Nonlocal strain gradient theory for damping vibration. In table 1, the properties of elastic section for steel and viscoelastic layer are presented. Mechanical response of beams of a nonlinear viscoelastic. We are now prepared to solve some simple stress problems involving a viscoelastic material. 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. Nonlinear inplane vibration of a viscoelastic cantilever beam.
Analysis of complex modal characteristics of fractional. A timoshenko functionally graded tfg imperfect microscale beam is considered and the coupled viscoelastic mechanics is analysed in a nonlinear regime. A viscoelastic higherorder beam finite element computational. The three viscoelastic polymers having the highest loss factor to shear modulus ratio were chosen and tested using a cantilever beam system. An anelastic material is a special case of a viscoelastic material. 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 beam model is then used to study the viscoelastic relaxation in rotary jet spinning. This paper investigates the dynamic behavior of nonlocal viscoelastic damped nanobeams. Operator based constitutive relationship is used to develop the general time domain, linear viscoelastic model. A dispersive equation for a viscoelastic timoshenko beam is given from the derived motion equation. Dispersion curves for a viscoelastic timoshenko beam with.
Free vibration analysis of viscoelastic sandwich beam. 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. Viscoelastic timoshenko beam theory, mechanics of time. Pdf viscoelastic timoshenko beam theory researchgate. Pdf timoshenko beam theorybased dynamic analysis of. 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.
The basic mechanical models of viscoelasticity, the maxwell and kelvin models, are introduced in section 10. 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. Pdf dynamic analysis of a viscoelastic timoshenko beam. The dynamic model of the system is derived using gibbsappell formulation and assumed mode method. A finite element model has been developed for the three layer viscoelastic sandwich beam. Higher order equation of motion is obtained based on eulerbernoulli and timoshenko beam theory. 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. 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 equation of motion for the viscoelastic sandwich beam is derived by using the hamiltons principle. 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 buckling of eulerbernoulli and timoshenko beams under time variant general loading conditions. Governing equilibrium equations are obtained by considering an element of micro beam.
The governing equations are developed based on eulerbernoulli beam theory and kelvinvoigt viscoelastic model. Freed nasa glenn research center, polymers branch, ms 493, 21 0000 brookdark road, brook park, ohio 445, usa a. The concept of elastic timoshenko shear coefficients is used as a guide for linear viscoelastic eulerbernoulli beams. We use the nonlinear eulerbernoulli beam theory to obtain the governing equations. The normal and the shear stressstrains are constituted by the kelvin model with different viscosity parameters. 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. Timedependent behavior of laminated functionally graded. The theory of timoshenko beam was developed early in the twentieth century by the ukrainianborn scientist stephan timoshenko. The timoshenko beam theory is adopted in the derivation of the governing equation. Viscoelastic response is often used as a probe in polymer science, since it is sensitive to. A viscoelastic internal variable constitutive theory is applied to a higherorder elastic beam theory and finite element formulation. Passive viscoelastic constrained layer damping application. 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.
Vibration of nonlocal kelvinvoigt viscoelastic damped timoshenko beams y. Viscoelastic timoshenko beam theory viscoelastic timoshenko beam theory hilton, harry 20090301 00. The constitutive equation is used in a study of the pure bending of beams. The mechanical behavior of each layer is described by the firstorder.
The standard linear solid model is employed to simulate the viscoelastic characteristics of the interlayer, in which the memory effect of strains is considered. This study is intended to analyze dynamic behavior of beams on pasternaktype viscoelastic foundation subjected to timedependent loads. Virtual work principle is also derived and applied to some case studies. Formulation for static behavior of the viscoelastic euler. A viscoelastic beam theory of polymer jets with application. Research article nonlinear dynamic analysis of a timoshenko beam resting on a viscoelastic foundation and traveled by a moving mass ahmadmamandi 1 andmohammadh. The formulation assumes linear viscoelastic material properties and is applicable to problems involving small strains and moderate rotations. Applicability of burger model in predicting the response of viscoelastic soil beds free download as pdf file. Using the eulerbernoulli beam theory, we present in our paper the governing equation for a simply supported viscoelastic beam under a. The numerical approach is based on the combination of the nonlinear cosserat beam theory and a viscoelastic model based on fractional derivatives. 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.
