Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 5432516, 22 pages

http://dx.doi.org/10.1155/2016/5432516

## Nonlinear Dynamic Behavior Analysis of Pressure Thin-Wall Pipe Segment with Supported Clearance at Both Ends

School of Mechanical Engineering & Automation, Northeastern University, Shenyang, Liaoning 110819, China

Received 24 March 2016; Revised 30 May 2016; Accepted 31 May 2016

Academic Editor: Francesco Tornabene

Copyright © 2016 Chaofeng Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An analysis of nonlinear behaviors of pressure thin-wall pipe segment with supported clearance at both ends was presented in this paper. The model of pressure thin-wall pipe segment with supported clearance was established by assuming the restraint condition as the work of springs in the deformation directions. Based on Sanders shell theory, Galerkin method was utilized to discretize the energy equations, external excitation, and nonlinear restraint forces. And the nonlinear governing equations of motion were derived by using Lagrange equation. The displacements in three directions were represented by the characteristic orthogonal polynomial series and trigonometric functions. The effects of supporting stiffness and supported clearance on dynamic behavior of pipe wall were discussed. The results show that the existence of supported clearance may lead to the changing of stiffness of the pipe vibration system and the dynamic behaviors of the pipe system show nonlinearity and become more complex; for example, the amplitude-frequency curve of the foundation frequency showed hard nonlinear phenomenon. The chaos and bifurcation may emerge at some region of the values of stiffness and clearance, which means that the responses of the pressure thin-wall pipe segment would be more complex, including periodic motion, times periodic motion, and quasiperiodic or chaotic motions.

#### 1. Instruction

Pressure pipe is commonly used as structural member or mechanical component in many engineering applications and chemical equipment, such as centrifugal compressor. For the inlet/outlet pipe of a centrifugal compressor, the pipe wall is commonly subjected to pulsed gas excitations and its restraint condition is always complex, which might cause nonlinear vibration of pipe. Because there is high pressure gas flowing through the pipe, strong vibration might cause pipe wall to break and it is a threat for air tightness and safety. It might lead to chemical gas leak and even an explosion. So the study of dynamic behavior of pressure pipe segment has an important signification for designing and maintenance of pipe system.

In actual work, the restraint condition of compressor pipe would be very complex. Firstly, installation error exists inevitably. Secondly, the long-playing vibration of the pipe system may lead to the deformation of pipe clamp and cause supported clearance between pipe wall and pipe clamp. The complex boundary condition would have effect on the dynamic characteristic of pipe. Up to now, researches on the nonlinear dynamic behaviors of pipe with nonlinear support were always based on pipe beam model. Beam model is often applied to the pipe which has large ratio of length to radius and large ratio of thickness to radius. And only the lateral movement of the pipe was studied by considering small circumferential mode. However, for the inlet/outlet pressure pipe of a centrifugal compressor, the ratio of length to radius and the ratio of thickness to radius are relatively small. They are always thin-wall pipes and their vibration is more complex. It would be more accurate to choose shell model to study the dynamic behaviors of pipe vibration system.

