Set Of Linear Springs Research Articles

Novel Design and Control of a Crank-Slider Series Elastic Actuated Knee Exoskeleton for Compliant Human–Robot Interaction

The lower-limb assist exoskeleton plays the role of torque assiting and compliant tracking for wearers to perform tasks. Accurate torque generation, backdrivability performance, low output impedance, and hardware compactness are essential factors for lower-limb exoskeleton to achieve better compliant physical interaction. This research studies a crank-slider series elastic actuator (CS-SEA) that can be used as a compact exoskeleton joint module. The device has a unique crank slider mechanism, and a set of linear springs are equipped inside the slider to guarantee the nonlinear stiffness of its physical impedance so that the torque effect can be improved and a high level of transparency can be achieved. The RBF-based sliding mode controller is chosen as the output torque controller of the exoskeleton, and the adaptive neuro-fuzzy sliding mode control law is designed and its stability is verified. The precise output force control performance of CS-SEA is verified by experiments. The actuator is incorporated into a knee exoskeleton prototype and was worn by the subjects. The experimental results demonstrate the precision of the compliant transparent and torque assisting control while interacting with the human wearer.

A solution method for free vibration of intact and cracked polygonal thin plates using the Ritz method and Jacobi polynomials

A method for solving for the free vibration of intact and straight through-cracked polygonal thin plates with arbitrary boundary conditions is proposed. To obtain the expressions for the energy integral, the polygonal domain of integration is divided into triangular and trapezoidal domains by a set of parallel lines passing through the vertexes. The energy integral can then be calculated analytically in closed form using the properties of Jacobi orthogonal polynomials, which are the admissible functions of the Ritz method in the framework of the Kirchhoff plate theory. The boundary conditions are modeled using linear springs to restrain the plate edges. The model for solving polygonal thin plate vibration for arbitrary boundary conditions is thus obtained. Accordingly, the cracked polygonal plate is decomposed into several polygonal subdomains based on the crack by applying the technique of domain decomposition. Each subdomain is solved as an intact plate. The subdomains are then connected by a set of linear springs with variable stiffness, and the crack or rigid connection is determined by setting the spring stiffness to zero or infinity, respectively. The convergence and accuracy of the proposed method are verified by comparing the numerical examples with results available in the literature. The proposed method is also applied to determine the natural frequencies of a concave pentagonal plate and cracked regular hexagonal plates. The effect of crack length and inclination on the natural frequencies and mode shapes of cracked regular hexagonal plates with several boundary conditions are studied. The natural frequencies of cracked regular hexagonal plates tabulated are the first published vibration data, and can be used as a reference for future studies.

Analytical approach for circular‐jointed shield tunnel lining based on the state space method

SummaryThis paper presents a new analytical solution to the circular segmental tunnel lining subjected to overburden and surrounding earth pressures. The governing equations are derived by adopting the curved Euler beam theory and the principle of minimum potential energy. Based on the state space method, the displacements and the relevant energy‐conjugated internal forces are treated as the fundamental unknown state vector and can be obtained by solving the state function. The inter‐segment joints on the lining are modeled by a set of linear springs, including shear, compression, and rotation. The presented method allows for the arbitrary distribution of the segmented joints and loads along the circumferential direction. The most striking advantages of the proposed method include the rigid body displacement treatment, lining‐displacement‐dependent soil reaction, and internal force direction dependency of the joint stiffness. Using this method, the displacements and internal forces of the entire lining can be obtained conveniently and simultaneously under the arbitrary loading and joint distribution conditions. The verification of the analytical solutions is provided by several examples.

Power Generation Using Kites in a GroundGen Airborne Wind Energy System: A Numerical Study