In this study correspondence principle is used for deriving constitutive equations for viscoelastic material based on the modified couple stress theory. Governing equations in terms of the displacements eulerbernoulli and. Passive viscoelastic constrained layer damping application for a small aircraft landing gear system by craig a. Then, introducing dimensionless quantities to differential equations and boundary conditions, the generalized complex eigenvalue equations of the system are obtained by the. When the beam is short in length direction, shear deformation is a factor that may have significant effects on system dynamic. An analytical solution of stresses and deformations for twolayer timoshenko beams glued by a viscoelastic interlayer under uniform transverse load is presented. Stochastic pbifurcation in a nonlinear viscoelastic beam. The differential constitutive law is then combined with the higherorder beam theory and finite element formulation of tessler 11providing viscoelastic capability for thick beams. Determination of viscoelastic core material properties using.
Beam theory ebt is based on the assumptions of 1straightness, 2inextensibility, and 3normality jn reddy z, x x z dw dx. Beams are often used as structural elements in many of those structures, like rotor blades, transmission shafts, frames and robotic arms. A twodimensional 2d elasticity solution is developed to investigate the timedependent response of laminated functionally graded beam with viscoelastic interlayer. On boundary feedback stabilisability of a viscoelastic beam. Herrick laboratories, school of mechanical engineering, purdue university, 140 s. Fractional viscoelastic eulerbernoulli beam sciencedirect. Closedform solution for the static deflection of simply supported micro beam is presented. So, in modeling, the assumption of timoshenko beam theory. Transverse vibration of viscoelastic timoshenko beam columns is investigated. Mar 01, 2009 viscoelastic timoshenko beam theory viscoelastic timoshenko beam theory hilton, harry 20090301 00.
A semianalytical method is developed to obtain the dynamic response of laterally loaded piles in a multilayered soil. Viscoelastic response is often used as a probe in polymer science, since it is sensitive to thematerial s chemistry andmicrostructure. 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. Kochersberger mechanical engineering department abstract the main purpose of this report was to test several common viscoelastic polymers. Beam theory ebt is based on the assumptions of 1straightness, 2inextensibility, and. 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. Kinematic and dynamic modeling of viscoelastic robotic. Theconcepts andtechniques presentedhereare importantforthispurpose,buttheprincipalobjectiveofthisdocumentistodemonstratehow linearviscoelasticity canbeincorporatedintothegeneraltheoryofmechanicsofmaterials, so. Research article nonlinear dynamic analysis of a timoshenko. It is shown that the corresponding timoshenko viscoelastic functions now depend. Fractional viscoelastic eulerbernoulli beam request pdf. Leonov the university ofakron, department of polymer engineering, akron, ohio 443250301, usa abstract the present series of three consecutive papers develops a general theory.
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. Then, introducing dimensionless quantities to differential equations and boundary conditions, the generalized complex eigenvalue equations of the system are obtained. Viscous materials, like water, resist shear flow and strain linearly with time when a stress is applied. Viscoelastic sandwich beam consists of three layers with viscoelastic material as a core layer, the face layers are isotropic and linear elastic material. 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. Leonov the university ofakron, department of polymer engineering, akron, ohio 443250301, usa abstract. Elementary viscoelastic stress analysis for bars and beams. Thus present study concentrates on exploring the dynamic behavior of damped cantilever beam with single open crack. Longtime behavior of a viscoelastic timoshenko beam. A viscoelastic higherorder beam finite element ntrs nasa. Viscoelastic mechanics of timoshenko functionally graded. It usually happens when the deformations are large or if the material changes its properties under deformations. In this paper free vibrations of fixed free sandwich beam with different configurations are investigated analytically. It is thus a special case of timoshenko beam theory.
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. The theory generalizes the classical eulerbernoulli theory to account for finite deformation and material incompressibility. This is to certify that the thesis entitled, vibration analysis of viscoelastic sandwich beam using finite element method submitted by mr. 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. Governing equation of fractional viscoelastic eulerbernoulli beam. Sol mech course text feb10 solid mechanics at harvard. This paper formulates a firstorder beam theory for nonlinear viscoelastic material.
