A Polynomial Homotopy Formulation of the Inverse Static Analysis of Planar Compliant Mechanisms

2005 ◽  
Vol 128 (4) ◽  
pp. 776-786 ◽  
Author(s):  
Hai-Jun Su ◽  
J. Michael McCarthy
Keyword(s):  
Potential Energy ◽  
Static Analysis ◽  
Compliant Mechanisms ◽  
Potential Energy Function ◽  
Equilibrium Equations ◽  
Equilibrium Configurations ◽  
Force Moment ◽  
Four Bar Linkage ◽  
Derivatives Of ◽  
Raphson Iteration

This paper formulates the inverse static analysis of planar compliant mechanisms in polynomial form. The goal is to find the equilibrium configurations of the system in response to a known force/moment applied to the mechanism. The geometric constraint of the linkage defines a set of kinematics equations which are combined with equilibrium equations obtained from partial derivatives of the potential-energy function. In order to apply polynomial homotopy solver to these equations, we approximate the linear torsion spring torque at each joint by using sine and cosine functions. The results obtained from the homotopy solver are then refined using Newton-Raphson iteration. To demonstrate the analysis steps, we study two example planar compliant mechanisms, a four-bar linkage with two torsional springs, and a parallel platform supported by three linear springs. Numerical examples are provided together with plots of the potential energy during a movement between selected equilibrium positions.

Inverse Static Analysis for a Planar Compliant Platform Mechanism

2004 ◽  
Author(s):  
Hai-Jun Su ◽  
J. Michael McCarthy
Keyword(s):  
Potential Energy ◽  
Static Analysis ◽  
Parallel Mechanism ◽  
Equilibrium Points ◽  
Potential Energy Function ◽  
System Behavior ◽  
Equilibrium Conditions ◽  
Equilibrium Configurations ◽  
Moving Platform ◽  
Derivatives Of

This paper studies the inverse static analysis of a planar parallel mechanism with compliant limbs. A known force and moment is applied to the moving platform, and it is required to determine the assembly configurations, or equilibrium points. Partial derivatives of the potential energy function yields the equilibrium conditions. The geometric and static constraints lead to a system of ten polynomials with ten unknowns. We use polynomial homotopy method to find that there are as many as 70 equilibrium configurations. Two examples with equilateral geometry are provided. We also examine the system behavior during a movement between selected equilibrium positions.

Synthesis of Compliant Mechanisms With Specified Equilibrium Positions

2005 ◽  
Author(s):  
Hai-Jun Su ◽  
J. Michael McCarthy
Keyword(s):  
Compliant Mechanisms ◽  
Equilibrium Constraints ◽  
Equilibrium Equations ◽  
Equilibrium Configurations ◽  
Form Design ◽  
Design Requirements ◽  
Synthesis Procedure ◽  
Position Synthesis ◽  
Four Bar Linkage ◽  
Sine And Cosine Functions

This paper presents a synthesis procedure for a compliant four-bar linkage with three specified equilibrium configurations. The finite position synthesis equations are combined with equilibrium constraints at the flexure pivots to form design equations. These equations are simplified by modeling the joint angle variables in the equilibrium equations using sine and cosine functions. Solutions to these design equations were computed using a polynomial homotopy solver. In order to provide a design specification, we first compute the six equilibrium configurations of a known compliant four-bar mechanism. We use these results as design requirements to synthesize a compliant four-bar. The solver obtained eight real solutions which we refined using a Newton-Raphson technique. A numerical example is provided to verify the design methodology.

Synthesis of Bistable Compliant Four-Bar Mechanisms Using Polynomial Homotopy

2006 ◽  
Vol 129 (10) ◽  
pp. 1094-1098 ◽  
Author(s):  
Hai-Jun Su ◽  
J. Michael McCarthy
Keyword(s):  
Compliant Mechanisms ◽  
Homotopy Continuation ◽  
Equilibrium Constraints ◽  
Equilibrium Equations ◽  
Kinematic Synthesis ◽  
Equilibrium Configurations ◽  
Form Design ◽  
Four Bar Linkage ◽  
Sine And Cosine Functions ◽  
Cosine Functions

In this paper we formulate and solve the synthesis equations for a compliant four-bar linkage with three specified equilibrium configurations in the plane. The kinematic synthesis equations as for rigid-body mechanisms are combined with equilibrium constraints at the flexure pivots to form design equations. These equations are simplified by modeling the joint angle variables in the equilibrium equations using sine and cosine functions. Polynomial homotopy continuation is applied to compute all of the design candidates that satisfy these design equations, which are refined using a Newton-Raphson technique. A numerical example demonstrates design methodology in which the homotopy solver obtained eight real solutions. Two of them provide two stable and one unstable equilibrium, and hence, can be used as the prototype of bistable compliant mechanisms.

