honeycomb Cores Based on Radial Basis Function X. Dai 1,2 & X. Shao 2 & C. Ma 1 & H. Yun 1 & F. Yang 2 & D. Zhang 1

Received: 21 February 2017 / Accepted: 27 October 2017 # The Society for Experimental Mechanics, Inc 2017

Abstract The vibration characteristic of the sandwich plate with a honeycomb core was investigated by experimental measurements and numerical calculation based on radial basis function (RBF). RBF method was used not only in the meshless approach but also in the post-processing of the experimental data. During the experiment, amplitudefluctuation electronic speckle pattern interferometry was applied to access the resonant frequencies and the corresponding vibration mode shapes simultaneously. Then RBF method was used to improve the quality of patterns and reconstruct the out-of-plane vibration amplitude after fringe analysis. As for numerical calculation, the modal parameters were numerically predicted using the first-order shear deformation theory. The computation approach was based on collocation with multi-quadric radial basis function. To understand the influence of the thickness of face sheet on dynamic behaviors, three types of specimens with different thickness were tested and analyzed. Of particular interest was that the vibration modes show veering due to the thickness increment. Furthermore, the numerical predicted results were compared with the experimental measurements for the first five modes. They are in good agreement with each other for resonant frequencies, mode shapes and relative out-of-plane amplitudes.

* X. Dai [email protected]

1

School of Transportation and Vehicle Engineering, Shandong University of Technology, Zibo 255049, People’s Republic of China

2

Department of Engineering Mechanics, Southeast University, Nanjing 210096, People’s Republic of China

Keywords Vibration . Honeycomb . Electronic speckle pattern interferometry . First shear deformation theory . Radial basis function

Introduction Sandwich plates with honeycomb cores have been received much attention in recent years because of their high strength to weight and stiffness to weight ratios, excellent heat resistance and favorable energy-absorbing capacity. Of the various materials used in honeycomb cells, glass-fiber honeycomb structures have been widely applied in the aerospace, automotive, marine and light weight structures. Usually, such kinds of structures are subjected to dynamic loadings. Furthermore, in many cases, the large amplitude excited at the resonant frequency can lead to the catastrophic failure of structures. Therefore, it is necessary to understand the dynamic behaviors of the glass-fiber honeycomb laminated plates. Vibration testing can provide information on prevention of failure. There are some methods that can be used for vibration testing, such as transducer method [1], holography [2, 3], digital image correlation (DIC) [4], finge projection [5], Laser Doppler Vibrometer [6] and electronic speckle pattern interferometry (ESPI) [7]. Among all these methods, ESPI is one of the most popular methods for vibration testing because of its outstanding features of non-contact, non-destructive, full-field and high spatial sensitivity. There are three different basic techniques that can be used in ESPI for vibration measurement: the time-averaged method, the stroboscopic method, and the pulsed laser. However, because the measurement system is low in cost and easy to set up, the time-averaged method is most widely used for vibration

Exp Tech

measurement [8, 9].There are three different imageprocessing methods based on the time-averaged technique: the video-signal-addition method, the videosignal-subtraction method, and the amplitude-fluctuation method. And the amplitude-fluctuation ESPI (AF-ESPI) can produce a better visibility and higher resolution of fringe pattern than the other two methods [10]. In this research, we employ the optical method based on AFESPI to study the vibration of glass-fiber honeycomb laminated plates. Numerical calculation is another way to understand the vibration characteristic of the laminated plates. Several theories have been proposed to describe the deformation of laminated plates. The classical laminate theory (CLT) is a simple one which neglects the transverse shear deformation. The first-order shear deformation theory (FSDT) is a typical theory which was developed from the theory of Mindlin [11]. And it has been applied for different kinds of composite structures [12, 13]. The higher-order shear deformation theories (HSDTs) are also well-known and have been widely used for composite laminates [14, 15]. Another popular set of theories are the layerwise theory [16, 17] and the zig-zag theory [18–20], which have been applied on multilayered plates and shells. In order to study the static and dynamic responses, different numerical tools were developed for the analysis of laminated composite plates. Finite element methods (FEM) are well-known. Also, the analysis of plates by FEM has been fully established [21]. However, element based approaches are usually sensitive to element shapes and

Fig. 1 Specimens with different thickness: (a) Geometric dimension and configuration of specimens, (b) real specimens

Table 1 Material properties

Properties

Values

Density (kg/m3) Elastic modulus (GPa)

