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.
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 [
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 [
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. [
Winkler and Pasternak foundation is also a topic researchers care about. Bakhtiari-Nejad and Mousavi Bideleh [
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,
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
Simplified system dynamic model of pressure pipe with supported clearance at both ends.
Massless springs were introduced to describe the constraint condition of pipe wall from pipe clamp, and the constraint conditions at a point are represented as four sets of independent springs placed at the ends, including three sets of linear springs (
As shown in Figure
So, the spring forces which represented the restraint forces are nonlinear forces and can be defined as
The clearance model of the spring forces.
The kinetic energy for vibration of pressure pipe is given by
The strain energy
And the pipe suffered uniform lateral gas pressure
In the course of work, the pipe wall may suffer the external loads on one point. Assume that the external loads of the unit area of pipe
The damping force can be denoted as
In order to use the Rayleigh-Ritz method, the displacements
Then the kinetic energy, potential energy, and forces can be expressed in terms of the generalized coordinates and displacement shape functions. Substitute (
Substitute (
Substituting (
Substituting (
Substituting (
Lagrange equation with the Rayleigh-Ritz method will be used to determine the equation of motion of the pressure pipe. Lagrange equation is written by
Substituting (
Therefore, the model of pressure pipe segment with supported clearance at both ends is a nonlinear equation with piecewise linearity.
To verify the accuracy and reliability of the present method, comparisons are made available in open literature through several numerical examples under the boundary condition without clearance, which are shown in Tables
Comparison of values of natural frequency parameter
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|
| ||
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Present | Reference [ |
Reference [ | ||
0.05 | 1 | 0.0161028 | 0.0161065/0.02 | 0.0161064/0.02 |
2 | 0.0392714 | 0.0393038/0.08 | 0.0393035/0.08 | |
3 | 0.1098115 | 0.1098527/0.04 | 0.1098468/0.03 | |
4 | 0.2102772 | 0.2103446/0.03 | 0.2103419/0.03 | |
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||||
0.002 | 1 | 0.0161011 | 0.0161011/0.00 | 0.0161023/0.01 |
2 | 0.0054530 | 0.0054532/0.00 | 0.0054547/0.03 | |
3 | 0.0050415 | 0.0050419/0.01 | 0.0050427/0.02 | |
4 | 0.0085339 | 0.0085341/0.00 | 0.0085344/0.01 |
Comparison of values of natural frequency parameter
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|
|
||||
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Present | Reference [ |
Difference (%) | Present | Reference [ |
Difference (%) | |
1 | 2479.2 | 2479.3 | 0.01 | 4830.2 | 4830.6 | 0.01 |
2 | 269.4 | 269.3 | 0.04 | 277.5 | 278.58 | 0.38 |
3 | 761.9 | 761.0 | 0.12 | 773.2 | 771.62 | 0.20 |
4 | 1460.9 | 1458.6 | 0.15 | 1473.7 | 1469.3 | 0.30 |
5 | 2362.4 | 2358.6 | 0.16 | 2376.3 | 2369.0 | 0.30 |
Firstly, what can be deduced is that the cylindrical shell is simply supported when
Usually, the restraint is not the classical boundary condition, so the case with elastic support was considered and the natural frequencies were listed in Table
Natural frequencies (Hz) for an elastically supported cylindrical shell (
Mode |
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|
|
|
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|
228.18 | 228.19 | 228.89 | 247.94 | 249.44 | 249.44 |
|
228.71 | 228.72 | 230.82 | 376.01 | 406.03 | 406.03 |
|
318.05 | 318.05 | 319.91 | 557.74 | 642.86 | 642.86 |
To study the nonlinear dynamic behaviors of pressure pipe segment with supported clearance at both ends, amplitude-frequency responses of pipe wall vibration were analyzed and the effects of supported clearance and supporting stiffness on amplitude-frequency characteristics were discussed in detail.
Figure
A harmonic amplitude-frequency response curve.
For pressure pipe segment, supported clearance is an important parameter which has a great influence on the system dynamic characteristics. This section gives the analysis of the amplitude-frequency characteristics of the pipe wall by changing the value of the supported clearance and keeping other parameters invariable. The curves in Figure
A harmonic amplitude-frequency response curve with different supported clearances.
The third harmonic amplitude-frequency response curve with different supported clearances.
For pressure pipe, the supporting stiffness is also an important parameter which affects the amplitude-frequency response characteristics of pipe vibration system. In this part, the amplitude-frequency characteristics of the pipe wall were analyzed by only changing the system supporting stiffness and keeping other parameters invariable. Figure
The first harmonic amplitude-frequency response curve with different supporting stiffness.
The third harmonic amplitude-frequency response curve with different supporting stiffness.
To fully analyze nonlinear dynamic behaviors of the pressure pipe vibration system with supported clearance at both ends, response analysis, parameter comparison, and other aspects were investigated. Through numerical simulating and calculating, the three-dimensional waterfall spectrogram was obtained. And time domain response graph, frequency spectrum plot, displacement-velocity phase diagram, and Poincaré section were also given to analyze the dynamic characteristics under different frequencies.
