Abstract

Nonlinear vibrations of a heat-exchanger tube modeled as a simply supported Euler–Bernoulli beam under axial load and cross-flow have been studied. The compressive axial loads are a consequence of thermal expansion, and tensile axial loads can be induced by design (prestress). The fluid forces are represented using an added mass, damping, and a time-delayed displacement term. Due to the presence of the time-delayed term, the equation governing the dynamics of the tube becomes a partial delay differential equation (PDDE). Using the modal-expansion procedure, the PDDE is converted into a nonlinear delay differential equation (DDE). The fixed points (zero and buckled equilibria) of the nonlinear DDE are found, and their linear stability is analyzed. It is found that stability can be lost via either supercritical or subcritical Hopf bifurcation. Using Galerkin approximations, the characteristic roots (spectrum) of the DDE are found and reported in the parametric space of fluid velocity and axial load. Furthermore, the stability chart obtained from the Galerkin approximations is compared with the critical curves obtained from analytical calculations. Next, the method of multiple scales (MMS) is used to derive the normal-form equations near the supercritical and subcritical Hopf bifurcation points for both zero and buckled equilibrium configurations. The steady-state amplitude response equation, obtained from the MMS, at Hopf bifurcation points is compared with the numerical solution. The coexistence of multiple limit cycles in the parametric space is found, and has implications in the fatigue life calculations of the heat-exchanger tubes.

References

References
1.
Mitra
,
D. R.
,
2005
, “
Fluid-Elastic Instability in Tube Arrays Subjected to Air-Water and Steam-Water Cross-Flow
,”
Ph.D. thesis
, University of California, Los Angeles, CA.https://search.proquest.com/openview/178a5ab8288a75cd12eeb9130d55732d/1?pq-origsite=gscholar&cbl=18750&diss=y
2.
Chen
,
S.
,
1983
, “
Instability Mechanisms and Stability Criteria of a Group of Circular Cylinders Subjected to Cross-Flow—Part I: Theory
,”
ASME J. Vib. Acoust.
,
105
(
1
), pp.
51
58
.10.1115/1.3269066
3.
Price
,
S. J.
, and
Paidoussis
,
M. P.
,
1986
, “
A Single-Flexible-Cylinder Analysis for the Fluidelastic Instability of an Array of Flexible Cylinders in Cross-Flow
,”
ASME J. Fluids Eng.
,
108
(
2
), pp.
193
199
.10.1115/1.3242562
4.
Khalifa
,
A.
,
Weaver
,
D.
, and
Ziada
,
S.
,
2012
, “
A Single Flexible Tube in a Rigid Array as a Model for Fluidelastic Instability in Tube Bundles
,”
J. Fluids Struct.
,
34
, pp.
14
32
.10.1016/j.jfluidstructs.2012.06.007
5.
Roberto
,
B. W.
,
1962
, “
Low Frequency, Self-Excited Vibration in a Row of Circular Cylinders Mounted in an Airstream
,” Ph.D. thesis, University of Cambridge.
6.
Connors
,
H. J.
,
1970
, “
Fluidelastic Vibration of Tube Arrays Excited by Cross Flow
,”
ASME Symposium on Flow-Induced Vibration in Heat Exchanger, Winter Annual Meeting, pp. 42–47.
7.
Blevins
,
R. D.
,
1974
, “
Fluid Elastic Whirling of a Tube Row
,”
ASME J. Pressure Vessel Technol.
,
96
(
4
), pp.
263
267
.10.1115/1.3454179
8.
Lever
,
J. H.
, and
Weaver
,
D. S.
,
1986
, “
On the Stability of Heat Exchanger Tube Bundles—Part I: Modified Theoretical Model
,”
J. Sound Vib.
,
107
(
3
), pp.
375
392
.10.1016/S0022-460X(86)80114-6
9.
