PSI - Issue 71

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

481

In this work, we employ the Rayleigh-Ritz method in conjunction with perturbation method to derive the closed form formula for natural frequency. In general shape function form is written as ( ) = ∑ 6 =1 ( ) ( ) represents the j th mode shape of straight uniform cantilever beam, represents undetermined constants. The general form for ( ) is given as ( ) = ( ) − ℎ ℎ ( ) − ( ( ) + ℎ ℎ ( ) ( ) + ℎ ℎ ( ) ) ( ( ) − ℎ ℎ ( ) ) The values of first six modes are reported by S. S. Rao. It is an admissible shape function for the tapered beam since the boundary conditions are identical to the uniform beam. The assumed form of ( ) is substituted in equation (10). Differentiating the Lagrangian with respect to constants i.e. ( 1, 2, 3, 4, 5, 6) six algebraic equations are obtained. Non-trivial solution of the corresponding matrix equation is obtained by equating the determinant to zero. Expanding the determinant, a polynomial in terms of is obtained. The roots of the polynomial give the natural frequencies of the system. We seek to derive the perturbation formula for tapered beam as a correction over the well known case of uniform cantilever beam. Thus, the formula should have the following form: Natural Frequency of Tapered beam = Natural Frequency of Straight Beam + Correction due to Taper in form of perturbation Therefore, substituting appropriately as = 1 + 2 (12) where, is square of natural frequency of system, 1 is fourth power of natural frequency of straight uniform cantilever beam, 2 is the correction factor for taper ratio. This expression is substituted in the determinant of the coefficient matrix. The large expression thus obtained is not given for brevity. Now, we collect the zeroth order terms O(0) of and equate the resulting equation to zero, fourth power of non dimensional natural frequencies ( 1 ) of a uniform straight cantilever beam are obtained, which is expected. Similarly, collecting first order terms O(1) of equating these terms to zero and substituting the value of 1 , correction factor 2 is obtain for the corresponding value of 1 . The values of 1 and 2 for first three modes are tabulated in Table 1.

Table 1. Numerical values for correction factors for first three modes. Mode Numbers First Mode

1 2

Second Mode

Third Mode

12.362 2.7742

485.48 -302.97

3806.43

Correction Factors

-3325.7016

2.2 Solution of Tapered Beam To validate the correctness of the derived formula, a high-fidelity FEM model is solved in ANSYS. The natural frequencies obtained from the FEM simulation are appropriately non-dimensionalized. The tapered beam was modelled using the BEAM189 element. The taper was introduced using a macro which assigned the appropriate cross sectional area and moment of inertia to each element. The convergence of the solution was ensured through a mesh convergence study. The kinematic boundary condition of a cantilever beam was imposed at the appropriate node, and modal analysis was performed in ANSYS APDL. Table 2 compares the natural frequencies obtained from the closed-