ρ E1

Shearing modulus of elasticity (GPa) Poisson’s ratio

G13

1.68 × 103 10.920 0.545 1.173 1.173

G23

0.218

E2 G12

0.25

types, and they normally spend the majority of its time in creating the mesh. Meshless methovhds can establish a system of algebraic equations for the whole problem domain without meshing [22]. Recently, radial basis function (RBF) methods have been suggested for the analysis of laminated plates as alternative methods [23]. RBF is a famous means for scattered data interpolation and multivariate approximation. Inasmuch as RBFs involve a single independent variable regardless of the dimension of the problem [24] and the models have a good stability in some cases, they have been applied extensively in many fields [25–27]. There are different types of radial basis functions, such as the multi-quadric (MQ), the polyharmonic spline, the Gaussian and many others. Among these functions, the Hardy’s MQ method has exponential convergence for approximation of functions and gives high accuracy with a relatively small number of grid nodes. For these reasons, the MQ method was applied to engineering problems by researchers. And it

Exp Tech Fig. 2 Schematic of fabrication process of honeycomb sandwich panels

was also used with success to the analysis of laminated plates [12]. Further, a number of mathematical theories were proposed to investigate this method on error estimate, optimal shape parameter, traditional and effective condition numbers and so on. In the present work, RBF method was used not only in the meshless approach but also in the post-processing of the experimental data. During the experiment, RBF method was used to improve the quality of patterns and reconstruct the out-of-plane vibration amplitude after fringe analysis. As for numerical calculation, the FSDT and MQ-based RBF method were combined to analyze the vibration characteristic of honeycomb laminated plates.

To understand the influence of the thickness of face sheet on dynamic behaviors, three types of specimens with different thickness were tested and analyzed. Moreover, the numerical results were presented and compared with the data obtained by AF-ESPI. They are in good agreement in terms of resonant frequencies, mode shapes for the out-ofplane vibration mode. The paper is organized as follows: radial basis function method and the first-order shear deformation theory are briefly introduced in the next sections; in forth section specimen is presented and experimental techniques are described in detail; in fifth section results are discussed and conclusions are presented in the last section.

Fig. 3 Measurement system: (a) modal testing arrangement, (b) AF-ESPI set-up for out-of-plane displacement, (c) real experimental set-up

Exp Tech M where f f i gM i¼1 are the data at nodes fX i gi¼1 . Once coefficients αi are found, equation (1) can be used to estimate value of the function at any point.

Radial Basis Function Interpolation Theory Radial basis function interpolation is a powerful tool to approximate multivariate functions. The basic idea is to interpolate a function by given values on a grid. To interpolate a n function by given values on nodes fX i gM i¼1 , Xi ∈ ℝ , where n is some positive integer, one uses an ansatz of a linear combination of shifted RBFs centered in each grid point, i.e. s ðX Þ ¼ ∑M i¼1 ai ϕ ðk X −X i kÞ

ð1Þ

where αi are the unknown RBF weighting coefficients and ∥X − Xi∥denotes the Euclidean distance between every two points of X and Xi. ϕ(∥X − Xi∥) is a single radial symmetric function and called radial basis function. There are several commonly used RBFs: the multi-quadric pffiffiffiffiffiffiffiffiffiffiffiffiffiffi (MQ) ϕðrÞ ¼ r2 þ c2 ; the thin plate spline (TPS) ϕ(r) = r2 log r, the Gaussian ϕ(r) = exp(−r2/c2),the bi-harmonic ϕ(r) = r and the tri-harmonic ϕ(r) = r 3 ,where r = ∥ X − X i ∥ is Euclidean distance, and c is the shape parameter. MQ, which is one of the most effective basis functions for two variables, was used in this research. The coefficients αi can be determined by 2 32 3 2 3 ϕ11 ϕ12 ⋯ ϕ1M f1 α1 ⋯ ϕ2M 76 α2 7 6 f 2 7 6 ϕ21 ϕ22 ¼ ð2Þ 4 ⋮ ⋮ ⋯ ⋮ 54 ⋮ 5 4 ⋮ 5 αM fM ϕM 1 ϕM 2 ⋯ ϕMM

Fig. 4 Measurement system and experimental procedure: (a) original vibration fringe pattern, (b) fringe pattern after RBF-filter, (c) central lines of bright and dark fringes, (d) relationship between interference fringes and the amplitude of vibration, and (e) continuous amplitude after reconstruction

The Eigenproblem and its Solution The Kansa’s unsymmetric collocation method, which has shown to give high accuracy with a relatively small number of grid nodes, was used in this paper. Consider a linear elliptic boundary-valued problem with a domain Ω ⊂ ℝn Lu ðxÞ ¼ f ðxÞ in Ω

ð3Þ

Bu ðxÞ j∂Ω ¼ g ðxÞ on ∂Ω

ð4Þ

where u(x) is the displacement in the global system, L and B are linear differential operators in the domain Ω and on the boundary ∂Ω, respectively. And f and g represent the external forces applied on the plate and the boundary conditions applied along the perimeter of the plate, respectively. For an eigenvalue problem, the operator equations in equations (3) and (4) take the following alternative forms: Lu þ λu ¼ 0 in Ω

ð5Þ

Bu ¼ 0 on ∂Ω

ð6Þ

where λ is called the eigenvalue, which is to be determined along with the eigenvector u.

