Abstract
The linear stability of a heatexchanger tube modeled as a singlespan cantilever beam subjected to crossflow has been studied with two parameters: (1) varying stiffness of the bafflecladding at the free end and (2) varying flow velocity. A mathematical model incorporating the motiondependent fluid forces acting on the beam is developed using the Euler–Bernoulli beam theory, under the inextensible condition. The partial delay differential equation governing the dynamics of the continuous system is discretized to a set of finite, nonlinear delay differential equations through a Galerkin method in which a single mode is considered. Unstable regions in the parametric space of dimensionless cladding stiffness and flow velocity are identified, along with the magnitude of damping in the stable region. This information can be used to determine the cladding stiffness at which the system should be operated to achieve maximum damping at a known operational flow velocity. Furthermore, the system is found to lose stability by Hopf bifurcation and the method of multiple scales is used to analyze its postinstability behavior. Stable and unstable limit cycles are observed for different values of the linear component of the dimensionless cladding stiffness. A global bifurcation analysis indicates that the number of limit cycles decreases with increasing linear cladding stiffness. An optimal range for the linear cladding stiffness is recommended where tube vibrations would either diminish to zero or assume a relatively low amplitude associated with a stable limit cycle.
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Abbreviations
 L :

Tube length (m)
 \(\zeta\) :

Damping ratio
 \(\rho\) :

Fluid density (kg m\(^{3}\))
 D :

Outer diameter of tube (m)
 t :

Dimensionless time
 X :

Axial spatial coordinate (m)
 T :

Time (s)
 \(\mu\) :

Flow retardation parameter
 \({\mathcal {O}}\) :

Order of magnitude
 x :

Dimensionless axial spatial coordinate
 w :

Dimensionless transverse displacement of tube
 \(\tilde{f}\delta (XX_b)\) :

Restraining force due to spring at \(X=X_b\) (N m\(^{1}\))
 A :

Crosssectional area of cylinder (\(\hbox {m}^2\))
 W :

Transverse displacement of tube (m)
 \(\tilde{U}\) :

Crossflow velocity (m s\(^{1}\))
 \(f\delta (xx_b)\) :

Dimensionless spring force
 E :

Young’s modulus of tube material (Pa)
 \(C_{D}\) :

Drag coefficient
 \(C_{L}\) :

Lift coefficient
 M :

Mass per unit length of tube (kg m\(^{1}\))
 C :

Viscous damping coefficient (N s m\(^{2}\))
 I :

Second moment of inertia of tube crosssection (m\(^{4}\))
 \(\delta _1\) :

Modal logarithmic decrement
 \(C_{ma}\) :

Addedmass coefficient
 \(\varDelta T\) :

Time delay (s) \(\left( \frac{\mu D}{ \tilde{U}}\right)\)
 m :

Tube massparameter \(\left( \frac{M}{\rho D^{2}}\right)\)
 U :

Dimensionless flowvelocity \(\left( \frac{2\pi \tilde{U}}{D\varOmega _1}\right)\)
 \(\omega _{cr}\) :

Dimensionless critical angular frequency
 \(\tau\) :

Dimensionless time delay due to flow retardation \(\left( \frac{2\pi }{U}\right)\)
 q :

Dimensionless tube vibration response
 \(\varOmega _1\) :

Frequency of first mode (\(\hbox {s}^{1}\))
 \(\lambda _1\) :

Dimensionless eigenvalue of first mode
 F :

Crossflowinduced force per unit length (N m\(^{1}\))
 \(K_1\) :

Linear spring stiffness (N m\(^{1}\))
 \(K_2\) :

Cubic spring stiffness (N m\(^{3}\))
 \(k_1\) :

Dimensionless linear spring stiffness \(\left( \frac{K_1L^4}{\lambda _1^4EID}\right)\)
 \(k_1^{cr}\) :

Dimensionless critical linear spring stiffness
 \(k_2\) :

Dimensionless cubic spring stiffness \(\left( \frac{K_2L^6}{\lambda _1^4EID}\right)\)
 P :

Pitch (m)
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Acknowledgements
The authors gratefully acknowledge the Department of Science and Technology for funding this research through the Inspire Fellowship (DST/INSPIRE/04/2014/000972).
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Vourganti, V., Desai, A., Samukham, S. et al. Effect of nonlinear cladding stiffness on the stability and Hopf bifurcation of a heatexchanger tube subject to crossflow. Meccanica (2020). https://doi.org/10.1007/s1101201901114z
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Keywords
 Heatexchanger tube
 Linear stability
 Nonlinear cladding stiffness
 Delay differential equation
 Method of multiple scales
 Hopf bifurcation