Free vibration analysis of plate/shell coupled structures by the method of reverberationray matrix
Dong Tang^{1} , Xiongliang Yao^{2} , Guoxun Wu^{3}
^{1, 2, 3}College of Shipbuilding Engineering, Harbin Engineering University, Harbin, 150001, China
^{1}Corresponding author
Journal of Vibroengineering, Vol. 18, Issue 5, 2016, p. 31173137.
https://doi.org/10.21595/jve.2016.16950
Received 4 March 2016; received in revised form 14 May 2016; accepted 28 June 2016; published 15 August 2016
JVE Conferences
This paper is concerned with free vibration analysis of plate/shell coupled structures with two opposite edges simply supported by the method of reverberationray matrix. The equations of motion of the flat plate and the open circular cylindrical shell, respectively based on the classical thin plate theory and the Flügge thin shell theory, are introduced. Analytical solutions of the combination of a traveling wave form along the circumferential direction and a standing wave form along the axial direction are obtained. The method of reverberationray matrix is applied to derive the equation of the natural frequencies for the plate/shell coupled structures. The semianalytical natural frequencies are obtained with the employment of the golden section search algorithm. The semianalytical calculation results of three typical plate/shell coupled structures are presented and the results are compared with those obtained by the finite element method. The comparison shows that the calculation results obtained in this paper are of high accuracy and that the formulation presented in this manuscript are validated for free vibration analysis of plate/shell coupled structures.
Keywords: plate/shell coupled structures, method of reverberationray matrix, free vibration analysis, Flügge thin shell theory, analytical wave form solution.
1. Introduction
Thin plates and thin shells are extensively used in civil, mechanical and aeronautical engineering as well as in naval architecture and ocean engineering. Most of the practical engineering structures, such as the fuselages of aircraft, the ship hulls and the ocean platforms, etc., involve plate/shell coupled structures. The vibration behaviors of such coupled structures attract much attention from engineers in their practical designs. Quite a few experimental and analytical studies have been conducted on vibration analysis of plates and shells, but not so many researches are concerned with the plate/shell coupled structures.
Vibration of coupled structures with plate components has been studied by many researchers in the past decades. Vibration behaviors of folded plates are analyzed in literature [19]. Vibration characteristics of and power flow transmission through the Lshaped plate are studied in [1020]. Vibration behaviors of boxtype structures are investigated by a few researchers. Dickinson and Warburton [21] analyzed the free flexural vibrations of open and closed rectangular boxes and presented a theoretical solution using a sine series. Popplewell [22] studied the free vibration of a boxtype structure and the natural frequencies and normal modes are presented. Handa [23] analyzed the inplane vibration of boxtype structures by a finite element method. By considering the spatial properties of distributed forces in terms of their Fourier components and hypothesizing that the uniform component is dominant, Fulford and Petersson [24, 25] accounted for the spatially distributed wavefield at the connections of the builtup structures and the vibratory power for the boxlike structure supported by an infinite platelike recipient were considered. Lee and Wooh [26] presented the free vibration analysis of folded structures and box beams made of composite materials using a fournoded Lagrangian and Hermite finite element that incorporates high order transverse shear deformation and rotary inertia and the significance of the high order plate theory in analyzing folded structures is enunciated. Lin and Pan studied the vibration characteristics of a boxtype structure using the finite element method [27], and subsequently investigated the sound radiation characteristics of a boxtype structure with the employment of the finite element and boundary element methods [28]. More recently, Chen et al. [29] developed an analytical approach to investigate the vibration behaviors of a boxtype builtup structure and energy transmission through the structure.
