Vibration analysis of Euler–Bernoulli nanobeams by using finite element method

In the present study, an efficient finite element model for vibration analysis of a nonlocal Euler–Bernoulli beam has been reported. Nonlocal constitutive equation of Eringen is proposed. Equations of motion for a nonlocal Euler–Bernoulli are derived based on varitional statement. The finite element method is employed to discretize the model and obtain a numerical approximation of the motion equation. The model has been verified with the previously published works and found a good agreement with them. Vibration characteristics, such as fundamental frequencies, are illustrated in graphical and tabulated form. Numerical results are presented to figure out the effects of nonlocal parameter, slenderness ratios, rotator inertia, and boundary conditions on the dynamic characteristics of the beam. The above mention effects play very important role on the dynamic behavior of nanobeams.     

Stabilization of transmission coupled wave and Euler–Bernoulli equations on Riemannian manifolds by nonlinear feedbacks

We consider the transmission system of coupling wave equations with Euler–Bernoulli equations on Riemannian manifolds. By introducing nonlinear boundary feedback controls, we establish the exponential and rational energy decay rate for the problem. Our proofs rely on the geometric multiplier method.     

Free vibration characteristics of a functionally graded beam by finite element method

This paper presents the dynamic characteristics of functionally graded beam with material graduation in axially or transversally through the thickness based on the power law. The present model is more effective for replacing the non-uniform geometrical beam with axially or transversally uniform geometrical graded beam. The system of equations of motion is derived by using the principle of virtual work under the assumptions of the Euler–Bernoulli beam theory. The finite element method is employed to discretize the model and obtain a numerical approximation of the motion equation. The model has been verified with the previously published works and found a good agreement with them. Numerical results are presented in both tabular and graphical forms to figure out the effects of different material distribution, slenderness ratios, and boundary conditions on the dynamic characteristics of the beam. The above mention effects play very important role on the dynamic behavior of the beam.     

Exponential stability of uniform Euler–Bernoulli beams with non-collocated boundary controllers

We study the stability of a robot system composed of two Euler–Bernoulli beams with non-collocated controllers. By the detailed spectral analysis, we prove that the asymptotical spectra of the system are distributed in the complex left-half plane and there is a sequence of the generalized eigenfunctions that forms a Riesz basis in the energy space. Since there exist at most finitely many spectral points of the system in the right half-plane, to obtain the exponential stability, we show that one can choose suitable feedback gains such that all eigenvalues of the system are located in the left half-plane. Hence the Riesz basis property ensures that the system is exponentially stable. Finally we give some simulation for spectra of the system.     

A homotopy analysis solution to large deformation of beams under static arbitrary distributed load

In this article, homotopy analysis method (HAM) is employed to investigate non-linear large deformation of Euler–Bernoulli beams subjected to an arbitrary distributed load. Constitutive equations of the problem are obtained. It is assumed that the length of the beam remains constant after applying external loads. Different auxiliary parameters and functions of the HAM and the extra auxiliary parameter, which is applied to initial guess of the solution, are employed to procure better convergence rate of the solution. The results of the solution are obtained for two different examples including constant cross sectional beam subjected to constant distributed load and periodic distributed load. Special base functions, orthogonal polynomials e.g. Chebyshev expansion, are employed as a tool to improve the convergence of the solution. The general solution, presented in this paper, can be used to attain the solution of the beam under arbitrary distributed load and flexural stiffness. Ultimately, it is shown that small deformation theory overestimates different quantities such as bending moment, shear force, etc. for large deflection of the beams in comparison with large deformation theory. Finally, it is concluded that solution of small deformation theory is far from reality for large deflection of straight Euler–Bernoulli beams.     

Riesz basis property, exponential stability of variable coefficient Euler-Bernoulli beams with indefinite damping

We study damped Euler–Bernoulli beams that have nonuniformthickness or density. These nonuniformfeatures result in variablecoefficient beam equations. We prove that despite the nonuniformfeatures, the eigenfunctions of the beam form a Riesz basisand asymptotic behaviour of the beam system can be deduced withoutany restrictions on the sign of the damping. We also providean answer to the frequently asked question on damping: ‘howmuch more positive than negative should the damping be withoutdisrupting the exponential stability?’, and result ina criterion condition which ensures that the system is exponentiallystable.     

