Modeling of Flexural Beams Subjected to Arbitrary End Loads

2002 ◽  
Vol 124 (2) ◽  
pp. 223-235 ◽  
Author(s):  
Chris Kimball ◽  
Lung-Wen Tsai
Keyword(s):  
Rigid Body ◽  
Inflection Point ◽  
Compliant Mechanisms ◽  
Elliptic Integral ◽  
System Optimization ◽  
Beam Deflection ◽  
Beam Equation ◽  
Euler Beam ◽  
Single Degree Of Freedom ◽  
Rigid Body Model

The analysis of compliant mechanisms is often complicated due to the geometric nonlinearities which become significant with large elastic deflections. Pseudo rigid body models (PRBM) may be used to accurately and efficiently model such large elastic deflections. Previously published models have only considered end forces with no end moment or end moment acting only in the same direction as the force. In this paper, we present a model for a cantilever beam with end moment acting in the opposite direction as the end force, which may or may not cause an inflection point. Two pivot points are used, thereby increasing the model’s accuracy when an inflection point exists. The Bernoulli-Euler beam equation is solved for large deflections with elliptic integrals, and the elliptic integral solutions are used to determine when an inflection point will exist. The beam tip deflections are then parameterized using a different parameterization from previous models, which renders the deflection paths easier to model with a single degree of freedom system. Optimization is used to find the pseudo rigid body model which best approximates the beam deflection and stiffness. This model, combined with those models developed for other loading conditions, may be used to efficiently analyze compliant mechansims subjected to any loading condition.

A Simple and Accurate Method for Determining Large Deflections in Compliant Mechanisms Subjected to End Forces and Moments

1998 ◽  
Vol 120 (3) ◽  
pp. 392-400 ◽  
Author(s):  
A. Saxena ◽  
S. N. Kramer
Keyword(s):  
Rigid Body ◽  
Compliant Mechanisms ◽  
Compliant Mechanism ◽  
Elliptic Integral ◽  
Beam Equation ◽  
Large Deflections ◽  
Euler Beam ◽  
Body Model ◽  
Rigid Body Model ◽  
Analysis And Synthesis

Compliant members in flexible link mechanisms undergo large deflections when subjected to external loads. Because of this fact, traditional methods of deflection analysis do not apply. Since the nonlinearities introduced by these large deflections make the system comprising such members difficult to solve, parametric deflection approximations are deemed helpful in the analysis and synthesis of compliant mechanisms. This is accomplished by representing the compliant mechanism as a pseudo-rigid-body model. A wealth of analysis and synthesis techniques available for rigid-body mechanisms thus become amenable to the design of compliant mechanisms. In this paper, a pseudo-rigid-body model is developed and solved for the tip deflection of flexible beams for combined end loads. A numerical integration technique using quadrature formulae has been employed to solve the large deflection Bernoulli-Euler beam equation for the tip deflection. Implementation of this scheme is simpler than the elliptic integral formulation and provides very accurate results. An example for the synthesis of a compliant mechanism using the proposed model is also presented.

A Simple and Accurate Method for Determining Large Deflections in Complaint Mechanisms Subjected to End Forces and Moments

1998 ◽  
Author(s):  
A. Saxena ◽  
Steven N. Kramer
Keyword(s):  
Rigid Body ◽  
Numerical Integration ◽  
Compliant Mechanisms ◽  
Compliant Mechanism ◽  
Beam Equation ◽  
Integration Scheme ◽  
Large Deflections ◽  
Body Model ◽  
Rigid Body Model ◽  
Analysis And Synthesis

Abstract Compliant members in flexible link mechanisms undergo large deflections when subjected to external loads for which, traditional methods of deflection analysis do not apply Nonlinearities introduced by these large deflections make the system comprising such members difficult to solve Parametric deflection approximations are then deemed helpful in the analysis and synthesis of compliant mechanisms This is accomplished by seeking the pseudo-rigid-body model representation of the compliant mechanism A wealth of analysis and synthesis techniques available for rigid-body mechanisms thus become amenable to the design of compliant mechanisms In this paper, a pseudo-rigid-body model is developed and solved for the tip deflection of flexible beams for combined end loads with positive end moments A numerical integration technique using quadrature formulae has been employed to solve the nonlinear Bernoulli-Euler beam equation for the tip deflection Implementation of this scheme is relatively simpler than the elliptic integral formulation and provides nearly accurate results Results of the numerical integration scheme are compared with the beam finite element analysis An example for the synthesis of a compliant mechanism using the proposed model is also presented.