Most of the available researches on plate/shell coupled structures are concerned with closed circular cylindrical shells with end plates. Yamada et al. [30] presented the free vibration analysis of a circular cylindrical doubleshell system closed by end plates. Schlesinger [31] investigated the transmission of elastic waves from a cylinder to an attached flat plate with the wave approach. Tso and Hansen [32] also studied the transmission of vibration waves through cylinder/plate junctions. Stanley and Ganesan [33] determined the natural frequencies of cylindrical shells with a circular plate attached at arbitrary locations for various boundary conditions using the semianalytical finite element method. Tso and Hansen [34] presented a theoretical and experimental study of the transmission of vibration through a two element structure which consists of a cylindrical shell coupled to an end plate. Wu et al. [35] analyzed the vibroacoustic coupling between a finite circular cylindrical shell closed at each end by a piece of circular plate and its enclosed cavity by using the coveringdomain method, which transforms the calculation of the scattering sound field of a complicatedshaped close cavity to that of a series of simply regularshaped close shells. Wang et al. [36] formulated a substructure approach to investigate the power flow characteristics of a platecylindrical shell system subject to both conservative and dissipative coupling conditions. Liang and Chen [37] investigated the natural frequencies and mode shapes for a conical shell with an annular end plate or a round end plate by means of the transfer matrix method. Subsequently, Liang et al. [38] extended the transfer matrix method to analyze a composite laminated conicalplate shell. Recently, much attentions are paid to vibration analysis of joined cylindrical, conical or spherical shells [3950].
This paper presents an analytical formulation for the free vibration analysis of plate/shell coupled structures with two opposite edges simply supported. Firstly, the force and moment resultants in a thin plate and in an open circular cylindrical shell (OCCS) are presented and the equations of motion of the thin plate and the OCCS, respectively based on the classical thin plate theory and the Flügge thin shell theory, are introduced. Then, analytical solutions of the combination of a traveling wave form along the circumferential direction and a standing wave form along the axial direction are obtained for both of the thin plate and the OCCS with two opposite edges simply supported. Subsequently, the method of reverberationray matrix (MRRM) is employed to derive the equation of natural frequencies for plate/shell coupled structures and the golden section search algorithm is applied to find the semianalytical natural frequencies of the plate/shell coupled structures. Finally, the calculation results of three typical plate/shell coupled structures are presented and the results are compared with those obtained by the finite element method.
2. Formulation
According to the classical thin plate theory and the Flügge thin shell theory, the force and moment resultants and the governing differential equations for the basic components of the plate/shell coupled structures are presented at the beginning of this section. Analytical solutions of the combination form of a traveling wave along one direction and a standing wave along the other direction are obtained for both of the thin plate and the OCCS with two opposite edges simply supported. After that, the displacements and the force and moment resultants are expressed in matrix form to derive the scattering matrix, the phase matrix and the permutation matrix, which are subsequently used to formulate the reverberationray matrix and to obtain the equation of natural frequencies of the plate/shell coupled structures. Finally, the golden section search algorithm is applied to find the natural frequencies of the plate/shell coupled structures.
2.1. Force and moment resultants in a plate
The force and moment resultants in a thin plate are shown in Fig. 1, in which the positive directions are indicated. Based on the generalized Hooke’s law, the straindisplacement relations and the stressstrain relations of the element of the thin plate, the force and moment resultants in the plate can be expressed in terms of the inplane longitudinal, inplane shear and outplane displacements as follows:
where $u$, $v$ and $w$ denote the inplane longitudinal, inplane shear and outplane displacements along $x$, $y$ and $z$ directions, respectively. $C=Eh/(1{\mu}^{2})$ and $D=E{h}^{3}/12\left(1{\mu}^{2}\right)$ are the membrane stiffness and the bending stiffness of the plate, where $E$ is Young’s modulus, $\mu $ is Poisson’s ratio, and $h$ is the thickness of the plate. ${N}_{x}$ and ${N}_{y}$ denote the inplane normal forces, ${N}_{xy}$ and ${N}_{yx}$, the inplane shear forces, ${M}_{x}$ and ${M}_{y}$, the bending moment, ${M}_{xy}$ and ${M}_{yx}$, the torsional moment, and ${Q}_{xz}$ and ${Q}_{yz}$, the outplane shear forces, ${V}_{xz}$ and ${V}_{yz}$, represent the Kirchhoff effective shear force resultants of the first kind acting on the crosssections perpendicular to the $x$ and $y$ directions, respectively.
Fig. 1. Force and moment resultants in a thin plate
2.2. Governing differential equations and the solutions of the plate
According to the dynamic equilibrium of forces in the $x$, $y$ and $z$ directions and the relations of the force and moment resultants with the displacements defined by Eqs. (1)(10), the governing differential equations for the free vibration of a thin plate are obtained as follows:
where $\rho $ is the mass density of the plate.
Taking the Fourier transforms of Eqs. (11)(13), the governing differential equations can be expressed in the frequency domain as:
where a tide over a symbol represents the corresponding physical quantity in the frequency domain, ${k}_{L}=\omega {\left[\left(1{\mu}^{2}\right)\rho /E\right]}^{1/2}$ denotes the inplane longitudinal wave number of the thin plate, and $\omega $ is the circular frequency.
With respect to a thin plate simply supported at $x=$ 0 and $x={L}_{x}$, the inplane longitudinal, inplane shear, and outplane displacements can be expressed as the series summation of the products of the modal waves along $x$ direction and the traveling waves along $y$ direction [51]:
where ${k}_{x}=m\pi /{L}_{x}$ denotes the wave number in the $x$ direction, $m$ is the mode number and ${L}_{x}$ represents the length of the plate. ${k}_{y1}={\left({k}_{x}^{2}{k}_{F}^{2}\right)}^{1/2}$ and ${k}_{y2}={\left({k}_{x}^{2}+{k}_{F}^{2}\right)}^{1/2}$ are respectively the wave numbers along the $y$ direction for the propagating and evanescent waves corresponding to the outplane displacement. ${k}_{y3}={\left({k}_{x}^{2}{k}_{L}^{2}\right)}^{1/2}$ and ${k}_{y4}={\left({k}_{x}^{2}{k}_{S}^{2}\right)}^{1/2}$ are respectively the wave numbers along the $y$ direction for the propagating waves corresponding to the inplane displacements. ${k}_{F}={\left({\omega}^{2}\rho h/D\right)}^{1/4}$ is the in vacuo flexural wave number of the plate. ${k}_{S}=\omega {\left[2\left(1+\mu \right)\rho /E\right]}^{1/2}$ denotes the inplane shear wave number. Wave amplitudes corresponding to the arriving wave and the departing wave are respectively indicated by ${a}_{i}$ and ${d}_{i}$ ($i=$ 14), in which ($i=$ 1, 2) for flexural waves of the outplane displacement and ($i=$ 3, 4) for longitudinal waves and shear waves of the inplane displacements.
The rotation of the normal to the midplane of the plate about the $x$ direction is defined as:
According to Eq. (19), the abovementioned rotation can be expressed in frequency domain as:
Substituting Eqs. (17)19) into the Fourier transforms of Eqs. (2), (3), (5) and (10) yields the frequency domain expressions of the force and moment resultants of the plate:
For an arbitrary axial mode number $m$, Eqs. (17)(19) and (21) can be expressed in matrix form as:
where ${\mathbf{W}}_{d}$ denotes the displacement vector of the plate, ${\mathbf{H}}_{m}\left(x\right)$ indicates the axial mode matrix, and ${\mathbf{W}}_{d}^{*}$ represents the vector of the traveling wave solutions corresponding to the displacement vector. They are presented in detail as follows:
in which ${\mathbf{A}}_{d}$ and ${\mathbf{D}}_{d}$ are coefficient matrices, $\mathbf{a}$ and $\mathbf{d}$ are amplitude vectors corresponding to the arriving wave and the departing wave, respectively. ${\mathbf{P}}_{h}\left(y\right)$ represents the phase matrix. They are presented in detail as follows:
Similarly, for an arbitrary axial mode number $m$, Eqs. (22)(25) can be expressed in matrix form as:
where the physical significance and expression of ${\mathbf{H}}_{m}\left(x\right)$ are the same as those presented in Eq. (28). ${\mathbf{W}}_{f}$ denotes the force vector of the plate, and ${\mathbf{W}}_{f}^{*}$ represents the vector of the wave solutions corresponding to the force vector. They are presented in detail as follows:
in which the physical significances and expressions of ${\mathbf{P}}_{h}\left(y\right)$, $\mathbf{a}$ and $\mathbf{d}$ are the same as those defined in Eq. (29). ${\mathbf{A}}_{f}$ and ${\mathbf{D}}_{f}$ are coefficient matrices corresponding to the arriving wave and the departing wave of the force and moment resultants of the plate. They are presented in detail as follows:
where ${\mu}_{2}$ is a nondimensional parameter presented in Appendix A.1.
2.3. Force and moment resultants in an open circular cylindrical shell
The force and moment resultants in an OCCS are shown in Fig. 2, in which the positive directions are indicated. Based on the Flügge thin shell theory, the force and moment resultants in the OCCS can be expressed in terms of the axial, circumferential and radial displacements as follows:
where $u$, $v$ and $w$ denote the displacement components in the axial ($x$), circumferential ($\theta $), and radial ($z$) directions, respectively. $C=Eh/1{\mu}^{2}$ and $D=E{h}^{3}/12\left(1{\mu}^{2}\right)$ are the membrane stiffness and the bending stiffness of the shell, where $E$ is Young’s modulus, $\mu $ is Poisson’s ratio, $h$ is the thickness and $R$ is the radius of the shell. $\lambda ={h}^{2}/12{R}^{2}$ is a dimensionless parameter, which is related with the ratio of the shell thickness to the shell radius. ${\lambda}_{1}$, ${\lambda}_{2}$, ${\mu}_{1}$, ${\mu}_{2}$ and ${\mu}_{3}$ are dimensionless parameters presented in Appendix A.1. ${N}_{x}$ and ${N}_{\theta}$ denote the inplane normal forces, ${N}_{x\theta}$ and ${N}_{\theta x}$, the inplane shear forces, ${M}_{x}$ and ${M}_{\theta}$, the bending moment, ${M}_{x\theta}$ and ${M}_{\theta x}$, the torsional moment, and ${Q}_{xz}$ and ${Q}_{\theta z}$, the outplane shear forces acting on the crosssections perpendicular to the axial and circumferential directions, respectively. Besides, ${V}_{xz}$ and ${V}_{\theta z}$ indicate the Kirchhoff effective shear force resultants of the first kind, namely, the inplane shear force resultants, and ${F}_{x\theta}$ and ${F}_{\theta x}$, the Kirchhoff effective shear force resultants of the second kind, namely, the outplane shear force resultants acting on the crosssections perpendicular to the axial and circumferential directions, respectively.
