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

Authors: Cheng Feng; Yun-dong Li; Gui-yu Ou
Journal: Mathematical Problems in Engineering
 (2022)

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## 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.


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## Body

## 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 Figure1. 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.Figure 1
(a) The model of the controlled pipe conveying fluid system and (b) cross-section diagram of the pipe conveying fluid.
(a)(b)


## 2. Mathematical Model and Descretized Method

The nonlinear motion equation of the controlled system is derived from the literature [12].(1)aEI∂5y∂x4∂t+EI∂4y∂x4+m1U2+m1+mpL−xg∂2y∂x2+2m1U∂2y∂x∂t−m1+mpg∂y∂x+m1+mp∂2y∂t2+K1y+K2y3δx−xb=∂∂xCmVtδx−xu−δx−xd,where Cm is a constant related to the physical and geometric properties of piezoelectric materials. The expression of Cm can be referred to [12].As shown in Figure1, 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 φ can be found in the expression of Cm 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.Table 1
Summary of symbols.
NotationExplanationEIFlexural rigidity of the pipeaCoefficient of Kelvin–Voigt viscoelastic damping of the pipe materialLPipe lengthmpMass per unit lengthm1Mass of the conveying fluid per unit lengthUFlow velocityK1Stiffness of the spring of the elastic supportK2Stiffness of cubic spring which represents the effect of the motion constraintsxbLocation of the constraintsxuDistance between the fixed end of the piezoelectric plate and the upper end of the pipelinexdDistance between the fixed end of the piezoelectric plate and the lower end of the pipelinex3Position coordinates of the sensorφEnvelope half angle of a piezoelectric plategAcceleration due to gravityδ⋅Dirac delta functionyx,tLateral deflection of the pipeIntroducing the following dimensionless quantities and parameters:(2)η=yL,ξ=xL,τ=EIm1+mp1/2tL2,u=m1EI1/2UL,γ=m1+mpEIgL3,ξb=xbL,ξ1=xuL,ξ2=xdL,α=EIm1+mp1/2aL2,k1=K1L3EI,k2=K2L5EI,β=m1m1+mp,Cm⋆Va∗t∗=LCmEIVat.Equation (1) can be rewritten as the following dimensionless form:(3)αη˙⁗+η⁗+u2+1−ξγη″+2βuη˙′−γη′+η¨+k1η+k2η3δξ−ξb=Cm⋆Va∗t∗δ′ξ−ξ1−δ′ξ−ξ2,where ⋅=∂/∂t,′=∂/∂ξ. The following equations remove the asterisk for convenience.Equation (3) can be discretized by Galerkin’s technique as follows:(4)ηξ,τ=∑r=1Nφrξqrτ,where(5)φrξ=coshλrξ−cosλrξ−sinhλr−sinλrcoshλr+cosλrsinhλrξ−sinλrξ,are the eigenfunctions of the cantilever beam andqrτ 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:(6)δsrq¨r+αλr4δsr+2βubsrq˙r+λr4δsr+u2csr−γesrqr+k1φsξbφrξbqr+k2φsξbφr3ξbqr3=MsVat,where(7)bsr=∫01φsφr′dξ,csr=∫01φsφr″dξ,dsr=∫01φsξφr″dξ,esr=bsr−csr+dsr,Ms=Cmφs′ξ2−φs′ξ1,where s,r=1,2,…,N and λr represent the eigenvalues of the cantilever beam and δsr 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:(8)Vat=∑r=1N−kdφrξ3qrt−τ,where kd is the displacement controller feedback gains, ξ3 is the position coordinates of the sensor, and τ is the inherent system delay, which denoted as −Mskdφrξ3=ksr.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 Vat into equation (7), we obtain(9)q¨1+αλ14+2βub11q˙1+2βub12q˙2+λ14+u2c11−γe11q1+u2c12−γe12q2+k1φ1ξbφ1ξbq1+φ2ξbq2+k2φ1ξbφ1ξbq1+φ2ξbq23=k11q1τ+k12q2τ,q¨2+αλ24+2βub22q˙2+2βub21q˙1+λ24+u2c22−γe22q2+u2c21−γe21q1+k1φ2ξbφ1ξbq1+φ2ξbq2+k2φ2ξbφ1ξbq1+φ2ξbq23=k21q1τ+k22q2τ.Letx1=q1,x2=q2,x3=q˙1,x4=q˙2, after proper arrangement and transformation, we obtain(10)X˙=AXt+BXt−τ+FX,where(11)X=x1,x2,x3,x4T,FX=0,0,F3,F4T,A=00100001a1a2a3a4b1b2b3b4,B=00000000k11k1200k21k2200,a1=−λ14+u2c11−γe11+k1g11,a2=−u2c12−γe12+k1g12,a3=−αλ14+2βub11,a4=−2βub12,b1=−u2c21−γe21+k1g21,b2=−λ24+u2c22−γe22+k1g22,b3=−2βub21,b4=−αλ24+2βub22,F3=−k2φ1ξbφ1ξbx1+φ2ξbx23,F4=φ2ξbF3φ1ξbgsr=φsξbφrξbr,s=1,2.


