# Vibration and Reliability Analysis of Non-Uniform Composite Beam under Random Load
## 4 Conclusions  
In this paper, an iterative method is proposed to solve the forced vibration responses and reliability of non-uniform composite beam under random load. Considering the change of frequency and acceleration amplitude of base motion excitation in the iteration process, the vibration response of beam in each short time period/iteration step is calculated by ADM and Duhamel integral considering both the transient and steady response. Through numerical studies, the results reveal the following conclusions:  
(1) ADM is used to be combined with iteration progress solving non-uniform composite beam structural vibration under random load. The correctness of the mathematical model is proved. Comparing solutions under random and harmonic base motion, the vibration response under random load shows the randomness and non-frequency characteristics, and its vibration amplitude is larger than the one under the same level harmonic excitation.  
(2) Within the ranges of studied design parameters, the composite beam nature frequencies increase with the increment of \(h_{0}\), \(n\), \(m\) and decrease with the increment of \(h\). \(b_{0}\) has little effect on nature frequencies.  
(3) The higher mean value and variance of base motion excitation frequency, the smaller amplitude and maximum strain of the non-uniform composite beam can be excited.  
(4) The greater mean and variance of the base motion excitation amplitude, the greater vibration amplitude and strain of the non-uniform composite beam, and the lower the vibration reliability in the same time period.  
(5) The vibration reliability will increase or decrease with the increment of operation time considered, but the overall trend is downward if the operation time of the system is long enough.  
(6) The reliability of the non-uniform composite beam increases with the increment of \(n\) and decreases with the increment of \(m\). For the multiple layered composite beam, different design parameters lead to different reliability results. Reasonable design can enhance structural reliability.  
(7) However, the research results obtained within the range of parameters studied occur when the operation time is long enough. When the calculation/analysis time duration is very short, the results may change. The statement that the operation time is long enough means that the analyzed period should be enough to ensure that the excitation amplitude and frequency distribution law can be illustrated and judged from the data points of the random loads in the calculation process.  
Conceptualization, P.W. and N.W.; methodology, P.W.; software, P.W.; validation, P.W., H.L. and Z.S.; formal analysis, P.W.; investigation, P.W.; data curation, P.W.; writing--original draft preparation, P.W.; writing--review and editing, N.W.; supervision, N.W.; project administration, N.W. and Z.S.; funding acquisition, N.W. and Z.S. All authors have read and agreed to the published version of the manuscript.  
This research was funded by National Natural Science Foundation of China (No. 51775097) and the University of Manitoba, and the Natural Sciences and Engineering Research Council of Canada (NSERC). This research was also funded by the China Scholarship Council (File No. 202106080045).