# Nonlinear Dynamics of Cross-Flow Tubes Subjected to Initial Axial Load and Distributed Impacting Constraints

Authors: Mengdi Liu; Yikun Wang; Tao Qin; Jing Zhao; Yujin Du
Journal: Shock and Vibration
 (2021)

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## Abstract

The dynamics of cross-flow tubes were studied in consideration of initial axial load and distributed impacting constraints, modeled as cubic and trilinear spring constraints. The tubes were modeled as Euler–Bernoulli beams and supported at both ends, including the simply supported tube and clamped-clamped tube. The analytical model involves a time-delayed displacement term induced by the cross flow based on the quasi-steady theory. For simplicity, a single flexible supported beam in a rigid square array of cylinders was studied by using the damping-controlled mechanism. The mean extension of the tube was considered, and thus, it added another nonlinear term in the equation of motion. Results show that the tube loses stability by buckling and fluttering at various initial pressure loads and cross-flow velocities. An increase was observed for critical velocities and initial pressure loads. Chaotic oscillations were observed for the trilinear spring model. The distribution of the impacting forces was also calculated. Some of the fresh results obtained in the impact system are expected to be helpful in understanding and controlling the dynamic responses of fluid-conveying pipes.


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## Body

## 1. Introduction

Cross-flow heat exchanger tubes are found in many power generating industries, such as in steam generators, boilers, and nuclear reactors [1–4]. For the past 50 years, the cross-flow-induced dynamics of tube arrays has become an important topic worldwide. Studies on heat exchanger tube failures related to flow induced vibration began in the 1950s and have greatly progressed since the 1970s [5–7]. Tube arrays exhibit fluid elastic instabilities at sufficiently high cross-flow velocities. These instabilities are related to negative fluid damping caused by the flow-induced dynamic forces acting on the tubes. With negative damping, the surrounding fluid on the tubes exerts a positive input of work [3, 8–10]. These flow-induced instabilities may severely damage the tubes. For example, the resultant large displacement may cause the tube to impact onto the loose supports and always wear with loose supports [11]; the axial pressure from manufacturing may cause the tube to lose stability either by fluttering or buckling [12–14].Many studies have focused on the dynamics of cross-flow induced vibrations of heat exchanger tubes by using single degree of freedom models and complicated continuous models of flexible beams. Price [1] classified the fluid-elastic instability models into seven groups: “jet-switch,” “quasi-static,” “inviscid flow,” “quasi-steady,” “semianalytical,” “unsteady,” and “computational fluid dynamic.” In [15] and [16], the fluid force coefficients for a tube array and critical cross-flow velocities were determined through experiment observations and numerical results. Rzentkowski and Lever [17] conducted the stability analysis of tube arrays subjected to turbulence. Results showed that a nonlinear flexible tube moves in the transverse direction perpendicular to the direction of the cross flow. In their study, the other tubes surrounding the flexible tube were assumed to be rigid. The results showed that cross-flow turbulence may reduce the fluid-elastic instability. The displacement of the tube may become large when the cross-flow velocity increases after the onset of fluid-elastic instability. In this case, the nonlinear effects become important to some extent.Weaver et al. [18] discussed nonlinearity as an inherent property of the system, such as heat exchangers subjected to loose supports. Many studies reported such type of strong nonlinearities in cross-flow-induced vibration problems. Païdoussis and Li [19] discussed theoretically the chaotic dynamics of heat exchanger tubes affecting the loose baffle plates. The analytical model of the cross-flow induced force involves a time delay term. Chen et al. [20] performed a series of experiments to obtain the relation between tube displacements and cross-flow velocity for different impacting clearances. The experimental results were in good agreement with an analytical model based on the unsteady flow theory. Theoretical analysis was conducted on the fluid-elastic instability of nonlinearly supported tubes via the unsteady flow theory with a bilinear mathematical impacting model [21–25]. Wang and Ni [26] explored the Hopf bifurcation and chaotic vibrations of a tubular cantilever tube impacting on loose support. A new time-domain model for cross-flow induced forces of tubes with loose supports was recently formulated to study the flow-induced vibration and wear in tube arrays [27–29]. Xia and Wang [30] considered the nonlinearity that is either induced by loose support or associated with mean axial extension of the tubes. For supported tubes with loose support, the effect of mean axial extension could be neglected in the presence of loose supports and should be added in the absence of loose supports. Wang et al. [31] investigated the nonsmooth impacting oscillations of a two-segment rigid pipe conveying fluid subjected to a one-sided rigid stop. The structure parameters of the articulated pipe and the rigid stop were discussed via bifurcation diagrams, where periodic and chaotic vibro-impact oscillations were observed. The possibility of stick-slip motion relating to the nonsmooth characteristics was detected.Wang et al. [14] studied the instability and postinstability behavior of cross-flow tube arrays subjected to initial axial load. The initial load is assumed to be a constant force imposed externally and induced by temperature changes or an interference assembly stress. The results show that, with increasing initial axial load, the tube may still exhibit a symmetric limit cycle motion even beyond the critical cross-flow velocity. Huang et al. [13] examined the instability of mistuned tube arrays considering axial loads and obtained regions of stability. Results indicate that the axial load substantially changes the stability of the cross-flow tube array system with fluid coupling.The aforementioned investigations related to tube arrays subjected to cross flow have focused on only one or several loose supports on the tube. In heat exchangers, the tubes are multispanned and threaded through the baffle plates through holes [11, 27, 32–34]. The gap between the tube and the hole could be modeled as loose supports to permit thermal expansion. For single-span heat exchangers, the cross-flow induced vibration of the flexible tube would interact with the support walls, or in other words, the distributed constraints, under the assumption that the tubes surrounding the flexible tube are rigid and can be seen as support walls. Ni et al. [35] and Wang et al. [36] studied the nonlinear dynamics of a cantilevered and simply supported pipe conveying fluid subjected to distributed impacting constraints, respectively. They assumed that the impact would occur anywhere along the pipe. These studies are the motivation of the present work.In this paper, the nonlinear dynamics of cross-flow tubes subjected to initial axial load and distributed impacting constraints was analysed by considering the effect of axial extension of the tubes. The force of the distributed impact constraints was modeled as either a cubic spring or a smoothed trilinear spring, and the results were in good agreement with experiments [37]. The supported tube of hinged-hinged and clamped-clamped boundary conditions was also discussed. This work is the first exploration on such a system. The analytical model was validated by degenerating the problem to cross-flow tubes subjected to a single loose support, and the results were compared with those of Xia and Wang [30].