Mathematical modelling and dynamic response of a multi-straight-line path tracing flexible robot manipulator with rotating-prismatic joint

In this study, mathematical modelling and dynamic response of a flexible robot manipulator with rotating-prismatic joint are investigated. The tip end of the flexible robot manipulator traces a multi-straight-line path under the action of an external driving torque and an axial force. Considered robot manipulator consists of a rotating prismatic joint and a sliding flexible arm with a tip mass. Flexible arm is assumed to be an Euler–Bernoulli beam carrying an end-mass. Equations of motion of the flexible manipulator are obtained by using Lagrange’s equation of motion. Effect of rotary inertia, axial shortening and gravitation is considered in the analysis. Equations of motion are solved by using fourth order Runge–Kutta method. Numerical simulations obtained by using a developed computer program are presented and physical trend of the results are discussed.     

Nonlinear analysis of buckling,free vibration and dynamic stability for the piezoelectric functionally graded beams in thermal environment

Employing Euler–Bernoulli beam theory and the physical neutral surface concept, the nonlinear governing equation for the functionally graded material beam with two clamped ends and surface-bonded piezoelectric actuators is derived by the Hamilton’s principle. The thermo-piezoelectric buckling, nonlinear free vibration and dynamic stability for the piezoelectric functionally graded beams, subjected to one-dimensional steady heat conduction in the thickness direction, are studied. The critical buckling loads for the beam are obtained by the existing methods in the analysis of thermo-piezoelectric buckling. The Galerkin’s procedure and elliptic function are adopted to obtain the analytical solution of the nonlinear free vibration, and the incremental harmonic balance method is applied to obtain the principle unstable regions of the piezoelectric functionally graded beam. In the numerical examples, the good agreements between the present results and existing solutions verify the validity and accuracy of the present analysis and solving method. Simultaneously, validation of the results achieved by rule of mixture against those obtained via the Mori–Tanaka scheme is carried out, and excellent agreements are reported. The effects of the thermal load, electric load, and thermal properties of the constituent materials on the thermo-piezoelectric buckling, nonlinear free vibration, and dynamic stability of the piezoelectric functionally graded beam are discussed, and some meaningful conclusions have been drawn.     

Free vibration and stability of tapered Euler–Bernoulli beams made of axially functionally graded materials

The free vibration and stability of axially functionally graded tapered Euler–Bernoulli beams are studied through solving the governing differential equations of motion. Observing the fact that the conventional differential transform method (DTM) does not necessarily converge to satisfactory results, a new approach based on DTM called differential transform element method (DTEM) is introduced which considerably improves the convergence rate of the method. In addition to DTEM, differential quadrature element method of lowest-order (DQEL) is used to solve the governing differential equation, as well. Carrying out several numerical examples, the competency of DQEL and DTEM in determination of free longitudinal and free transverse frequencies and critical buckling load of tapered Euler–Bernoulli beams made of axially functionally graded materials is verified.     

Dynamic model of a flying manipulator with two highly flexible links

Nonlinear dynamic model of a flying manipulator with two revolute joints and two highly flexible links is obtained using Hamilton’s principle. Flying base of the manipulator is a rigid body. Stress is treated three dimensionally in the isotropic linearly-elastic links, but the in-plane and out-of-plane warpings of the links’ cross-sections are neglected. Although the links’ cross-sections undergo negligible elastic orientation, their models are more accurate than a nonlinear 3D Euler–Bernoulli beam. Tension, compression, twisting and spatial deflections of each link are coupled to each other by some nonlinear terms including two new ones. In the issue of flying flexible-link manipulators new terminologies, namely forward/inverse kinetics instead of forward/inverse kinematics are suggested, since determination of position and orientation of the end-effector is coupled to the partial differential motion equations.     

An inverse vibration problem in estimating the spatial and temporal-dependent external forces for cutting tools

An inverse forced vibration problem, based on the conjugate gradient method (CGM), (or the iterative regularization method), is examined in this study to estimate the unknown spatial and temporal-dependent external forces for the cutting tools by utilizing the simulated beam displacement measurements. The tool is represented by an Euler–Bernoulli beam. The accuracy of the inverse analysis is examined by using the simulated exact and inexact displacement measurements. The numerical experiments are performed to test the validity of the present algorithm by using different types of external forces, sensor arrangements and measurement errors. Results show that excellent estimations on the external forces can be obtained with any arbitrary initial guesses.     

