Introduction

The theoretical study of shell structure vibration was carried out earlier, and Love summarized and formed the earliest Love shell vibration theory [1].On the basis of the Love shell vibration theory, the classic theories such as Donnell shell theory [2], Novozhilov shell theory [3] and Flugge [4] shell theory were formed according to the actual problem. Leissa [5] made a systematic summary of the research on free vibration of shell structures, and elaborated the derivation process of various classical thin shell theories. The effects of different boundary conditions, additional mass, anisotropy, variable thickness, initial stress and other factors of the vibration characteristics of closed and open shell structure were discussed. Rayleigh and Lamb [6] first studied the stress wave transmission in isotropic material plates, and obtained the Rayleigh-Lamb dispersion characteristic equation for the plane vibration of a single-layer unimpeded isotropic plate. Lu [7] proposed a dynamic modeling method of bolted joint structure based on spectral finite element method. This method obtained the frequency domain response from the structure by deriving the dynamic stiffness matrix of the structure in the frequency domain and combining the boundary conditions of force and displacement.Mauro [8] conducted a force study on the free vibration characteristics of a thin-walled cylindrical shell structure under the shear boundary with both ends. Urak [9] studied the free vibration characteristics of the cylindrical shell structure. The study showed that the lowest order natural frequency is approximately proportional to the root mean square of the thickness, and the change of the shell thickness has a significant impact on the cylindrical shell mode shape. Farshidianfar [10] used different classical shell vibration theories to solve, compare and summarize the modal characteristics of cylindrical shells. Jin [11] designed a lightweight cylindrical honeycomb sandwiched structure and established an analytical model based on shell theory. Sofiyev [12] established the mathematical model of in-homogeneous orthotropic double-curved shallow shells. The influences of parameters on the forced vibration of orthotropic shell structures under external excitations are studied. Avey [13] studied the nonlinear vibration of moderately thick multilayer shell-type structural elements with double curvature by using shell theory. Marsick [14] studied the vitro-acoustic response of a cylindrical shell submerged near a free sea surface. Vinogradova [15] studied acoustic oscillations of a rigid spherical shell with two symmetric circular apertures by the Method of Analytical Regularization.Zhang [16] proposed a structural modeling method for submarine shock response analysis, which simplified the submarine structure into a beam of constant section and calculated the structural vibration frequency. Qin [17] proposed a general approach for the vibration analysis of a rotating cylindrical shell coupled with an annular plate.The effects of the geometric parameters and the boundary and coupling conditions on the vibration behavior of the coupled structure were evaluated. Zhu [18] presented a unified modeling approach for cylindrical shells with partial constrained layer damping (CLD) treatment under general boundary conditions. Qin [19] proposed a unified method to analyze the free vibration of graphene sheet (GPL) reinforced functionally graded laminate shallow shells under arbitrary boundary conditions.Liu [20] established the nonlinear coupled mechanical model and analyze nonlinear forced vibrations of rotating shells subjected to multi-harmonic excitations in thermal environment, and analyzed the influence of main factors on the nonlinear dynamic response of rotating shells.

The classical thin shell theory is difficult to analyze the vibration response of the pressure shell structure. The ribs on the actual submarine pressure hull are generally arranged periodically. Therefore, it is necessary to adopt a new method to analyze the vibration of the pressure shell structure.The spectral element method combines the dynamic stiffness matrix method and the fast Fourier transform. It is more efficient and convenient than the traditional finite element method. The analysis of geometry still faces some difficulties. It is necessary to combine other theories to realize the calculation of complex structures.The ribs on the actual submarine pressure hull are generally arranged periodically. This periodic arrangement will form a vibration band gap between the pressure hull structure. It is necessary to combine the periodic structure theory to study the band gap characteristics of the pressure hull structure.

The organization of this paper is as follows. In Sect. "Analytical calculation", the dynamic analysis of the pressure shell structure is carried out by using the spectral element method. The force and boundary conditions of the pressure shell structure are established, and the dynamic stiffness matrix of the pressure shell structure in the frequency domain is obtained. The kinematic characteristics of the pressure shell structure with periodic distribution of ring ribs are analyzed by using the Bloch theory of periodic structure. In Sect. 3, the validity and feasibility of the calculation method proposed are verified through the finite element software calculation. In Sect. 4, a test platform is built for the modal test of the pressure-resistant shell structure.

Analytical calculation

Model of a segmented pressure shell

A cylindrical coordinate system (y, \(\theta\), z)is established, and the midpoint of the axis is taken as the coordinate origin. The radial direction is y, the circumferential direction is \(\theta\), the axis direction is z, and the displacements in the corresponding directions are (u, g, w). The coordinate system is shown in Fig. 1.

