Hopf Bifurcation of a Standing Cantilever Pipe Conveying Fluid with Time Delay

Feng, Cheng; Li, Yun-dong; Ou, Gui-yu

doi:https://doi.org/10.1155/2022/3588068

Mathematical Problems in Engineering

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 3588068 | https://doi.org/10.1155/2022/3588068

Hopf Bifurcation of a Standing Cantilever Pipe Conveying Fluid with Time Delay

Cheng Feng,¹Yun-dong Li,¹and Gui-yu Ou²

Academic Editor: Lei Hou

Received29 Sept 2022

Revised24 Nov 2022

Accepted06 Dec 2022

Published24 Dec 2022

Abstract

In this article, the dynamical behavior of a standing cantilever pipe conveying fluid is investigated with time delay. By applying piezoelectric materials and considering the time delay of voltage, the motion equation is built with motion-limiting constraints and elastic support. The motion equation is discretized into ordinary differential equations by the Galerkin method. A stability analysis of the equilibrium point with three parameters is obtained. The system will lose stability at the equilibrium point and generate Hopf branches. The central manifold theorem and the canonical type theory are used to study the stability of Hopf branching directions and branching period solutions. Finally, numerical results validate the theoretical analysis.

1. Introduction

Pipes conveying fluid can be found in some engineering structures, such as submarine oil pipelines, fuel delivery pipelines for aircraft, and heat exchange systems for nuclear power plants. The fluid flowing in the pipe will cause vibration problems of pipe. Long-term vibration will cause safety problems for the structure. Based on the safety problems, the vibration of pipes conveying fluid is one of the hot spots studied by a lot of scholars. Therefore, there have been a number of studies on the stability of pipes conveying fluid in the past decades. By analyzing the linear model of pipe conveying fluid, Paidoussis [1] found two noteworthy instabilities: divergence instability and flutter instability. A pipe conveying fluid with support at both ends shows the divergence instability beyond a critical flow velocity, and a cantilever pipe conveying fluid has flutter instability beyond a critical flow velocity. Especially, vibration research of cantilever pipe conveying fluid has aroused great interest among some scholars. By considering constraint conditions in a linear model, Païdoussis et al. [2, 3] studied the stability and chaotic motion of a cantilevered pipe conveying fluid under three times nonlinear spring constraints. Jin [4] investigated the stability and dynamics of a cantilevered pipe conveying fluid with motion-limiting constraints and an elastic support and showed the effect of the spring constant and some other parameters on the dynamics of the system. Wang and Ni [5] developed the research results of Jin and presented the stability and chaotic motions of a standing pipe conveying fluid that is studied and further compared with that of a hanging system. Based on a novel extended version of the Lagrange equations for systems containing nonmaterial volumes, Stangl et al. [6] deduced the nonlinear equations of motion for cantilever pipe systems conveying fluid. An alternative to existing methods utilizing Newtonian balance equations or Hamilton’s principle is thus provided. Ghayesh et al. [7] studied the nonlinear planar dynamics of a fluid-conveying cantilevered pipe. The extended version of the Lagrange equations for systems containing nonmaterial volumes is employed to derive the equations of motion, resulting directly in a set of coupled nonlinear ordinary differential equations. The pseudoarclength continuation technique along with direct time integration is used to solve these equations. Liu et al. [8] explored the nonlinear forced vibrations of a cantilevered pipe conveying fluid under base excitations by means of the full nonlinear equation of motion. The self-excited vibration is briefly discussed, focusing on the effect of flow velocity on the stability and postflutter dynamical behavior of the pipe system with parameters close to those in previous experiments.

