Analytical Solution for the Free Vibration Analysis of Delaminated Timoshenko Beams

Abstract

This work presents a method to find the exact solutions for the free vibration analysis of a delaminated beam based on the Timoshenko type with different boundary conditions. The solutions are obtained by the method of Lagrange multipliers in which the free vibration problem is posed as a constrained variational problem. The Legendre orthogonal polynomials are used as the beam eigenfunctions. Natural frequencies and mode shapes of various Timoshenko beams are presented to demonstrate the efficiency of the methodology.

Natural vibration-induced parametric excitation in delaminated Kirchhoff plates

“…By setting the determinant of the matrix term to be equal to zero provides the critical load as N x,cr ¼ N x or N xy,cr ¼ N xy . The mode shapes are calculated by taking back the load control parameters (eigenvalues) into equation (56). It will be assumed that the eigenshape vectors for static and dynamic buckling are essentially the same.…”

“…It is important to note that the associated mode shapes are calculated from equation (56), in other words it is assumed that the dynamic and static buckling mode shapes are equivalent. The estimated dynamic buckling forces under the action of trigonometric widthwise forces are calculated by producting the critical dynamic forces by the trigonometric load multiplicator.…”

“…The critical value of the in-plane normal and shear forces (F d ¼ N x or N xy ) can be calculated by taking back any of the free vibration frequencies and equating the determinant of the matrix in equation (66) to zero. It is important to note that the associated mode shapes are calculated from equation (56), in other words it is assumed that the dynamic and static buckling mode shapes are equivalent. The estimated dynamic buckling forces under the action of trigonometric widthwise forces are calculated by producting the critical dynamic forces by the trigonometric load multiplicator.…”

“…38 This paper was inspired by the really huge literature related to the vibration analysis of delaminated composite beams including dynamic stiffness method, 39 traditional methods [40][41][42][43] and application even to multiple delaminated beams, [44][45][46][47] finite element (FE) analysis, 48,49 postbuckled plates, 50 and solution relative to buckled state. 51 Moreover, many papers deal with the vibration of delaminated beams with finite element method (FEM), [52][53][54][55] application of Timoshenko beams with delamination 56,57 including moving loads and masses [58][59][60][61][62][63] and elastic foundation. 64 The solution of the problems is also possible by the differential quadrature method.…”

Free vibration analysis of elastically restrained cantilever Timoshenko beam with attachments

“…Salarieh and Ghorashi [13] analysed the free vibration of a cantilever Timoshenko beam with rigid mass and compared with other beam theories. In the work by Jafari-Talookolaei and Abedi [14], a new method was presented to obtain the exact solution for the free vibration of a Timoshenko beam with different boundary conditions. The vibration analysis of a cantilever beam with an eccentric three dimensional object has been investigated by Kati and Gökdag [15].…”