Fig. 1
figure 1

The coordinate system

Full size image

The Donnell equation of motion [5] of the pressure shell structure is:

$${r}^{2}\frac{{\partial }^{2}u}{{\partial x}^{2}}+\frac{1-\lambda }{2}\frac{{\partial }^{2}u}{{\partial \theta }^{2}}+\frac{r\left(1+\lambda \right)}{2r}\frac{{\partial }^{2}g}{\partial x\partial \theta }+r\lambda \frac{\partial w}{\partial x}=\frac{{\rho r}^{2}\left(1-{\lambda }^{2}\right)}{E}\frac{{\partial }^{2}u}{{\partial t}^{2}}\frac{r\left(1+\lambda \right)}{2}\frac{{\partial }^{2}u}{\partial x\partial \theta }+\frac{{r}^{2}\left(1-\lambda \right)}{2}\frac{{\partial }^{2}g}{\partial {x}^{2}}+\frac{{\partial }^{2}g}{\partial {\theta }^{2}}+\frac{\partial w}{\partial \theta }=\frac{{\rho r}^{2}\left(1-{r}^{2}\right)}{E}\frac{{\partial }^{2}g}{{\partial t}^{2}}-r\lambda \frac{\partial u}{\partial x}-\frac{\partial g}{\partial \theta }-w-\frac{{d}^{2}}{12{r}^{2}}\left({r}^{4}\frac{{\partial }^{4}w}{{\partial x}^{4}}+2{r}^{2}\frac{{\partial }^{4}w}{{\partial x}^{2}{\partial \theta }^{2}}+\frac{{\partial }^{4}w}{{\partial \theta }^{4}}\right)=\frac{{\rho r}^{2}\left(1-{\lambda }^{2}\right)}{E}\frac{{\partial }^{2}w}{{\partial t}^{2}}$$
(1)

where, r is the radius of the pressure shell, d is the thickness, the material parameters E is Young's modulus, \(\lambda\) is Poisson's ratio, and \(\rho\) is material density.

When the pressure casing is only stressed at both ends, the form of the structural solution of the pressure casing can be set as:

$$\left\{\begin{array}{c}u\left(x,\theta ,t\right)={u}_{0}{e}^{-ikx}\mathrm{cos}\xi \theta {e}^{i\omega t}\\ g\left(x,\theta ,t\right)={v}_{0}{e}^{-ikx}\mathrm{sin}\xi \theta {e}^{i\omega t}\\ w\left(x,\theta ,t\right)={w}_{0}{e}^{-ikx}\mathrm{cos}\xi \theta {e}^{i\omega t}\end{array}\right.$$
(2)

Substituting Eq. (2) into Eq. (1), a homogeneous system of equation can be obtained:

$$\left[\left(k,\xi ,\omega \right)\right]\left[\begin{array}{c}{u}_{0}\\ {g}_{0}\\ {w}_{0}\end{array}\right]=0$$
(3)

where,

$$\left[\left(k,\xi ,\omega \right)\right]=\left[\begin{array}{ccc}{r}^{2}{k}^{2}+{\Omega }^{2}+\frac{1-\lambda }{2}{\xi }^{2}& -ik\xi \frac{r\left(1+\lambda \right)}{2r}& -ikr\lambda \\ ik\xi \frac{r\left(1+\lambda \right)}{2}& {k}^{2}\frac{{r}^{2}\left(1-\lambda \right)}{2}& -\xi \\ ikr\lambda & -\xi & -1-\frac{{d}^{2}}{12{r}^{2}}\left({r}^{4}{k}^{4}+2{r}^{2}{k}^{2}{\xi }^{2}+{\xi }^{4}\right)+{\Omega }^{2}\end{array}\right]$$

,

$${\Omega }^{2}={\upomega }^{2}\frac{\rho {r}^{2}\left(1-{\lambda }^{2}\right)}{E}$$

By solving the coefficient matrix, the solution of the lower wave number k of each mode can be obtained. The ratio between the amplitudes can be obtained from the Eq. (2).The pressure shell is a curved structure, so a degree of freedom \(\psi =\mathrm{d}w/\mathrm{d}x\) needs to be introduced to fully describe its nodal state. By setting \(\psi =\mathrm{d}w/\mathrm{d}x=0\), then the rest of the terms can be expressed as a function \({u}_{0}\), and the solution under the \({k}_{j}\) is:

$${\left\{\begin{array}{c}{u}_{0}\\ {g}_{0}\\ {w}_{0}\\ {\psi }_{0}\end{array}\right\}}_{j}={\left\{\begin{array}{c}1\\ {\phi }_{v}\\ {\phi }_{w}\\ {\phi }_{\psi }\end{array}\right\}}_{j}{u}_{0}={\left\{\Phi \right\}}_{j}{u}_{0}$$
(4)