## 2. Model Description

The model is a single flexible tube in an otherwise rigid array subjected to initial load and cross-flow normal to the tube axis, as shown in Figure1. The distributed impacting constraints were model by antivibration bars (AVBs) positioned along the tube axis. For simplicity, the motion of the tube was assumed to be planar, and the tube vibrates in the transverse direction perpendicular to the cross-flow velocity.Figure 1
Schematic of the cross-flow system. (a) Top view of an array of tubes; (b) isometric view of the simply supported tube; (c) isometric view of the clamped-clamped tube; (d) side view of the simply supported tube; (e) side view of the clamped-clamped tube.The system consists of a cylinder of lengthL, outer diameter D, cross-sectional area A, mass per unit length m, flexural rigidity EI, initial axial load P0, and damping coefficient c. The fluid is of density ρ with flow velocity U. In consideration of the initial axial load, the distributed impacting constraints, and the effect of axial extension, the modified equation of motion of the supported flexible cross-flow tube is given as follows:(1)EI∂4w∂x4+c∂w∂t+m∂2w∂t2−EA2L∂2w∂x2∫0L∂w∂x2dx+P0∂2w∂x2+f˜w=Fw,w˙,w¨,where w is the lateral displacement of the tube, fw is the nonlinear force of the distributed impacting constraints, with the Dirac delta function neglected compared to Xia and Wang [30], and Fw,w˙,w¨ is the cross-flow induced force on the tube, which could be obtained according to the quasi-steady theory [19]:(2)Fw,w˙,w¨=C1∂2wx,t∂t2+C2∂wx,t∂t+C3wx,t−Δt,where(3)C1=−π4ρD2Cma,C2=−12ρUDCD,C3=12ρU2D∂CL∂y,Δt=μDU,where CD and CL are the drag and lift coefficients, respectively, based on the flow velocity in the gap between the tubes, Cma is the added mass coefficient of the fluid surrounding the tube, and Δt is the time delay between tube motion and the simultaneously generated fluid dynamic forces.The dimensionless eigenvalueλ1 of the first mode was introduced to the hinged-hinged beam and clamped-clamped beam, and the following dimensionless quantities were obtained:(4)η=wD,ξ=xL,τ=λ12EImL4t=Ω1t,γ=AD22Iλ14,p0=P0L2EI,m˜=mρD2,ς=cΩ1m,β=11+4m˜/πCma,u=2πUDΩ1,Δτ=2πu,fη=1mΩ12f˜w.The dimensionless equation of motion may be obtained by substituting the dimensionless quantities into equations (1) and (2):(5)11−β∂2η∂τ2+ς+CDu4πm˜∂η∂τ+1λ14∂4η∂ξ4+p01λ14∂2η∂ξ2−u28π2m˜∂CL∂yηξ,τ−Δτ−γ∂2η∂ξ2∫01∂η∂ξ2dξ+fη=0.The dimensionless boundary conditions for two types of supported tubes are shown as follows:For the hinged-hinged tube,(6)η0=η1=0,η″0=η″1=0.For the clamped-clamped tube,(7)η0=η1=0,η′0=η′1=0.The nonlinear impacting force of the distributed impacting constraints can be represented by various mathematical models. In this work, two typical models, namely, a cubic spring model and a modified trilinear spring model, were introduced to approximate the impacting forces [37]. The dimensionless formulation of fη for the two models is listed below:For the cubic spring model,(8)fη=κcη3.For the modified trilinear spring model,(9)fη=κtη−12η+d−η−d3.In the above two equations,κc and κt are the dimensionless stiffness of the spring model and d is the gap between the tube and AVBs. With regard to the difference in the dimensionless quantities, the values of the parameters of the two spring models may be different from those in [37]. Instead, these values of the constraints were adopted from those in [30], i.e., κc=κt=1000 and d=0.5.The governing equation was discretized via Galerkin expansion and modal truncation techniques with the eigenfunctionsφjξ of hinged-hinged and clamped-clamped beams and corresponding generalized coordinates qjτ. Thus,(10)ηξ,τ=∑j=1Nφjξqjτ,where N is the mode number used in numerical calculation. Substituting equation (9) into equation (4), multiplying by φiξ, and integrating from 0 to 1 lead to(11)mijq¨j+cijq˙j+kijqj+kijhqjτ−Δτ+γijklqjqkql+fi=0,in which(12)mij=11−β∫01φiφjdξ=11−βδij,cij=δiνiπ+CDu4πm˜∫01φiφjdξ=δiνiπ+CDu4πm˜δij,kij=1λ14∫01φiφj⁗dξ+p0∫01φiφj″dξ=1λ14λi4δij+p0∫01φiφjdξ,kijh=−u28π2m˜∂CL∂η∫01φiφjdξ=−u28π2m˜∂CL∂ηδij,γijkl=−γ∫φiφj″∫01φk′φl′dξdξ,fi=∫01φifηξ,τdξ.Equation (11) was solved by using fourth-order Runge–Kutta integration algorithm with variable time step. The initial conditions were qi0=0.001 and q˙i0=0. The results were obtained with reference to the midpoint (ξ=0.5) on the tube.Current experiments related to cross-flow tubes subjected to loose support were performed via five-mode truncation, which shows good convergence and high accuracy. The distributed impacting model was degenerated to a small segment impacting model to simulate loose support at one point on the tube. The proposed distributed impacting model was validated via the degenerate system. The results of the degenerate model were compared with those of the foregoing study [36]. Ni et al. [35] performed an exploration on a similar impacting problem by focusing on the nonlinear dynamics of a cantilevered pipe conveying fluid subjected to distributed motion constraints. Strong nonlinearity was observed all over the tube for the impacting problem. The nonlinear equation of motion was discretized via a five-mode expansion and showed good accuracy compared with existing works. In the present paper, the mode number N=5 was adopted in the system analysis. The flowchart of the numerical scheme is plotted in Figure 2.Figure 2
Flowchart of the numerical calculation process.


