Viscoelastic Boundary Conditions Research Articles

Vibrations and energy distribution in inhomogeneous rods with elastic and viscous boundary conditions

Functionally graded materials have broad engineering applications including mechanical engineering, electronics, chemistry, and biomedical engineering. One notable advantage of such materials is that their stiffness distribution can be optimized to avoid stress concentration. A novel approach for solving the equations describing the longitudinal vibration of functionally graded rods with viscous and elastic boundary conditions is proposed. The characteristic equation of the system is derived for the solution of the undamped case for the constant stiffness rod. Then, a homotopy method is applied to compute the eigenvalues and mode shapes of graded rods for viscoelastic boundary conditions. The changes of the eigenvalues and mode shapes as function of the damping parameters are investigated. The optimal damping of the system is computed. It is shown that the qualitative behavior depends on the relation between the actual damping and the optimal damping of the system. The energy density distribution of graded rods is also discussed. An energy measure, the mean scaled energy density distribution is introduced to characterize the energy distribution along the rod in the asymptotic time limit. The significance of such a measure is that it reveals how the energy tends to distribute along the rod. It is shown that the energy distribution can be manipulated by changing the damping parameters. Qualitative changes depending on the relation between the actual damping and the optimal damping are highlighted.

Whirling analysis of unbalanced rotor using the vectors of solution coefficients

A prevalent issue in rotors is mass unbalance, which leads to synchronous lateral vibrations. In this paper, the critical whirling speeds and global responses of an unbalanced rotor-bearing system with viscoelastic boundary condition are computed using the relationships between the vectors of solution coefficients of differential equations governing the transverse displacements. The mathematical formulation is based on the matrix form of the aforementioned relationships and boundary conditions in the resonance case. The Timoshenko beam model, including the effects of gyroscopic moments is employed. It was found that the presented approach in comparison with the transfer matrix method reduces the matrix product when shaft segments have the same geometric and mechanical properties. A good agreement is found between the results of this study and the findings of studies available in the literature. Furthermore, the eigenvectors for free vibrations of the system were calculated by combining Natanson’s technique with the presented approach while considering the derived rotor’s characteristic equation after removing the forced term in the equation that describes the forced vibrations. Each obtained forward whirling mode shape describes perfectly the global response in the vicinity of the corresponding forward critical speed. The presented approach can, therefore, be applied to obtain the unbalance responses, whirling speeds, and associated mode shapes of rotor-bearing systems with complex boundary conditions.

A random non-uniform wave input method based on complex incident site waves

To improve the deficiency of the wave input method (WIM) in expressing complex incident site waves, in this paper, a random non-uniform wave input method (RNUWIM) is derived from the use of the random incident waves and the visco-elastic boundary condition. The random incident waves are generated and adjusted by the comparison of the horizontal ground motion’s response spectrum and the target spectrum, so that they are representative of the actual wavefield in project sites. Furthermore, the RNUWIM is implemented and verified by the finite element method. The result shows that the RNUWIM inherits the advantages of the WIM in simulating the wave propagation effect and the radiation damping effect. In the new method, the spatial variation of ground motions such as time delays, peak variation, and time-history shape differences can be simulated synthetically. The errors of RNUWIM in ground acceleration, velocity, and displacement at the reference point are respectively 3.23%, 0.66%, and 0.07% in one case, which proves the efficiency of the RNUWIM in seismic response analyses.

Effect of porosity and viscoelastic boundary conditions on FG sandwich beams in thermal environment: Buckling and vibration studies

The present study is carried out to investigate the combined effect of porosity and temperature on the buckling and vibration attributes of FG sandwich beams in the thermal environment using FE formulation. The modeled sandwich beam consisting of the viscoelastic core material is restrained by viscoelastic boundary conditions (VBCs). The FG face layers and core are subjected to temperature-dependent material properties. Complex stiffness model is adopted for VBCs. Porosity patterns such as H, V, X, and O are incorporated into FG face layers. The Lagrange equation is used to derive the sandwich beam’s equilibrium equations of motion in static and dynamic conditions. The derived equilibrium equations are solved for buckling and vibration of the beam using the FE solution. Lagrange and Hermite shape functions are assumed for axial and transverse displacements. Critical buckling temperature (CBT), natural frequency (NF), and loss factors (LF) are obtained for various temperatures and boundary stiffness values (BSVs). Transverse buckling and vibration mode shapes are extracted for changing BSVs. The behavior of NF and LF at buckling temperature is also discussed. The existence of porosities ameliorates the buckling characteristics of the sandwich beam. VBCs expedite the vibration damping of sandwich beams alongside the viscoelastic core. The natural frequency and loss factor reach zero and infinity, respectively, when the temperature reaches CBT.

Modelling and predicting the dynamic response of an axially graded viscoelastic core sandwich beam

The present study explored the influence of axial gradation of viscoelastic materials on the dynamic response of the sandwich beam for structural applications. The finite element (FE) formulations are used to model and investigate dynamic response of the sandwich beam. The classical beam theory is used to develop the FE formulations and Lagrange's approach is considered to obtain the equations of motion (EOM). FE code is developed and validated with the existing literature and also conducted the convergence study for the developed FE method. Further, the influence of different viscoelastic materials and boundary conditions on the dynamic response of the sandwich beam is investigated. Four different axial gradation configurations of viscoelastic materials are considered for the present work to explore the influence on natural frequency, loss factor and frequency response of the sandwich beam. The modeled axial gradation of viscoelastic material has displayed a considerable impact on the peak vibrational amplitude response of the sandwich beam for all the boundary conditions and these configurations improved the damping capabilities at different configurations for the structural applications.

Implementation of a viscoelastic boundary condition to Yade – open source DEM software

1. Brzeziński K., Gladky A., 2022, Clump breakage algorithm for DEM simulation of crushable aggregates, Tribology International, 173, 107661. CrossRef Google Scholar

Wave equation with viscoelastic acoustic boundary conditions and supercritical source term

This paper is concerning with the study of Hadamard well-posedness for a wave equation with nonlinear internal damping and acoustic boundary conditions for non-locally reacting boundary. On the boundary we consider the action of viscoelastic and frictional damping and also a supercritical source term. To prove the existence of solution we use semigroup theory. We also study the asymptotic behavior of the functional energy associated to the problem.

Natural Frequencies of a Viscoelastic Supported Axially Functionally Graded Bar With Tip Masses

The frequency parameters of an Axially Functionally Graded (AFG) bar for viscoelastic and point mass boundary conditions are studied in this paper. The structures with functionally graded materials under axial loads can be modelled as AFG bars. Those may be considered as support elements with spring and dashpot in the vibration isolation. Keeping the ratio of force frequency and natural frequency at a certain level is ensured to control the natural frequency therefore dynamic amplification factor with different material combinations. Various boundary conditions are attained by changing the spring and damping coefficients of viscoelastic support elements and the ratio of rod mass to tip mass. Researchers assume that the beam material properties in directions of length and thickness change exponentially, individually, or both in their studies. There are a few studies on the AFG rods in the existing studies. Analysis is carried out via the finite element method for the non-dimensional frequency parameters of the bar in MATLAB. The energy equations of the motion are obtained in the frame of axial bar theory considering the material properties of the bar vary longitudinally according to the power-law distribution. The effects of material distribution, spring, damping and tip mass values on the bar's frequency parameters and structural behaviour have been extensively investigated.

Mechanical behavior of high-pressure pipeline installed through horizontal directional drilling under seismic loads

The present study investigates the mechanical behavior of high-pressure pipelines installed through horizontal directional drilling (HDD), a crucial trenchless installation method, under seismic loads. Utilizing the finite element method and artificial viscoelastic boundary conditions, this paper analyzes the stress response of pipelines subjected to seismic loads in two directions. The seismic loads are applied using the response spectrum method, and the pipeline stress is analyzed in time history. A comparison of the stress in pipelines installed through the HDD method and the traditional open-cut method is conducted, and the impact of earthquake magnitude on pipeline stress is evaluated. The results indicate that the pipeline stress is greater when the seismic acceleration acts transversely, with bending deformation being the dominant mode of response. A 9.29% increase in maximum stress is observed for each increment in earthquake magnitude. Furthermore, the stress in pipelines installed through the HDD method is 48% lower than that of pipelines installed using the open-cut method when subjected to transverse seismic loads.

A stepwise artificial boundary condition for wave propagation in elasto-plastic media

Wave propagation in elasto-plastic media is often encountered in earthquake geotechnical engineering. However, dynamic boundary conditions in the numerical simulation of wave propagation problems are often developed based on linear elastic assumptions. In this study, the method of characteristics is utilized to propose a novel stepwise artificial boundary condition (S-ABC) for wave propagation in elasto-plastic media. The numerical implementation of S-ABC within a finite element procedure in time domain is developed. Typical numerical examples are chosen to verify the efficiency of this ABC, including: 1) a semi-infinite soil column model subjected to impulse load, where the influence of unloading wave front in elasto-plastic media is discussed; 2) an infinite and a finite soil column model under cyclic seismic wave input, where the proposed S-ABC is compared with the traditional viscoelastic artificial boundary condition (V-ABC). Numerical results demonstrate the significantly higher accuracy and robustness of the proposed S-ABC for nonlinear and elasto-plastic media compared to traditional V-ABC. The influence of unloading wave front on artificial boundary condition for elasto-plastic media is also revealed, which has not been reported previously.

Force relaxation of contact between a flat-ended cylindrical indenter and a poroviscoelastic layer

Interpreting the contact force relaxation between an indenter and a poroviscoelastic material at a fixed displacement to obtain the materials properties is a challenging task, because both fluid drainage and solid viscoelasticity contribute to the relaxation. In this research, the indentation of a poroviscoelastic layer caused by a flat-ended cylindrical indenter is formulated analytically by the displacement functions in the Laplace-transformed domain. The contact force of the indenter at a fixed displacement was expressed by a closed-form in the transformed domain and numerically inverse-transformed to the time domain. The force relaxation is analyzed for combinations of three drainage conditions prescribed on the layer top and two boundary conditions prescribed on the layer bottom. The closed-form solution can approach that of the flat-indentation on a poroviscoelastic half-space, when the layer is very thick compared to the indenter radius. In addition, finite element simulations of the flat-indentation on poroviscoelastic layers were carried out as a comparison for the semi-analytical results. The relaxation ratios at the limits of the drained-nonviscoelastic state, the nondrained-viscoelastic state and the drained-viscoelastic state are shown for different ratios of the layer thickness to the indenter radius, and verified with the corresponding elastic solutions. The effects of the viscoelastic characteristic time on the force relaxations are also presented for different viscoelastic properties and boundary conditions.

Bending and vibration studies of FG porous sandwich beam with viscoelastic boundary conditions: FE approach

Bending and vibration characteristics of FG porous sandwich beam with viscoelastic boundary conditions are investigated. Complex shear modulus and associated loss factor are considered for the viscoelastic interlayer. The beam is constrained by viscoelastic supports (VES) at either end. Complex stiffness model is adopted for VES. The transverse deflection, natural frequency, loss factors, and mode shapes are obtained by varying VES stiffness. Furthermore, the study is extended to sandwich beams with various (H, O, V, and X) porosity patterns. The results convey that VES contribution in vibration damping is more predominant when the supports are less stiff (more viscous).

An analytical framework for broadband dynamic analysis of plate built-up structures with uncertain viscoelastic boundary or connection conditions

This paper proposes an analytical stochastic spectral dynamic stiffness method (SSDSM) for free and forced vibration analysis of plate built-up structures subject to uncertain viscoelastic boundary or connection conditions (BCs or CCs). First, a recently developed spectral dynamic stiffness (SDS) theory for broadband vibration analysis of plate built-up structure with arbitrary spatially-varying viscoelastic BCs or CCs is extended to model deterministic viscoelastic BCs or CCs. Then, uncertain viscoelastic BCs or CCs are described by random fields in stiffness and damping, which are discretized by Karhunen–Loève expansion. By using the modified Fourier series as the shape functions for the BCs or CCs, the analytical SSDS matrices of the uncertain viscoelastic BCs or CCs are developed. Then, those SSDS matrices are superposed directly to the SDS matrix of the plate built-up structure. For the solution technique, the extended Wittrick–Williams algorithm is used for stochastic eigenvalue analysis, whereas two different methods are proposed for stochastic response analysis. Representative examples are chosen to validate and demonstrate the superiorities of the proposed method. The proposed method retains all the advantages of the SDS method which is highly efficient and accurate within the whole frequency range. Meanwhile, the proposed method also provides a feasible technique for stochastic broadband dynamic analysis of plate-like structures subject to uncertain boundary or connection conditions.

New general decay results for a multi-dimensional Bresse system with viscoelastic boundary conditions

<p style='text-indent:20px;'>In this paper we consider a multi-dimensional Bresse with memory-type boundary conditions. By assuming minimal conditions on the resolvent kernel, we establish an optimal explicit and general energy decay result. This result is new and substantially improves earlier results in the literature.</p>

Analytical approach to the stepped multi-span rotor-bearing system with isotropic elastic boundary conditions

This paper strives to derive the analytical solutions for the dynamic analysis of stepped multi-span rotor system, where the rotating shaft's internal damping effect is also considered. Specifically, the steady-state whirl analysis and free vibration analysis is given herein. For the steady-state whirl analysis, firstly, the stepped shaft is spilt to several uniform segments and their governing equations are established by Hamilton principle. Subsequently, by applying variable separation method and Laplace transform method to each segment's governing equation, their steady-state responses in terms of determinate interpolation functions multiplied by unknown boundary constants are derived. Then based on the transfer matrix method, each segment's boundary constants are determined with consideration of the compatibility conditions of each two adjacent segments and the end boundary conditions. For the free vibration analysis, one only need to remove the forced term in the steady-state form of transfer matrix and the rotor's characteristic equation will be obtained. Through the theoretical derivation, it is found that this analytical approach can be generalized to any isotropic elastic boundary conditions. To validate the proposed method, two case studies are proposed, where the finite element method is used as benchmark method. For the first one, the influence of internal damping on a uniform rotor's damped whirl characteristics and its stability is analyzed. The damped whirl characteristics analysis reveals that viscous internal damping result in the destabilization of forward modes as long as the spinning speed becomes higher than the critical speed. This phenomenon is also demonstrated by the time responses analysis. For the second one, the dynamic responses of a two-span three steps rotor-bearing system with isotropic viscoelastic boundary conditions are determined. All of simulated results show a great agreement between analytical results and FEM results.

Vibration suppression effect of porous graphene platelet coating on fiber reinforced polymer composite plate with viscoelastic damping boundary conditions resting on viscoelastic foundation

This work is devoted to the investigation of vibration suppression effects of the viscoelastic foundation, viscoelastic damping boundary condition and porous graphene platelet coating (PGPC) on the fiber reinforced polymer (FRP) composite plate. A dynamic model is built by combining the first-order shear deformation theory, the multi-segment partition technique, the virtual spring technology, the Rayleigh-Ritz approach, and the mode strain energy method. Afterwards, the structural natural frequencies, modal shapes and modal damping ratios are calculated, as well as the verification of the convergence and the correctness of the constructed model. Finally, the vibration suppression effect of the FRP-PGPC composite plate with different parameters, such as geometrical and material parameters, foundation and boundary condition types, are discussed in details.

Seismic response analysis of earth-rock fill dam on deep overburden under viscoelastic boundary condition

The dynamic response characteristics of an earth-rock fill dam on a deep overburden are the focus of seismic research. In particular, evaluating the influence of the earthquake safety of the dam. A dynamic response analysis of earth-rock fill dam on a deep overburden based on viscoelastic boundary conditions was used to study the influence of boundary conditions (including fixed boundary conditions and viscoelastic boundary conditions). The results show that the fixed boundary condition greatly improves the dynamic response level of the dam during an earthquake and has no obvious influence on the distribution of the acceleration response in the dam and the foundation. The difference in the calculation results under the two boundary conditions is related to the seismic input characteristics and dynamic deformation characteristics of the soil material. An analysis of the acceleration response spectrum shows that the influence of the boundary conditions on the calculation results is limited to the magnitude level of the acceleration response, while the spectral characteristics of the vibration of the dam and the foundation do not have a significant impact.

General decay of solutions of a thermoelastic Bresse system with viscoelastic boundary conditions

In this paper we consider a multidimensional thermoviscoelastic system of Bresse type where the heat conduction is given by Green and Naghdi theories. For a wider class of relaxation functions, We show that the dissipation produced by the memory eect is strong enough to produce a general decay results. We establish a general decay results, from which the usual exponential and polynomial decay rates are only special cases.

A New Beam Element for Static, Free and Forced Vibration Responses of Microbeams Resting on Viscoelastic Foundation Based on Modified Couple Stress and Third-Order Beam Theories

In this paper, a new beam element based on third-order Reddy beam theory and modified couple stress theory is introduced for static and dynamic analyses of microbeams. The beam is resting on Kelvin–Voigt-type viscoelastic foundation. Equilibrium equations are derived by using Hamilton’s principle. Galerkin finite element method is applied to the governing equations, and shape functions of the new beam element have been derived. As a result, applying only six elements are suitable for assessing beam behavior precisely, and it is beneficial due to the reduction in computation time. The proposed element is a two-node element with 4 degrees of freedom at each node. The effect of different stiffness and damping of viscoelastic foundation, boundary condition and length parameters on natural frequency and transient response of the microbeam is investigated.

Hamiltonian System and Viscoelastic Boundary Conditions

By using Laplace integral transformation, the viscoelastic stress-strain relationship in the time domain is described as an algebraic equation in Laplace space, which makes the Hamiltonian system method smoothly implemented. Thus, the basic control equations with displacement and stress as basic variables are established, and the viscoelastic problem is transformed into the eigenvalue and eigensolutions. The eigensolutions include a series of independent functions, in which the zero eigenvalue eigensolutions cover all the basic deformations such as tension and bending, and do not decay with the space coordinates.