PSI - Issue 71

Ishaan Tapas et al. / Procedia Structural Integrity 71 (2025) 477–483

482

form formula derived in this work with those obtained from ANSYS for various degrees of taper. It is observed that there is a good agreement between the natural frequencies obtained from the derived formula and FEM simulation.

Perturbation Formula: ( 1 + 2 ) 0.5 3.5936 20.6127

Table 2. Comparison of analytical and FEM results.

Taper Mode Number

ANSYS 3.5967 20.2167 3.6554 19.5285 3.7260 18.81051

Error (%)

1 2 1 2 1 2

-0.084

ε=0.2

1.95 -0.63 1.71 -1.50 1.46

3.6320 19.8642 3.67005 19.08646

ε=0.3

ε=0.4

ṁ Figure 2 compares the variation of first two natural frequencies obtained from the derived perturbation formula and ANSYS. It is observed that the formula shows good correlation with ANSYS up to the taper ratio of 0.55 (within ±2% error margin). As the taper parameter increases, the correction term becomes comparable to the order zero term. As a result, the perturbation formula becomes inaccurate. This problem can be mitigated by including second order correction terms in the perturbation formula. Conclusion The Lagrangian for the tapered beam is formulated using first principles. The Rayleigh-Ritz method is used to obtain the characteristic polynomial for the system. The mode shapes of a uniform straight cantilever beam were considered as shape functions for this purpose. Considering taper as a perturbation over a uniform straight beam, a closed-form formula using the perturbation method was derived. It was observed that the perturbation formula works efficiently upto the taper ratio of 0.6, wherein the results lie within the 2% error margin over the FEM results. This is expected as the perturbation theory works only for small perturbation terms. This is due to the correction term becoming comparable (in order) with the zeroth term. The accuracy of the formula can be improved by considering the next 3.