Figure
The three-dimensional spectrum diagram under the condition of
The following figures are the time domain response graph, frequency spectrum plot, displacement-velocity phase diagram, and Poincaré section of the vibration of the pressure pipe segment in several pulse frequencies which had been shown in Figure
When the pulse frequency is equal to 325 Hz in Figure
Kinetic characteristic curve under the condition
Time domain response graph
Frequency spectrum plot
Poincaré section
Under
Kinetic characteristic curve under the condition
Time domain response graph
Frequency spectrum plot
Poincaré section
From Figure
Kinetic characteristic curve under the condition
Time domain response graph
Frequency spectrum plot
Poincaré section
As shown in Figure
Kinetic characteristic curve under the condition
Time domain response graph
Frequency spectrum plot
Poincaré section
From Figure
Kinetic characteristic curve under the condition
Time domain response graph
Frequency spectrum plot
Poincaré section
Supporting stiffness is an important parameter for dynamic response of pressure pipe segment vibration. Different supporting stiffness would cause different dynamic behaviors. In this part, dynamic behaviors of pressure pipe segment would be analyzed by changing the value of supporting stiffness and keeping other parameters unchanged. The dimensional spectrum plots and bifurcation diagrams were given to study the effect of stiffness on dynamic characteristic and stability of pressure pipe segment. And then time domain response graph, frequency spectrum plot, displacement-velocity phase diagram, and Poincaré section were selected as tools to analyze in detail the nonlinear dynamic behaviors of the pipe wall vibration under several supporting stiffness values.
Figure
The dimensional spectrum under different stiffness.
To explain the dynamic behaviors in detail, arranging supporting stiffness as bifurcation parameter, which is in the range from
The dimensional spectrum and the bifurcation diagram under different stiffness.
The dimensional spectrum
The bifurcation diagram
As demonstrated in Figure
Different dynamic behaviors of pipe wall are shown in Figures
Kinetic characteristic curve under the condition
Time domain response graph
Frequency spectrum plot
Poincaré section
Kinetic characteristic curve under the condition
Time domain response graph
Frequency spectrum plot
Poincaré section
Kinetic characteristic curve under the condition
Time domain response graph
Frequency spectrum plot
Poincaré section
Kinetic characteristic curve under the condition
Time domain response graph
Frequency spectrum plot
Poincaré section
When
Eight local peaks can be found in the response curve under the condition
When
Supported clearance is also an important influence factor on dynamic response of pressure pipe segment vibration. In this part, the supported clearance was set to be the variable parameter to study dynamic behaviors of pressure pipe segment. The bifurcation diagram and the dimensional spectrum diagram, time domain response graph, frequency spectrum plot, displacement-velocity phase diagram, and Poincaré section were selected as tools to analyze in detail the nonlinear dynamic behavior of the pipe wall vibration at several supported clearance values.
Figure
The three-dimensional spectrum diagram under different supported clearance.
To analyze the form of motion, the bifurcation diagram about supported clearance is shown in Figure
The bifurcation diagram about supported clearance.
Kinetic characteristic curve under the condition
Time domain response graph
Frequency spectrum plot
Poincaré section
Kinetic characteristic curve under the condition
Time domain response graph
Frequency spectrum plot
Poincaré section
Kinetic characteristic curve under the condition
Time domain response graph
Frequency spectrum plot
Poincaré section
At the supported clearance
When
The steady time domain response curve has many random peaks under
The nonlinear dynamic behaviors of pressure pipe segment with supported clearance at both ends were presented. Lagrange’s equations had been written for the nonlinear vibration cases and by numerical calculation, three-dimensional spectrum, bifurcation diagram, time domain response graph, frequency spectrum plot,
By numerical simulation and calculation, the dynamics behaviors of pressure pipe segment with supported clearance at both ends show strong nonlinearity. Firstly, the amplitude-frequency curves under supported clearance skew to the right, which is called “hardening curve.” Supported clearance and supporting stiffness have a great effect on resonance amplitude and resonance frequency. The larger the value of supported clearance is, the larger the resonance amplitude is and the smaller the resonance frequency is. But the nonlinearity is more obvious under smaller clearance. The high harmonic which has the large amplitude indicates that high-order resonance may happen under the condition with supported clearance. Jump phenomena emerge obviously at some frequencies in the process of frequency change. They would cause large relative motion, which may produce so great stress that fatigue failure happens to reduce the working life of pipe. Secondly, the response is also very complex. The frequency components of the pipe response are very abundant and they are effected by pulse frequency, supporting stiffness, and supported clearance. The complex response causes the diversification of vibration mode of pipe segment. It can be deduced from the response curve that the response of pipe segment may be the periodic, quasiperiodic, or even chaotic motion under different parameters.
One has the following:
The authors declare that there is no conflict of interests regarding the publication of this paper.
The project was supported by Fundamental Research Funds for the Central Universities (nos. N140304002 and N140301001) and the China Natural Science Funds (no. 51575093).