Viscoelastic Beam Research Articles

Fugue effect on bending of nonlinear viscoelastic beams

Based on the Leaderman constitutive relations in nonlinear viscoelasticity and the linear geometrical assumption, a mathematical model for the bending of nonlinear viscoelastic beams was established in this paper. The Laplace transformation method and the Titchmarsh theorem were used to prove that some relations exist between solutions to bending problems of visco- and elastic beams, which reveals the fugue effect of viscoelastic materials. The high-order Galerkin approximate solution to the quasi-static response of nonlinear viscoelastic beams under a step load was obtained by using the new method suggested in this paper as well as the Mathematica software and the Newton iteration technique.

A thermoviscoelastic beam with a tip body

We present and investigate a model for the thermomechanical behaviour of a viscoelastic beam with an attached tip body. The model consists of a coupled energy–elastic system of equations for the beam together with an equation for the temperature of the tip body. We establish existence and uniqueness of a weak solution to the system. We also propose a finite element algorithm for solutions to the system and present numerical simulations based on the algorithm.

Stochastic stability of non-gyroscopic viscoelastic systems

In this paper the dynamic stability of non-gyroscopic viscoelastic systems under multiple parametric excitations is investigated. The largest Lyapunov exponent as an indicator of the almost-sure asymptotic stability of the system is obtained by applying the stochastic averaging method together with Khasminskii's technique. The integral term arising from the viscoelastic effect is averaged by making use of Larianov's method. As an application, the flexural–torsional instability of a deep rectangular viscoelastic beam under stochastically fluctuating central load and end moments applied simultaneously is investigated. Both cases of follower and non-follower central fluctuating load are included in this analysis.

Nonlinear equations for a thin beam

Finite oscillations of a thin beam are analyzed, taking into account large deflections and rotations with small strains. It is shown that, depending on the assumptions, problems for an elastic and viscoelastic beam lead to solving hyperbolic or parabolic nonlinear partial differential equations. As an example, a cantilever beam is considered. The solutions for nonlinear vibrations are obtained by means of the Galerkin method, and numerical results are presented graphically for four material models.

Longtime behavior of a viscoelastic Timoshenko beam

We consider a Timoshenko model of a viscoelastic beam fixed at the endpoints and subject to nonlinear external forces. The model consists of two coupled second order linear integrodifferential hyperbolic equations that govern the evolution of the lateral displacement $u$ and the total rotation angle $\phi$. We prove that these equations generate a dissipative dynamical system, whose trajectories are eventually confined in a uniform absorbing set, the dissipativity being due to the memory mechanism solely. This fact allows us to state the existence of a uniform compact attractor.

Control of lateral motion in moving webs

A partial differential equation for the lateral motion of a web conveyance system is derived by modeling the web as a viscoelastic beam under axial tension. This model treats the web position between rollers as a function of both time and space, assumes that there is no slip between the web material and the rollers, and incorporates the web material's viscoelastic damping property. A finite-difference approximation of the model is used to simulate a typical two-span web system. The finite-difference approximation is validated by comparing its frequency responses with those of an analytical frequency domain model. The analytical frequency domain model is used to design feedback compensation strategies that make the two-span web system less sensitive to upstream disturbances. The results show that, using a transverse vibration model incorporating viscoelasticity to design even simple classical controllers, it is possible to make the web system less sensitive to upstream disturbances at the sensor location.

The mixed finite element for quasi-static and dynamic analysis of viscoelastic circular beams

The quasi-static and dynamic responses of a linear viscoelastic circular beam on Winkler foundation are studied numerically by using the mixed finite element method in transformed Laplace-Carson space. This element VCR12 has 12 independent variables. The solution is obtained in transformed space and Schapery, Dubner, Durbin and Maximum Degree of Precision (MDOP) transform techniques are employed for numerical inversion. The performance of the method is presented by several quasi-static and dynamic example problems.

A numerical method for fractional integral with applications

A new numerical method for the fractional integral that only stores part history data is presented, and its discretization error is estimated. The method can be used to solve the integro-differential equation including fractional integral or fractional derivative in a long history. The difficulty of storing all history data is overcome and the error can be controlled. As application, motion equations governing the dynamical behavior of a viscoelastic Timoshenko beam with fractional derivative constitutive relation are given. The dynamical response of the beam subjected to a periodic excitation is studied by using the separation variables method. Then the new numerical method is used to solve a class of weakly singular Volterra integro-differential equations which are applied to describe the dynamical behavior of viscoelastic beams with fractional derivative constitutive relations. The analytical and unmerical results are compared. It is found that they are very close.