Abstract A computational model of a massless kite that produces power in an airborne wind energy (AWE) system is presented. AWE systems use tethered kites at high altitudes to extract energy from the wind and are being considered as an alternative to wind turbines since the kites can move in high-speed cross-wind motions over large swept areas to increase power production. In our model, the kite completes successive power-retraction cycles where the kite angle of attack is altered as required to vary the resultant aerodynamic forces on the kite. The flow field is found in a two-dimensional domain near the flexible kite by solving the full Navier–Stokes equations using an Eulerian grid together with a Lagrangian representation of the kite. The flow solver is a finite volume projection method using a non-uniform mesh on a staggered grid and corrector–predictor technique to ensure a second-order accuracy in time. The two-dimensional kite shape is modeled as a slightly cambered immersed boundary that moves with the flow. The flexible kite is modeled with a set of linear springs following Hooke’s law. The unstretched length of each elastic tether at a given time step is controlled using periodic triangular wave shapes to achieve the required power-retraction phases. A study was conducted in which the wave shape amplitude, frequency, and phase (between two tethers) were adjusted to achieve a suitably high net power output. The results are in good agreement with predictions for Loyd’s simple kite in two-dimensional motion. Aerodynamic coefficients for the kite, tether tensions, tether reel-out and reel-in speeds, and the vorticity fields in the kite wake are also determined.

Thermal Buckling Analysis of Circular Bilayer Graphene Sheets Resting on an Elastic Matrix Based on Nonlocal Continuum Mechanics

In this article, the thermal buckling behavior of orthotropic circular bilayer graphene sheets embedded in the Winkler–Pasternak elastic medium is scrutinized. Using the nonlocal elasticity theory, the bilayer graphene sheets are modeled as a nonlocal double–layered plate that contains small scale effects and van der Waals (vdW) interaction forces. The vdW interaction forces between the layers are simulated as a set of linear springs using the Lennard–Jones potential model. Using the principle of virtual work, the set of equilibrium equations are obtained based on the first-order shear deformation theory (FSDT) and nonlocal differential constitutive relation of Eringen. Differential quadrature method (DQM) is employed to solve the governing equations for simply-supported and clamped boundary conditions. Finally, the effects of the small scale parameter, vdW forces, aspect ratio, elastic foundation, and boundary conditions are considered in detail.

Preliminary Validation of a Cable-Driven Powered Ankle–Foot Orthosis With Dual Actuation Mode

Motor weakness at the ankle joint is a major contributor to locomotor deficits in stroke, as the ankle provides the largest propulsive moment during walking, compared to the knee and the hip joints. In this paper, a novel lightweight powered ankle–foot orthosis (AFO), named Stevens ankle–foot electromechanical (SAFE) orthosis, is presented and evaluated. The prominent feature of the SAFE orthosis relies on its two alternative actuation modes (AMs): 1) fully active and 2) semi-active. The former mode utilizes two BLDC motors to provide ankle plantar- and dorsi-flexion torques, whereas the latter replaces the motor on the dorsiflexion side with a set of linear springs. This design allows direct investigation of how the introduction of a passive antagonistic element may affect the performance of a powered AFOs. Experimental results from both bench-top tests and human walking tests indicate that the performance of the two AMs are comparable in terms of device transparency and torque tracking. Implications of these results for the design of future powered AFOs are discussed in this paper.

Closed form solution for the first natural frequency of offshore wind turbine jackets supported on multiple foundations incorporating soil-structure interaction

Offshore Wind Turbines (OWTs) are dynamically sensitive structures and as a result estimating the natural frequency of the whole system taking into effect the flexibility of the foundation is one of the major design considerations. The natural frequency is necessary to predict the long-term performance as well as the fatigue life. Currently, jackets supported on multiple foundations (such as piles or suction caissons) are being considered to support WTG (Wind Turbine Generators) for deeper water developments. This paper presents a practical method to compute the natural frequency of a jacket supporting WTG by incorporating Soil-Structure-Interaction (SSI) based on closed form solutions. The formulation presented can be easily programmed in a spreadsheet type program and can serve as a convenient way to obtain the natural frequency with the least amount of input. The basis of this method is the Euler-Bernoulli beam theory where foundations are idealized with a set of linear springs. In this method, a 3-Dimensional jacket is first converted into a 2-Dimensional problem along the orthogonal planes of vibration which are essentially the principal axes of the foundation geometry. Subsequently, the jacket is converted into an equivalent beam representing it's stiffness and a formulation is presented to find an equivalent beam for the entire tower-jacket system. Using energy methods, an equivalent mass of the RNA (Rotor Nacelle Assembly)-tower-jacket system is also calculated and the fixed base frequency of the jacket is estimated. To consider the flexibility effects of the foundation, a formulation for an equivalent rotational spring of the foundation is developed. A method to incorporate the mass of the transition piece is also presented. Finally, a step-by-step application of the methodology is presented by taking example problems from literature which also serves the purpose of validation and verification.