Exp Tech

To solve the eigenproblem by RBF, the approximation of displacement u(x) is defined as ~uðxÞ ¼ ∑Nj¼1 α j ψ ðxÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  2  N x−x j þ y−y j þ c2 ¼ ∑ j¼1 α j

ð7Þ

NI nodes in the interior of the domain and NB nodes on the boundary are selected as sampling points during the solution, then the total numbers of nodes N = NI + NB. For the points in the domain, the eigenproblem can be defined as ∑Nj¼1 α j Lψðxi Þ ¼ λ~uðxi Þ;

i ¼ 1; ⋯; N I

ð8Þ

The problem can also be expressed in matrix form as follows:  I  I A L α¼λ α ð10Þ B 0 w h e r e LI ¼ Lψ½ð∥xN I −yi ∥2 ÞN I N , B ¼ Bψ½ð∥xN I þ1 −yi ∥2 ÞN B N , AI ¼ ψ½ð∥xN I −yi ∥2 ÞN I N , and α = [α1, α2, ⋯, αN]T. Then the generalized eigenvalues and eigenvectors of the matrix can be calculated based on the corresponding algorithm.

First-Order Shear Deformation Theory

Similarly, we can get the following function for the boundary points

The first-order shear deformation theory is based on the displacement field

∑Nj¼1 α j Bψðxi Þ ¼ 0; i ¼ N I þ 1; ⋯; N B

u ¼ u0 þ zϕx ; v ¼ v0 þ zϕy ;

Fig. 5 Mode shapes and frequencies of specimen H0 obtained from AF-ESPI and FSDT

ð9Þ

w ¼ w0

ð11Þ

Exp Tech

where (u0, v0, w0) are the displacements of a point on the reference surface, and (ϕx, ϕy) are the rotations of the transverse normal about the positive y-axis and negative x-axis, respectively. The equilibrium equations of the plate subjected to a transverse distributed load are defined as 8 > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > :

ð17Þ ∂N x ∂N xy ∂2 u ∂2 ϕ þ ¼ I 0 2 þ I 1 2x ∂x ∂y ∂t ∂t 2 2 ∂ ϕy ∂N y ∂N xy ∂v þ ¼ I0 2 þ I1 2 ∂y ∂x ∂t ∂t ∂Qx ∂Qy ∂2 w þ þ q ¼ I0 2 ∂x ∂y ∂t ∂M x ∂M xy ∂2 u ∂2 ϕ þ −Qx ¼ I 1 2 þ I 2 2x ∂x ∂y ∂t ∂t ∂2 ϕy ∂M xy ∂M y ∂2 v þ −Qy ¼ I 1 2 þ I 2 2 ∂x ∂y ∂t ∂t

For free vibration problems we set q = 0 and assume harmonic solutions in terms of displacements w, ϕx, ϕy in the form ð12Þ

Qx Qy

¼ k ½ A55

9 2 8 A11 < Nx = Ny ¼ 4 A12 ; : N xy A16

A12 A22 A26

9 8 ∂w > > = < ϕx þ ∂x A44  ∂w > > ; : ϕy þ ∂y 8 9 ∂u0 > > > > > 3> 2 > > ∂x > > A16 < B11 = ∂v0 A26 5 þ 4 B12 ∂y > > > A66 > B16 > > ∂u0 ∂v0 > > > > : ; þ ∂y ∂x

ð13Þ

B12 B22 B26

8 9 ∂ϕx > > > > > > > 3> > > ∂x > B16 > < = ∂ϕy B26 5 ∂y > > > B66 > > > ∂ϕx ∂ϕy > > > > > > þ : ; ∂y ∂x

ð14Þ 8 9 2 B11 < Mx = My ¼ 4 B12 : ; M xy B16

B12 B22 B26

8 9 ∂u0 > > > > > 3> 2 > > ∂x > > B16 < D11 = ∂v0 B26 5 þ 4 D12 ∂y > > > B66 > D16 > > ∂u0 ∂v0 > > > > : ; þ ∂y ∂x

D12 D22 D26

8 9 ∂ϕx > > > > > > > > 3> > ∂x > D16 > < = ∂ϕy D26 5 ∂y > > > D66 > > > ∂ϕx ∂ϕy > > > > > > þ : ; ∂y ∂x

