A timoshenko functionally graded tfg imperfect microscale beam is considered and the coupled viscoelastic mechanics is analysed in a nonlinear regime. Beams are often used as structural elements in many of those structures, like rotor blades, transmission shafts, frames and robotic arms. So, in modeling, the assumption of timoshenko beam theory. A viscoelastic beam theory of polymer jets with application to rotary jet spinning. Closedform solution for the static deflection of simply supported micro beam is presented. The phase velocity increases as the fractional order approaches 0, and. Higher order equation of motion is obtained based on eulerbernoulli and timoshenko beam theory. The constitutive equation is used in a study of the pure bending of beams. Sol mech course text feb10 solid mechanics at harvard. Then, introducing dimensionless quantities to differential equations and boundary conditions, the generalized complex eigenvalue equations of the system are obtained by the.
This paper investigates the dynamic behavior of nonlocal viscoelastic damped nanobeams. Then, introducing dimensionless quantities to differential equations and boundary conditions, the generalized complex eigenvalue equations of the system are obtained. The theory of timoshenko beam was developed early in the twentieth century by the ukrainianborn scientist stephan timoshenko. Scribd is the worlds largest social reading and publishing site. Consequently, it is necessary to directly solve the coupled viscoelastic beam governing relations for bending and twisting deflections by using appropriate solution protocols as discussed herein. In this paper, we study the stochastic pbifurcation problem for axially moving of a bistable viscoelastic beam with fractional derivatives of high order nonlinear terms under gaussian white noise excitation. Thus present study concentrates on exploring the dynamic behavior of damped cantilever beam with single open crack.
The elastic modulus in each functionally graded layer varies through the thickness following an exponential function, and the mechanical property of each layer is described by the exact 2d. The theory generalizes the classical eulerbernoulli theory to account for finite deformation and material incompressibility. Mechanical response of beams of a nonlinear viscoelastic material alan wineman and raymond kolberg department of mechanical engineering and applied mechanics university of michigan ann arbor, michigan 48 1 09 a constitutive equation for nonlinear viscoelasticity is used to model the me chanical response of solid polymers such as polycarbonate. The sandwich beam is modelled using linear displacement field at face layer and nonlinear displacement field at core layer. A viscoelastic beam theory of polymer jets with application. In this article, free vibration of functionally graded fg viscoelastic nanobeams embedded in viscoelastic foundation exposed to hygrothermal loading is investigated based on nonlocal strain gradient elasticity theory and a higher order refined beam theory which captures shear deformation influences without the need for any shear correction factor. Starting from the local fractional viscoelastic relationship between axial stress and axial strain, it is shown that bending moment, curvature, shear, and the gradient of curvature involve fractional operators. The concept of elastic timoshenko shear coefficients is used as a guide for linear viscoelastic eulerbernoulli beams. Leonov the university ofakron, department of polymer engineering, akron, ohio 443250301, usa abstract. On boundary feedback stabilisability of a viscoelastic beam. Mar 01, 2009 viscoelastic timoshenko beam theory viscoelastic timoshenko beam theory hilton, harry 20090301 00. Governing equations in terms of the displacements eulerbernoulli and.
It is shown that a cantilevered beam with weak viscoelastic damping of boltzmanntype can be uniformly stabilised by velocity feedback applied as a shearing force at the free end of the beam. Mechanical response of beams of a nonlinear viscoelastic material polymer engineering and science, february 1995, vol. In the analysis, the soil is modeled as a threedimensional viscoelastic continuum with frequencyindependent hysteretic material damping and the pile as a circular elastic timoshenko beam. Dynamic analysis of beams on viscoelastic foundation. The normal and the shear stressstrains are constituted by the kelvin model with different viscosity parameters. Vibration of nonlocal kelvinvoigt viscoelastic damped timoshenko beams y. Theconcepts andtechniques presentedhereare importantforthispurpose,buttheprincipalobjectiveofthisdocumentistodemonstratehow linearviscoelasticity canbeincorporatedintothegeneraltheoryofmechanicsofmaterials, so. This is to certify that the thesis entitled, vibration analysis of viscoelastic sandwich beam using finite element method submitted by mr. Friswell b a college of aerospace and material engineering, national university of defence technology, changsha, hunan 410073, pr china b college of engineering, swansea university, singleton park, swansea sa2 8pp, uk article info article history. Kochersberger mechanical engineering department abstract the main purpose of this report was to test several common viscoelastic polymers. It is shown that the corresponding timoshenko viscoelastic functions now depend. Mechanical response of beams of a nonlinear viscoelastic. This paper presents an investigation into the development of modeling of nviscoelastic robotic manipulators. Estimates for the viscoelastic energy are derived using the energy multiplier method.
