Dynamic Stability Analysis Of Nanobeams Based On High-order Beam Theor

Due to their superior performance,nanostructure components such as nanobeams,nanofilms,and nanoplates have attracted the attention of researchers worldwide,especially nanobeams which have been widely used in nanomechanical systems in recent years.Currently,the issue of vibration of nanobeams under external loads has been extensively studied,but research on the dynamic stability of nanobeams is very limited.Dynamic stability is an important indicator for evaluating the safety of structures and plays a crucial role in ensuring their safety.However,due to the influence of size features and non-local effects of nanostructures,it is often difficult to obtain the dynamic stability information of nanobeams through direct measurement.In addition,the complexity of nanostructures also leads to certain errors in calculating the dynamic stability of nanobeams based on Euler beam theory and Timoshenko beam theory.Currently,although research on the dynamic stability of beam structures based on Euler beam theory and Timoshenko beam theory is relatively mature,there is very limited research on the dynamic stability of nanobeams under high-order beam theory.This article establishes the motion equation of axially excited high-order nanobeams supported on elastic foundations under non local effects based on Reddy beam theory,Levinson beam theory,and non local theory.Using Bolotin theory,the dynamic stability control equation and critical frequency equation of high-order nanobeams are derived.A calculation method for the dynamic stability of high-order nanobeams is proposed,and the influence of material and geometric parameters of nanobeams on their dynamic stability is studied.This research will further improve the theory of nanomechanics,provide theoretical guidance for the dynamic stability and safety analysis of nano beams in nano electromechanical systems,and help promote the industrial development of nano materials.The main research work of this article is as follows:(1)Based on the displacement and strain fields of Reddy beam theory,as well as the constitutive relationship of nanostructures under non local action,the motion equations of the simply supported nanobeam model at both ends were established based on the Hamilton principle and virtual displacement principle.The dynamic stability control equation and critical frequency equation of the Reddy nanobeam model were derived using the Bolotin principle,and the boundary of the dynamic instability domain was obtained by solving the corresponding eigenvalue problem.The influence of material and geometric parameters of nanobeams on their dynamic stability was studied through parameter analysis.(2)The equation of motion of the Reddy nanobeam structure with fixed ends is further established,and the integral equation method is used to solve the problem of different mode functions of each order in the high-order deformation.The corresponding boundary of the dynamic instability region is obtained by using the Bolotin method,and the influence of various parameters on the dynamic stability of the Reddy nanobeam structure with fixed ends is studied.Research has shown that the influence of cross-sectional width on the dynamic stability of Reddy nanobeams with two different boundaries is opposite.Compared to simply supported boundary conditions,the influence of small-scale coefficient,elastic modulus,material length,and Poisson’s ratio on the dynamic stability of Reddy nanobeams with fixed ends is reduced.(3)Based on Levinson beam theory and nonlocal theory,the motion equation of a simply supported nanobeam model at both ends was established.The dynamic stability control equation and critical frequency equation of the Levinson nanobeam model were derived using the Bolotin principle,and the dynamic instability domain boundary was obtained by solving the corresponding eigenvalue problem.The influence of various parameters on its dynamic stability was studied through parameter analysis.(4)Based on Levinson beam theory and integral equation method,the motion equation and dynamic stability control equation of Levinson nano beam structure fixed at both ends are further established.The corresponding dynamic instability region boundary is obtained by using the Bolotin method,and the influence of various parameters on the dynamic stability of Levinson nano beam structure fixed at both ends is studied.Research has shown that the influence of cross-sectional width on the dynamic stability of two different types of nanobeam structures with different boundaries is opposite.Compared to the simply supported boundary conditions,the influence of small-scale coefficient,elastic modulus,material length,and Poisson’s ratio on the dynamic stability of Levinson nanobeams with fixed ends is reduced.(5)The control equation for the dynamic stability domain of nano beams based on Euler Bernoulli and Timoshenko’s low order classical beam theory was derived.The unstable domain of low order beams was compared with the results of high order beams,due to the correlation between high-order strain and the nth power of section height h,the main reason for the error in the results of low-order beams comes from the section height h.Through parameter analysis,it has been verified that increasing the section height can gradually overlap the unstable region of low-order beams with the results of high-order beams,reducing the error.