# Vibration and Reliability Analysis of Non-Uniform Composite Beam under Random Load
###### Abstract
Differential transformation method and transfer matrix method (TMM) were also used to study the free vibration of an FGM beam in references [34; 35]. [PERSON] et al. developed an analytical solution method using [PERSON]'s procedure to study the dynamic behavior of carbon nanotube reinforced functional gradient (FG) beams on two parameter elastic foundation under a moving load [36]. [PERSON] et al. studied and analyzed the free vibration and dynamic response of an S-shaped FG Timoshenko beam model under a moving load based on [PERSON]'s method [37]. There is also some research work on the non-uniform FGM beam. [PERSON] used the differential transformation (DT) based dynamic stiffness approach to study the free vibration of axially functionally graded non-uniform beams under different boundary conditions [38]. [PERSON] and [PERSON] described the effect of different profile variations on vibrational properties of non-uniform beams made of graded porous materials [39]. Under the framework of nonlocal strain gradient theory, [PERSON] and [PERSON] used the Lagrange interpolation method, Gaussian quadrature method and Wilson multiplier method to study the bending, buckling and vibration behavior of depth functionally graded in depth direction non-uniform nanobeams [40]. Adomian Decomposition Method (ADM) has been used to analyze the free and forced vibration of general non-uniform cone beams with axial functionally graded material by [PERSON] et al. [41; 42; 43]. Based on dynamic modeling and analysis of different structures, their reliability analysis under random load excitation can be conducted. [PERSON] and [PERSON] carried out nonlinear dynamic analysis of the platform for response calculation. The response history obtained has been used for the fatigue reliability analysis of Tension leg platform (TLP) tethers under long crested random sea and associated wind [44]. [PERSON] et al. used a standard gradient-based algorithm with a line search to solve the reliability optimization problem of a structural system under random load [45]. [PERSON] and [PERSON] deduced the calculation expression of the mean value of structural dynamic reliability under stationary random excitation, and conveniently obtained the dynamic reliability analysis results of random structures under random excitation [46]. [PERSON] et al. determined the fatigue reliability of an automotive crankshaft by stochastically inducing random loads [47]. In summary, there have been many studies on the dynamic analysis of composite structures and non-uniform beams, and the vibration and reliability analysis of uniform structures and mechanical parts with certain design under random loads is also a common and well-studied problem in engineering. However, to the best of the authors' knowledge, there is no research on the vibration reliability analysis of a non-uniform composite beam under a random load. Considering the various mechanical properties as well as unknown response and strain/stress distribution of the non-uniform structures made from multiple composite layers, the reliability of such structures can be different from traditional uniform structures as novel research [48; 49], especially under different random dynamic loads. There is no explicit expression for the random load whose excitation frequency and amplitude are changing, so the analytical solution of forced vibration at each time point needs to be solved by iterative method. To deal with this challenge, a model combining ADM and iteration numerical process is proposed to solve the vibration response and dynamic stress-distribution of the non-uniform composite beams for reliability analysis in this research. In this paper, the vibration responses and reliability of a general non-uniform beam with tapered composite layers and host beam under random load are calculated and analyzed. The cross-section rigidity and mass distribution can be adjusted by changing the volume proportion of different materials of the non-uniform laminated composite beam.