It should be noted that a Hopf bifurcation would arise when the cantilever pipe conveying fluid system is unstable. Hopf bifurcation is an essential part of the qualitative theory of dynamical systems, which is widely used in nature and engineering, such as biology, mechanics, and artificial intelligence. Many scholars have contributed to studying the Hopf bifurcation problem in cantilevered pipes conveying fluid. Bajaj et al. [9] investigated the self-induced planar nonlinear oscillations of a cantilever tube carrying an incompressible fluid. When the zero solution becomes unstable by a pair of complex conjugate eigenvalues of the linearized system crossing the imaginary axis, the system satisfies conditions for Hopf bifurcation and the zero solution bifurcates into periodic orbits. The bifurcated periodic solutions are determined using center-manifold and averaging ideas. In the literature [10–13], the nonlinear response characteristics of a cantilevered pipe conveying fluid with a spring support and nonlinear motion constraint and the bifurcation behavior of the system near the double degradation point were investigated. Modarres-Sadeghi et al. [14] investigated the two-dimensional (2-D) and three-dimensional (3-D) flutter of cantilevered pipes conveying fluid. Yamashita et al. [15] investigated the dynamics of a pipe with a spring support and an end mass. They focus on Hopf-Hopf interactions and conduct nonlinear analyses to clarify the evolutions of the amplitudes of two unstable modes. In a certain range of the parameters, there are mixed-mode self-excited vibrations, which are produced by the Hopf-Hopf interactions.

Active control of the vibration is necessary to reduce the hazards associated with cantilever pipe conveying fluid. With the development of material science, the preparation technology of smart materials such as piezoelectric materials tends to mature, making the cost of vibration control based on piezoelectric gradually reduced. Therefore, discussions on piezoelectrically driven cantilever pipes conveying fluid have drawn serious attention. Lin and Chu [16] used state-of-the-art piezoelectric ceramics as the actuators for active flutter suppression of a cantilever tube transporting high-speed moving fluid. The flutter vibration of the cantilever tube can be effectively suppressed with the moderate driving control voltage fed into the piezoelectric actuators. Tsai and Lin [17] designed a model reference adaptive control method in modal space for flutter control of a cantilevered pipe conveying fluid. The control input is provided by a pair of surface-mounted piezoelectric actuators which are driven 180° out of phase to provide an equivalent bending moment acting on the controlled system. Active flutter control of a thin-walled functionally graded piezoelectric (FGP) cantilever pipe via optimal control law was investigated by Fazelzadeh and Yazdanpanah [18]. A linear quadratic regulator (LQR) controller is employed in order to suppress the unwanted vibration caused by the system’s flutter. Abbasnejad et al. [19] presented the stability analysis of a fluid-conveying micropipe axially loaded with a pair of piezoelectric layers located at its top and bottom surfaces. Taking into account clamped-free boundary conditions with and without intermediate support, the stability of the system is investigated to demonstrate the influence of flow velocity as well as the voltage of the piezoelectric layers on the flow-induced flutter instability. Wang and Shen [20] developed a nonlinear model for a cantilevered piezoelectric pipe conveying fluid that included geometric nonlinearity and electromechanical coupling. Due to the presence of piezoelectric materials, the critical flutter velocity depends on resistive piezoelectric damping and electromechanical coupling. Dai et al. [21] employed a kind of large deformations induced by nonlinear vibrations of a soft pipe conveying fluid to design an underwater bio-inspired snake robot that consists of a rigid head and a soft tail. The research provided a new thought to design driving devices by using nonlinear flow-induced vibrations.