An analytical dynamic model for single-cracked beams including bending,axial stiffness,rotational inertia,shear deformation and coupling effects

This paper develops an analytical dynamic model for cracked beams including bending, axial stiffness, rotational inertia, shear deformation and the coupling of the last two effects. The damage is modelled using a rotational spring that simulates the crack based on fracture mechanics theory. The developed model is used to predict variations on natural frequencies for several crack sites and damage magnitude along the beam. The importance of this work lies in the development of an analytical model that has no approximation due to discretization of the displacement field. This initial theoretical approach describes the expected behaviour for changes in the natural frequencies for simply-supported and clamped-free beams with the precision that only analytical methods allow. The results provide a useful benchmark to compare with approximate numerical methods that can be used to model and analyse the problem. The model showed similar results for long span beams, but the inclusion of rotational inertia and shear deformation effects rendered improvements in the dynamic behaviour mainly in the case of slender and short span beams when compared with the simplified Euler–Bernoulli model.     

Analytical analysis of buckling and post-buckling of fluid conveying multi-walled carbon nanotubes

In this work, buckling and post-buckling analysis of fluid conveying multi-walled carbon nanotubes are investigated analytically. The nonlinear governing equations of motion and boundary conditions are derived based on Eringen nonlocal elasticity theory. The nanotube is modeled based on Euler–Bernoulli and Timoshenko beam theories. The Von Karman strain–displacement equation is used to model the structural nonlinearities. Furthermore, the Van der Waals interaction between adjacent layers is taken into account. An analytical approach is employed to determine the critical (buckling) fluid flow velocities and post-buckling deflection. The effects of the small-scale parameter, Van der Waals force, ends support, shear deformation and aspect ratio are carefully examined on the critical fluid velocities and post-buckling behavior.     

Stabilization of the dynamic elastica only by damping torque

We consider a system of partial differential equations called the dynamic elastica, which describes the motion of a geometrically nonlinear (i.e., largely deflecting) elastic beam and is derived based on Euler–Bernoulli’s assumption. The aim of the paper is to examine the effect of damping torque (i.e., external torque generated as a negative feedback of the angular velocity of the beam’s centerline) on the stability of the elastica.     

具有动态边界的非均匀Euler-Bernoulli梁的边界反馈镇定

本讨论一端带有重物的Euler-Bernoulli梁的边界反馈镇定问题。在生物的质量忽略不计而只考虑重物的转动惯量的情况下，证明了同时在梁的自由端施加力和力矩反馈，闭环系统的能量可被指数镇定。进而对于系统只有力反馈或只有力矩反馈的情况，得到了闭环系统（指数）稳定的充分必要条件。     

Exact boundary controllability of vibrating non-classical Euler–Bernoulli micro-scale beams

This study investigates the exact controllability problem for a vibrating non-classical Euler–Bernoulli micro-beam whose governing partial differential equation (PDE) of motion is derived based on the non-classical continuum mechanics. In this paper, it is proved that via boundary controls, it is possible to obtain exact controllability which consists of driving the vibrating system to rest in finite time. This control objective is achieved based on the PDE model of the system which causes that spillover instabilities do not occur.     

Static analysis of composite beams with weak shear connection

The objective of the present paper is to analyse the static behaviour of elastic two-layer beams with interlayer slip. The Euler–Bernoulli hypothesis is assumed to hold for each layer separately, and a linear constitutive equation between the horizontal slip and the interlaminar shear force is considered. The applied loads act in the plane of symmetry of the composite beam, and the material and geometrical properties do not depend on the axial coordinate. Closed-form solutions for displacements and interlayer slips are developed. A second order differential equation is derived for the interlayer slip whose solution is used to determine the deflections and slopes. Examples illustrate the application of the method presented.     

Modelling of the forced motions of an elastic beam using the method of integrodifferential relations

A variational approach to the numerical modelling of forced lateral motions of an Euler–Bernoulli elastic beam is developed for a number of linear boundary conditions using the method of integrodifferential relations. A class of linear boundary actions is considered. A family of quadratic functionals, connecting the displacement field of points of the beam with the bending-moment functions in the cross section and the momentum density is proposed. Variational formulations of the original initial-boundary value problem on the motion of the beam are given and the necessary conditions for the functionals introduced to be stationary are analysed. The integral and local quality characteristics of the admissible approximate solutions are determined. The relation between the variational problems, formulated for the beam model, with the classical Hamilton–Ostrogradskii variational principles is demonstrated. An algorithm for constructing approximate systems of ordinary differential equations is developed, the solution of which yields stationary (minimum) values of the functionals introduced on a specified set of displacement fields, moments and momenta. Examples of calculations of the displacements for an elastic beam and an analysis of the quality of the numerical solutions obtained are presented.