Flexural vibration band gaps in periodic stiffened plate structures

Due to their high rigidity-to-weight ratio and economical cost, stiffened plate and shell structures are used extensively in various engineering applications such as bridges, ship hulls and decks, and aircraft structures. In the past few decades, many researchers have discussed the performance of stiffened plates under dynamic loading, which may lead to a wide implementation in the field of vibration and noise control. Determining the free vibration characteristics of a structural system is a fundamental task in dynamic analysis. Recently, the propagation of elastic or acoustic waves in artificial periodic composite structures known as phononic crystals (PCs) has received a great deal of attention [1-4]. One of the most attractive characteristics of PCs is that the propagation of sound and other vibrations is forbidden in their elastic wave band gaps. PCs are essentially periodic structures, and they have inherent relations with periodic structures widely used in traditional engineering. Introducing the theoretical and calculation methods of PCs into the investigation of the dynamic behavior of periodic structures in engineering will provide a new technique for the control of vibration and noise. So far, vibration band gaps in periodic beam [5], grid [6, 7], and plate [8] structures have been researched. These studies are theoretically significant, but the structures studied are less practical than those structures widely used in engineering, such as stiffened plate structures. Moreover, the theoretical methods for calculating the band gaps of PCs such as the plane wave expansion (PWE) method, the Finite-difference-time-domain (FDTD) method, the multiple-scattering theory (MST) method, and the lumped-mass method, are valid for a periodic beam or plate structure. However, by using the above methods, it is difficult to calculate the band gaps of more complicated engineering structures, such as the stiffened plate. In the late 1980s, Mead and his collaborators[9] studied propagating wave motion in regularly stiffened plates and stiffened cylindrical shells using the hierarchical finite element method. Mead et al. [10] modelled the beams as simple line supports and analysed free vibration of an orthogonally stiffened flat plate. Later, they [11, 12] also determined the propagation frequencies of elastic waves by computing phase constant surfaces for a number of different cylinder-stiffener configurations. Cheng Wei and Zhu Dechao [13] analysed the characteristics of wave propagation in a periodic plate reinforced by regular orthogonal stiffeners and discussed the effect of the ratio of length to width, the parameters of the stiffeners, and boundary conditions. However, there are two main problems with these studies. First, the stiffened plate structures must be divided into small finite elements to ensure that the stiffeners are always located on the boundaries of these elements. As a result, the number of the finite elements will dramatically increase as the number of stiffeners with different orientations increases or those with a small spacing between them increases. Second, the theory to predict the vibration response of the stiffened plate structures has been primarily applied to analyse periodic structures as pass band and stop band. However, the physical mechanism of the band gaps has not been exhaustively explained [9]. The present work presents an improved finite element model for periodic stiffened plate structures with any number or orientation of stiffeners. Using the model, we analyse flexural vibration band gaps and study the physical mechanism for their formation in these periodic structures. This paper is organized as follows. In section 2, we summarized the fundamentals of the technique used to analyse the propagation of waves in 2-D periodic structures. Moreover, we established a finite element model of the stiffened plate by considering the unit cell of the infinite periodic structure discussed. Within the model, any number of stiffeners is allowed to take on an arbitrary orientation, and they need not necessarily follow the nodal lines of the mesh division. In section 3, the flexural vibration band gaps of the periodic grid structure and the periodic stiffened plate structures with different skin thicknesses are calculated and comparatively analysed. Finally, the main conclusions of this work are discussed in section 4.