The time delay inevitably exists in active control systems. In the control system of pipe conveying fluid, the time delay may be utilized to enhance the critical flow velocity. The study of time-delay dynamical systems has essential theoretical and practical significance. Some dynamical phenomena can only be explained reasonably when the time delay is considered. For example, when negative feedback is introduced in a cantilever beam to increase the damping of the system, it is not the case that the greater the feedback gain, the better the stability of the system. When the feedback gain reaches a specific amount, the system will exhibit strong self-excited vibration under a specific initial disturbance. This phenomenon is difficult to explain without considering the effect of time delay. On the other hand, the time delay can also be exploited to improve the performance of control systems. For example, time-delay feedback control is one of the main methods to control chaos in dynamical systems. Sugimoto et al. [22] added a compensator to the time-delay controller, which made the stability and other performance of the time-delay controller better than the traditional time-delay controller. The phase lag caused by the delay link is compensated to some extent. For dual-input dual-output linear time-delay systems, Mirkin and Gutman [23] proposed an output feedback control scheme and adaptive rate based on MRAC, which can make the error asymptotically approach zero. Qi and Xu [24] investigated the relationship between time delay and critical flow velocity of cantilevered pipe conveying fluid under a time-delay control strategy. They achieved the purpose of increasing the critical flow velocity by actively adjusting the time delay. Wang and Ni [25] investigated the Hopf bifurcation and chaotic motions of a tubular cantilever impacting on loose support by using an analytic model that involves delay differential equations. By using the damping-controlled mechanism, a single flexible cantilever in an otherwise rigid square array of cylinders is analyzed.

From the previously mentioned works, it is confirmed that the studies on the cantilevered pipe conveying fluid with time delay of voltage are rather limited. In this article, based on Jin’s model, a time-delay control model of a standing cantilever pipe conveying fluid is developed by applying piezoelectric materials and considering the time delay of voltage, as shown in Figure 1. The stability of the trivial equilibrium of the nonlinear controlled system and the corresponding Hopf bifurcations appearing on the stability boundary is investigated. It should be noted that the theory of the linear part of the system is the same as the one studied by Qi and Xu [24]. The purpose of this paper is to investigate the effect of parameters such as time delay on stability and bifurcation behavior of the system after the system changes from a conventional dynamical system to a more complex time-delay dynamical system.

(a)

(b)

2. Mathematical Model and Descretized Method

The nonlinear motion equation of the controlled system is derived from the literature [12].where is a constant related to the physical and geometric properties of piezoelectric materials. The expression of can be referred to [12].

As shown in Figure 1, the left and right surfaces of the pipe are symmetrically arranged with piezoelectric patches. It is assumed that the piezoelectric patches are ideally attached to the pipe, regardless of the influence of the mass and stiffness of the piezoelectric patches. The piezoelectric patches are surface bonded on the pipe, piezoelectric patches produce geometric deformation under controlled voltage, and equivalent bending moments are generated, which is opposite to the vibration direction of the pipe. The explanation of each notation of the system is shown in the following table. In Table 1, the parameter can be found in the expression of which can be seen in Ref [12]. The central manifold theorem and the canonical type theory are used to study the stability of Hopf branching directions and branching period solution about a standing cantilever pipe conveying fluid.

Introducing the following dimensionless quantities and parameters:

Equation (1) can be rewritten as the following dimensionless form:where . The following equations remove the asterisk for convenience.

Equation (3) can be discretized by Galerkin’s technique as follows:where

are the eigenfunctions of the cantilever beam and are the corresponding generalized coordinate. Substituting (4) into (3) and using the Galerkin method, we obtain a four-dimensional ordinary differential equation of motion as follows:wherewhere and represent the eigenvalues of the cantilever beam and is Kronecker’s delta function.

In this article, the driving voltage considering piezoelectric excitation adopts the displacement delay feedback strategy [26], which is denoted as follows:where is the displacement controller feedback gains, is the position coordinates of the sensor, and is the inherent system delay, which denoted as .

In system (1), the flutter instability occurs in the second mode, so the first two modes of the system are intercepted by Galerkin’s method. Substituting into equation (7), we obtain

Let , after proper arrangement and transformation, we obtainwhere

3. Stability of Equilibrium with Time Delay

In the previous section, we built the model of the standing cantilever pipe conveying fluid with time delay. In this section, the stability of the system will be discussed by the eigenvalue method. The discussion of the linear part has been carried out by Qi and Xu [24]. In order to describe the Hopf bifurcation of the nonlinear systems in the next section, a simple derivation of the linear part is presented.

