Finite Stiffness Research Articles

A microscopic Kapitza pendulum

Pyotr Kapitza studied in 1951 the unusual equilibrium features of a rigid pendulum when its point of suspension is under a high-frequency vertical vibration. A sufficiently fast vibration makes the top position stable, putting the pendulum in an inverted orientation that seemingly defies gravity. Kapitza’s analytical method, based on an asymptotic separation of fast and slow variables yielding a renormalized potential, has found application in many diverse areas. Here we study Kapitza’s pendulum going beyond its typical idealizations, by explicitly considering its finite stiffness and the dissipative interaction with the surrounding medium, and using similar theoretical methods as Kapitza. The pendulum is realized at the micrometre scale using a colloidal particle suspended in water and trapped by optical tweezers. Though the strong dissipation present at this scale prevents the inverted pendulum regime, new ones appear in which the equilibrium positions are displaced to the side, and with transitions between them determined either by the driving frequency or the friction coefficient. These new regimes could be exploited in applications aimed at particle separation at small scales.

Analysis of a rotor supported in bearing with gyroscopic effects

Rotors with one or more offset disks and supported on bearings are common in high speed turbomachinery. During start up, these rotors have to cross through several critical speeds before reaching their operating speed. At critical speeds, the spin speed of the rotor matches with one of its natural frequency leading to the condition of resonance and large vibration amplitude. In rotating systems, these natural frequencies depend on the support stiffness. Also, they are a function on the spin speed because of the phenomenon of gyroscopic effect. The gyroscopic effects on critical speeds of a rotor system supported in bearing can be studied by means of a Campbell diagram, which has been used in the design of turbines. In this paper, gyroscopic effects of a rotor with an offset disk and supported on bearing is studied by means of Campbell diagrams. The numerical study is carried out by modelling of the rotor-bearing system using finite element mass, stiffness and gyroscopic matrices. The solution is obtained by solving the assembled equations of motion; following application of geometric boundary conditions and representing the second order differential equations of motion in statespace form. The results of critical speed obtained are compared with those of the results obtained through modal analysis using FE tool (ANSYS). The analysis of the results can be used to extend the study for a multi-disk rotor with different bearing supports.

Accelerating coupled finite element-kinetic Monte Carlo models: 200 $$\times $$ × speedup of shear transformation zone dynamics simulations

Shear transformation zone dynamics models of metallic glass deformation access experimentally-relevant time scales by using the kinetic Monte Carlo method to simulate small, fast, often discrete events, while the finite element method calculates macroscopic shape change and continuum-level interactions within samples. The most time-consuming portion of these models is the finite element method calculation on each step. However, in cases where the finite element mesh geometry and element elastic properties do not change from step to step, the finite element stiffness matrix (and its Cholesky factors) from previous steps can be reused. This strategy improves the asymptotic complexity of these models and in practice accelerates their execution by nearly 200 $$\times $$ . This enables simulation of larger samples in more reasonable time. A set of three-dimensional shear transformation zone dynamics simulations, with larger length scales than any currently in the literature, illustrates the utility of this approach.

Reducing uncertainties in energy dissipation measurements in atomic force spectroscopy of molecular networks and cell-adhesion studies

Atomic force microscope (AFM) based single molecule force spectroscopy (SMFS) is a valuable tool in biophysics to investigate the ligand-receptor interactions, cell adhesion and cell mechanics. However, the force spectroscopy data analysis needs to be done carefully to extract the required quantitative parameters correctly. Especially the large number of molecules, commonly involved in complex networks formation; leads to very complicated force spectroscopy curves. One therefore, generally characterizes the total dissipated energy over a whole pulling cycle, as it is difficult to decompose the complex force curves into individual single molecule events. However, calculating the energy dissipation directly from the transformed force spectroscopy curves can lead to a significant over-estimation of the dissipated energy during a pulling experiment. The over-estimation of dissipated energy arises from the finite stiffness of the cantilever used for AFM based SMFS. Although this error can be significant, it is generally not compensated for. This can lead to significant misinterpretation of the energy dissipation (up to the order of 30%). In this paper, we show how in complex SMFS the excess dissipated energy caused by the stiffness of the cantilever can be identified and corrected using a high throughput algorithm. This algorithm is then applied to experimental results from molecular networks and cell-adhesion measurements to quantify the improvement in the estimation of the total energy dissipation.

