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.

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.

The Effect of Two Rigid Spherical Inclusions on the Stresses in an Infinite Elastic Solid

1966 ◽  
Vol 33 (1) ◽  
pp. 68-74 ◽  
Author(s):  
Joseph F. Shelley ◽  
Yi-Yuan Yu
Keyword(s):  
Stress Field ◽  
Uniaxial Tension ◽  
Elastic Solid ◽  
Displacement Function ◽  
Hydrostatic Tension ◽  
Series Form ◽  
Spherical Inclusions ◽  
Spherical Cavities ◽  
In Series ◽  
The Common

Presented in this paper is a solution in series form for the stresses in an infinite elastic solid which contains two rigid spherical inclusions of the same size. The stress field at infinity is assumed to be either hydrostatic tension or uniaxial tension in the direction of the common axis of the inclusions. The solution is based upon the Papkovich-Boussinesq displacement-function approach and makes use of the spherical dipolar harmonics developed by Sternberg and Sadowsky. The problem is closely related to, but turns out to be much more involved than, the corresponding problem of two spherical cavities solved by these authors.

Nonlinear Dynamics of an Imperfect Microbeam Under an Axial Load and Electric Excitation

2011 ◽  
Author(s):  
Laura Ruzziconi ◽  
Mohammad I. Younis ◽  
Stefano Lenci
Keyword(s):  
Nonlinear Dynamics ◽  
Microelectromechanical Systems ◽  
Complex Dynamics ◽  
Nonlinear Behavior ◽  
Single Mode ◽  
Phase Portraits ◽  
Nonlinear Phenomena ◽  
Theoretical Point ◽  
Initial Shape ◽  
Electric Excitation

This study is motivated by the growing attention, both from a practical and a theoretical point of view, toward the nonlinear behavior of microelectromechanical systems (MEMS). We analyze the nonlinear dynamics of an imperfect microbeam under an axial force and electric excitation. The imperfection of the microbeam, typically due to microfabrication processes, is simulated assuming the microbeam to be of a shallow arched initial shape. The device has a bistable static behavior. The aim is that of illustrating the nonlinear phenomena, which arise due to the coupling of mechanical and electrical nonlinearities, and discussing their usefulness for the engineering design of the microstructure. We derive a single-mode-reduced-order model by combining the classical Galerkin technique and the Pade´ approximation. Despite its apparent simplicity, this model is able to capture the main features of the complex dynamics of the device. Extensive numerical simulations are performed using frequency response diagrams, attractor-basins phase portraits, and frequency-dynamic voltage behavior charts. We investigate the overall scenario, up to the inevitable escape, obtaining the theoretical boundaries of appearance and disappearance of the main attractors. The main features of the nonlinear dynamics are discussed, stressing their existence and their practical relevance. We focus on the coexistence of robust attractors, which leads to a considerable versatility of behavior. This is a very attractive feature in MEMS applications. The ranges of coexistence are analyzed in detail, remarkably at high values of the dynamic excitation, where the penetration of the escape (dynamic pull-in) inside the double well may prevent the safe jump between the attractors.

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.

Compensation for the Residual Error of the Voltage Drive of the Charge Control of a Piezoelectric Actuator

2018 ◽  
Vol 140 (7) ◽  
Author(s):  
Shih-Tang Liu ◽  
Jia-Yush Yen ◽  
Fu-Cheng Wang
Keyword(s):  
Piezoelectric Actuator ◽  
Voltage Drop ◽  
Tracking Error ◽  
Nonlinear Behavior ◽  
Piezo Actuator ◽  
Charge Control ◽  
In Series ◽  
Series Of Experiments ◽  
The One ◽  
Positional Control

One very effective approach to suppress hysteresis from the piezoelectric actuator is to use the charge control across the associated capacitance. The charge driver often uses an additional capacitor connected to the piezo-actuator in series for the charge sense feedback control. When this charge sense is used with a voltage drive for the charge control, the applied voltage will include two parts. The one is the voltage drop across the useful electro-mechanical part and effectively converted to the driving force, whereas the other part indicates the equivalent voltage drop due to the hysteresis. In our research, we noticed that it is possible to use a simple estimator as the hysteresis voltage observer and use it to precompensate for the voltage drop. Comparing to the conventional hysteresis suppression achieved by the closed-loop positional control, we show significant improvement of the control performance. For dynamic applications, we also proposed a combination of the Preisach model with the hysteresis estimator to better suppress the nonlinear behavior. A series of experiments were conducted to demonstrate the performance improvement of the proposed compensator. A 10 nm and 25 nm maximum tracking error can be maintained while tracking a staircase reference and a 30 Hz sinusoidal signal, respectively.