## 3. Results and Discussion

### 3.1. Degenerate System

In this section, the distributed impacting constraint was degenerated to a single loose support on the tube to validate the model application in studying the impacting system. A small segment of the tube was selected, and loose support was imposed on this segment. The length of the impacting segment was set at the microscale, sayδL=0.001, to allow for a good approximation of a loose support at a certain point on the tube. The parameters of the degenerate model were the same as those used by Xia and Wang [30]. The bifurcation diagram for the system with cubic spring loose support at γ=0.015 is presented in Figure 3. A Hopf bifurcation occurred at approximately uH=1.78, leading to periodic oscillations. The obtained results were in agreement with those obtained by Xia and Wang [30], thus verifying the effectiveness of the distributed impacting model.Figure 3
Bifurcation diagram for the degenerate system with the cubic spring impacting model.


### 3.2. Hinged-Hinged Tube with Impacting Constraints

The dynamics of the hinged-hinged tube subjected to axial loads and distributed cubic spring impacting model and trilinear spring impacting model was discussed. In this work, a positivep0 leads to a compressive force imposed on the tube and a negative p0 leads to a tensile force imposed on the tube. Wang et al. [14] stated that, for sufficiently large initial pressure load, the destabilizing force may overcome the flexural force and lead to divergence (buckling) instability. Therefore, the tube may exhibit either buckling or fluttering instabilities in the system under cross flow. Any part of the tube may be in contact with the distributed impacting constraints for the vibrations induced by the cross flow [35]. When the amplitude of the lateral displacement of the tube satisfies the impacting condition, reactive forces will be exerted on the tubes. From this point of view, the tube may behave similar to impacting.Bifurcation diagrams for the two systems with the cubic impacting model and modified trilinear impacting model under various initial loads are calculated and listed in Figures4 and 5, respectively.Figure 4
Bifurcation diagrams for the hinged-hinged tube subjected to the distributed cubic impacting model at various pressure loads. (a)p0 = −15. (b) p0 = −9. (c) p0 = 9. (d) p0 = 15.
(a)(b)(c)(d)Figure 5
Bifurcation diagrams for the hinged-hinged tube subjected to the distributed trilinear impacting model at various pressure loads. (a)p0 = −15. (b) p0 = −9. (c) p0 = 9. (d) p0 = 15.
(a)(b)(c)(d)


