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Research Papers

Stability of Vertically Traveling, Pre-tensioned, Heavy Cables

[+] Author and Article Information
Abhinav Ravindra Dehadrai

Mechanics & Applied Mathematics Group,
Department of Mechanical Engineering,
Indian Institute of Technology Kanpur,
Kanpur 208016, India
e-mail: abhinavd@iitk.ac.in

Ishan Sharma

Mechanics & Applied Mathematics Group,
Department of Mechanical Engineering,
Indian Institute of Technology Kanpur,
Kanpur 208016, India
e-mail: ishans@iitk.ac.in

Shakti S. Gupta

Mechanics & Applied Mathematics Group,
Department of Mechanical Engineering,
Indian Institute of Technology Kanpur,
Kanpur 208016, India
e-mail: ssgupta@iitk.ac.in

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received January 6, 2018; final manuscript received May 11, 2018; published online June 18, 2018. Assoc. Editor: Katrin Ellermann.

J. Comput. Nonlinear Dynam 13(8), 081003 (Jun 18, 2018) (9 pages) Paper No: CND-18-1005; doi: 10.1115/1.4040344 History: Received January 06, 2018; Revised May 11, 2018

We study the stability of a pre-tensioned, heavy cable traveling vertically against gravity at a constant speed. The cable is modeled as a slender beam incorporating rotary inertia. Gravity modifies the tension along the traveling cable and introduces spatially varying coefficients in the equation of motion, thereby precluding an analytical solution. The onset of instability is determined by employing both the Galerkin method with sine modes and finite element (FE) analysis to compute the eigenvalues associated with the governing equation of motion. A spectral stability analysis is necessary for traveling cables where an energy stability analysis is not comprehensive, because of the presence of gyroscopic terms in the governing equation. Consistency of the solution is checked by direct time integration of the governing equation of motion with specified initial conditions. In the stable regime of operations, the rate of change of total energy of the system is found to oscillate with bounded amplitude indicating that the system, although stable, is nonconservative. A comprehensive stability analysis is carried out in the parameter space of traveling speed, pre-tension, bending rigidity, external damping, and the slenderness ratio of the cable. We conclude that pre-tension, bending rigidity, external damping, and slenderness ratio enhance the stability of the traveling cable while gravity destabilizes the cable.

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References

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Figures

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Fig. 1

Schematic of a traveling cable at time t. The deflections of the cable are exaggerated for clarity.

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Fig. 2

Comparison of the first eigenvalues for cables modeled as Euler–Bernoulli beam: (—-•—-) represents the vertically traveling cable for which the parameter ρ=10, while (—-°—-) represents the cable traveling horizontally for which ρ=0. For both the cases, the other parameters μ = 1, SR−2=0 and c = 0.

Grahic Jump Location
Fig. 3

Stability curves for Euler–Bernoulli beams traveling horizontally (–°–) and vertically (–•–) are obtained by plotting the nondimensional critical speed v¯crit as a function of nondimensional end tension μ. The parameter ρ=0 and 10 for the horizontally and vertically traveling beams, respectively, while the other parameters c = 0 and SR−2=0.

Grahic Jump Location
Fig. 4

Stability curves obtained by plotting nondimensional critical speed v¯crit as a function of slenderness ratio SR for horizontally traveling Rayleigh (—) and Euler–Bernoulli (- - -) beams. The other parameters ρ=0, μ = 1, and c = 0.

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Fig. 5

Real and imaginary part of the first mode's eigenvalue of an undamped (c = 0), vertically traveling Rayleigh beam as functions of speed for ρ=10, μ = 1 and SR=20. The inset shows the existence of positive real part at all nondimensional speeds v¯>0, while the bifurcation (as in Fig. 2) happens much later when v¯=v¯bif=2.36.

Grahic Jump Location
Fig. 6

Real and imaginary parts of the first mode's eigenvalue for a damped, vertically traveling, Rayleigh beam as function ofnondimensional speed v¯ for nondimensional viscosity c = 0 (—), 5 (–°–) and 10 (–•–). The other parameters ρ=10, SR=20 and μ = 1. We observe that with damping, the critical speed v¯crit increases from 0 for c = 0, to 2.375 for c = 5, and 2.435 for c = 10. Also, Re(ω) bifurcates at speeds v¯bif<v¯crit for c = 5 and 10, in contrast to when c = 0.

Grahic Jump Location
Fig. 7

Stability curve (—) obtained by plotting nondimensional critical speed v¯crit as a function of nondimensional damping c for vertically traveling Rayleigh beam, while its bifurcation speed (- - -) vanishes monotonically with increase in c, thus, marking a boundary within the subcritical (stable) region inside which the cable undergoes underdamped oscillations. The inset shows the steep initial rise in the stability curve due to the inclusion of damping. The other parameters ρ=10, SR=20 and μ = 1.

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Fig. 8

Nondimensional critical speed v¯crit as a function ofscaled end tension μs for horizontally (—°—) and vertically (—•—) traveling strings. The parameter ρ=0 and 10 for the horizontally and vertically traveling strings, respectively, while the parameter c = 0.

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Fig. 9

Plot (a) compares the errors in critical speed v¯critN obtained from the N-term Galerkin projection and the N-element FE solution for horizontally (ρ=0) and vertically (ρ=10) traveling cables. In each case μ = 10, c = 0 and SR−2=0. For horizontally traveling cable the critical speed v¯critref=10.48, as found by Wickert and Mote [10]. For vertically traveling cable, the critical velocity calculated using N = 399 terms is taken as the reference. Plot (b) compares errors in the calculation of the first mode's eigenvalue ω at subcritical speed v¯subref=5.

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Fig. 10

Temporal evolution of the transverse nondimensional displacement of the horizontally traveling cable at x¯=0.5 (x=L/2) and the rate of change of total mechanical energy (nondimensional). The subcritical and super-critical nondimensional speeds are v¯sub=3.26 and v¯sup=3.297, respectively, while the parameters ρ=0, μ = 1, c = 0 and SR−2=0. The inset semilog plot shows that y¯sup (◻) grows modestly for some time following which the growth is exponential (—) at a rate exp(ωsupt¯)=exp {Re(ωsup)t¯}, with ωsup=0.068.

Grahic Jump Location
Fig. 11

Temporal evolution of the transverse nondimensional displacement of the vertically traveling cable at x¯=0.5 (x=L/2) and the rate of change of total mechanical energy (nondimensional). The subcritical and super-critical nondimensional speeds are v¯sub=2.37 and v¯sup=2.38723, respectively, while the parameters ρ=10, μ = 1, c = 0 and SR−2=0. The inset semilog plot shows that y¯sup (◻) grows exponentially, at a rate exp(ωsupt¯)=exp {Re(ωsup)t¯} with ωsup=0.068, after a rapid initial rise in its amplitude.

Grahic Jump Location
Fig. 12

Stability curves for Euler–Bernoulli beams traveling horizontally (–°–) and vertically (–•–) are obtained by plotting the nondimensional critical speed v¯crit as a function of nondimensional bending rigidity θ. The parameter ρ=0 and 10 for the horizontally and vertically traveling beams, respectively, while the other parameters c = 0 and SR=0. At θ = 0, the critical points HS (v¯crit=1) and VS (v¯crit=0) are the critical speeds of horizontally and vertically traveling strings, respectively.

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