ð15Þ h 2



 1; z; z2 ρðzÞdz

8 < wðx; y; t Þ ¼ W ðx; yÞeiωt ϕ ðx; y; t Þ ¼ Ψ x ðx; yÞeiωt : x ϕy ðx; y; t Þ ¼ Ψ y ðx; yÞeiωt

ð18Þ

where ω is the frequency of natural vibration. As for honeycomb panels, equation (17) may be rewritten as [29]

where Nx, Ny and Nxy are the in-plane forces, Qx and Qy are the transverse shear forces, Mx, My and Mxy are the bending moments, (I0, I1, I2) are the inertia terms. They are respectively defined as

8  ∂2 ϕy ∂2 ϕ x ∂2 ϕx ∂w ∂2 ϕ > > > −kA D þ D þ ð D þ D Þ ϕ þ ¼ I 2 2x 11 66 12 66 55 x > 2 2 > ∂x ∂y ∂x∂y ∂t ∂x > >  < ∂2 ϕ y ∂2 ϕy ∂2 ϕ y ∂2 ϕx ∂w −kA44 ϕy þ D22 2 þ D66 2 þ ðD12 þ D66 Þ ¼ I2 2 > ∂y ∂x ∂x∂y ∂t ∂y > >   > 2 2 > ∂ϕ ∂ϕ ∂ w ∂ w ∂2 w > y x > : þ 2 þ kA44 þ 2 þ qðx; yÞ ¼ I 0 2 kA55 ∂x ∂y ∂x ∂y ∂t

2

∂ 6 ∂x 6 6 ∂ 6 6 −ka44 6 ∂y 6 4 ∂2 ∂2 ka55 2 þ ka44 2 ∂y ∂x 2 3 W 4 Ψx 5 Ψy −ka55

3 ∂ ∂2 ∂2 þ d −ka ð d þ d Þ 66 55 12 66 7 ∂y2 ∂x∂y ∂x2 7 7 ∂2 ∂2 ∂2 7 d 22 2 þ d 66 2 −ka44 7 ðd 12 þ d 66 Þ 7 ∂x∂y ∂y ∂x 7 5 ∂ ∂ ka44 ka55 ∂x ∂y 2 32 3 2 3 W i0 0 0 0 −ω2 4 0 i2 0 5 4 Ψ x 5 ¼ 4 0 5 Ψy 0 0 i2 0

d 11

ð19Þ where aij, dij and (i0,i2) are the non-dimensionalized stiffness A and inertial parameters, which can be expressed as aij ¼ Ehij ,   D I0 I0 ; ρh . d ij ¼ Ehij3 , (i0,i2 Þ ¼ ρh Let us now apply the unsymmetrical radial basis function collocation method, the kinematic unknowns may be then written as 8 w 9 8 9 > = < αj > : ; ; : ϕy > Ψy αj

ð16Þ

where ψj(x) represents the jth radial basis function and   ϕ ϕ αwj ; α j x ; α j y are the jth kinematic unknown coefficients

where k is the shearing correction factor, Aij , B ij and Dij are the in-plane (or membrane) and transverse shear stiffness, coupling stiffness and bending stiffness components, which can be calculated according to Ref. [28]. For the case of a symmetric cross-ply laminated sandwich plate, the equations governing the transverse behavior are uncoupled from those governing the in-plane behavior. Substituting equations (13)-(15) into equation (12), the governing differential equations can be expressed as

of the approximation. Substituting equation (20) into equation (19), this equation can be discretized with radial basis functions as

ðI 0 ; I 1 ; I 2 Þ ¼ ∫− h 2

2 2 ∂ 2 ψ j ð xi Þ ϕ ∂ ψ j ðxi Þ ϕ ∂ ψ j ðxi Þ − þ d 66 ∑Nj¼1 α j x þ ðd 12 þ d 66 Þ∑Nj¼1 α j y 2 2 ∂y ∂x∂y  ∂x ∂ψ ð x Þ i j ϕ ϕ ka55 ∑Nj¼1 α j x ψ j ðxi Þ þ ∑Nj¼1 αwj ¼ −i2 ω2 ∑Nj¼1 α j x ψ j ðxi Þ ∂x

d 11 ∑Nj¼1 αwj

ð21Þ

Exp Tech 2 2 ∂ 2 ψ j ð xi Þ ϕ ∂ ψ j ð xi Þ ϕ ∂ ψ j ð xi Þ þ d 66 ∑Nj¼1 α j y þ ðd 12 þ d 66 Þ∑Nj¼1 α j x − 2 2 ∂x ∂x∂y  ∂y ∂ψ j ðxi Þ ϕ ϕ ¼ −i2 ω2 ∑Nj¼1 α j y ψ j ðxi Þ ka44 ∑Nj¼1 α j y ψ j ðxi Þ þ ∑Nj¼1 αwj ∂y ϕ

The mass matrix is given as

d 22 ∑Nj¼1 α j y

ð22Þ ka55

∂ 2 ψ j ð xi Þ ϕ ∂ψ j ðxi Þ ∑Nj¼1 α j x þ ∑Nj¼1 αwj ∂x ∂x2 ϕ

þka44 ∑Nj¼1 α j y

!