#### 3.2.1. Subjected to Cubic Impacting Constraint

For the cubic spring impacting model, the tube lost stability by flutter atuH=2.88, 2.52, and 0.64 for p0=−15, −9, and 9, respectively, as shown in Figures 4(a)–4(c). A negative p0 means a tensile force on the tube that relatively enlarges the flexural force; hence, the critical velocity is larger than that with a positive value of p0. If the pressure load is sufficiently large, the flexural force may be overcome by the pressure load. Then, the tube would lose stability either by buckling at low cross-flow velocities or by fluttering at high flow velocities as depicted in Figure 4(d). With increase in cross-flow velocity, period-3 oscillation occurs for three different initial pressure loads, 6.26<u<7.28 for p0=−15, 6.10<u<7.14 for p0=−9, 5.58<u<6.68 for p0=9, and 5.40<u<6.52 for p0=15, respectively. Once the cross-flow velocity exceeds the upper bound of the velocity range for period-3, the motion oscillation jumps from period-3 to quasi-periodic for negative pressure loads or period-1 motion for positive pressure loads. Afterward, the system with the cubic spring impacting model consistently undergoes period-1 motion for the considered velocities. Time histories, phase portraits, and power spectral density diagrams are presented in Figures 6 and 7 for some values of p0 and u. Vibration modes of the system can be seen from these numerical results.Figure 6
Time histories and phase portraits for several particular values ofp0 and u. (a) Buckling p0 = 15 and u = 0.5; (b) period-1 motion p0 = 9 and u = 4; (c) period-3 motion p0 = −9 and u = 6.8; (d) quasi-periodic motion p0 = −15 and u = 7.6.
(a)(b)(c)(d)Figure 7
Power spectral density (PSD) diagram for (a) period-1p0 = 9 and u = 4; (b) quasi-periodic motion p0 = −15 and u = 7.6.
(a)(b)


#### 3.2.2. Subjected to Trilinear Impacting Constraints

For the trilinear spring impacting model, the system shows the same critical cross-flow velocities in Section3.2.1, as shown in Figures 6(a)–6(c). Some differences were noted when comparing the two types of impacting models. Figures 6(a)–6(c) show that the amplitude of the tube increases quickly when the cross-flow velocity is just beyond the critical velocity, signifying the contact between the tube and impacting constraints. For negative pressure loads, i.e., tensile loads, the tube always exhibits periodic oscillation after Hopf bifurcation, such as the period-1, period-3, and period-4 motions in Figures 6(a) and 6(b). For positive pressure loads, i.e., compressive loads, the tube may behave periodic, quasi-periodic, and chaotic oscillations, as shown in Figure 6(c). For sufficiently large values of initial pressure load, the tube loses stability by buckling at low cross-flow velocities. For a small range of flow velocity, the tube vibrates periodically on one side of the constraints. This vibration leads to chaotic oscillations as presented in Figure 6(d). Sample results of time histories, phase portraits, and power spectral density diagrams reflecting several vibration modes for different initial loads and cross-flow velocities are presented in Figures 8 and 9.Figure 8
Time histories and phase portraits for several particular values ofp0 and u. (a) Buckling p0 = 15 and u = 0.5; (b) period-1 motion p0 = −15 and u = 3.2; (c) period-4 motion p0 = −9 and u = 4.6; (d) chaotic motion p0 = 15 and u = 3.5.
(a)(b)(c)(d)Figure 9
Power spectral density (PSD) diagram for (a) period-4p0 = −9 and u = 4.6; (b) chaotic motion p0 = 15 and u = 3.5.
(a)(b)The lateral displacement of the midpoint on the tube exceeds the gap between the tube and constraints. The gap isd=0.5, and the maximum lateral displacement is approximately 0.6 to 0.8 due to the flexibility of the impacting constraints. The power of the formulation of the trilinear spring model used by Xia and Wang [30] is 1, indicating that the trilinear stiffness is continuous but not analytical at the moment of impact. In the current paper, the power is 3, reflecting that the stiffness is continuous, analytical, and soft at the moment of impact. This phenomenon explains why the lateral displacement is larger than the gap.


