Damage and Crack Detection Methods Based on the Vibrational Characteristics of Damaged and Cracked Cantilever Beams
Creators: name: Aladiev, Valery
Publisher: name: Maryland Shared Open Access Repository
Year: 2013
Subjects: Beam
Crack
Damage
FEM
Frequencies
Vibrations

description: Damage detection methods utilizing the vibration characteristics of a component or structure have been emerged over the last few decades as viable methods in assessing and monitoring their structural health.  Such methods often utilize physics based models that are capable of predicting the frequency and modal response of the structure in the presence of localized damage including cracks.  They are often easy to use and do not necessarily require access to the full structure since they may rely on local component measurements and response. This work aims at advancing damage detection methods through the development of a new family of such physics based models.  More specifically, beam model theories capable of predicting the modal characteristics of beam structures are revisited for the purpose of establishing their range of validity when compared to 2D Finite Elements.  The governing equations of a vibrating beam are presented within the Euler-Bernoulli as well as Timoshenko assumptions.  The requisite eigenvaTRAClue problem is then formulated and solved.  For comparison purposes, a requisite 2D finite element model as well as one dimensional beam finite element models are developed through which the frequencies and related mode shapes, as well as their slope and curvature profiles are obtained.  Broad parametric studies comparing the Euler-Bernoulli, Timoshenko and finite element predictions are presented and validity envelopes for the beam theories are established. The Euler-Bernoulli, Timoshenko and 2D Finite Element beam models are further developed to simulate the presence of a localized damage zone of reduced stiffness within a cantilever beam. The models allow for the damage zone to be placed at a prescribed location.  A modulus of elasticity reduction profile is introduced simulating a gradient damage characteristic within the prescribed damage zone. The results suggest that evidence of the presence, extent and degree of damage can be seen in the predicted frequency changes as well as in the localized changes exhibited by all mode shapes.  More pronounced evidence of the existence of damage is exhibited by the mode shape slope and particular by the mode shape curvature profiles.  Thus, appreciable changes in mode shape curvatures relative to their healthy counterparts are predicted offering a viable alternative to detecting the presence, extent and severity of localized dispersed damage in beam and most likely other structures. In further expanding the available family of models for damage detection, a 2D finite element model was developed aimed at studying the modal response and related mechanical characteristics of a cantilever beam with a horizontal sharp crack.  Broad parametric studies are reported exploring the effects of beam aspect ratio, crack center location and crack length on the frequency and modal characteristics of such cracked cantilever structures. Similarities in the predictions between models are discussed while also establishing correlations between the discrete crack geometry and the profile characteristics of the stiffness reduction zones employed by previous models. The spatial and temporal response similar cracked beams were also studied via the method of finite elements.  The finite element methodology used is developed along with the numerical finite difference integration scheme employed in solving the time dependent boundary and initial value problem.  An impulse force and time dependent sinusoidal forcing functions at the upper right corner of the beam are used to model the time varying applied loading.  As in the previous models, a broad range of beam length to height ratios ranging from 5 to 50 has been considered. Time profiles of selected beam displacement are presented and compared to known beam solutions.  The presence of the crack is shown to cause a noticeable time shift in the beam deflection periodical response.  Discrete Fourier Transform profile of the beam's tip displacement and related frequencies also show evidence of the presence of the crack. The simulation results obtained by the above new generation of models suggest that the presence of local zone of reduced elastic stiffness or the presence of a crack induced a shift in the transverse beam frequencies most likely due to the increase beam compliance induced by the damage zone or the crack.  In fact, beams containing cracks of larger length were shown to vibrate at lower frequencies when compared to those with shorter cracks and of course when compared to a healthy structure.  In addition, distinct deviations from an otherwise smooth profile exhibited by the healthy beam were predicted in the damage or crack regions along the beam's free surface for the mode shape slope and curvatures with the latter exhibiting much higher sensitivity to the presence of damage or crack when compared to the slopes.  The frequency reduction induced by the damage or a crack has also been validated through Discrete Fourier Transform displacement profiles which showed that larger frequency reduction is predicted for beams containing larger cracks.  The above findings establish the foundation for the development of new novel model based and non-model methods for damage detection.