3.1. Solution of the Equilibrium

The equilibrium can be obtained from the following equation:

According to Jin’s method [4], three equilibriums of the system can be obtained as follows:where

It is clear that the solution (0, 0, 0, 0) always exists. The nonzero equilibrium exists when c > 0 as shown in Figure 2. The parameters are set as .

3.2. Stability Analysis of the Equilibrium

At the zero equilibrium, we have . Substituting the trial solution into the linear part of (10), we obtain the characteristic equation as follows:where , , , , and .

3.2.1. Stability of Systems without Delay

When , (15) becomes

Let

Then, the Routh–Hurwitz criterion is given by

If (18) holds, the equilibrium is locally asymptotically stable.

The stability boundaries of the equilibrium on the plane without delay are shown in Figure 3. When the flow velocity crosses the boundary from left to right, the system loses its stability and a Hopf bifurcation occurs. Figure 4 shows the curve of an eigenvalue changing with flow velocity for the system with and . It can be seen that when increases to 5.2, the system undergoes a Hopf bifurcation.

3.2.2. Stability of System with Delay

By multiplying on both sides of (15), we getwhere . By using the method of Qi and Xu [24], the corresponding critical value of time delay can be determined bywhere , , , , and , .

The real and imaginary part of at is given bywhere

If , the critical roots of (19) cross the axis from left to right. If , the critical roots of (19) cross the axis from right to left.

We set and the other parameters are the same as in Figure 2. We obtain the stability region of the equilibrium on the plane according to the continuity and stability switches analysis, as shown in Figure 5.

Taking the velocity as an example, the positive real roots of (20) and (21) can be calculated as and the corresponding critical delays can be calculated as follows:

Then, the sign of can be obtained as and , which means the characteristic roots cross the imaginary axis from right to left at and from left to right at .

The distribution of the eigenvalue for the delayed control system can be simulated by the MATLAB BIFTOOL package, as shown in Figure 6.(1)When , the system has only four eigenvalues. Two of them are on the right half of the complex plane, indicating that the system is unstable, as shown in Figure 6(a).(2)As the delay increases, it can be seen from Figures 6(b) and 6(c) that the delayed control system is unstable when . When , the system undergoes the first stability switch and becomes stable.(3)As the delay continues to increase, it can be seen from Figures 6(d) and 6(e) that the delayed control system is stable when . When , the system undergoes the stability switch again and becomes unstable.

(a)

(b)

(c)

(d)

(e)

Time history of the controlled system with and without time delay when u = 5.3 is shown in Figure 7. It is found that delayed control makes the previously unstable system stable.

(a)

(b)

The stability region of the equilibrium on the plane is shown in Figure 8. By comparing Figures 5 and 8, it can be seen that can also affect the critical flow velocity of the system. When and , the system is stable in the delay interval [0.19, 0.41] and [0.74, 0.89]. When and , the system is stable in the full delay interval.

It should be noted that a Hopf bifurcation would occur on the stability boundary (the red lines in Figure 5). When the time delay crosses the stability boundaries shown in Figure 5, a periodic solution may appear near the equilibrium point. The stability and bifurcating direction of these periodic solutions, as well as the period of the periodic solutions, will be further analyzed in the next section.

4. Direction and Stability of the Hopf Bifurcation

In the previous section, there exists one pair of purely imaginary eigenvalues on the stability boundary of the equilibrium. Consequently, a Hopf bifurcation will arise at . In this section, the direction of Hopf bifurcation and the stability of bifurcation periodic solutions will be analyzed. Our approach is the paradigm theory and central flow type theorem proposed by Hassan et al. [27]. For simplicity, let . Then, is the corresponding Hopf bifurcation point.

For , we assume thatwhere matrices and are defined in (10). According to the Riesz representation theorem, there exists a bounded variation function such thatwhere we choose the following expression to satisfy the equation:

For , we definethen, (10) can be rewritten as follows:where and .

