Generalized Synthesis of Nonlinear Springs for Prescribed Load-Displacement Functions

2006 ◽  
Author(s):  
Christine M. Vehar ◽  
Sridhar Kota
Keyword(s):  
Shape Function ◽  
General Scheme ◽  
Stress Constraints ◽  
Displacement Function ◽  
Algorithm Optimization ◽  
Load Range ◽  
Element Analysis ◽  
Nonlinear Load ◽  
Nonlinear Springs ◽  
Load Displacement

A spring’s nonlinear load-displacement function is described by three factors, the (i) shape function, (ii) load-range, and (iii) displacement-range. The shape function encompasses the nonlinear relationship between the load and displacement, and therefore, is the most difficult factor to match. In this paper, we present a general scheme for topology, size, and shape optimization of nonlinear springs for prescribed load–displacement shape functions, while simultaneously meeting manufacturing, space, and stress constraints. This paper presents the objective function and a novel, floating point parametric model used within a genetic algorithm optimization scheme. The nonlinear springs all undergo large deformations and are evaluated by nonlinear finite element analysis. Two examples are included to demonstrate the effectiveness of the methodology in synthesizing nonlinear springs that match a prescribed load-displacement shape function.

Design of Nonlinear Springs for Prescribed Load-Displacement Functions

2008 ◽  
Vol 130 (8) ◽  
Author(s):  
Christine Vehar Jutte ◽  
Sridhar Kota
Keyword(s):  
Displacement Function ◽  
Curved Beams ◽  
Element Analysis ◽  
Nonlinear Load ◽  
Nonlinear Spring ◽  
Nonlinear Springs ◽  
Bending Stresses ◽  
Bending Loads ◽  
Load Displacement ◽  
Synthesis Methodology

A nonlinear spring has a defined nonlinear load-displacement function, which is also equivalent to its strain energy absorption rate. Various applications benefit from nonlinear springs, including prosthetics and microelectromechanical system devices. Since each nonlinear spring application requires a unique load-displacement function, spring configurations must be custom designed, and no generalized design methodology exists. In this paper, we present a generalized nonlinear spring synthesis methodology that (i) synthesizes a spring for any prescribed nonlinear load-displacement function and (ii) generates designs having distributed compliance. We introduce a design parametrization that is conducive to geometric nonlinearities, enabling individual beam segments to vary their effective stiffness as the spring deforms. Key features of our method include (i) a branching network of compliant beams used for topology synthesis rather than a ground structure or a continuum model based design parametrization, (ii) curved beams without sudden changes in cross section, offering a more even stress distribution, and (iii) boundary conditions that impose both axial and bending loads on the compliant members and enable large rotations while minimizing bending stresses. To generate nonlinear spring designs, the design parametrization is implemented into a genetic algorithm, and the objective function evaluates spring designs based on the prescribed load-displacement function. The designs are analyzed using nonlinear finite element analysis. Three nonlinear spring examples are presented. Each has a unique prescribed load-displacement function, including a (i) “J-shaped,” (ii) “S-shaped,” and (iii) constant-force function. A fourth example reveals the methodology’s versatility by generating a large displacement linear spring. The results demonstrate the effectiveness of this generalized synthesis methodology for designing nonlinear springs for any given load-displacement function.

Design of Single, Multiple, and Scaled Nonlinear Springs for Prescribed Nonlinear Responses

2009 ◽  
Vol 132 (1) ◽  
Author(s):  
Christine Vehar Jutte ◽  
Sridhar Kota
Keyword(s):  
Microelectromechanical Systems ◽  
Solution Space ◽  
Nonlinear Behavior ◽  
Stress Constraints ◽  
Displacement Function ◽  
Load Range ◽  
Vibration Absorption ◽  
Nonlinear Springs ◽  
Nonlinear Responses ◽  
In Series

Nonlinear springs enhance the performance of many applications including prosthetics, microelectromechanical systems devices, and vibration absorption systems. This paper describes a comprehensive approach to developing compliant elements of prescribed nonlinear stiffness. It presents a generalized methodology for designing a single planar nonlinear spring for a prescribed load-displacement function. The spring’s load-range, displacement-range, and nonlinear behavior are matched using this methodology, while also addressing stress, material, stability, and space constraints. Scaling guidelines are included within the optimization to relax the constraints on the solution space. Given the nonlinear nature of the spring designs, this paper further investigates their function in new configurations. Compliant structures with customized elastic properties are constructed by exploiting symmetry and by arranging nonlinear springs in series and/or in parallel. Scaling guidelines are used to meet new design specifications. The guidelines allow adjustment of load-range, displacement-range, material, and the overall footprint while preserving the spring’s nonlinear behavior without violating stress constraints. Various examples are provided throughout the paper to demonstrate the implementation and merit of these design approaches.

