• Journal of the Korean Society for Precision Engineering
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• v.16 no.11
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• pp.204-212
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• 1999

In this paper, the vibrational motion of a flexible beam clamped on a translating base and carrying a moving mass is investigated. The equations of motion which describe the total dynamics of the beam-mass-cart system are derived and the coupled dynamic equations are solved by unconstrained modal analysis. In modal analysis, the exact normal mode solutions corresponding to the eigenfrequencies for the position of the moving mass and the ratios of the mass of the flexible beam, the moving mass and the base cart are used. Proper transformations of the time solutions between the normal modes for a position and those for the next position of the moving mass are also adopted. Numerical simulations are carried out to obtain the open-loop responses of the system in tracking the pre-designed path of the moving mass.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.23 no.11 s.170
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• pp.1940-1951
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• 1999

In this paper, the equations of motion of a Bernoulli-Euler cantilever beam fixed on a moving cart and carrying a lumped mass concentrated at an arbitrary position along the beam is derived. The motion of the beam-mass-cart system is analyzed through unconstrained modal analysis, and a unified characteristic equation for calculating the natural frequencies of the system is obtained. The changes of natural frequencies and the corresponding mode shapes with respect to the changes in mass ratios of the system and to the concentrated position of the lumped mass are investigated with the frequency equation, which can be generally applied to this kind of systems. The exact and assumed-mode solutions including the dynamics of the base cart are obtained, and the open-loop responses of the system by arbitrarily designed forcing function are given by numerical simulations. The results match well with physical phenomena even at the extreme cases where the concentrated mass is attached to the bottom and to the top of the beam.