Geometrically Non-Linear Frequency Response of Axially Functionally Graded Beams Resting on Elastic Foundation Under Harmonic Excitation

This article presents geometrically nonlinear forced vibration analysis of an axially functionally graded (AFG) non-uniform beam resting on an elastic foundation. The mathematical formulation is displacement based and derivation of governing equations is accomplished following Hamilton's principle. The foundation has been mathematically incorporated into the analysis as a set of linear springs. According to the basic assumption of the present method force equilibrium condition is satisfied at a maximum excitation amplitude value. Thus, the dynamic problem is equivalently represented as a static one, which is solved by following a numerical implementation of the Broyden method. It is a method that utilizes the Jacobian matrix and subsequent correction of the initial Jacobian to solve a system of nonlinear equations. The large amplitude dynamic behaviour of the system in terms of non-dimensional frequency response curves is validated against established results and new results are furnished for a parabolic tapered AFG beam on a linear elastic foundation.

Vortex-induced vibration of floating circular cylinders with very low aspect ratio

Experiments regarding vortex-induced vibration (VIV) on floating circular cylinders with low aspect ratio were carried out in a recirculation water channel. The floating circular cylinders were elastically supported by a set of linear springs. Eight different aspect ratios were tested, namely L/D= 0.2, 0.3, 0.4, 0.5, 0.75, 1.0, 1.5 and 2.0. These aspect ratios were selected to cover the aspect ratio range of the main offshore circular platforms, such as spar and monocolumn. The aim was to understand the vortex-induced motions (VIM) of such platforms; due to this, the cylinders were floating, or m∗=1. The range of Reynolds number covered 2800 < Re < 55,400. The amplitude results showed a decrease in amplitude with decreasing aspect ratio in the in-line and in the transverse directions. The frequency results confirm a different behavior for cylinders with L/D≤0.5; in these cases, the cylinder free-end effects were predominant. The resonant behavior was no longer observed for L/D≤0.2. The decrease in Strouhal number with decreasing aspect ratio is also verified, as well as for drag and lift forces. The added mass results help to identify the end of resonance for both directions, in-line and transverse. The amplitude results for the vertical direction, roll, pitch and yaw did not affect the VIV behavior.

Buckling analysis of biaxially compressed double-layered graphene sheets with various boundary conditions based on nonlocal elasticity theory

This paper investigates the buckling behavior of biaxially compressed orthotropic double-layered graphene sheet (DLGS) embedded in an elastic medium based on nonlocal elasticity theory. The DLGS is modeled as a nonlocal double-layered plate which contains small scale effect and van der Waals interaction forces. The van der Waals interactions between the graphene layers are simulated as a set of linear springs using the Lennard-Jones potential model. Using the principle of virtual work, equilibrium equations are derived based on first order shear deformation theory and the nonlocal differential constitutive relations of Eringen. Differential quadrature method is employed to solve the governing equations for various combinations of free, simply supported or clamped boundary conditions. The present formulation and method of solution are validated by comparing the results, in the limit cases, with those available in the open literature. Finally, the effects of nonlocal parameter, geometric properties, elastic medium, compression ratio, mode numbers and boundary conditions are investigated on the buckling load of DLGS in detail. It is observed that in DLGSs with width of 20 nm and lengthes larger than 30 nm, the buckling load is independent of aspect ratios and boundary conditions.

Negative Poisson’s ratio in hollow sphere materials

This paper presents a study on the auxetic behaviour of the materials composed of hollow spheres. Although, as a newly developed material, the hollow sphere structures have gained a great deal of scientific attention, few studies have acknowledged its auxetic performance. Auxetic materials or materials with a negative Poisson’s ratio have been a subject of intensive research due to their unusual and, in many extents, superior mechanical properties. A number of different auxetic systems that have been developed commonly based on designing special geometry of the material microstructure. However, in the hollow sphere structures, this mechanism is essentially different and the auxetic effect is attributed to the peculiarities of deformation of a thin-wall hollow sphere and to the interaction between the spheres in the assembly. In this study it is shown that the negative Poisson’s ratio in this material is due to the high ratio between tangential to normal stiffnesses of a thin-wall hollow sphere. To evaluate the mechanical properties of these materials a homogenisation procedure based on the Cosserat 3D continuum theory is adopted. In this procedure, an assembly of the flexible thin-wall spheres is modelled as a stack of rigid spheres connected by a set of linear springs. The developed continuum model is validated against the direct finite element analysis of a regular assembly of the hollow spheres. Influence of the various design parameters on the macroscopic mechanical behaviour of the hollow sphere materials is investigated. The numerical study is focused on the auxetic behaviour of various kinds of packing arrangements of the hollow spheres.