Estimation of the state vector and identification of the complex modulus of a beam

A non-uniform viscoelastic beam traversed by flexural waves was considered. Methods based on Timoshenko's model were established for (i) estimation of its state (shear force, transverse velocity, bending moment and angular velocity) at an arbitrary section on the basis of at least four independent measurements, and (ii) identification of its complex modulus, parametric as well as non-parametric, on the basis of at least five independent measurements. From the estimated state, related useful quantities such as strain, stress and power transmission can be obtained. Experimental tests were carried out with beams made of polymethyl methacrylate and polypropylene and instrumented with pairs of strain gauges at eight non-uniformly distributed sections. Estimation of strain at one instrumented section was based on measured strains at five to seven surrounding sections, while identification of the complex modulus was based on measured strains at five to eight sections. Generally, the identified complex moduli showed fair agreement with previous results from tests involving extensional waves, while the estimated strains were in good accord with measured strains. No significant improvement in the quality of results was achieved when the number of measured strains was increased to more than five for the identification of the complex modulus and six for the estimation of state.

Application of Galerkin Method to Dynamical Behavior of Viscoelastic Timoshenko Beam with Finite Deformation

The motion equations governing the dynamical behavior of a viscoelasticTimoshenko beam with finite deformation are derived and simplified byGalerkin method. The viscoelastic material is assumed to obey thethree-dimensional fractional derivative constitutive relation. Thedynamical behaviors of the simplified systems with order 1 and order 2are numerically computed and compared by using the computational methodpresented by the authors. The dynamical behaviors of the systems areuniform qualitatively, but there is a little deviation quantitatively.And the truncated system with order 1 is safer than the one of order 2.It is also shown that the lower order system is reasonable. Theinfluences of the load parameter and the fractional derivative parameter(material parameter) on the deflection of the beam are consideredrespectively. The numerical methods in nonlinear dynamics, such as phasediagram, and Poincare section, are applied to reveal dynamical behaviorsof the nonlinear viscoelastic Timoshenko beam. There are plenty ofdynamical behaviors, such as periodicity, bifurcation, quasi-periodicityand chaos in the dynamical system.

Analysis and simulations of vibrations of a beam with a slider

A model for vibrations of a beam with a slider is derived, analysed and numerically simulated. It describes a viscoelastic beam that is clamped at one end to a vibrating device, while the other end moves between two stops attached to a slider. The contact is described by the normal compliance or by the Signorini conditions. The existence of weak solutions is established using the theory of set-valued pseudomonotone operators. The model is discretized using fourth-order spatial discretization, the solutions are numerically simulated and their results presented. The dynamics of the vibrations are depicted and so are the noise characteristics of the system.

Viscoelastic Beam Damping And Piezoelectric Control Of Deformations, Probabilistic Failures And Survival Times – Analytical And Massively Parallel Computational Simulations

The effects of controlling viscoelastic beam creep deformations through applications of piezoelectric devices is investigated and solved by finite element methods. It is shown that delamination failure probabilities can be retarded and life times extended by the proper imposition of voltages. Computational performance evaluation studies for CPU times and speed ups are undertaken for distinct number of processors and evaluated. It is shown that the introduction of piezo devices and viscoelastic effects not only decrease temporal probabilities of failure, but also retard the structure’s survival times. These results are also of importance in inverse problems, where voltages may be input to generate controlled displacements. Thus, such voltages, either generated and dissipated through resistors or input fi-om external sources, can be used to control viscoelastic deformations, failure probability envelopes and survival times of composite and elevated temperature metal structural elements. It also shown that the use of LS-DYNA3DTk’~ significantly improves CPU times and speed ups compared to previous usage of ABAQUSTY~.