We define the corresponding adjoint operator of the operator and a bilinear form of vectorswhere .

According to the discussion in the previous section, ± iω are the eigenvectors of and , respectively. Next, we need to figure out the eigenvector of the operator and the eigenvector of . Based on the definition of and , we obtainwhere

From , we obtain .

According to the algorithm of Hassard et al. [27], we calculate the coordinate quantity describing the central manifold when . We assume that is a solution of (30) and assume that

On the central manifold , the following expression holds:where and are the local coordinates of the central manifold along the direction and , respectively. Known that is the solution of (30) and since , we have

Equation (38) can be rewritten as follows:where

Notice that , we obtain

According to the literature [27], the critical properties of bifurcation periodic solutions are determined by the following parameters:

Obviously, we need to calculate the expressions of , , , and . Substituting and (41) into (38) and comparing the same order of terms, we obtain the expressions of as follows:

According to the analysis, the following theorem can be obtained:

Theorem 1. [27]The direction of the Hopf bifurcation and the stability of the bifurcation period solution are determined by the signs of and in (42). If ), the Hopf bifurcation is supercritical (subcritical). Moreover, if , the bifurcation period solution is stable (unstable). The value of determines the period of the bifurcated periodic solution, and if , the period increases (decreases).

In order to verify the results of the theorem, taking the velocity as an example, the positive real roots of (20) and (21) can be calculated as and the corresponding critical delays can be calculated as follows:

As shown in Figure 5, when the time delay increases to the critical value in the case of , which is marked as the point , the equilibrium gains its stability. The corresponding parameter values in (42) are calculated as , , and . According to Theorem 3.1, the Hopf bifurcation is subcritical, and the bifurcating periodic solution for is stable. We set , the other parameters are the same as in Figure 5, and we obtain the corresponding bifurcation diagram near the Hopf bifurcation point , as shown in Figure 9.

The fourth-order Runge–Kutta method is used to numerically simulate the periodic orbit amplitudes and compare with the theoretical predictions, and the results are in good agreement. The average radius of the bifurcating periodic orbit [28] can be theoretically determined as . Here, . When the delay exceeds the critical value , which is marked as the point , the equilibrium loses its stability. The corresponding parameter values in (42) are calculated as , , and . According to Theorem 3.1, the Hopf bifurcation is supercritical and the bifurcating periodic solution for is stable. The phase diagram and response diagram before and after the bifurcation are shown in Figure 10. Figure 10(a) shows that the vibration of the pipe conveying fluid before the bifurcation is stable. Figure 10(b) is the phase diagram corresponding to Figure 10(a), and it can be seen that the pipe conveying fluid tends to a stable equilibrium. Figure 10(c) shows that the pipe conveying fluid after bifurcation is in stable periodic motion. Moreover, the stable bifurcating periodic orbit is shown in Figure 10(d).

(a)

(b)

(c)

(d)

5. Conclusion

In this article, a time-delay-controlled strategy was used to actively control the flutter of a standing cantilever pipe conveying fluid subject to linear springs and cubic nonlinear constraints. The motion equation is built with motion-limiting constraints and elastic support by applying piezoelectric materials and considering the time delay of voltage. The motion equation is discretized into ordinary differential equations by the Galerkin method. A stability analysis of the equilibrium point with three parameters is obtained. The stability analysis of the equilibrium of the controlled system was performed by studying the characteristic equation and stability switches. The central manifold theorem and the canonical type theory are used to study the stability of Hopf branching directions and branching period solutions. The results show that the delayed feedback control strategy can effectively improve the critical velocity of the pipe and effectively control the flutter of the nonlinear system. The results between the theoretical analysis and the numerical simulation show a good agreement, indicating the effectiveness of the control strategy.

Data Availability

The data used to support the findings of this study can be obtained from the corresponding author upon reasonable request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The research was partially supported by the National Natural Science Foundation of China (grant number 11872043) and the Opening Project of Sichuan Province University Key Laboratory of Bridge Nondestruction Detecting and Engineering Computing (2022QYY06). The authors thank the anonymous reviewers for their helpful suggest.