From the coefficient matrix of Eq. (3), there are 8 solutions to the k of the mode can be obtained.The 8 solutions are opposite numbers to each other, which means that there are waves propagating in the forward direction. The \(+{k}_{j}\) represents the forward propagating wave, and \(-{k}_{j}\) represents the reverse propagating wave. For the finite-length pressure-resistant shell structure, it represents the boundary reflected wave.Then the displacement solution \(\widetilde{u}\) of the pressure-resistant shell structure with length L can be written as:

$$\widetilde{u}\left(x\right)={A}_{1}{\phi }_{11}{e}^{-i{k}_{1}x}+{A}_{2}{\phi }_{12}{e}^{-i{k}_{2}x}+{A}_{3}{\phi }_{13}{e}^{-i{k}_{3}x}+{A}_{4}{\phi }_{14}{e}^{-i{k}_{4}x}+{A}_{5}{\phi }_{15}{e}^{-i{k}_{1}\left(L-x\right)}+{A}_{6}{\phi }_{16}{e}^{-i{k}_{2}\left(L-x\right)}+{A}_{7}{\phi }_{17}{e}^{-i{k}_{3}\left(L-x\right)}+{A}_{8}{\phi }_{18}{e}^{-i{k}_{4}\left(L-x\right)}$$
(5)

For each mode, its amplitude \({A}_{j}\) is undetermined.Therefore, only the value of the amplitude coefficient needs to be solved, and the displacement solution of the pressure shell structure can be completely expressed:

$$\left[\begin{array}{c}\widetilde{u}\left(x\right)\\ \widetilde{g}\left(x\right)\\ \widetilde{w}\left(x\right)\\ \widetilde{\psi }\left(x\right)\end{array}\right]=\left[{\Phi }_{\mathrm{A}}\right]\left[e\left(x\right)\right]\left[\begin{array}{c}{\mathrm{A}}_{1}\\ {\mathrm{A}}_{2}\\ {\mathrm{A}}_{3}\\ {\mathrm{A}}_{4}\end{array}\right]+\left[{\Phi }_{\mathrm{B}}\right]\left[e\left(L-x\right)\right]\left[\begin{array}{c}{\mathrm{A}}_{5}\\ {\mathrm{A}}_{6}\\ {\mathrm{A}}_{7}\\ {\mathrm{A}}_{8}\end{array}\right]$$
(6)

where, \(\left[{\Phi }_{\mathrm{A}}\right]=\left[{\left\{\begin{array}{c}1\\ {\phi }_{v}\\ {\phi }_{w}\\ {\phi }_{\psi }\end{array}\right\}}_{1}{\left\{\begin{array}{c}{\phi }_{u}\\ 1\\ {\phi }_{w}\\ {\phi }_{\psi }\end{array}\right\}}_{2}{\left\{\begin{array}{c}{\phi }_{u}\\ {\phi }_{v}\\ 1\\ {\phi }_{\psi }\end{array}\right\}}_{3}{\left\{\begin{array}{c}{\phi }_{u}\\ {\phi }_{v}\\ 1\\ {\phi }_{\psi }\end{array}\right\}}_{4}\right],\left[{\Phi }_{A}\right]=\left[{\Phi }_{B}\right]\).

$$\left[e\left(x\right)\right]=\left[\begin{array}{cccc}{e}^{-i{k}_{1}x}& 0& 0& 0\\ 0& {e}^{-i{k}_{2}x}& 0& 0\\ 0& 0& {e}^{-i{k}_{3}x}& 0\\ 0& 0& 0& {e}^{-i{k}_{4}x}\end{array}\right]$$
(7)

The displacement on both sides of the pressure shell structure is:

$$\begin{array}{c}\left\{{\widetilde{u}}_{1}\right\}={\left[\widetilde{u}\left(0\right) \widetilde{g}\left(0\right) \widetilde{w}\left(0\right) \widetilde{\psi }\left(0\right)\right]}^{T}\\ \left\{{\widetilde{u}}_{2}\right\}={\left[\widetilde{u}\left(L\right) \widetilde{g}\left(L\right) \widetilde{w}\left(L\right) \widetilde{\psi }\left(L\right)\right]}^{T}\end{array}$$
(8)

For the convenience of writing, let:

$$\begin{array}{c}\left\{\mathrm{A}\right\}={\left[{\mathrm{A}}_{1} {\mathrm{A}}_{2} {\mathrm{A}}_{3} {\mathrm{A}}_{4}\right]}^{T}\\ \left\{\mathrm{B}\right\}={\left[{\mathrm{A}}_{5} {\mathrm{A}}_{6} {\mathrm{A}}_{7} {\mathrm{A}}_{8}\right]}^{T}\end{array}$$
(9)

