Free Vibrations Analysis of Cracked Micro Cantilever Beams Based on the Modified Strain Gradient Theory

This paper explores the lateral vibration behavior of a micro cantilever beam with an open edge crack under axial load using the modified strain gradient theory (msgt). a concentrated mass, incorporating its rotational inertia, is positioned at the beam's free end. the open edge crack is represented using the dirac delta function. by employing msgt, hamilton's principle, and the dirac delta function, the governing equations for system motion and relevant boundary conditions are derived to examine the size-dependence effects. analytical solutions for the first and second natural frequencies of the cracked cantilever beam are provided, validated through finite element modeling. the study further investigates the impact of various system parameters, including material length scale parameters, crack depth, crack location, cantilever beam length, axial load, and the presence of the concentrated mass, on the natural frequencies. the findings demonstrate that the crack depth, crack location, and material length scale parameters considerably influence the lateral vibration characteristics of the system. notably, increasing the values of 𝑙𝑖/ℎ from 0 to 0.25 leads to an approximate 40% rise in the natural frequency.