## 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:(12)a1x1+a2x2+k11x1t−τ+k12x2t−τ+F3=0,b1x1+b2x2+k21x1t−τ+k22x2t−τ+F4=0,x3=0,x4=0.According to Jin’s method [4], three equilibriums of the system can be obtained as follows:(13)0,0,0,0≡0,kc,c,0,0≡+N,−kc,−c,0,0≡−N,where(14)c=a1k+a2+k11k+k12b,b=k2g112k+g12g113,k=a2+k12−eb2−ek22eb1+ek21−a1−k11=eλ24+u2ec22−c12−γee22−e12+k12−ek22λ14+u2c11−ec21−γe11−ee21+ek21−k11.It is clear that the solution (0, 0, 0, 0) always exists. The nonzero equilibrium exists whenc > 0 as shown in Figure 2. The parameters are set as α=0.005,β=0.2,γ=10,ξb=0.82,ξ1=0,ξ2=0.2,ξ3=0.8,Cm=0.178.Figure 2
Existence region of nonzero equilibrium. The domain marked by the gray shadows indicates the existence of nonzero equilibria.


### 3.2. Stability Analysis of the Equilibrium

At the zero equilibrium, we haveO=0,0,0,0. Substituting the trial solution X=X¯eλt into the linear part of (10), we obtain the characteristic equation as follows:(15)λ4+d1λ3+d2λ2+d3λ+d4+d5λ2+d6λ+d7e−λτ+d8e−2λτ=0,where d1=−a3−b4,d2=−a1−b2+a3b4−a4b3, d3=a1b4−a2b3+a3b2−a4b1, d4=a1b2−a2b1,d5=−k11−k22, d6=a3k22−a4k21−b3k12+b4k11, and d7=a1k22−a2k21−b1k12+b2k11,d8=k11k22−k12k21.


#### 3.2.1. Stability of Systems without Delay

Whenτ=0, (15) becomes(16)λ4+d1λ3+d2+d5λ2+d3+d6λ+d4+d7+d8=0.Let(17)D=d1100d3+d6d2+d5d110d4+d7+d8d3+d6d2+d5000d4+d7+d8.Then, the Routh–Hurwitz criterion is given by(18)D1>0,D2>0,D3>0,D4>0.If (18) holds, the equilibrium O0,0,0,0 is locally asymptotically stable.The stability boundaries of the equilibriumx∗ on the u,k1 plane without delay are shown in Figure 3. When the flow velocity u crosses the boundary BH 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 τ=0 and k1=20. It can be seen that when u increases to 5.2, the system undergoes a Hopf bifurcation.Figure 3
Stable regions of the equilibriumO on the u,k1 plane without delay.Figure 4
Eigenvalue evolutions of the equilibriumO for k1=20.