Substituting Eq. (8) into Eq. (6), we can get:

$$\left\{\begin{array}{c}\left\{{\widetilde{u}}_{1}\right\}\\ \left\{{\widetilde{u}}_{2}\right\}\end{array}\right\}=\left[\begin{array}{cc}{\left\{\left[{\Phi }_{\mathrm{A}}\right]\left[e\left(0\right)\right]\right\}}_{4\times 4}& {\left\{\left[{\Phi }_{\mathrm{B}}\right]\left[e\left(L\right)\right]\right\}}_{4\times 4}\\ {\left\{\left[{\Phi }_{\mathrm{A}}\right]\left[e\left(L\right)\right]\right\}}_{4\times 4}& {\left\{\left[{\Phi }_{\mathrm{B}}\right]\left[e\left(0\right)\right]\right\}}_{4\times 4}\end{array}\right]\left\{\begin{array}{c}\left\{A\right\}\\ \left\{B\right\}\end{array}\right\}$$
(10)

where \(\left[e\left(0\right)\right]\), \(\left[e\left(L\right)\right]\), \(\left[{\Phi }_{\mathrm{A}}\right]\) and \(\left[{\Phi }_{\mathrm{B}}\right]\) is determined by the structure of the pressure-resistant shell, \(\widetilde{u}\left(0\right)\), \(\widetilde{g}\left(0\right)\), \(\widetilde{w}\left(0\right)\), \(\widetilde{\psi }\left(0\right)\), \(\widetilde{u}\left(L\right)\), \(\widetilde{v}\left(L\right)\), \(\widetilde{w}\left(L\right)\) and \(\widetilde{\psi }\left(L\right)\) is the displacement of the pressure shell at the node.

For the finite-length pressure shell structure, the displacement solution can be expressed linearly by the normalized displacement. So the dynamic stiffness relationship of the structure can be expressed as:

$$\left\{\begin{array}{c}{\left\{F\right\}}_{1}\\ {\left\{F\right\}}_{2}\end{array}\right\}=\left[K\left(\omega ,\xi \right)\right]\left[\begin{array}{cc}{\left\{\left[{\Phi }_{\mathrm{A}}\right]\left[e\left(0\right)\right]\right\}}_{4\times 4}& {\left\{\left[{\Phi }_{\mathrm{B}}\right]\left[e\left(L\right)\right]\right\}}_{4\times 4}\\ {\left\{\left[{\Phi }_{\mathrm{A}}\right]\left[e\left(L\right)\right]\right\}}_{4\times 4}& {\left\{\left[{\Phi }_{\mathrm{B}}\right]\left[e\left(0\right)\right]\right\}}_{4\times 4}\end{array}\right]\left\{\begin{array}{c}\left\{A\right\}\\ \left\{B\right\}\end{array}\right\}$$
(11)

For pressure-resistant shell structures, four forces or displacement boundary conditions are satisfied at each end. Therefore, there are eight unknown forces and displacements on the left side of Eq. (10) and Eq. (11).Since only the amplitude coefficients need to be solved, the equations containing unknown forces and displacements can be discarded. By solving the simplified equations, the solutions to the amplitude coefficients can be obtained.Then, we can obtain the displacement solution to this section of the pressure shell by substituting the solution to Eq. (6).

Boundary Condition Analysis

When the pressure-resistant shell structure is excited to vibrate, the ring rib on the pressure-resistant shell will also vibrate with the vibration of the pressure-resistant shell structure. The vibration differential equation of the ring-rib structure can be written as follows:

$$\begin{array}{c}\frac{{E}_{b}{I}_{1}}{{r}_{b}^{4}}\left(\frac{{\partial }^{4}{w}^{*}}{\partial {\theta }^{4}}-\frac{{\partial }^{3}{g}^{*}}{\partial {\theta }^{3}}\right)+\frac{{E}_{b}{A}_{b}}{{r}_{b}^{2}}\left({w}^{*}-\frac{\partial {g}^{*}}{\partial \theta }\right)+{\rho }_{b}{A}_{b}\frac{{\partial }^{2}{w}^{*}}{\partial {t}^{2}}={F}_{w}\left(\theta ,t\right)\\ \frac{{E}_{b}{I}_{1}}{{r}_{b}^{4}}\left(\frac{{\partial }^{3}{w}^{*}}{\partial {\theta }^{3}}-\frac{{\partial }^{2}{g}^{*}}{\partial {\theta }^{2}}\right)+\frac{{E}_{b}{A}_{b}}{{r}_{b}^{2}}\left(\frac{\partial {w}^{*}}{\partial \theta }+\frac{{\partial }^{2}{g}^{*}}{\partial {\theta }^{2}}\right)+{\rho }_{b}{A}_{b}\frac{{\partial }^{2}{g}^{*}}{\partial {t}^{2}}=0 \\ \frac{{E}_{b}{\overline{I} }_{1}}{{r}_{b}^{4}}\left(\frac{{\partial }^{4}{u}^{*}}{\partial {\theta }^{4}}-\frac{{\partial }^{2}\left({r}_{b}\varphi \right)}{\partial {\theta }^{2}}\right)+\frac{{G}_{b}J}{{r}_{b}^{4}}\left(\frac{{\partial }^{2}{u}^{*}}{\partial {\theta }^{2}}+\frac{\left({r}_{b}\varphi \right)}{{\theta }^{2}}\right)+{\rho }_{b}{A}_{b}\frac{{\partial }^{2}{u}^{*}}{\partial {t}^{2}}=0\\ \frac{{E}_{b}{\overline{I} }_{1}}{{r}_{b}^{4}}\left({r}_{b}\varphi -\frac{{\partial }^{2}{u}^{*}}{\partial {\theta }^{2}}\right)+\frac{{G}_{b}J}{{r}_{b}^{2}}\left(\frac{{\partial }^{2}{u}^{*}}{\partial {\theta }^{2}}+\frac{{\partial }^{2}\left({r}_{b}\varphi \right)}{\partial {\theta }^{2}}\right)+{\rho }_{b}{I}_{PI}\frac{{\partial }^{2}\left({r}_{b}\varphi \right)}{\partial {t}^{2}}={R}_{b}{F}_{\varphi }\left(\theta ,t\right)\end{array}$$
(12)

