Based on the nonlocal continuum theory, the nonlinear vibration of an embedded single-walled carbon nanotube (SWCNT) subjected to a harmonic load is in- vestigated. In the present study, the SWCNT is assumed to be a...Based on the nonlocal continuum theory, the nonlinear vibration of an embedded single-walled carbon nanotube (SWCNT) subjected to a harmonic load is in- vestigated. In the present study, the SWCNT is assumed to be a curved beam, which is unlike previous similar work. Firstly, the governing equations of motion are derived by the Hamilton principle, meanwhile, the Galerkin approach is carried out to convert the nonlinear integral-differential equation into a second-order nonlinear ordinary differ- ential equation. Then, the precise integration method based on the local linearzation is appropriately designed for solving the above dynamic equations. Besides, the numerical example is presented, the effects of the nonlocal parameters, the elastic medium constants, the waviness ratios, and the material lengths on the dynamic response are analyzed. The results show that the above mentioned effects have influences on the dynamic behavior of the SWCNT.展开更多
Free vibration response of functionally graded material (FGM) beams is studied based on the Levinson beam theory (LBT). Equations of motion of an FGM beam are derived by directly integrating the stress-form equati...Free vibration response of functionally graded material (FGM) beams is studied based on the Levinson beam theory (LBT). Equations of motion of an FGM beam are derived by directly integrating the stress-form equations of elasticity along the beam depth with the inertial resultant forces related to the included coupling and higherorder shear strain. Assuming harmonic response, governing equations of the free vibration of the FGM beam are reduced to a standard system of second-order ordinary differential equations associated with boundary conditions in terms of shape functions related to axial and transverse displacements and the rotational angle. By a shooting method to solve the two-point boundary value problem of the three coupled ordinary differential equations, free vibration response of thick FGM beams is obtained numerically. Particularly, for a beam with simply supported edges, the natural frequency of an FGM Levinson beam is analytically derived in terms of the natural frequency of a corresponding homogenous Euler-Bernoulli beam. As the material properties are assumed to vary through the depth according to the power-law functions, the numerical results of frequencies are presented to examine the effects of the material gradient parameter, the length-to-depth ratio, and the boundary conditions on the vibration response.展开更多
Facing the lateral vibration problem of a machine rotor as a beam on elastic supports in bending,the authors deal with the free vibration of elastically restrained Bernoulli-Euler beams carrying a finite number of con...Facing the lateral vibration problem of a machine rotor as a beam on elastic supports in bending,the authors deal with the free vibration of elastically restrained Bernoulli-Euler beams carrying a finite number of concentrated elements along their length.Based on Rayleigh’s quotient,an iterative strategy is developed to find the approximated torsional stiffness coefficients,which allows the reconciliation between the theoretical model results and the experimental ones,obtained through impact tests.The mentioned algorithm treats the vibration of continuous beams under a determined set of boundary and continuity conditions, including different torsional stiffness coefficients and the effect of attached concentrated masses and rotational inertias, not only in the energetic terms of the Rayleigh’s quotient but also on the mode shapes,considering the shape functions defined in branches.Several loading cases are examined and examples are given to illustrate the validity of the model and accuracy of the obtained natural frequencies.展开更多
This paper deals with non-linear vibration of rectangular reticulated shallow shells by applying non-linear elastic theory of such structures established by the author .Us-ing the assumed (generalized)Fourier series s...This paper deals with non-linear vibration of rectangular reticulated shallow shells by applying non-linear elastic theory of such structures established by the author .Us-ing the assumed (generalized)Fourier series solutions for transverse deflection (latticejoint transverse displacement )and force function,weighted means of the trial functions lead to the relations among the coefficients related to the solutions and vibration equ-ation which determines the unknown time function,and then the amplitude -frequeney relations for free vibration and forced vibration due to harmonic force are derived withthe aid of the regular perturbation method and Galerkin procedure,respectively.Nu-merical examples are given as well.展开更多
This paper investigates a highly efficient and promising control method for forced vibration control of an axially moving beam with an attached nonlinear energy sink(NES).Because of the axial velocity,external force...This paper investigates a highly efficient and promising control method for forced vibration control of an axially moving beam with an attached nonlinear energy sink(NES).Because of the axial velocity,external force and external excitation frequency,the beam undergoes a high-amplitude vibration.The Galerkin method is applied to discretize the dynamic equations of the beam–NES system.The steady-state responses of the beams with an attached NES and with nothing attached are acquired by numerical simulation.Furthermore,the fast Fourier transform(FFT)is applied to get the amplitude–frequency responses.From the perspective of frequency domain analysis,it is explained that the NES has little effect on the natural frequency of the beam.Results confirm that NES has a great potential to control the excessive vibration.展开更多
基金Project supported by the National Basic Research Program of China (No. 2011CB610300)the National Natural Science Foundation of China (Nos. 10972182, 11172239, and 10902089)+3 种基金the 111 Project of China (No. B07050)the Ph. D. Programs Foundation of Ministry of Education of China (No. 20106102110019)the Open Foundation of State Key Laboratory of Structural Analysis of Industrial Equipment (No. GZ0802)the Doctorate Foundation of Northwestern Polytechnical University (No. CX201224)
文摘Based on the nonlocal continuum theory, the nonlinear vibration of an embedded single-walled carbon nanotube (SWCNT) subjected to a harmonic load is in- vestigated. In the present study, the SWCNT is assumed to be a curved beam, which is unlike previous similar work. Firstly, the governing equations of motion are derived by the Hamilton principle, meanwhile, the Galerkin approach is carried out to convert the nonlinear integral-differential equation into a second-order nonlinear ordinary differ- ential equation. Then, the precise integration method based on the local linearzation is appropriately designed for solving the above dynamic equations. Besides, the numerical example is presented, the effects of the nonlocal parameters, the elastic medium constants, the waviness ratios, and the material lengths on the dynamic response are analyzed. The results show that the above mentioned effects have influences on the dynamic behavior of the SWCNT.
