The equations of motion governing the quasistatic and dynamical behavior of a viscoelastic timoshenko beam are derived. Firstly, the equations of equilibrium are presented and then the classical beam theories based on bernoullieuler and timoshenko beam kinematics are derived. Two coupled partial differential equations which describe the motion of a viscoelastic kelvinvoigt type timoshenko beam are formulated with the complementarity conditions. Dynamic response analysis of timoshenko beam on viscoelastic foundation under an arbitrary distributed harmonic moving load 2 002 proceeding of the 5th international conference on structural dynamics. The concept of elastic timoshenko shear coefficients is used as a guide for linear viscoelastic eulerbernoulli beams subjected to simultaneous bending and twisting. The consistent linearization of the fully geometrically nonlinear beam theory is used in the derivation.
A viscoelastic timoshenko beam with dynamic frictionless impact. The concept of elastic timoshenko shear coefficients is used as a guide for linear viscoelastic eulerbernoulli beams subjected to simultaneous bending and it is shown that the corresponding timoshenko viscoelastic functions now depend not only on material properties and geometry as they do in elasticity, but also additionally on stresses and their time histories. Hilton1 aerospace engineering department ae technology research, education and commercialization. In the present paper, we present not only the correct equation for the impact response of the elastic timoshenko beam, but the deduction and analysis of equations describing the behaviour of a viscoelastic timoshenkotype beam impacted by an elastic sphere are given. Finite element methods for timoshenko beams learning outcome a. Viscoelastic timoshenko beam theory, mechanics of time. Elastic beams in three dimensions aalborg universitet.
Impact response of a timoshenkotype viscoelastic beam. A major concern is to pursue an investigation into conservation of energy or energy. Forced vibration analysis of damped beam structures with. Vibration of nonlocal kelvinvoigt viscoelastic damped timoshenko. In the analysis, the timoshenko beam theory, which includes the transverse shear and rotatory inertia effect and conventional beam theory, are used to solve this problem. Dynamic response of timoshenko beam resting on non. Figure 3 shows a portion of the deformed centerline of the beam. As its name indicates, the model is based on the standard timoshenko beam model 27, 33. In this analysis the field equation for viscoelastic material is used. Viscoelastic timoshenko beam theory article pdf available in mechanics of timedependent materials 1. For instance, when dealing with viscoelastic behavior, shear deformations play a fundamental role.
In timoshenko beam theory, the displacement field is assumed to be. In the analysis, the soil is modeled as a threedimensional viscoelastic continuum, and the pile as an elastic timoshenko beam with solid circular crosssection. Employing the timoshenko beam theory implies that both the shear deformation and the rotary inertia not only of the viscoelastic layers but of the elastic layers are taken into consideration. Rajesh1 dept of mechanical engineering, faculty of science and technology, ifhe university, hyderabad 501203, india j. This paper investigates the dynamic behavior of nonlocal viscoelastic damped nanobeams. A viscoelastic timoshenko beam with dynamic frictionless. The excitation form is the concentrated force, and its random characteristic is the ideal white noise in the time domain. Stationary random vibration of a viscoelastic timoshenko. The kelvinvoigt viscoelastic model,velocitydependent external damping and timoshenko beam theory are employed toestablish the governing equations and boundary condit ions for the bending vibratio n of nanotubes. Viscoelastic buckling of eulerbernoulli and timoshenko. For eg the euler beam theory assumes beams whose plane c. Vibration of nonlocal kelvinavoigt viscoelastic damped. Fractional viscoelastic timoshenko beam deflection via single equation. Vibration of nonlocal kelvinvoigt viscoelastic damped.