Damping of laminated composite beams with multiple viscoelastic layers. Mechanical response of beams of a nonlinear viscoelastic material polymer engineering and science, february 1995, vol. Vibration analysis of elastic beams with unconstrained. Pdf damping of laminated composite beams with multiple. Analytical solution is carried out using eulerbernoulli beam theory to find the. Thus the elastic simplicity and generality is lost and hence rendering the use of viscoelastic timoshenko shear functions as highly impractical. 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. While plastic behavior is essentially nonlinear piecewise linear at best, viscoelasticity, like elasticity, permits a linear theory. Elastic materials strain when stretched and immediately return to their original state once the stress is removed.
Highlights the elastic timoshenko beam equation is extended to a viscoelastic timoshenko beam equation. Hyer department of mechanical engineering and mechanics, old dominion university, norfolk, virginia 23508, u. A viscoelastic beam theory of polymer jets with application to rotary jet spinning. Anderson department of aerospace engineering, university of michigan and r. Unlike the eulerbernoulli beam, the timoshenko beam model for shear deformation and rotational inertia effects. Timoshenko beam theorybased dynamic analysis of laterally.
Pdf analytical solution for an infinite eulerbernoulli. It covers the case for small deflections of a beam that are subjected to lateral loads only. Scribd is the worlds largest social reading and publishing site. Damping of elasticviscoelastic beams rit scholar works. Quasistatic and dynamic analysis for viscoelastic beams with the. 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. The concept of elastic timoshenko shear coefficients is used as a guide for linear viscoelastic eulerbernoulli beams subjected to simultaneous bending and twisting.
Mechanical response of beams of a nonlinear viscoelastic material. Free vibration analysis of viscoelastic sandwich beam using. The sandwich beam is modelled using linear displacement field at face layer and nonlinear displacement field at core layer. Engineering viscoelasticity david roylance department of materials science and engineering. An efficient solution methodology to study the response of a. 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. Time domain modeling and simulation of nonlinear slender. 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. Aim of this paper is the response evaluation of fractional viscoelastic eulerbernoulli beam under quasistatic and dynamic loads. This paper presents an investigation into the development of modeling of nviscoelastic robotic manipulators. 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. Viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Kargarnovin 2 department of mechanical engineering, parand branch, islamic azad university, tehran, iran department of mechanical engineering, sharif university of technology, tehran, iran. There is viscoelastic material between two mentioned plates, which is made of nbr with thickness of 0.
The kinematics derived is then combined with the oldroydb model to derive the constitutive equations of a nonlinear viscoelastic beam. 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. Stochastic pbifurcation of a bistable viscoelastic beam. 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. It is assumed that the classical assump tion of beam theory is valid, i. Quasistatic and dynamic analysis for viscoelastic beams with. 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. Pdf the concept of elastic timoshenko shear coefficients is used as a guide for linear viscoelastic eulerbernoulli beams subjected to. The behavior of the viscous material in the beam is. Knowledge of the viscoelastic response of a material is based on measurement.
The phase velocity increases as the fractional order approaches 0, and. Applicability of burger model in predicting the response. 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. Due to the complex nature of the derived frequency equation and other factors contributing to the. In order to answer how the vel length and thickness affect the modal parameters and dynamic response, both free and forced vibration. Bernoulli beam theory, we present in our paper the governing equation for a simply supported viscoelastic beam under a uniform distributed load, see. Quasistatic and dynamic analysis for viscoelastic beams. Nonlinear viscoelasticity is when the function is not separable. In load module, we apply 1 n load on the beam end, and close al l freedom degrees at the. 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.
Only the first mode of a viscoelastic timoshenko beam converged to the rayleigh wave velocity. Analytical solution for an infinite eulerbernoulli beam on a viscoelastic foundation subjected to arbitrary dynamic loads. 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. Analytical solution of deformations for twolayer timoshenko. 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. Vibration of nonlocal kelvinvoigt viscoelastic damped.
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