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Modeling and dynamic analysis of bolted joined cylindrical shell

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Abstract

Based on Sanders shell theory, modeling and dynamic analysis of bolted joined cylindrical shell were studied in this paper. When subjected to external excitations, contact state such as stick, slip and separation may occur at those locations of bolts. Considering these three contact states, an analytical model of cylindrical shell with a piecewise-linear boundary was established for the bolted joined cylindrical shell. First, the model was verified by the simplified line system, and the effects of stiffness in connecting interface and the number of bolts on natural frequency and mode shape were investigated. Then, through the response under instantaneous excitation, damping characteristic of the system was proved which is caused by the friction model. Last, the effects of external load frequency, response location, excitation amplitude and connecting parameters including stiffness and preload were numerically investigated and fully explained by 3-D frequency spectrum. The results indicated that periodic motion, times periodic motion and even chaotic motion were observed based on different parameters.

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Abbreviations

u,v,w :

Displacement in the x, \(\theta \), z directions

H :

Thickness of the shell

L :

Length of the shell

R :

Radius of the shell

\(\rho \) :

Mass density

\(\mu \) :

Poisson’s ratio

E :

Young’s modulus

\(k_u ,k_v ,k_w ,k_\theta \) :

Stiffness of axial, circumferential, radial, rotational spring

\(N_\mathrm{B} \) :

Number of bolts

\(v_c , w_c \) :

Displacement of damper in the \(\theta \), z direction

\({\tilde{u}},{\tilde{v}},{\tilde{w}}\) :

Dimensionless displacement in the x, \(\theta \), z direction

\({\tilde{l}}\) :

Dimensionless length of the shell

\({\tilde{h}}\) :

Dimensionless thickness of the shell

\({\tilde{\omega }}\) :

Dimensionless frequency

\({\tilde{t}}\) :

Dimensionless time

\(\xi \) :

Dimensionless scale in the x direction

\({\tilde{k}}_u ,{\tilde{k}}_v ,{\tilde{k}}_w ,{\tilde{k}}_\theta \) :

Dimensionless stiffness of axial, circumferential, radial, rotational spring

F :

The amplitude of the external load

n :

The circumferential wave number

\(A_n ,B_n ,C_n \) :

The amplitude parameters of displacement

NT:

The number of terms for orthogonal polynomials

\({{\varvec{U}}}_{\theta _s } ,{{\varvec{V}}}_{\theta _s } ,{{\varvec{W}}}_{\theta _s } \) :

The generalized vector of modal functions for the location \((\xi =0,\theta =\theta _S )\)

\(u_{\theta _s } ,v_{\theta _s } ,w_{\theta _s } \) :

the displacement the location \((\xi =0,\theta =\theta _S )\)

M, C, K :

Mass matrix, damping matrix, stiffness matrix

\({{\varvec{F}}}_f \) :

The generalized vector of the external load

\({{\varvec{F}}}_\mathrm{bnon} \) :

The generalized vector of the nonlinear contact force

\({{\varvec{K}}}_\mathrm{bolt} \) :

Stiffness matrix of bolt joints in the simplified linear model

\(N_\mathrm{axial} \) :

The normal force on the contact surface

\(f_\mathrm{ax} \) :

The axial force on the contact surface

\(f_\mathrm{tan} \) :

The circumferential force on the contact surface

\(f_\mathrm{rad} \) :

The radial force on the contact surface

M :

The bending moment on the contact surface

\(F_{\mathrm{pre},\theta _s } \) :

The preload of bolts at \((\xi =0,\theta =\theta _S )\)

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Acknowledgements

The project was supported by the Fundamental Research Funds for the Central Universities (Nos. N160306004 and N160313001), the China Natural Science Funds (No. 51575093) and Liaoning Province Natural Science Funds (No. 2015020153).

Author information

Authors and Affiliations

  1. School of Mechanical Engineering and Automation, Northeastern University, Shenyang, 110819, China

    Qiansheng Tang, Chaofeng Li, Houxin She & Bangchun Wen

  2. Key Laboratory of Vibration and Control of Aero-Propulsion System Ministry of Education, Northeastern University, Shenyang, 110819, China

    Chaofeng Li & Bangchun Wen

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  1. Qiansheng Tang
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Correspondence to Chaofeng Li.

