Modeling Natural Frequencies of Vibration of Three Dimensional Frames Under Two Dimensional Loading

Abstract

The aim of this study was to determine the relationship between natural frequency of vibration, the height of the structure, the stiffnesses of members and number bays of a structure. The relationship was to be developed based on data obtained using two methods. The methods were theoretical whereby Computer Modeling was undertaken based on structural theory, and experimental, whereby physical prototypes of structures were subjected to free vibrations. In the theoretical method, a matrix approach to analysis was adopted to develop a computer program which generated structural models. A horizontal force would be applied at the topmost joint of each model and deflection at the centre of mass was calculated, which was the amplitude of vibration. The overall stiffness of the structure was calculated using the structural amplitude obtained. The overall stiffness was then used to calculate the frequency of vibration for each structural model. In the experimental method, physical models of miniature structures were built with different heights, member stiffnesses and number of bays. Each model was subjected to free vibrations and the deflections against time were measured. To simulate a free vibration in a model, a measured horizontal force was applied at a joint located at the top of the structure to produce an initial deflection and then the force was withdrawn to allow free vibrations. The deflections at the centre of gravity for free vibrations were measured against time. The equipment used to measure the free vibrations was horizontal motion transducer. The instrument has a probe which gets depressed when anobject is pushed against it. The transducer was attached to a TDS 302 data-logger which printed deflection against time. The data obtained was analysed graphically. It was found that the theoretical values were very close to the experimental values, with very high positive correlation coefficients.A relationship between the natural frequency of vibration and the various parameters was developed. The relationship obtained will enable the engineering design of tall buildings against dangerous resonance with the forces they are subjected to. The relationship will be used to calculate the natural frequency of a proposed structure. If the frequency is found to the same of that of the forces expected to act on the structure, a change will be made in the dimensions of the structure to change its frequency. This way catastrophic resonance will be avoided.