Development of a Methodology for Pseudo-Rigid-Body Models of Compliant Beams With Inserts, and Experimental Validation

2015 ◽  
Author(s):  
Ashok Midha ◽  
Raghvendra S. Kuber ◽  
Sushrut G. Bapat
Keyword(s):  
Rigid Body ◽  
Compliant Mechanisms ◽  
Elliptic Integral ◽  
Creep Recovery ◽  
Design Problems ◽  
Body Model ◽  
Current State ◽  
Material Usage ◽  
Innovative Solutions ◽  
Rigid Body Model

Compliant mechanisms have shown a great deal of potential, in just a few decades of its development, in providing innovative solutions to design problems. However, their use has been limited due to challenges associated with the materials. With ever increasing focus on the applications of compliant mechanisms, it is necessary to find alternatives to the existing material usage and methods of prototyping. This paper presents a methodology for the design of compliant segments and compliant mechanisms with improved creep resistance and fatigue life properties using the current state-of-the-art materials. The methodology proposes using a stronger material at the core of a softer casing. The paper provides an equivalent pseudo-rigid-body model and a closed-form elliptic integral formulation for a fixed-free compliant segment with an insert. The equivalent pseudo-rigid-body model is verified experimentally for the prediction of beam end point displacements. The paper also presents experimental results that show improvements obtained in the creep recovery properties as expected using the proposed design philosophy.

Closed-Form Elliptic Integral Solution of Initially-Straight and Initially-Curved Small-Length Flexural Pivots

2014 ◽  
Author(s):  
Ashok Midha ◽  
Raghvendra Kuber
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Compliant Mechanisms ◽  
Elliptic Integral ◽  
Small Length ◽  
Body Model ◽  
Rigid Body Model ◽  
Elliptic Integral Method ◽  
Integral Solutions ◽  
Made In

Compliant mechanisms gain some or all of their mobility from the deflection of their flexible members. The pseudo-rigid-body model (PRBM) concept allows compliant mechanisms to be modeled using existing knowledge of rigid-body mechanisms, thereby, simplifying the design process. A pseudo-rigid-body model represents a compliant segment with two or more rigid-body segments, connected by pin joints or characteristic pivots. A compliant segment that is small in length, compared to the relatively rigid segments between which it is affixed, is termed a small-length flexural pivot (SLFP). This paper presents closed-form deflection solutions using the elliptic integral method for initially-straight and initially-curved SLFPs. The assumptions made in modeling the small-length flexural pivots in a PRBM are validated by means of the elliptic integral solutions.

Pseudo-Rigid-Body Model for the Flexural Beam With an Inflection Point in Compliant Mechanisms

2017 ◽  
Vol 9 (3) ◽  
Author(s):  
Shun-Kun Zhu ◽  
Yue-Qing Yu
Keyword(s):  
Rigid Body ◽  
Inflection Point ◽  
Compliant Mechanisms ◽  
Angular Displacement ◽  
Flexible Beam ◽  
Body Model ◽  
Numerical Result ◽  
Characteristic Radius ◽  
Rigid Body Model ◽  
Flexural Beam

The pseudo-rigid-body model (PRBM) used to simulate compliant beams without inflection point had been well developed. In this paper, two types of PRBMs are proposed to simulate the large deflection of flexible beam with an inflection point in different configurations. These models are composed of five rigid links connected by three joints added with torsional springs and one hinge without spring representing the inflection point in the flexural beam. The characteristic radius factors of the PRBMs are determined by solving the objective function established according to the relative angular displacement of the two rigid links jointed by the hinge via genetic algorithm. The spring stiffness coefficients are obtained using a linear regression technique. The effective ranges of these two models are determined by the load index. The numerical result shows that both the tip locus and inflection point of the flexural beam with single inflection can be precisely simulated using the model proposed in this paper.

Modeling Flexible Segments With Force and Moment End Loads via the Pseudo-Rigid-Body Model

2000 ◽  
Author(s):  
Scott M. Lyon ◽  
Larry L. Howell ◽  
Gregory M. Roach
Keyword(s):  
Rigid Body ◽  
Cantilever Beam ◽  
Inflection Point ◽  
Compliant Mechanisms ◽  
Body Model ◽  
Rigid Body Model

Abstract This paper presents the development of a new pseudo-rigid-body model to model the deflection path of flexible segments with force and moment loads. Three separate loading cases are presented including: a cantilever beam with applied end-force and moment in the same direction, a cantilever beam with the applied end-force and moment in opposite directions with no inflection point being produced in the beam, and a cantilever beam with the applied end-force and moment in opposite directions which produces an inflection point in the beam. These types of segments are common in compliant mechanisms so it is important to have a method for their design and analysis.