#### 3.2.2. Stability of System with Delay

By multiplyingeλτ on both sides of (15), we get(19)Pλeλτ+Qλ+Rλe−λτ=0,where Pλ=λ4+d1λ3+d2λ2+d3λ+d4,Qλ=d5λ2+d6λ+d7,Rλ=d8. By using the method of Qi and Xu [24], the corresponding critical value of time delay τk can be determined by(20)cosω0τk=−PR−RRQR+PI−RIQIPR2+PI2−RR2−RI2,(21)sinω0τk=PI+RIQR−PR+RRQIPR2+PI2−RR2−RI2,where PR=RePiω=ω4−d2ω2+d4, PI=ImPiω=−d1ω3i+d3ωi, QR=ReQiω=−d5ω2+d7, QI=ImQiω=d6ωi, and RR=ReRiω=d8, RI=ImRiω=0.The real and imaginary part ofdλ/dτ at λ=iω0 is given by(22)Redλdττ=τk=U1V1+U2V2V12+V22,Imdλdττ=τk=−U1V2+U2V1V12+V22,where(23)U1=−d1ω4−d3ω2cosωτk−−ω5+d2ω3−d4ω−d8ωsinωτ,U2=−ω5−d2ω3+d4ω−d8ωcosωτ−d1ω4−d3ω2sinωτ,V1=−3d1ω2+d3+ω4τk−d2ω2τk+d4τk−d8τkcosωτk+4ω3−2d2ω+d1ω3τk−d3ωτksinωτk+d6,V2=−4ω3+2d2ω−d1ω3τk+d3ωτkcosωτk+−3d1ω2+d3+ω4τk−d2ω2τk+d4τk+d8τksinωτk+2d5ω.IfRedλ/dττ=τk>0, the critical roots of (19) cross the axis from left to right. If Redλ/dττ=τk<0, the critical roots of (19) cross the axis from right to left.We setk1=20 and the other parameters are the same as in Figure 2. We obtain the stability region of the equilibrium on the u,τ plane according to the continuity and stability switches analysis, as shown in Figure 5.Figure 5
Stable regions of the equilibriumx∗. The stable regions are marked by the gray shadows, and the points H1 and H2 represent the generic Hopf bifurcation point.Taking the velocityu=5.3 as an example, the positive real roots of (20) and (21) can be calculated as ω1≈11.777491,ω2≈13.116553 and the corresponding critical delays τ1k,τ2k can be calculated as follows:(24)τ10≈0.141761,τ11≈0.675252⋯,τ20≈0.456237,τ21≈0.935264⋯.Then, the sign ofRedλ/dττ=τk can be obtained as Redλ/dττ=τ10<0 and Redλ/dττ=τ20>0, which means the characteristic roots cross the imaginary axis from right to left at τ10 and from left to right at τ20.The distribution of the eigenvalue for the delayed control system can be simulated by the MATLAB BIFTOOL package, as shown in Figure6.(1)
Whenτ=0, 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 Figures6(b) and 6(c) that the delayed control system is unstable when τ=0.14. When τ=0.15, the system undergoes the first stability switch and becomes stable.(3)
As the delay continues to increase, it can be seen from Figures6(d) and 6(e) that the delayed control system is stable when τ=0.45. When τ=0.46, the system undergoes the stability switch again and becomes unstable.Figure 6
Distribution of eigenvalues for the delayed control system.
(a)(b)(c)(d)(e)Time history of the controlled system with and without time delay whenu = 5.3 is shown in Figure 7. It is found that delayed control makes the previously unstable system stable.Figure 7
Time history of the controlled system. (a)u,τ=5.3,0 with the initial values (x10, x20, x30, and x40) = 0.01,0,0,0 and (b)u,τ=5.3,0.15 with the initial values (x10, x20, x30, and x40) = 0.1,0,0,0.
(a)(b)The stability region of the equilibrium on thek1,τ plane is shown in Figure 8. By comparing Figures 5 and 8, it can be seen that k1 can also affect the critical flow velocity of the system. When u=5.5 and k1=20, the system is stable in the delay interval [0.19, 0.41] and [0.74, 0.89]. When u=5.5 and k1=35, the system is stable in the full delay interval.Figure 8
Stable regions of the equilibriumx∗ when u=5.5. The stable regions are marked by the gray shadows.It should be noted that a Hopf bifurcation would occur on the stability boundary (the red lines in Figure5). When the time delay τ 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.


