Modeling of Initially Curved Beam Structures for Design of Multistable MEMS

2011 ◽  
Vol 79 (1) ◽  
Author(s):  
Matthew D. Williams ◽  
Fred van Keulen ◽  
Mark Sheplak
Keyword(s):  
Microelectromechanical Systems ◽  
Practical Importance ◽  
Curved Beam ◽  
Normal Flow ◽  
Simultaneous Solution ◽  
Solution Approach ◽  
Loading Pattern ◽  
Equilibrium Configurations ◽  
Flow Algorithm ◽  
Nonlinear Equilibrium

This article describes a mathematical model and two solution methodologies for efficiently predicting the equilibrium paths of an arbitrarily shaped, precurved, clamped beam. Such structures are common among multistable microelectromechanical systems (MEMS). First, a novel polynomial-based solution approach enables simultaneous solution of all equilibrium configurations associated with an arbitrary mechanical loading pattern. Second, the normal flow algorithm is used to negotiate the particularly complex nonlinear equilibrium paths associated with electrostatic loading and is shown to perform exceptionally well. Overall, the techniques presented herein provide designers with general and efficient computational frameworks for studying the effects of loading, shape, and imperfections on beam behavior. Sample problems motivated from switch and actuator applications in the literature demonstrate the methodologies’ utility in predicting the nonlinear equilibrium paths for structures of practical importance.

Solution for Geometrically Non-Linear Elastic Deformation of Simple Frames by a Shooting Method

2011 ◽  
Vol 462-463 ◽  
pp. 668-673
Author(s):  
Shi Rong Li ◽  
Ya Dong Hu
Keyword(s):  
Large Deformation ◽  
Shooting Method ◽  
Frame Structure ◽  
Curved Beams ◽  
Equilibrium Equations ◽  
Nonlinear Bending ◽  
Linear Elastic ◽  
Equilibrium Configurations ◽  
Nonlinear Equilibrium ◽  
Plane Frame

Based on an exact geometric nonlinear theory for plane curved beams, geometrically nonlinear equilibrium equations and boundary conditions governing the nonlinear bending of a simple plane frame structure subjected distributed loads were derived. By using the shooting method to numerically solve the boundary value problem of nonlinear ordinary differential equations, large deformation equilibrium configurations of a simple frame with both straight and the curved beam elements subjected uniformly distributed load were obtained. The theory and methodology presented can be used to analyze large deformation of plane simple frames with a variety of geometries and loadings.

The Optimal Control Approach to Dynamical Inverse Problems

2012 ◽  
Vol 134 (2) ◽  
Author(s):  
Wolfgang Steiner ◽  
Stefan Reichl
Keyword(s):  
Optimal Control ◽  
Inverse Problems ◽  
Practical Importance ◽  
Differential Algebraic Equations ◽  
Performance Measure ◽  
State Variables ◽  
Algebraic Equations ◽  
State Equations ◽  
Solution Approach ◽  
Control Approach

This paper considers solution strategies for “dynamical inverse problems,” where the main goal is to determine the excitation of a dynamical system, such that some output variables, which are derived from the system’s state variables, coincide with desired time functions. The paper demonstrates how such problems can be restated as optimal control problems and presents a numerical solution approach based on the method of steepest descent. First, a performance measure is introduced, which characterizes the deviation of the output variables from the desired values, and which is minimized by the solution of the inverse problem. Second, we show, how the gradient of this error functional can be computed efficiently by applying the theory of optimal control, in particular by following an idea of Kelley and Bryson. As the major contribution of this paper we present a modification of this method which allows the application to the case where the state equations are given by a set of differential algebraic equations. This situation has great practical importance since multibody systems are mostly described in this way. For comparison, we also discuss an approach which bases an a direct transcription of the optimal control problem. Moreover, other methods to solve dynamical inverse problems are summarized.

Modeling of Initially Curved Beam Structures for Design of Multistable MEMS

2009 ◽  
Author(s):  
Matthew D. Williams ◽  
Fred van Keulen ◽  
Mark Sheplak
Keyword(s):  
Mathematical Model ◽  
Microelectromechanical Systems ◽  
Curved Beam ◽  
Beam Structures ◽  
The Many ◽  
Dead Loading ◽  
Solution Methodology

This article describes a mathematical model and solution methodology for efficiently predicting the many equilibrium paths of an arbitrarily-shaped, pre-curved, clamped beam subjected to arbitrary dead loading. Such structures are common among multistable microelectromechanical systems (MEMS), and the presented model provides designers with an easy way to study the effects of loading, shape, and imperfections on beam behavior. Results are presented for switch and actuator applications motivated from the literature.

Simultaneous solution approach to model-based experimental design

AIChE Journal ◽  
2013 ◽  
Vol 59 (11) ◽  
pp. 4169-4183 ◽  
Author(s):  
M. D. Hoang ◽  
T. Barz ◽  
V. A. Merchan ◽  
L. T. Biegler ◽  
H. Arellano-Garcia
Keyword(s):  
Experimental Design ◽  
Simultaneous Solution ◽  
Solution Approach ◽  
Model Based
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