Finite element buckling analysis of multi-layered graphene sheets on elastic substrate based on nonlocal elasticity theory

Graphene-polymer nano-composites are one of the most applicable engineering nanostructures with superior mechanical properties. In the present study, a finite element (FE) approach based on the size dependent nonlocal elasticity theory is developed for buckling analysis of nano-scaled multi-layered graphene sheets (MLGSs) embedded in polymer matrix. The van der Waals (vdW) interactions between the graphene layers and graphene-polymer are simulated as a set of linear springs using the Lennard-Jones potential model. The governing stability equations for nonlocal classical orthotropic plates together with the weighted residual formulation are employed to explicitly obtain stiffness and buckling matrices for a multi-layered super element of MLGS. The accuracy of the current finite element analysis (FEA) is approved through a comparison with molecular dynamics (MD) and analytical solutions available in the literature. Effects of nonlocal parameter, dimensions, vdW interactions, elastic foundation, mode numbers and boundary conditions on critical in-plane loads are investigated for different types of MLGS. It is found that buckling loads of MLGS are generally of two types namely In-Phase (INPH) and Out-of-Phase (OPH) loads. The INPH loads are independent of interlayer vdW interactions while the OPH loads depend on vdW interactions. It is seen that the decreasing effect of nonlocal parameter on the OPH buckling loads dwindles as the interlayer vdW interactions become stronger. Also, it is found that the small scale and polymer substrate have noticeable effects on the buckling loads of embedded MLGS.

On the equilibrium of a rigid body suspended by a set of linear springs

In this paper an approach is described for determining equilibrium states of a rigid body suspended elastically in space by a set of linear springs. This system is considered as a two-terminal generalized spring with terminal across (translational and rotational velocities, vG, ωG) and terminal through (terminal force and moment, fG, mG) variables. The algorithmic approach used for the solution of six nonlinear and coupled equilibrium equations consists of two major steps. The first step is to assign an initial orientation to the rigid body which is represented by the transformation (rotation) matrix T(θ, n) and reduce the problem to the solution of force equations only through a computer program. This yields the position vector ξ of a preselected point G on the rigid body. Although the terminal force fG becomes zero at this position, the calculated terminal moment mG, in general, is not equal to zero. The second step is to try to determine the correct orientation of the rigid body based on an argument that the terminal moment should vanish. The same argument is also used for the solution of force equilibrium equations. These two steps are repeated several times until both fG and mG vanish simultaneously yielding an equilibrium state (ξ, T(θ, n)). Application of the approach is illustrated through various examples. It is observed that, if there are nonstable equilibrium states of the system, then sometimes all possible physical equilibrium states may not be obtained with this approach. © 2000 John Wiley & Sons, Inc.

Efficient modelling and analysis for crankshaft three-dimensional vibrations under firing conditions

The three-dimensional vibrations of an engine crankshaft system under firing conditions were investigated by simple modelling and analysis using the dynamic stiffness matrix method (DSMM). In the analyses, the authors took account of the dynamic behaviour of cylinder block, oil film and torsional damper. To simplify the analyses, the crankshaft was idealized by a set of jointed structures consisting of simple round rods and simple beam blocks of rectangular cross-section. The main journal bearings were idealized by a set of linear springs and dash-pots, and the flywheel was modelled by finite elements. Then the dynamic stiffness matrix (DSM) was derived in closed form for each constituent member. To eliminate the complicated finite element method (FEM) analysis for a cylinder block of complicated structure [1,2], the authors derived the dynamic stiffness matrix from the inverse matrix of the compliance matrix. Here, the compliance matrix was derived in analytical form from the modal parameters obtained from a series of hammering tests. Finally, the dynamic stiffness matrix was constructed for the total engine system consisting of the crankshaft system and the cylinder block. The three-dimensional vibrations of the crankshaft system and cylinder block surface vibrations near the main bearings under firing conditions were calculated for an automobile diesel engine in which five kinds of solid pulley, each with different masses and moments of inertia, and a torsional damper were attached to the crankshaft. The calculated results were compared with the experimental results.