where, \({u}^{*}\) is axial displacement, \({g}^{*}\) is circumferential displacement,and \({w}^{*}\) is radial displacement; \({I}_{1}\) is axial principal moment of inertia, \({\overline{I} }_{1}\) is radial principal moment of inertia,and \({I}_{PI}\) is polar moment of inertia; \({r}_{b}=r-e\) is the centroid radius of the ring rib section;e is the eccentricity; \({A}_{b}\) is ring rib cross-sectional area, \(J\) is torsional constant of ring rib section; \({F}_{w}\) is external force acting on centroid of ring rib section per unit length, and \({F}_{\varphi }\) is torque; \(\varphi =\partial w/\partial x\) is corner;\({\rho }_{b}\),\({E}_{b}\), \({G}_{b}\) are the density, elastic modulus and shear modulus of the ring rib, respectively.

The force generated by the pressure shell structure of the ring rib can be calculated from the Eq. (12).By taking the action of the ring rib on the pressure shell structure as a force boundary condition acting on the pressure shell structure, the vibration response to the pressure shell structure can be solved and calculated.

Vibration characteristics analysis of periodically ribbed shell structure

When the rib of the pressure shell structure is distributed periodically, it can be regarded as a one-dimensional periodic structure. The kinematic characteristic of the structure is analyzed by the periodic structure Bloch theory.When analyzing the features of the periodic structure, the structure in a period can be studied. Then combined with the existing periodic structure theory, the overall structural characteristics can be calculated. In a single cycle, the dynamic equation of the structure can be described by the following formula.

$$\left\{\begin{array}{c}{f}_{L1}\\ {f}_{R1}\end{array}\right\}=\left[\begin{array}{cc}{\widehat{k}}_{11}\left({a}_{1}\right)& {\widehat{k}}_{12}\left({a}_{1}\right)\\ {\widehat{k}}_{21}\left({a}_{1}\right)& {\widehat{k}}_{22}\left({a}_{1}\right)\end{array}\right]\left\{\begin{array}{c}{u}_{L1}\\ {u}_{R1}\end{array}\right\}$$
(13)

Similarly, the dynamic equation of the structure in the second period can be obtained:

$$\left\{\begin{array}{c}{f}_{L2}\\ {f}_{R2}\end{array}\right\}=\left[\begin{array}{cc}{\widehat{k}}_{11}\left({a}_{2}\right)& {\widehat{k}}_{12}\left({a}_{1}\right)\\ {\widehat{k}}_{21}\left({a}_{2}\right)& {\widehat{k}}_{22}\left({a}_{1}\right)\end{array}\right]\left\{\begin{array}{c}{u}_{L2}\\ {u}_{R2}\end{array}\right\}$$
(14)

By displacement continuity conditions are: \({u}_{R1}={u}_{L2}={u}_{I}\). The kinetic equation for a structure involving two periods can then be written as,

$$\left\{\begin{array}{c}{f}_{L}\\ 0\\ {f}_{R}\end{array}\right\}=\left[\begin{array}{ccc}{\widehat{k}}_{11}\left({a}_{1}\right)& {\widehat{k}}_{12}\left({a}_{1}\right)& 0\\ {\widehat{k}}_{21}\left({a}_{1}\right)& {\widehat{k}}_{22}\left({a}_{1}\right)+{\widehat{k}}_{11}\left({a}_{2}\right)& {\widehat{k}}_{12}\left({a}_{2}\right)\\ 0& {\widehat{k}}_{21}\left({a}_{2}\right)& {\widehat{k}}_{22}\left({a}_{2}\right)\end{array}\right]\left\{\begin{array}{c}{u}_{L}\\ {u}_{I}\\ {u}_{R}\end{array}\right\}$$
(15)

The Eq. (15) can be simplified to obtain an expression containing only the displacement and force at the left and right ends.

