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The nonlinear resonant responses, mode interactions, and multitime periodic and chaotic oscillations of the cantilevered pipe conveying pulsating fluid are studied under the harmonic external force in this research. According to the nonlinear dynamic model of the cantilevered beam derived using Hamilton’s principle under the uniformly distributed external harmonic excitation, we combine Galerkin technique and the method of multiple scales together to obtain the average equation of the cantilevered pipe conveying pulsating fluid under 1 : 3 internal resonance and principal parametric resonance. Based on the average equation in the polar form, several amplitude-frequency response curves are obtained corresponding to the certain parameters. It is found that there exist the hardening-spring type behaviors and jumping phenomena in the cantilevered pipe conveying pulsating fluid. The nonlinear oscillations of the cantilevered pipe conveying pulsating fluid can be excited more easily with the increase of the flow velocity, external excitation, and coupling degree of two order modes. Numerical simulations are performed to study the chaos of the cantilevered pipe conveying pulsating fluid with the external harmonic excitation. The simulation results exhibit the existence of the period, multiperiod, and chaotic responses with the variations of the fluid velocity or excitation. It is found that, in the cantilevered pipe conveying pulsating fluid, there are the multitime nonlinear vibrations around the left-mode and the right-mode positions, respectively. We also observe that there exist alternately the periodic and chaotic vibrations of the cantilevered pipe conveying pulsating fluid in the certain range.

Pipes conveying fluid are widely utilized in many engineering fields, such as aeronautic, astronautic, and mechanical engineering systems. It is extremely important for us to ensure the efficient utilization and safe operation of the pipe conveying fluid system, and its stable and safe operations are closely related to all aspects of the personal life and industrial production. However, the applications of the pipes conveying pulsating fluid are particularly challenging because they undergo the large deformations and significant stresses. The large deformations often lead to the nonlinear vibrations of the pipes conveying pulsating fluid. One of the main reasons for the nonlinear vibrations of the pipes conveying pulsating fluid is the time-varying flow speed and external harmonic excitation. Pulsating flow due to the pump operation can cause a parametric excitation loading in the pipes conveying fluid. The nonlinear oscillations of the pipes conveying pulsating fluid will lead to the structure damages. As we all know, there are three typical types of nonlinear oscillations in the structures and systems, namely, the periodic, quasi-periodic, and chaotic oscillations. In fact, the chaotic oscillations of the pipes conveying pulsating fluid are dangerous because the amplitudes of the chaotic oscillations are larger than those of the periodic oscillations, which have been the object of increasing attention in engineering applications. However, there is less research on the nonlinear oscillations of the cantilevered pipe conveying pulsating fluid with 1 : 3 internal resonance when the fluid is transported at a critical speed through the pipe. Therefore, it is of great significance for us to study the nonlinear oscillations of the cantilevered pipe conveying pulsating fluid under the case of 1 : 3 internal resonance.

The pipe conveying fluid mainly consists of three important elements: pipeline, fluid, and external environment. It is necessary to establish a mathematical model for obtaining a reasonable description of the pipes conveying fluid. The beam model is usually used for analysis of the vibration when the pipe diameter is much smaller than the length. The nonlinear dynamic study of the pipe conveying fluid system began in 1980s. Researches for the pipes conveying pulsating fluid have become a hot field of engineering and science [

In addition, some scholars also provided several different mathematical models to investigate the vibrations of the pipes conveying fluid. According to the nonlinear Novozhilov shell theory for the isotropic materials, Tubaldi et al. [

After the establishment of the rational mathematical model, the further researches on the vibrations of the pipes conveying fluid are mainly focused on three significant aspects. Firstly, the vibration characteristics and instability conditions of the pipes conveying fluid are studied, which have different boundary conditions, material properties, and functions. Secondly, the problems of the nonlinear dynamics are studied to understand the nonlinear vibration characteristics of the pipes conveying fluid because the pipes can be regarded as the complex nonlinear dynamical systems. Thirdly, the transfer mechanism of the energy between two modes and the internal resonance are studied to avoid the chaotic vibrations of the pipes conveying fluid.

More and more researches about model and analysis have been published with the continuous development of modern computing and analytical techniques. Bajaj et al. [

Yoshizawa et al. [

Based on the fluid-structure interaction, Liang et al. [

Several papers researched the nonlinear dynamic characteristics of the pipes conveying fluid. Li and Paidoussis [

For high-dimensional nonlinear dynamical systems, due to the existence of the modal interactions, there exist the relationships among several types of internal resonances, which can lead to different nonlinear oscillations [

To avoid the damage of the pipes conveying pulsating fluid caused by chaotic oscillations, we study the nonlinear oscillations of the pipes conveying pulsating fluid under 1 : 3 internal resonance. Based on the nonlinear dynamic model of the cantilevered beam, the nonlinear dynamic equations of motion for the cantilevered pipe conveying pulsating fluid under the uniformly distributed harmonic excitation and average equations are obtained through using the combination of the multiple-scale method and Galerkin technique under 1 : 3 internal resonance and principal parametric resonance. We analyze the nonlinear resonant responses and the mode interactions of the cantilevered pipe conveying pulsating fluid. Moreover, numerical simulations are performed to study the multitime periodic and chaotic oscillations of the cantilevered pipe conveying pulsating fluid under the external harmonic excitation. It is found that, in the cantilevered pipe conveying pulsating fluid, there are the multitime nonlinear and chaotic oscillations around the left-mode and the right-mode positions, respectively.