Evolution of callus tissue behavior during stable distraction osteogenesis.

Multiple studies have sought to characterize the mechanical behavior of callus tissue in vivo during distraction osteogenesis. The aims of such studies are to understand the mechanobiology of distraction and elucidate the complex viscoelasticity and evolution of the tissue. The former objective has direct clinical relevance to surgical technique and process control while the latter is necessary for the calibration and validation of the predictive healing models. Such models seek to reduce the researcher's dependence on animal studies and prospectively allow improved surgical planning. To date, no study has been capable of controlling the mechanical conditions sufficiently enough to decouple the distraction process from the secondary mechanical stimulation associated with the finite stiffness of the fixation constructs employed. It is the goal of this work to understand the mechanobiology of pure distraction as well as characterize viscoelastic tissue behavior under precisely defined mechanical conditions. This is achieved using a novel lateral distraction model. The structural integrity of the bone is maintained, allowing the collection of force relaxation data due to a stepwise distraction process without the superimposed influence of secondary mechanical stimulation. The average instantaneous modulus increases from approximately 2 kPa to approximately 1100 kPa while the equilibrium modulus increases from approximately 0 kPa to 200 kPa over the distraction period.

Effect of Non-Uniform Torsion on Elastostatics of a Frame of Hollow Rectangular Cross-Section

Abstract In this paper, results of numerical simulations and measurements are presented concerning the non-uniform torsion and bending of an angled members of hollow cross-section. In numerical simulation, our linear-elastic 3D Timoshenko warping beam finite element is used, which allows consideration of non-uniform torsion. The finite element is suitable for analysis of spatial structures consisting of beams with constant open and closed cross-sections. The effect of the secondary torsional moment and of the shear forces on the deformation is included in the local finite beam element stiffness matrix. The warping part of the first derivative of the twist angle due to bimoment is considered as an additional degree of freedom at the nodes of the finite elements. Standard beam, shell and solid finite elements are also used in the comparative stress and deformation simulations. Results of the numerical experiments are discussed, compared, and evaluated. Measurements are performed for confirmation of the calculated results.

Effective second moment of area of uniform built‐up members

AbstractDedicated to Prof. Dr.‐Ing. habil. Joachim Lindner on the occasion of his 80th birthdayFor built‐up compression members, Eurocode 3 in part EN 1993‐1‐1 [1] includes the shear stiffness to account for the effect of shear deformations directly into the amplification factor when determining second‐order forces. Built‐up members loaded in bending are not treated at all in EN 1993‐1‐1 [1]. This article is based on a study [2] performed at Eindhoven University of Technology and introduces the concept of effective second moment of area to account for the influence of the finite shear stiffness of built‐up members. This article is largely based on [3] but is an updated and improved version. The effective second moment of area is checked by numerical models. The shear stiffness is derived for a simple configuration of a battened built‐up member and for more complex configurations and for laced built‐up columns shear stiffnesses are presented. The design rules for built‐up compression members in EN 1993‐1‐1 [1] are analysed and suggestions for modifications are made to include the concept of the effective second moment of area. It is advantageous that the concept of the effective second moment of area can be used for both built‐up columns and beams. For built‐up beams, the effective second moment of area including the shear stiffness allows the deflections to be calculated with the standard rules for flexural bending.