∂ψ j ðxi Þ ∂2 ψ j ðxi Þ þ ∑Nj¼1 αwj ∂y ∂y2

! ¼ −i0 ω2 ∑Nj¼1 αwj ψ j ðxi Þ

ð23Þ We can obtain the following homogeneous algebraic equations, leading to a standard eigenvalue problem for free vibration

−1  ð24Þ ℐ S−ω2 I α ¼ 0 where ℐ, S are the system mass and stiffness matrices, I is the identity matrix, and α is the vibration mode associated with the natural frequency ω. 2

∂ −ka55 6 ∂x 6 6 6 ∂ s¼6 −ka44 6 ∂y 6 4 ∂2 ∂2 ka55 2 þ ka44 2 ∂x ∂y

2

ℐ1 0 ℐ2 6 0 ℐ ¼4 ⋮ ⋱ ⋯ 0

Specimen and Experimental Techniques Specimen The vibration behaviors of honeycomb structures were experimentally investigated and numerically analyzed herein. The geometrical configuration of the five-layered glass-fiber panel with honeycomb core is shown in Fig. 1(a). The length and the width of the panel are 240 mm and 78 mm, respectively. Considering the influence of the thickness of face sheet on dynamic behaviors, three types of specimens with different thickness were selected to be analyzed. As depicted in Fig. 1(a), the specimens are named H0, H1 and H2. The thickness of the top and bottom face sheets is different, while the thickness of the middle face sheet and the honeycombs is the same. Figure 1(b) shows the real testing specimens. The honeycomb core is made with the same glass-fiber of the facesheets. And the material properties of glass-fiber are listed in Table 1. During the experiments, one side of the specimen was clamped and other sides were free for the boundary condition.

3 0 ⋮ 7 0 5 ℐN

ð25Þ

2

3 i0 0 0 where ℐ j ¼ 4 0 i 2 0 5. 0 0 i2 The stiffness matrix is given as 2

sψ1 ðx1 Þ 6 sψ1 ðx2 Þ S¼4 ⋮ sψ1 ðxN Þ

sψ2 ðx1 Þ sψ2 ðx2 Þ ⋮ sψN ðx2 Þ

⋯ ⋯ ⋮ ⋯

3 sψN ðx1 Þ sψN ðx2 Þ 7 5 ⋮ sψN ðxN Þ

ð26Þ

where

∂2 ψ j ðxi Þ ∂2 ψ j ð x i Þ þ d −ka55 d 11 66 ∂x2 ∂y2 ∂2 ψ j ð x i Þ ðd 12 þ d 66 Þ ∂x∂y ∂ψ j ðxi Þ ka55 ∂x

Then the natural frequencies and the modes of vibration can be obtained by solving the generalized eigenproblem given in equation (24)

⋯ ⋱ ⋱ 0

3 ∂2 ψ j ð x i Þ ðd 12 þ d 66 Þ 7 ∂x∂y 7 7 2 2 ∂ ψ j ð xi Þ ∂ ψ j ð xi Þ 7 7 d 22 þ d −ka 66 44 2 2 7 ∂y ∂x 7 5 ∂ψ j ðxi Þ ka44 ∂y

Figure 2 sketches the components and the fabrication process of honeycomb sandwich panels. The glass-fiber face sheet is used to bear the structural loads. The adhesive layer plays a role in bonding at interface, while the honeycomb core is used to reduce the total weight, provide the shear stress transmission, absorb the energy and mitigate the shock. Firstly, polyurethane adhesives were painted on the glassfiber honeycomb cores to provide core-to-facing bonds. Then the orthotropic cross ply glass fiber face sheet was stacked on both sides of the honeycomb core with the polyurethane material having an adhesive property placed in between, and the polyurethane material and the face sheet were heated at the curing temperature of the polyurethane material to cause in initial hardening. Subsequently, the glass fiber face sheet was impregnated with adhesives and the adhesives impregnated into the sheet were hardened by hot-pressing an entire assembly thus prepared under specific conditions. According to the procedure, the glass-fiber honeycomb sandwich panel can be manufactured. AF-ESPI Time-averaged ESPI is a widely used method for vibration investigation. One reference image is recorded before

