## DYNAMIC ANALYSIS OF BEAMS BY THE MODIFICATION OF THE SYSTEM’S STIFFNESS DISTRIBUTION

SOURCE:

Faculty: **Engineering**

Department: **Civil**

CONTRIBUTORS:

**Okonkwo, O. V.
Aginam, C. H.
Nwaiwu, C. M. O. **

ABSTRACT:

For structural systems with complex geometries approximate solutions can be obtained by the lumping of continuous masses. The lumping of continuous mass has the effect of reducing the degrees of freedom of the system from infinity to a finite value. However the mass distribution of the structure is altered by lumping thus introducing error in the dynamic analysis. In this work lumped mass beams and frames were analysed dynamically by the modification of the structure’s stiffness distribution. This was achieved by formulating the force equilibrium equations for beams of different end conditions under vibration in order to determine the equations of the inherent forces causing vibration. The lumped mass structures were simulated with the equations of motion. The equations for the fixed end forces {F} and nodal displacements {D} were formulated for any arbitrary segment of a vibrating beam at time t = 0. This was substituted into the element force equilibrium equations to obtain the vector of nodal forces {P} that was causing the vibration. The obtained nodal forces and displacement were substituted into the equations of motion to obtain the modified stiffness values as functions of a set of stiffness modification factors. By employing the Lagrange equations to lumped massed beams using these modification factors, we were able to predict accurately the fundamental frequency of beams irrespective of the position or number of lumped mass introduced. The predicted values of natural frequencies for frames in this study when compared with those from experimental studies and the finite element software ANSYS, were observed to be exact for beams and nearly exact for columns (error < 3%), which is very much acceptable in engineering design. The results of the present study also gave a statistically significant (at 95% confidence level) improvement over predictions made without the modification of the structure’s stiffness distribution.