Introduction
Due to their high rigidity-to-weight ratio and economical cost, stiffened plate and shell structures are used extensively in various engineering applications such as bridges, ship hulls and decks, and aircraft structures.In the past few decades, many researchers have discussed the performance of stiffened plates under dynamic loading, which may lead to a wide implementation in the field of vibration and noise control.Determining the free vibration characteristics of a structural system is a fundamental task in dynamic analysis.
Recently, the propagation of elastic or acoustic waves in artificial periodic composite structures known as phononic crystals (PCs) has received a great deal of attention [1][2][3][4].One of the most attractive characteristics of PCs is that the propagation of sound and other vibrations is forbidden in their elastic wave band gaps.PCs are essentially periodic structures, and they have inherent relations with periodic structures widely used in traditional engineering.Introducing the theoretical and calculation methods of PCs into the investigation of the dynamic behavior of periodic structures in engineering will provide a new technique for the control of vibration and noise.So far, vibration band gaps in periodic beam [5], grid [6,7], and plate [8] structures have been researched.These studies are theoretically significant, but the structures studied are less practical than those structures widely used in engineering, such as stiffened plate structures.Moreover, the theoretical methods for calculating the band gaps of PCs such as the plane wave expansion (PWE) method, the Finite-difference-time-domain (FDTD) method, the multiple-scattering theory (MST) method, and the lumped-mass method, are valid for a periodic beam or plate structure.However, by using the above methods, it is difficult to calculate the band gaps of more complicated engineering structures, such as the stiffened plate.
In the late 1980s, Mead and his collaborators [9] studied propagating wave motion in regularly stiffened plates and stiffened cylindrical shells using the hierarchical finite element method.Mead et al. [10] modelled the beams as simple line supports and analysed free vibration of an orthogonally stiffened flat plate.Later, they [11,12] also determined the propagation frequencies of elastic waves by computing phase constant surfaces for a number of different cylinder-stiffener configurations.Cheng Wei and Zhu Dechao [13] analysed the characteristics of wave propagation in a periodic plate reinforced by regular orthogonal stiffeners and discussed the effect of the ratio of length to width, the parameters of the stiffeners, and boundary conditions.
However, there are two main problems with these studies.First, the stiffened plate structures must be divided into small finite elements to ensure that the stiffeners are always located on the boundaries of these elements.As a result, the number of the finite elements will dramatically increase as the number of stiffeners with different orientations increases or those with a small spacing between them increases.Second, the theory to predict the vibration response of the stiffened plate structures has been primarily applied to analyse periodic structures as pass band and stop band.However, the physical mechanism of the band gaps has not been exhaustively explained [9].The present work presents an improved finite element model for periodic stiffened plate structures with any number or orientation of stiffeners.Using the model, we analyse flexural vibration band gaps and study the physical mechanism for their formation in these periodic structures.
This paper is organized as follows.In section 2, we summarized the fundamentals of the technique used to analyse the propagation of waves in 2-D periodic structures.Moreover, we established a finite element model of the stiffened plate by considering the unit cell of the infinite periodic structure discussed.Within the model, any number of stiffeners is allowed to take on an arbitrary orientation, and they need not necessarily follow the nodal lines of the mesh division.In section 3, the flexural vibration band gaps of the periodic grid structure and the periodic stiffened plate structures with different skin thicknesses are calculated and comparatively analysed.Finally, the main conclusions of this work are discussed in section 4.

Analysis of free wave motion
A generic 2-D periodic structure is assembled by identically connecting a base unit or cell along the x-y plane.According to Bloch's theorem, a wave propagating in a 2-D periodic structure can be described by the motion of a single cell and by a propagation vector defining the wave amplitude and phase change from one cell to the next [14].A schematic of the stiffened plate structure configuration and associated unit cell is shown in Fig. 1.
Wave motion in the 2-D periodic structure can be expressed as follows where w is the generalized displacement of point r belonging to the unit cell at location n, while w 0 describes the motion of the unit cell.Also, The behaviour of the unit cell can be conveniently described by defining the cell's interaction with its neighbours and using a discretized equation of motion.For plane harmonic waves at frequency ω, the equation of motion for the unit cell can be written as where the matrices K and M denote the assembled global stiffness and mass matrix of the unit cell respectively.The vectors δ and F are the nodal displacements and forces respectively.From Bloch's theorem, the following relationships are obtained for the unit cell interfaces [7,15] ,, where the subscripts l, r, b, t, rb, lb, rt, lt, and i respectively indicate the generalized displacements at the left, right, bottom, top, right-bottom, left-bottom, right-top, left-top, and internal nodes of the unit cell, as shown in Fig. 2.
Using the above relationships, one can define the following transformation where δ red denotes the displacement of the nodes in the Bloch reduced coordinates defined by and T 1 is a linear transformation parameterized by μ x and μ y .
For free wave motion, Eq. ( 2) would be written as where the superscript H denotes the Hermitian transpose and K red and M red are the reduced stiffness and mass matrices according to Bloch's theorem and defined by Eq. ( 7) is the desired eigenvalue problem parameterized by ω and defines the dispersion relations of the infinite periodic structure.

