Random Eigenvalue Characterization for Free Vibration of Axially Loaded Euler-Bernoulli Beams

Typically, numerical or approximate methods are used for the free vibration analysis of axially loaded Euler-Bernoulli beams because the governing differential equation does not yield an exact solution, even for uniform beams. However, for certain polynomial variations of the mass and stiffness, there exists a fundamental closed-form solution to the fourth-order governing differential equation, which is used to design axially loaded, cantilever beams having a prespecified fundamental natural frequency. In the presence of uncertainty, this flexural stiffness is treated as a spatial random field. For several known probability distributions (normal, log-normal, Weibull, Gumbel, and exponential) of the natural frequencies, the corresponding distributions of this field are determined analytically. These analytical expressions can serve as benchmark solutions for different statistical simulation tools to find the probabilistic nature of the stiffness distribution for known probability distributions of the frequencies. The analytical solution corresponding to the normal distribution is further used to derive the coefficient of variation of the stiffness distribution for a rotating cantilever beam, which is further used as an example problem to optimize the beam profile to maximize the allowable tolerance during manufacturing. An example of such a beam with a rectangular cross section is provided. The effects of the length of the beam and the uniform rotation speed on the optimal beam profile are also studied.