Analytical modeling for vibration analysis of thin rectangular orthotropic/functionally graded plates with an internal crack

Abstract An analytical model is presented for vibration analysis of a thin orthotropic and general functionally graded rectangular plate containing an internal crack located at the center. The continuous line crack is parallel to one of the edges of the plate. The equation of motion of the orthotropic plate is derived using the equilibrium principle. The crack terms are formulated using Line Spring Model. The effect of location of crack along the thickness of the plate on natural frequencies is analyzed using appropriate crack compliance coefficients in the Line Spring Model. By using the Berger formulation for in-plane forces, the derived equation of motion of the cracked plate is transformed into a cubic nonlinear system. Applying the Galerkin׳s method, the equation is converted into well known Duffing equation. The peak amplitude is obtained by employing Multiple Scales perturbation method. The effect of nonlinearity is also established by deriving frequency response equation for the cracked plate using method of multiple scales. The influence of crack length, boundary conditions and crack location along the thickness, on the natural frequencies of a square and rectangular plate is demonstrated. It is found that the vibration characteristics are affected by the length and location of crack along the thickness of the plate. It is thus deduced that the natural frequencies are minimum when crack is internal and its depth is symmetric about the mid-plane of the plate for all the three boundary conditions considered. Further, it is concluded that the presence of crack across the fibers decreases the frequency more as compared to crack along the fibers. The Effect of varying elasticity ratio on the fundamental frequencies of the cracked plate is also established.