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.

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 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.

An Exact Three-Dimensional Beam Element With Nonuniform Cross Section

2010 ◽  
Vol 77 (6) ◽  
Author(s):  
M. Jafari ◽  
M. J. Mahjoob
Keyword(s):  
Cross Section ◽  
Stiffness Matrix ◽  
Degrees Of Freedom ◽  
Direct Method ◽  
Three Dimensional ◽  
Curved Beams ◽  
Element Analysis ◽  
Shear Loads ◽  
Line Passing ◽  
Exact Stiffness Matrix

In this paper, the exact stiffness matrix of curved beams with nonuniform cross section is derived using direct method. The considered element has two nodes and 12 degrees of freedom, with three forces and three moments applied at each node. The noncoincidence effect of shear center and center of area is also considered in this element. The deformations of the beam are due to bending, torsion, tensile, and shear loads. The line passing through center of area is a general three-dimensional curve and the cross section properties may change arbitrarily along it. The method is extended to deal with distributed loads on the curved beams. The stiffness matrix of some selected types of beams is determined by this method. The results are compared (where possible) with previously published results, simple beam finite element analysis and analytic solution. It is shown that the determined stiffness matrix is exact and that any type of beam can be analyzed by this method.

scholarly journals A Novel Method for Tooth Bending Stress Calculation of Gears With Asymmetric Teeth

2021 ◽  
Author(s):  
Oguz DOGAN ◽  
Celalettin YUCE ◽  
Fatih KARPAT
Keyword(s):  
Finite Element ◽  
Finite Element Model ◽  
Experimental Studies ◽  
Element Model ◽  
Bending Stress ◽  
Element Analysis ◽  
Stress Calculation ◽  
Bending Stresses ◽  
New Equation ◽  
Asymmetric Factor

Abstract Today, gear designs with asymmetric tooth profiles offer essential solutions in reducing tooth root stresses of gears. Although numerical, analytical, and experimental studies are carried out to calculate the bending stresses in gears with asymmetric tooth profiles a standard or a simplified equation or empirical statement has not been encountered in the literature. In this study, a novel bending stress calculation procedure for gears with asymmetric tooth profiles is developed using both the DIN3990 standard and the finite element method. The bending stresses of gears with symmetrical profile were determined by the developed finite element model and was verified by comparing the results with the DIN 3990 standard. Using the verified finite element model, by changing the drive side pressure angle between 20° and 30° and the number of teeth between 18 and 100, 66 different cases were examined and the bending stresses in gears with asymmetric profile were determined. As a result of the analysis, a new asymmetric factor was derived. By adding the obtained asymmetric factor to the DIN 3390 formula, a new equation has been derived to be used in tooth bending stresses of gears with asymmetric profile. Thanks to this equation, designers will be able to calculate tooth bending stresses with high precision in gears with asymmetric tooth profile without the need for finite element analysis.

scholarly journals Topology Optimization of Spiral Bevel Gear for Differential

2020 ◽  
Vol 9 (3) ◽  
pp. 1205-1209
Keyword(s):  
Taguchi Method ◽  
Optimal Parameter ◽  
Bevel Gear ◽  
Spiral Bevel Gear ◽  
Element Analysis ◽  
Differential Gear ◽  
Bending Stresses ◽  
Spiral Bevel ◽  
Time Required ◽  
Sufficient Strength

The Spiral Bevel gear used in differential should be enough stiff to resist the vibrations and stresses encountered during its operation. The gear must also have sufficient strength to bear the bending stresses occurring in the differential assembly in its course of operation. This research is typically focused in designing a differential gear with least weight and minimal stresses. The model of the gear is designed in the Solidworks version 2015 while its analysis is carried in ANSYS 14.5. The number of parameters and levels involved in designing are more; the number of probable models is too many. To choose the optimal parameter among the list of choices, TAGUCHI method along with Finite Element Analysis (FEA) is used. By application of TAGUCHI method, not only the time required to design all the probable models is reduced, but also the time required to analyze all the models is cut down. Orthogonal Array has been incorporated to change the parameters necessary for reducing the weight of the gear. To get the best possible model of gear, FEA is then performed on the designed models. This process not only saves production time, but also prevents material wastage and production cost.

Finite Element Analysis for a CAD of a Concrete Mixer Drum

1994 ◽  
Author(s):  
Hossam S. Badawi ◽  
Sherif A. Mourad ◽  
Sayed M. Metwalli
Keyword(s):  
Finite Element ◽  
Computer Aided Design ◽  
Three Dimensional ◽  
Plain Concrete ◽  
Element Analysis ◽  
Shell Elements ◽  
Bending Stresses ◽  
Loading Effects ◽  
Concrete Mixer ◽  
Aided Design

Abstract For a Computer Aided Design of a concrete truck mixer, a six cubic meter concrete mixer drum is analyzed using the finite element method. The complex mixer drum structure is subjected to pressure loading resulting from the plain concrete inside the drum, in addition to its own weight. The effect of deceleration of the vehicle and the rotational motion of the drum on the reactions and stresses are also considered. Equivalent static loads are used to represent the dynamic loading effects. Three-dimensional shell elements are used to model the drum, and frame elements are used to represent a ring stiffener around the shell. Membrane forces and bending stresses are obtained for different loading conditions. Results are also compared with approximate analysis. The CAD procedure directly used the available drafting and the results were used effectively in the design of the concrete mixer drum.

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.

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