A 3D Pseudo-Rigid-Body Model for Large Spatial Deflections of Rectangular Cantilever Beams

2006 ◽  
Author(s):  
Nathan O. Rasmussen ◽  
Jonathan W. Wittwer ◽  
Robert H. Todd ◽  
Larry L. Howell ◽  
Spencer P. Magleby
Keyword(s):  
Rigid Body ◽  
Cross Sections ◽  
Compliant Mechanisms ◽  
Circular Path ◽  
Cantilever Beams ◽  
Spherical Joint ◽  
3 Dimensional ◽  
Body Model ◽  
Rigid Body Model ◽  
Out Of Plane

The design of compliant mechanisms has been aided by the development of pseudo-rigid-body models to predict the motion of flexible members undergoing large displacements. Many of these models are based on the fact that the end of a cantilever beam follows a near-circular path when planar loads are applied. This paper shows that the application of 3-dimensional end-loading causes a beam to follow a near-spherical path, even for beams with non-circular cross-sections. A 3D pseudo-rigid-body model is presented that allows the motion of an end-loaded rectangular beam to be predicted using a rigid link and a spherical joint. Two sets of deflection limits for 0.5% error are presented and shown to be dependent upon the aspect ratio of the cross-section of the beam. The model has the potential for aiding in the design of spatial compliant mechanisms and analysis of planar compliant mechanisms undergoing large out-of-plane motions.

scholarly journals A 3-D Pseudo-Rigid Body Model for Rectangular Cantilever Beams With an Arbitrary Force End-Load

2014 ◽  
Author(s):  
Jairo Chimento ◽  
Craig Lusk ◽  
Ahmad Alqasimi
Keyword(s):  
Rigid Body ◽  
Cross Sections ◽  
Degrees Of Freedom ◽  
Three Dimensional ◽  
Neutral Axis ◽  
Spatial Behavior ◽  
Model Parameters ◽  
Cantilever Beams ◽  
Body Model ◽  
Rigid Body Model

This paper presents the first three-dimensional pseudo-rigid body model (3-D PRBM) for straight cantilever beams with rectangular cross sections and spatial motion. Numerical integration of a system of differential equations yields approximate displacement and orientation of the beam’s neutral axis at the free-end, and curvatures of the neutral axis at the fixed-end. This data was used to develop the 3-D PRBM which consists of two torsional springs connecting two rigid links for a total of 2 degrees of freedom (DOF). The 3-D PRBM parameters that are comparable with existing 2-D model parameters are characteristic radius factor (means: γ = 0.8322), bending stiffness coefficient (means: KΘ = 2.5167) and parametric angle coefficient (means: cΘ = 1.2501). New parameters are introduced in the model in order to capture the spatial behavior of the deflected beam including two parametric angle coefficients (means: cΨ = 1.0714; cΦ = 1.0087).

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.

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.

Compliant Pantographs via the Pseudo-Rigid-Body Model

1998 ◽  
Author(s):  
Andrew J. Nielson ◽  
Larry L. Howell
Keyword(s):  
Rigid Body ◽  
Compliant Mechanisms ◽  
Compliant Mechanism ◽  
Test Results ◽  
Rigid Link ◽  
Body Model ◽  
Design Flexibility ◽  
Model Predictions ◽  
Rigid Body Model ◽  
Classical Mechanism

Abstract This paper uses a familiar classical mechanism, the pantograph, to demonstrate the utility of the pseudo-rigid-body model in the design of compliant mechanisms to replace rigid-link mechanisms, and to illustrate the advantages and limitations of the resulting compliant mechanisms. To demonstrate the increase in design flexibility, three different compliant mechanism configurations were developed for a single corresponding rigid-link mechanism. The rigid-link pantograph consisted of six links and seven joints, while the corresponding compliant mechanisms had no more than two links and three joints (a reduction of at least four links and four joints). A fourth compliant pantograph, corresponding to a rhomboid pantograph, was also designed and tested. The test results showed that the pseudo-rigid-body model predictions were accurate over a large range, and the mechanisms had displacement characteristics of rigid-link mechanisms in that range. The limitations of the compliant mechanisms included reduced range compared to their rigid-link counterparts. Also, the force-deflection characteristics were predicted by the pseudo-rigid-body model, but they did not resemble those for a rigid-link pantograph because of the energy storage in the flexible segments.