Fig. 2. Force and moment resultants in an open circular cylindrical shell
2.4. Governing differential equations and solutions of the open circular cylindrical shell
The governing differential equations for the free vibration of an OCCS based on the Flügge thin shell theory can be written as:
$+\mu R\frac{\partial w}{\partial x}{R}^{2}\frac{\left(1{\mu}^{2}\right)\rho}{E}\frac{{\partial}^{2}u}{\partial {t}^{2}}=0,$
$+2\lambda {R}^{2}\frac{{\partial}^{4}w}{\partial {x}^{2}\partial {\theta}^{2}}+\lambda \frac{{\partial}^{4}w}{\partial {\theta}^{4}}+2\lambda \frac{{\partial}^{2}w}{\partial {\theta}^{2}}+{\lambda}_{1}w+{R}^{2}\frac{\left(1{\mu}^{2}\right)\rho}{E}\frac{{\partial}^{2}w}{\partial {t}^{2}}=0.$
Taking the Fourier transforms of Eqs. (53)(55), the governing differential equations can be expressed in the frequency domain as:
$+2\lambda {R}^{2}\frac{{\partial}^{4}\stackrel{~}{w}}{\partial {x}^{2}\partial {\theta}^{2}}+\lambda \frac{{\partial}^{4}\stackrel{~}{w}}{\partial {\theta}^{4}}+2\lambda \frac{{\partial}^{2}\stackrel{~}{w}}{\partial {\theta}^{2}}+{\lambda}_{1}\stackrel{~}{w}{R}^{2}{k}_{L}^{2}\stackrel{~}{w}=0.$
With respect to an OCCS simply supported at $x=$ 0 and $x={L}_{x}$, the axial, circumferential and radial displacements can be expressed as the series summation of the products of the mode waves along the axial direction and the traveling waves along the circumferential direction:
where ${k}_{x}=m\pi /{L}_{x}$ denotes the wave number in the x direction, $m$ is the mode number and ${L}_{x}$ represents the length of the shell. ${a}_{i}$ and ${d}_{i}$ ($i=$ 14) represent wave amplitudes corresponding to the arriving wave and the departing wave, respectively. ${\alpha}_{i}$ and ${\beta}_{i}$($i=$ 14) are amplitude coefficients of the axial and circumferential waves. The expressions of ${\alpha}_{i}$ and ${\beta}_{i}$ are defined as follows:
and ${k}_{\theta i}$ ($i=$ 14) are circumferential wave numbers defined by the following equation:
$+\frac{\left({\xi}_{6}^{2}{k}_{\theta}^{2}+{\xi}_{7}^{2}{\xi}_{8}^{2}+{\xi}_{2}^{2}{\xi}_{9}^{2}\right)\left({\mu}_{1}\lambda {k}_{\theta}^{4}{\xi}_{10}^{2}{k}_{\theta}^{2}+{\xi}_{3}^{2}{\xi}_{8}^{2}\right)}{\lambda}=0,$
where ${\mu}_{i}$ ($i=$ 13), ${\lambda}_{i}$ ($i=$ 12) and ${\xi}_{i}$ ($i=$ 110) are nondimensional parameters presented in detail in Appendix A1 and Appendix A2.
The rotation of the normal to the midsurface of the shell about the axial direction is defined as:
According to Eqs. (60) and (61), the abovementioned rotation can be expressed in frequency domain as:
Meanwhile, substitution of Eqs. (59)(61) into the Fourier transforms of Eqs. (40), (44), (50) and (52) yields the frequency domain expressions of the force and moment resultants in the OCCS:
For an arbitrary axial mode number $m$, Eqs. (59)(61) and (66) can be expressed in matrix form as:
where ${\mathbf{W}}_{d}$ denotes the displacement vector of the OCCS, ${\mathbf{H}}_{m}\left(x\right)$ indicates the axial mode matrix and ${\mathbf{W}}_{d}$^{*} represents the vector of the circumferential traveling wave solutions corresponding to the displacement vector. They are presented in detail as follows:
in which ${\mathbf{P}}_{h}\left(\theta \right)$ denotes the phase matrix. $\mathbf{a}$ and $\mathbf{d}$ are amplitude vectors corresponding to the arriving wave and the departing wave of the displacements of the OCCS, respectively. They are presented in detail as follows:
and ${\mathbf{A}}_{d}$ and ${\mathbf{D}}_{d}$ are coefficient matrices corresponding to the arriving wave and the departing wave of the displacements of the OCCS, respectively. The elements of the coefficient matrices ${\mathbf{A}}_{d}$ and ${\mathbf{D}}_{d}$ are listed as follows:
where $j=$ 1, 2, 3, 4.
Similarly, for an arbitrary axial mode number $m$, Eqs. (67)(70) can be expressed in matrix form as:
where the physical significance and expression of ${\mathbf{H}}_{m}\left(x\right)$ are the same as those presented in Eq. (71). ${\mathbf{W}}_{f}$ denotes the force vector of the shell, and ${\mathbf{W}}_{f}^{*}$ represents the vector of the circumferential wave solutions corresponding to the force vector. They are presented in detail as follows:
in which ${\mathbf{A}}_{f}$ and ${\mathbf{D}}_{f}$ are coefficient matrices corresponding to the arriving wave and the departing wave of the force and moment resultants of the OCCS, respectively. The elements of the coefficient matrices ${\mathbf{A}}_{f}$ and ${\mathbf{D}}_{f}$ are listed as follows:
where $j=$ 1, 2, 3, 4.
2.5. Equation of natural frequencies for the plate/shell coupled structures
Taking advantage of the unidirectional wave form solutions of matrix form for the thin plate obtained in Subsection 2.2 and for the OCCS obtained in Subsection 2.4, the MRRM is introduced to derive the equation of the natural frequencies of the plate/shell coupled structures.