We consider a cantilevered pipe conveying pulsating fluid, where

The dynamic model of a cantilevered pipe conveying pulsating fluid is given.

The basic assumptions of the cantilevered pipe and the fluid are made as follows:

The fluid is incompressible.

The diameter of the pipe is small compared to its length. Therefore, the pipe behaves like an Euler-Bernoulli beam.

The vibration of the pipe is planar, and the deflections of the pipe are large.

The rotatory inertia and shear deformation are neglected.

The pipe centerline has inextensible property in the case of a cantilevered pipe.

Based on researches given by Semler et al. [

In this paper, we assume that the cantilevered pipe conveying pulsating fluid is made of Kelvin-Voigt type viscoelastic material. Therefore, we have

In order to obtain the dimensionless governing equation of motion for the cantilevered pipe conveying pulsating fluid, the transformations of the variables and the parameters are introduced as

For simplicity, the notation

It is assumed that the flow velocity

We find that there exist the gyroscopic terms in equation (

For the perturbation analysis, equation (

The method of multiple scales [

Then, the derivatives with respect to

Substituting equations (

The boundary conditions are given as

In order to simplify the cantilevered pipe conveying pulsating fluid to the finite dimension by using Galerkin discretization, the modal function is selected as a beam function:

The general solution of equation (

Substituting equation (

According to the geometries and the material properties of the cantilevered pipe conveying pulsating fluid, the parameters are chosen as

The Campbell diagram of the cantilevered pipe conveying pulsating fluid is obtained in the case of 1 : 3 internal resonance.

We only consider the case of 1 : 3 internal resonance, principle parameter resonance, and 1/2 subharmonic resonance for equation (

Substituting equations (

The solution exists when the nonhomogeneous equation corresponding to equation (

Let

Substituting equation (

In order to obtain the averaged equations in the Cartesian form, we express

Based on the same way as the aforementioned analysis, the averaged equations in the Cartesian form are obtained for the cantilevered pipe conveying pulsating fluid:

Equations (

The resonant response curves are the important basis for judging and studying the nonlinear oscillations. They include many complex nonlinear dynamic phenomena. The practical problems can be solved better by analyzing these phenomena. Based on the averaged equations (

Two cases of the amplitude-frequency response curves [

There is no coupling effect between the first-order and third-order oscillation modes. We only consider the decoupled case and set

We let an amplitude change when another is fixed because the amplitude is much smaller than that in the case of weak coupling; namely, set

We obtain the amplitude-frequency response curves in the decoupled and coupled cases based on equations (

For both the decoupled and coupled cases, when the velocity parametric excitation is

The amplitude-frequency response curves of the cantilevered pipe conveying pulsating fluid are obtained when flow velocity is

In the weak coupling cases, the influences of the flow velocity

The amplitude-frequency response curves of the cantilevered pipe conveying pulsating fluid are obtained in the coupling case with different parameters. (a) Amplitude-frequency response curves when flow velocities

The effects of the flow velocity and external excitation on the amplitude-frequency response curves are investigated for the cantilevered pipe conveying pulsating fluid in Figures

The amplitude-frequency response curves of the cantilevered pipe conveying pulsating fluid are obtained. (a) Amplitude-response curves on flow velocity; (b) amplitude-response curves on external excitation.

We explore the influences of different parameters on the amplitude-frequency and force-amplitude response curves in the cantilevered pipe conveying pulsating fluid, respectively. In Figures

The influences of different parameters on the amplitude-frequency response curves and force-amplitude response curves are given for the cantilevered pipe conveying pulsating fluid. (a) Amplitude-frequency response curves on flow velocity when

Based on the analyses of the amplitude-frequency response curves, it can be found that the hardening-spring type behaviors and jumping phenomena are exhibited for the cantilevered pipe conveying pulsating fluid. The jumping phenomena also occur in the force-amplitude response curves versus the flow velocity and external excitation. Moreover, it is known that the flow velocity, external excitation, and coupling degree of two oscillation modes can affect the nonlinear oscillations of the cantilevered pipe conveying pulsating fluid under the external harmonic force. The nonlinearity bends the amplitude-frequency response curves to the right when the cantilevered pipe conveying pulsating fluid has the hardening-spring type behaviors. The bending of the amplitude-frequency response curves leads to the occurrence of the jumping phenomena. The hardening-spring type behavior and jumping phenomena mean that the oscillation amplitudes of the cantilevered pipe conveying pulsating fluid increase firstly and change abruptly. The energy of the fluid motion is transferred into the energy of the cantilevered pipe when the flow velocity increases. Thus, the fluid-structure interaction happens for the cantilevered pipe conveying pulsating fluid. The nonlinear resonance of the large amplitude for the cantilevered pipe conveying pulsating fluid can be stimulated with the increase of these factors.