A numerical method for static analysis of pseudo-rigid-body model of compliant mechanisms

2014 ◽  
Vol 228 (17) ◽  
pp. 3170-3177 ◽  
Author(s):  
Mohui Jin ◽  
Xianmin Zhang ◽  
Benliang Zhu
Keyword(s):  
Finite Elements ◽  
Potential Energy ◽  
Rigid Body ◽  
Numerical Method ◽  
Compliant Mechanisms ◽  
Potential Energy Function ◽  
Automated Analysis ◽  
Optimization Techniques ◽  
Body Model ◽  
Rigid Body Model

This paper presents a numerical method for analyzing the pseudo-rigid-body model of compliant mechanisms based on finite elements and the principle of minimum potential energy. The proposed method represents the links of pseudo-rigid-body model with truss elements. As a result, the pseudo-rigid-body model is modeled into a compliant system that consists of finite elements and springs. The static equilibrium position of the pseudo-rigid-body model can be obtained by minimizing the potential energy function of this compliant system. A comparison between the proposed method and the MinPE method is presented. Lastly, a case study is provided to demonstrate the application of this method in the automated analysis of pseudo-rigid-body models. This numerical method paves the way for introducing the topology optimization techniques into the synthesis of flexure-based compliant mechanisms.

Stability of Cohesive Crack Model: Part I—Energy Principles

1995 ◽  
Vol 62 (4) ◽  
pp. 959-964 ◽  
Author(s):  
Z. P. Bazˇant ◽  
Yuan-Neng Li
Keyword(s):  
Potential Energy ◽  
Crack Length ◽  
Crack Opening ◽  
Cohesive Crack ◽  
Crack Model ◽  
Cohesive Crack Model ◽  
Equilibrium Equations ◽  
Complementary Energy ◽  
Maximum Displacement ◽  
Derivatives Of

The paper deals with a cohesive crack model in which the cohesive (crack-bridging) stress is a specified decreasing function of the crack-opening displacement. Under the assumption that no part of the crack undergoes unloading, the complementary energy and potential energy of an elastic structure which has a cohesive crack and is loaded by a flexible elastic frame is formulated using continuous influence functions representing compliances or stiffnesses relating various points along the crack. By variational analysis, in which the derivatives of the compliance or stiffness functions with respect to the crack length are related to the crack-tip stress intensity factors due to various unit loads, it is shown that the minimizing conditions reduce to the usual compatibility or equilibrium equations for the cohesive cracks. The variational equations obtained can be used as a basis for approximate solutions. Furthermore, the conditions of stability loss of a structure with a growing cohesive crack are obtained from the condition of vanishing of the second variation of the complementary energy or the potential energy. They have the form of a homogeneous Fredholm integral equation for the derivatives of the cohesive stresses or crack opening displacements with respect to the crack length. Loadings with displacement control, load control, or through a flexible loading frame are considered. Extension to the analysis of size effect on the maximum load or maximum displacement are left to a subsequent companion paper.

Analysis of a Gravity Compensated Four-Bar Linkage Mechanism With Linear Spring Suspension

2007 ◽  
Vol 130 (1) ◽  
Author(s):  
Theeraphong Wongratanaphisan ◽  
Matthew O. T. Cole
Keyword(s):  
Potential Energy ◽  
High Speed ◽  
Vertical Plane ◽  
Potential Energy Function ◽  
Linkage Mechanism ◽  
Wide Range ◽  
Spring Suspension ◽  
Linear Spring ◽  
Four Bar Linkage ◽  
Load Mass

This paper presents the analysis of a gravity compensated four-bar linkage mechanism with zero-free-length linear spring suspension. The objective of the study is to seek the possibility of employing the four-bar linkage or similar mechanisms for assisting vertical planar motion of a load mass in a gravitational field. The analysis is based on the system potential energy framework. Firstly, an arrangement of springs for gravity compensation in a four-bar linkage mechanism is proposed. It is then shown that for a four-bar linkage with symmetric geometric and mass properties the potential energy of the system has interesting and useful characteristics near the configuration at which the middle link is horizontal: an ideal operating configuration. The study also covers more practical cases where there is asymmetry in the mass distribution. The potential use of the mechanism in these cases is validated through a study of the sensitivity of the system potential energy function around the equilibrium point. Finally, based on the results obtained a novel mechanism is proposed for achieving gravity compensated vertical plane motion of a load mass. The proposed mechanism can have a wide range of travel and has significant potential for use not only in low-speed mechanical systems but also in high-speed heavy automated systems, where operating accelerations are of the order of 1g or less.

Nonlinear Static Analysis of Deep Ocean Mining Pipe—Part I: Modeling and Formulation

1981 ◽  
Vol 103 (1) ◽  
pp. 11-15 ◽  
Author(s):  
C. A. Felippa ◽  
J. S. Chung
Keyword(s):  
Static Analysis ◽  
Deep Ocean ◽  
Static Equilibrium ◽  
Nonlinear Static Analysis ◽  
Load Parameter ◽  
Equilibrium Equations ◽  
Tangent Stiffness Matrix ◽  
Equilibrium Configurations ◽  
Nonlinear Static ◽  
External Forces

A static analysis procedure is formulated and implemented for the numerical determination of nonlinear static equilibrium configurations of deep ocean risers or mining pipes. This implementation involves selection of a finite element model, modeling of structure, surface and subsurface environment and external forces, and solution of nonlinear equilibrium equations. The riser is modeled by three-dimensional beam finite elements which include axial, bending, and torsional deformations. These deformations are coupled through geometrically nonlinear effects. The resulting tangent-stiffness matrix includes three contributions identified as linear, geometric (initial-stress) and initial-displacment stiffness matrices. For the solution, a combination of load-parameter incrementation, state updating of fluid properties, and corrective Newton-Raphson iteration is used. The resulting riser configuration reflects the realistic modeling of environments and external forces. The static equilibrium solution can be used as initial condition for vibration or transient analysis. Numerical studies are presented in Part II of this paper.

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