## 3.1. Solution of the Equilibrium

The equilibrium can be obtained from the following equation:(12)a1x1+a2x2+k11x1t−τ+k12x2t−τ+F3=0,b1x1+b2x2+k21x1t−τ+k22x2t−τ+F4=0,x3=0,x4=0.According to Jin’s method [4], three equilibriums of the system can be obtained as follows:(13)0,0,0,0≡0,kc,c,0,0≡+N,−kc,−c,0,0≡−N,where(14)c=a1k+a2+k11k+k12b,b=k2g112k+g12g113,k=a2+k12−eb2−ek22eb1+ek21−a1−k11=eλ24+u2ec22−c12−γee22−e12+k12−ek22λ14+u2c11−ec21−γe11−ee21+ek21−k11.It is clear that the solution (0, 0, 0, 0) always exists. The nonzero equilibrium exists whenc > 0 as shown in Figure 2. The parameters are set as α=0.005,β=0.2,γ=10,ξb=0.82,ξ1=0,ξ2=0.2,ξ3=0.8,Cm=0.178.Figure 2
Existence region of nonzero equilibrium. The domain marked by the gray shadows indicates the existence of nonzero equilibria.


## 3.2. Stability Analysis of the Equilibrium

At the zero equilibrium, we haveO=0,0,0,0. Substituting the trial solution X=X¯eλt into the linear part of (10), we obtain the characteristic equation as follows:(15)λ4+d1λ3+d2λ2+d3λ+d4+d5λ2+d6λ+d7e−λτ+d8e−2λτ=0,where d1=−a3−b4,d2=−a1−b2+a3b4−a4b3, d3=a1b4−a2b3+a3b2−a4b1, d4=a1b2−a2b1,d5=−k11−k22, d6=a3k22−a4k21−b3k12+b4k11, and d7=a1k22−a2k21−b1k12+b2k11,d8=k11k22−k12k21.


### 3.2.1. Stability of Systems without Delay

Whenτ=0, (15) becomes(16)λ4+d1λ3+d2+d5λ2+d3+d6λ+d4+d7+d8=0.Let(17)D=d1100d3+d6d2+d5d110d4+d7+d8d3+d6d2+d5000d4+d7+d8.Then, the Routh–Hurwitz criterion is given by(18)D1>0,D2>0,D3>0,D4>0.If (18) holds, the equilibrium O0,0,0,0 is locally asymptotically stable.The stability boundaries of the equilibriumx∗ on the u,k1 plane without delay are shown in Figure 3. When the flow velocity u crosses the boundary BH 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 τ=0 and k1=20. It can be seen that when u increases to 5.2, the system undergoes a Hopf bifurcation.Figure 3
Stable regions of the equilibriumO on the u,k1 plane without delay.Figure 4
Eigenvalue evolutions of the equilibriumO for k1=20.


