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Research Papers

The Instability Mechanism of a Confined Rod Under Axial Vibrations

[+] Author and Article Information
Jia Liu

John A. Paulson School of Engineering and
Applied Sciences,
Harvard University,
Cambridge, MA 02138

Tianxiang Su, Nathan Wicks, Jahir Pabon

Schlumberger-Doll Research,
One Hampshire Street, MD-B353,
Cambridge, MA 02139

Katia Bertoldi

John A. Paulson School of Engineering and
Applied Sciences,
Harvard University,
Cambridge, MA 02138;
Kavli Institute,
Harvard University,
Cambridge, MA 02138

Manuscript received September 4, 2015; final manuscript received September 29, 2015; published online October 20, 2015. Editor: Yonggang Huang.

J. Appl. Mech 83(1), 011005 (Oct 20, 2015) (10 pages) Paper No: JAM-15-1473; doi: 10.1115/1.4031710 History: Received September 04, 2015; Revised September 29, 2015

We studied the stability of a confined rod under axial vibrations through a combination of analytical and numerical analysis. We find that the stability of the system is significantly different than in the static case and that both the frequency and magnitude of the applied vibrational force play an important role. In particular, while larger vibrational forces always tend to destabilize the system, our analysis indicates that the effect of the frequency is not obvious and monotonic. For certain frequencies, a very small force is sufficient to trigger an instability, while for others the rod is stable even for large forces. Furthermore, we find that the stability of the confined rod is significantly enhanced by the presence of frictional contact and that in this case also the magnitude of the perturbation affects its response.

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Figures

Grahic Jump Location
Fig. 1

Schematic of a confined rod in a cylindrical channel. Two identical axial forces with opposite direction, F = Fave + dF sin(ωt), are applied to its ends. When buckling occurs, the rod takes a wavy configuration.

Grahic Jump Location
Fig. 2

Stability map obtained by solving Eq. (3) numerically. (a) The markers indicate the stable configurations as determined by solving numerically the PDE. (b)–(d) First 10 modes for a Nitinol rod superimposed on the stability map, when we varied (b) ω, (c) Fave, and (d) dF. The shape of the markers indicates a specific set of loading conditions, while their color represents the mode number.

Grahic Jump Location
Fig. 3

(a)–(d) Numerically obtained stability maps for a Nitinol rod when (a) dF = 0.1Fcr, (b) dF = 0.2Fcr, (c) dF = 0.3Fcr, and (d) dF = 0.5Fcr. All stable configurations are indicated by dots. (e) and (f) Evolution of the unstable modes as a function of Fave for dF = 0.2Fcr. Results for both (e) ω = 85 rad/s and (f) ω = 150 rad/s are reported.

Grahic Jump Location
Fig. 4

Numerical top-view snapshots showing the configurations of the rod at t = 1.5 s, as obtained from the dynamic simulations for (a) ω = 85 rad/s and (b) ω = 150 rad/s. All simulated loading conditions are indicated by a marker in Fig. 3(b).

Grahic Jump Location
Fig. 5

Stability maps obtained by solving Eq. (14) numerically. The friction term γ is kept constant while the perturbation ϵ is varied from 0.005 to 0.5.

Grahic Jump Location
Fig. 6

(a)–(d) Numerically obtained stability maps for a Nitinol rod in the presence of frictional contact (μ = 0.1) when (a) dF = 0.5Fcr and vθ(0) = 30 mm/s, (b) dF = 0.7Fcr and vθ(0) = 30 mm/s, (c) dF = 0.5Fcr and vθ(0) = 50 mm/s, and (d) dF = 0.7Fcr and vθ(0) = 50 mm/s. All stable configurations are indicated by dots. (e) and (f) Evolution of the unstable modes as a function of Fave for dF = 0.7Fcr and vθ(0) = 50 mm/s. Results for both (e) ω = 85 rad/s and (f) ω = 150 rad/s are reported.

Grahic Jump Location
Fig. 7

Numerical top-view snapshots showing the configurations of the rod at t = 4 s, as obtained from the dynamic simulations for (a) ω = 85 rad/s and (b) ω = 150 rad/s. All simulated loading conditions are indicated by a marker in Fig. 6(d).

Grahic Jump Location
Fig. 8

Stability maps obtained by solving Eq. (3) analytically and numerically. The shaded areas indicate the stable domains as predicted by the analytical solution obtained using (a) the method of Hill’s determinants and (b) the simplified method in which the sinusoidal functions are approximated as square waves, while the markers indicate the stable configurations as determined by solving numerically the PDE.

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