Generalised beam theory (GBT) formulation to analyse the vibration behaviour of thin-walled steel frames

This paper reports the results of an investigation on the use of Generalised Beam Theory (GBT) to analyse the vibration behaviour of thin-walled steel frames. After an overview of the main concepts and procedures involved in performing a GBT vibration analysis, the formulation and implementation of a GBT-based beam finite element are presented, namely the determination of the finite element and frame linear stiffness and mass matrices (incorporating the influence frame joint configurations). Finally, in order to illustrate the application and capabilities of the proposed GBT-based finite element formulation, numerical results concerning the local, distortional and global vibration behaviour of (i) a symmetric portal frame and (ii) an “L-shaped” frame exhibiting four commonly used joint configurations are presented and discussed. For validation purposes, some GBT results (frequency values and vibration mode shapes) are compared with values yielded by Ansys shell finite element analyses.

Parametric analysis of in-plane rigid-elastic coupled non-contact tire model for heavily loaded radial tire

Parametric analysis of in-plane vibration characteristic for unloaded heavily loaded radial tire is put forward and researched utilizing in-plane rigid-elastic coupled model. Coupled vibration characteristic between the flexible tread and circumferential sidewall is investigated with theoretical modeling and experimental modal method. In-plane analytical vibration feature is modeled with flexible beam on modified elastic foundation tire model. The rigid-elastic coupled tire model is derived with finite difference method and stiffness matrix and mass matrix are presented analytically with the geometrical and structural parameters. Structural parameters identification is implemented with genetic algorithm based on in-plane experimental modal parameter. The in-plane transfer functions with different structural parameters are compared and the parametric effect of structural parameters on in-plane transfer function is discussed. Experimental and theoretical result shows that the in-plane rigid-elastic coupled tire model can achieve the higher precision on predicting the transfer function and vibration feature of heavily loaded tire within the frequency band of 300 Hz.

Analysis of the capability of non-specific simulation software for studying the dynamic interaction between pantograph and rigid overhead conductor rail

Present paper is based on the generation of simulation models of different lengths for the study of the dynamic interaction between pantograph and OCR. This project is meant to analyse the capability of a CAE software, used mainly as a teaching tool, to make highly complex dynamic studies which usually require the use of specific software. Given the features of the software and the complexity of the problem, a key factor of this work has been the definition of the optimal technique to model the different elements that an OCR system contains. Regarding the pantograph, a lumped mass model has been used. The finite element model of the OCR structure has been based on one-dimensional elements for the contact rail and finite vertical stiffness elements for the supports. After analysing the acquired results, it can be concluded that this procedure is feasible, accurate and fast when a reduced model is simulated. Nevertheless, when working with longer models, there are several limitations in comparison with the use of specific simulation software.

Comparative analysis of efficiency of use of finite elements of different dimensionality in the analysis of the stress-strain state of thin shells

Relevance. To determine the stress-strain state (SSS) of thin-walled shells due to the complexity of obtaining numerical results, the theory of thin shells was developed with the introduction of the direct normal hypothesis to reduce the three-dimensional SSS to the two-dimensional one. With the modern development of digital technology and numerical methods of calculation, in particular the finite element method (FEM), it became possible to obtain numerical results without the use of the direct normal hypothesis, namely on the basis of the theory of elasticity in three-dimensional formulation even for thin shells. Aims. The aim of this work is to compare the efficiency of algorithms for the use of finite element stiffness matrices obtained on the basis of the theory of thin shells with the hypothesis of a straight normal and on the basis of the relations of the three-dimensional theory of elasticity. Methods. The results of comparative analysis of finite element calculations of thin shells using a two-dimensional sampling element in the form of a quadrangular fragment of the middle surface and a three-dimensional element in the form of an eight-node six-face are presented. The components of the displacement vector and their first derivatives were chosen as the nodal variable parameters. The functions of the form for both types of discretization elements were represented by products of Hermite polynomials of the third degree. Results. On the example of calculation of the cylindrical shell clamped at the ends it is shown that the two-dimensional statement in calculations of thin shells is adequate and allows to receive acceptable results at optimum costs of machine time.

