• Journal of KSNVE
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• v.7 no.1
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• pp.91-97
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• 1997

An analysis is presented on the stability of an elastic cantilever column having the elastic restraints at its free end, carrying an added tip mass, and subjected to uniformly distributed follower forces. The elastic restraints are formed by both a translational spring and a rotatory spring. For this purpose, the governing equations and boundary conditions are derived by using Hamilton's principle, and the critical flutter loads and frequencies are obtained from the numerical evaluation of the eigenvalue functions of this elastic system. The added tip mass increases as a whole the critical flutter load of the elastic cantilever column, but the presence of its moment of inertia of mass has a destabilizing effect. The existence of the translational and rotatory springs at the free end increases the critical flutter load of the elastic cantilever column. Nevertheless, their effects on the critical flutter load are not uniform because of their coupling. The translational spring restraining the free end of the cantilever column decreases the critical flutter load by coupling with a large value of tip mass, while by coupling with the moment of inertia of tip pass its effect on the critical flutter load is contrary. The rotatory spring restraining the free end of the cantilever column increases the critical flutter load by coupling with the tip mass, but decreases it by coupling with the moment of inertia of the tip mass.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 1996.04a
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• pp.309-315
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• 1996

An analysis is presented on the stability of elastic cantilever column subjected to uniformly distributed follower forces as to the influence of the elastic restraints and a tip mass at the free end. The elastic restraints are formed by both the translational and the rotatory springs. For this purpose, the governing equations and boundary conditions are derived by using Hamilton's principle, and the critical flutter loads and frequencies are obtained from the numerical evaluation of the eigenvalue functions of this elastic system. The added tip mass increases as a whole the critical flutter load in this system, but the presence of its moment of inertia of mass has a destabilizing effect. The existence of the translational and rotatory spring at the free end increases the critical flutter load of the elastic cantilever column. Nevertheless their effects on the critical flutter load are not uniform because of their coupling. The translational spring restraining the end of cantilever column decreases the critical flutter load by coupling with a large value of tip mass, while by coupling with the moment of inertia of tip mass its effect on the critical flutter load is contrary. The rotatory spring restraining the free end of cantilever column increases the critical flutter load by coupling with the tip mass, but decreases it by coupling with the moment of inertia of tip mass.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2005.11a
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• pp.903-906
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• 2005

The purpose of this paper is to investigate free vibrations and critical loads of the uniform Beck's columns with a tip spring, carrying a tip mass. The ordinary differential equation governing free vibrations of such Beck's column subjected to a follower force is derived based on the Bernoulli-Euler beam theory. Both the divergence and flutter critical loads are calculated from the load-frequency curves that are obtained by solving the differential equation numerically. The critical loads are presented in the figures as functions of various non-dimensional system parameters such as the mass moment of inertia and spring parameter.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.19 no.1 s.71
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• pp.85-92
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• 2006

This paper investigates critical loads of the tapered Beck's columns with clamped and spring supports, subjected to a subtangential follower force. The linearly tapered columns with the solid rectangular cross-section is adopted as the column taper. The differential equation governing free vibrations of such Beck's columns is derived using the Bemoulli-Euler beam theory. Both divergence and flutter critical loads are calculated from the load-frequency curves which are obtained by solving the differential equation. The critical loads are presented as functions of various non-dimensional system parameters: the taper type, the subtangential parameter and the spring stiffness.

• Journal of the Korean Society for Precision Engineering
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• v.21 no.11
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• pp.179-185
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• 2004

The effect of viscous damping on stability of beam resting on an elastic foundation subjected to a dry friction force is analytically studied. The beam resting on an elastic foundation subjected to dry friction force is modeled for simplicity into a beam resting on Kelvin-Voigt type foundation subjected to distributed follower load. In particular, the effects of four boundary conditions (clamped-free, clamped-pinned, pinned-pinned, clamped-clamped) on the system stability are considered. The critical value and instability type of columns on the elastic foundation subjected to a distributed follower load is investigated by means of finite element method for four boundary conditions. The elastic foundation modulus, viscous damping coefficient and boundary conditions affect greatly both the instability type and critical load. Also, the increase of damping coefficient raises the critical flutter load (stabilizing effect) but reduces the critical divergence load (destabilizing effect).