### 3.3. Clamped-Clamped Tube with Impacting Constraints

The dynamics of the simply supported tube subjected to various initial pressure loads and distributed impacting constraints was previously studied. In this section, the same load cases were exerted onto the clamped-clamped tube under the cross flow. Numerical results were calculated, and bifurcation diagrams are shown in Figures10 and 11 for the cubic spring impacting model and modified trilinear impacting model, respectively. Similar responses of the clamped-clamped system were compared with those of the simply supported system. The critical velocity decreases with the increase in initial pressure load. Tensile force stabilizes the system, whereas compressive force destabilizes the system. The numerical values of the critical velocity for cubic and trilinear impacting models are uH=2.14, 2.02, 1.64, and 1.48 for p0=−15, −9, 9, and 15, respectively. Different from Figures 4 and 5, no buckling instability for the parameters is observed for the clamped-clamped system.Figure 10
Bifurcation diagrams for the clamped-clamped tube subjected to the distributed cubic impacting model at various pressure loads. (a)p0 = −15. (b) p0 = −9. (c) p0 = 9. (d) p0 = 15.
(a)(b)(c)(d)Figure 11
Bifurcation diagrams for the clamped-clamped tube subjected to the distributed trilinear impacting model at various pressure loads. (a)p0 = −15. (b) p0 = −9. (c) p0 = 9. (d) p0 = 15.
(a)(b)(c)(d)The responses of the clamped-clamped system subjected to cubic impacting constraints are quite similar to those of the simply supported system. Period-1 motion occurs first, then period-3 motion, followed by small lateral amplitude of quasi-periodic motion at a limited range of cross-flow velocity, and finally, the system goes back to period-1 motion. The responses of the clamped-clamped system subjected to trilinear impacting constraints are different from those of the simply supported system. Even for negative pressure loads, i.e., tensile stress, the system exhibits chaotic oscillations. Multiperiod oscillations and quasi-period occur at a wide range of cross-flow velocities. No buckling instability is observed.Sample results of time histories, phase portraits, and power spectral density diagrams are omitted in this section. The interested reader can get these responses easily referring to Section3.2.


### 3.4. Vibration Shapes and Impacting Forces

Vibration shapes of the tube and impacting forces induced by the interaction between the tube and impacting constraints are presented in this section. Typical results of the envelope of impacting forces and vibration shapes for the simply supported tube are shown in Figures12 and 13. The case of the clamped-clamped tube could be easily calculated via the same strategy. Substantial differences are discovered between the cubic impacting model and trilinear impacting model. For the cubic impacting model, impacting force occurs once the lateral displacement is not zero, and the force is distributed all along the tube axis, as shown in Figures 12(a) and 13(a). For the trilinear impacting model, the impacting force becomes zero when the lateral displacement is smaller than the gap between the tube and constraints. Effective impacting force occurs once the lateral displacement exceeds the gap, i.e., the tube impacts the constraints as depicted by Figures 12(b) and 13(b). From this point of view, the modified trilinear impacting model is better than the cubic impacting model as a mathematical approximation to simulate impacting behaviors.Figure 12
Distribution of impacting force and vibration shape for period-1 motion of the simply supported tube subjected to the two kinds of impacting models. (a) Cubic modelp0 = 9 and u = 4. (b) Trilinear model p0 = −15 and u = 3.2.
(a)(b)Figure 13
Distribution of impacting force and vibration shape for period-4 and quasi-periodic motion of the simply supported tube subjected to the two kinds of impacting models. (a) Cubic modelp0 = −15 and u = 7.6. (b) Trilinear model p0 = −9 and u = 4.6.
(a)(b)


## 3.1. Degenerate System

In this section, the distributed impacting constraint was degenerated to a single loose support on the tube to validate the model application in studying the impacting system. A small segment of the tube was selected, and loose support was imposed on this segment. The length of the impacting segment was set at the microscale, sayδL=0.001, to allow for a good approximation of a loose support at a certain point on the tube. The parameters of the degenerate model were the same as those used by Xia and Wang [30]. The bifurcation diagram for the system with cubic spring loose support at γ=0.015 is presented in Figure 3. A Hopf bifurcation occurred at approximately uH=1.78, leading to periodic oscillations. The obtained results were in agreement with those obtained by Xia and Wang [30], thus verifying the effectiveness of the distributed impacting model.Figure 3
Bifurcation diagram for the degenerate system with the cubic spring impacting model.