Exp Tech Fig. 6 Mode shapes and frequencies of specimen H1 obtained from AF-ESPI and FSDT

vibration, and it is continuously subtracted from the incoming images after vibration. The vibration fringe can be observed directly. To improve the interferometric fringe visibility and the resolution, AF-ESPI which combines the advantages of the time-averaged and subtraction methods was proposed. A typical out-of-plane vibration measurement system based on AF-ESPI consists of two parts: modal testing system and optical arrangement, which are shown in (Fig. 3(a) and (b)), respectively. A specific period signal is generated by a signal generator, and the input signal is transmitted to an exciter to produce force after being amplified by a power amplifier. A Michelson-type optical set-up is used for out-of-plane vibration measurement. The light from a laser source (50 mW, 533 nm) is expanded by an expander and separated into two with a beam splitter. The transmitted and the reflected beams fall on a flat mirror and the surface of the specimen, respectively. Then the beams are reflected back to the beam splitter, where the two beams are recombined along a common path and directed into a CCD video camera (1280(H) × 1024(V),

USB 2.0 uEye RE). The real experimental set-up is given in Fig. 3(c). The light intensity of the fringe pattern obtained from the test arrangement can be expressed as   2   1 pffiffiffiffiffiffiffiffiffi 4π 4π  I¼ I 0 I R cosðΔϕÞ ΔΑ J 0 Α  ð27Þ   2 λ λ where I0 is the object light intensity, IR is the reference light intensity, Δϕ is the phase difference between object and reference light, λ is the wavelength of laser, Α is the vibration amplitude and ΔΑ is the amplitude increment. It can be found that the amplitude obtained by AF-ESPI is controlled by a zero-order Bessel function of the first kind J0. The quantitative relationship between the fringe and the vibration amplitude is illustrated in Fig. 4(d). Thus the mode shape and the vibration amplitude can be obtained using AF-ESPI system, and the correspondent resonant frequency can be measured from the signal generator simultaneously.

Exp Tech Fig. 7 Mode shapes and frequencies of specimen H2 obtained from AF-ESPI and FSDT

Results and Discussion The resonant frequencies and mode shapes of the glass-fiber panel with honeycomb core are investigated and numerical analyzed in this section. Figure 4 illustrates the detailed experimental procedure of AF-ESPI. First, a reference image was taken after the specimen was excited by exciter connected with the power amplifier. The second image was recorded subsequently. Then the real time vibration fringe patterns can be observed on the monitor by means of the real-time image subtraction operations. In case the vibrating frequency approached to the resonant frequency, fringe patterns would be stationary and regular. The clear fringe patterns can be obtained by tuning the signal generator carefully. Thus, the resonant frequency and the fringe of correspondent mode shape can be obtained simultaneously. Then pre-filtering RBF method was applied to improve the image-quality of fringe patterns, followed by the skeleton line extracting.

Figure 4(a) and (b) present the original and RBF-filtered fringe patterns of the first mode shape of a vibrating honeycomb plate with an excitation frequency of 103 Hz, respectively. The extracted locating central lines of bright and dark fringes are shown in Fig. 4(c). Following the relationship between amplitude and intensity given in Fig. 4(d), the continuous amplitudes can be rebuilt using RBF interpolation method. The amplitude reconstruction is shown in Fig. 4(e). RBF-based FSDT was applied to determine the mode shapes and the natural frequencies. Also, the numerical calculation data were compared with the experimental results. In FSDT, the shear correction factor was set as 1, which is appropriate to the multi-skin sandwich structures [30]. As for RBF, the shape parameter was selected according to the algorithm based on a convergence analysis [22]. The first eight mode shapes of three types of specimens obtained from AFESPI and FSDT are shown in Figs. 5, 6, and 7. The AF-ESPI experimental results are presented as the pattern of alternating

Exp Tech Table 3

Rate of frequency increment obtained by AF-ESPI 1st

2nd

3rd

4th

5th

6th

7th

8th

rate1, 0(%) 15.38 19.55 3.12 23.54 18.67 6.10 12.91 9.13 rate2, 1(%) 1.90 43.79 26.62 13.87 24.13 9.93 19.58 23.92

Fig. 8 Veering between modes: (a) AF-ESPI, (b) FSDT

bright and dark lines (intereference fringes). The brightest fringes represent the nodal lines of the structure in resonance, and the rest of the fringes are contours of constant vibrating amplitude. And the fringes at different positions have different orders. The further away from the nodal line the fringe is, the order is higher. Furthermore, the fringes become more irregular as the thickness of the specimen is increased, associated with the stiffness. For the FSDT results, the line also denotes the displacement. It can be seen from the figures that the experimental fringe patterns and numerical calculations agree

