PSI - Issue 71

480 Ishaan Tapas et al. / Procedia Structural Integrity 71 (2025) 477–483 where ρ is the density and ω is the natural frequency of the beam. As mentioned, in a tapered beam, cross section area varies linearly along the length as shown in the Fig. 1. Let, b be the width of the beam, 2 is the height of the beam at root, 1 is the height of the beam at tip, height of the beam at any x is given by ( ) = 1 + 2 ( − ) (5) Therefore, the equations for area and moment of inertia become, ( ) = ( ) = ( 1 + 2 ( − )) = 0 (1 − ) (6) where, 0 = 2 , is the cross-sectional area at the root and =1− 1 2 isthe perturbation parameter. Similarly, we get ( ) = 1 2 ( 1 + 2 ( − )) 3 = 23 12 ( 1 2 + 1 − ) 3 = 0 (1 − ) 3 (7) where, 0 = 1 1 2 23 is the moment of inertia at the root. The Lagrangian is given by: = − = 2 2 ∫ 0 ( ) 2 − 12 ∫ 0 ( 2 2 ) 2 (8) Substituting appropriate expressions, the Lagrangian becomes, = 2 2 ∫ 0 ( ( 1 + 2 ( − ))) 2 − 12 ∫ 0 (1 2 ( 1 + 2 ( − )) 3 )( 2 2 ) 2 (9) We define the following parameters in order to non-dimensionalize the Lagrangian: = 1 − 1 2 , = , = , = 2 0 4 0 where is the perturbation parameter for taper, U are the non-dimensional displacements, is the non-dimensional length coordinate and is the square of non-dimensional natural frequency. Using the above parameters in equation (9), the Lagrangian becomes, = 2 ∫ 1 0 (1 − ) 2 − 12 ∫ 1 0 (1 − ) 3 ( 2 2 ) 2 (10) It can be noted that in the case of =0 we obtain the Lagrangian for a straight uniform beam. The boundary conditions can be derived from the Lagrangian. The kinematic boundary conditions are in terms of prescribed displacement and slope at the fixed end. They have the following form, =0, ( ) =0 =0 (11) These boundary conditions are same as of a straight Euler-Bernoulli uniform cantilever beam. In this part it is assumed that the dynamics of a tapered beam is valid for small value of tapered parameter. It will be shown that this simplifying assumption leads to a simple formula for evaluating natural frequencies of tapered beam.