## 3.2. Hinged-Hinged Tube with Impacting Constraints

The dynamics of the hinged-hinged tube subjected to axial loads and distributed cubic spring impacting model and trilinear spring impacting model was discussed. In this work, a positivep0 leads to a compressive force imposed on the tube and a negative p0 leads to a tensile force imposed on the tube. Wang et al. [14] stated that, for sufficiently large initial pressure load, the destabilizing force may overcome the flexural force and lead to divergence (buckling) instability. Therefore, the tube may exhibit either buckling or fluttering instabilities in the system under cross flow. Any part of the tube may be in contact with the distributed impacting constraints for the vibrations induced by the cross flow [35]. When the amplitude of the lateral displacement of the tube satisfies the impacting condition, reactive forces will be exerted on the tubes. From this point of view, the tube may behave similar to impacting.Bifurcation diagrams for the two systems with the cubic impacting model and modified trilinear impacting model under various initial loads are calculated and listed in Figures4 and 5, respectively.Figure 4
Bifurcation diagrams for the hinged-hinged tube subjected to the distributed cubic impacting model at various pressure loads. (a)p0 = −15. (b) p0 = −9. (c) p0 = 9. (d) p0 = 15.
(a)(b)(c)(d)Figure 5
Bifurcation diagrams for the hinged-hinged tube subjected to the distributed trilinear impacting model at various pressure loads. (a)p0 = −15. (b) p0 = −9. (c) p0 = 9. (d) p0 = 15.
(a)(b)(c)(d)


### 3.2.1. Subjected to Cubic Impacting Constraint

For the cubic spring impacting model, the tube lost stability by flutter atuH=2.88, 2.52, and 0.64 for p0=−15, −9, and 9, respectively, as shown in Figures 4(a)–4(c). A negative p0 means a tensile force on the tube that relatively enlarges the flexural force; hence, the critical velocity is larger than that with a positive value of p0. If the pressure load is sufficiently large, the flexural force may be overcome by the pressure load. Then, the tube would lose stability either by buckling at low cross-flow velocities or by fluttering at high flow velocities as depicted in Figure 4(d). With increase in cross-flow velocity, period-3 oscillation occurs for three different initial pressure loads, 6.26<u<7.28 for p0=−15, 6.10<u<7.14 for p0=−9, 5.58<u<6.68 for p0=9, and 5.40<u<6.52 for p0=15, respectively. Once the cross-flow velocity exceeds the upper bound of the velocity range for period-3, the motion oscillation jumps from period-3 to quasi-periodic for negative pressure loads or period-1 motion for positive pressure loads. Afterward, the system with the cubic spring impacting model consistently undergoes period-1 motion for the considered velocities. Time histories, phase portraits, and power spectral density diagrams are presented in Figures 6 and 7 for some values of p0 and u. Vibration modes of the system can be seen from these numerical results.Figure 6
Time histories and phase portraits for several particular values ofp0 and u. (a) Buckling p0 = 15 and u = 0.5; (b) period-1 motion p0 = 9 and u = 4; (c) period-3 motion p0 = −9 and u = 6.8; (d) quasi-periodic motion p0 = −15 and u = 7.6.
(a)(b)(c)(d)Figure 7
Power spectral density (PSD) diagram for (a) period-1p0 = 9 and u = 4; (b) quasi-periodic motion p0 = −15 and u = 7.6.
(a)(b)


### 3.2.2. Subjected to Trilinear Impacting Constraints

For the trilinear spring impacting model, the system shows the same critical cross-flow velocities in Section3.2.1, as shown in Figures 6(a)–6(c). Some differences were noted when comparing the two types of impacting models. Figures 6(a)–6(c) show that the amplitude of the tube increases quickly when the cross-flow velocity is just beyond the critical velocity, signifying the contact between the tube and impacting constraints. For negative pressure loads, i.e., tensile loads, the tube always exhibits periodic oscillation after Hopf bifurcation, such as the period-1, period-3, and period-4 motions in Figures 6(a) and 6(b). For positive pressure loads, i.e., compressive loads, the tube may behave periodic, quasi-periodic, and chaotic oscillations, as shown in Figure 6(c). For sufficiently large values of initial pressure load, the tube loses stability by buckling at low cross-flow velocities. For a small range of flow velocity, the tube vibrates periodically on one side of the constraints. This vibration leads to chaotic oscillations as presented in Figure 6(d). Sample results of time histories, phase portraits, and power spectral density diagrams reflecting several vibration modes for different initial loads and cross-flow velocities are presented in Figures 8 and 9.Figure 8
Time histories and phase portraits for several particular values ofp0 and u. (a) Buckling p0 = 15 and u = 0.5; (b) period-1 motion p0 = −15 and u = 3.2; (c) period-4 motion p0 = −9 and u = 4.6; (d) chaotic motion p0 = 15 and u = 3.5.
(a)(b)(c)(d)Figure 9
Power spectral density (PSD) diagram for (a) period-4p0 = −9 and u = 4.6; (b) chaotic motion p0 = 15 and u = 3.5.
(a)(b)The lateral displacement of the midpoint on the tube exceeds the gap between the tube and constraints. The gap isd=0.5, and the maximum lateral displacement is approximately 0.6 to 0.8 due to the flexibility of the impacting constraints. The power of the formulation of the trilinear spring model used by Xia and Wang [30] is 1, indicating that the trilinear stiffness is continuous but not analytical at the moment of impact. In the current paper, the power is 3, reflecting that the stiffness is continuous, analytical, and soft at the moment of impact. This phenomenon explains why the lateral displacement is larger than the gap.