Table 2

Relative error between AF-ESPI and FSDT 1st

2nd

3rd

4th

5th

6th

7th

8th

errorH0(%) 3.30 3.15 3.70 9.07 7.89 11.85 16.67 11.11 errorH1(%) 3.81 3.57 10.21 9.83 11.66 5.46 19.47 13.09 errorH2(%) 6.54 9.67 10.46 8.75 6.46 11.09 19.74 10.09

well with each other for the specimen H0 and the modes with lower frequencies for H1 and H2. Different from the numerical results, which are all highly symmetrical, fringe patterns are deflected for the mode shapes with higher frequencies of specimens H1 and H2. This is due to the complex geometry of the honeycomb core and to the high transverse anisotropy. Furthermore, higher modal shapes are more sensitive to the configuration and characteristics than lower modal ones. It should be specially mentioned that a very interesting aspect found from the comparison is the variations in the mode shapes. In order to see the veering regions more closely, and to see the variations, representative cases are shown in Fig. 8(a) and (b). This is why the fringe patterns do not match with each other for the modes 6, 7 and 8 between H1 and H2. As illustrated in Fig. 8(a), the veering between the modes 6 and 7 exhibits in the specimens H1 and H2. The main reason of the mode-shape-jumping is that the mass and the stiffness increased nonlinearly due to the thickness increment of the front and rear face sheets. As for vibration mode shapes, the generalized eigenvalues were sorted numerically. And the eigenvalues change with the increment of the mass and the stiffness, followed by the presence of the frequency veering. Then the mode jumping takes place because the mode shapes associated with the natural frequencies switch their orders. However, different from the experimental result obtained by AF-ESPI, the loci veering occurs between modes 4 and 5 for the numerical calculation data. This can be explained by considering that the mass matrix and the stiffness matrix changed with the thickness. Moreover, there are some differences between the theoretical model and the real specimen structure. The results show that the thickness plays an important role in the vibrational response of a honeycomb sandwich panel, it not only changes the mode frequencies but also shifts the mode order. The resonant frequencies for the honeycomb plate measured by AF-ESPI and FSDT are also given in Figs. 5, 6, and 7. To describe the differences between the results obtained from two methods quantitatively, the relative error function was used, which can be expressed as errorHi ¼

 FSDT AFESPI  f  −f Hi

Hi

f AFESPI Hi

 100%; i ¼ 0; 1; 2

ð28Þ

where Hi represents the ith specimen, fFSDT and fAFESPI are the resonant frequencies obtained from FSDT and AF-ESPI,

Exp Tech Fig. 9 Relative amplitudes obtained from AF-ESPI (left) and FSDT (right): (a-1),(b-1) 1st mode; (a-2),(b-2) 2nd mode; (a3),(b-3) 3rd mode; (a-4),(b-4) 4th mode; (a-5),(b-5) 5th mode

Exp Tech

respectively. The relative errors of the first eight modes are given in Table 2. Table 2 indicates that the maximum relative error is less than 11% for the first four modes, and it is also less than 14% in general, except for the 7th mode. For the case of stiffer specimens, the errors on the higher-order modes frequency are noticeable. The difference between experimental and numerical analysis is mainly due to the uncertainty of the

material constants of adhesive between the sheet and the honeycomb core used for numerical computations. In addition, the use of simplification assumptions while building the mathematical model of the structure may cause the predictive models to become distant from the real fabricated specimens. To estimate the impacts of the thickness change on the resonant frequency, the rate of frequency increment

Fig. 10 Comparison of the vibration displacements along the central line: (a) 1st mode, (b) 2nd mode, (c) 3rd mode, (d) 4th mode, (e) 5th mode

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between the neighboring specimens was introduced, which is defined as  n   f − f n iþ1 i n rateiþ1;i ¼  100%; n ¼ 1; 2; ⋯; 8; i f ni ¼ 0; 1:

ð29Þ

where f ni is the nth resonant frequency of the ith specimen. The rate of frequency increment between specimens H0 and H1 and the rate between H1 and H2 are listed in Table 3, respectively. Table 3 indicates that the resonant frequencies increase with the thickness of the face sheet. It also demonstrates that, in comparison with thickness increment of one face sheet, the increasing range of the specimen H2, which both of front and rear sheets thicken, is higher. The reason is that the effect of the specimen stiffness prevails on the specimen mass thus leading to an increase of the natural frequencies by augmenting of the thickness increment of the front and rear face sheets. As for AF-ESPI method, the amplitudes of vibration displacement can be evaluated from the number of fringes based on the relationship between amplitude and intensity. So, the fringe patterns can provide qualitative observations and fullfield quantitative measurement of vibration amplitude. It can be found from Eq. (27) that the intensity of the fringe patterns is proportional to the first kind zero-order Bessel function of amplitude Α. The related out-of-plane vibration displacements of each dark and bright fringe in the mode shapes can be quantitatively calculated by the equation  J0