Paidoussis
,
M. P.
, and
Li
,
G. X.
,
1992
, “
Cross-Flow-Induced Chaotic Vibrations of Heat-Exchanger Tubes Impacting on Loose Supports
,”
J. Sound Vib.
,
152
(
2
), pp.
305
326
.10.1016/0022-460X(92)90363-3
10.
Tanaka
,
H.
, and
Takahara
,
S.
,
1981
, “
Fluid Elastic Vibration of Tube Array in Cross Flow
,”
J. Sound Vib.
,
77
(
1
), pp.
19
37
.10.1016/S0022-460X(81)80005-3
11.
Dalton
,
C.
, and
Helfinstine
,
R. A.
,
1971
, “
Potential Flow Past a Group of Circular Cylinders
,”
ASME J. Fluids Eng.
,
93
(
4
), pp.
636
642
.10.1115/1.3425320
12.
Païdoussis
,
M. P.
,
Price
,
S. J.
, and
De Langre
,
E.
,
2010
,
Fluid-Structure Interactions: Cross-Flow-Induced Instabilities
,
Cambridge University Press
New York.
13.
Bazilevs
,
Y.
,
Takizawa
,
K.
, and
Tezduyar
,
T. E.
,
2013
,
Computational Fluid-Structure Interaction: Methods and Applications
,
Wiley
, Hoboken, NJ.
14.
Wang
,
L.
, and
Ni
,
Q.
,
2010
, “
Hopf Bifurcation and Chaotic Motions of a Tubular Cantilever Subject to Cross Flow and Loose Support
,”
Nonlinear Dyn.
,
59
(
1–2
), pp.
329
338
.10.1007/s11071-009-9542-8
15.
Xia
,
W.
, and
Wang
,
L.
,
2010
, “
The Effect of Axial Extension on the Fluidelastic Vibration of an Array of Cylinders in Cross-Flow
,”
Nucl. Eng. Des.
,
240
(
7
), pp.
1707
1713
.10.1016/j.nucengdes.2010.03.024
16.
Wang
,
L.
,
Dai
,
H. L.
, and
Han
,
Y. Y.
,
2012
, “
Cross-Flow-Induced Instability and Nonlinear Dynamics of Cylinder Arrays With Consideration of Initial Axial Load
,”
Nonlinear Dyn.
,
67
(
2
), pp.
1043
1051
.10.1007/s11071-011-0047-x
17.
Price
,
S. J.
, and
Paidoussis
,
M. P.
,
1984
, “
An Improved Mathematical Model for the Stability of Cylinder Rows Subject to Cross-Flow
,”
J. Sound Vib.
,
97
(
4
), pp.
615
640
.10.1016/0022-460X(84)90512-1
18.
Granger
,
S.
, and
Paidoussis
,
M.
,
1996
, “
An Improvement to the Quasi-Steady Model With Application to Cross-Flow-Induced Vibration of Tube Arrays
,”
J. Fluid Mech.
,
320
(
1
), pp.
163
184
.10.1017/S0022112096007495
19.
Li
,
H.
, and
Mureithi
,
N.
,
2017
, “
Development of a Time Delay Formulation for Fluidelastic Instability Model
,”
J. Fluids Struct.
,
70
, pp.
346
359
.10.1016/j.jfluidstructs.2017.01.020
20.
Mahon
,
J.
, and
Meskell
,
C.
,
2013
, “
Estimation of the Time Delay Associated With Damping Controlled Fluidelastic Instability in a Normal Triangular Tube Array
,”
ASME J. Pressure Vessel Technol.
,
135
(
3
), p.
030903
.10.1115/1.4024144
21.
Sawadogo
,
T.
, and
Mureithi
,
N.
,
2014
, “
Fluidelastic Instability Study on a Rotated Triangular Tube Array Subject to Two-Phase Cross-Flow—Part II: Experimental Tests and Comparison With Theoretical Results
,”
J. Fluids Struct.
,
49
, pp.
16
28
.10.1016/j.jfluidstructs.2014.04.013
22.
Sandström
,
S.