Finite element modeling of the unit cell
The unit cell of an infinite periodic stiffened plate structure consists of a base structure forming the "skin" as well as local reinforcement elements called "stiffeners" which are periodically collocated.Therefore, based on Mindlin plate theory and Timoshenko beam theory, the dynamic characteristic of the unit cell can be accurately described by establishing an efficient finite element model of the stiffened plate using eight-node isoparametric plate bending elements and three-node isoparametric beam elements.
According to the Mindlin plate theory, the stiffness and mass matrices of the eight-node isoparametric plate bending element are given [16,17] where J is the Jacobian matrix, B p is the strain-displacement relationship matrix, D p is the constitutive matrix, N p is the mapped shape functions of the plate element, and m p is defined as where ρ p is the density of the plate material and h p is the thickness of the plate element.The Timoshenko beam element has three nodes and each node has three degrees of freedom, (w b , θ bs , θ bt ).The stiffness and mass matrices of the isoparametric beam element are then given by where ρ b is the density of the beam material, A b is the cross-sectional area of the beam, I bz is the second moment of the beam cross-sectional area about the z-axis, and J b is the polar moment of inertia of the beam.The displacement field of the stiffener element is expressed using the plate element degrees of freedom [18,19] as where Λ is the orientation matrix, φ is the stiffener inclination with respect to the plate x-axis, T 2 is the transformation matrix of the nodal displacements from the beam element nodes into the plate element nodes, and (N r I) i (i = 1, 2, 3) are the shape functions of the eight-node plate elements defined at the 3 points (m, n, p) in the ξ-η coordinate system, as shown in Fig. 3.The stiffener node coordinates in the ξ-η coordinate system are obtained using the plate shape functions themselves, and the solution can easily be calculated using Newton's iteration method.
Finally, the combined stiffness and mass matrices of the stiffened plate for the unit cell are calculated by adding those of the plate elements and of the stiffener elements.Using above approach, any number or orientation of stiffeners within the unit cell can be easily modelled without the need to change the ground mesh of the plate.

