Vibration Analysis Of Beams Based On A Higher-order Beam Theory And A Couple Stress Theory

With the development of science and engineering,particularly with wide application of MEMS,the more accuracy prediction of frequencies is significant in the higher-mode flexural vibration of micro-beams,especially when the wavelength approaches to height of beams.For instance,for a micro-cantilever probe with a small length to height ratio,its flexural vibration wavelength could approach to height of the beam although the vibration mode is not very high.Consequently,it is extremely important for the design of MEMS to accurately predict the frequency of higher-order vibration of micro-beams.However,the prediction of the frequencies in the higher-mode flexural vibration given by the popular engineering beam theories is not very accurate.By using an improved third-order shear deformation beam theory and Hamiltonian principle,this thesis presents a new set of equations of motion for the flexural vibration of shear deformable beams,in which the kinetic energy of the rigid rotation of micro-elements is taken into account by a couple stress theory.The resulting differential equations of motion are defined in terms of the deflection and the transverse shear strain of beam cross-section and contain a characteristic length.The differential orders of these equations are six in space domain and four in time domain.And the rotatory inertia effect in variational consistent boundary condition is discussed.A feasible value of the characteristic length for beam flexural vibrations is determined by matching the resulting phase velocity to the elasticity solution when the wavelength approaches to zero.The analytical solutions of the phase velocities of infinite long beams are solved and compared with the elasticity solutions.A comparative study of the methods for the determination of the intrinsic material length is also conducted.A two-noded beam element accounting for couple stress effect is developed for the free vibration of beams by employing the present equations of motion and the quasi-conforming element technique.The accuracy and efficiency of the new beam element is evaluated by numerical examples.The numerical results indicate that the present beam element with couple stress effect can be used to accurately compute the higher-mode flexural frequencies of beams.