Spatial and Temporal Quantification of Community Resilience: Gotham City under Attack

AbstractMitigating the impact of disasters on communities requires not only a deep understanding of the essential features of infrastructure, social, and economical components that make a community resilient; but also the development of mathematical models that can seamlessly integrate these features. This article lays the foundation for an integrative model that captures interaction between these components. The underlying fundamentals of the proposed model hinges on the principle of a damped harmonic oscillator by assuming the behavior of a community in response to a hazard is equivalent to the response of a vibrating mass of finite stiffness and damping. We implemented the dynamic model by developing a novel finite element formulation capable of quantifying resilience both temporally and spatially. We then used the developed model to devise a suitable hazard‐agnostic definition of community resilience. This was realized through a set of demonstration and logical verification tests conducted on Gotham City, the fictional city of the infamous character, Batman. It was observed that the model can be used to identify sensitive and vulnerable areas in a community, provide a spatial and temporal quantification of community resilience, and accommodate various types of hazards such as physical disruptions, economic downtimes, and even social disorders.

DNA Nanotubes as a Versatile Tool to Study Semiflexible Polymers.

Mechanical properties of complex, polymer-based soft matter, such as cells or biopolymer networks, can be understood in neither the classical frame of flexible polymers nor of rigid rods. Underlying filaments remain outstretched due to their non-vanishing backbone stiffness, which is quantified via the persistence length (lp), but they are also subject to strong thermal fluctuations. Their finite bending stiffness leads to unique, non-trivial collective mechanics of bulk networks, enabling the formation of stable scaffolds at low volume fractions while providing large mesh sizes. This underlying principle is prevalent in nature (e.g., in cells or tissues), minimizing the high molecular content and thereby facilitating diffusive or active transport. Due to their biological implications and potential technological applications in biocompatible hydrogels, semiflexible polymers have been subject to considerable study. However, comprehensible investigations remained challenging since they relied on natural polymers, such as actin filaments, which are not freely tunable. Despite these limitations and due to the lack of synthetic, mechanically tunable, and semiflexible polymers, actin filaments were established as the common model system. A major limitation is that the central quantity lp cannot be freely tuned to study its impact on macroscopic bulk structures. This limitation was resolved by employing structurally programmable DNA nanotubes, enabling controlled alteration of the filament stiffness. They are formed through tile-based designs, where a discrete set of partially complementary strands hybridize in a ring structure with a discrete circumference. These rings feature sticky ends, enabling the effective polymerization into filaments several microns in length, and display similar polymerization kinetics as natural biopolymers. Due to their programmable mechanics, these tubes are versatile, novel tools to study the impact of lp on the single-molecule as well as the bulk scale. In contrast to actin filaments, they remain stable over weeks, without notable degeneration, and their handling is comparably straightforward.

Analysis of beams with piezo-patches by node-dependent kinematic finite element method models

This article presents a family of one-dimensional finite element method models with node-dependent kinematics for the analysis of beam structures with piezo-patches. The models proposed are built by applying Carrera unified formulation. Carrera unified formulation permits to obtain finite element method stiffness matrices through so-called fundamental nuclei whose form is independent of the assumptions made for the displacement/electrical field over the cross section of a beam. In the previous works, uniform kinematic assumptions have been applied to all the nodes within the same element. The present contribution proposes to use different kinematics on different nodes, leading to node-dependent kinematic finite element method formulations. In such an approach, non-uniform cross sections introduced by piezo-patches can be considered. With the help of layer-wise models, piezoelectric and mechanical domains each can possess individual constitutive relations. Meanwhile, node-dependent kinematics can integrate equivalent single layer models and layer-wise models to reach an optimal balance between accuracy and use of computational resources. Static governing equations for beam elements with node-dependent kinematics accounting for electromechanical effects are derived from the principle of virtual displacements. The competence of the proposed approach is validated by comparing the obtained results with solutions taken from the literature and ABAQUS three-dimensional modelling. Both extension and shear actuation mechanisms are considered.