Operator based constitutive relationship is used to develop the general time domain, linear viscoelastic model. A shaft finite element for analysis of viscoelastic. Sizedependent couple stress timoshenko beam theory arxiv. Cheng, quasistatic and dynamical analysis for viscoelastic timoshenko beam with fractional derivative constitutive relation, appl. Refinement of timoshenko beam theory for composite and. Beam theory ebt straightness, inextensibility, and normality. Timoshenkos beam for transversely vibrations uniform viscoelastic. This variationally consistent theory is derived from the virtual work principle and employs a novel piecewise linear zigzag function that provides a more realistic representation of the. A timoshenko beam element of length l mounted on threeparameter viscoelastic pasternak foundation is shown in fig. The kelvinvoigt viscoelastic model, velocitydependent external damping and timoshenko beam theory are employed to establish the governing equations and boundary conditions for the bending vibration of nanotubes. Dynamic response to a moving load of a timoshenko beam resting on a nonlinear viscoelastic foundation.
From the kinematics of a shear flexible beam theory based on the timoshenko beam theory. Finite element modeling of thick rotor deep rotor requires the use of the timoshenko element, which takes care of shear deformation. Viscoelastic timoshenko beams with occasionally constant. Dynamics of timoshenko beams on pasternak foundation under moving load. Timoshenko beam theory or higherorder theories should be adopted to analyze dynamic problems, especially for the high frequency response and wave. Transverse vibration of viscoelastic timoshenko beamcolumns is investigated. An analytical solution of stresses and deformations for twolayer timoshenko beams glued by a viscoelastic interlayer under uniform transverse load is presented. However, in some cases, euler bernoulli theory, which neglects the effect of transversal shear deformation, yields unacceptable results. Pdf the concept of elastic timoshenko shear coefficients is used as a guide for linear viscoelastic eulerbernoulli beams subjected to. Nonlinear dynamic analysis of a timoshenko beam resting on.
Viscoelastic timoshenko beams with occasionally constant relaxation functions viscoelastic timoshenko beams with occasionally constant relaxation functions tatar, nassereddine 20120801 00. Mean square responses of a viscoelastic timoshenko. The dynamic response of a timoshenko beam with immovable ends resting on a nonlinear viscoelastic foundation and subjected to motion of a traveling mass moving with a constant velocity is studied. A timoshenko functionally graded tfg imperfect microscale beam is considered and the coupled viscoelastic mechanics is analysed in a nonlinear regime.
The analytical solution of the problem is derived based on the approximate timoshenko beam theory for a general continuous loading acting on the upper beam face over the whole beam width and perpendicular to the beam axis. Higher order equation of motion is obtained based on eulerbernoulli and timoshenko beam theory. Buckling analysis of micro and nanorodstubes based on. Timoshenko beam under the act of a moving mass is conducted. Dynamic analysis of laterally loaded endbearing piles in. Viscoelastic timoshenko beam theory viscoelastic timoshenko beam theory hilton, harry 20090301 00. Based on the aforesaid, the further numerical treatment presented in vershinin occurs to be invalid. This chapter gives an introduction is given to elastic beams in three dimensions. In contrast, timoshenko theory is not so much used by engineers. One of its components the rotational displacement is viscoelastic described by.
Viscoelastic mechanics of timoshenko functionally graded. This dynamic impact problem is considered a boundary thin obstacle problem. Transverse vibration of viscoelastic timoshenko beamcolumns. The material of the beam studied is assumed linear orthotropic viscoelastic. An investigation is performed on the viscoelastic nonlinear vibrations of functionally graded imperfect timoshenko beams. The timoshenko beam theory was developed by stephen timoshenko early in the 20th century. Primarily, the beams nonlinear governing coupled pdes of motion for the lateral and longitudinal displacements as well as the beams crosssectional rotation are. It deals with displacement and force on a beam when acted upon by a force. Dynamic response of timoshenko beam resting on non linear viscoelastic foundation carrying any number of spring mass systems.