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Appendices

Appendix A

\(\psi _1^u \left( \xi \right) , \quad \psi _1^v \left( \xi \right) \) and \(\psi _1^w \left( \xi \right) \) are given as the first terms of the polynomials in the process of generating characteristic orthogonal polynomials \(\varphi _i^u \left( \xi \right) \), \(\varphi _i^v \left( \xi \right) \) and \(\varphi _i^w \left( \xi \right) \) using Gram–Schmidt method and they are satisfied according to boundary condition. \(\psi _1^u \left( \xi \right) \) is applied to generate \(\varphi _i^u \left( \xi \right) \) as an example.

$$\begin{aligned} \psi _2^u \left( \xi \right)= & {} \left( {\xi -B_1^u } \right) \psi _1^u \left( \xi \right) \\ B_1^u= & {} \frac{\int _0^1 {\xi \left[ {\psi _1^u \left( \xi \right) } \right] ^{2}\hbox {d}\xi } }{\int _0^1 {\left[ {\psi _1^u \left( \xi \right) } \right] ^{2}\hbox {d}\xi } }\\ \psi _{k+1}^u \left( \xi \right)= & {} \left( {\xi -B_k^u } \right) \psi _k^u \left( \xi \right) -C_k \psi _{k-1}^u \left( \xi \right) , k\ge 2\\ B_k^u= & {} \frac{\int _0^1 {\xi \left[ {\psi _k^u \left( \xi \right) } \right] ^{2}\hbox {d}\xi } }{\int _0^1 {\left[ {\psi _k^u \left( \xi \right) } \right] ^{2}\hbox {d}\xi } },\nonumber \\ C_k= & {} \frac{\int _0^1 {\xi \psi _k^u \left( \xi \right) \psi _{k-1}^u \left( \xi \right) \hbox {d}\xi } }{\int _0^1 {\left[ {\psi _{k-1}^u \left( \xi \right) } \right] ^{2}\hbox {d}\xi } }\\ \varphi _k^u \left( \xi \right)= & {} \frac{\psi _k^u \left( \xi \right) }{\int _0^1 {\left[ {\psi _k^u \left( \xi \right) } \right] ^{2}\hbox {d}\xi } } \end{aligned}$$

It should be noted that the polynomials constructed satisfy the orthogonality condition

$$\begin{aligned} \int _0^1 {\xi \varphi _k^u \left( \xi \right) \varphi _l^u \left( \xi \right) \hbox {d}\xi } =\left\{ {\begin{array}{l} 0,k\ne l \\ 1,k=l \\ \end{array}} \right. \end{aligned}$$

The same operation can be done for \(\psi _1^v \left( \xi \right) \) and \(\psi _1^w \left( \xi \right) \).