For shell model, many researches have been done in the literature review. In most of the literatures, boundary condition of shell is a much-discussed topic. Interrelated researches began from those about shells with classical boundaries (simply supported (S), clamped (C), free (F), and so on). And many methods were used to study free vibration and forced vibration of cylindrical shells based on different theories. Lee and Kwak [1] constructed a dynamic model for the free vibration analysis of a circular cylindrical shell by using the Rayleigh-Ritz method and compared the results based on different theories such as Donnell-Mushtari theory, Sanders theory, Flügge theory, Vlasov theory, Love-Timoshenko theory, and Reissner theory and the results under different boundary conditions. Lam and Loy [2] studied the effects of boundary conditions on the free vibration characteristic for a multilayered cylindrical shell based on Love’s first approximation theory using Ritz method. The displacements were represented as combination beam functions with trigonometric functions. And nine kinds of classical boundaries were discussed. Based on three-dimensional theory of elasticity, the boundary conditions of thick cylindrical shell, S-S and C-C, were studied by Loy and Lam [3, 4] using an energy minimization principle. And extensive frequency parameters were presented for a wide range of thickness to radius and thickness to length ratios. An improved version of the differential quadrature method was applied to solutions of cylindrical shell problems and the frequencies and fundamental frequencies for S-S, C-C, and C-S boundary conditions were determined [5]. Wave propagation method was also used in the vibration analysis of cylindrical shells by Zhang et al. [6]. This method is a noniterative method and the axial wave number of standing wave is determined approximately by the wave number of an equivalent beam with similar boundary conditions as the shell. Harr wavelet method was used by Xie et al. [7] for free vibrations of thin cylindrical shells with various boundary conditions based on Goldenveizer-Novozhilov shell theory and the displacements were represented by Haar wavelet series and their integrals in the axial direction and the Fourier series in the circumferential direction. Using the Flügge shell theory, Xie [8] employed transfer matrix method to analyze the natural characteristic of cylindrical shells and the precise integration technique was adopted in numerical calculation. Ebrahimi and Najafizadeh [9] used the methods of generalized differential quadrature (GDQ) and generalized integral quadrature (GIQ) to study the free vibration of a two-dimensional functionally graded circular cylindrical shell based on Love’s first approximation classical shell theory. Sun et al. [10] used Fourier series expansion method, which is derived by Stokes, transformation, to analyze the vibration characteristics of thin rotating cylindrical shells with several classical boundary conditions. Lopatin and Morozov [11] presented an analytical formula to calculate the fundamental frequencies for the laminated composite cylindrical shell with clamped edges based on the Fourier decomposition method. Wang et al. [12] studied the vibration of a cantilever cylindrical shell suffered a moving harmonic excitation and the method of averaging was used to analyze the nonlinear traveling wave responses. Dey and Ramachandra [13] presented the prebuckling analysis of a SS-SS composite cylindrical shell under periodic partial edge loadings by employing the Galerkin and Bolotin methods based on Donnell’s nonlinear shallow shell theory and studied the influence of dynamic load amplitude on the nonlinear response.

The main work of compressor pipe is conveying gas, which causes high pressure on pipe wall. So the lateral gas pressure is an important loading condition for compressor pipe and can also impact the dynamic behavior of pipe. Pressure shell was also a popular object for researchers. Isvandzibaei et al. studied natural frequency characteristics of a thin-walled multiple layered cylindrical shell under lateral pressure with symmetric [14] and asymmetric [15] boundary conditions using energy method. And the effects of geometrical parameter and pressure on natural frequencies were investigated. They [16] also employed this method into FGM cylindrical shell with rings support under symmetric uniform interior pressure distribution and ten different boundary conditions were discussed to study the natural characteristics. Based on Donnell-type classical shell theory, Forouzesh and Jafari [17] used The Hamilton’s principle, differential quadrature, and Newmark method to solve the radial vibration problem of simply supported pseudoelastic shape memory alloy cylindrical shells under time-dependant internal pressure.