Firstly, the plate/shell coupled structure is discretized into basic components such as flat plates and OCCSs. A dual local coordinate system is established for each of the components at both of the ends, where the cartesian coordinate for a flat plate and the circular cylindrical coordinate for an OCCS. Then, the local scattering matrix is derived according to the continuity conditions of the displacements and the equilibrium conditions of the internal forces and moments at each of the joints of the plate/shell coupled structure. Meanwhile, the local phase matrix is obtained according to the inherent relations of the harmonic waves in the dual local coordinate system. With all of the local scattering matrices and all of the local phase matrices respectively assembled into a global scattering matrix and a global phase matrix, the global scattering equation and the global phase equation are obtained. When keeping the two global amplitude vectors of the arriving wave the same, the two global amplitude vectors of the departing wave contain the same scalar state variables arranged in different sequential orders. Therefore, a permutation equation can be obtained from the relation between the two global amplitude vectors of the departing wave. Subsequently, the reverberationray matrix can be obtained from the simultaneous equations of the global scattering equation, the global phase equation and the permutation equation. Finally, the equation of the natural frequencies can be derived by equating the determinant of the coefficient matrix of the global amplitude vector of the departing wave to zero.
With the derivation procedure mentioned above, the equations of the natural frequencies of the three plate/shell coupled structures including a boxtype structure, a racetrack cylindrical shell and a ship hull structure will be obtained in the following discussions in this subsection. However, since the three independent derivation procedures are quite similar to each other, for simplicity, one of them will be taken as an example and the other two will be omitted. Note that the solutions obtained in Subsection 2.2 are applied for flat plate components while the solutions obtained in Subsection 2.4 are applied for OCCS components.
Next, the derivation for equation of the natural frequencies of the racetrack cylindrical shell shown in Fig. 3, is presented as follows.
Fig. 3. Dual local coordinate systems for the racetrack cylindrical shell
2.5.1. Scattering matrix
The continuity conditions for the plate and the OCCS at Node Line 1 are presented as follows:
which can be rewritten in matrix form as:
where ${\mathbf{T}}_{d}^{1}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left\{1111\right\}$ denotes the transfer matrix corresponding to the displacement vector at Node Line 1. The variables with superscript $IJ$ ($I$, $J=$ 14) indicate the physical quantities of the substructure bounded by Node Line $I$ and Node Line $J$, this will not be mentioned repeatedly in the following discussions.
Since the local coordinates are established at the node lines, the phase matrix turns to be a unit matrix at the node lines. Taking into account of this condition and substituting Eqs. (26) and (71) into Eq. (90) yields:
Meanwhile, the equilibrium conditions for the plate and the OCCS at Node Line 1 are presented as:
which can be rewritten in matrix form as:
where ${\mathbf{T}}_{f}^{1}={\mathbf{T}}_{d}^{1}$ denotes the transfer matrix corresponding to the force vector at Node Line 1.
Substitution of Eqs. (34) and (82) into Eq. (93) results in:
Eqs. (91) and (94) can be combined and expressed in a single matrix form as:
where ${\mathbf{d}}^{1}={\left\{{\left({\mathbf{d}}^{14}\right)}^{T}{\left({\mathbf{d}}^{12}\right)}^{T}\right\}}^{T}$ and ${\mathbf{a}}^{1}={\left\{{\left({\mathbf{a}}^{14}\right)}^{T}{\left({\mathbf{a}}^{12}\right)}^{T}\right\}}^{T}$ are amplitude coefficients of the departing wave and arriving wave at Node Line 1, respectively. ${\mathbf{S}}^{1}$, the scattering matrix at Node Line 1, is defined as:
In the same manner, the scattering relations and scattering matrices for the rest node lines can be obtained. For simplicity, the derivation procedures are omitted and the results are given straightforwardly as follows.
The scattering relations at an arbitrary node line $J$ can be presented as:
where ${\mathbf{d}}^{J}={\left\{{\left({\mathbf{d}}^{JI}\right)}^{T}{\left({\mathbf{d}}^{JK}\right)}^{T}\right\}}^{T}$ and ${\mathbf{a}}^{J}={\left\{{\left({\mathbf{a}}^{JI}\right)}^{T}{\left({\mathbf{a}}^{JK}\right)}^{T}\right\}}^{T}$ are amplitude coefficients of the departing wave and arriving wave at Node Line $J$, respectively. ${\mathbf{S}}^{J}$, the scattering matrix at Node Line $J$, is defined as:
in which $J=$ 2, 3, 4. As $J=$ 2 and 3, $I=J1$ and $K=J+1$. However, as $J=$ 4, $I=$ 3 and $K=$ 1.
Assembling all of the local scattering equations for Node Line 14 by stacking ${\mathbf{d}}^{1}$${\mathbf{d}}^{4}$ and ${\mathbf{a}}^{1}$${\mathbf{a}}^{4}$ into two column vectors $\mathbf{d}$ and $\mathbf{a}$, the global scattering equation can be obtained as follows:
where $\mathbf{d}$ and $\mathbf{a}$ are global amplitude vectors of the departing wave and the arriving wave, and $\mathbf{S}$ is the global scattering matrix. They are presented in detail as follows:
2.5.2. Phase matrix
The phase relations of harmonic waves in the dual coordinate system provide additional equations for solving the unknown amplitude vectors. Note that the departing wave from Node Line 1 (3) is exactly the arriving wave to Node Line 2 (4), and vice versa. Therefore, the amplitudes of the departing wave and the arriving wave differ with each other by a phase factor.
With the employment of the solutions for the plate obtained in Subsection 2.2, the relations between the amplitudes of the departing wave and the arriving wave between Node Line 1 (3) and Node Line 2 (4) are presented as:
where $IJ=$ 12 or 34 and $JI=$ 21 or 43.
Similarly, with the employment of the solutions for the OCCS obtained in Subsection 2.4, the relations between the amplitudes of the departing wave and the arriving wave between Node Line 2 (4) and Node Line 3 (1) are presented as:
where $IJ=$ 23 or 41 and $JI=$ 32 or 14.
Assembling all of the local phase equations defined by Eqs. (103)(106) results in the global phase equation:
where the physical significance and expression of $\mathbf{a}$ are the same as the one presented in Eq. (99). ${\mathbf{d}}^{*}$ is a rearranged global amplitude vector of the departing wave, and $\mathbf{P}$ is the global phase matrix. They are presented in detail as follows:
where ${\theta}_{0}$ denotes the included angle of the OCCS, and ${L}_{y}$ represents the length of the plate in y direction.
2.5.3. Permutation matrix
A comparison of the global amplitude vectors of the departing wave $\mathbf{d}$ and ${\mathbf{d}}^{*}$ indicates that the two amplitude vectors contain the same scalar state variables arranged in different sequential orders. The relation between $\mathbf{d}$ and ${\mathbf{d}}^{*}$ is:
where $\mathbf{U}$ is the permutation matrix, which is presented in detail as follows:
in which ${0}_{4}$ and ${\mathbf{I}}_{4}$ are respectively zero matrix and unit matrix of fourth order.
2.5.4. Reverberationray matrix and the equation of natural frequencies
Substitution of Eqs. (107) and (110) into Eq. (99) yields:
where $\mathbf{I}$ is a unit matrix of 32nd order, and $\mathbf{R}=\mathbf{S}\mathbf{P}\mathbf{U}$ is defined as the reverberationray matrix of the racetrack cylindrical shell.
To obtain a nontrivial solution of the global amplitude vector of the departing wave, the determinant of $\left(\mathbf{I}\mathbf{R}\right)$ must be zero, namely:
which is the equation of natural frequencies of the racetrack cylindrical shell.
2.6. Searching algorithm for natural frequencies
As the equation of natural frequencies of the plate/shell coupled structure is obtained, the problem at hand is to solve the equation for the natural frequencies. It is obvious that the left hand side of Eq. (113) is a function of frequency, and the natural frequencies are zeros of the function. Unfortunately, for most of the frequencies, the function values are complex numbers. Therefore, finding the zeros of the function needs to search the common zeros of the real part and the imaginary part of the function. A good idea is to search the zeros or the minimal values of the absolute value of the function. This simple approach is adopted in this paper and the golden section search algorithm is introduced to determine the natural frequencies of the plate/shell coupled structure. The procedure for determining the natural frequencies of the plate/shell coupled structure is same to the one for free vibration analysis of open and closed circular cylindrical shell by MRRM presented in [5254].
3. Results and discussions
In this section, free vibration analysis of plate/shell coupled structures with two opposite edges simply supported is presented to verify the validity and accuracy of the present method. The natural frequencies of the three plate/shell coupled structures are calculated by MRRM and by FEM, and the comparison results are presented in tabular form. It is particularly pointed out that all the results obtained by FEM in the following discussions are calculated with the commercial software ANSYS.
3.1. The boxtype structure
Studies on the free vibration of the boxtype structure, as shown in Fig. 4, with two opposite edges simply supported are conducted in this subsection. The material properties of the boxtype structure are: Young’s modulus $E=$ 2.1×10^{11} Pa, Poisson’s ratio $\mu =$ 0.3, mass density $\rho =$ 7800 kg m^{−3}, and the geometrical parameters of the boxtype structure are: ${L}_{x}=$ 10 m, ${L}_{y1}={L}_{y2}=$ 4 m, $h=$ 0.01 m. The results for natural frequencies of the boxtype structure obtained by MRRM are compared with those obtained by FEM in Table 1, where $m$ denotes the axial mode number, $n$ represents the mode number in the circumferential direction, and PError indicates the percentage error between the results obtained by MRRM and FEM.
Fig. 4. Geometry and notations of a boxtype structure
Table 1. Comparison of natural frequencies obtained by MRRM and FEM for the boxtype structure
Mode number