### 3.2.2. Stability of System with Delay

By multiplyingeλτ on both sides of (15), we get(19)Pλeλτ+Qλ+Rλe−λτ=0,where Pλ=λ4+d1λ3+d2λ2+d3λ+d4,Qλ=d5λ2+d6λ+d7,Rλ=d8. By using the method of Qi and Xu [24], the corresponding critical value of time delay τk can be determined by(20)cosω0τk=−PR−RRQR+PI−RIQIPR2+PI2−RR2−RI2,(21)sinω0τk=PI+RIQR−PR+RRQIPR2+PI2−RR2−RI2,where PR=RePiω=ω4−d2ω2+d4, PI=ImPiω=−d1ω3i+d3ωi, QR=ReQiω=−d5ω2+d7, QI=ImQiω=d6ωi, and RR=ReRiω=d8, RI=ImRiω=0.The real and imaginary part ofdλ/dτ at λ=iω0 is given by(22)Redλdττ=τk=U1V1+U2V2V12+V22,Imdλdττ=τk=−U1V2+U2V1V12+V22,where(23)U1=−d1ω4−d3ω2cosωτk−−ω5+d2ω3−d4ω−d8ωsinωτ,U2=−ω5−d2ω3+d4ω−d8ωcosωτ−d1ω4−d3ω2sinωτ,V1=−3d1ω2+d3+ω4τk−d2ω2τk+d4τk−d8τkcosωτk+4ω3−2d2ω+d1ω3τk−d3ωτksinωτk+d6,V2=−4ω3+2d2ω−d1ω3τk+d3ωτkcosωτk+−3d1ω2+d3+ω4τk−d2ω2τk+d4τk+d8τksinωτk+2d5ω.IfRedλ/dττ=τk>0, the critical roots of (19) cross the axis from left to right. If Redλ/dττ=τk<0, the critical roots of (19) cross the axis from right to left.We setk1=20 and the other parameters are the same as in Figure 2. We obtain the stability region of the equilibrium on the u,τ plane according to the continuity and stability switches analysis, as shown in Figure 5.Figure 5
Stable regions of the equilibriumx∗. The stable regions are marked by the gray shadows, and the points H1 and H2 represent the generic Hopf bifurcation point.Taking the velocityu=5.3 as an example, the positive real roots of (20) and (21) can be calculated as ω1≈11.777491,ω2≈13.116553 and the corresponding critical delays τ1k,τ2k can be calculated as follows:(24)τ10≈0.141761,τ11≈0.675252⋯,τ20≈0.456237,τ21≈0.935264⋯.Then, the sign ofRedλ/dττ=τk can be obtained as Redλ/dττ=τ10<0 and Redλ/dττ=τ20>0, which means the characteristic roots cross the imaginary axis from right to left at τ10 and from left to right at τ20.The distribution of the eigenvalue for the delayed control system can be simulated by the MATLAB BIFTOOL package, as shown in Figure6.(1)
Whenτ=0, 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 Figures6(b) and 6(c) that the delayed control system is unstable when τ=0.14. When τ=0.15, the system undergoes the first stability switch and becomes stable.(3)
As the delay continues to increase, it can be seen from Figures6(d) and 6(e) that the delayed control system is stable when τ=0.45. When τ=0.46, the system undergoes the stability switch again and becomes unstable.Figure 6
Distribution of eigenvalues for the delayed control system.
(a)(b)(c)(d)(e)Time history of the controlled system with and without time delay whenu = 5.3 is shown in Figure 7. It is found that delayed control makes the previously unstable system stable.Figure 7
Time history of the controlled system. (a)u,τ=5.3,0 with the initial values (x10, x20, x30, and x40) = 0.01,0,0,0 and (b)u,τ=5.3,0.15 with the initial values (x10, x20, x30, and x40) = 0.1,0,0,0.
(a)(b)The stability region of the equilibrium on thek1,τ plane is shown in Figure 8. By comparing Figures 5 and 8, it can be seen that k1 can also affect the critical flow velocity of the system. When u=5.5 and k1=20, the system is stable in the delay interval [0.19, 0.41] and [0.74, 0.89]. When u=5.5 and k1=35, the system is stable in the full delay interval.Figure 8
Stable regions of the equilibriumx∗ when u=5.5. The stable regions are marked by the gray shadows.It should be noted that a Hopf bifurcation would occur on the stability boundary (the red lines in Figure5). When the time delay τ 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.


## 3.2.1. Stability of Systems without Delay

Whenτ=0, (15) becomes(16)λ4+d1λ3+d2+d5λ2+d3+d6λ+d4+d7+d8=0.Let(17)D=d1100d3+d6d2+d5d110d4+d7+d8d3+d6d2+d5000d4+d7+d8.Then, the Routh–Hurwitz criterion is given by(18)D1>0,D2>0,D3>0,D4>0.If (18) holds, the equilibrium O0,0,0,0 is locally asymptotically stable.The stability boundaries of the equilibriumx∗ on the u,k1 plane without delay are shown in Figure 3. When the flow velocity u crosses the boundary BH 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 τ=0 and k1=20. It can be seen that when u increases to 5.2, the system undergoes a Hopf bifurcation.Figure 3
Stable regions of the equilibriumO on the u,k1 plane without delay.Figure 4
Eigenvalue evolutions of the equilibriumO for k1=20.