Flexural vibration band gaps of periodic grid structures
In order to compare with the flexural vibration band gap characteristics of periodic stiffened plate structures, it is necessary to analyse the characteristics of periodic grid structures.
The grid structure has the same geometry and material properties of the stiffened plate structures but does not have the same skin.The material parameters used are ρ steel = 7780 kg/m 3 , E steel = 21.06×10 10 Pa and ν steel = 0.3.The lattice constant of the unit cell is 0.25 m, and the grid section parameters are b = 0.015 m and h = 0.015 m.The flexural vibration band structure of the grid structure using the above finite element method (FEM) is shown in Fig. 4, a.It is well known that the frequency response function (FRF) of vibration can be used to effectively describe the band gaps.The FRF of the flexural vibration was determined with the FEM software, MSC Nastran, for a 16×16 grid and is illustrated in Fig. 4, b.There are no complete band gaps in the band structure or any large attenuation in the FRF.The material parameters of the skin in the periodic stiffened plate structures are ρ epoxy = 180 kg/m 3 , E epoxy = 4.35×109 Pa, ν epoxy = 0.3679, and d epoxy = 0.02.The thickness parameters of the skin are varied to analyse the influence of the skin thickness on flexural vibration band gaps of the periodic stiffened plate structures.Fig. 5 shows the flexural vibration band structure and the finite structure FRF of the periodic stiffened plate structure with a 1 mm-thick skin.It can be clearly seen from the band structure in Fig. 5, a that there are numerous dispersion curves which are almost flat along all three boundaries of the irreducible Brillouin zone.The frequencies of the flat bands are approximately the eigenvalues of the localized vibration modes of the stiffener-surrounded skin (which can be considered a four-sides-clamped plate) especially in the 0~0.4 kHz region, as shown in Fig. 6.
An enlarged figure of the dashed rectangular region in Fig. 5, a is shown in Fig. 7.The mode shapes of the stiffeners and the skin corresponding to the four points In fact, if the flat bands are ignored, the dispersion curves for the system with a 1 mm-thick skin are quite similar to those of the periodic grid structure.In other words, the vibrations of the skin and the stiffeners are uncoupled, and the vibration of the stiffeners plays a major role in the stiffened plate system.Similar to the periodic grid structure, there also is no complete band gap in the band structure or any large drop in the FRF.
The band structure and the FRF corresponding to the 8 mm-thick skin are shown in Fig. 9. Due to the increase of the skin to stiffener thickness ratio, the number of flat curves is relatively smaller in the range of 0 ~ 3 kHz and the four-sides-clamped boundary condition is weakened such that the curves are no longer flat.This means that the vibration coupling between the skin and the stif-Frequency, f (kHz) Wake vector, k   There is a series of peaks within the other two large drops.These peaks are primarily a result of two phenomena.The first is that the vibration of the stiffeners couples with the skin and the amplitudes of the mode shapes of the skin and the stiffeners are both larger.For example, the mode shapes of the skin and the stiffeners at points marked E and F are shown in Fig. 10.The second is that the FRF is not that of an infinite periodic structure.
The results measured at the 14th period and the 16th period of the 16×16 structure are shown in Fig. 9, b as the dashed line and the solid line respectively.It is clearly seen that the attenuation at the 14th period is larger than that at the 16th period.That is to say, the effect of boundary condition of finite structure at the 14th period is smaller than that at the 16th period.
The corresponding results of the stiffened plate structure, which show that the thickness of the skin is the same as those of the stiffeners, are given in Fig. 11, a and c.The natural frequencies of the four-sides-clamped plate are plotted in Fig. 11, b.In this case, the boundary conditions for the four-sides-clamped plate are no longer applicable for the stiffeners-surrounded skin and all of the dispersive The flexural vibration band gaps of the stiffened plate structures in which the thickness of the skin varies from 1 mm to 15 mm are calculated, and the frequency ranges of the former six complete band gaps are plotted in Fig. 12.It can be seen that, while the skin to stiffener thickness ratio increases, several complete band gaps appear and when the thickness of the skin is close to that of the stiffener, the complete band gaps gradually disappear.

Conclusion
An improved finite element model of the periodic stiffened plate structures is established and used to analyze flexural wave propagation in periodic stiffened plate structures.The model demonstrates that the skin and the stiffeners are vibration coupled.When the height of the cross section of the stiffeners is much larger than the thickness of the skin, the coupling between them is relatively weak and the stiffeners play a major role in the band gap characteristics of the stiffened-plate system.As the thickness ratio between the skin and stiffeners increases, the four-sides-clamped boundary condition weakens.Therefore, the vibration coupling between the skin and the stiffeners is strengthened and several complete band gaps are generated.These band gaps primarily correspond to the frequency ranges of vibration attenuation.When the thickness of the skin is equal to that of the stiffeners, the vibrations of the skin and the stiffeners are strongly coupled and the homogeneous skin weakens the impedance matching of the periodic stiffeners so that the complete band gaps vanish.These results show that the vibration coupling between the skin and stiffeners influence the formation of the complete band gap.By optimizing the thickness ratio of the skin and the stiffeners, large attenuation can be expected in various frequency ranges.

