Research On Vibration Characteristics Of Functionally Graded Material Cracked Beams With Scale Effect

Functionally graded material microbeams are widely used in MEMS and other engineering fields.The crack effect and boundary relaxation effect will affect the mechanical properties of the structure,which directly affects the measurement results.At present,for the vibration problem of functionally graded material beams,the existing research generally includes only one or two of the crack effect,boundary relaxation effect and microscale effect,and almost no research includes all the above effects.Therefore,it is significant to study the vibration characteristics of functionally graded material cracked beams with scale effect under arbitrary boundary conditions.In this paper,the torsional spring model is used to simulate the crack,and the virtual spring technology is used to simulate the boundary conditions.Considering simultaneously the crack effect,boundary relaxation effect and microscale effect,the vibration characteristics of functionally graded material cracked beams with scale effect under arbitrary boundary conditions are studied.Based on the Euler beam theory and the torsion spring model,the vibration control equation of the homogeneous cracked beams is established.After the displacement function of simply supported beams is obtained,the characteristic equation is solved by Ritz method.The accuracy and effectiveness of the method are verified.Based on this,the influence of crack parameters and slenderness ratio on the natural frequency of homogeneous simply supported beams with cracks is discussed.The boundary relaxation effect is added,and the Euler beam theory is changed to the first-order shear deformation theory,and the transverse vibration control equation of cracked beams under arbitrary boundary conditions is established.The Jacobi polynomial is used to uniformly represent the displacement trial function of the cracked beam under arbitrary boundary conditions,and the characteristic equation is solved by the Ritz method.After verifying the method,the influence of boundary relaxation effect on the natural frequency of homogeneous cracked beams is discussed.It is found that when the boundary springs stiffness kθL or k_{w L} is only10⁷～10^{1 0},the boundary relaxation will reduce the natural frequency of the cracked beams.On the basis of the above,replacing the homogeneous beams with the functionally graded material beams,and the vibration control equation of the functionally graded material cracked beams under arbitrary boundary conditions is established.The characteristic equation is still solved by Ritz method.After verifying the method,the effects of three gradient power laws and crack parameters on the natural frequency of FGM cracked beams are discussed.The analysis results found that when n=0.2,the crack parameters have the least influence on the natural frequency of FGM cracked beams,but the effect is the largest when n=5.The crack effect,boundary relaxation effect and microscale effect are considered at the same time,and the displacement trial function of the beams under arbitrary boundary conditions is uniformly expressed by Jacobi polynomial.Based on the first-order shear deformation theory,the modified couple stress theory and the torsion spring model,the transverse vibration control equation of the functionally graded material cracked beams with microscale effect under arbitrary boundary conditions is established.Finally,the Ritz method is used to solve the characteristic equation.The validity and accuracy of the method are verified.Based on this method,the effects of characteristic scale parameters,crack parameters,beam parameters,gradient power law and boundary conditions on the natural frequency of micro-scale FGM cracked beams are discussed.The results show that when the size of micro-scale FGM cracked beams is in the micron level,the larger the characteristic scale parameter is,the larger the natural frequency is,and it is very obvious.The micro-scale effect will expand the influence of cracks on the natural frequency of micro-scale FGM cracked beams and reduce it faster.