## 3.2.2. Stability of System with Delay

By multiplyingeλτ on both sides of (15), we get(19)Pλeλτ+Qλ+Rλe−λτ=0,where Pλ=λ4+d1λ3+d2λ2+d3λ+d4,Qλ=d5λ2+d6λ+d7,Rλ=d8. By using the method of Qi and Xu [24], the corresponding critical value of time delay τk can be determined by(20)cosω0τk=−PR−RRQR+PI−RIQIPR2+PI2−RR2−RI2,(21)sinω0τk=PI+RIQR−PR+RRQIPR2+PI2−RR2−RI2,where PR=RePiω=ω4−d2ω2+d4, PI=ImPiω=−d1ω3i+d3ωi, QR=ReQiω=−d5ω2+d7, QI=ImQiω=d6ωi, and RR=ReRiω=d8, RI=ImRiω=0.The real and imaginary part ofdλ/dτ at λ=iω0 is given by(22)Redλdττ=τk=U1V1+U2V2V12+V22,Imdλdττ=τk=−U1V2+U2V1V12+V22,where(23)U1=−d1ω4−d3ω2cosωτk−−ω5+d2ω3−d4ω−d8ωsinωτ,U2=−ω5−d2ω3+d4ω−d8ωcosωτ−d1ω4−d3ω2sinωτ,V1=−3d1ω2+d3+ω4τk−d2ω2τk+d4τk−d8τkcosωτk+4ω3−2d2ω+d1ω3τk−d3ωτksinωτk+d6,V2=−4ω3+2d2ω−d1ω3τk+d3ωτkcosωτk+−3d1ω2+d3+ω4τk−d2ω2τk+d4τk+d8τksinωτk+2d5ω.IfRedλ/dττ=τk>0, the critical roots of (19) cross the axis from left to right. If Redλ/dττ=τk<0, the critical roots of (19) cross the axis from right to left.We setk1=20 and the other parameters are the same as in Figure 2. We obtain the stability region of the equilibrium on the u,τ plane according to the continuity and stability switches analysis, as shown in Figure 5.Figure 5
Stable regions of the equilibriumx∗. The stable regions are marked by the gray shadows, and the points H1 and H2 represent the generic Hopf bifurcation point.Taking the velocityu=5.3 as an example, the positive real roots of (20) and (21) can be calculated as ω1≈11.777491,ω2≈13.116553 and the corresponding critical delays τ1k,τ2k can be calculated as follows:(24)τ10≈0.141761,τ11≈0.675252⋯,τ20≈0.456237,τ21≈0.935264⋯.Then, the sign ofRedλ/dττ=τk can be obtained as Redλ/dττ=τ10<0 and Redλ/dττ=τ20>0, which means the characteristic roots cross the imaginary axis from right to left at τ10 and from left to right at τ20.The distribution of the eigenvalue for the delayed control system can be simulated by the MATLAB BIFTOOL package, as shown in Figure6.(1)
Whenτ=0, 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 Figures6(b) and 6(c) that the delayed control system is unstable when τ=0.14. When τ=0.15, the system undergoes the first stability switch and becomes stable.(3)
As the delay continues to increase, it can be seen from Figures6(d) and 6(e) that the delayed control system is stable when τ=0.45. When τ=0.46, the system undergoes the stability switch again and becomes unstable.Figure 6
Distribution of eigenvalues for the delayed control system.
(a)(b)(c)(d)(e)Time history of the controlled system with and without time delay whenu = 5.3 is shown in Figure 7. It is found that delayed control makes the previously unstable system stable.Figure 7
Time history of the controlled system. (a)u,τ=5.3,0 with the initial values (x10, x20, x30, and x40) = 0.01,0,0,0 and (b)u,τ=5.3,0.15 with the initial values (x10, x20, x30, and x40) = 0.1,0,0,0.