Design of Planar Nonlinear Springs for Prescribed Load-Displacement Functions

2007 ◽  
Author(s):  
Christine V. Jutte ◽  
Sridhar Kota
Keyword(s):  
Parametric Model ◽  
Nonlinear Finite Element ◽  
Algorithm Optimization ◽  
Optimization Scheme ◽  
Element Analysis ◽  
Single Plane ◽  
Nonlinear Spring ◽  
Vehicle Suspensions ◽  
Nonlinear Springs ◽  
Load Displacement

Nonlinear springs can simplify and improve the performance of a variety of devices, including prosthetics, MEMS, and vehicle suspensions. Each nonlinear spring application has unique load-displacement specifications that do not correspond to one general spring design. This limits the use of nonlinear springs and thus compromises the performance of these applications. This paper presents a generalized methodology, including topology, size, and shape optimization, for creating nonlinear springs with prescribed load-displacement functions. The methodology includes a new parametric model that represents nonlinear springs as a single-plane, ‘fractal’-like network of splines. The parametric model and the objective function are incorporated into a genetic algorithm optimization scheme. Nonlinear finite element analysis evaluates the large displacements of each spring design. Three nonlinear spring examples, each having uniquely prescribed load-displacement functions including a “J”-shaped, an “S”-shaped, and a constant-force function, generate designs that demonstrate the methodology’s effectiveness in designing nonlinear springs.

A Closed-Form Nonlinear Model for the Constraint Characteristics of Symmetric Spatial Beams

2013 ◽  
Vol 135 (3) ◽  
Author(s):  
Shiladitya Sen ◽  
Shorya Awtar
Keyword(s):  
Closed Form ◽  
Virtual Work ◽  
Practical Interest ◽  
Element Analysis ◽  
Nonlinear Load ◽  
Flexure Mechanism ◽  
Spatial Beams ◽  
Spatial Beam ◽  
Load Displacement ◽  
Constraint Model

The constraint-based design of flexure mechanisms requires a qualitative and quantitative understanding of the constraint characteristics of flexure elements that serve as constraints. This paper presents the constraint characterization of a uniform and symmetric cross-section, slender, spatial beam—a basic flexure element commonly used in three-dimensional flexure mechanisms. The constraint characteristics of interest, namely stiffness and error motions, are determined from the nonlinear load–displacement relations at the beam end. Appropriate assumptions are made while formulating the strain and strain energy expressions for the spatial beam to retain relevant geometric nonlinearities. Using the principle of virtual work, nonlinear beam governing equations are derived and subsequently solved for general end loads. The resulting nonlinear load–displacement relations capture the constraint characteristics of the spatial beam in a compact, closed-form, and parametric manner. This constraint model is shown to be accurate using nonlinear finite element analysis, within a load and displacement range of practical interest. The utility of this model lies in the physical and analytical insight that it offers into the constraint behavior of a spatial beam flexure, its use in design and optimization of 3D flexure mechanism geometries, and its elucidation of fundamental performance tradeoffs in flexure mechanism design.

Geometric Design of a Slotted Disc Spring for a Prescribed Load-Displacement Function

2010 ◽  
Author(s):  
Noor Fawazi ◽  
Ji-Hyun Yoon ◽  
Jae-Eung Oh ◽  
Jung-Youn Lee
Keyword(s):  
Geometric Parameter ◽  
Displacement Function ◽  
Geometric Design ◽  
Geometric Parameters ◽  
Parameter Design ◽  
Target Point ◽  
Nonlinear Load ◽  
Design Algorithm ◽  
Load Displacement ◽  
Disc Spring