Mechanical switching of ferroelectric domains beyond flexoelectricity

The resurgence of interest in flexoelectricity has prompted discussions on the feasibility of switching ferroelectric domains ‘non-electrically’. In this work, we perform three-dimensional thermodynamic simulations in combination with ab initio calculations and effective Hamiltonian simulations to demonstrate the great effects of surface screening and surface bonding on ferroelectric domain switching triggered by local tip loading. A three-dimensional simulation scheme has been developed to capture the tip-induced domain switching behavior in ferroelectric thin films by adequately taking into account the surface screening effect and surface bonding effect of the ferroelectric film, as well as the finite elastic stiffness of the substrate and the electrode layers. The major findings are as follows. (i) Compared with flexoelectricity, surface effects can be overwhelming and lead to much more efficient mechanical switching caused by tip loading. (ii) The surface-assisted mechanical switching can be bi-directional without the necessity of reversing strain gradients. (iii) A mode transition from local to propagating domain switching occurs when the screening below a critical value. A ripple effect of domain switching appears with the formation of concentric loop domains. (iv) The ripple effect can lead to ‘domain interference’ and a deterministic writing of confined loop domain patterns by local excitations. Our study reveals the hidden switching mechanisms of ferroelectric domains and the possible roles of surface in mechanical switching. The ripple effect of domain switching, which is believed to be general in dipole systems, broadens our current knowledge of domain engineering.

Skeletal Microstructure and Estimated Bone Strength Improve Following Parathyroidectomy in Primary Hyperparathyroidism.

High-resolution peripheral quantitative computed tomography (HRpQCT) is a noninvasive imaging technology that can provide insight into skeletal microstructure and strength. In asymptomatic primary hyperparathyroidism (PHPT), HRpQCT imaging has demonstrated both decreased cortical and trabecular indices, consistent with evidence for increased fracture risk. There are limited data regarding changes in HRpQCT parameters postparathyroidectomy. To evaluate changes in skeletal microstructure by HRpQCT in subjects with PHPT after parathyroidectomy. We studied 29 subjects with PHPT (21 women, 8 men) with HRpQCT at baseline and 6, 12, 18, and 24 months postparathyroidectomy. Volumetric bone mineral density, microarchitectural indices, and finite element analysis at the distal radius and tibia. At both the radius and tibia, there were significant improvements in total, cortical, and trabecular volumetric bone density as early as 6 months postparathyroidectomy (24-month values for total volumetric bone density, radius: +2.8 ± 4%, tibia: +4.4 ± 4%; P < 0.0001 for both), cortical thickness (radius: +1.1 ± 2%, tibia: +2.0 ± 3%; P < 0.01 for both), and trabecular bone volume (radius: +3.8 ± 5%, tibia: +3.2 ± 4%; P < 0.0001 for both). At both sites, by finite element analysis, stiffness and failure load were improved starting at 6 months postparathyroidectomy (24-month values for failure load, radius: +6.2 ± 6%, tibia: +4.8 ± 7%; P < 0.0001 for both). These results provide information about skeletal microarchitecture in subjects with PHPT followed through 2 years after parathyroidectomy. Estimated bone strength is improved, consistent with data showing decreased fracture risk postparathyroidectomy.

13.18: Advanced analysis of braced steel frames: Modelling accounting for the post‐limit behaviour of compressed angle braces

ABSTRACTThis paper presents a method of advanced analysis applied to steel plane load bearing systems like trusses as well as unbraced and braced frames. The effects of both material and geometrical nonlinearities on the structural behaviour are studied together with tracing the whole range of truss member behaviour. In the braced frameworks, the truss bracing model accounts for the effect of post‐limit behaviour on the inelastic redistribution of forces in frame members. Details of finite element incremental stiffness matrices for frame members and truss bracing members are briefly presented and followed by the discussion how the effects of gradual yielding, buckling and semi‐rigid joints are included. The implementation into engineering computer software is summarized and followed by certain aspects of truss and frame structural systems design according to Eurocode 3. A simple two‐bar arch truss sensitive to the elastic snap‐through buckling for which the closed form solution exists is considered as an example. Simulations of the behaviour of this truss show how the nonlinear behaviour of a single member affects the global instability of the truss. Conclusions are formulated with reference to the application of advanced analysis in direct design of steel braced frameworks.