Dynamic Characteristics of Planar Compliant Parallel-Guiding Mechanisms

2008 ◽  
Author(s):  
Wenjing Wang ◽  
Yueqing Yu
Keyword(s):  
Numerical Methods ◽  
Rigid Body ◽  
Natural Frequency ◽  
Dynamic Modeling ◽  
Dynamic Characteristics ◽  
Compliant Mechanisms ◽  
Design Parameters ◽  
Dynamic Effects ◽  
Body Model ◽  
Rigid Body Model

Dynamic effects are very important to improving the design of compliant mechanisms. An investigation on the dynamic characteristics of planar compliant parallel-guiding mechanism is presented. Based on the pseudo-rigid-body model, the dynamic model of planar compliant parallel-guiding mechanisms is developed using the numerical methods at first. The natural frequency is then calculated, and frequency characteristics of this mechanism are studied. The numerical results show the accuracy of the proposed method for dynamic modeling of compliant mechanisms, and the relationships between the natural frequency and design parameters are analyzed clearly.

Classification of Compliant Mechanisms and Determination of the Degrees of Freedom Using the Concepts of Compliance Number and Pseudo-Rigid-Body Model

2019 ◽  
Author(s):  
Pratheek Bagivalu Prasanna ◽  
Ashok Midha ◽  
Sushrut G. Bapat
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Compliant Mechanisms ◽  
Compliant Mechanism ◽  
Body Model ◽  
Active Compliance ◽  
Kinematic Behavior ◽  
Rigid Body Model

Abstract Understanding the kinematic properties of a compliant mechanism has always proved to be a challenge. A concept of compliance number offered earlier emphasized the development of terminology that aided in its determination. A method to evaluate the elastic degrees of freedom associated with the flexible segments/links of a compliant mechanism using the pseudo-rigid-body model (PRBM) concept is provided. In this process, two distinct classes of compliant mechanisms are developed involving: (i) Active Compliance and (ii) Passive Compliance. Furthermore, these also aid in a better characterization of the kinematic behavior of a compliant mechanism. A more lucid interpretation of the significance of compliance number is provided. Applications of this method to both active and passive compliant mechanisms are exemplified. Finally, an experimental procedure that aids in visualizing the degrees of freedom as calculated is 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.

Design and Evaluation of Statically Balanced Compliant Mechanisms for Haptic Interfaces

2010 ◽  
Author(s):  
Levi C. Leishman ◽  
Daniel J. Ricks ◽  
Mark B. Colton
Keyword(s):  
Mathematical Model ◽  
Rigid Body ◽  
Virtual Environment ◽  
Compliant Mechanisms ◽  
Haptic Interface ◽  
Design Element ◽  
Haptic Interfaces ◽  
Body Model ◽  
Dynamic Forces ◽  
Rigid Body Model

Compliant mechanisms have the potential to increase the performance of haptic interfaces by reducing the friction and inertia felt by the user. The net result is that the user feels the dynamic forces of the virtual environment, without feeling the dynamics of the haptic interface. This “transparency” typically comes at a cost — compliant mechanisms exhibit a return-to-zero behavior that must be compensated in software. This paper presents a step toward improving the situation by using statically balanced compliant mechanisms (SBCMs), which are compliant devices that do not exhibit the return-to-zero behavior typical with most compliant mechanisms. The design and construction of a prototype haptic device based on SBCMs is presented, along with its mathematical model derived using the pseudo-rigid body model (PRBM) approach. Experimental results indicate that SBCMs effectively eliminate the return-to-zero behavior and are a feasible design element in haptic interfaces.

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