The elastic modulus in each functionally graded layer varies through the thickness following an exponential function, and the mechanical property of each layer is described by the exact 2d elasticity equations. The dynamic characteristics of damped viscoelastic nonlocal beams are studied in this. An efficient solution methodology to study the response of a. The concept of elastic timoshenko shear coefficients is used as a guide for linear viscoelastic eulerbernoulli beams subjected to simultaneous bending and twisting. A ross, kerwin, and ungar analysis was used to predict the loss factor of the cantilever beam system with applied treatment and the predictions were compared to experimental data. The kinematics derived is then combined with the oldroydb model to derive the constitutive equations of a nonlinear viscoelastic beam. Beam theory ebt is based on the assumptions of 1straightness, 2inextensibility, and. In this paper, we study the stochastic pbifurcation problem for an axially moving bistable viscoelastic beam with fractional derivatives of highorder nonlinear terms under colored noise excitatio. Timoshenko beam theorybased dynamic analysis of laterally.
Nonlinear inplane vibration of a viscoelastic cantilever beam. Quasistatic and dynamic analysis for viscoelastic beams. Time domain modeling and simulation of nonlinear slender. Timedependent behavior of laminated functionally graded. The behavior of the viscous material in the beam is. The concept of elastic timoshenko shear coefficients is used as a guide for linear viscoelastic eulerbernoulli beams subjected to simultaneous. A finite element model has been developed for the three layer viscoelastic sandwich beam. Virtual work principle is also derived and applied to some case studies. Nonlinear viscoelasticity is when the function is not separable. In this paper free vibrations of fixed free sandwich beam with different configurations are investigated analytically. Stochastic pbifurcation in a nonlinear viscoelastic beam. This paper formulates a firstorder beam theory for nonlinear viscoelastic material. In load module, we apply 1 n load on the beam end, and close al l freedom degrees at the. In order to answer how the vel length and thickness affect the modal parameters and dynamic response, both free and forced vibration.
First, using the principle for minimum mean square error, we show that the fractional derivative term is equivalent to a linear combination of the damping force and. Governing equilibrium equations are obtained by considering an element of micro beam. Unlike the eulerbernoulli beam, the timoshenko beam model for shear deformation and rotational inertia effects. Kinematic and dynamic modeling of viscoelastic robotic. Elastic materials strain when stretched and immediately return to their original state once the stress is removed. 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. An anelastic material is a special case of a viscoelastic material. Viscoelastic timoshenko beam theory, mechanics of time. Mechanical response of beams of a nonlinear viscoelastic material. Aim of this paper is the response evaluation of fractional viscoelastic eulerbernoulli beam under quasistatic and dynamic loads. Determination of viscoelastic core material properties using sandwich beam theory and modal experiments 1999011677 damping material for automotive structures is often quantified in terms of composite loss factor or damping ratio by using astmsae beam or modal tests. Applicability of burger model in predicting the response of viscoelastic soil beds free download as pdf file.
It is shown that the corresponding timoshenko viscoelastic functions now. Passive viscoelastic constrained layer damping application. Analytical solution is carried out using eulerbernoulli beam theory to find the. A viscoelastic internal variable constitutive theory is applied to a higherorder elastic beam theory and finite element formulation. Longtime behavior of a viscoelastic timoshenko beam. Viscous materials, like water, resist shear flow and strain linearly with time when a stress is applied. The formulation assumes linear viscoelastic material properties and is applicable to problems involving small strains and moderate rotations. A viscoelastic higherorder beam finite element computational. Free vibration analysis of viscoelastic sandwich beam. Knowledge of the viscoelastic response of a material is based on measurement. 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. There is viscoelastic material between two mentioned plates, which is made of nbr with thickness of 0. A viscoelastic higherorder beam finite element ntrs nasa.