Geometric parameter design is an important stage in any product design. For example, by varying any of its geometric parameters, a slotted disc spring will show various defined nonlinear load-displacement behaviors. Therefore, these geometric parameters must be precisely designed to ensure the output spring design possesses a nonlinear load-displacement behavior that satisfies particular nonlinear criteria. More importantly, various engineering designs benefit from nonlinear behavior in order to meet certain engineering design requirements. Since each nonlinear spring application requires a unique load-displacement function, the spring geometric parameters must be precisely custom designed. However, there is no specific algorithm available to calculate such geometric design parameterization. The aim of this study is to propose a generalized algorithm for a slotted disc spring geometric design that ensures the output design exhibits identical load-displacement function with any prescribed one. A predicted geometric design algorithm for a slotted disc spring is proposed in this study. The design is characterized by a prescribed load-displacement function obtained from numerical model in the previous literature. The key feature of our proposed algorithm is that, the identified meeting point, which is defined from a prescribed function, can be used as a target point to match the predicted function with the prescribed function. Our proposed algorithm manipulates the slope characteristics of the established slotted disc spring numerical formulation to tune the predicted nonlinear function. This enables a geometric parameter design to be achieved. Improvements to the proposed geometric parameters were done by searching the best combination of optimum variables that produce minimum least mean square error between the prescribed and proposed nonlinear functions. The obtained numerical results demonstrate the effectiveness of the proposed algorithm to parameterize the geometric parameters for a slotted disc spring design.

scholarly journals Prediction of Vehicle Crashworthiness Parameters Using Piecewise Lumped Parameters and Finite Element Models

Designs ◽  
2018 ◽  
Vol 2 (4) ◽  
pp. 43 ◽  
Author(s):  
Bernard B. Munyazikwiye ◽  
Dmitry Vysochinskiy ◽  
Mikhail Khadyko ◽  
Kjell G. Robbersmyr
Keyword(s):  
Finite Element ◽  
Severity Index ◽  
Finite Element Models ◽  
Element Analysis ◽  
Computational Costs ◽  
Nonlinear Springs ◽  
Crash Testing ◽  
Lumped Parameters ◽  
Crashworthiness Analysis ◽  
Vehicle Crashworthiness

Estimating the vehicle crashworthiness experimentally is expensive and time-consuming. For these reasons, different modelling approaches are utilised to predict the vehicle behaviour and reduce the need for full-scale crash testing. The earlier numerical methods used for vehicle crashworthiness analysis were based on the use of lumped parameters models (LPM), a combination of masses and nonlinear springs interconnected in various configurations. Nowadays, the explicit nonlinear finite element analysis (FEA) is probably the most widely recognised modelling technique. Although informative, finite element models (FEM) of vehicle crash are expensive both in terms of man-hours put into assembling the model and related computational costs. A simpler analytical tool for preliminary analysis of vehicle crashworthiness could greatly assist the modelling and save time. In this paper, the authors investigate whether a simple piecewise LPM can serve as such a tool. The model is first calibrated at an impact velocity of 56 km/h. After the calibration, the LPM is applied to a range of velocities (40, 48, 64 and 72 km/h) and the crashworthiness parameters such as the acceleration severity index (ASI) and the maximum dynamic crush are calculated. The predictions for crashworthiness parameters from the LPM are then compared with the same predictions from the FEA.

scholarly journals Computation of Deflections for PC Box Girder Bridges with Corrugated Steel Webs considering the Effects of Shear Lag and Shear Deformation

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Wei Ji ◽  
Kui Luo ◽  
Jingwei Zhang
Keyword(s):  
Shear Deformation ◽  
Prestressed Concrete ◽  
Displacement Function ◽  
Shear Lag ◽  
Longitudinal Displacement ◽  
Box Girders ◽  
Element Analysis ◽  
Box Girder ◽  
Bridge Model ◽  
Bridge Models

Prestressed concrete (PC) girders with corrugated steel webs (CSWs) have received considerable attention in the past two decades due to their light self-weight and high prestressing efficiency. Most previous studies were focused on the static behavior of CSWs and simple beams with CSWs. The calculation of deflection is an important part in the static analysis of structures. However, very few studies have been conducted to investigate the deflection of full PC girders or bridges with CSWs and no simple formulas are available for estimating their deflection under static loads. In addition, experimental work on full-scale bridges or scale bridge models with CSWs is very limited. In this paper, a formula for calculating the deflection of PC box girders with CSWs is derived. The longitudinal displacement function of PC box girders with CSWs, which can consider the shear lag effect and shear deformation of CSWs, is first derived. Based on the longitudinal displacement function, the formula for predicting the deflection of PC box girders with CSWs is derived using the variational principle method. The accuracy of the derived formula is verified against experimental results from a scaled bridge model and the finite element analysis results. Parametric studies are also performed, and the influences of shear lag and shear deformation on the deflection of the box girder with CSWs are investigated by considering different width-to-span ratios and different girder heights. The present study provides an effective and efficient tool for determining the deflection of PC box girders with CSWs.

