Mathematical Modeling of the Effect of Temperature on the Dynamic Characteristics of a Cantilever Beam with Flexible Root

This study presents the development of an analytical solution for the dynamic response of a cantilever beam with a flexible root taking into account the influence of temperature. The investigated cantilever beam has a uniform rectangular cross-section with finite lengths. The dynamic response of the cantilever was investigated under three conditions, namely, rigid root, resilient root, and resilient root accompanied by different surrounding temperatures. The selected lengths for the beam were 0.3175, 0.1588, 0.1058, 0.0794, 0.0635, 0.0529, 0.0454, 0.0397, 0.0353, and 0.03175 m. The chosen linear spring coefficients were 0.01, 0.1, 100, and ∞ N/m while rotational spring coefficients were 0.01, 0.1, 100, and ∞ N·m/rad. The surrounding temperatures for the third condition were −100, 25, 100, and 200°C. A MATLAB code was developed to calculate the fundamental natural frequency under different surrounding temperatures and spring coefficients. The proposed mathematical solution was validated with real experimental data and the verification findings revealed a good match between them. For the rigid condition, the finding revealed good matching between the analytical model and experimental results, particularly at the length range of 0.3175−0.1058 m. For the resilient condition, the fundamental natural frequencies were found to be highly affected by decreasing beam length and increased at 100 N/m and 100 N·m/rad and higher coefficients. Finally, there was a reduction in the calculated natural frequencies with increasing temperature.