Typically, numerical or approximate methods are used for the free vibration analysis of rotating Euler-Bernoulli beams as the governing differential equation does not yield an exact solution, even for uniform beams. But, 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 rotating cantilever beams having a pre-specified fundamental natural frequency. In presence of uncertainty, this flexural stiffness is treated as a spatial random field. For known probability distributions of the natural frequencies, the corresponding distribution of this field is determined analytically. This analytical solution can serve as a benchmark solution for different statistical simulation tools to find the probabilistic nature of the stiffness distribution for known probability distributions of the frequencies. The analytical solution is also used to derive the coefficient of variation of the stiffness distribution, which is further used to optimize the beam profile to maximize the allowable tolerance during manufacturing. Example of such a beam with rectangular cross-section is provided. The effect of the length of the beam and the uniform rotation speed on the optimal beam profile is also studied.
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