Appendix B

$$\begin{aligned} {{\varvec{M}}}_1= & {} \int _0^1 {\int _0^{2{\uppi }} {\left[ {{\tilde{l}}^{2}{{\varvec{UU}}}^{\mathrm{T}}} \right] } } \hbox {d}\xi \hbox {d}\theta ,{{\varvec{M}}}_2\\= & {} \int _0^1 {\int _0^{2{\pi }} {\left[ {{{\varvec{VV}}}^{\mathrm{T}}} \right] } } d\xi d\theta ,\\ M_3= & {} \int _0^1 {\int _0^{2{\uppi }} {\left[ {{\tilde{h}}^{2}{{\varvec{WW}}}^{\mathrm{T}}} \right] } } \hbox {d}\xi \hbox {d}\theta \\ {{\varvec{K}}}_1= & {} \int _0^1 \int _{0}^{2\pi } \left[ \frac{\partial {{\varvec{U}}}}{\partial \xi } \frac{\partial {{\varvec{U}}}^{\mathrm{T}}}{\partial \xi }+\frac{\left( {1-\mu } \right) {\tilde{l}}^{2}}{2}\frac{\partial {{\varvec{U}}}}{\partial \theta }\frac{\partial {{\varvec{U}}}^{\mathrm{T}}}{\partial \theta }\right. \\&\left. +\,\frac{{(1 - \mu ){{\tilde{l}}^2}{{\tilde{h}}^2}}}{{96}}\frac{{\partial {{\varvec{U}}}}}{{\partial \theta }} \frac{{\partial {{{\varvec{U}}}^\mathrm{{T}}}}}{{\partial \theta }} \right] \mathrm{{d}}\xi \mathrm{{d}}\theta \\ {{\varvec{K}}}_2= & {} \int _0^1 \int _{0}^{2\pi } \left[ 2\mu \frac{\partial {{\varvec{U}}}}{\partial \xi }\frac{\partial {{\varvec{V}}}^{\mathrm{T}}}{\partial \theta }+\left( {1-\mu } \right) \frac{\partial {{\varvec{U}}}}{\partial \theta }\frac{\partial {{\varvec{V}}}^{\mathrm{T}}}{\partial \xi }\right. \\&\left. -\,\frac{(1-\mu ){\tilde{h}}^{2}}{16}\frac{\partial {{\varvec{U}}}}{\partial \theta } \frac{\partial {{\varvec{V}}}^{\mathrm{T}}}{\partial \xi } \right] \hbox {d} \xi \hbox {d}\theta \\ {{\varvec{K}}}_3= & {} \int _0^1 {\int _0^{2\pi } {\left[ {2\mu {\tilde{h}}\frac{\partial {{\varvec{U}}}}{\partial \xi }{{\varvec{W}}}^{\mathrm{T}}+\frac{\left( {1-\mu } \right) {\tilde{h}}^{3}}{12} \frac{\partial {{\varvec{U}}}}{\partial \theta }\frac{\partial ^{2}{{\varvec{W}}}^{\mathrm{T}}}{\partial \xi \partial \theta }} \right] } } \hbox {d}\xi \hbox {d}\theta \\ {{\varvec{K}}}_4= & {} \int _0^1 \int _0^{2\pi } \left[ \frac{\partial {{\varvec{V}}}}{\partial \theta } \frac{\partial {{\varvec{V}}}^{\mathrm{T}}}{\partial \theta }+\frac{\left( {1-\mu } \right) }{2{\tilde{l}}^{2}}\frac{\partial {{\varvec{V}}}}{\partial \xi }\frac{\partial {{\varvec{V}}}^{\mathrm{T}}}{\partial \xi }\right. \\&\left. +\,\frac{{\tilde{h}}^{2}}{12} \frac{\partial {{\varvec{V}}}}{\partial \theta } \frac{\partial {{\varvec{V}}}^{\mathrm{T}}}{\partial \theta }\right. \\&\left. +\,\frac{3(1-\mu ){\tilde{h}}^{2}}{32{\tilde{l}}^{2}}\frac{\partial {{\varvec{V}}}}{\partial \xi }\frac{\partial {{\varvec{V}}}^{\mathrm{T}}}{\partial \xi } \right] \hbox {d}\xi \hbox {d}\theta \\ {{\varvec{K}}}_5= & {} \int _0^1 \int _0^{2\pi } \left[ 2{\tilde{h}}\frac{\partial {{\varvec{V}}}}{\partial \theta }{{\varvec{W}}}^{\mathrm{T}}-\frac{{\tilde{h}}^{3}}{6}\frac{\partial {{\varvec{V}}}}{\partial \theta }\frac{\partial ^{2}{{\varvec{W}}}^{\mathrm{T}}}{\partial \theta ^{2}}\right. \\&\left. -\,\frac{\left( {1-\mu } \right) {\tilde{h}}^{3}}{4{\tilde{l}}^{2}} \frac{\partial {{\varvec{V}}}}{\partial \xi }\frac{\partial ^{2}{{\varvec{W}}}^{\mathrm{T}}}{\partial \xi \partial \theta }\right. \\&\left. -\,\frac{\mu {\tilde{h}}^{3}}{6{\tilde{l}}^{2}}\frac{\partial {{\varvec{V}}}^{\mathrm{T}}}{\partial \theta }\frac{\partial ^{2}{{\varvec{W}}}}{\partial \xi ^{2}} \right] \hbox {d}\xi \hbox {d}\theta \\ {{\varvec{K}}}_6= & {} \int _0^1 \int _0^{2\pi } \left[ {\tilde{h}}^{2}{{\varvec{WW}}}^{\mathrm{T}}+ \frac{{\tilde{h}}^{4}}{12{\tilde{l}}^{4}}\frac{\partial ^{2}{{\varvec{W}}}}{\partial \xi ^{2}} \frac{\partial ^{2}{{\varvec{W}}}^{\mathrm{T}}}{\partial \xi ^{2}}\right. \\&\left. +\,\frac{\mu {\tilde{h}}^{4}}{6{\tilde{l}}^{2}}\frac{\partial ^{2}{{\varvec{W}}}}{\partial \xi ^{2}}\frac{\partial ^{2} {{\varvec{W}}}^{\mathrm{T}}}{\partial \theta ^{2}}+\frac{{\tilde{h}}^{4}}{12} \frac{\partial ^{2}{{\varvec{W}}}}{\partial \theta ^{2}}\frac{\partial ^{2}{{\varvec{W}}}^{\mathrm{T}}}{\partial \theta ^{2}}\right. \\&\left. +\,\frac{(1-\mu ){\tilde{h}}^{4}}{6{\tilde{l}}^{2}} \frac{\partial ^{2}{{\varvec{W}}}}{\partial \xi \partial \theta }\frac{\partial ^{2}{{\varvec{W}}}^{\mathrm{T}}}{\partial \xi \partial \theta } \right] \hbox {d}\xi \hbox {d}\theta \\ {{\varvec{F}}}_\mathrm{bnon}= & {} \left[ {{\begin{array}{lll} {{\tilde{{{\varvec{F}}}}}_{bu} }&{} {{\tilde{{{\varvec{F}}}}}_{bv} }&{} {{\tilde{{{\varvec{F}}}}}_{bw} } \\ \end{array} }} \right] ^{\mathrm{T}} \end{aligned}$$
$$\begin{aligned} {\tilde{{{\varvec{F}}}}}_{bu}= & {} \sum _{s=1}^{N_\mathrm{B} } {\int _0^{2{\uppi }} {\left( {{\tilde{k}}_u {\tilde{l}}^{2}{{\varvec{U}}}_{\theta _s }^T {{\varvec{p}}}+{\tilde{l}} {\tilde{F}}_{\mathrm{pre},\theta _s } } \right) {{\varvec{U}}}_{\theta _s }^T \delta \left( {\theta =\theta _s } \right) d\theta } }\\ {\tilde{{{\varvec{F}}}}}_{bv}= & {} \sum _{s=1}^{N_\mathrm{B} } \left\{ {{\begin{array}{l@{\quad }ll} {\int _0^{2{\uppi }} {{\tilde{k}}_{v1} {{\varvec{V}}}_{\theta _s }^\mathrm{T} +{\tilde{k}}_{v2} \left( {{\varvec{V}}_{\theta _s }^\mathrm{T} q -{\tilde{\nu }}_{c,\theta _s } } \right) {\varvec{V}}_{\theta _s }^\mathrm{T} \delta \left( {\theta =\theta _s } \right) \hbox {d}\theta } } &{} \left| {{\tilde{k}}_{v2} \left( {{\tilde{v}}_{\theta _s } -{\tilde{v}}_{c,\theta _s } } \right) } \right| \le \nu \left| {{\tilde{F}}_{\mathrm{pre},\theta _s } -{\tilde{k}}_u {\tilde{l}}{\tilde{u}}_{\theta _s } } \right| &{}\hbox {for stick} \\ {\int _0^{2{\uppi }} {{\tilde{k}}_{v 1} {{\varvec{V}}}_{\theta _s }^\mathrm{T} +\nu \left| {{\tilde{F}}_{\mathrm{pre},\theta _s } -{\tilde{k}}_u {\tilde{l}} {{\varvec{U}}}_{\theta _s }^\mathrm{T} {{\varvec{p}}}} \right| \hbox {sgn}({\dot{{{\varvec{v}}}}}_{\theta =\theta _b } ) {{\varvec{V}}}_{\theta _s }^\mathrm{T} \delta \left( {\theta =\theta _s } \right) \hbox {d} \theta } }&{} \left| {{\tilde{k}}_{v2} \left( {{\tilde{v}}_{\theta _s } -{\tilde{v}}_{c,\theta _s } } \right) } \right|>\nu \left| {{\tilde{F}}_{\mathrm{pre},\theta _s } -{\tilde{k}}_u {\tilde{l}} {\tilde{u}}_{\theta _s } } \right| &{} \hbox {for slip} \\ \int _0^{2{\uppi }} {\tilde{k}}_{v1} {{\varvec{V}}}_{\theta _s }^\mathrm{T}\delta (\theta = \theta _{s})\hbox {d}\theta &{} &{} \hbox {for separation}\\ \end{array} }} \right. \\ {\tilde{{{\varvec{F}}}}}_{bw}= & {} \sum _{s=1}^{N_\mathrm{B} }\left\{ \begin{array}{l@{\quad }ll} \int _0^{2{\uppi }}\left( {\tilde{k}}_{w1} {\tilde{h}}^{2} {{\varvec{W}}}_{\theta _s }^\mathrm{T} +{\tilde{k}}_{w2} {\tilde{h}}^{2}\left( {{{\varvec{W}}}_{\theta _s }^\mathrm{T} {{\varvec{r}}}-{\tilde{w}}_{c,\theta _s } } \right) {{\varvec{W}}}_{\theta _s }^\mathrm{T}\right) \delta \left( {\theta =\theta _s } \right) d\theta &{} \left| {{\tilde{k}}_{w2} {\tilde{h}} \left( {{\tilde{w}}_{\theta _s } -{\tilde{w}}_{c,\theta _s } } \right) } \right| \le \nu \left| {{\tilde{F}}_{\mathrm{pre},\theta _s } -{\tilde{k}}_u {\tilde{l}} {\tilde{u}}_{\theta _s } } \right| &{}\hbox {for stick} \\ {{\int _0^{2{\uppi }} {{\tilde{k}}_{w1} {\tilde{h}}^{2} {{\varvec{W}}}_{\theta _s }^\mathrm{T} +\nu {\tilde{h}}\left| {{\tilde{k}}_u {\tilde{l}}{{\varvec{U}}}_{\theta _s }^\mathrm{T} {{\varvec{p}}}+{\tilde{F}}_{\mathrm{pre},\theta _s } } \right| \hbox {sgn}\left( {\dot{w}_{\theta _s } } \right) {{\varvec{W}}}_{\theta _s }^\mathrm{T} \delta \left( {\theta =\theta _s } \right) d\theta } } }&{} \left| {{\tilde{k}}_{w2} {\tilde{h}}\left( {{\tilde{w}}_{\theta _s } -{\tilde{w}}_{c,\theta _s } } \right) } \right| >\nu \left| {{\tilde{F}}_{\mathrm{pre},\theta _s } -{\tilde{k}}_u {\tilde{l}}{\tilde{u}}_{\theta _s } } \right| &{} \hbox {for slip} \\ \int _0^{2{\uppi }} {\tilde{k}}_{w1} {\tilde{h}}^{2} {{\varvec{W}}}_{\theta _s }^\mathrm{T} \delta (\theta =\theta _{s})\hbox {d}\theta &{} &{}\hbox {for separation}\\ \end{array}\right. \\ {{\varvec{K}}}_\mathrm{bolt}= & {} \hbox {diag}\left( \sum _{s=1}^N {{\tilde{k}}_u {\tilde{l}}^{2} {{\varvec{U}}}_{\theta _s } {{\varvec{U}}}_{\theta _s }^\mathrm{T} }, \sum _{s=1}^N {{\tilde{k}}_v {{\varvec{V}}}_{\theta _s } {{\varvec{V}}}_{\theta _s }^\mathrm{T} } , \sum _{s=1}^N {{\tilde{k}}_w {\tilde{h}}^{2} {{\varvec{W}}}_{\theta _s } {{\varvec{W}}}_{\theta _s }^\mathrm{T} } +\sum _{s=1}^N {{\tilde{k}}_\theta \frac{{\tilde{h}}^{2}}{{\tilde{l}}^{2}} \frac{\partial {{\varvec{W}}}_{\theta _s } }{\partial \xi }\frac{\partial {{\varvec{W}}}_{\theta _s }^\mathrm{T} }{\partial \xi }} \right) . \end{aligned}$$

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Tang, Q., Li, C., She, H. et al. Modeling and dynamic analysis of bolted joined cylindrical shell. Nonlinear Dyn 93, 1953–1975 (2018). https://doi.org/10.1007/s11071-018-4300-4

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