,
1987
, “
Vibration Analysis of a Heat Exchanger Tube Row With ADINA
,”
Comput. Struct.
,
26
(
1–2
), pp.
297
305
.10.1016/0045-7949(87)90260-4
23.
Azizian
,
R.
, and
Mureithi
,
N.
,
2014
, “
A Simple Empirical Model for Tube–Support Normal Impact Interaction
,”
ASME J. Pressure Vessel Technol.
,
136
(
5
), p.
051303
.10.1115/1.4027797
24.
Saffarian
,
M. R.
,
Fazelpour
,
F.
, and
Sham
,
M.
,
2019
, “
Numerical Study of Shell and Tube Heat Exchanger With Different Cross-Section Tubes and Combined Tubes
,”
Int. J. Energy Environ. Eng.
,
10
(
1
), pp.
33
46
.10.1007/s40095-019-0297-9
25.
Wahi
,
P.
, and
Chatterjee
,
A.
,
2005
, “
Galerkin Projections for Delay Differential Equations
,”
ASME J. Dyn. Syst. Meas. Control
,
127
(
1
), pp.
80
87
.10.1115/1.1870042
26.
Vyasarayani
,
C. P.
,
Subhash
,
S.
, and
Kalmár-Nagy
,
T.
,
2014
, “
Spectral Approximations for Characteristic Roots of Delay Differential Equations
,”
Int. J. Dyn. Control
,
2
(
2
), pp.
126
132
.10.1007/s40435-014-0060-2
27.
Das
,
S. L.
, and
Chatterjee
,
A.
,
2002
, “
Multiple Scales Without Center Manifold Reductions for Delay Differential Equations Near Hopf Bifurcations
,”
Nonlinear Dyn.
,
30
(
4
), pp.
323
335
.10.1023/A:1021220117746
28.
Yi
,
S.
,
Nelson
,
P. W.
, and
Ulsoy
,
A. G.
,
2010
,
Time-Delay Systems: Analysis and Control Using the Lambert W Function
,
World Scientific
,
Hackensack, NJ
.
29.
Asl
,
F. M.
, and
Ulsoy
,
A. G.
,
2003
, “
Analysis of a System of Linear Delay Differential Equations
,”
J. Dyn. Syst. Meas. Control
,
125
(
2
), pp.
215
223
.10.1115/1.1568121
30.
Jarlebring
,
E.
, and
Damm
,
T.
,
2007
, “
The Lambert W Function and the Spectrum of Some Multidimensional Time-Delay Systems
,”
Automatica
,
43
(
12
), pp.
2124
2128
.10.1016/j.automatica.2007.04.001
31.
Yi
,
S.
,
Nelson
,
P. W.
, and
Ulsoy
,
A. G.
,
2007
, “
Survey on Analysis of Time Delayed Systems Via the Lambert W Function
,”
Advances in Dynamical Systems
, 14(S2), pp.
296
301
.https://www.researchgate.net/publication/237231748_Survey_on_Analysis_of_Time_Delayed_Systems_via_the_Lambert_W_Function
32.
Wahi
,
P.
, and
Chatterjee
,
A.
,
2005
, “
Asymptotics for the Characteristic Roots of Delayed Dynamic Systems
,”
ASME J. Appl. Mech.
,
72
(
4
), pp.
475
483
.10.1115/1.1875492
33.
Vyasarayani
,
C. P.
,
2012
, “
Galerkin Approximations for Higher Order Delay Differential Equations
,”
ASME J. Comput. Nonlinear Dyn.
,
7
(
3
), p.
031004
.10.1115/1.4005931
34.
Sadath
,
A.
, and
Vyasarayani
,
C. P.
,
2015
, “
Galerkin Approximations for Stability of Delay Differential Equations With Distributed Delays
,”
ASME J. Comput. Nonlinear Dyn.
,
10
(
6
), p.
061024
.10.1115/1.4030153
35.
Insperger
,
T.
