A Numerical Method for Position Analysis of Compliant Mechanisms With More Degrees of Freedom Than Inputs

2010 ◽  
Author(s):  
Quentin T. Aten ◽  
Shannon A. Zirbel ◽  
Brian D. Jensen ◽  
Larry L. Howell
Keyword(s):  
Potential Energy ◽  
Rigid Body ◽  
Equilibrium Position ◽  
Degrees Of Freedom ◽  
Compliant Mechanisms ◽  
Virtual Work ◽  
Compliant Mechanism ◽  
Loop Equation ◽  
Body Model ◽  
Rigid Body Model

An under-actuated or underconstrained compliant mechanism may have a determined equilibrium position because its energy storage elements cause a position of local minimum potential energy. The minimization of potential energy (MinPE) method is a numerical approach to finding the equilibrium position of compliant mechanisms with more degrees of freedom (DOF) than inputs. Given the pseudo-rigid-body model of a compliant mechanism, the MinPE method finds the equilibrium position by solving a constrained optimization problem: minimize the potential energy stored in the mechanism, subject to the mechanism’s vector loop equation(s) being equal to zero. The MinPE method agrees with the method of virtual work for position and force determination for under-actuated 1-DOF and 2-DOF pseudo-rigid-body models. Experimental force-deflection data is presented for a fully compliant constant-force mechanism. Because the mechanism’s behavior is not adequately modeled using a 1-DOF pseudo-rigid-body model, a 13-DOF pseudo-rigid-body model is developed and solved using the MinPE method. The MinPE solution is shown to agree well with non-linear finite element analysis and experimental force-displacement data.

A Numerical Method for Position Analysis of Compliant Mechanisms With More Degrees of Freedom Than Inputs

2011 ◽  
Vol 133 (6) ◽  
Author(s):  
Quentin T. Aten ◽  
Shannon A. Zirbel ◽  
Brian D. Jensen ◽  
Larry L. Howell
Keyword(s):  
Potential Energy ◽  
Equilibrium Position ◽  
Degrees Of Freedom ◽  
Compliant Mechanisms ◽  
Virtual Work ◽  
Compliant Mechanism ◽  
Numerical Approach ◽  
Element Analysis ◽  
Loop Equation ◽  
Body Model

An underactuated or underconstrained compliant mechanism may have a determined equilibrium position because its energy storage elements cause a position of local minimum potential energy. The minimization of potential energy (MinPE) method is a numerical approach to finding the equilibrium position of compliant mechanisms with more degrees of freedom (DOF) than inputs. Given the pseudorigid-body model of a compliant mechanism, the MinPE method finds the equilibrium position by solving a constrained optimization problem: minimize the potential energy stored in the mechanism, subject to the mechanism’s vector loop equation(s) being equal to zero. The MinPE method agrees with the method of virtual work for position and force determination for underactuated 1-DOF and 2-DOF pseudorigid-body models. Experimental force-deflection data are presented for a fully compliant constant-force mechanism. Because the mechanism’s behavior is not adequately modeled using a 1-DOF pseudorigid-body model, a 13-DOF pseudorigid-body model is developed and solved using the MinPE method. The MinPE solution is shown to agree well with nonlinear finite element analysis and experimental force-displacement data.

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.

A Methodology for Determining Static Mode Shapes of a Compliant Mechanism Using the Pseudo-Rigid-Body Model (PRBM) Concept and the Degrees-of-Freedom Analysis

2019 ◽  
Author(s):  
Sushrut G. Bapat ◽  
Pratheek Bagivalu Prasanna ◽  
Ashok Midha
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Compliant Mechanisms ◽  
Compliant Mechanism ◽  
Mode Shapes ◽  
Model Concept ◽  
Static Mode ◽  
Displacement Boundary ◽  
Body Model ◽  
Rigid Body Model

Abstract Traditionally, the deflected configuration of compliant segments is determined through rigorous mathematical analysis using Newtonian mechanics. Application of these principles in evaluating the deformed configuration of compliant mechanisms, containing a variety of segment types, becomes cumbersome. This paper introduces a methodology to determine the expected deflected configuration(s) of a compliant mechanism, for a given set of load and/or displacement boundary conditions. The method utilizes the principle of minimum total potential energy, in conjunction with the degrees-of-freedom analysis and the pseudo-rigid-body model concept. The static mode shape(s) of compliant segments are integrated in identifying the possible functional configuration(s) of a given compliant mechanism’s structural configuration. The methodology, in turn, also facilitates the in situ determination of the deformed configuration of the constituent compliant segments. It thus assists in the identification of an appropriate pseudo-rigid-body model for design and analysis of a compliant mechanism.

