# Vibration and Reliability Analysis of Non-Uniform Composite Beam under Random Load
## 3 Numerical Studies, Results and Discussion
### Influences of Different Ranges of Base Motion Excitation Angular Frequency on Vibration Responses  
During the vibration process under random base motion excitation, the influences of exact amplitude and frequency with different means and variances on vibration characteristics are different. Firstly, the influences of base motion excitation frequency on vibration responses were studied. The amplitude of acceleration of base motion accord with Gaussian distribution with \(A_{a}\sim N\big{(}50,10^{2}\big{)}\). The amplitude of base motion is \(Y=\frac{A_{a}}{\phi^{2}}\). Four groups of base motion excitation angular frequencies with Gaussian distribution and standard deviation of \(1/15\) of the average value were selected for analysis. The specific base motion excitation frequency variation ranges are \(\varphi_{1}\sim N\big{(}200,(200/15)^{2}\big{)}\) rad/s, \(\varphi_{2}\sim N\big{(}440,(440/15)^{2}\big{)}\) rad/s, \(\varphi_{3}\sim N\big{(}3600,(3600/15)^{2}\big{)}\) rad/s, and \(\varphi_{4}\sim N\big{(}11000,(11000/15)^{2}\big{)}\) rad/s. The first three natural frequencies of non-uniform composite beam with \(b_{0}=0.005\) m, \(h_{0}=0.001\) m, \(n=2\), \(m=2.5\) and \(h=1.5\) are 439.2 rad/s, 3555.4 rad/s and 10832.3 rad/s. The frequency ranges of \(\varphi_{2}\), \(\varphi_{3}\) and  
Figure 2: Vibration responses of non-uniform composite beam at free end under base motion with random frequency and amplitude of acceleration and harmonic base motion: (**a**) vibration responses under random frequency and amplitude of base motion acceleration; (**b**) vibration responses under constant frequency and amplitude of base motion acceleration; (**c**) PSD of time domain signal under random frequency and amplitude of base motion acceleration; (**d**) PSD of time domain signal under constant frequency and amplitude of base motion acceleration. \(\varphi_{4}\) can cover the first, second and third beam natural frequency, respectively. In addition, another four groups of base motion excitation angular frequencies with Gaussian distribution and different means and variances were selected. The specific value ranges are \(\varphi_{5}\sim N\big{(}200,20^{2}\big{)}\) rad/s, \(\varphi_{6}\sim N\big{(}440,44^{2}\big{)}\) rad/s, \(\varphi_{7}\sim N\big{(}2810,518^{2}\big{)}\) rad/s and \(\varphi_{8}\sim N\big{(}8360,1628^{2}\big{)}\) rad/s. The values of \(\varphi_{6}\), \(\varphi_{7}\) and \(\varphi_{8}\) can ensure the following relationships \(1/2\omega_{1}\leq\varphi_{6}\leq 3/2\omega_{1}\), \(1/2\omega_{1}\leq\varphi_{7}\leq 3/2\omega_{2}\), \(1/2\omega_{1}\leq\varphi_{8}\leq 3/2\omega_{3}\) and cover the first one, two and three beam natural frequencies, respectively. The maximum vibration amplitude and strain at all locations along the length of the beam with different variation ranges of base motion excitation frequency are shown in Figure 3.  
As can be seen from Figure 3, when the angular frequencies of several groups of random loads cover the first, second and third and the first one, two and three order natural frequencies, the larger the mean and variance of base motion excitation frequency, the  
Figure 3: Maximum vibration amplitude and strain with different ranges of frequency: (**a**) covering the first, second and third natural frequencies; (**b**) covering the first one, two and three natural frequencies. smaller the maximum amplitude and strain of beam in the same period of time can be excited. In engineering practice, the vibration amplitude of the structure can be controlled by changing the base motion excitation frequency. From the iteration numbers with different excitation frequency, we can know that when the base motion excitation frequency only covers the third order natural frequency, the high-frequency component accounts for the largest proportion in base motion excitation frequency within the same period of time.