(a)(b)The stability region of the equilibrium on thek1,τ plane is shown in Figure 8. By comparing Figures 5 and 8, it can be seen that k1 can also affect the critical flow velocity of the system. When u=5.5 and k1=20, the system is stable in the delay interval [0.19, 0.41] and [0.74, 0.89]. When u=5.5 and k1=35, the system is stable in the full delay interval.Figure 8
Stable regions of the equilibriumx∗ when u=5.5. The stable regions are marked by the gray shadows.It should be noted that a Hopf bifurcation would occur on the stability boundary (the red lines in Figure5). When the time delay τ 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±iω on the stability boundary of the equilibrium. Consequently, a Hopf bifurcation will arise at τ=τk. 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 τ=τk+μ,μ∈R. Then, μ=0 is the corresponding Hopf bifurcation point.Forϕ∈C=C−τ,0,R4, we assume that(25)Lμϕ=Aϕ0+Bϕ−τ,where matrices A and B are defined in (10). According to the Riesz representation theorem, there exists a bounded variation function ηθ,μ such that(26)Lμϕ=∫−τ0dηθ,μϕθforϕ∈C−τ,0⟶R4,where we choose the following expression to satisfy the equation:(27)ηθ,μ=A,θ=0,0,θ∈−τ,0,−B,θ=−τ.Forϕ∈C=C−τ,0,R4, we define(28)Sμϕθ=dϕθdθ,θ∈−τ,0,∫−τ0dηθ,μϕθθ=0,(29)Tμϕθ=0,θ∈−τ,0,Fϕ0,θ=0,then, (10) can be rewritten as follows:(30)X˙t=SμXt+TμXt,where Xtθ=Xt+θ and θ∈−τ,0.We define the corresponding adjoint operatorS∗ of the operator T(31)S∗μψs=−dψsds,s∈0,τ,∫−\tau0dηTs,μψ−s,s=0,and a bilinear form of vectors(32)ψs,ϕθ=ψ¯T0ϕ0−∫θ=−τ0∫ξ=0θψ¯Tξ−θdηθϕξdξ,where ηθ=ηθ,0.According to the discussion in the previous section, ±iω are the eigenvectors of S0 and S∗0, respectively. Next, we need to figure out the eigenvector of the operator S0 and the eigenvector of S∗0. Based on the definition of S0 and S∗0, we obtain(33)qθ=1v1v2v3Teiωθ,θ∈−τ,0.(34)q∗s=G1v1∗v2∗v3∗Teiωs,s∈0,τ,where(35)v1=−a1+a3ωi+ω2+k11e−iωτka2+a4ωi+k12e−iωτk,v2=iω,v3=iωv1,v1∗=ω3−a3ω2i−b4ω2i−a1b4i+a4b1i−a3b4ω+a4b3ω+a4k21eiωτki−b4k11eiωτki+k11ωeiωτka1b3i−a3b1i+b1ω−a3k21eiωτki+b3k11eiωτk+k21ωeiωτk,v2∗=−b1−b3ωi+k21eiωτka3b1−a1b3+b1ωi+a3k21eiωτk−b3k11eiωτk+k21ωeiωτki,v3∗=a1+k11eiωτk+ω2−a3ωia3b1−a1b3+b1ωi+a3k21eiωτk−b3k11eiωτk+k21ωeiωτki.Fromq∗s,qθ=1, we obtain G¯=1+v¯1∗v1+v¯2∗v2+v¯3∗v3+k11v¯2∗+k21v¯3∗τke−iωτk+k12v¯2∗+k22v¯3∗v1τke−iωτk−1.According to the algorithm of Hassard et al. [27], we calculate the coordinate quantity describing the central manifold C0 when μ=0. We assume that Xt is a solution of (30) and assume that(36)zt=q∗,Xt,Wt,θ=Xtθ−zq−z¯q.¯On the central manifoldC0, the following expression holds:(37)Wz,z¯,θ=W20θz22+W11θzz¯+W02θz¯22+⋯,where z and z¯ are the local coordinates of the central manifold C0 along the direction q∗ and q¯∗, respectively. Known that Xt is the solution of (30) and since μ=0, we have(38)z˙t=q∗,X˙t=iωz+q¯∗T0Fz,z¯,Equation (38) can be rewritten as follows:(39)z˙t=iωz+gz,z¯,where(40)gz,z¯=g20z22+g11zz¯+g02z¯22+g21z2z¯2+⋯.Notice thatXtθ=zqθ+z¯q¯θ+Wt,θ, we obtain(41)x1t0=z+z¯+W2010z22+W1110zz¯+W0210z¯22+⋯x2t0=zv1+z¯v¯1+W2020z22+W1120zz¯+W0220z¯22+⋯x3t0=zv2+z¯v¯2+W2030z22+W1130zz¯+W0230z¯22+⋯x4t0=zv2+z¯v¯3+W2040z22+W1140zz¯+W0240z¯22+⋯.According