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Research Papers

Nonlinear Oscillations Induced by Follower Forces in Prestressed Clamped Rods Subjected to Drag

[+] Author and Article Information
Soheil Fatehiboroujeni

Department of Mechanical Engineering,
University of California,
Merced, CA 95343
e-mail: sfatehiboroujeni@ucmerced.edu

Arvind Gopinath

Department of Bioengineering,
Health Science Research Institute,
University of California,
Merced, CA 95343
e-mail: agopinath@ucmerced.edu

Sachin Goyal

Department of Mechanical Engineering,
Health Science Research Institute,
University of California,
Merced, CA 95343
e-mail: sgoyal2@ucmerced.edu

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received March 19, 2018; final manuscript received September 30, 2018; published online October 29, 2018. Assoc. Editor: Anindya Chatterjee.

J. Comput. Nonlinear Dynam 13(12), 121005 (Oct 29, 2018) (8 pages) Paper No: CND-18-1111; doi: 10.1115/1.4041681 History: Received March 19, 2018; Revised September 30, 2018

Elastic-driven slender filaments subjected to compressive follower forces provide a synthetic way to mimic the oscillatory beating of biological flagella and cilia. Here, we use a continuum model to study the dynamical, nonlinear buckling instabilities that arise due to the action of nonconservative follower forces on a prestressed slender rod clamped at both ends and allowed to move in a fluid. Stable oscillatory responses are observed as a result of the interplay between the structural elastic instability of the inextensible slender rod, geometric constraints that control the onset of instability, energy pumped into the system by the active follower forces, and motion-driven fluid dissipation. Initial buckling instabilities are initiated by the effect of the follower forces and inertia; fluid drag subsequently allows for the active energy pumped into the system to be dissipated away and results in self-limiting amplitudes. By integrating the equations of equilibrium and compatibility conditions with linear constitutive laws, we compute the critical follower forces for the onset of oscillations, emergent frequencies of these solutions, and the postcritical nonlinear rod shapes for two forms of the drag force, namely linear Stokes drag and quadratic Morrison drag. For a rod with fixed inertia and drag parameters, the minimum (critical) force required to initiate stable oscillations depends on the initial slack and weakly on the nature of the drag force. Emergent frequencies and the amplitudes postonset are determined by the extent of prestress as well as the nature of the fluid drag. Far from onset, for large follower forces, the frequency of the oscillations can be predicted by evoking a power balance between the energy input by the active forces and the dissipation due to fluid drag.

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Figures

Grahic Jump Location
Fig. 1

(a) Schematic representation of a rod of unstressed length L with fixed–fixed boundary condition (clamped at both ends). The end-to-end distance when buckled is Lee < L. (b) The motion of material points comprising the cross section of the rod at arc-length position, s and at time t is determined by tracking the transformations of the body-fixed frame âi(s,t) with respect to the inertial frame of reference êi. (c) The shape (top) and prestress (bottom) in the buckled state for different values of Lee/L. The dashed line corresponds to the unbuckled case Lee/L = 1.0. Prestress here is defined as the component of the internal force in the direction of cross-sectional normal vector â3(s,t) i.e., f3. Here, we use the tension along the filament, f3, to characterize prestress. For the shapes we study in this paper, there is a one-to-one correspondence between the two.

Grahic Jump Location
Fig. 2

(Left) A measure of the prestress may be gained by examining the plot of f3(L) versus 1 − Lee/L. The results of this paper focus on the range 0.05 < 1 − Lee/L < 0.5 shown between the two markers (red circles). (Right) Critical load for onset of oscillations Fcr versus scaled decrease in end-to-end distance 1 − Lee/L for both Stokes [S] drag and Morrison [M] drag. We note that the critical loads are roughly the same for the two forms of fluid drag and deviate little from the nondrag value. This is not surprising for the Morrison drag [M] as it is nonlinear and hence does not enter the linear stability equations at leading order for small perturbations. Given that the Stokes [S] values are close to the no drag values, we surmise that the onset of vibrations and onset of flutter are very close to one another for the parameter range investigated. For 0.05 < 1 − Lee/L < 0.5, the critical force Fcr increases as 1 − Lee/L, or prestress, increases. For the region 1 − Lee/L < 0.05, we are unable to obtain well-defined results since the slack is very small in this region and inextensibility constraint makes the problem very stiff necessitating very small time intervals. Since the focus of our paper is to investigate the role of slack (prestress), we choose to study values of Lee/L < 0.9.

Grahic Jump Location
Fig. 3

Configurations of the oscillating rod when |F|=15 N/m, when the drag force is of the Stokes [S] form. We show kymographs of the curvature in the top column, as well as the shapes over a period (labeled sequentially from 1–10) in the bottom column. We see increasingly sharper shapes (higher values of |κ2|) for smaller values of Lee/L.

Grahic Jump Location
Fig. 4

Configurations of the oscillating rod when |F|=15 N/m, when the drag force is of the Morrison [M] form. We show kymographs of the curvature in the top column, as well as the shapes over a period (labeled sequentially from 1–10) in the bottom column. As in the [S] case, we see increasingly sharper shapes (higher values of |κ2|) for smaller values of Lee/L. Furthermore, the shapes are smoother and the oscillations involve lower amplitudes than the corresponding shapes for the [S] case.

Grahic Jump Location
Fig. 5

Fourier transform (in the time domain) of the shear force at the midspan length of the rod shows that higher harmonics (insets show the raw data) are damped in the case of the nonlinear Morrison drag ((a) and (b)) more effectively than for the linear Stokes drag ((c) and (d)). The right column of the picture corresponds to Lee/L = 0.9 while the left column corresponds to Lee/L = 0.7. The ratio of drag coefficients for both cases is 3.18. Intuitively, we expect this ratio to affect the extent of dampening. The magnitude of |F| for all cases is 14 N/m.

Grahic Jump Location
Fig. 6

Frequency of oscillations for rods as a function of |F| for various values of Lee when the fluid drag is of the Stokes [S] form. The form of the curves suggest possible transitions that may be related to the activation or deactivation of higher order mode shapes (typical shapes are shown alongside the curves). Sudden jumps in the frequency–force curve, for example from point A to point B, are due to the suppression of oscillation modes with smaller wavelength that also have higher energy level. Higher follower force densities yield increasingly steep shapes with localized curvature variations. When plotted on logarithmic axes, we find that the frequencies do not correlate well with the ω∼|F|4/3 power law form for an inertialess cantilever with distributed follower forces, oscillating with a single dominant wavelength and frequency.

Grahic Jump Location
Fig. 7

(a) Frequency for the Morrison [M] drag plotted as a function of the force density |F| plotted in logarithmic scales to illustrate two salient features—(i) as the follower force increases to values much larger than the critical values, the effect of the prestress diminishes, and (ii) the frequencies in the limit |F|≫Fcr scale roughly as ω∼|F|5/6 consistent with our theoretical prediction. (b) Emergent frequency plotted as a function of the scaled end-to-end distance showing nonmonotonic behavior at fixed values of |F|. However, we note that as |F| increases, the effect of the prestress and slack becomes decreasingly important for the range of Lee/L investigated.

Grahic Jump Location
Fig. 8

Fourier transform (in the time domain) of the shear force at the midspan length of the rod subject to Stokes drag shows that higher harmonics are less pronounced as follower load, |F|, increases from |F|=Fcr=11.5 N/m (point A) to |F|=13 N/m (point B) for Lee/L = 0.85. This correlates with the increase of steady-state frequency of the oscillations from point A to pointB.

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