4π Α λ

¼0

ð30Þ

Then the amplitudes can be obtained according to the local minimum and maximum roots. During the testing, the wavelength of the light source was 533 nm. Take the first mode of the specimen H0 as an example. As shown in Fig. 5, the amplitudes of the first six dark fringes are 103, 235, 368, 501, 634 and 767 nm, respectively. The amplitudes of the first six bright fringes are 164, 299, 433, 566, 700 and 833 nm, respectively. The brightest fringe is the nodal line, and its outof-plane displacement equals to zero. Generally, the first five modes are considered to be very important for the structure. Hereafter, the comparison between experimental results and numerical data was made on these five modes. Following the procedure mentioned above, the continuous amplitudes can be rebuilt using RBF interpolation method. For the 1st mode, the maximum and minimum amplitudes are 1300 nm and zero, respectively. And they are located at the free and fixed ends, respectively. The maximum out-ofplane displacements in two directions of the 2nd mode to the

5th mode are 566 nm and -566 nm, 427 nm and -570 nm, 425 nm and -424 nm, 417 nm and -412 nm, respectively. It is generally known that, during AF-ESPI experiments, the vibration amplitude depends heavily on the excited voltage which is difficult to be simulated for numerical calculation. And the voltage ranges of 0.5–5 V were used to excite the out-of-plane resonant vibration. To eliminate the effect of the voltages, the normalized amplitude was used for comparison. The reconstructed amplitudes of the first five modes are shown in Fig. 9. It can be seen from the comparison that good agreement of relative amplitudes is indicated in both experimental and numerical values. And the numerical results meet the better symmetry than experimental results. The reason is that there are certain disparities in comparing with the ideal ones during testing, such as the boundary condition. The constraint was not a strict fixed support although the bottom end of specimen was clamped by a fixture. To give a detail comparison, a typical position was selected and the data from experimental measurements and numerical calculation were expressed in the same figures. Figure 10 presents the vibration displacements along the central line of the specimen in resonance for the first five modes. The blue dashed line and the red solid line indicate the data obtained from AF-ESPI and FSDT, respectively. It can be found that the FSDT method provides flatter and smoother results than AF-ESPI. Further, the numerical results have a better linearity for the 1st, 2nd and the 5th modes and have a better symmetry for the 3rd and the 4th modes. This is because the noises in the fringe patterns were not removed completely by pre-filtering method. And the amplitudes were contaminated by these small amounts of unfiltered noises during reconstruction. Still, good agreements for the results obtained from AFESPI and FSDT are found.

Conclusion In this paper AF-ESPI was applied to determine the resonant frequencies and corresponding mode shapes of honeycomb plates. The RBF-based pre-filtering method was used to improve the quality of vibration fringe patterns. Out-of-plane amplitudes were reconstructed based on RBF after the extraction of central fringe lines. The MQ-RBF collocation method was also used to analyze the vibration characteristics based on the first-order shear deformation theory. To understand the influence of the thickness of face sheet on dynamic behaviors, the experimental and numerical modal analysis was performed and compared on three types of specimens with different thickness of sheet. In particular, the veering phenomena for the modes of the three specimens were investigated by AFESPI and FSDT. The results show that the thickness plays an important role in the vibrational response of the honeycomb sandwich panel, it not only changes the mode frequencies but

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also shifts the mode order. The experimental results were compared with the numerical calculations. It was shown that the resonant frequencies increase with the thickness of the face sheet and the characteristics of veering are affected to some extent by the increment of the mass and the stiffness. The predicted data of FSDT show a good accordance with the results of AF-ESPI in the first five mode shapes, the relative out-of-plane amplitudes and resonant frequencies. Although the maximum relative error of the resonant frequency is less than 12% in general, the accuracy still needs to be improved. Considering the heterogeneous stacking sequences (three facesheets and two honeycomb cores), analysis on the plates with honeycomb cores using higher-order theories, whose accuracy do not depend on the shear correction factor, and the radial basis function collocation technique can be expected in the future work. Acknowledgements The work is supported by the National Natural Science Foundation of China (under Grant Nos. 11672167, 11472081). We are grateful to the anonymous reviewers for the constructive and helpful comments.

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