Simulation of the vibrations of a non-uniform beam loaded with both a transversely and axially eccentric tip mass

Abstract

The main purpose of this work is to employ the Adomian modified decomposition method (AMDM) to calculate free transverse vibrations of non-uniform cantilever beams carrying a transversely and axially eccentric tip mass. The effects of the variable axial force are taken into account here, and Hamilton's principle and Timoshenko beam theory are used to obtain a single governing non-linear partial differential equation of the system as well as the appropriate boundary conditions. Two product non-linearities result from the analysis and the respective Cauchy products are computed using Adomian polynomials. The use of AMDM to make calculations for such a cantilever beam/tip mass arrangement has not, to the authors' knowledge, been used before. The obtained analytical results are compared with numerical calculations reported in the literature and good agreement is observed. The qualitative and quantitative knowledge gained from this research is expected to enable the study of the effects of an eccentric tip mass and beam non-uniformity on the vibration of beams for improved dynamic performance.

Elzaki Transform Method for Natural Frequency Analysis of Euler-Bernoulli Beams

“…Consideration of boundary conditions for simply supported beams reduced the problem to the homogeneous algebraic eigenvalue problem in Equation (24). The frequency equation obtained for simply supported beams is determined as Equation ( 25) which upon expansion gave the eigenequation in Equation (26). The solution of Equation (26) gives the eigenvalues given in Equation ( 27) and the exact natural frequencies which are given in Table 1.…”

“…The frequency equation obtained for simply supported beams is determined as Equation ( 25) which upon expansion gave the eigenequation in Equation (26). The solution of Equation (26) gives the eigenvalues given in Equation ( 27) and the exact natural frequencies which are given in Table 1. Table 1 shows exact agreement with previous works of Avcar [6], Hurty and Rubinstein [12] and Ike [35].…”

“…They applied their formulation to determine the natural frequencies for prismatic and non-prismatic thin beams for varieties of boundaries and developed satisfactory eigenfrequencies solutions that agreed with exact eigenfrequencies. Adair and Jaeger [26], used the Adomian modified decomposition method (AMDM) to solve the transverse vibration problems of non-prismatic cantilever Timoshenko beams subjected to a transversely and axially eccentric tip mass. Torabi et al [27], utilized the Variational Iteration Methodology (VIM) to study linear and non-linear transverse vibration problems of thin beams.…”

Effect of the Cross-Sectional Shape on the Dynamic Response of a Cantilever Steel Beam Using Three Modal Analysis Methods

“…Ghaemdoust et al [7] used the analytical and the finite element method to determine the fundamental natural frequency of unbonded pre-stressed beams. Adair and Jaeger [8] calculated non-uniform cantilever beams with a free transverse vibration caused by an axially and transversely eccentric tip mass. Here, the effects of the varying axial force are considered.…”

Effect of Thrust on the Structural Vibrations of a Nonuniform Slender Rocket

“…Lai et al [21] and Lai et al [22] employed both the AMDM and Adomian decomposition methods as an innovative eigenvalue solver to determine the free vibration of an Euler-Bernoulli beam under various supporting conditions. Yaman [23] used the Adomian decomposition method to investigate the influence of the orientation effect on the natural frequency of a cantilever beam carrying a tip mass, while Adair and Jaeger [24] investigated the vibrations of a beam with both a transversely and axially eccentric tip mass present using the AMDM.…”