Non-uniform Timoshenko Beam Research Articles

Nonlinear locally distributed feedback stabilization

The stabilization problem of a nonuniform Timoshenko beam with nonlinear locally distributed feedback controls is considered. By means of nonlinear semigroup theory, energy-perturbation approach and piecewise multiplier method it is shown that the energy of the closed loop system decays exponentially or in the rate of negative power of time.

Exponential stability of the linearized nonuniform Timoshenko beam

The exponential stability of the nonuniform Timoshenko beam with one internal control is investigated in this paper under the Neumann–Dirichlet Boundary conditions.The proof is based on multipliers techniques.

Exact dynamic stiffness method for non-uniform Timoshenko beam vibrations and Bernoulli–Euler column buckling

The exact dynamic stiffness method is further extended, using a recently developed approach for vibration of Bernoulli–Euler members, to flexural free vibration of non-uniform Timoshenko beams with gradual or stepwise non-uniformity of geometric and/or material properties and to Euler buckling of similarly non-uniform columns. Two key strategies are emphasized: (i) formulation of the governing ordinary differential equations (ODE) for dynamic stiffnesses and their derivatives and the solution of the ODE problem by standard ODE solvers; and (ii) establishment of mesh generation rules for the two problems. Extension of the method to three-dimensional frames with non-uniform members poses no major theoretical hurdles. Numerical examples, including challenging problems, are given to show the effectiveness, efficiency, accuracy and reliability of the proposed method which, unlike the finite element method, is exact and so can be iterated until any preset accuracy is achieved.

Free transverse vibration of an axially loaded non-uniform spinning twisted Timoshenko beam using differential transform

This study investigates the vibration problems of an axially loaded non-uniform spinning twisted Timoshenko beam. First, using the Timoshenko beam theory and Hamilton's principle, we derive the governing equations and boundary conditions of the beam. Secondly, the differential transform method is used to solve these equations with appropriate boundary conditions. Finally, the effects of the twist angle, spinning speed, and axial force on the natural frequencies of a non-uniform Timoshenko beam are investigated and discussed.

Stabilization of the nonuniform Timoshenko beam

In this paper we consider a Timoshenko beam with variable physical parameters, we prove that the model can be stabilize by one control force for both internal and boundary cases.

Boundary feedback stabilization of the undamped Timoshenko beam with both ends free

In this paper, we study a boundary feedback system of a class of nonuniform undamped Timoshenko beam with both ends free. We give some sufficient conditions and some necessary conditions for the system to have exponential stability. Our method is based on the operator semigroup technique, the multiplier technique, and the contradiction argument of the frequency domain method.

A numerical method for static and free-vibration analysis of non-uniform Timoshenko beam–columns

A generalized numerical method is proposed to derive the static and dynamic stiffness matrices and to handle the nodal load vector for static analysis of non-uniform Timoshenko beam–columns under several effects. This method presents a unified approach based on effective utilization of the Mohr method and focuses on the following arbitrarily variable characteristics: geometrical properties, bending and shear deformations, transverse and rotatory inertia of mass, distributed and (or) concentrated axial and (or) transverse loads, and Winkler foundation modulus and shear foundation modulus. A successive iterative algorithm is developed to comprise all these characteristics systematically. The algorithm enables a non-uniform Timoshenko beam–column to be regarded as a substructure. This provides an important advantage to incorporate all the variable characteristics based on the substructure. The buckling load and fundamental natural frequency of a substructure subjected to the cited effects are also assessed. Numerical examples confirm the efficiency of the numerical method.Key words: non-uniform, Timoshenko, substructure, elastic foundation, geometrical nonlinearity, stiffness, stability, free vibration.

Stabilization of coupled nonuniform Timoshenko beam system

In this paper, we consider the stabilization problem of a coupled nonuniform Timoshenko beam system using feedback control. It is shown that with linear boundary force and moment feedback controls simultaneously applied at the hinged junction and at one end of the beam, the energy of the closed loop system exponentially decays to zero. Furthermore, the closed loop system can be proven to be asymptotically stable with comparatively fewer feedback controls.