AN EXACT METHOD FOR THE FREE VIBRATION ANALYSIS OF TIMOSHENKO–KELVIN BEAMS WITH OSCILLATORS

In this paper, a modern exact method is proposed for solving the problem of free vibrations of a Timoshenko-type viscoelastic beam with discrete rigid bodies, connected to the beam by means of viscoelastic constraints. The phenomenon of free vibrations of this discrete–continuous system is described by a set of three partial and two subsystem ordinary differential equations with generalized boundary conditions and initial conditions. Vector notation of the equations allows one to identify the self-adjoint linear operators of inertia, stiffness and damping. In this case, these operators are not homothetic hence a separation of variables in this set of equations is possible only in a complex Hilbert space. Such separation of variables leads to ordinary differential equations of motion with respect to time as well as to a set of three ordinary differential equations with respect to a spatial variable and two subsystem algebraical equations. Solution of the boundary-value problem was carried out in the classical way, but its results are complex conjugated. Using these results and the fundamental principle, describing the orthogonality property of complex eigenvectors, the problem of free vibrations of the system with arbitrary initial conditions has been finally solved exactly.

Development of a curvilinear viscoelastic constitutive relationship for time dependent materials. Part B: Example problems

AbstractThe boundary value problem of a two‐dimensional viscoelastic beam clamped at two ends and subjected to both a dead load and normal surface traction is solved using a finite element method here in Part B. The constitutive relationship between stress and strain developed in Part A uses a convected coordinate system and body tensor mathematics. Maxwell elements (springs and dashpots) describe the material for both fluid and solid characterizations. The overall finite element results calculated were physically reasonable. However, the assumed boundary conditions produce a stress singularity at the boundary, a singularity that is also seen in small deformation elastic theory.

Attractors of a rotating viscoelastic beam

We investigate the non-linear oscillations of a rotating viscoelastic beam with variable pitch angle. The governing equations of motion are two coupled partial differential equations for the longitudinal and transversal displacements. Using a perturbation technique and Galerkin's projection, we reduce the equations of motion to a non-autonomous ordinary differential equation. Our regular perturbation technique is based on the expansion of longitudinal displacement and the amplitude of first transversal mode in terms of a small parameter. We numerically generate the Poincaré maps of the reduced equations and reveal that the system exhibits regular and chaotic attractors. The regular attractors are stable limit-cycles that are relevant to stable, short-period oscillations of the beam. A bifurcation analysis has also been performed when the pitch angle is constant.

Asymptotic behaviour of a semilinear viscoelastic beam model

The asymptotic behaviour of a model for a viscoelastic beam with nonlinear load is studied. Under the assumptions that the energy is coercive and the solution set of the stationary problem is discrete, the convergence of the solutions of the dynamical problem to a steady state are shown. To prove this statement, a method known for problems in finite dimensions is combined with results about the corresponding linear problem, energy estimates for the nonlinear problem, and harmonic analysis of vector-valued functions.

Elastic beam in adhesive contact

Abstract Dynamic and quasistatic processes of contact with adhesion between an elastic or viscoelastic beam and a foundation are considered. The contact is modeled with the Signorini condition when the foundation is rigid, and with normal compliance when it is deformable. The adhesion is modeled by introducing the bonding function β, the evolution of which is described by an ordinary differential equation. The existence and uniqueness of the weak solution for each of the problems is established using the theory of variational inequalities, fixed point arguments and the existence and uniqueness result in Commun. Contemp. Math. 1(1) (1999) 87–123. The numerical approximations of the quasistatic problem with normal compliance are considered, based on semi-discrete and fully discrete schemes. The convergence of the solutions of the discretized schemes is proved and error estimates for these approximate solutions are derived.

The use of the surface location approach in the modeling of periodically nonhomogeneous slender visco-elastic beams

The use of the surface location approach in the modeling of periodically nonhomogeneous slender visco-elastic beams

Influence of transverse shear on stochastic instability of viscoelastic beam

The stochastic instability problem associated with an axially loaded Timochenko beam made of viscoelastic material is formulated. The beam is treated as Voigt–Kelvin body compressed by time-dependent deterministic and stochastic forces. By using the direct Liapunov method, bounds of the almost sure instability of beams as a function of retardation time, variance of the stochastic force, mode number, section shape factor and intensity of the deterministic component of axial loading, are obtained. Calculations are performed for the Gaussian process with a zero mean and variance σ 2 as well as for harmonic process with an amplitude A.

Nonlinear viscoelastic beam bending with piezoelectric control – analytical and computational simulations

Bending of nonlinear viscoelastic beams with small or large deformations is analyzed in the presence of viscoelastic piezoelectric devices placed on the upper and lower outer beam fibers. Probabilities of failure and corresponding ultimate structural survival times for delamination onset are derived and evaluated.