基金supported by the National Natural Science Foundation of China(No.11272278)
文摘Free vibration response of functionally graded material (FGM) beams is studied based on the Levinson beam theory (LBT). Equations of motion of an FGM beam are derived by directly integrating the stress-form equations of elasticity along the beam depth with the inertial resultant forces related to the included coupling and higherorder shear strain. Assuming harmonic response, governing equations of the free vibration of the FGM beam are reduced to a standard system of second-order ordinary differential equations associated with boundary conditions in terms of shape functions related to axial and transverse displacements and the rotational angle. By a shooting method to solve the two-point boundary value problem of the three coupled ordinary differential equations, free vibration response of thick FGM beams is obtained numerically. Particularly, for a beam with simply supported edges, the natural frequency of an FGM Levinson beam is analytically derived in terms of the natural frequency of a corresponding homogenous Euler-Bernoulli beam. As the material properties are assumed to vary through the depth according to the power-law functions, the numerical results of frequencies are presented to examine the effects of the material gradient parameter, the length-to-depth ratio, and the boundary conditions on the vibration response.
基金supported by the Portuguese Foundation for Science and Tech-nology(FCT),under the project POCI 2010 and the PhD grant SFRH/BD/44696/2008
文摘Facing the lateral vibration problem of a machine rotor as a beam on elastic supports in bending,the authors deal with the free vibration of elastically restrained Bernoulli-Euler beams carrying a finite number of concentrated elements along their length.Based on Rayleigh’s quotient,an iterative strategy is developed to find the approximated torsional stiffness coefficients,which allows the reconciliation between the theoretical model results and the experimental ones,obtained through impact tests.The mentioned algorithm treats the vibration of continuous beams under a determined set of boundary and continuity conditions, including different torsional stiffness coefficients and the effect of attached concentrated masses and rotational inertias, not only in the energetic terms of the Rayleigh’s quotient but also on the mode shapes,considering the shape functions defined in branches.Several loading cases are examined and examples are given to illustrate the validity of the model and accuracy of the obtained natural frequencies.
文摘This paper deals with non-linear vibration of rectangular reticulated shallow shells by applying non-linear elastic theory of such structures established by the author .Us-ing the assumed (generalized)Fourier series solutions for transverse deflection (latticejoint transverse displacement )and force function,weighted means of the trial functions lead to the relations among the coefficients related to the solutions and vibration equ-ation which determines the unknown time function,and then the amplitude -frequeney relations for free vibration and forced vibration due to harmonic force are derived withthe aid of the regular perturbation method and Galerkin procedure,respectively.Nu-merical examples are given as well.
基金supported by the National Natural Science Foundation of China (project nos.11772205 , 11202140 , 11402151 , 11572182 , 51305421)the funding support from the Natural Science Foundation of Liaoning Province (201501708)
文摘This paper investigates a highly efficient and promising control method for forced vibration control of an axially moving beam with an attached nonlinear energy sink(NES).Because of the axial velocity,external force and external excitation frequency,the beam undergoes a high-amplitude vibration.The Galerkin method is applied to discretize the dynamic equations of the beam–NES system.The steady-state responses of the beams with an attached NES and with nothing attached are acquired by numerical simulation.Furthermore,the fast Fourier transform(FFT)is applied to get the amplitude–frequency responses.From the perspective of frequency domain analysis,it is explained that the NES has little effect on the natural frequency of the beam.Results confirm that NES has a great potential to control the excessive vibration.