The Influence of Nonlinear Boundary Conditions on the Nonplanar Autoparametric Responses of an Inextensible Beam

2001 ◽  
Author(s):  
Haider N. Arafat ◽  
Ali H. Nayfeh
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Nonlinear Behavior ◽  
Nonlinear Boundary Conditions ◽  
Nonlinear Springs ◽  
Out Of Plane

Abstract The nonplanar responses of a beam clamped at one end and restrained by nonlinear springs at the other end is investigated under a primary resonance base excitation. The beam’s geometry and the springs’ linear stiffnesses are such that the system possesses a one-to-one autoparametric resonance between the nth in-plane and out-of-plane modes. The beam is modeled using Euler-Bernoulli theory and includes cubic geometric and inertia nonlinearities. The objective is to assess the influence of the nonlinear boundary conditions on the beam’s oscillations. To this end, the method of multiple scales is directly applied to the integral-partial-differential equations of motion and associated boundary conditions. The result is a set of four nonlinear ordinary-differential equations that govern the slow dynamics of the system. Solutions of these modulation equations are then used to characterize the system’s nonlinear behavior.

A review of the most important failure, reliability and nonlinearity aspects in the development of microelectromechanical systems (MEMS)

2017 ◽  
Vol 34 (1) ◽  
pp. 9-21 ◽  
Author(s):  
Peyman Rafiee ◽  
Golta Khatibi ◽  
Michael Zehetbauer
Keyword(s):  
Dynamic Response ◽  
Frequency Shift ◽  
Mode Coupling ◽  
Microelectromechanical Systems ◽  
Nonlinear Behavior ◽  
Nonlinear Phenomena ◽  
Mems Devices ◽  
Mechanical Loads ◽  
Content Type ◽  
Proper Operation

Purpose The purpose of this paper is to provide an overview of the major reliability issues of microelectromechanical systems (MEMS) under mechanical and environmental loading conditions. Furthermore, a comprehensive study on the nonlinear behavior of silicon MEMS devices is presented and different aspects of this phenomenon are discussed. Design/methodology/approach Regarding the reliability investigations, the most important failure aspects affecting the proper operation of the MEMS components with focus on those caused by environmental and mechanical loads are reviewed. These studies include failures due to fatigue loads, mechanical vibration, mechanical shock, humidity, temperature and particulate contamination. In addition, the influence of squeeze film air damping on the dynamic response of MEMS devices is briefly discussed. A further subject of this paper is discussion of studies on the nonlinearity of silicon MEMS. For this purpose, after a description of the basic principles of nonlinearity, the consequences of nonlinear phenomena such as frequency shift, hysteresis and harmonic generation and their effects on the device performance are reviewed. Special attention is paid to the mode coupling effect between the resonant modes as a result of energy transfer because of the nonlinearity of silicon. For a better understanding of these effects, the nonlinear behavior of silicon is demonstrated by using the example of Si cantilever beams. Findings It is shown that environmental and mechanical loads can influence on proper operation of the MEMS components and lead to early fracture. In addition, it is demonstrated that nonlinearity modifies dynamic response and leads to new phenomena such as frequency shift and mode coupling. Finally, some ideas are given as possible future areas of research works. Originality/value This is a review paper and aimed to review the latest manuscripts published in the field of reliability and nonlinearity of the MEMS structures.

SECOND-ORDER ANALYSIS AND DESIGN OF CABLES AND CABLE-FRAMES

2005 ◽  
Vol 05 (04) ◽  
pp. 521-537
Author(s):  
GANPING SHU ◽  
SIU LAI CHAN ◽  
ZHITAO LÜ
Keyword(s):  
Integrated Design ◽  
Nonlinear Behavior ◽  
Initial Imperfection ◽  
Displacement Function ◽  
Tangent Stiffness Matrix ◽  
Analysis And Design ◽  
Order Polynomial ◽  
Analysis Computer ◽  
Analysis Computer Program ◽  
Fourth Order Polynomial

Cable structures are lightweighted, simple to fabricate and reusable. They provide effective solutions for large-span structures. Analysis of cables is complex because of their highly geometrically nonlinear behavior. Based on the Lagrangian formulation and a fourth-order polynomial displacement function, the tangent stiffness matrix for a five-node curved cable element is derived and statically condensed to a simple form readily for incorporation into a frame analysis computer program. The program uses the pointwise–equilibrium–polynomial (PEP) element with initial imperfection and the "Nonlinear Integrated Design and Analysis (NIDA)" method for design and nonlinear analysis of cabled structures. Numerical examples demonstrate the robustness and practicality of the proposed method.

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