A Method for a More Accurate Calculation of the Stiffness Coefficient in a Pseudo-Rigid-Body Model (PRBM) of a Fixed-Free Beam Subjected to End Forces

2014 ◽  
Author(s):  
Ashok Midha ◽  
Raghvendra S. Kuber ◽  
Vivekananda Chinta ◽  
Sushrut G. Bapat
Keyword(s):  
Rigid Body ◽  
Inflection Point ◽  
Compliant Mechanisms ◽  
Load Factor ◽  
Stiffness Coefficient ◽  
Body Angle ◽  
Analysis And Design ◽  
Body Model ◽  
Body Parameter ◽  
Rigid Body Model

The pseudo-rigid-body model (PRBM) concept allows compliant mechanisms to be modeled using existing knowledge of rigid-body mechanisms, thereby considerably simplifying their analysis and design. The PRBMs represent the compliant segments with two or more rigid-body segments, connected using pin joints (characteristic pivots). The beam compliance is modeled using a torsional spring placed at the characteristic pivot, whose spring constant K is evaluated using a pseudo-rigid-body parameter termed as the beam stiffness coefficient. This paper presents a method to more accurately calculate the beam stiffness coefficient for a fixed-free compliant beam subjected to a combination of horizontal and vertical forces. The improved stiffness coefficient (KΘ) expressions are derived as a function of the pseudo-rigid-body angle, Θ and the load factor, n. To exemplify the application of the improved results, the expressions derived are successfully implemented in modeling a fixed-guided beam with an inflection point, allowing it to be modeled as two fixed-free beams pinned at the inflection point.

Quantifying Uncertainty for Planar Pseudo-Rigid Body Models

2011 ◽  
Author(s):  
Craig P. Lusk
Keyword(s):  
Rigid Body ◽  
Phase Portrait ◽  
Single Phase ◽  
Compliant Mechanisms ◽  
Euler Beam ◽  
Simple Version ◽  
Body Model ◽  
Beam Equations ◽  
Rigid Body Model ◽  
Quantifying Uncertainty

The Pseudo-Rigid-Body Model (PRBM) is an important technique for analyzing and synthesizing compliant mechanisms. This paper describes an approach for generalizing the PRBM and for quantifying its accuracy. The Bernoulli-Euler beam equations are solved and the solution is represented using a phase-portrait. It is shown that the slope-curvature relations for all uniform fixed free-beams can be represented on a single phase portrait. This representation allows for data about PRBM representations for each cantilever beam to be efficiently presented, and for broad conclusions to be drawn, e.g. that the relative position error associated with a simple version of the PRBM is never greater than 7%.

Evaluation of Equivalent Spring Stiffness for Use in a Pseudo-Rigid-Body Model of Large-Deflection Compliant Mechanisms

1994 ◽  
Author(s):  
Larry L. Howell ◽  
Ashok Midha
Keyword(s):  
Rigid Body ◽  
Large Deflection ◽  
Compliant Mechanisms ◽  
Compliant Mechanism ◽  
Spring Stiffness ◽  
Model Concept ◽  
Analysis And Design ◽  
Body Model ◽  
Rigid Body Model ◽  
Body Joints

Abstract Compliant mechanisms gain some or all of their mobility from the flexibility of their members rather than from rigid-body joints only. More efficient and usable analysis and design techniques are needed before the advantages of compliant mechanisms can be fully utilized. In an earlier work, a pseudo-rigid-body model concept, corresponding to an end-loaded geometrically nonlinear, large-deflection beam, was developed to help fulfill this need. In this paper, the pseudo-rigid-body equivalent spring stiffness is investigated and new modeling equations are proposed. The result is a simplified method of modeling the force/deflection relationships of large-deflection members in compliant mechanisms. Flexible segments which maintain a constant end angle are discussed, and an example mechanism is analyzed. The resulting models are valuable in the visualization of the motion of large-deflection systems, as well as the quick and efficient evaluation and optimization of compliant mechanism designs.

Parametric Deflection Approximations for Initially Curved, Large-Deflection Beams in Compliant Mechanisms

1996 ◽  
Author(s):  
Larry L. Howell ◽  
Ashok Midha
Keyword(s):  
Rigid Body ◽  
Large Deflection ◽  
Compliant Mechanisms ◽  
Curved Beam ◽  
Analysis And Design ◽  
Body Model ◽  
Rigid Body Model ◽  
Fundamental Type ◽  
Applied Forces ◽  
Flexible Member

Abstract The analysis of systems containing highly flexible members is made difficult by the nonlineararities caused by large deflections of the flexible members. The analysis and design of many such systems may be simplified by using pseudo-rigid-body approximations in modeling the flexible members. The pseudo-rigid-body model represents flexible members as rigid links, joined at pin joints with torsional springs. Appropriate values for link lengths and torsional spring stiffnesses are determined such that the deflection path and force-deflection relationships are modeled accurately. Pseudo-rigid-body approximations have been developed for initially straight beams with externally applied forces at the beam end. This work develops approximations for another fundamental type of flexible member, the initially curved beam with applied force at the beam end. This type of flexible member is commonly used in compliant mechanisms. An example of the use of the resulting pseudo-rigid-body approximations in compliant mechanisms is included.

Sign in / Sign up

Export Citation Format

Share Document