Dynamic response of a cracked beam under moving load using adomian decomposition method

Some engineering applications cover beam members such as bridges, train rails, aircraft wings and etc. They can be modelled in terms of the beam. Once, the beam model is defined, then its governing equations are obtained to do dynamic analysis. These equations are complicated in general. Many solution techniques can be adopted to solve these equations. One of these techniques is Adomian methods which have begun to be used to solve complex differential equations recently. In this study, the Adomian Decomposition Method (ADM) as well as its modified (MADM) and improved (IADM) forms, is utilized to carry out the vibration response of the Euler-Bernoulli beam with a crack under a moving load. The model consists of a moving load acting on the two continuous segments where the crack was modelled as a sectional flexibility rotational spring. The equations of motion of the spans become the fourth order, non-homogenous partial differential equations which are transformed to a second-order, non-homogeneous, ordinary differential equation using modal analysis, considering initial and boundary conditions. Modal displacement of the beam is represented in a series expansion using ADM. These infinite series are generalized using Binomial and Taylor series expansions. The analytical solution of dynamic response was obtained for cracked simply supported beam and it revealed the effects of crack with different crack location, crack size, amplitude and velocity of acting load and beam length. In addition, the influence of crack on natural frequency was studied. Some numerical results and of the midpoint deflection of cracked beam were obtained by using MATLAB software and the midpoint deflection was presented as graph. The solution method was verified with some reference studies. The solution method was verified with some reference studies.