References

M. P. Païdoussis and N. T. Issid, “Dynamic stability of pipes conveying fluid,” Journal of Sound and Vibration, vol. 33, no. 3, pp. 267–294, 1974.
View at: Publisher Site | Google Scholar
M. P. Païdoussis, G. X. Li, and F. C. Moon, “Chaotic oscillations of the autonomous system of a constrained pipe conveying fluid,” Journal of Sound and Vibration, vol. 135, no. 1, pp. 1–19, 1989.
View at: Publisher Site | Google Scholar
M. P. Païdoussis, G. X. Li, and R. H. Rand, “Chaotic motions of a constrained pipe conveying fluid: comparison between simulation, analysis, and experiment,” Journal of Applied Mechanics, vol. 58, no. 2, pp. 559–565, 1991.
View at: Publisher Site | Google Scholar
J. D. Jin, “Stability and chaotic motions of a restrained pipe conveying fluid,” Journal of Sound and Vibration, vol. 208, no. 3, pp. 427–439, 1997.
View at: Publisher Site | Google Scholar
L. Wang and Q. Ni, “A note on the stability and chaotic motions of a restrained pipe conveying fluid,” Journal of Sound and Vibration, vol. 296, no. 4-5, pp. 1079–1083, 2006.
View at: Publisher Site | Google Scholar
M. Stangl, J. Gerstmayr, and H. Irschik, “An alternative approach for the analysis of nonlinear vibrations of pipes conveying fluid,” Journal of Sound and Vibration, vol. 310, no. 3, pp. 493–511, 2008.
View at: Publisher Site | Google Scholar
M. H. Ghayesh, M. P. Païdoussis, and M. Amabili, “Nonlinear dynamics of cantilevered extensible pipes conveying fluid,” Journal of Sound and Vibration, vol. 332, no. 24, pp. 6405–6418, 2013.
View at: Publisher Site | Google Scholar
Z. Y. Liu, L. Wang, and X. P. Sun, “Nonlinear forced vibration of cantilevered pipes conveying fluid,” Acta Mechanica Solida Sinica, vol. 31, no. 1, pp. 32–50, 2018.
View at: Publisher Site | Google Scholar
A. K. Bajaj, P. R. Sethna, and T. S. Lundgren, “Hopf bifurcation phenomena in tubes carrying a fluid,” SIAM Journal on Applied Mathematics, vol. 39, no. 2, pp. 213–230, 1980.
View at: Publisher Site | Google Scholar
J. D. Jin, Y. Lin, and G. S. Zou, “Flutter and chaotic motions of a cantilevered pipe conveying fluid,” Journal of Vibration Engineering, vol. 10, no. 3, pp. 68–74, 1997.
View at: Google Scholar
J. D. Jin, G. S. Zou, and Y. F. Zhang, “Bifurcations and chaotic motions of a cantilevered pipe conveying fluid,” Theoretical and Applied Mechanics, vol. 34, no. 6, pp. 863–873, 2002.
View at: Google Scholar
G. S. Zou, J. D. Jin, and B. C. Wen, “Adaptive vibration control of a cantilevered pipe conveying fluid,” China Mechanical Engineering, vol. 16, no. 7, pp. 626–629, 2005.
View at: Google Scholar
J. D. Jin and G. S. Zou, “Bifurcations and chaotic motions in the autonomous system of a restrained pipe conveying fluid,” Journal of Sound and Vibration, vol. 260, no. 5, pp. 783–805, 2003.
View at: Publisher Site | Google Scholar
Y. Modarres-Sadeghi, M. P. Païdoussis, and C. Semler, “Three-dimensional oscillations of a cantilever pipe conveying fluid,” International Journal of Non-linear Mechanics, vol. 43, no. 1, pp. 18–25, 2008.
View at: Publisher Site | Google Scholar
K. Yamashita, N. Nishiyama, K. Katsura, and H. Yabuno, “Hopf-Hopf interactions in a spring-supported pipe conveying fluid,” Mechanical Systems and Signal Processing, vol. 152, Article ID 107390, 2021.