to the literature [27], the critical properties of bifurcation periodic solutions are determined by the following parameters:(42)c1kc,τk=i2ωg20g11−2g112−g0223+g212,μ2=−Rec1τkReλ′τk,β2=2Rec1τk,T2=−Imc1τk+μ2Imλ′τkω.Obviously, we need to calculate the expressions ofg20, g11, g02, and g21. Substituting FX and (41) into (38) and comparing the same order of terms, we obtain the expressions of gij′s as follows:(43)g20=g11=g02=0,g21=2G¯−k2φ1ξb3φ12ξbφ2ξbv¯1+2v2+3φ1ξbφ22ξbv12+2v1v¯1+3φ13ξb3φ23ξbv12v¯1v¯2∗−k2φ2ξb3φ12ξbφ2ξbv¯1+2v2+3φ1ξbφ22ξbv12+2v1v¯1+3φ13ξb+3φ23ξbv12v¯1v¯3∗.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 μ2 and β2 in (42). If μ2>0μ2<0), the Hopf bifurcation is supercritical (subcritical). Moreover, if β2<0β2>0, the bifurcation period solution is stable (unstable). The value of T2 determines the period of the bifurcated periodic solution, and if T2>0T2<0, the period increases (decreases).In order to verify the results of the theorem, taking the velocityu=5.6 as an example, the positive real roots of (20) and (21) can be calculated as ω1≈11.377791,ω2≈12.997710 and the corresponding critical delays τ1k,τ2k can be calculated as follows:(44)τ10≈0.213505,τ11≈0.765738⋯τ20≈0.397255,τ21≈0.880662⋯.As shown in Figure5, when the time delay increases to the critical value τ10 in the case of u=5.6, which is marked as the point H1, the equilibrium gains its stability. The corresponding parameter values in (42) are calculated as μ2≈−5.7, β2≈−127.7, and T2≈−1.6. According to Theorem 3.1, the Hopf bifurcation is subcritical, and the bifurcating periodic solution for τ<τ10 is stable. We set k2=100, the other parameters are the same as in Figure 5, and we obtain the corresponding bifurcation diagram near the Hopf bifurcation point H1, 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 r≈δ−Reλ′τkτ−τk/Rec1τk. Here, =1/2π∫02πeiθ+e−iθ2+eiθv1+e−iθv¯12+eiθv2+e−iθv¯22+eiθv3+e−iθv¯32dθ . When the delay exceeds the critical value τ20, which is marked as the point H2, the equilibrium loses its stability. The corresponding parameter values in (42) are calculated as μ2≈3.6, β2≈−67.3, and T2≈−2.8. According to Theorem 3.1, the Hopf bifurcation is supercritical and the bifurcating periodic solution for τ>τ20 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).Figure 9
Bifurcation diagram near the Hopf bifurcation pointH1 in Figure 6. The Y-axis r represents the average radius of the periodic orbit, i.e., r=x12+x22+x32+x42.Figure 10
The phase diagram and response diagram near the generic Hopf bifurcation pointH2 are shown in Figure 6. (a)u,τ=5.6,0.38 with the initial values (x10, x20, x30, and x40) = 0.1,0,0,0, (b) the phase diagram corresponding to (a), (c)u,τ=5.6,0.42 with the initial values (x10, x20, x30, and x40) =0.001,0,0,0, and (d) the phase diagram corresponding to (c).
(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.



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