Estimation of transverse tensile behavior of Zircaloy pressure tubes using ring-tensile test and finite element analysis

2012 ◽  
Vol 227 (6) ◽  
pp. 1177-1186 ◽  
Author(s):  
MK Samal ◽  
KS Balakrishnan ◽  
J Parashar ◽  
GP Tiwari ◽  
S Anantharaman
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Tensile Tests ◽  
Tensile Specimen ◽  
Displacement Curve ◽  
Stress Strain ◽  
Element Analysis ◽  
Transverse Tensile ◽  
Pressure Tubes ◽  
Load Displacement

Determination of transverse mechanical properties from the ring type of specimens directly machined from the nuclear reactor pressure tubes is not straightforward. It is due to the presence of combined membrane as well as bending stresses arising in the loaded condition because of the curvature of the specimen. These tubes are manufactured through a complicated process of pilgering and heat treatment and hence, the transverse properties need to be determined in the as-manufactured condition. It may not also be possible to machine small miniaturized specimen in the circumferential direction especially in the irradiated condition. In this work, we have performed ring-tensile tests on the un-irradiated ring tensile specimen using two split semi-cylindrical mandrels as the loading device. A three-dimensional finite element analysis was performed in order to determine the material true stress–strain curve by comparing experimental load–displacement data with those predicted by finite element analysis. In order to validate the methodology, miniaturized tensile specimens were machined from these tubes and tested. It was observed that the stress–strain data as obtained from ring tensile specimen could describe the load–displacement curve of the miniaturized flat tensile specimen very well. However, it was noted that the engineering stress–strain as directly obtained from the experimental load–displacement curves of the ring tensile tests were very different from that of the miniaturized specimen. This important aspect has been resolved in this work through the use of an innovative type of 3-piece loading mandrel.

Effect of Fixed Axes of Rotation on the Varus-Valgus and Torsional Load-Displacement Characteristics of the In-Vitro Human Knee

1979 ◽  
Vol 101 (2) ◽  
pp. 134-140 ◽  
Author(s):  
J. Rastegar ◽  
R. L. Piziali ◽  
D. A. Nagel ◽  
D. J. Schurman
Keyword(s):  
Least Squares ◽  
Cubic Spline ◽  
Nonlinear Load ◽  
Human Knee ◽  
Torsional Load ◽  
Axis Of Rotation ◽  
Second Derivatives ◽  
Load Displacement ◽  
Coupled Loads

The effects of fixed axes of rotation on the varus-valgus and torsional load-displacement characteristics of the human knee have been determined. The location of the axes of varus-valgus and torsional rotations resulting in minimum resisting loads are also determined, and it is shown that they correspond to minimal coupled load levels. The coupled loads are seen to be sensitive to the location of the axis of rotation. The nonlinear load-displacement data is fitted with a four interval least-squares cubic spline with matching first and second derivatives at nodes. The data from two fresh human knees are presented.

scholarly journals Compressive Creep Measurements of Fired Magnesia Bricks at Elevated Temperatures Including Creep Law Parameter Identification and Evaluation by Finite Element Analysis

Ceramics ◽  
2020 ◽  
Vol 3 (2) ◽  
pp. 210-222 ◽  
Author(s):  
Guenter Unterreiter ◽  
Daniel R. Kreuzer ◽  
Bernd Lorenzoni ◽  
Hans U. Marschall ◽  
Christoph Wagner ◽  
...  
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Creep Behavior ◽  
Elevated Temperatures ◽  
Cell Model ◽  
Experimental Investigations ◽  
Load Range ◽  
Element Analysis ◽  
Compressive Creep ◽  
Creep Deformations

Creep behavior is very important for the selection of refractory materials. This paper presents a methodology to measure the compressive creep behavior of fired magnesia materials at elevated temperatures. The measurements were carried out at 1150–1500 °C and under compression loads from 1–8 MPa. Creep strain was calculated from the measured total strain data. The obtained creep deformations of the experimental investigations were subjected to detailed analysis to identify the Norton-Bailey creep law parameters. The modulus of elasticity was determined in advance to simplify the inverse estimation process for finding the Norton-Bailey creep parameters. In the next step; an extended material model including creep was used in a finite element analysis (FEA) and the creep testing procedure was reproduced numerically. Within the investigated temperature and load range; the creep deformations calculated by FEA demonstrated a good agreement with the results of the experimental investigations. Finally; a finite element unit cell model of a quarter brick representing a section of the lining of a ferrochrome (FeCr) electric arc furnace (direct current) was used to assess the thermo-mechanical stresses and strains including creep during a heat-up procedure. The implementation of the creep behavior into the design process led to an improved prediction of strains and stresses.

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