On boundary feedback stabilization of Timoshenko beam with rotor inertia at the tip

The feedback stabilization problem of a nonuniform Timoshenko beam system with rotor inertia at the tip of the beam is studied. First, as a special kind of linear boundary force feedback and moment control is applied to the beam' s tip, the strict mathematical treatment,a suitable state Hilbert space is chosen, and the well-poseness of the corresponding closed loop system is proved by using the semigroup theory of bounded linear operators. Then the energy corresponding to the closed loop system is shown to be exponentially stable. Finally, in the special case of uniform beam,some sufficient and necessary conditions for the corresponding closed loop system to be asymptotically stable are derived.

A general solution for dynamic response of axially loaded non-uniform Timoshenko beams

A dynamic investigation method for the analysis of Timoshenko beams which takes into account the shearing deformation and the rotating inertia is proposed. The solution of the problem is obtained through the iterative variational Rayleigh–Ritz method and assuming as test functions an appropriate class of orthogonal polynomials which respect the essential conditions only. The procedure, applied to tapered beams for which a closed form solution is not known, is an alternative approach to the usual FEM methodologies used in literature. The small number of Lagrangean parameters needed for the analysis allows the use of strict symbolic calculation programs obtaining an high numerical accuracy with a relative short computer time. The work ends with the analysis of a few numerical examples and the results are compared with the ones obtained from other authors mentioned in bibliography.

Nonlinear Boundary Stabilization of Nonuniform Timoshenko Beam

In this paper, the stabilization problem of nonuniform Timoshenko beam by some nonlinear boundary feedback controls is considered. By virtue of nonlinear semigroup theory, energy-perturbed approach and exponential multiplier method, it is shown that the vibration of the beam under the proposed control action decays exponentially or in negative power of time t as t → ∞.

A DQEM for vibration of shear deformable nonuniform beams with general boundary conditions

A differential quadrature element method (DQEM) for free vibration analysis of arbitrary non-uniform Timoshenko beams with attachments, i.e. concentrated mass and rotary inertia and resting on elastic supports is proposed. Using the Hamilton’s principle the governing equation and the natural compatibility conditions at the interface of adjacent elements are devised in a systematic manner. The differential quadrature (DQ) analogs to the governing equation, the compatibility conditions and the external boundary conditions implementations are explicitly formulated. The versatility, accuracy and efficiency of the presented DQEM for free vibration analysis of Timoshenko beams are tested against other solution procedures. The examples include; stepped non-uniform Timoshenko beams, non-uniform beams with attached heavy masses and supported elastically in transverse directions, and also non-uniform beams on elastic foundations. Accurate solutions are archived via few grid points for all the cases considered.

THE FORCED VIBRATION AND BOUNDARY CONTROL OF PRETWISTED TIMOSHENKO BEAMS WITH GENERAL TIME DEPENDENT ELASTIC BOUNDARY CONDITIONS

The governing differential equations and the general time-dependent elastic boundary conditions for the coupled bending–bending forced vibration of a pretwisted non-uniform Timoshenko beam are derived by Hamilton's principle. By introducing a general change of dependent variable with shifting functions, the original system is transformed into a system composed of four non-homogeneous governing differential equations and eight homogeneous boundary conditions. The transformed system is proved to be self-adjoint. Consequently, the method of separation of variables can be used to solve the transformed problem. The physical meanings of these shifting functions are explored. The orthogonality condition for the eigenfunctions of a pretwisted non-uniform beam with elastic boundary conditions is also derived. The relation between the shifting functions and the stiffness matrix is derived. The boundary control of a pretwist Timoshenko beam is studied. The effects of the total pretwist angle, the position of loading and the boundary spring constants on the energy required to control the performance of a pretwisted beam are investigated.

Resonance Interaction of Bending and Shear Modes in a Nonuniform Timoshenko Beam

Propagation of modal solutions in a smoothly inhomogeneous Timoshenko beam is considered. An asymptotic description is given to the interaction of high-frequency bending and shear modes that occurs if their phase velocities intersect at some point at a nonzero angle. A similar problem on the propagation of discontinuities is also considered. The results are presented in the generalized form, so that they have applications to problems on the interaction of other types of waves, e.g., water waves, electromagnetic waves, etc. Bibliography: 19 titles.

