Elastodynamic Model Research Articles

A Screw Theory-Based Semi-Analytical Approach for Elastodynamics of the Tricept Robot

Taking the well-known Tricept robot as an example, this paper presents a semi-analytical approach for elastodynamic modeling of five or six degrees of freedom (DOF) hybrid robots composed of a 3-DOF parallel mechanism plus a 2- or 3-DOF wrist. Drawing heavily on screw theory combined with structural dynamics, the kinetic and elastic potential energies of the parallel mechanism and of the wrist are formulated using the dual properties of twist/wrench systems and a static condensation technique. This results in a 9-DOF dynamic model that enables the lower-order dynamic behavior over the entire workspace to be estimated in a very efficient and accurate manner. The lower-order natural frequencies and mode shapes estimated by the proposed approach are shown to have very good agreement with those obtained by a full-order finite element (FE) model. It thus provides a very time-effective tool for optimal design within a virtual prototyping framework for hybrid robot-based machine tools.

Axial and Torsional Free Vibrations of Elastic Nano-Beams by Stress-Driven Two-Phase Elasticity

Size-dependent longitudinal and torsional vibrations of nano-beams are examined by two-phase mixture integral elasticity. A new and efficient elastodynamic model is conceived by convexly combining the local phase with strain- and stress-driven purely nonlocal phases. The proposed stress-driven nonlocal integral mixture leads to well-posed structural problems for any value of the scale parameter. Effectiveness of stress-driven mixture is illustrated by analyzing axial and torsional free vibrations of cantilever and doubly clamped nano-beams. The local/nonlocal integral mixture is conveniently replaced with an equivalent differential law equipped with higher-order constitutive boundary conditions. Exact solutions of fundamental natural frequencies associated with strain- and stress-driven mixtures are evaluated and compared with counterpart results obtained by strain gradient elasticity theory. The provided new numerical benchmarks can be effectively employed for modelling and design of Nano-Electro-Mechanical-Systems (NEMS).

Two Elastodynamic Incremental Models: The Incremental Theory of Diffraction and a Huygens Method.

The elastodynamic geometrical theory of diffraction (GTD) has proved to be useful in ultrasonic nondestructive testing (NDT) and utilizes the so-called diffraction coefficients obtained by solving canonical problems, such as diffraction from a half-plane or an infinite wedge. Consequently, applying GTD as a ray method leads to several limitations notably when the scatterer contour cannot be locally approximated by a straight infinite line: when the contour has a singularity (for instance, at a corner of a rectangular scatterer), the GTD field is, therefore, spatially nonuniform. In particular, defects encountered in ultrasonic NDT have contours of complex shape and finite length. Incremental models represent an alternative to standard GTD in the view of overcoming its limitations. Two elastodynamic incremental models have been developed to better take into consideration the finite length and shape of the defect contour and provide a more physical representation of the edge diffracted field: the first one is an extension to elastodynamics of the incremental theory of diffraction (ITD) previously developed in electromagnetism, while the second one relies on the Huygens principle. These two methods have been tested numerically, showing that they predict a spatially continuous scattered field and their experimental validation is presented in a 3-D configuration.

Tunable phononic structures using Lamb waves in a piezoceramic plate

Phononic crystals made of piezoceramic materials allow for frequency band structure tunability due to the ease of electric command use. In this paper, we develop a full elastodynamic model, taking into account piezoelectric coupling effects for the band-structure calculation of Lamb eigenmodes of a phononic crystal consisting of a piezoceramic plate with a one-dimensional periodic array of electrodes set on both sides. The dispersion-relation characteristics of these eigenmodes are nondestructively tuned via external circuit impedance loads coupled to the phononic plate, e.g., through inductance loads inducing tunable resonant flat bands that hybridize with the Lamb modes of the elastic plate, thus opening up hybridization gaps. It is shown that additional control of the shape of these resonant bands can be achieved by incorporating an impedance interconnecting two adjacent electrodes of the same side, the whole structure forming an electric quadripole. This behavior can be easily predicted with the help of our formalism combined with a simplified electric line model of the phononic crystal plate.

Elastodynamic modeling and parameter sensitivity analysis of a parallel manipulator with articulated traveling plate

This paper deals with the elastodynamic modeling and parameter sensitivity analysis of a parallel manipulator with articulated traveling plate (PM-ATP) for assembling large components in aviation and aerospace. In the elastodynamic modeling, the PM-ATP is divided into four levels, i.e., element, part, substructure, and the whole mechanism. Herein, three substructures, including translation, bar, and ATP, are categorized according to the composition of the PM-ATP. Based on the kineto-elastodynamic (KED) method, differential motion equations of lower levels are formulated and assembled to build the elastodynamic model of the upper level. Degrees of freedom (DoFs) at connecting nodes of parts and deformation compatibility conditions of substructures are considered in the assembling. The proposed layer-by-layer method makes the modeling process more explicit, especially for the ATP having complex structures and multiple joints. Simulations by finite element software and experiments by dynamic testing system are carried out to verify the natural frequencies of the PM-ATP, which show consistency with the results from the analytical model. In the parameter sensitivity analysis, response surface method (RSM) is applied to formulate the surrogate model between the elastic dynamic performances and parameters. On this basis, differentiation of performance reliability to the parameter mean value and standard variance are adopted as the sensitivity indices, from which the main parameters that greatly affect the elastic dynamic performances can be selected as the design variables. The present works are necessary preparations for future optimal design. They can also provide reference for the analysis and evaluation of other PM-ATPs.

Elastodynamic Optimization of a 5-DoF Parallel Kinematic Machine Considering Parameter Uncertainty

Geometric errors, vibration, and elastic deformation are the main causes for inaccuracy of parallel kinematic machines (PKMs). Instead of tackling these inaccuracies after the prototype has been built, this paper proposes a design optimization method to minimize vibration and deformation considering the effects of geometric errors before constructing the PKM. In this paper, geometric errors are described as parameter uncertainty because they are unknown in design stage. A five degree-of-freedom (DoF) PKM is taken to exemplify this method. Elastodynamic model is first formulated by a step-by-step strategy. On this basis, dynamic performances, including natural frequency, elastic deformation, and maximum stress, are analyzed. These analytical results are verified by finite-element simulation and experiment. Then, the necessity of concerning parameter uncertainty in optimization is addressed. Next, parameter uncertainty is added to the formulation of objectives and constraints by Monte Carlo simulation and response surface method. Finally, elastodynamic optimization of the 5-DoF PKM is implemented to rebuild a prototype which is robust to geometric errors and has minimal vibration and deformation. The proposed method can also be applied to accuracy improvement of any machines in practical applications.

Relevance of dopamine D2-agonist cabergoline in modulation of the rewarding effects of alcohol

Two modified full-field Digital Gradient Sensing (DGS) methods of higher measurement sensitivity, transmission-reflection DGS (tr-DGS) and double-transmission DGS (t2-DGS), are extended in this work to examine their feasibility to study fast fracture events in soda-lime glass by measuring stress gradients. Full-field optical measurement of deformations and stresses in such high-stiffness and low-toughness brittle material is very challenging. Extremely high crack speeds (~1400 m/s), low failure strain (εf < 0.1%), and highly localized sub-micron scale deformations are among the challenges to overcome in this material system to succeed in this regard. The higher measurement sensitivity of tr-DGS and t2-DGS make them particularly suitable for studying transparent glasses and ceramics relative to previous versions of DGS method. These two new methods are demonstrated for studying dynamically growing cracks in soda-lime glass plates subjected to dynamic impact loading in a modified Hopkinson pressure bar device and ultrahigh-speed photography (1 million fps). Previously proposed reflection DGS (r-DGS) is also implemented to comparatively show the benefits of higher measurement sensitivity offered by tr-DGS and t2-DGS methods. The measured stress gradients are numerically integrated to estimate σxx+σyy stress fields around the crack tip vicinity. The optical measurements are also used to obtain fracture parameter histories of crack speed, stress intensity factors and energy release rates under mode-I conditions. The energy release rates increase rapidly when the crack speed approaches ~1400 m/s. The measured fracture parameters are also assessed using a complementary elasto-dynamic finite element model up to crack initiation. The agreement between measurements and simulations supports the feasibility of these two optical methods for studying dynamic fracture mechanics of this challenging material.

Dynamic stress distribution around the wellbore influenced by surge/swab pressure

The calculation of dynamic stress distribution around the wellbore under surge/swab pressure is an unsolved issue. The conventional method to calculate wellbore stress is only applicable to the steady state without considering the coupled effect of deformation-diffusion. In this paper, an elastodynamics model and a poroelastodynamics model have been established to overcome these deficiencies. The numerical solution is provided by the implicit finite difference method, and it is verified by the analytical solution achieved from Laplace transform and Crump inversion. Through sensitivity analysis, it shows that the period of the sine function (inner boundary condition), Biot coefficient, hydraulic diffusivity, and rock types could influence the stress distribution. Then the dynamic surge/swab pressure predicted based on Lubinski's method is implemented into these two models. Two field cases with different depths have been analyzed, and it demonstrates that the surge/swab pressure is more significant for a shallow formation because the static formation pressure and in-situ stresses are relatively small. If the tripping velocity is high, the stress distribution will be significantly changed. Finally, the stress calculated by these two models is compared with the stress obtained using the conventional solution. The comparison shows that the poroelastodynamics solution is different from the elastodynamics solution because pore pressure is significant in the poroelastodynamics solution. The poroelastodynamics solution is also significantly distinguishable from the conventional solution, which demonstrates the importance of the poroelastodynamics model. Overall, it fills the gap in calculating dynamic wellbore stress under tripping conditions, which can be applied to wellbore stability analysis and contribute to a reduction in both tripping time and drilling costs.

Elastodynamic-model-based stiffness analysis of a 6-RSS PKM

An elastodynamic model for a 6-RSS parallel kinematic machine (PKM) is established by combining the virtual joint method (VJM) and the finite element method (FEM). According to the proposed elastodynamic model, stiffness matrix of the system is derived, based on which static stiffness of the system is analyzed. In order to validate the proposed model, finite element analysis is conducted. Results show that the proposed model is relatively accurate and can be used to evaluate rigidity of the 6-RSS PKM. Two stiffness indices are proposed for the convenience of analysis. Stiffness distributions as well as parametric analysis are conducted to offer significant guidance on the trajectory planning and structural optimization at the early design stage.

Existence for elastodynamic Griffith fracture with a weak maximal dissipation condition

We consider a model of elastodynamics with fracture evolution, based on energy-dissipation balance and a maximal dissipation condition. We prove an existence result in the case of planar elasticity with a free crack path, where the maximal dissipation condition is satisfied among suitably regular competitor cracks.

First measurement of the full elastic constants of Ni-based superalloy René 88DT

The full elastic constants of Ni-based superalloy René 88DT were measured for the first time using a recently developed method that matches the computed surface acoustic wave (SAW) velocities from an elastodynamic model with experimental velocities of SAWs that are generated and detected with a femtosecond laser. The experimental measurements were performed on several grains of a polycrystalline sample without the need of growing a single crystal. The computed Young's modulus from the resultant elastic constants agrees with the literature values, lending confidence on our measured elastic constants: C11=267.1, C12=170.5, and C44=107.6 (GPa).

Numerical convergence of nonlinear nonlocal continuum models to local elastodynamics

SummaryWe quantify the numerical error and modeling error associated with replacing a nonlinear nonlocal bond‐based peridynamic model with a local elasticity model or a linearized peridynamic model away from the fracture set. The nonlocal model treated here is characterized by a double‐well potential and is a smooth version of the peridynamic model introduced in the work of Silling. The nonlinear peridynamic evolutions are shown to converge to the solution of linear elastodynamics at a rate linear with respect to the length scale ε of nonlocal interaction. This rate also holds for the convergence of solutions of the linearized peridynamic model to the solution of the local elastodynamic model. For local linear Lagrange interpolation, the consistency error for the numerical approximation is found to depend on the ratio between mesh size h and ε. More generally, for local Lagrange interpolation of order p≥1, the consistency error is of order hp/ε. A new stability theory for the time discretization is provided and an explicit generalization of the CFL condition on the time step and its relation to mesh size h is given. Numerical simulations are provided illustrating the consistency error associated with the convergence of nonlinear and linearized peridynamics to linear elastodynamics.

An approach for elastodynamic modeling of hybrid robots based on substructure synthesis technique

Parallel kinematic machines exhibit strong pose-dependent static and dynamic behaviors, which makes accurate and rapid compliance analysis over the entire workspace an important issue in the design optimization. This paper presents a general approach for elastodynamic modeling of parallel kinematic machines using substructure synthesis technique. Firstly, the whole system is decomposed into two groups of substructures, i.e. component substructures and joint substructures. Then, the degrees of freedom of component substructures are reduced using modal reduction technique, and joint substructures are modeled by virtual springs with contact stiffness between adjacent component substructures. Finally, two threads are merged at junction surfaces to offer the equations of motion of the system as a whole, allowing the static and dynamic performances to be rapidly predicted with sufficient accuracy. The dynamic model of a 5-DOF hybrid robot is developed using the proposed method and the computational results show that rigidity and lower mode natural frequencies over the workspace match well with those obtained by the full finite element analysis.

Dependence of equilibrium Griffith surface energy on crack speed in phase-field models for fracture coupled to elastodynamics

Phase-field models for crack propagation enable the simulation of complex crack patterns without complex and expensive tracking and remeshing as cracks grow. In the setting without inertia, the crack evolution is obtained from a variational energetic starting point, and leads to an equation for the order parameter coupled to elastostatics. Careful mathematical analysis has shown that this is consistent with the Griffith model for fracture. Recent efforts to include inertia in this formulation have replaced elastostatics by elastodynamics. In this brief note, we examine the elastodynamic augmentation, and find that it effectively causes the Griffith surface energy to depend on the velocity of the crack. That is, considering two identical specimens that are each fractured by a single crack that grows at different velocities in the two specimens, it is expected that the final equilibrium configurations are nominally identical; however, the phase-field fracture models augmented with elastodynamics achieve final configurations—in particular, the Griffiths surface energy contributions—that depend on the crack velocity. The physical reason is that the finite relaxation time for the stresses in the elastodynamic setting enables the cracked region to widen, beyond the value observed in the quasistatic setting. Once the crack widens, the “no-healing” condition prevents it from relaxing even after the specimen reaches equilibrium. In phase-field models, crack width in the reference configuration is unrelated to the physical opening of the crack but is instead a measure of Griffiths surface energy. This observation suggests that elastodynamic phase-field fracture models should not be used in settings where the crack velocity is large.

Local radial basis function collocation method for bending analyses of quasicrystal plates

• Meshless formulations are developed for static and dynamic bending of QC plates. • The phonon-phason coupling effects are studied numerically. • The influence of elastic foundation and variable plate thickness is investigated. • The response of QC plates to impact loading is simulated. The local radial basis function collocation method (LRBFCM) is proposed for plate bending analysis in orthorhombic quasicrystals (QCs) under static and transient dynamic loads. Three common types of the plate bending problems are considered: (1) QC plates resting on Winkler foundation (2) QC plates with variable thickness and (3) QC plates under a transient dynamic load. According to the Reissner–Mindlin plate bending theory, there is allowed to simulate the behavior of the two excitations in QC plates, phonon and phason, by 2D strong formulations for the system of governing equations. The governing equations, which describe the phason displacements, are based on Agiasofitou and Lazar elastodynamic model. Numerical results demonstrate the effect of the elastic foundation, as well as plate thickness on the phonon and phason characteristics in this paper. For the transient dynamic analysis, the influence of the phason friction coefficients on the responses of QC plate to transient dynamic loads is also studied.

Strain-mediated 180° switching in CoFeB and Terfenol-D nanodots with perpendicular magnetic anisotropy

A micromagnetic and elastodynamic finite element model is used to compare the 180° out-of-plane magnetic switching behavior of CoFeB and Terfenol-D nanodots with perpendicular magnetic easy axes. The systems simulated here consist of 50 nm diameter nanodots on top of a 100 nm-thick PZT (Pby[ZrxTi1-x]O3) thin film, which is attached to a Si substrate. This allows voltage pulses to induce strain-mediated magnetic switching in a magnetic field free environment. Coherent and incoherent switching behaviors are observed in both CoFeB and Terfenol nanodots, with incoherent flipping associated with larger or faster applied switching voltages. The energy to flip a Terfenol-D memory element is an ultralow 22 aJ, which is 3–4 orders more efficient than spin-transfer-torque. Consecutive switching is also demonstrated by applying sequential 2.8 V voltage pulses to a CoFeB nanodot system with switching times as low as 0.2 ns.

Horizontal Stiffness and Damping of Piles in Inhomogeneous Soil

AbstractA practically oriented analytical procedure for determining the dynamic stiffness and damping (impedance coefficients) of a laterally loaded pile in soil exhibiting different types of inhomogeneity with depth, is presented. To this end, an energy method based on the Winkler model of soil reaction in conjunction with pertinent shape functions for the deflected shape of the pile are employed. A new elastodynamic model for the wave field around a pile is also introduced. The method is self-standing and free of empirical formulas or constants. Dimensionless closed-form solutions are derived for (1) the distributed (Winkler) springs and dashpots along the pile; (2) dynamic stiffness and damping coefficients at the pile head; (3) active length, beyond which the pile can be treated as infinitely long; and (4) relative contributions to the overall head stiffness and damping of the soil and the pile media. Swaying, rocking, and cross swaying-rocking impedances are considered for parabolic, exponential, and...

A hybridizable discontinuous Galerkin method for modeling fluid–structure interaction

This work presents a novel application of the hybridizable discontinuous Galerkin (HDG) finite element method to the multi-physics simulation of coupled fluid–structure interaction (FSI) problems. Recent applications of the HDG method have primarily been for single-physics problems including both solids and fluids, which are necessary building blocks for FSI modeling. Utilizing these established models, HDG formulations for linear elastostatics, a nonlinear elastodynamic model, and arbitrary Lagrangian–Eulerian Navier–Stokes are derived. The elasticity formulations are written in a Lagrangian reference frame, with the nonlinear formulation restricted to hyperelastic materials.With these individual solid and fluid formulations, the remaining challenge in FSI modeling is coupling together their disparate mathematics on the fluid–solid interface. This coupling is presented, along with the resultant HDG FSI formulation. Verification of the component models, through the method of manufactured solutions, is performed and each model is shown to converge at the expected rate. The individual components, along with the complete FSI model, are then compared to the benchmark problems proposed by Turek and Hron [1]. The solutions from the HDG formulation presented in this work trend towards the benchmark as the spatial polynomial order and the temporal order of integration are increased.

Multimodal non-rigid image registration based on elastodynamics

In this paper, we present a new multimodal image registration technique established on elastodynamics notion. The main idea behind this concept is the progression of waves on an elastic body as soon as it is disturbed from its initial rest state. We propose to solve the multimodal registration problem by modeling the non-linear deformations as elastic waves and iteratively solving the elastodynamics wave equation to estimate the transformation. The inertial force in elastodynamics model is computed as the gradient of mutual information which considers the statistical relationship between the intensities of the images acquired using different imaging modalities. We tested our method on T1–T2 weighted MR brain image pairs and MR-CT brain image pairs. The proposed registration technique was compared against a variant of demons method proposed for multimodal images. The registration results were analyzed by examining the overlay images and by computing the normalized mutual information. The qualitative and quantitative analysis proved that our proposed method registers the images better than the compared method.

A Timoshenko-like model for the study of three-dimensional vibrations of an elastic ring of general cross-section

The present paper proposes analytical formulations of the eigenvalues and eigenfunctions (frequencies and modes) of vibrating rings of any cross-section shape, so as to be applied to engineering problems, like design optimization or life duration improvement. This is done by means of the Timoshenko beam theory accounting for the curved metric through new constitutive laws. An original non-dimensionalization reduces the number of independent parameters to only four. The motion is governed by a system of six differential equations applied to six independent variables. However, the reference curvature is found to induce fundamental changes to the mode structure. In that sense, a classical series decomposition is not efficient to provide analytical expressions of the mode shapes. The Chapman–Enskog method is proposed and explained to overcome this difficulty. Then, all modes (flexural, shear, torsional and longitudinal) are formulated explicitly, showing the coupling order of each of the six degrees of freedom of the cross-section. These results are compared to computations using a 3D elastodynamic model in order to validate the model and to point out its limitations. The latter is finally discussed with respect to other models proposed in the literature.