However, classical boundary could not represent general boundary conditions. Researchers shifted their attention to elastic boundary conditions. Massless springs were introduced to represent the interaction between the pipe end and the frame. And at each point the restraint condition is represented as four sets of independent springs, including three sets of linear springs and one set of rotational springs, and different boundary conditions can be obtained by setting different spring stiffness. Jin et al. [18–20] developed a series of methods to investigate free vibration of different cylindrical shells with general elastically restrained boundaries, such as composite laminated cylindrical shells and a three-layered passive constrained layer damping (PCLD) cylindrical shell, considering elastic restraints and intermediate ring supports. Qu et al. [21] applied the method of domain decomposition to the analysis of cylindrical shells. Free vibration of the stepped shells under different combinations of free, simply, clamp, and elastic support boundaries was discussed and force vibration response of stepped cylindrical shells was also studied with the influence of structural damping, stepped thickness, and boundary conditions. Based on Flügge’s theory, Dai et al. [22] provided some numerical examples for the vibration analysis of circular cylindrical shells with various boundary conditions using the elastic equations. Chen et al. [23] investigated the vibration of cylindrical shell with nonuniform elastic boundary constraints using improved Fourier series method. Varying stiffness of boundary springs, point supported and partially supported boundary conditions were studied. Zhou et al. [24] analyzed free vibration of cylindrical shells with elastic boundary conditions by using the method of wave propagations based on Flügge classical thin shell theory. Shi et al. [25] used a spectro-geometric-Ritz method to study free vibration of open and closed shells with arbitrary boundary conditions. And the displacement components were also represented as a standard Fourier cosine series and several auxiliary functions. Sun and Liu [26, 27] employed the Rayleigh-Ritz method to study the effect of the variations of restraint stiffness on the natural characteristics of rotating cylindrical shells with and without ring-support based on the Sanders’ shell theory and the displacements were considered as the characteristic orthogonal polynomial series.

Winkler and Pasternak foundation is also a topic researchers care about. Bakhtiari-Nejad and Mousavi Bideleh [28] investigated nonlinear free vibration analysis of prestressed circular cylindrical shells placed on Winkler and Pasternak foundation and the effects of prestressed condition and elastic foundation on natural frequencies under various boundary conditions were analyzed. The Rayleigh-Ritz procedure and Perturbation methods are used to study amplitude-frequency characteristics of prestressed circular cylindrical shells. Shah et al. [29] employed wave propagation method and the approximate eigenvalues of characteristic beam functions to study the vibrations of functionally graded cylindrical shells based on the Winkler and Pasternak foundations.

As far as the study of circular cylindrical shell was considered, most of the existing works were limited to linear boundary conditions. However, nonlinear boundary conditions, such as those with supported clearance, could be encountered in many engineering applications. In present work, a short thin-wall cylindrical shell model with supported clearance at both ends was established for the pressure pipe of centrifugal compressor. Based on Sanders theory, Lagrange’s equations had been written for the nonlinear vibration differential equations. In the analytical formulation, the Rayleigh-Ritz method with a set of displacement shape functions was used to deduce mass, damping, stiffness, and force matrices of the pipe system. The displacements in three directions were represented by the characteristic orthogonal polynomial series and trigonometric functions. By numerical calculation, dimensional spectrum, bifurcation diagram, time domain response graph, frequency spectrum plot,* s-v* phase diagram, and Poincaré section with different parameters were obtained to analyze the effect of pulsed frequency, supporting stiffness, and supported clearance on dynamic characteristic of pressure pipe.

#### 2. Theory Formulation

##### 2.1. Simplified Dynamic Model of Pressure Thin-Wall Pipe Segment System

In actual work, due to poor installation quality or long time vibrating of the export pipe segment used in compressor, the pipe clamp may have a deformation or have the phenomenon of expansion, which can result in the supported clearance between pipe wall and pipe clamp, and it does harm to the pipe system. Considering the structure characteristic of export pipe of compressor that the ratio of length to diameter is little and the ratio of diameter to thickness is large, a pipe segment can be simplified as a cylindrical shell model shown in Figure 1. According to Sanders shell theory, the movement of each point on the pipe can be expressed by the movement of the points on the middle surface of pipe. So the cylindrical coordinate is established as illustrated in Figure 1(a), where , , and represent the axial direction, circumferential direction, and the radial direction, respectively. And the deformation of the cylindrical pipe segment with references to this coordinate system are denoted by , , and in , , and directions. And the pipe wall suffered uniform lateral gas pressure ; the mode of the pressure is in Figure 1(b). In addition, the thickness, the length, and the mean radius of the cylinder are denoted by , , and , respectively. The material of the pipe is assumed to be isotropic with the mass density , Poisson’s ratio , and Young’s modulus .