doi:10.1016/j.jsv.2004.07.029 
Copyright © 2004 Elsevier Ltd All rights reserved.
Short Communication
Analysis of buckled and pre-bent fixed-end columns used as vibration isolators
R.H. Plauta, 
, 
, J.E. Sidburya and L.N. Virginb
aDepartment of Civil and Environmental Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0105, USA
bDepartment of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708-0300, USA
Received 19 February 2004;
revised 18 June 2004;
accepted 19 July 2004.
Available online 16 December 2004.
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Abstract
The use of a buckled or pre-bent column with fixed ends as a vibration isolator is analyzed. The column is designed to have a high axial stiffness under the weight that it supports, so that the static displacement of the weight is not excessive, and then to have a low stiffness during excitation. The base of the column is assumed to have an axial motion which is simple harmonic or a linear combination of two simple harmonic functions. The column is modeled as an elastica. First the equilibrium shape under the supported weight is determined. Then small steady-state vibrations about the equilibrium configuration are obtained numerically using a shooting method. The inertia of the supported weight and the transverse and axial inertias of the column are included. The axial displacement transmissibility is computed, and the effects of external and internal damping, column stiffness, supported weight, and initial curvature are investigated. For the two-frequency excitation, the effects of the relative amplitudes and frequencies of the excitation components are considered.
Fig. 1. Geometry of buckled or pre-bent column subjected to base motion and to static and inertia forces of supported weight, in nondimensional terms.
Fig. 2. Free body diagram of column element, including inertia and external damping forces.
Fig. 3. Variation of transmissibility with ω√r (proportional to dimensional frequency); 
. 
, r=0.1; - - - -, r=1; ——, r=10.
Fig. 4. Maximum transmissibility vs. stiffness parameter; c=1,γ=0,a0=0. - - - -, p0=40; ——, p0=41.
Fig. 5. Transmissibility curves; p0=40,r=0.1,c=0,a0=0. ——, γ=0; - - - -, γ=0.001; ·, γ=0.01; —·—, γ=0.1.
Fig. 6. Transmissibility curves; r=1,c=1,γ=0,a0=0. - - - -, p0=40; ——, p0=41.
Fig. 7. Transmissibility curves; p0=40,r=1,c=1,γ=0. - - - -, a0=0; ·, a0=0.01; — —, a0=0.05; ——, a0=0.1.
Fig. 8. Transmissibility curves; p0=10,r=1,c=1,γ=0. ·, a0=0.01; - - - -, a0=0.05; ——, a0=0.1.
Fig. 9. Weight vs. peak frequency; r=1,c=1,γ=0. — —, a0=0.01; - - - -, a0=0.05; ——, a0=0.1.
Fig. 10. Transmissibility curves with 0<ω<300;r=1,c=0.1,γ=0,a0=0.05. ——, p0=10; - - - -, p0=20; — —, p0=30; —·—, p0=40.
Fig. 11. Transmissibility curves with 0<ω<300 and 0<TR<2;r=1,c=0.1,γ=0,a0=0.05. - - - -, p0=20; ——, p0=40.
Fig. 12. Transmissibility curves; p0=40,r=1,c=1,γ=0,a0=0,rf=1.5. ··, ra=0.5; - - - -, ra=1.0; ——, ra=1.5.
Fig. 13. Variation of transmissibility with ω√r (proportional to dimensional frequency); 
. ·, r=0.1; - - - -, r=1; ——, r=10.
Fig. 14. Transmissibility curves; p0=40,r=0.1,c=1,γ=0,a0=0. —— ra=0 (single-frequency excitation); - - - - ra=1 and rf=2 (two-frequency excitation).
Table 1.
Central displacement and end shortening for various supported weights, in nondimensional terms
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