Vibration of nonlocal kelvinvoigt viscoelastic damped. It usually happens when the deformations are large or if the material changes its properties under deformations. Herrick laboratories, school of mechanical engineering, purdue university, 140 s. The differential constitutive law is then combined with the higherorder beam theory and finite element formulation of tessler 11providing viscoelastic capability for thick beams. Viscoelastic response is often used as a probe in polymer science, since it is sensitive to. It covers the case for small deflections of a beam that are subjected to lateral loads only. Elementary viscoelastic stress analysis for bars and beams. Freed nasa glenn research center, polymers branch, ms 493, 21 0000 brookdark road, brook park, ohio 445, usa a. Analysis of complex modal characteristics of fractional. The numerical approach is based on the combination of the nonlinear cosserat beam theory and a viscoelastic model based on fractional derivatives. The latter activities are, of course, the domain of engineering and many important modern sub fields of solid mechanics have been actively developed by engineering scientists concerned, for example, with mechanical, structural, materials, civil or aerospace engineering.
The standard linear solid model is employed to simulate the viscoelastic characteristics of the interlayer, in which the memory effect of strains is considered. It is assumed that the classical assump tion of beam theory is valid, i. Pdf timoshenko beam theorybased dynamic analysis of. While plastic behavior is essentially nonlinear piecewise linear at best, viscoelasticity, like elasticity, permits a linear theory.
The beam model is then used to study the viscoelastic relaxation in rotary jet spinning. Quasistatic and dynamic analysis for viscoelastic beams with the. Dispersion curves for a viscoelastic timoshenko beam with. Research article nonlinear dynamic analysis of a timoshenko beam resting on a viscoelastic foundation and traveled by a moving mass ahmadmamandi 1 andmohammadh. 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. Viscoelastic response is often used as a probe in polymer science, since it is sensitive to thematerial s chemistry andmicrostructure. It is thus a special case of timoshenko beam theory. Applicability of burger model in predicting the response. Highlights the elastic timoshenko beam equation is extended to a viscoelastic timoshenko beam equation. Kargarnovin 2 department of mechanical engineering, parand branch, islamic azad university, tehran, iran department of mechanical engineering, sharif university of technology, tehran, iran. Tessler computational structures branch nasa langley research center, ms 240 hampton, va 23681 abstract a viscoelastic internal variable constitutive theory is applied to a higherorder elastic beam theory and finite element formulation. Pdf the concept of elastic timoshenko shear coefficients is used as a guide for linear viscoelastic eulerbernoulli beams subjected to. Transverse vibration of viscoelastic timoshenko beam columns is investigated.
Nonlocal strain gradient theory for damping vibration. When the beam is short in length direction, shear deformation is a factor that may have significant effects on system dynamic. Viscoelastic buckling of eulerbernoulli and timoshenko beams under time variant general loading conditions. Review simple beam theory generalize simple beam theory to three dimensions and general cross sections consider combined e ects of bending, shear and torsion study the case of shell beams 7. The timoshenko beam theory is adopted in the derivation of the governing equation. Viscoelastic timoshenko beam theory viscoelastic timoshenko beam theory hilton, harry 20090301 00. The mathematical formulation of viscoelasticity theory is presented in the following chapters with the aim of enabling prediction of the material response to arbitrary load histories. A dispersive equation for a viscoelastic timoshenko beam is given from the derived motion equation.
Only the first mode of a viscoelastic timoshenko beam converged to the rayleigh wave velocity. 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. Fractional viscoelastic eulerbernoulli beam request pdf. Quasistatic and dynamic analysis for viscoelastic beams with. Determination of viscoelastic core material properties using. Analytical solution for an infinite eulerbernoulli beam on a viscoelastic foundation subjected to arbitrary dynamic loads. Fractional viscoelastic eulerbernoulli beam sciencedirect. The three viscoelastic polymers having the highest loss factor to shear modulus ratio were chosen and tested using a cantilever beam system. A twodimensional 2d elasticity solution is developed to investigate the timedependent response of laminated functionally graded beam with viscoelastic interlayer. 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. Viscoelastic materials are widely used for passive damping in a variety of engineering structures due to the need for structural stability and durability. The governing equations are developed based on eulerbernoulli beam theory and kelvinvoigt viscoelastic model. The equation of motion for the viscoelastic sandwich beam is derived by using the hamiltons principle. In this study correspondence principle is used for deriving constitutive equations for viscoelastic material based on the modified couple stress theory.