View at: Publisher Site | Google Scholar
Y. H. Lin and C. L. Chu, “Active flutter control of a cantilever tube conveying fluid using piezoelectric actuators,” Journal of Sound and Vibration, vol. 196, no. 1, pp. 97–105, 1996.
View at: Publisher Site | Google Scholar
Y. K. Tsai and Y. H. Lin, “Adaptive modal vibration control of a fluid-conveying cantilever pipe,” Journal of Fluids and Structures, vol. 11, no. 5, pp. 535–547, 1997.
View at: Publisher Site | Google Scholar
S. A. Fazelzadeh and B. Yazdanpanah, “Active Flutter Suppression of Thin-Walled Cantilever Functionally Graded Piezoelectric Pipes Conveying Fluid,” in Proceedings of the 20th Annual International Conference on Mechanical Engineering-Isme2012, pp. 1–4, Shiraz University, School of Mechanical Engineering, Shiraz, Iran, May, 2012.
View at: Google Scholar
B. Abbasnejad, R. Shabani, and G. Rezazadeh, “Stability analysis of a piezoelectrically actuated micro-pipe conveying fluid,” Microfluidics and Nanofluidics, vol. 19, no. 3, pp. 577–584, 2015.
View at: Publisher Site | Google Scholar
G. Wang and J. Shen, “Flutter instabilities of cantilevered piezoelectric pipe conveying fluid,” Journal of Intelligent Material Systems and Structures, vol. 30, no. 4, pp. 606–617, 2019.
View at: Publisher Site | Google Scholar
H. Dai, Y. He, K. Zhou, Z. Peng, L. Wang, and P. Hagedorn, “Utilization of nonlinear vibrations of soft pipe conveying fluid for driving underwater bio-inspired robot,” Applied Mathematics and Mechanics, vol. 43, no. 7, pp. 1109–1124, 2022.
View at: Publisher Site | Google Scholar
H. Sugimoto, A. Horiuchi, and K. Kawasaki, “A new repetitive control and its characteristics,” in Proceedings of the SICEAnnual Conference, vol. 2, pp. 2261–2266, Fukui, Japan, August, 2003.
View at: Google Scholar
B. M. Mirkin and P. O. Gutman, “Adaptive control of linear time delay systems,” in Proceedings of the 2004 American Control Conference, vol. 2, pp. 1241–1246, Massachusetts: Boston, June, 2004.
View at: Google Scholar
H. H. Qi and J. Xu, “Delayed feedback control for flutter in the pipe conveying fluid,” Journal of Vibration Engineering, vol. 22, no. 6, pp. 576–582, 2009.
View at: Google Scholar
L. Wang and Q. Ni, “Hopf bifurcation and chaotic motions of a tubular cantilever subject to cross flow and loose support,” Nonlinear Dynamics, vol. 59, no. 1-2, pp. 329–338, 2010.
View at: Publisher Site | Google Scholar
J. Peng, G. Zhang, M. Xiang, H. Sun, X. Wang, and X. Xie, “Vibration control for the nonlinear resonant response of a piezoelectric elastic beam via time-delayed feedback,” Smart Materials and Structures, vol. 28, no. 9, Article ID 095010, 2019.
View at: Publisher Site | Google Scholar
B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981.
S. Deng, J. Ji, S. Yin, and G. Wen, “Multistability in the centrifugal governor system under a time-delay control strategy,” Journal of Computational and Nonlinear Dynamics, vol. 14, no. 11, Article ID 111010, 2019.
View at: Publisher Site | Google Scholar

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Copyright © 2022 Cheng Feng 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.

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