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