The Development of Force-Deflection Relationships for Compliant Mechanisms

1994 ◽  
Author(s):  
Larry L. Howell ◽  
Ashok Midha
Keyword(s):  
Rigid Body ◽  
Compliant Mechanisms ◽  
Virtual Work ◽  
Compliant Mechanism ◽  
Flexible Link ◽  
Model Concept ◽  
Body Model ◽  
Design Flexibility ◽  
Rigid Body Model ◽  
Manufacturing Time

Abstract The advantages of compliant or flexible link mechanisms include increased design flexibility and reduction in manufacturing time and cost. The analysis of such mechanisms may be difficult and time consuming due to the nonlinearities introduced by large deflections. Also, unlike rigid-body mechanisms, the type and form of motion of a compliant mechanism is dependent on the location and magnitude of applied loads. The pseudo-rigid-body model concept has been developed to simplify the analysis of compliant mechanisms by allowing them to be modeled as rigid-link mechanisms with springs. This work uses the principle of virtual work and the pseudo-rigid-body model concept to develop force-deflection relationships for compliant mechanisms. Several examples are presented, and general design equations are derived for pseudo-rigid-body four-bar and slider-crank mechanisms.

A Methodology for Determining Static Mode Shapes of a Compliant Mechanism Using the Pseudo-Rigid-Body Model Concept and the Degrees-Of-Freedom Analysis

2020 ◽  
Vol 12 (2) ◽  
Author(s):  
Pratheek Bagivalu Prasanna ◽  
Sushrut G. Bapat ◽  
Ashok Midha ◽  
Vamsi Lodagala
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Compliant Mechanisms ◽  
Compliant Mechanism ◽  
Mode Shapes ◽  
Model Concept ◽  
Static Mode ◽  
Displacement Boundary ◽  
Body Model ◽  
Rigid Body Model

Abstract Traditionally, the deflected configuration of compliant segments is determined through rigorous mathematical analysis using Newtonian mechanics. Application of this approach in evaluating the deformed configuration of compliant mechanisms, containing a variety of segment types, becomes cumbersome. This paper introduces a methodology to determine the possible deflected configuration(s) of a compliant mechanism, for a given set of load and/or displacement boundary conditions. The methodology utilizes the principle of minimum potential energy, in conjunction with the degrees-of-freedom analysis and the pseudo-rigid-body model concept. The static mode shape(s) of compliant segments are integrated in identifying the possible deflected configuration(s) of a given compliant mechanism. The methodology facilitates the in situ determination of the possible deformed configuration(s) of the compliant mechanism and its constituent segments. This, in turn, assists in the important task of identifying an appropriate pseudo-rigid-body model for the design and analysis of a compliant mechanism. The proposed methodology is illustrated with examples, and supported with experimental validation.

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.

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.

On a Generalized Approach for Design of Compliant Mechanisms Using the Pseudo-Rigid-Body Model Concept

2014 ◽  
Author(s):  
Sushrut G. Bapat ◽  
Ashok Midha ◽  
Ashish B. Koli
Keyword(s):  
Rigid Body ◽  
Compliant Mechanisms ◽  
Compliant Mechanism ◽  
Motion Generation ◽  
Model Concept ◽  
Finite Element Software ◽  
Body Model ◽  
Wide Range ◽  
Rigid Body Model ◽  
Torque Values

This paper provides a generalized approach for the design of compliant mechanisms. The paper discusses the implicit uncoupling, between the kinematic and energy/torque equations, enabled by the pseudo-rigid-body model concept, and utilizes it for designing a variety of compliant mechanism types for a wide-range of user specifications. Pseudo-rigid-body four-bar mechanisms, with one to four torsional springs located at the revolute joints, are considered to demonstrate the design methodology. Mechanisms are designed for conventional tasks, such as function, path and motion generation, and path generation with prescribed timing, with energy/torque specified at the precision-positions. State-of-the-art rigid-body synthesis techniques are applied to the pseudo-rigid-body model to satisfy the kinematic requirements. Energy/torque equations are then used to account for the necessary compliance according to the user specifications. The approach utilizes a conventional, simple yet efficient optimization formulation to solve energy/torque equations that allow a designer to i) achieve realistic solutions, ii) specify appropriate energy/torque values, and iii) reduce the sensitivities associated with the ‘synthesis with compliance’ approach. A variety of examples are presented to demonstrate the applicability and effectiveness of the approach. All of the examples are verified with the finite element software ANSYS®.

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