Study On Nonlinear Dynamics Of Cantilever Plate With Side Crack

The crack has the characteristics of being difficult to find, easy to expansion andstrong damage. Many catastrophic accidents in real life are caused by structural fracturewhich the crack propagation has not been inhibited. If the structure has cracks in it, thephysical characteristics, such as the damping and stiffness, will often be changed, thusleading to changes in dynamic characteristics and structural behavior. Considering thecantilever plate with side crack, this research study the vibration characteristic of the crackplate comprehensively from three aspects, theory, experiment and numerical simulation toanalyze the effects of different crack parameters on the natural frequency and modalfunction of cantilever plate. The nonlinear dynamic behavior of crack plate has also beeninvestigated and the relationship of nonlinear dynamical behavior changing from differentcrack parameters has been got. The basic research contents are as follows.(1) Firstly, we format the mode shape function of cantilever plate. The mode shapefunction is composed of two parts, one part is the ideal mode shape of intact cantileverplate obtained by the beam function combination method, and the another part is thecorner function of displacement near the crack which using the vertex singularity plate.The corner function actually uses the properties of half-angle trigonometric function toconstruct the expression of deflection and rotation discontinuity along the crack. Themode shape function of crack cantilever plate can provide theoretical basis for solving thenatural frequency, modal function and nonlinear dynamics.(2) The first third order natural frequencies and modal functions of the crackcantilever plate can be obtained by the symbol calculation software Maple. The calculatedresults are shown by the form of graph intuitively, at the same time, the effects of differentcrack length, crack position and crack angle on the first third order natural frequency ofthe crack plate.(3) The results of theory have been compared and verified by vibration experimentand finite element simulation. Vibration experiment uses the hammer method to measurethe natural frequency and modal of cantilever plate with side crack, and the experimentaldata was acquired through the vibration test system of LMS. The finite element simulationapplies the analysis software Ansys to model and solve the vibration problems ofcantilever crack plate. The verified results show the theoretical method to solve the naturalfrequency and mode shape function of cantilever plate with side crack is accurate.(4) Aiming at the large deflection nonlinear vibration problems of the cracked plate,the dynamic governing equations are established with Hamilton principle based on theclassical thin plate theory. And taking the first third order modal functions for Galerkin method, the partial differential equations are transformed into ordinary differentialequations. The numerical simulation is carried out by Runge-Kutta method in the Matlabprogram, and then the vibration waveform, the phase portraits and the bifurcation diagramare obtained. Through investigation of the different nonlinear dynamic behaviors causedby different crack length, position and angle, we analyze and compare the rules. Andcomparing to the intact plate, the crack plate is prone to have the more the complexnonlinear dynamic behavior, such as multiple periodic, quasi-periodicity, bifurcation,chaos.Considering the cantilever rectangular plate with side crack in this paper, the firstthird order natural frequencies and mode shape functions of the cantilever crack plate withdifferent crack parameters are calculated, and the effects of crack parameters on thenatural frequencies are investigated. At the same time, through analyzing and studying thenonlinear dynamic behavior of cracked plate, we summarize a rule which the nonlineardynamical behaviors change with the different crack parameters. From the research of thispaper, it can be seen that, the existence of the crack will cause the changes of vibrationcharacteristics and nonlinear dynamic behaviors of a cantilever plate, and the influence tothem is different from crack parameters (crack length, crack location and angle).