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2007.11a
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• pp.344-347
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• 2007

In this paper, the stability of a cracked cantilever beam subjected to follower force is presented. In addition, an analysis of the flutter instability(flutter critical follower force) of a cracked cantilever beam subjected to a follower compressive load is presented. Based on the Timoshenko beam theory. The vibration analysis on such cracked beam is conducted to identify the critical follower force for flutter instability based on the variation of the first two resonant frequencies of the beam. Besides, the effect of the crack's intensity and location on the flutter follower force is studied. The crack section is represented by a local flexibility matrix connecting two undamaged beam segments. The crack is assumed to be in the first mode of fracture and to be always opened during the vibrations. Generally, the critical follower force for flutter is proportional to the crack depth.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 2006.04a
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• pp.949-954
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• 2006

The purpose of this paper is to investigate the stability of tapered columns with clamped one end and carrying a tip mass of rotatory inertia with translational elastic support at the other end. The linearly tapered columns with the solid rectangular cross-section is adopted as the column taper. The differential equation governing free vibrations of such Beck columns is derived using the Bernoulli-Euler beam theory. Both the divergence and flutter critical loads are calculated from the load-frequency curves which are obtained by solving the differential equation. The critical loads are presented as functions of various non-dimensional system parameters: the taper type, the subtangential parameter, mass ratio and spring stiffness.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2006.11a
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• pp.801-804
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• 2006

The purpose of this paper is to investigate the stability of stepped cantilever columns with a tip mass of rotatory inertia and a translational spring at one end. The column model is based on the Bernoulli-Euler theory which neglects the effects of rotatory inertia and shear deformation. The governing differential equation for the free vibration of columns with stepwise variable cross-section and subjected to a subtangential follower force is solved numerically using the corresponding boundary conditions. And the bisection method is used to calculate the critical divergence/flutter load. The frequency and critical divergence/flutter load for the stepped column with a single step are presented as functions of various non-dimensional system parameters: the segmental length parameter, the section ratio, the subtangential parameter, the mass, the moment of inertia of the mass, and the spring parameter.

• Journal of Korean Society of Steel Construction
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• v.17 no.6 s.79
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• pp.699-705
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• 2005

This paper investigates critical loads of tapered cantilever columns with a tip mass, subjected to a follower force. The linearly tapered solid rectangular cross-sections are adopted as the column taper. The differential equation governing free vibrations of such columns, also called Beck's columns, is derived using the Bernoulli-Euler beam theory. Both divergence and flutter critical loads are calculated from the load-frequency curves that are obtained by solving the differential equation. The critical loads are presented as functions of various non-dimensional system parameters, namely, the taper type, the subtangential parameter, and the mass ratio.

• International Journal of Aeronautical and Space Sciences
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• v.15 no.1
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• pp.20-31
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• 2014

This paper deals with the development of a realistic shape optimization of damaged columns that are subjected to conservative and non-conservative forces, using the Genetic Algorithm (GA). The analysis is based on the design of the most optimized shape of the column under the constraint of constant weight, considering the Static, Vibrational, and Flutter characteristics. Under the action of conservative and non-conservative longitudinal forces, an elastic column loses its stability. A numerical analysis based on FEM has been performed on a uniform damaged column, to compute the fundamental buckling load, vibration frequency, and flutter load, under various end restraints. An optimization search based on the Genetic Algorithm is then executed, to find the optimal shape design of the column. The optimized column references the one having the highest buckling load, highest vibration frequency, and highest flutter load, among all the possible shapes of the column, for a given volume. A comparison is then made between the values obtained for the optimized damaged column, and those obtained for the optimized undamaged column. The comparison reveals that the incorporation of damage in the column alters its optimal shape to only a certain extent. Also, the critical load and frequency values for the optimized damaged column are comparatively low, compared with those obtained for the optimized undamaged column. However, these results hold true only for moderate-intensity damage cases. For high intensity damage, the optimal shape may not remain the same, and may vary, according to the severity of damage.