, and
Stépán
,
G.
,
2011
,
Semi-Discretization for Time-Delay Systems: Stability and Engineering Applications
, Vol.
178
,
Springer Science & Business Media
,
New York
.
36.
Butcher
,
E. A.
,
Ma
,
H.
,
Bueler
,
E.
,
Averina
,
V.
, and
Szabo
,
Z.
,
2004
, “
Stability of Linear Time-Periodic Delay-Differential Equations Via Chebyshev Polynomials
,”
Int. J. Numer. Methods Eng.
,
59
(
7
), pp.
895
922
.10.1002/nme.894
37.
Breda
,
D.
,
Maset
,
S.
, and
Vermiglio
,
R.
,
2005
, “
Pseudospectral Differencing Methods for Characteristic Roots of Delay Differential Equations
,”
SIAM J. Sci. Comput.
,
27
(
2
), pp.
482
495
.10.1137/030601600
38.
Wu
,
Z.
, and
Michiels
,
W.
,
2012
, “
Reliably Computing All Characteristic Roots of Delay Differential Equations in a Given Right Half Plane Using a Spectral Method
,”
J. Comput. Appl. Math.
,
236
(
9
), pp.
2499
2514
.10.1016/j.cam.2011.12.009
39.
Sun
,
J.-Q.
,
2009
, “
A Method of Continuous Time Approximation of Delayed Dynamical Systems
,”
Commun. Nonlinear Sci. Numer. Simul.
,
14
(
4
), pp.
998
1007
.10.1016/j.cnsns.2008.02.008
40.
Song
,
B.
, and
Sun
,
J.-Q.
,
2011
, “
Low pass Filter-Based Continuous-Time Approximation of Delayed Dynamical Systems
,”
J. Vib. Control
,
17
(
8
), pp.
1173
1183
.10.1177/1077546310378432
41.
Vyhlidal
,
T.
, and
Zítek
,
P.
,
2009
, “
Mapping Based Algorithm for Large-Scale Computation of Quasi-Polynomial Zeros
,”
IEEE Trans. Autom. Control
,
54
(
1
), pp.
171
177
.10.1109/TAC.2008.2008345
42.
Olgac
,
N.
, and
Sipahi
,
R.
,
2002
, “
An Exact Method for the Stability Analysis of Time-Delayed Linear Time-Invariant (LTI) Systems
,”
IEEE Trans. Autom. Control
,
47
(
5
), pp.
793
797
.10.1109/TAC.2002.1000275
43.
Pekař
,
L.
, and
Gao
,
Q.
,
2018
, “
Spectrum Analysis of LTI Continuous-Time Systems With Constant Delays: A Literature Overview of Some Recent Results
,”
IEEE Access
,
6
, pp.
35457
35491
.10.1109/ACCESS.2018.2851453
44.
Wahi
,
P.
, and
Chatterjee
,
A.
,
2004
, “
Averaging Oscillations With Small Fractional Damping and Delayed Terms
,”
Nonlinear Dyn.
,
38
(
1–4
), pp.
3
22
.10.1007/s11071-004-3744-x
45.
Molnar
,
T. G.
,
Insperger
,
T.
, and
Stepan
,
G.
,
2019
, “
Closed-Form Estimations of the Bistable Region in Metal Cutting Via the Method of Averaging
,”
Int. J. Non-Linear Mech.
,
112
, pp.
49
56
.10.1016/j.ijnonlinmec.2018.09.005
46.
Kalmár-Nagy
,
T.
,
Stépán
,
G.
, and
Moon
,
F. C.
,
2001
, “
Subcritical Hopf Bifurcation in the Delay Equation Model for Machine Tool Vibrations
,”
Nonlinear Dyn.
,
26
(
2
), pp.
121
142
.10.1023/A:1012990608060
47.
Nayfeh
,
A. H.
, and
Balachandran
,
B.
,
2008
,
Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods
,
Wiley
, Hoboken, NJ.
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