Natural frequencies and percentage errors


$n$

$m$

1

2

3

4

5

6

7

8

1

FEM

1.787

2.526

3.758

5.484

7.703

10.416

13.621

17.320

MRRM

1.788087

2.528058

3.761279

5.487782

7.707571

10.420647

13.627011

17.326663


PError

0.06 %

0.08 %

0.09 %

0.07 %

0.06 %

0.04 %

0.04 %

0.04 %


2

FEM

2.596

3.206

4.305

5.926

8.067

10.723

13.886

17.552

MRRM

2.597011

3.207152

4.307759

5.928857

8.070860

10.726757

13.890259

17.556971


PError

0.04 %

0.04 %

0.06 %

0.05 %

0.05 %

0.04 %

0.03 %

0.03 %


3

FEM

3.635

4.116

5.053

6.523

8.548

11.117

14.217

17.835

MRRM

3.633189

4.116464

5.053479

6.524894

8.550082

11.119675

14.219630

17.838747


PError

0.05 %

0.01 %

0.01 %

0.03 %

0.02 %

0.02 %

0.02 %

0.02 %


4

FEM

6.411

7.149

8.380

10.104

12.321

15.032

18.236

21.934

MRRM

6.412601

7.152497

8.385686

10.112165

12.331930

15.044982

18.251319

21.950942


PError

0.02 %

0.05 %

0.07 %

0.08 %

0.09 %

0.09 %

0.08 %

0.08 %


5

FEM

8.015

8.662

9.768

11.357

13.445

16.039

19.142

22.752

MRRM

8.012426

8.664084

9.772218

11.363087

13.452738

16.048779

19.153304

22.765631


PError

0.03 %

0.02 %

0.04 %

0.05 %

0.06 %

0.06 %

0.06 %

0.06 %


6

FEM

9.798

10.384

11.374

12.824

14.768

17.224

20.203

23.706

MRRM

9.817337

10.387455

11.377389

12.827979

14.772323

17.230424

20.210830

23.714801


PError

0.20 %

0.03 %

0.03 %

0.03 %

0.03 %

0.04 %

0.04 %

0.04 %


7

FEM

14.118

14.855

16.085

17.807

20.023

22.732

25.934

29.629

MRRM

14.115270

14.858683

16.092494

17.819161

20.038987

22.752050

25.958374

29.657972


PError

0.02 %

0.02 %

0.05 %

0.07 %

0.08 %

0.09 %

0.09 %

0.10 %


8

FEM

16.484

17.169

18.300

19.901

21.984

24.560

27.633

31.209

MRRM

16.503369

17.175778

18.307301

19.910498

21.997010

24.575867

27.653055

31.231947


PError

0.12 %

0.04 %

0.04 %

0.05 %

0.06 %

0.06 %

0.07 %

0.07 %

It can be found from Table 1 that, the natural frequencies obtained by MRRM and FEM agree well with each other. The maximum of the percentage errors is 0.20 %, and most of the percentage errors are no larger than 0.10 %. The small discrepancies in the results should be attributed to the approximation of FEM. Therefore, it indicates that MRRM is validate and of high precision for free vibration analysis of plate coupled structures such as the boxtype structure.
3.2. The racetrack cylindrical shell
Consider an isotropic, racetrack cylindrical shell composed of thin plates and OCCSs with axial length ${L}_{x}=$ 10 m, uniform thickness $h=$ 0.01 m, circumferential length of the plate ${L}_{y}=$ 4 m, middle surface radius of the OCCS $R=$ 2 m, as shown in Fig. 5. The material properties of the racetrack cylindrical shell are the same as those defined for the boxtype structure. The results for the natural frequencies of the racetrack cylindrical shell obtained by MRRM are compared with those obtained by FEM in Table 2, in which the notations are of the same meaning as those in Table 1.
Table 2 shows that the natural frequencies obtained by MRRM and FEM agree well with each other. The maximum of the percentage errors is 1.34%, and most of the percentage errors are no larger than 1.00 %. The difference between the results obtained by MRRM and FEM may be caused by the approximation of FEM and the different shell theories adopted by FEM and this paper. Therefore, it indicates that MRRM is validate for free vibration analysis of plateshell coupled structures such as the racetrack cylindrical shell, and the results are of high precision.
Table 2. Comparison of natural frequencies obtained by MRRM and FEM for the racetrack cylindrical shell
Mode number