In order to study the nonlinear dynamic properties of the cantilevered pipe conveying pulsating fluid, the influences of the velocity parametric excitation and external excitation on the nonlinear oscillations of the pipe are investigated. In this section, the average equations (

To reveal the nonlinear dynamic behaviors of the cantilevered pipe conveying pulsating fluid, the bifurcation diagrams, maximum Lyapunov exponents, phase portraits, waveforms, and Poincare map are depicted. Figures

The bifurcation diagrams and maximum Lyapunov exponents of the cantilevered pipe conveying pulsating fluid are given for the parametric excitations of the flow velocities when the external excitation is

The bifurcation diagrams and maximum Lyapunov exponents of the cantilevered pipe conveying pulsating fluid are obtained for the external excitations when the flow velocity is

In order to study the influences of the flow velocities on the multitime nonlinear oscillations of the cantilevered pipe conveying pulsating fluid, we set the external excitation while other parameters remain unchanged. The bifurcation diagrams and maximum Lyapunov exponents of the cantilevered pipe conveying pulsating fluid with the change of the flow velocity are shown in Figure

Based on numerical simulations corresponding to the aforementioned analyses, we further investigate the effects of the external excitations on the nonlinear dynamics of the cantilevered pipe conveying pulsating fluid when the flow velocity is

We present a variety of figures to confirm the vibrations of the cantilevered pipe conveying pulsating fluid corresponding to different flow rates and different external excitations. In Figures

The periodic oscillation of the cantilevered pipe conveying pulsating fluid is obtained when the parametric excitation of the flow velocity is

The period-2 oscillation of the cantilevered pipe conveying pulsating fluid is obtained when the parametric excitation of the flow velocity is

The multiperiodic oscillation of the cantilevered pipe conveying pulsating fluid is obtained when the parametric excitation of the flow velocity is

The chaotic oscillations of the cantilevered pipe conveying pulsating fluid are obtained when the parametric excitation of the flow velocity is

The period-1 oscillation of the cantilevered pipe conveying pulsating fluid is obtained when the parametric excitation of the flow velocity is

The multiperiodic oscillation of the cantilevered pipe conveying pulsating fluid is obtained when the external excitation is

The multitime periodic oscillation of the cantilevered pipe conveying pulsating fluid is obtained when the external excitation is

The quasi-periodic oscillation of the cantilevered pipe conveying pulsating fluid is obtained when the external excitation is

The chaotic oscillations of the cantilevered pipe conveying pulsating fluid are obtained when the external excitation is

According to the characteristics of each figure for different motions, there exist the periodic, multitime quasi-periodic, and multitime chaotic oscillations of the cantilevered pipe conveying pulsating fluid under different velocity parametric excitations, as shown in Figures

Figure

Figure

We also find the periodic, multitime periodic, and multitime chaotic oscillations of the cantilevered pipe conveying pulsating fluid under different external excitations, as shown in Figures

Figure

Moreover, the oscillations corresponding to different velocity parametric excitations and external excitations are consistent with the bifurcation diagrams of the cantilevered pipe conveying pulsating fluid under the external harmonic force, as shown in Figures

The nonlinear resonant responses and multitime chaotic dynamics of the cantilevered pipe conveying pulsating fluid are investigated under the external harmonic force. Based on the nonlinear partial differential governing equation of motion for the cantilevered pipe conveying pulsating fluid derived by using Hamilton’s principle, the 1 : 3 internal resonance and primary parametric resonance-1/2 subharmonic resonance are considered. A combination method of the method of multiple scales and Galerkin technique is utilized to obtain four-dimensional nonlinear averaged equations. Several amplitude-frequency response curves are obtained corresponding to the certain parameters. From the analysis of the amplitude-frequency response curves, it is found that there exist the hardening-spring type behaviors and the jumping phenomena. The jumping phenomena also occur in the amplitude-force response curves versus the flow velocity and external force.

Moreover, we find that the flow velocity, external force, and coupling degree of two order modes can affect the nonlinear vibrations of the cantilevered pipe conveying pulsating fluid under the external harmonic force. The nonlinear vibrations of the cantilevered pipe conveying pulsating fluid can be excited more easily with the increase of the flow velocity, external force, and coupling degree of two order modes. It is known that the nonlinear dynamic behaviors of the cantilevered pipe conveying pulsating fluid under the external harmonic force will be affected due to the flow rate and external excitation under the case of 1 : 3 internal resonance. It is observed that the multitime chaotic vibrations will occur for the cantilevered pipe conveying pulsating fluid when the velocity parametric excitation or external excitation reaches a certain value. From Figures

All data generated or analyzed during this study are included in this published article.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors gratefully acknowledge the support of the National Natural Science Foundation of China (NNSFC) through Grant nos. 11672188, 11832002, and 11427801 and the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHRIHLB).