Dynamic Finite Element Approach Research Articles

Running-specific prosthesis' performance characterization by dynamic finite element approach.

Composite running-specific prostheses (RSP) are widely used by athletes with lower limb amputations to simulate the spring-like behavior of biological legs. However, the effect of these devices on the biomechanics of athletes with transtibial amputations remains uncertain. To address this issue, this study proposes a time-dependent finite element model that uses angles and dynamic loads during ground contact to evaluate RSP performance parameters such as stiffness and energy efficiency. The study also examines the impact of running speed and RSP geometry on performance. The in-silico characterization approach used in this study takes into account both the intrinsic characteristics of the RSP and the athlete's biomechanics to identify the influence of fundamental geometric variables of the RSP on performance. The model is verified by comparing its results with experimental data. The study found that as running speed increases, RSP stiffness, vertical ground reaction force (vGRF), and contact time decrease. The force-displacement profiles of RSP are nonlinear, but a linear function can be used to accurately represent their behavior at high sprinting speeds. Using higher RSP reduces energy efficiency and vGRF due to their lower stiffness. J-curve RSP result in higher stiffness, vGRF, and strain energy, while C-curve RSP are associated with longer contact times and higher energy efficiency.

Undamped Free Vibration Analysis of Functionally Graded Beams: A Dynamic Finite Element Approach

A Dynamic Finite Element (DFE) method for coupled axial–flexural undamped free vibration analysis of functionally graded beams is developed and subsequently used to investigate the system’s natural frequencies and mode shapes. The formulation is based on the Euler–Bernoulli beam theory and material grading is assumed to follow a power law variation through the thickness direction. Using the closed-form solutions to the uncoupled segments of the system’s governing differential equations as the basis functions of approximation space, the dynamic, frequency-dependent, trigonometric interpolation functions are developed. The interpolation functions are used with the weighted residual method to develop the DFE of the system. The resulting nonlinear eigenvalue problem is then solved to determine the coupled natural frequencies. Example elements using DFE, Finite Element Method (FEM) and the Dynamic Stiffness Method (DSM) are implemented in MATLAB for testing, verification, and validation. Good agreement was observed and the DFE formulation exhibited superior convergence performance compared to the FEM.

Broadband dynamic elastic moduli of honeycomb lattice materials: A generalized analytical approach

A generic analytical framework is proposed to obtain the dynamic elastic moduli of lattice materials under steady-state vibration conditions. The dynamic deformation behaviour of the individual beam elements of a lattice is distinct from the behaviour under a static condition. This leads to a completely different global deformation pattern of the lattice material and subsequently opens up a tremendous opportunity to modulate amplitude and phase of the elastic properties of lattices as a function of the ambient vibration. The dynamic stiffness approach proposed in this article precisely captures the sub-wavelength scale dynamics of the periodic network of beams in a lattice material using a single beam-like member. Here the dynamic stiffness matrix of a damped beam element based on the Timoshenko beam theory along with axial stretching is coupled with the unit cell-based approach to derive the most general closed-form analytical formulae for the elastic moduli of lattice materials across the whole frequency range. It is systematically shown how the general expressions of dynamic elastic moduli can be reduced to different special cases by neglecting axial and shear deformations under dynamic as well as classical static conditions. The significance of developing the dynamic stiffness approach compared to conventional dynamic finite element approach is highlighted by presenting detailed analytical derivations and representative numerical results. Further, it is shown how the analytical framework can be readily extended to lattices with non-prismatic beam elements with any spatial variation in geometry and intrinsic material properties. In general, research activities in the field of lattice metamaterials dealing with elastic properties revolve around intuitively designing the microstructural geometry of the lattice structure. Here we propose to couple the physics of deformation as a function of vibrating frequency along with the conventional approach of designing microstructural geometry to expand the effective design space significantly. The stretching-enriched physics of deformation in the lattice materials in addition to the bending and shear deformations under dynamic conditions lead to complex-valued elastic moduli due to the presence of damping in the constituent material. The amplitude, as well as the phase of effective elastic properties of lattice materials, can be quantified using the proposed approach. The dependence of Poisson's ratio on the intrinsic material physics in case of a geometrically regular lattice is found to be in contrary to the common notion that Poisson's ratios of perfectly periodic lattices are only the function of microstructural geometry. The generic analytical approach for analysing the elastic moduli is applicable to any form of two- or three-dimensional lattices, and any profile of the constituent beam-like elements (different cross-sections as well as spatially varying geometry and intrinsic material properties) through a wide range of frequency band. The closed-form expressions of elastic moduli derived in this article can be viewed as the broadband dynamic generalisation of the well-established classical expressions of elastic moduli under static loading, essentially adding a new exploitable dimension in the metamaterials research in terms of dynamics of the intrinsic material.

Determination of cavitation load spectra—Part 2: Dynamic finite element approach

Cavitation erosion is a well-known problem in fluid machineries which occurs due to repeated hydrodynamic impacts caused by cavitation bubble collapse. Cavitation pitting test is often used for the quantification of flow aggressiveness required for lifetime prediction of hydraulic equipment. Understanding the response of the target material under such hydrodynamic impact is essential for correctly interpreting the results obtained by cavitation pitting test. Moreover the proper knowledge of cavitation pitting mechanism would enable us to design new materials more resistant to cavitation erosion. In this paper, the dynamic behavior of three materials 7075 Aluminum alloy, 2205 duplex stainless steel and Nickel–Aluminum Bronze under cavitation hydrodynamic impact has been studied in details by using finite element simulations. The applied load due to hydrodynamic impact is represented by a Gaussian pressure field which has a peak stress and, space and time evolution of Gaussian type. Mechanism of cavitation pit formation and the effect of inertia and strain rate sensitivity of the materials have been discussed. It is found that if the impact duration is very short compared to a characteristic time of the material based on its natural frequency, no pit would form into the material even if the impact stress is very high. It is also found that strain rate sensitivity reduces the size of the deformed region and thereby could enhance the cavitation erosion resistance of the material.

Frequency domain analysis of nonlocal rods embedded in an elastic medium

A novel dynamic finite element method is carried out for a small-scale nonlocal rod which is embedded in an elastic medium and undergoing axial vibration. Eringen's nonlocal elasticity theory is employed. Natural frequencies are derived for general boundary conditions. An asymptotic analysis is carried out. The stiffness and mass matrices of the embedded nonlocal rod are obtained using the proposed finite element method. Nonlocal rods embedded in an elastic medium have an upper cut-off natural frequency which is independent of the boundary conditions and the length of the rod. Dynamic response for the damped case has been obtained using the conventional finite element and dynamic finite element approaches. The present study would be helpful for developing nonlocal finite element models and study of embedded carbon nanotubes for future nanocomposite materials.

Dynamic crack nucleation, propagation, and interactions with crystalline secondary phases in aluminum alloys subjected to large deformations

The major objective of this work was to model within a continuum framework the dynamic nucleation and evolution of failure surfaces in aluminum alloys with complex microstructures, using a recently developed compatibility-based fracture criterion for large deformations. Computational analyses were conducted to understand how Mn-bearing dispersoids, Ω and θ′ precipitates affect dynamic fracture processes in an Al–Cu–Mg–Ag alloys (2139-Al). High strain-rate simulations were based on a rate-dependent dislocation-density-based crystalline plasticity formulation and a nonlinear explicit dynamic finite-element approach. Results indicate that the fracture criterion elucidated how dispersoids and precipitates have a dominant role in dynamic crack blunting, branching and arrest. Rationally orientated precipitates result in overall dynamic microstructural strengthening and enhanced uniformity of deformation. These precipitates, however, accelerated unstable crack propagation, and this is amplified in the presence of a pre-crack. In contrast, dispersoids decreased microstructural toughness and ductility, but greatly improved dynamic damage tolerance, especially in the presence of a pre-crack. It can also be predicted that low angle boundaries can change the propagation direction of ductile cracks, and contribute to damage tolerance without crack initiation. Collectively, rationally oriented precipitates and dispersoids can significantly improve the combined dynamic strength, toughness and damage tolerance of crystalline aluminum alloys.

The Use Of Trigonometric Interpolation Functions For Vibration AnalysisOf Beam Structures - Bridging Gap Between FEM And Exact Formulations

The small linear harmonic oscillations of linearly elastic coupled beam structures are addressed. The Finite Element Method (FEM), conventional Hermite beam formulation, and the resulting element matrices are then briefly discussed. The so-called exact Dynamic Stiffness Matrix (DSM) formulation for the longitudinal vibration of axially loaded beams is presented. Finally, the Dynamic Finite Element (DFE) approach is introduced and its application to the axial vibration of beams is displayed. The comparison is made between the standard static and (frequency dependent) Dynamic beam shape functions. The DFE formulation, combines the generality of the FEM and the high precision provided by DSM methods. The weighting functions and shape functions are evaluated referring to the appropriate exact DSM formulation. The DFE approach can be advantageously extended to more complex cases which distinguishes this method from the DSM method.

Dynamic finite element approach for analyzing stress and distortion in multilevel devices

Stress analysis is of crucial importance in the design of components and systems in the electronic industry. This paper presents a dynamic finite element approach dynamic finite element approach for following the evolution of residual thermal stress during pulsed exposure of multilevel structures. A numerical example is given to demonstrate the application of this approach. Its concern the transient thermal stress modeling of multilevel semiconductor devices undergoing short pulses heating. We describe the elastic deformation of device due to thermal pulsed loading. The main advantage of proposed approach is that it allows one to account for thermomechanical properties and conditions changes during repetitive exposure heating, as well as the historical stresses and distortions introduced at each pulse. A three-dimensional finite element model was constructed to simulate multilevel devices under short pulsed loading. The temperature distributions obtained were used as input in order to compute thermal stress and mechanical deformations of the multilevel structures.

An Impedance Method for Dynamic Analysis of Active Material Systems

This paper describes a new approach to analyzing the dynamic response of active material systems with integrated induced strain actuators, including piezoelectric, electrostrictive, and magnetostrictive actuators. This approach, referred to as the impedance method, has many advantages compared with the conventional static approach and the dynamic finite element approach, such as pin force models and consistent beam and plate models. The impedance approach is presented and described using a simple example, a PZT actuator-driven one-degree-of-freedom spring-massdamper system, to demonstrate its ability to capture the physics of adaptive material systems, which is the impedance match between various active components and host-structures, and its utility and importance by means of an experimental example and a numerical case study. The conventional static and dynamic finite element approaches are briefly summarized. The impedance methodology is then discussed in comparison with the static approach. The basic elements of the impedance method, i.e., the structural impedance corresponding to actuator loading and the dynamic output characteristics of PZT actuators, are addressed. The advantages of using the impedance approach over conventional approaches are discussed using a simple numerical example. A comparison of the impedance method with the static and the dynamic finite element approaches are provided at the conclusion of this paper.

An Impedance Method for Dynamic Analysis of Active Material Systems

This paper describes a new approach to analyzing the dynamic response of active material systems with integrated induced strain actuators, including piezoelectric, electrostrictive, and magnetostrictive actuators. This approach, referred to as the impedance method, has many advantages compared with the conventional static approach and the dynamic finite element approach, such as pin force models and consistent beam and plate models. The impedance approach is presented and described using a simple example, a PZT actuator-driven one-degree-of-freedom spring-mass-damper system, to demonstrate its ability to capture the physics of adaptive material systems, which is the impedance match between various active components and host-structures, and its utility and importance by means of an experimental example and a numerical case study. The conventional static and dynamic finite element approaches are briefly summarized. The impedance methodology is then discussed in comparison with the static approach. The basic elements of the impedance method, i.e., the structural impedance corresponding to actuator loading and the dynamic output characteristics of PZT actuators, are addressed. The advantages of using the impedance approach over conventional approaches are discussed using a simple numerical example. A comparison of the impedance method with the static and the dynamic finite element approaches are provided at the conclusion of this paper.

Dynamic finite-element calculation of excitation modes in periodic and disordered Lucite–steel composite systems

Recent experiment by L. Ye etal. on Lucite–steel composite system [Phys. Rev. Lett. 69, 3080 (1992)] has demonstrated the localization of bending waves. This is particularly surprising on first sight because the huge acoustic impedance mismatch between Lucite and steel would seem to imply that Lucite should have very little effect on the steel. A dynamic finite-element approach was used to calculate the excitation modes in a 2-D periodic structure consisting of a steel plate decorated by a checkerboard pattern of Lucite blocks. The results show that the coupling between the steel plate bending wave and the flexural resonances of the Lucite blocks can drastically alter the dispersion relation of the bending wave, whereas the effect of periodicity is minimal. Since the Lucite flexural resonance frequencies depend on the Lucite block height, its effect on the bending wave can be tuned in terms of the frequency as demonstrated experimentally by Ye etal. This talk will also report the results of calculations on a random 1-D Lucite–steel system. In particular, the localization characteristics will be emphasized.

Special topics on soil-structure interaction

In the design of nuclear power plants, the topic of soil-structure interaction has traditionally been one of great theoretical and practical controversy for several reasons. In the beginning of the era of nuclear power, the science of soil-structure interaction was in its infancy and very little theory or observational data were available to the engineering profession. At the same time, the state of the art of structural dynamics, however, was already developed and engineers had a considerable amount of data on actual dynamic performance of structures, so that a great amount of confidence in the ability to perform a dynamic structural analysis was already in existence. It was only natural to extend the basic fundamentals of dynamic structural modeling to soil-structure interaction. Consequently, many ‘approaches’ and ‘theories’ of how to account for soil-structure interaction were erroneously developed without providing real consideration of basic fundamentals of interaction of a foundation with an elastic continuum, commonly referred to as elastic half-space or continuum solutions. At the same time, while great confusion was developing on how to apply elastic half-space theory to earthquake analysis problems, the dynamic finite element approach was born and was thought by many to be the final panacea to the problem of soil-structure interaction. Of course, as is usually the case, there is no perfect or single best approach to any problem and the end result is that each of the many approaches to account for soil-structure interaction, has its strong and weak points depending on site or structural conditions. This paper is intended to discuss some of the important details of soil-structure interaction theory to provide a common means of comparison and to introduce some new approaches to simplify the solutions of deeply embedded foundations. A review of recent literature on soil-structure interaction reveals several important facts. First, conflicting comparisons between the lumped parameter and finite element solutions apparently exist. In some cases, both approaches give similar results while for others the results vary widely essentially because different models are used for comparison of both methods. Secondly, fundamental errors are committed on finite element mesh size parameters and boundary conditions. Thirdly, misunderstandings on soil mass and foundation or structural modal damping lead to gross errors in the lumped parameter approach. Finally, the limitations of the various approaches are not always understood. It is noted that both the finite element and lumped parameter approaches should yield similar results if they are appropriately used to solve the same problems. A summary of the advantages and limitations of both approaches are presented and discussed with a short presentation regarding the state of the art in the determination of soil stiffness and material damping characteristics. Furthermore, the paper will illustrate one type of analysis technique which uses a hybrid approach of both finite element results and lumped parameter solutions. Such an approach is developed to account more accurately for the influence of embedment. Results using both pure finite element solutions and lumped parameter models show that the influence of embedment can be accurately considered even for deeply embedded structures (depth to width ratio equal to 1.0). The key to the approach lies in establishing the coupling between horizontal and rocking modes of foundation vibration. Once the coupling parameter is accounted for, it is then possible to develop a lumped parameter models that account for the variation of soil motions below the ground sufrace. Previous lumped parameter models have not accounted for these variations from the surface to the depth of the foundation. Details of the lumped parameter approach for embedded foundations and an illustration with numerical examples are provided. Recommendations are then presented on a procedure for soil-structure interaction of deeply embedded foundations. The primary advantages are that: (1) the model is more easily generated than a finite element model, (2) the model is less susceptible to modeling errors, such as mesh size, model size, and boundary influences, (3) parametric studies may be easily conducted since parameters such as damping may be more directly controlled, and (4) the computer costs for analysis are significantly reduced.