Vibration analysis of elastic beams with unconstrained. Governing equation of fractional viscoelastic eulerbernoulli beam. An analytical solution of stresses and deformations for twolayer timoshenko beams glued by a viscoelastic interlayer under uniform transverse load is presented. The basic mechanical models of viscoelasticity, the maxwell and kelvin models, are introduced in section 10. Research article nonlinear dynamic analysis of a timoshenko. We are now prepared to solve some simple stress problems involving a viscoelastic material. Pdf analytical solution for an infinite eulerbernoulli. Formulation for static behavior of the viscoelastic euler. Operator based constitutive relationship is used to develop the general time domain, linear viscoelastic model. Using the eulerbernoulli beam theory, we present in our paper the governing equation for a simply supported viscoelastic beam under a. A semianalytical method is developed to obtain the dynamic response of laterally loaded piles in a multilayered soil. Using the eulerbernoulli beam theory, we present in our paper the governing equation for a simply supported viscoelastic beam under a uniform distributed load, see 19.
Hyer department of mechanical engineering and mechanics, old dominion university, norfolk, virginia 23508, u. Viscoelastic mechanics of timoshenko functionally graded. Thus the elastic simplicity and generality is lost and hence rendering the use of viscoelastic timoshenko shear functions as highly impractical. Stochastic pbifurcation of a bistable viscoelastic beam. Viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. The mechanical behavior of each layer is described by the firstorder.
In table 1, the properties of elastic section for steel and viscoelastic layer are presented. The subject of our study will be a structure, that is, a deformable body of known shape to which external forces, the loads, are applied. Beam theory ebt is based on the assumptions of 1straightness, 2inextensibility, and 3normality jn reddy z, x x z dw dx. Passive viscoelastic constrained layer damping application for a small aircraft landing gear system by craig a. Leonov the university ofakron, department of polymer engineering, akron, ohio 443250301, usa abstract the present series of three consecutive papers develops a general theory. As with pressure vessels, the geometry of the beam, and the specific type of loading which will be considered, allows for approximations to be made to the full threedimensional linear elastic stressstrain relations. Free vibration analysis of viscoelastic sandwich beam using. Damping of elasticviscoelastic beams rit scholar works.
The dynamic model of the system is derived using gibbsappell formulation and assumed mode method. Bernoulli beam theory, we present in our paper the governing equation for a simply supported viscoelastic beam under a uniform distributed load, see. Damping of laminated composite beams with multiple viscoelastic layers. We consider the nonlinear response of a slender isotropic viscoelastic cantilever beam with lumped mass m at the tip, subject to harmonic transverse base excitation, v b see figures1and2. The beam model is then used to study the viscoelastic relaxation. Engineering viscoelasticity david roylance department of materials science and engineering. This study is intended to analyze dynamic behavior of beams on pasternaktype viscoelastic foundation subjected to timedependent loads. For the transverse vibration problem of a fractional derivative viscoelastic rotating beam, the differential equation of the system is obtained based on the eulerbernoulli beam theory and hamilton principle. We use the nonlinear eulerbernoulli beam theory to obtain the governing equations. In this article, free vibration of functionally graded fg viscoelastic nanobeams embedded in viscoelastic foundation exposed to hygrothermal loading is investigated based on nonlocal strain gradient elasticity theory and a higher order refined beam theory which captures shear deformation influences without the need for any shear correction. Pdf viscoelastic timoshenko beam theory researchgate. Viscoelastic sandwich beam consists of three layers with viscoelastic material as a core layer, the face layers are isotropic and linear elastic material. Pdf dynamic analysis of a viscoelastic timoshenko beam. Pdf damping of laminated composite beams with multiple.
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