Backward wellposedness of nonuniform Timoshenko beam equation

In this paper, we consider the Timoshenko equation of a nonuniform beam, with clamped boundary condition at one end and with feedback controls at the other end. It is proved that the system is backward wellposedness when the feedback controls are weak enough.

VIBRATION AND STABILITY ANALYSIS OF NON-UNIFORM TIMOSHENKO BEAMS UNDER AXIAL AND DISTRIBUTED TANGENTIAL LOADS

Two sets of governing equations for transverse vibration of non-uniform Timoshenko beam subjected to both axial and tangential loads have been presented. In the first set, the axial and tangential loads were taken perpendicular to the shearing force, i.e., normal to the cross-section inclined at an angle ψ, while in the second set, the axial force is assumed to be tangential to the axis of the beam-column. For each case, there exist a pair of differential equations coupled in terms of the flexural displacement and the angle of rotation due to bending. The two coupled second order governing differential equations were combined into one fourth order ordinary differential equation with variable coefficients. The parameters of the frequency equation were determined for different boundary conditions. The exact fundamental solutions could be found by expressing the coefficients of the reduced differential equation in a polynomial form before applying the Frobenius method. Several illustrative examples of uniform and non-uniform beams with various boundary conditions such as clamped supported, elastically supported, and free end mass and pinned end mass, have been presented. The stability analysis, for the variation of the natural frequencies of the uniform and non-uniform beams with the axial force, has also been investigated. Moreover, the present work illustrates the frequency behavior of the beam under a tangential load.

Exponential stabilization of nonuniform Timoshenko beam with locally distributed feedbacks

The stabilization of the Timoshenko equation of a nonuniform beam with locally distributed feedbacks is considered. It is proved that the system is exponentially stabilizable. The frequency domain method and the multiplier technique are applied.

Boundary stabilization of nonuniform Timoshenko beam

In this paper, the boundary stabilization of the Timoshenko equation of a nonuniform beam, with clamped boundary condition at one end and with bending moment and shear force controls at the other end, is considered. It is proved that the system is exponentially stabilizable when the bending moment and shear force controls are simultaneously applied. The frequency domain method and the multiplier technique are used.

Dynamic Analysis of Rotating Nonuniform Timoshenko Beams With an Elastically Restrained Root

A systematic solution procedure for studying the dynamic response of a rotating nonuniform Timoshenko beam with an elastically restrained root is presented. The partial differential equations are transformed into the ordinary differential equations by taking the Laplace transform. The two coupled governing differential equations are uncoupled into two complete fourth-order differential equations with variable coefficients in the flexural displacement and in the angle of rotation due to bending, respectively. The general solution and the generalized Green function of the uncoupled system are derived. They are expressed in terms of the four corresponding linearly independent homogenous solutions, respectively. The shifting relations of the four homogenous solutions of the uncoupled governing differential equation with constant coefficients are revealed. The generalized Green function of an nth order ordinary differential equation can be obtained by using the proposed method. Finally, the influence of the elastic root restraints, the setting angle, and the excitation frequency on the steady response of a beam is investigated.

NON-UNIFORM TIMOSHENKO BEAMS WITH TIME-DEPENDENT ELASTIC BOUNDARY CONDITIONS

A solution procedure for studying the dynamic responses of a non-uniform Timoshenko beam with general time-dependent boundary conditions is developed by generalizing the method of Mindlin–Goodman and utilizing the exact solutions of non-uniform Timoshenko beam vibration given by Lee and Lin. A general change of dependent variable with shifting functions is introduced and the physical meanings of these shifting functions are further explored. The orthogonality condition for the eigenfunctions of a non-uniform Timoshenko beam with elastic boundary conditions is also derived. Several limiting cases and their corresponding procedures are revealed. Finally, the influence of the spring constant on the steady response of a beam subjected to a harmonic based excitation is investigated.