Fig. 1 a
Fig. 1 a -schematic of the periodic stiffened plate structure; b -corresponding unit cell where B b is the strain-displacement relationship matrix, D b is the constitutive matrix, N b is the mapped shape functions of the beam element, and m b is defined as

Fig. 3
Fig.3Mapping of the stiffened plate element from the global coordinate system to the local coordinate system Furthermore, the stiffness and mass matrices of a stiffener element can be expressed in terms of the plate nodal degrees of freedom as

FrequencyFig. 4 a
Fig. 4 a -flexural wave band structure of the periodic grid structure, b -corresponding FRF 3.2.The influence of the skin thickness on flexural vibration band gaps of periodic stiffened plate structures

FrequencyFig. 5 a
Fig. 5 a -flexural wave band structure of the periodic stiffened plate structure; the skin thickness is 1 mm; b -corresponding FRF Frequency, f (kHz)

Fig. 7
Fig.7The enlarged drawing of the specified dashed rectangular region in Fig.5, a

Fig. 8
Fig. 8 The mode shapes of the skin and the stiffeners: a and b refer to point A; c and d refer to point B; e and f refer to point C; and g and h refer to point D in Fig. 7 feners is strengthened.Moreover, there are large drops in the FRF.The frequency ranges, 0.520 ~ 0.540 kHz, 0.786 ~ 1.351 kHz, and 1.781 ~ 2.740 kHz are labelled as I, II, and III and shown in Fig. 9, b.The first drop agrees with the first complete band gap (0.534 ~ 0.564 kHz) in band structure.There is a series of peaks within the other two large drops.These peaks are primarily a result of two phenomena.The first is that the vibration of the stiffeners couples with the skin and the amplitudes of the mode shapes of the skin and the stiffeners are both larger.For example, the mode shapes of the skin and the stiffeners at points marked E and F are shown in Fig.10.The second is that the FRF is not that of an infinite periodic structure.The results measured at the 14th period and the 16th period of the 16×16 structure are shown in Fig.9, b as the dashed line and the solid line respectively.It is clearly seen that the attenuation at the 14th period is larger than that at the 16th period.That is to say, the effect of boundary condition of finite structure at the 14th period is smaller than that at the 16th period.The corresponding results of the stiffened plate structure, which show that the thickness of the skin is the same as those of the stiffeners, are given in Fig.11, a and c.The natural frequencies of the four-sides-clamped plate are plotted in Fig.11, b.In this case, the boundary conditions for the four-sides-clamped plate are no longer applicable for the stiffeners-surrounded skin and all of the dispersive

Fig. 9 aFig. 10 Fig. 11 a
Fig. 9 a -flexural wave band structure of the periodic stiffened plate structure; the skin thickness is 8 mm; b -corresponding FRF

Fig. 12
Fig. 12 Variation of the frequency ranges of the former six complete band gaps with skin thickness from 1 mm to 15 mm Jianwei Wang, Gang Wang, Jihong Wen, Xisen Wen FLEXURAL VIBRATION BAND GAPS IN PERIODIC STIFFENED PLATE STRUCTURES S u m m a r y Based on Mindlin plate theory, Timoshenko beam theory and Bloch's theorem, an improved finite element model for periodic stiffened plate structures with any number or orientation of stiffeners is developed to describe the propagation of elastic waves in the periodic structures undergoing flexural vibration.Using the finite element model, the flexural vibration band gaps of a periodic grid structure and several periodic stiffened plate structures with different skin thicknesses are compared and analyzed.The results indicate that the vibration coupling between the skin and the stiffeners influences the formation of flexural vibration band gaps.