$$\left\{\begin{array}{c}{f}_{L}\\ {f}_{R}\end{array}\right\}=\left[\begin{array}{cc}{\widehat{k}}_{11}& {\widehat{k}}_{12}\\ {\widehat{k}}_{21}& {\widehat{k}}_{22}\end{array}\right]\left\{\begin{array}{c}{u}_{L}\\ {u}_{R}\end{array}\right\}$$
(16)

where,

$$\begin{array}{c}{\widehat{k}}_{11}={\widehat{k}}_{11}\left({a}_{1}\right)-{\widehat{k}}_{12}\left({a}_{1}\right){\left({\widehat{k}}_{2}\left({a}_{1}\right)+{\widehat{k}}_{11}\left({a}_{2}\right)\right)}^{-1}{\widehat{k}}_{21}\left({a}_{1}\right)\\ {\widehat{k}}_{12}=-{\widehat{k}}_{12}\left({a}_{1}\right){\left({\widehat{k}}_{22}\left({a}_{1}\right)+{\widehat{k}}_{11}\left({a}_{2}\right)\right)}^{-1}{\widehat{k}}_{12}\left({a}_{2}\right)\\ {\widehat{k}}_{21}=-{\widehat{k}}_{21}\left({a}_{2}\right){\left({\widehat{k}}_{22}\left({a}_{1}\right)+{\widehat{k}}_{11}\left({a}_{2}\right)\right)}^{-1}{\widehat{k}}_{21}\left({a}_{1}\right)\\ {\widehat{k}}_{22}={\widehat{k}}_{22}\left({a}_{2}\right)-{\widehat{k}}_{21}\left({a}_{2}\right){\left({\widehat{k}}_{22}\left({a}_{1}\right)+{\widehat{k}}_{11}\left({a}_{2}\right)\right)}^{-1}{\widehat{k}}_{12}\left({a}_{2}\right)\end{array}$$
(17)

According to Bloch's theorem for periodic structures, the boundary displacement and force vector of a cell satisfied the following relationship.

$$\begin{array}{c}{u}_{R}={e}^{-iqL}{u}_{L}\\ {f}_{R}={e}^{-iqL}{f}_{L}\end{array}$$
(18)

Substituting Eq. (18) into Eq. (16), the characteristic equation containing only displacement is obtained.

$$\left(\left[\begin{array}{cc}{\widehat{k}}_{21}& {\widehat{k}}_{22}\\ 0& I\end{array}\right]-{e}^{iqL}\left[\begin{array}{cc}-{\widehat{k}}_{11}& -{\widehat{k}}_{12}\\ I& 0\end{array}\right]\right)\left\{\begin{array}{c}{u}_{L}\\ {u}_{R}\end{array}\right\}=\left\{\begin{array}{c}0\\ 0\end{array}\right\}$$
(19)

Solving the eigenvalues of Eq. (19), two pairs of Bloch wave vector solutions \(\pm {q}_{1}\), \(\pm {q}_{2}\) of the same size and opposite to each other and the corresponding eigenvector \({e}^{-iqL}\) can be obtained.Taking the eigenvector of \({u}_{L}\) corresponding to \({e}^{-i{q}_{1}L}\), \({e}^{-i{q}_{2}L}\), \({e}^{i{q}_{1}L}\), \({e}^{i{q}_{2}L}\), and denote them as \({\varphi }_{1}\), \({\varphi }_{2}\), \({\varphi }_{3}\), \({\varphi }_{4}\).Then \({u}_{L}\) can be obtained as

$${u}_{L}={a}_{1}{\varphi }_{1}+{a}_{2}{\varphi }_{2}+{a}_{3}{\varphi }_{3}+{a}_{4}{\varphi }_{4}$$
(20)

Using Eq. (16) and Bloch's theorem (18), we can get

$${f}_{L}=\left({\widehat{k}}_{11}+{e}^{-iqL}{\widehat{k}}_{12}\right){u}_{L}={a}_{1}{\Phi }_{1}+{a}_{2}{\Phi }_{2}+{a}_{3}{\Phi }_{3}+{a}_{4}{\Phi }_{4}$$
(21)

where, \({\lambda }_{j}\) is the eigenvalue corresponding to \(\left(j=\mathrm{1,2},\mathrm{3,4}\right)\), then there are,

$${\Phi }_{j}=\left({\widehat{k}}_{11}+{\lambda }_{j}{\widehat{k}}_{12}\right){\varphi }_{j},\left(j=\mathrm{1,2},\mathrm{3,4}\right)$$
(22)

For a beam structure with finite period, its period number is recorded as N, from Bloch's theorem we can get,

$${u}_{NR}={a}_{1}{e}^{-iN{q}_{1}L}{\varphi }_{1}+{a}_{2}{e}^{-iN{q}_{2}L}{\varphi }_{2}+{a}_{3}{e}^{iN{q}_{1}L}{\varphi }_{3}+{a}_{4}{e}^{iN{q}_{2}L}{\varphi }_{4}$$
(23)