## 3.2.1. Subjected to Cubic Impacting Constraint

For the cubic spring impacting model, the tube lost stability by flutter atuH=2.88, 2.52, and 0.64 for p0=−15, −9, and 9, respectively, as shown in Figures 4(a)–4(c). A negative p0 means a tensile force on the tube that relatively enlarges the flexural force; hence, the critical velocity is larger than that with a positive value of p0. If the pressure load is sufficiently large, the flexural force may be overcome by the pressure load. Then, the tube would lose stability either by buckling at low cross-flow velocities or by fluttering at high flow velocities as depicted in Figure 4(d). With increase in cross-flow velocity, period-3 oscillation occurs for three different initial pressure loads, 6.26<u<7.28 for p0=−15, 6.10<u<7.14 for p0=−9, 5.58<u<6.68 for p0=9, and 5.40<u<6.52 for p0=15, respectively. Once the cross-flow velocity exceeds the upper bound of the velocity range for period-3, the motion oscillation jumps from period-3 to quasi-periodic for negative pressure loads or period-1 motion for positive pressure loads. Afterward, the system with the cubic spring impacting model consistently undergoes period-1 motion for the considered velocities. Time histories, phase portraits, and power spectral density diagrams are presented in Figures 6 and 7 for some values of p0 and u. Vibration modes of the system can be seen from these numerical results.Figure 6
Time histories and phase portraits for several particular values ofp0 and u. (a) Buckling p0 = 15 and u = 0.5; (b) period-1 motion p0 = 9 and u = 4; (c) period-3 motion p0 = −9 and u = 6.8; (d) quasi-periodic motion p0 = −15 and u = 7.6.
(a)(b)(c)(d)Figure 7
Power spectral density (PSD) diagram for (a) period-1p0 = 9 and u = 4; (b) quasi-periodic motion p0 = −15 and u = 7.6.
(a)(b)


## 3.2.2. Subjected to Trilinear Impacting Constraints

For the trilinear spring impacting model, the system shows the same critical cross-flow velocities in Section3.2.1, as shown in Figures 6(a)–6(c). Some differences were noted when comparing the two types of impacting models. Figures 6(a)–6(c) show that the amplitude of the tube increases quickly when the cross-flow velocity is just beyond the critical velocity, signifying the contact between the tube and impacting constraints. For negative pressure loads, i.e., tensile loads, the tube always exhibits periodic oscillation after Hopf bifurcation, such as the period-1, period-3, and period-4 motions in Figures 6(a) and 6(b). For positive pressure loads, i.e., compressive loads, the tube may behave periodic, quasi-periodic, and chaotic oscillations, as shown in Figure 6(c). For sufficiently large values of initial pressure load, the tube loses stability by buckling at low cross-flow velocities. For a small range of flow velocity, the tube vibrates periodically on one side of the constraints. This vibration leads to chaotic oscillations as presented in Figure 6(d). Sample results of time histories, phase portraits, and power spectral density diagrams reflecting several vibration modes for different initial loads and cross-flow velocities are presented in Figures 8 and 9.Figure 8
Time histories and phase portraits for several particular values ofp0 and u. (a) Buckling p0 = 15 and u = 0.5; (b) period-1 motion p0 = −15 and u = 3.2; (c) period-4 motion p0 = −9 and u = 4.6; (d) chaotic motion p0 = 15 and u = 3.5.
(a)(b)(c)(d)Figure 9
Power spectral density (PSD) diagram for (a) period-4p0 = −9 and u = 4.6; (b) chaotic motion p0 = 15 and u = 3.5.
(a)(b)The lateral displacement of the midpoint on the tube exceeds the gap between the tube and constraints. The gap isd=0.5, and the maximum lateral displacement is approximately 0.6 to 0.8 due to the flexibility of the impacting constraints. The power of the formulation of the trilinear spring model used by Xia and Wang [30] is 1, indicating that the trilinear stiffness is continuous but not analytical at the moment of impact. In the current paper, the power is 3, reflecting that the stiffness is continuous, analytical, and soft at the moment of impact. This phenomenon explains why the lateral displacement is larger than the gap.


