Nonlinear Dynamics

, Volume 93, Issue 4, pp 1953–1975 | Cite as

Modeling and dynamic analysis of bolted joined cylindrical shell

  • Qiansheng Tang
  • Chaofeng LiEmail author
  • Houxin She
  • Bangchun Wen
Original Paper


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.


Bolted joined cylindrical shell Slip Nonlinear dynamic Chaotic motion Period motion 

List of symbols


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


Thickness of the shell


Length of the shell


Radius of the shell

\(\rho \)

Mass density

\(\mu \)

Poisson’s ratio


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


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


Dimensionless length of the shell


Dimensionless thickness of the shell

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

Dimensionless frequency


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


The amplitude of the external load


The circumferential wave number

\(A_n ,B_n ,C_n \)

The amplitude parameters of displacement


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


The bending moment on the contact surface

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

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



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).

Compliance with ethical standards

Conflicts of interests

The authors declare that there is no conflict of interests regarding the publication of this article.


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Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  • Qiansheng Tang
    • 1
  • Chaofeng Li
    • 1
    • 2
    Email author
  • Houxin She
    • 1
  • Bangchun Wen
    • 1
    • 2
  1. 1.School of Mechanical Engineering and AutomationNortheastern UniversityShenyangChina
  2. 2.Key Laboratory of Vibration and Control of Aero-Propulsion System Ministry of EducationNortheastern UniversityShenyangChina

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