Equation (23) is written in matrix form as:

$$\left\{\begin{array}{c}{u}_{L}\\ {u}_{NR}\end{array}\right\}=\left[\varphi \right]\left[a\right]$$
(24)

where,

$$\left\{\begin{array}{c}\left[\varphi \right]=\left[\begin{array}{cccc}{\varphi }_{1}& {\varphi }_{2}& {\varphi }_{3}& {\varphi }_{4}\\ {e}^{-iN{q}_{1}L}{\varphi }_{1}& {e}^{-iN{q}_{2}L}{\varphi }_{2}& {e}^{iN{q}_{1}L}{\varphi }_{3}& {e}^{iN{q}_{2}L}{\varphi }_{4}\end{array}\right]\\ \left[a\right]={\left[\begin{array}{cccc}{a}_{1}& {a}_{2}& {a}_{3}& {a}_{4}\end{array}\right]}^{T}\end{array}\right.$$
(25)

The same can be obtained

$$\left\{\begin{array}{c}{f}_{L}\\ {f}_{NR}\end{array}\right\}=\left[\Phi \right]\left[a\right]$$
(26)

where,

$$\left[\Phi \right]=\left[\begin{array}{cccc}{\Phi }_{1}& {\Phi }_{2}& {\Phi }_{3}& {\Phi }_{4}\\ -{e}^{-iN{q}_{1}L}{\Phi }_{1}& -{e}^{-iN{q}_{2}L}{\Phi }_{2}& -{e}^{iN{q}_{1}L}{\Phi }_{3}& -{e}^{iN{q}_{2}L}{\Phi }_{4}\end{array}\right]$$
(27)

Combining Eq. (24) and Eq. (26), we can get:

$$\left\{\begin{array}{c}{f}_{L}\\ {f}_{NR}\end{array}\right\}=\left[{K}_{N}\right]\left\{\begin{array}{c}{u}_{L}\\ {u}_{NR}\end{array}\right\}$$
(28)

where, \(\left[{K}_{N}\right]=\left[\Phi \right]{\left[\varphi \right]}^{-1}=\left[\begin{array}{cc}{K}_{N11}& {K}_{N12}\\ {K}_{N21}& {K}_{N22}\end{array}\right]\).

Simulation analysis

Natural frequency calculation

In order to verify the accuracy of the calculation method for the vibration of the pressure shell structure, firstly, the structural model of unstiffened shell is established.The basic model's structure parameters are following:the shell is 2 m long,the diameter is 0.5 m,and the pressure shell thickness is 0.003 m.The shell is a steel structure, and the material parameter elastic modulus is 210 GPa, density is 7900 kg/m3, Poisson's ratio is 0.3, and damping coefficient is 0.01.On the one hand, the vibration frequency response curve of the structure is calculated by the spectral element method. The modal frequencies of each order of the structure can be obtained according to the peak value of the frequency response curve.On the other hand, the finite element software calculates the model's various order modes and their corresponding modal frequencies.Modal analysis is one of the main methods to study the dynamic characteristics of the structure. The natural frequency of the structure can be obtained through modal analysis. The finite element software can calculate the modal frequency, and the modal frequency of different orders can be obtained through the modal deformation. Figure 2 is the vibration frequency response curve of the pressure housing calculated by the spectral element method. Figure 3 shows the first three modes calculated by the finite element software.

Fig. 2
figure 2

The vibration spectrum of cylindrical shell calculated by spectral analysis

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Fig. 3
figure 3

The first third order mode calculated by finite element method

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It can be seen from Fig. 2 that the first three-order modal frequencies calculated by the spectral element method are 144 Hz, 188 Hz, and 212 Hz, respectively.The first three modal frequencies calculated by the finite element software are 144 Hz, 190 Hz and 214 Hz.The results are listed in Table 1. The error between the spectral element method and the finite element calculation is tiny.

Table 1 The results between spectral analysis and finite element method
Full size table

Figure 4 shows the acceleration distribution on the surface of the pressure-resistant shell calculated by the spectral element method when the excitation force frequency is 144 Hz.The calculation results are consistent with the first-order mode shown in Fig. 3, and it can be concluded that the spectral element method can be used to calculate the pressure-resistant shell structure.