Natural frequencies and percentage errors


$n$

$m$

1

2

3

4

5

6

7

8

1

FEM

2.151

2.953

4.090

5.715

7.854

10.508

13.670

17.336

MRRM

2.151035

2.957795

4.099758

5.727604

7.869848

10.525766

13.689947

17.357884


PError

0.00 %

0.16 %

0.24 %

0.22 %

0.20 %

0.17 %

0.15 %

0.13 %


2

FEM

5.383

6.984

8.461

10.232

12.415

15.064

18.200

21.832

MRRM

5.456198

7.031901

8.506268

10.281368

12.471784

15.128039

18.271764

21.911835


PError

1.34 %

0.68 %

0.53 %

0.48 %

0.46 %

0.42 %

0.39 %

0.36 %


3

FEM

5.444

13.162

15.027

17.038

19.382

22.133

25.328

28.989

MRRM

5.514263

13.195591

15.093706

17.133938

19.500229

22.270141

25.483314

29.162474


PError

1.27 %

0.25 %

0.44 %

0.56 %

0.61 %

0.62 %

0.61 %

0.59 %


4

FEM

10.458

20.291

23.281

25.846

28.524

31.492

34.835

38.598

MRRM

10.460565

20.468996

23.452468

26.038717

28.741440

31.738403

35.110916

38.903664


PError

0.02 %

0.87 %

0.73 %

0.74 %

0.76 %

0.78 %

0.79 %

0.79 %


5

FEM

10.684

20.334

33.382

36.659

39.751

43.017

46.583

50.513

MRRM

10.687016

20.525417

33.560942

36.926488

40.084378

43.403142

47.016813

50.994129


PError

0.03 %

0.93 %

0.53 %

0.72 %

0.83 %

0.89 %

0.92 %

0.94 %


6

FEM

13.803

27.932

33.447

48.481

52.618

56.447

60.381

64.577

MRRM

13.927559

27.962842

33.633721

48.880722

53.107713

57.012575

61.015628

65.278488


PError

0.89 %

0.11 %

0.56 %

0.82 %

0.92 %

0.99 %

1.04 %

1.07 %


7

FEM

15.378

28.755

42.048

48.529

66.309

71.352

75.927

80.554

MRRM

15.492372

28.837613

42.200925

48.948598

66.815182

72.081826

76.777865

81.507231


PError

0.74 %

0.29 %

0.36 %

0.86 %

0.76 %

1.01 %

1.11 %

1.17 %


8

FEM

15.955

30.057

42.961

57.665

72.451

84.556

92.219

97.881

MRRM

15.956436

30.175983

43.324195

57.735824

72.588720

84.589523

93.167426

99.063747


PError

0.01 %

0.39 %

0.84 %

0.12 %

0.19 %

0.04 %

1.02 %

1.19 %

Fig. 5. Geometry and notations of a racetrack cylindrical shell
Fig. 6. Geometry and notations of a ship hull structure
3.3. The ship hull structure
In this subsection, the natural frequencies of the ship hull structure, as shown in Fig. 6, with two opposite edges simply supported are calculated. The material properties of the ship hull structure are the same as those defined for the boxtype structure. The geometrical parameters are: ${L}_{x}=$ 10 m, ${L}_{y1}=$ 2 m, ${L}_{y2}=$ 4m, $R=$ 2 m and $h=$ 0.01 m. The results for the natural frequencies of the boxtype structure obtained by MRRM are compared with those obtained by FEM in Table 3, where the notations are of the same meaning as those in Tables 1 and 2.
Table 3. Comparison of natural frequencies obtained by MRRM and FEM for the ship hull structure
Mode number

Natural frequencies and percentage errors


$n$

$m$

1

2

3

4

5

6

7

8

1

FEM

2.562

5.713

6.575

8.847

13.875

16.349

16.377

20.440

MRRM

2.574674

5.803015

6.651149

8.818441

13.925635

16.342406

17.035003

20.525130


PError

0.49 %

1.55 %

1.14 %

0.32 %

0.36 %

0.04 %

3.86 %

0.41 %


2

FEM

3.217

7.027

8.241

9.973

16.702

21.496

22.775

28.663

MRRM

3.222129

7.072771

8.300801

9.991594

16.703286

21.734920

23.026966

27.617563


PError

0.16 %

0.65 %

0.72 %

0.19 %

0.01 %

1.10 %

1.09 %

3.79 %


3

FEM

4.318

8.407

9.681

11.209

18.061

24.346

25.892

30.900

MRRM

4.322056

8.441562

9.737088

11.237360

18.073568

24.512647

26.124280

31.037877


PError

0.09 %

0.41 %

0.58 %

0.25 %

0.07 %

0.68 %

0.89 %

0.44 %


4

FEM

5.934

10.159

11.372

12.757

19.754

26.778

28.435

32.886

MRRM

5.937682

10.193966

11.432079

12.791735

19.773413

26.937417

28.661965

33.011183


PError

0.06 %

0.34 %

0.53 %

0.27 %

0.10 %

0.59 %

0.79 %

0.38 %


5

FEM

8.071

12.362

13.460

14.723

21.882

29.349

31.044

35.145

MRRM

8.075624

12.400127

13.527708

14.764371

21.906555

29.520271

31.286598

35.270387


PError

0.06 %

0.31 %

0.50 %

0.28 %

0.11 %

0.58 %

0.78 %

0.36 %


6

FEM

10.725

15.043

16.010

17.168

24.480

32.245

33.918

37.775

MRRM

10.729149

15.085841

16.086390

17.215377

24.509296

32.435677

34.185502

37.908113


PError

0.04 %

0.28 %

0.47 %

0.28 %

0.12 %

0.59 %

0.78 %

0.35 %


7

FEM

13.886

18.212

19.052

20.122

27.566

35.541

37.152

40.826

MRRM

13.891362

18.261305

19.137747

20.174461

27.599471

35.755749

37.449705

40.969879


PError

0.04 %

0.27 %

0.45 %

0.26 %

0.12 %

0.60 %

0.79 %

0.35 %


8

FEM

17.552

21.874

22.599

23.598

31.149

39.276

40.800

44.330

MRRM

17.557387

21.930540

22.694437

23.653557

31.186065

39.516188

41.130030

44.484581


PError

0.03 %

0.26 %

0.42 %

0.23 %

0.12 %

0.61 %

0.80 %

0.35 %

It can be observed from Table 3 that, the natural frequencies obtained by MRRM and FEM agree well with each other. Except for certain mode numbers, the maximum of the percentage errors is 1.55 %, and most of the percentage errors are no larger than 1.00 %. The difference between the results obtained by MRRM and FEM may be caused by the approximation of FEM and the different shell theories adopted by FEM and this paper. Therefore, it indicates that MRRM is applicable for free vibration analysis of plateshell coupled structures such as the ship hull structure, and the results are of high precision.
4. Conclusions
This paper presents a semianalytical solution procedure and accurate calculation results for plate/shell coupled structures with two opposite edges simply supported. The validity and applicability of the MRRM for free vibration analysis of plate/shell coupled structures are verified. It has been proved that the results obtained by MRRM are in excellent agreement with those obtained by FEM and that high precision of MRRM has been shown. MRRM is advantageous in its simple and uniform formulation as well as accurate results for dynamic response analysis of coupled structures.
It should be pointed out that the method of reverberationray matrix only applies to plate/shell coupled structures with two opposite edges simplysupported at present. This limitation may be breached by replacing the standing wave form solution with the combination of the trigonometric function and the tangent or cotangent functions. The effects of the structural parameters, boundary conditions and connection forms on the vibration characteristics of the plate/shell coupled structures will be discussed in the subsequent researches.
Acknowledgements
This work is financially supported by National Natural Science Foundation of China (Grant Nos. 51479041, 51279038). The authors would like to express their profound thanks for the financial support and sincerely thank Miss Jingjing Yu for the scientific discussions and suggestions and the anonymous reviewers for the critical and constructive comments on this paper.
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