## 3.3. Clamped-Clamped Tube with Impacting Constraints

The dynamics of the simply supported tube subjected to various initial pressure loads and distributed impacting constraints was previously studied. In this section, the same load cases were exerted onto the clamped-clamped tube under the cross flow. Numerical results were calculated, and bifurcation diagrams are shown in Figures10 and 11 for the cubic spring impacting model and modified trilinear impacting model, respectively. Similar responses of the clamped-clamped system were compared with those of the simply supported system. The critical velocity decreases with the increase in initial pressure load. Tensile force stabilizes the system, whereas compressive force destabilizes the system. The numerical values of the critical velocity for cubic and trilinear impacting models are uH=2.14, 2.02, 1.64, and 1.48 for p0=−15, −9, 9, and 15, respectively. Different from Figures 4 and 5, no buckling instability for the parameters is observed for the clamped-clamped system.Figure 10
Bifurcation diagrams for the clamped-clamped tube subjected to the distributed cubic impacting model at various pressure loads. (a)p0 = −15. (b) p0 = −9. (c) p0 = 9. (d) p0 = 15.
(a)(b)(c)(d)Figure 11
Bifurcation diagrams for the clamped-clamped tube subjected to the distributed trilinear impacting model at various pressure loads. (a)p0 = −15. (b) p0 = −9. (c) p0 = 9. (d) p0 = 15.
(a)(b)(c)(d)The responses of the clamped-clamped system subjected to cubic impacting constraints are quite similar to those of the simply supported system. Period-1 motion occurs first, then period-3 motion, followed by small lateral amplitude of quasi-periodic motion at a limited range of cross-flow velocity, and finally, the system goes back to period-1 motion. The responses of the clamped-clamped system subjected to trilinear impacting constraints are different from those of the simply supported system. Even for negative pressure loads, i.e., tensile stress, the system exhibits chaotic oscillations. Multiperiod oscillations and quasi-period occur at a wide range of cross-flow velocities. No buckling instability is observed.Sample results of time histories, phase portraits, and power spectral density diagrams are omitted in this section. The interested reader can get these responses easily referring to Section3.2.


## 3.4. Vibration Shapes and Impacting Forces

Vibration shapes of the tube and impacting forces induced by the interaction between the tube and impacting constraints are presented in this section. Typical results of the envelope of impacting forces and vibration shapes for the simply supported tube are shown in Figures12 and 13. The case of the clamped-clamped tube could be easily calculated via the same strategy. Substantial differences are discovered between the cubic impacting model and trilinear impacting model. For the cubic impacting model, impacting force occurs once the lateral displacement is not zero, and the force is distributed all along the tube axis, as shown in Figures 12(a) and 13(a). For the trilinear impacting model, the impacting force becomes zero when the lateral displacement is smaller than the gap between the tube and constraints. Effective impacting force occurs once the lateral displacement exceeds the gap, i.e., the tube impacts the constraints as depicted by Figures 12(b) and 13(b). From this point of view, the modified trilinear impacting model is better than the cubic impacting model as a mathematical approximation to simulate impacting behaviors.Figure 12
Distribution of impacting force and vibration shape for period-1 motion of the simply supported tube subjected to the two kinds of impacting models. (a) Cubic modelp0 = 9 and u = 4. (b) Trilinear model p0 = −15 and u = 3.2.
(a)(b)Figure 13
Distribution of impacting force and vibration shape for period-4 and quasi-periodic motion of the simply supported tube subjected to the two kinds of impacting models. (a) Cubic modelp0 = −15 and u = 7.6. (b) Trilinear model p0 = −9 and u = 4.6.
(a)(b)


## 4. Conclusions

The impacting dynamics of cross-flow supported tubes (simply supported tube and clamped-clamped tube) subjected to initial axial load and distributed impacting constraints was first explored. This distributed impacting constraint may be viewed as the interactions induced by the surrounding tubes. The impacting forces are modeled by two types of springs, namely, the cubic spring model and the modified trilinear spring model, which have been widely used in many articles. Forces induced by the cross-flow were introduced and have a time delay term based on the quasi-steady theory. Mean axial extension was also considered in the mathematical formulation of the model. Interesting results were obtained by numerically solving the equations of motion.(1)
With the use of the cubic impacting model, the tube exhibits apparent Hopf bifurcation at particular initial axial load for the simply supported tube and clamped-clamped tube. Buckling, limit circle motion, and quasi-periodic motion were observed with corresponding initial axial loads and cross-flow velocities. No chaotic oscillation occurred within the cross-flow velocity range. Impacting force distribution revealed that once the tube begins oscillating, the nonzero displacement would always induce impacting forces along the tube axis.(2)
With the use of the trilinear impacting model, Hopf bifurcation occurs quickly as soon as the cross-flow velocity exceeds the critical velocity. Buckling, limit circle motion, quasi-period motion, and chaotic motion were observed with this kind of impacting constraints for the simply supported tube and clamped-clamped tube. Different from that, in the cubic model, impacting behavior happens only for amplitude of the lateral displacement that exceeds the gap.(3)
Numerical results show that the tube may undergo chaotic oscillations in a wide range of cross-flow velocities. With regard to the initial axial loads, negative pressure loads increase the critical velocity, and positive pressure loads decrease the critical velocity. For sufficiently large axial pressure loads, the tube would lose stability by buckling.These achievements are of practical value and imply the importance of the effects of initial axial pressure loads and distributed impacting constraints on the nonlinear dynamics of the supported tube.



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*Source: 2359090-2021-09-07.xml*