Fig. 4
figure 4

Normal acceleration distribution of cylindrical shell at the frequency of 144 Hz

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Structural Response Calculation of Pressure Shells

According to the theoretical calculation method established above, the structure model of the pressure shell is launched. The center of the symmetry axis of the pressure shell is taken as the origin. The axis direction is z, as shown in Fig. 5.The material properties and geometric dimensions of the pressure shell structure are defined as follows: the pressure shell and the ring rib reinforcement are all composed of the same steel material, the elastic modulus is 210GPa, the density is 7900 kg/m3, the Poisson's ratio is 0.3, and the damping coefficient is 0.01, the inner diameter of the pressure shell is 1.5 m, the shell thickness is 0.006 m, and the length is 1.56 m.There are 12 ring ribs evenly distributed in the pressure shell, the rib spacing is 0.12 m, the rib thickness is 0.005 m, and the rib height is 0.045 m. Each excitation force applied is a standard unit force radiating 1N, acting on the inner surface\(\left({\theta }_{0},{z}_{0}\right)\).

Fig. 5
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The pressure hull structure model

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The accuracy and validity of the pressure shell structure's vibration response are calculated to verify the established calculation theory. According to the theoretical calculation method shown above, the frequency response curve of the pressure shell structure under excitation is calculated.The finite element model of the pressure shell structure is established by finite element software.The frequency response curve of the pressure shell structure under excitation is obtained through software calculation. The frequency response curves calculated by the two methods are shown in Fig. 6.

Fig. 6
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Velocity level comparison of theoretical calculations and finite element calculations

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It can be seen from Fig. 6 that the vibration response of the pressure shell structure obtained by the theoretical calculation is in good agreement with the response curve calculated by the finite element model. There is a specific deviation between the theoretical analysis and finite element calculation near the natural frequency, which is caused by simplifying the annular rib of the pressure shell structure to the form of acting force in the process of theoretical modeling.It can be seen that the difference between the two curves is small.The results prove the effectiveness of the calculation theory.

Experimental Research

Because the structure of the submarine is complex and the diameter is relatively large, it is difficult to measure the effective vibration signal in the position far from the excitation.Therefore, when carrying out the structural vibration characteristics test of the structural shell, the whole ship's suction and exhaust fans in the power cabin are used as the excitation source.The diameter of the pressure shell in the test area is 6.9 m, the thickness is 35 mm, the rib spacing is 600 mm, and the cross-section of the ring rib is T-section. The measuring point distribution on the pressure hull is shown in Fig. 7. Figure 8 is a partial view of the test area.

Fig. 7
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Schematic diagram of measuring point distribution on the pressure hull

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Fig. 8
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Partial view of the pressure shell

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The experimental principle is shown in Fig. 9. The experiment consists of a test bench, a data acquisition module, and a computer. The acceleration sensor is used to collect acceleration signals at different positions of the structure. The data acquisition module is used to input signals to the computer.The computer is used to process the collected signals.

Fig. 9
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The experimental principle

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Since the contribution of different order modes to noise is different when the structure is vibrating, this modal test is carried out to find out the charge that has a more significant contribution to the radiated noise of the hull. The number of measurement points depends on the selected frequency range, the desired number of modes, the area of interest of the test object, the number of sensors available and the time. This experiment uses a simplified model, and the distribution of measurement points in this test is shown in Fig. 10.

Fig. 10
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The modal test experimental model

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Through the test, the calculated acceleration admittance of the submarine pressure hull is shown in Fig. 11. In the frequency range of 10 ~ 315 Hz, the acceleration admittance of the structure has two evident stop bands. The bandwidth of the first-stop band is much smaller than that of the second-stop band. This is a typical band gap characteristic of the periodic structure, which verifies the applicability and feasibility of the periodic structure theory on the pressure shell with periodic distribution ribs.

Fig. 11
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Acceleration admittance of exhaust fan

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This modal test adopts the hammering method, and the test method is single-point excitation and multi-point response. The sensor adopts the integrated circuit piezoelectric acceleration sensor of PCB company. The sensitivity of the sensor is 9.8mv/g. Due to the limitation of the on-site knocking position, only the vertical direction is knocked during the working modal test.

The frequency response function is measured by the MODAL TESTING module.The frequency response functions of all points are averaged 5 times, and the hull’s modal parameters and related mode shapes are obtained by ensemble analysis of the frequency response functions of 24 measuring points.The parameters, such as the first two modal frequencies and the damping ratio of the hull, are shown in Table 2. The first-order mode shape diagram is shown in Fig. 12.The result of the first-order mode shape diagram is 149.8HZ, close to the theoretical calculation result.

Table 2 Modal test result
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Fig. 12
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The first order mode

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Conclusions

In this paper, the spectral dynamic stiffness matrix of the pressure shell structure is established based on the spectral element method. Combined with the periodic structure Bloch theory, a theoretical analysis method for the vibration characteristics of the pressure shell structure based on the spectral element method and the periodic structure theory is proposed.Finally, the actual boat test experiment is carried out.In the frequency range of 10 ~ 315 Hz, there are two obvious stop bands for the acceleration admittance. The bandwidth and attenuation effect of the first-stop band are much smaller than those of the second-stop band, which is a very typical band gap characteristic of the periodic structure. The applicability and feasibility of the periodic structure theory on the pressure shell with the periodic distribution of the rib are presented.