Finite Difference Approach to Stability and Vibration Analysis of Line Continuum
Okpara, U. O.
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This research work presents Stability and Vibration Analysis of Line Continuum with various boundary conditions using Finite Difference Method developed from Polynomial series expansion, truncated at the fourth term. The boundary conditions considered were pin – roller support (P – R), clamped – clamped support (C – C) and clamped – roller support (C – R). Considering the three boundary conditions stated above, three cases of line continuum were studied using the various boundary conditions which are Line continuum with three flexible nodes, Line continuum with five flexible nodes and Line continuum with seven flexible nodes. The Finite Difference pattern derived from polynomial expansion series was substituted into each case of line continuum to obtain matrices as well as Eigen values for stability and vibration analysis respectively and the natural frequency and buckling load were then sought. The values from this study were compared to the exact result from previous work. For Stability analysis, the results of critical buckling load obtained for P – R support has a percentage difference ranging from -5.03% to 1.31% as compared to exact result. For C – C support, it gave a percentage difference of -18.94% to -5.03% while that of C – R support gave a percentage difference of -11.19% to -2.88%, all when compared to exact result. For Vibration analysis, the natural frequency obtained for P – R support gave a percentage difference ranging from -2.75% to -0.65%. For C – C support , it gave a percentage difference of -13.1% to -3.35% and C – R support revealed a percentage difference ranging from -7.51% to -1.84%, all when compared to values from exact result. The percentage difference obtained gave negative differences which depicts Finite Difference method provides a lower bound solution. These results showed that this Finite Difference Method, truncated at the fourth – order gave approximate solutions close to the exact results from previous works and hence, can be applied in structural engineering.