Influence of the Cylinder Block Dynamic Stiffness and the Crankshaft Damper Characteristics on the Three-Dimensional Vibrations of the Crankshaft-Systems.

Three-dimensional vibrations of the crankshaft system under running conditions were inves- tigated for an automobile engine, taking account of the dynamic behavior of cylinder block and torsional damper. To simplify the analyses, the crankshaft was idealized by a set of jointed structures consisting of simple round rods and simple beam blocks of rectangular cross-section. The main journal bearings were modeled by a set of linear springs and dash-pots. For the flywheel, the dynamic stiffness matrix was derived from FEM. For the cylinder block, the dynamic stiffness matrix was constructed from the modal parameters obtained from the experimental modal analysis because of the complicated structure. Then the dynamic stiffness matrix was constructed for the crankshaft-cylinder block system, by the principle of superposition. The three-dimensional vibra- tions of the crankshaft system were analyzed for the two cases in which a solid pulley and a crankshaft damper were attached to the crankshaft. The calculated results were compared with the experimental results. In the analysis, influence of the vibration amplitude and temperature of damper on the damper characteristics were taken into account. The influences of the dynamic stiffness of the crankshaft thrust bearings and the oil film on the vibrations behavior of the crankshaft system were also investigated.

Deformation of flexible anisotropic elastic systems of shells of revolution on a nonlinear foundation

UDC 539.3 The problem of the interaction of the elastic elements of structures and their foundation is of central importance in the design of machines, space facilities, and engineering structures. A large number of studies have analyzed the work of structures without allowance for the elastic foundation. There have also been numerous investigations that have made allowance for same. However, most of them have used linear models both for the shells (the structural elements) and the foundation. For the latter, the model has usually been the classical Winkler model. The Winkler model presumes that the intensity of the reaction of the foundation is proportional to the deformation in the structure. The foundation is represented as a set of linear springs without mass that are deformed only in the normal direction [1, 2]. Also, researchers generally examine isotropic or orthotropic plates and cylindrical or spherical shells. At the same time, experience shows that in the case of large elastic displacements, the interaction of the structure with the foundation is described by a nonlinear function. This article analyzes the interaction of anisotropic shell systems with a nonlinear elastic foundation on the basis of the geometrically nonlinear theory of thin shells. We will examine the stress-strain state of elastic systems composed of a series of coaxial closed mnltilayered shells of revolution. The shells themselves are made up of an arbitrary number of anisotropic layers having a thickness that varies along the meridian. There is one plane of elastic symmetry at each point of the shell, this plane being parallel to its coordinate surface. The elastic characteristics do not change along the parallel. The elastic system is subjected to surface and contour loads such that it deforms while retaining its axial symmetry. We also assume that the system interacts with the elastic foundation without slip or separation and that the intensity of the reaction of the foundation is determined by the formula

Simple Modeling and Analysis for Crankshaft Three-Dimensional Vibrations, Part 1: Background and Application to Free Vibrations

To simplify the analysis of the three-dimensional vibrations of automobile engine crankshafts under firing conditions, the crankshaft was idealized by a set of jointed structures consisting of simple round rods and simple beam blocks of rectangular cross-section. The front pulley, timing gear, and the fly-wheel were idealized by a set of masses and moments of inertia. The main journal bearings were idealized by a set of linear springs and dash-pots. For each constituent member, the dynamic stiffness matrix was derived (in closed form) from the transfer matrix. Then the dynamic stiffness matrix for the total crankshaft system was constructed, and the natural frequencies and mode shapes were calculated. The modeling and analysis procedures were applied to the analysis of free vibrations of four kinds of crankshafts: single cylinder, three-cylinder in-line, four-cylinder in-line, and V-six engines. The different coupling behavior of the three-dimensional vibrations in the planar-structure and the solid-structure crankshaft is discussed, and the influence of the bearing oil film stiffness on the crankshaft natural frequency is also analyzed.