Propulsion via flexible flapping in granular media.

Biological locomotion in nature is often achieved by the interaction between a flexible body and its surrounding medium. The interaction of a flexible body with granular media is less understood compared with viscous fluids partially due to its complex rheological properties. In this work, we explore the effect of flexibility on granular propulsion by considering a simple mechanical model in which a rigid rod is connected to a torsional spring that is under a displacement actuation using a granular resistive force theory. Through a combined numerical and asymptotic investigation, we characterize the propulsive dynamics of such a flexible flapper in relation to the actuation amplitude and spring stiffness, and we compare these dynamics with those observed in a viscous fluid. In addition, we demonstrate that the maximum possible propulsive force can be obtained in the steady propulsion limit with a finite spring stiffness and large actuation amplitude. These results may apply to the development of synthetic locomotive systems that exploit flexibility to move through complex terrestrial media.

A boundary condition correction for the clamped constraint of elastic plate/beam theory

Elastic plate/beam theory is used by mechanical and civil engineers to model a variety of structures which are flat in two/one dimension(s) and thin in the other direction(s). If the plate is clamped at an end by being sandwiched between two rigid surfaces, it is customary to model this constraint (i.e. boundary condition) as a rigid constraint which does not permit either displacement or rotation at that edge. Here it is shown that an elastic plate/beam, with an end perfectly bonded at its top and bottom surfaces to a rigid constraint, deflects an additional amount due to elastic rotation effects within the support. It is shown that this correction can be accommodated with a simple modified boundary condition to standard plate theory without the need for a more complicated theory which includes a warping function. The analysis presented here is valid for plate bending theory but is equally valid for beam theory with minor modification.The finite rotational stiffness within the support is determined by applying a distributed load (statically equivalent to a couple moment) to such a constrained infinite elastic strip. The displacement profile is determined analytically and approximated by a straight line. The ratio of the applied moment to the slope of this line provides the rotational stiffness. Examples are given for a cantilever and a clamped-clamped plate/beam as well as for a clamped circular plate. For a beam we use Timoshenko beam theory, while for a plate Reissner-Mindlin is used. For each of these cases the correction due to the effect of rotational compliance is often greater than the correction due to shear deformation. Finite element analyzes, using two-dimensional elasticity, of these configurations show excellent agreement with the analytical results of this modified plate/beam theory.

Vibration mitigation in partially liquid-filled vessel using passive energy absorbers

We consider possible solutions for vibration mitigation in reduced-order model (ROM) of partially filled liquid tank under impulsive forcing. Such excitations may lead to strong hydraulic impacts applied to the tank inner walls. Finite stiffness of the tank walls is taken into account. In order to mitigate the dangerous internal stresses in the tank walls, we explore both linear (Tuned Mass Damper) and nonlinear (Nonlinear Energy Sink) passive vibration absorbers; mitigation performance in both cases is examined numerically. The liquid sloshing mass is modeled by equivalent mass-spring-dashpot system, which can both perform small-amplitude linear oscillations and hit the vessel walls. We use parameters of the equivalent mass-spring-dashpot system for a well-explored case of cylindrical tanks. The hydraulic impacts are modeled by high-power potential and dissipation functions. Critical location in the tank structure is determined and expression of the corresponding local mechanical stress is derived. We use finite element approach to assess the natural frequencies for specific system parameters. Numerical evaluation criteria are suggested to determine the energy absorption performance.