Finite Stiffness Research Articles

Research on Time Delay Feedback Control of Lateral Vibration Axially Moving Viscoelastic Beam

The nonlinear dynamic behavior of a viscoelastic beam in axially variable motion under time delay is studied. Considering that the axial velocity pulsation and radial tension change periodically with time under the action of time delay, the viscous damping and finite support stiffness are taken into account, and the viscoelastic constitutive relationship adopts Kelvin model. A mathematical model of lateral vibration of an axially moving viscoelastic beam with changes in axial velocity and tension under time delay is established, and a partial differential-integral control equation describing the lateral nonlinear vibration of an axially moving beam is given. Based on the Galerkin truncation method, the control equations established based on the mechanical model are discretized, so that the direct multi-scale method obtains the numerical solution of the lateral nonlinear vibration of the axially moving beam under the action of time delay, and determines the nonlinear dynamic behavior of the system under the action of time delay. By analyzing the numerical solution of the vibration displacement and velocity at the midpoint of the beam, the whole process of the average velocity of the axially moving structure along the axis, the amplitude of the disturbance tension and the change of the viscoelastic coefficient is simulated. It provides a numerical theoretical basis for the study of the nonlinear dynamic behavior of axially moving beams under time delay.Through the research in this article, the introduction of time delay is studied, and the time delay is used to analyze the axial force that changes along the radial direction of the beam caused by the speed pulsation, and the theoretical framework for nonlinear vibration analysis of the axially moving viscoelastic beam is constructed to expand the time delay. The application scope of the nonlinear vibration theory and nonlinear dynamics theory, the development and improvement of the approximate analysis and numerical solution of the nonlinear continuum (especially the gyro body) under the action of time delay, to provide a theoretical basis for the analysis and design of the corresponding engineering system and technical reserves.

Clogging and avalanches in quasi-two-dimensional emulsion hopper flow.

We experimentally and computationally study the flow of a quasi-two-dimensional emulsion through a constricting hopper shape. Our area fractions are above jamming such that the droplets are always in contact with one another and are in many cases highly deformed. At the lowest flow rates, the droplets often clog and thus exit the hopper via intermittent avalanches. At the highest flow rates, the droplets exit continuously. The transition between these two types of behaviors is a fairly smooth function of the mean strain rate. The avalanches are characterized by a power-law distribution of the time interval between droplets exiting the hopper, with long intervals between the avalanches. Our computational studies reproduce the experimental observations by adding a flexible compliance to the system (in other words, a finite stiffness of the sample chamber). The compliance results in continuous flow at high flow rates, and allows the system to clog at low flow rates leading to avalanches. The computational results suggest that the interplay of the flow rate and compliance controls the presence or absence of the avalanches.

Crack propagation analysis in masonry structures via an inter-element cohesive fracture approach: assessment of mesh dependency issues

Inter-element cohesive methodologies have proved to be very effective to simulate complex cracking phenomena in quasi-brittle materials, by virtue of their capability of naturally handling multiple crack nucleation and propagation along unknown paths, also including crack bifurcation and coalescence. Nevertheless, in several computational implementations, such methodologies suffer from a certain mesh bias, since crack propagation is restricted to occur along inter-element mesh boundaries. Besides, the resulting structural response is inevitably affected by the additional compliance provided by the embedded cohesive elements with finite stiffness even prior to the crack onset. In this work, the diffuse inter-element cohesive method has been applied to masonry structures, focusing the attention on the role of related mesh dependency issues. In particular, the aim of this work is twofold. First, a novel calibration approach of the initial cohesive stiffness is developed to reduce the aforementioned artificial compliance effect. Second, an effective strategy is proposed to avoid the occurrence of multiple solutions, which are typical of diffuse inter-element cohesive approaches, especially in structured meshes. To assess the validity of the proposed method, numerical simulations of the failure behavior of simple masonry specimens are performed. The results show that a rational calibration of the cohesive properties (i.e., stiffness and strength) of diffuse embedded interfaces is useful to ensure numerical stability and accuracy, in terms of both load-bearing capacity and related crack pattern.

INVESTIGATION OF CONTACT INTERACTION WITH STRIPE WITH INITIAL STRESSES IF SHE REINFORCED BY THE ELASTIC THIN FINITE STRINGER

In this paper, within the framework of the linearized theory of elasticity, the contact problem of reinforcement of a prestressed elastic band by a finite elastic stringer is considered. Elastic strip, free from reinforcement and load face, rigidly clamped to the base. The problem is solved for the cases of equal and unequal roots of the defining equation in the case of elastic potentials of an arbitrary structure. The research is carried out in general for the theory of large initial deformations and two variants of the theory of small initial deformations within the linearized theory of elasticity at arbitrary structure of elastic potential. It is assumed that the initial stress state of the elastic stringer and the band in the contact area is homogeneous. The research is carried out in the coordinates of the initial deformed state, which are related to the Lagrangian coordinates (natural state). In addition, the external concentrated load causes small perturbations of the corresponding values of the ground stress-strain state. The general solutions of the basic differential equations of the linearized theory of elasticity are given. Assuming that the stringer is loaded with both vertical and horizontal forces, it should be noted that the elastic stringer bends in the vertical direction like a normal beam, and in the horizontal direction is compressed or stretched like a normal rod with finite stiffness, which is in uniaxial stress. deformed state, the problem is mathematically formulated as a system of integro-differential equations with respect to unknown contact stresses. Using the Fourier transform, the system is solved in a closed form. Ultimately, the expressions for the contact voltages are represented as Fourier integrals. In the case of uneven roots for chemically active rubber SKU-6 and Treloir potential (neogukov type material), the results of numerical analysis are presented, presented in the form of a graph illustrating a fairly significant effect of initial stresses. Therefore, the effect of initial stresses on the stressstrain state of the elastic band is that the initial stresses in the strip lead to a decrease in stress in the contact area in the case of compression, and in the case of tension to increase them. As for the movements in the area of contact of the stringer with the strip, the opposite is true.

Stress state of hydrotechnical facilities resistant to natural disasters

The equations of the theory of the stress-strain state of an elastic body for practical engineering structures of the agro-industrial complex do not have analytical solutions. Therefore, the development of numerical methods for determining the strength parameters of agricultural facilities is an urgent task. Among the numerical methods of strength calculations, the finite element method is currently widely used. In a Cartesian coordinate system, a finite element of a hexahedral shape is used to determine the stress-strain state of an elastic body under bulk loading in two formulations: in the formulation of the method of displacements with nodal unknowns in the form of displacements and their derivatives, and in a mixed formulation with nodal unknowns in the form of displacements and stresses. Approximation of displacements through nodal unknowns in obtaining the finite element stiffness matrix was performed using a form function, the elements of which were Hermite polynomials of the third degree. When obtaining the deformation matrix, displacements and stresses of the internal points of the finite element were approximated in terms of nodal unknowns using bilinear functions. The calculation example shows a significant advantage of using a finite element in a mixed formulation.

Feasibility of Active Material Vibration Suppression in a Large Stroke Flexure Hinge

Flexure mechanisms are widely used in precision applications. The finite support stiffness results in parasitic resonance frequencies, typically with low damping, that can be detrimental for positioning performance. Moreover, these resonances may vary over the workspace for mechanisms with moderate or large deflection. In this paper we consider a model of a large stroke flexure hinge that requires accurate end-effector positioning. The feasibility of damping the parasitic resonances using active material is investigated. It is shown that, by placing piezoelectric patches at suitable locations on the flexures, the problematic parasitic resonances can be sensed and actuated. Using modal decoupling the vibration modes are separated, whereafter positive position feedback control is used to suppress the vibration modes. By changing the controller resonance frequency as a function of the nominal deformation, vibration suppression over the entire workspace is achieved. The improved damping in the mechanism is shown using transient simulations.

Scaling of current-voltage characteristics without and with finite superfluid phase stiffness below [formula omitted

Superfluid phase stiffness (SPS) of NCBCO superconductor as a function of T has been extracted by using the Ambegaokar–Halperin–Nelson–Siggia (AHNS) theory at the zero magnetic field. The Berezinskii–Kosterlitz–Thouless (BKT) phase transition temperature, TBKT has been extracted to be 54.0 K. The idea of the vortex glass (VG) transition has been applied to the resistive states of the BKT phase transition observed at 54.0 K. We have extracted the static exponent required in the Fisher, Fisher and Huse (FFH) scaling equation.The scaling of the current-voltage (IV) curves around 54.0 K with and without finite SPS is possible within the framework of the FFH theory.

A transient and time lag deformation alternating-coupling micro elastohydrodynamic lubrication model

As transient and time lag deformations occur alternately and influence each other for viscoelastic materials of dynamically loaded journal bearings, the time lag deformation per unit time is defined based on the finite element stiffness equation and the standard linear solid model, accumulating it by loop and superposing transient deformation to obtain actual deformation at every moment. Thereby, a novel temporal association deformation equation of bearing is constructed. It is used to improve the film thickness calculation method of the traditional micro elastohydrodynamic lubrication (MEHD) model. Finally, a novel transient and time lag deformation alternating-coupling micro elastohydrodynamic lubrication (TTL-MEHD) model is established. The comparison between lubrication performance parameters calculated by the new model and the MEHD model under the eccentric motion of the journal proves the new model is correct, and can effectively analyze the knocking and rough rubbing characteristics at the early stage of bearing instability. The new model is implemented to explore the effect of the time lag degree of material deformation on mixed lubrication performance. The application of the new model in connecting rod bearing of internal-combustion engine proves that it can effectively compensate for the impact lag deviation caused by the complete elasticity assumption.

Effect of elastic support on the linear buckling response of quasi-isotropic cylindrical shells under axial compression

Cylindrical shells under compressive loading are highly sensitive to boundary conditions. Considering that these structures are connected by surrounding structural components with finite stiffness, an accurate evaluation of the effects of their boundary stiffness is crucial in their design. As such, this work investigates the effect of elastic boundary conditions on the linear buckling behaviour of cylindrical shells under compressive loading. To achieve this goal, a virtual testing investigation on the effect of translational and rotational constraints to the linear buckling response of a quasi-isotropic cylinder subjected to axial compression is performed. Subsequently, the effect of many kinds of constraints on linear buckling behaviour is discussed and interesting insights regarding a significant coupling effect between the radial and tangential translational constraints are given. Results obtained from virtual testing show that seven recurrent buckling mode shapes occur with seven corresponding similar linear buckling loads. Therefore, based on these similarities, seven groups of classical boundary conditions are introduced to classify all possible linear buckling behaviours exhibited by the cylinder under consideration. Finally, these findings can support the development of theoretical models for cascade, or flange, designs of multiple connecting cylinders.

Approximating inverse FEM matrices on non-uniform meshes with {mathcal{H}}-matrices

We consider the approximation of the inverse of the finite element stiffness matrix in the data sparse {mathcal{H}}-matrix format. For a large class of shape regular but possibly non-uniform meshes including algebraically graded meshes, we prove that the inverse of the stiffness matrix can be approximated in the {mathcal{H}}-matrix format at an exponential rate in the block rank. Since the storage complexity of the hierarchical matrix is logarithmic-linear and only grows linearly in the block-rank, we obtain an efficient approximation that can be used, e.g., as an approximate direct solver or preconditioner for iterative solvers.

Modeling multi-fracturing fibers in fiber networks using elastoplastic Timoshenko beam finite elements with embedded strong discontinuities — Formulation and staggered algorithm

To model fiber failures in random fiber networks, we have developed an elastoplastic Timoshenko beam finite element with embedded discontinuities. The method is based on the theory of strong discontinuities where the generalized displacement field is enhanced by a jump. The continuum mechanics formulation accounts for a fracture process zone and a bulk material while retaining traction continuity across the discontinuity. The additional degrees of freedom that are associated with the discontinuity are represented by a midpoint node, which is statically condensed to enable the implementation in commercial software through the user element interface. We propose a quasi-brittle fracture model, where the failure-related deformation is uncoupled from the plastic deformation in the bulk material. To retain the positive definite finite element stiffness matrix of the bulk material, we neglect the fracture-related softening of the discontinuity and employ a modified Newton iteration in the strain softening domain. Our implementation facilitates the integration into commercial finite element software and examples illustrate the robustness of the method. The FORTRAN source code is freely available to benchmark our model. We show that fiber failures contribute to the nonlinear stress–strain response of paper. Together with fiber–fiber bond failures, they can potentially explain the nonlinear stress–strain response of paper and nanopaper.

An Differential Quadrature Finite Element and the Differential Quadrature Hierarchical Finite Element Methods for the Dynamics Analysis of on Board Shaft

In this paper the dynamic analysis of a shaft rotor whose support is mobile is studied. For the calculation of kinetic energy and stiffness energy, the beam theory of Euler Bernoulli was used, and the matrices of elements and systems are developed using two methods derived from the differential quadrature method (DQM). The first method is the Differential Quadrature Finite Element Method (DQFEM) systematically, as a combination of the Differential Quadrature Method (DQM) and the Standard Finite Element Method (FEM), which has a reduced computational cost for problems in dynamics. The second method is the Differential Quadrature Hierarchical Finite Element Method (DQHFEM) which is used by expressing the matrices of the hierarchical finite element method in a similar form to that of the Differential Quadrature Finite Element Method and introducing an interpolation basis on the element boundary of the hierarchical finite element method. The discretization element used for both methods is a three-dimensional beam element. In the differential quadrature finite element method (DQFEM), the mass, gyroscopic and stiffness matrices are simply calculated using the weighting coefficient matrices given by the differential quadrature (DQ) and Gauss-Lobatto quadrature rules. The sampling points are determined by the Gauss-Lobatto node method. In the Differential Quadrature Hierarchical Finite Element Method (DQHFEM) the same approaches were used, and the cubic Hermite shape functions and the special Legendre polynomial Rodrigues shape polynomial were added. The assembly of the matrices for both methods (DQFEM and DQHFEM) is similar to that of the classical finite element method. The results of the calculation are validated with the h- and hp finite element methods and also with the literature.

Vibration and Stability Analysis of a Bearing–Rotor System with Transverse Breathing Crack and Initial Bending

This paper focuses on the stability and nonlinear response of a bearing-rotor system affected by a transverse crack and initial bending which was thought to be part of an unbalance or had been neglected before. The differences of breathing functions for the transverse breathing crack caused by initial bending is presented here, and the calculation of time-varying finite elements stiffness matrix of the cracked shaft is improved by replacing traditional the approximate crack segment with an exact area. After establishing the dynamic model of the cracked rotor with initial bending, vibrational characteristics such as amplitude-speed diagram, frequency spectrogram and bifurcations are investigated in detail. The eigenvalues of the transition matrix are calculated and analyzed as an indicator of dynamic stability with the growths of crack depth and initial bending. Many differences are found between the two cases of dynamic response of rotor system by numerical simulation. The frequency change with the growth of initial bending is opposite to the change with the growth of crack depth, and the shapes of amplitude-speed also having great different features. Stable regions are reduced and extended laterally by initial bending. All these results obtained in this paper will contribute to identify the bending fault and assess the stability of the bearing-rotor systems.

Effect of interfacial stiffness on the stretchability of metal/elastomer bilayers under in-plane biaxial tension

Flexible electronic devices are often subjected to large and repeated deformation, so that their functional components such as metal interconnects need to sustain strains up to tens of percent, which is far beyond the intrinsic deformability of metal materials (~1%). To meet the stringent requirements of flexible electronics, metal/elastomer bilayers, a stretchable structure that consists of a metal film adhered to a stretchable elastomer substrate, have been developed to improve the stretch capability of metal interconnects. Previous studies have predicted that the metal/elastomer bilayers are much more stretchable than freestanding metal films. However, these investigations usually assume perfect bonding between the metal and elastomer layers. In this work, the effect of the metal/elastomer interface with a finite interfacial stiffness on the stretchability of bilayer structures is analyzed. The results show that the assumption of perfect interface (with infinite interfacial stiffness) may lead to an overestimation of the stretchability of bilayer structures. It is also demonstrated that increased adhesion between the metal and elastomer layers can enhance the stretchability of the metal layer.

Effect of surface tension on hysteresis losses during approach-retraction of spherical adhesive contact

ABSTRACT In the conventional adhesive contact mechanics, the value of the Tabor parameter μ provides criterion for the applicability of the JKR and the DMT models. While the JKR model is an appropriate approximation for a large Tabor parameter (μ > 5), Ciavarella et al. demonstrated that the JKR theory overestimates the hysteresis loss during an approach-retraction period, even though the Tabor parameter is high enough. The present study investigates how surface tension affects the hysteretic loss in an approach-retraction cycle of spherical adhesive contact. With surface tension, Wu’s solution is invalid in terms of the prediction of the jump-in position. Based on Civarella’s research, we developed a hybrid model to approximate the hysteretic loss. This model used direct numerical solution from the Lennard-Jones potential Law. The same method can also be generalized to the spherical contact system with finite stiffness, in terms of hysteretic loss determination.

Phase Behavior of Lyotropic Liquid-Crystalline Polymers under Varying Solvent Conditions: Effect of External Fields on the Phase Diagram.

In this work, we report a Density Functional Theory based study of phase behavior of lyotropic liquid-crystalline polymers under both good and varying solvent conditions in the presence of external electric or magnetic field. Our microscopic model for the good solvent case is based on the tangent hard-sphere chain with bond-bending potential to account for the chain stiffness; the variable solvent quality is modeled by adding attractive monomer-monomer interactions. The phase diagrams are constructed in three intensive variables (temperature, pressure, and field strength), and are characterized by the presence of critical and triple lines, which originate from the critical and triple points of the corresponding zero-field case. The merging of critical and triple lines results in the appearance of the "double critical" and "critical triple" points, already known from the earlier studies of the phase behavior of spin fluids in magnetic fields. The important difference of the present model from the spin fluids is due to the finite stiffness of the polymer chains (characterized by their persistence length), which adds an additional parameter controlling the morphology of the phase diagrams.

Investigation of the modular building metal frame nodal connection rotational stiffness

The research subject is the rotational stiffness of the modular building metal frame nodal connection. The stiffness of nodal connections has a significant impact on the results of calculating the load-bearing frame of a modular building. It is known that absolutely rigid connections exist only in theory. In practice, each nodal connection has a finite stiffness, as evidenced by numerous studies of the researchers. When designing the real objects, the studies on the stiffness of nodal connections of a certain design should be carried out. As a rule, when designing real objects, there is no time to carry out the research. The designer operates with well-known and reliably proven recommendations. It is important for him to have the ready-made solutions for implementation in design practice. Such solutions can be provided on the basis of preliminary studies of the joints’ behavior of the structural elements under load. The article presents a solution to such a problem. The study of the stiffness of the nodal connection with the connected elements parameters variation has been carried out. The purpose of the study is to find the boundaries of design solutions at which the connection can be considered rigid.

Computation of the influence of the surcharge on the state of stress of the round plate on a bumping foundation

The problem of calculating a round plate located on the surface of a combined foundation is being considered in the paper. An analytical solution is deduced for a plate of finite stiffness and a stiff plate (round stamp) under the influence of an axisymmetric bending. The influence of an axially symmetric ring load applied to the surface of the base outside the slab is analyzed.

Учет изгиба килевой балки в расчетах общей и местной прочности корабля при доковании в сухом доке

This paper presents an approximate method for taking into account finite stiffness of bottom grillages in strength calculations of ship installed on one keel track in the dry dock. It compares the approximate method and FEM results for the docking of a 600 t barge platform. The results have shown good correlation. It has been demonstrated that local stresses in the keel near transverse bulkheads might become quite significant.

Mode I debonding under large deformation conditions including notes on cleavage-peeling transition

Peeling and cleavage are common load cases for studying and evaluating adhesion between materials. The former is suitable for testing adhesion of thin or highly flexible materials for which the bending energy is negligible while the latter uses the rate of released bending energy as a measure of fracture energy. Their data reduction schemes are very different, which could lead to situations where peel-cleavage transition cases are misinterpreted. In this work, two analytical frameworks allowing transition between peel and cleavage configurations to be accounted for, are exploited. The first is an elliptic integral based solution, which in the present study is enhanced by the elastic foundation formulation to mimic interfacial reaction forces to external loading. The second is built on a pseudo-linear equivalent system, which allows representation of geometrically non-linear problem through a number of simple, linear segments. Both solutions are successfully confronted against each other and a geometrically non-linear finite element formulation. In addition to the analytical framework, a number of phenomenological insights into the transition are gained. These are presented as master curves relating geometrical and material parameters to the shape of the steady-state fracture response and the rotation of the load application point through the spectrum from cleavage to peeling. The disclosed results could prove valuable when designing, evaluating or analyzing mode I fracture of materials and structures of finite and non-zero bending stiffness.