In this paper, we consider the buckling of an Euler-Bernoulli graphene beam due to an axial compressive load. We formulate the problem as a non-linear (eigenvalue) two-point boundary value problem, prove some properti...In this paper, we consider the buckling of an Euler-Bernoulli graphene beam due to an axial compressive load. We formulate the problem as a non-linear (eigenvalue) two-point boundary value problem, prove some properties of the eigenpairs and introduce a suitable numerical shooting method scheme for approximating them. We present the perturbation and the numerical approximations of the first and second buckling loads and the corresponding shapes.展开更多
A lumped-parameter nonlinear spring-mass model which takes into account the third-order elastic stiffness constant is considered for modeling the free and forced axial vibrations of a graphene sheet with one fixed end...A lumped-parameter nonlinear spring-mass model which takes into account the third-order elastic stiffness constant is considered for modeling the free and forced axial vibrations of a graphene sheet with one fixed end and one free end with a mass attached. It is demonstrated through this simple model that, in free vibration, within certain initial energy level and depending upon its length and the nonlinear elastic constants, that there exist bounded periodic solutions which are non-sinusoidal, and that for each fixed energy level, there is a bifurcation point depending upon material constants, beyond which the periodic solutions disappear. The amplitude, frequency, and the corresponding wave solutions for both free and forced harmonic vibrations are calculated analytically and numerically. Energy sweep is also performed for resonance applications.展开更多
In this work, we present a priori error estimates of finite element approximations of the solution for the equilibrium equation of an axially loaded Ramberg-Osgood bar. The existence and uniqueness of the solution to ...In this work, we present a priori error estimates of finite element approximations of the solution for the equilibrium equation of an axially loaded Ramberg-Osgood bar. The existence and uniqueness of the solution to the associated nonlinear two point boundary value problem is established and used as a foundation for the finite element analysis.展开更多
In this paper, a new Kelvin-Voigt type beam model of a microelectromechanical resonator made of power-law materials taking into account internal strain-rate damping is proposed and the corresponding lumped-parameter m...In this paper, a new Kelvin-Voigt type beam model of a microelectromechanical resonator made of power-law materials taking into account internal strain-rate damping is proposed and the corresponding lumped-parameter model is derived. Analytical formulas of the lumped parameters in the model are presented. And the pull-in solution is analyzed based on the lumped-parameter model. It is demonstrated analytically and numerically that the internal damping plays an important role in the pull-in solution as well as in determination of the amplitudes and frequencies of the resonator. The hysteresis loops are provided for this model with initial conditions using numerical simulations. The approximation of the electrostatic force in the lumped-parameter model can describe the relations between amplitudes and frequencies with different values of the stiffness and damping coefficients quite well.展开更多
We extend the well-known concept and results for lumped parameters used in the spring-like models for linear materials to Hollomon’s power-law materials.We provide the generalized stiffness and effective mass coeffic...We extend the well-known concept and results for lumped parameters used in the spring-like models for linear materials to Hollomon’s power-law materials.We provide the generalized stiffness and effective mass coefficients for the power-law Euler-Bernoulli beams under standard geometric and load conditions.In particular,our mass-spring lumped parameter models reduce to the classical models when Hollomon’s law reduces to Hooke’s law.Since there are no known solutions to the dynamic power-law beam equations,solutions to our mass lumped models are compared to the low-order Galerkin approximations in the case of cantilever beams with circular and rectangular cross-sections.展开更多
文摘In this paper, we consider the buckling of an Euler-Bernoulli graphene beam due to an axial compressive load. We formulate the problem as a non-linear (eigenvalue) two-point boundary value problem, prove some properties of the eigenpairs and introduce a suitable numerical shooting method scheme for approximating them. We present the perturbation and the numerical approximations of the first and second buckling loads and the corresponding shapes.
文摘A lumped-parameter nonlinear spring-mass model which takes into account the third-order elastic stiffness constant is considered for modeling the free and forced axial vibrations of a graphene sheet with one fixed end and one free end with a mass attached. It is demonstrated through this simple model that, in free vibration, within certain initial energy level and depending upon its length and the nonlinear elastic constants, that there exist bounded periodic solutions which are non-sinusoidal, and that for each fixed energy level, there is a bifurcation point depending upon material constants, beyond which the periodic solutions disappear. The amplitude, frequency, and the corresponding wave solutions for both free and forced harmonic vibrations are calculated analytically and numerically. Energy sweep is also performed for resonance applications.
文摘In this work, we present a priori error estimates of finite element approximations of the solution for the equilibrium equation of an axially loaded Ramberg-Osgood bar. The existence and uniqueness of the solution to the associated nonlinear two point boundary value problem is established and used as a foundation for the finite element analysis.
基金This work was supported by the Nazarbayev University research,rapid response fixed astronomical telescope for gamma ray bust observation(Grant OPCRP2020002).
文摘In this paper, a new Kelvin-Voigt type beam model of a microelectromechanical resonator made of power-law materials taking into account internal strain-rate damping is proposed and the corresponding lumped-parameter model is derived. Analytical formulas of the lumped parameters in the model are presented. And the pull-in solution is analyzed based on the lumped-parameter model. It is demonstrated analytically and numerically that the internal damping plays an important role in the pull-in solution as well as in determination of the amplitudes and frequencies of the resonator. The hysteresis loops are provided for this model with initial conditions using numerical simulations. The approximation of the electrostatic force in the lumped-parameter model can describe the relations between amplitudes and frequencies with different values of the stiffness and damping coefficients quite well.
文摘We extend the well-known concept and results for lumped parameters used in the spring-like models for linear materials to Hollomon’s power-law materials.We provide the generalized stiffness and effective mass coefficients for the power-law Euler-Bernoulli beams under standard geometric and load conditions.In particular,our mass-spring lumped parameter models reduce to the classical models when Hollomon’s law reduces to Hooke’s law.Since there are no known solutions to the dynamic power-law beam equations,solutions to our mass lumped models are compared to the low-order Galerkin approximations in the case of cantilever beams with circular and rectangular cross-sections.
