Vibrations of cracked Euler-Bernoulli beams using Meshless Local Petrov-Galerkin (MLPG) method

Structural health monitoring techniques based on vibration data have received increasing attention in recent years. Since the measured modal characteristics and the transient motion of a beam exhibit low sensitivity to damage, numerical techniques for accurately computing vibration characteristics are needed. Here we use a Meshless Local Petrov-Galerkin (MLPG) method to analyze vibrations of a beam with multiple cracks. The trial and the test functions are constructed using the Generalized Moving Least Squares (GMLS) approximation. The smoothness of the GMLS basis functions requires the use of special techniques to account for the slope discontinuities at the crack locations. Therefore, a set of Lagrange multipliers is introduced to model the spring effects at the crack locations and relate motions of the intact beam segments. The method is applied to study static and transient deformations of a cracked beam and to determine its modal properties (frequencies and mode shapes). Numerical results obtained for a simply supported beam are compared with experimental findings, analytical predictions and finite element solutions.