Dynamic Response of Beams Under Random Loads

In engineering, the study of the dynamic response of structures subjected to non-deterministically variable loads is particularly important, especially when considering the damage that such loads can cause due to fatigue phenomena. This is the case, for example, of the vibrations that a satellite must withstand during the launch phase. In the preliminary design phases, it is very useful to have semi-analytical calculation methodologies that are sufficiently reliable but, at the same time, simple. In the technical literature, there are numerous publications that deal with the study of the random dynamic response of beam models. In general, the presented studies are rather complex, and the dynamic solutions are often obtained in the time domain. The case of a linear elastic uniform cantilever beam model is considered here, for which the analytical expressions of the transfer functions for acceleration, displacement, bending moment, and bending stress are calculated, taking as input the acceleration assigned to the root section or an external lateral load. Knowing the spectral density of the input loads, the spectral densities of all the above-mentioned variables are calculated along the beam axis, assuming stationary and ergodic random processes. Using the spectral density of each output variable, the effective value (RMS) is obtained via integration, which allows for a preliminary estimate of the severity of the working conditions of the beam. The spectral density of the responses also allows us to quickly highlight the contribution of each natural vibration mode as the spectrum of the load varies. The results were obtained using simple spreadsheets available to the reader.