Free vibration analysis of viscoelastic sandwich beam. Pdf timoshenko beam theorybased dynamic analysis of. It is shown that the corresponding timoshenko viscoelastic functions now depend not only on material properties and geometry as they do in elasticity, but also additionally on stresses and their time. Longtime behavior of a viscoelastic timoshenko beam. Pdf dynamic analysis of a viscoelastic timoshenko beam. In this paper, the stochastic properties of a uniform timoshenko cantilever beam are investigated systematically. Parametrically excited vibration of a timoshenko beam on. Based on the timoshenko beam theory, incorporating geometric imperfections, the kelvinvoigt method is used for internal viscosity, the rotary inertia is automatically generated due to the.
Antonina pirrotta1,2,, stefano cutrona1, salvatore di lorenzo1. The standard linear solid model is employed to simulate the viscoelastic characteristics of the interlayer, in which the memory effect of strains is considered. Linear free vibration analysis of tapered timoshenko. Timoshenko beam abstract this paper investigates the dynamic behavior of nonlocal viscoelastic damped nanobeams. Timoshenko beam theory based dynamic analysis of laterally loaded piles in multilayered viscoelastic soil article pdf available in journal of engineering mechanics 1449 july 2018 with 262. Recently in anewanalyticalsolution to alleviate di culties related to fourier analysis and numerical integration for vibration analysis of an in nite. Nonlinear dynamic analysis of shear deformable beamcolumns on nonlinear threeparameter viscoelastic foundation. Timoshenko beam theory are of the form specify w or q. The concept of elastic timoshenko shear coefficients is used as a guide for linear viscoelastic eulerbernoulli beams subjected to simultaneous. The procedure combines a threedimensional material viscoelastic model with a threedimensional beam theory 9, 10. Based on the external viscous damping and kelvinvoigt viscoelastic damping, the partial differential equations of the timoshenko beam subjected to random excitation are derived.
Boundary control of the timoshenko beam siam journal on. Suresh kumar2 dept of mechanical engineering, jntu college of engineering, hyderabad500085, india. The hybrid laplace transformfinite element method applied. Abstract pdf 310 kb 2002 exponential decay rate of the energy of a timoshenko beam with. The model takes into account shear deformation and rotational bending effects, making it suitable for describing the behaviour of thick beams, sandwich composite beams, or beams subject to highfrequency excitation when the wavelength approaches the thickness of the beam. Free vibration analysis of viscoelastic sandwich beam using euler bernoulli theory ch. Formulations of viscoelastic constitutive laws for beams.
These are all extension of elasticity theories and have different assumptions. Thus present study concentrates on exploring the dynamic behavior of damped cantilever beam with single open crack. Generally, the visco elastic coefficients of bending and shear deformations. Modeling a rotor whose cross section varies can be done using uniform stepped cylindrical elements, but the number of elements required for the converged results is enormous. Eulerbernoulli beam theory also known as engineers beam theory or classical beam theory is a simplification of the linear theory of elasticity which provides a means of calculating the loadcarrying and deflection characteristics of beams. It is shown that the corresponding timoshenko viscoelastic functions now depend. In this paper, considering the moment of inertia and shear deformation, the partial differential equation sets governing the vibration of viscoelastic timoshenko beam subjected to random excitations were derived.
Material property distributions follow the moritanaka model. Understanding of the basic properties of the timoshenko beam problem and ability to derive the basic formulations related to the problem b. It covers the case for small deflections of a beam that are subjected to lateral loads only. Basic knowledge and tools for solving timoshenko beam problems by finite element methods. The equations of motion for rotating timoshenko beam are derived by the dalembert principle and the virtual work principle. Beams centerline stretching is the cause of geometric nonlinearities. The normal and the shear stressstrains are constituted by the kelvin model with different viscosity parameters. Kinematics of timoshenko beam theory undeformed beam. The internal viscosity is incorporated using the kelvinvoigt scheme. A new refined theory for laminatedcomposite and sandwich beams that contains the kinematics of the timoshenko beam theory as a proper baseline subset is presented.
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