Compliant mechanisms utilize the deformation of the elastic members to achieve the desired motion. Currently, design and analysis of compliant mechanisms rely on several commercial dynamics and finite element simulation tools. However, these tools do not implement the most recently developed theories in compliant mechanism research. In this article, we present DAS-2D (Design, Analysis and Synthesis), a conceptual design tool which integrates the recently developed pseudo-rigid-body models and kinetostatic analysis/synthesis theories for compliant mechanisms. Coded in Matlab, the software features a kinematic solver for general rigid-body mechanisms, a kinetostatic solver for compliant mechanisms and a fully interactive graphical user interface. The implementation details of all modules of the program are presented and demonstrated with four different case studies. This tool can be beneficial to classroom teaching as well as engineering practices in design of compliant mechanisms.

Rigid-body mechanisms

The PRB model approach substitutes compliant links with a series of
(typically two to four) rigid segments joined with torsion or linear springs.
Howell and Midha

In terms of computer-aided design of mechanisms, there has been a number of
static solvers that have been developed throughout the years and KINMAC and
STATMAC by

WorkingModel 2D is a commercial software for multi-body dynamic simulation of
planar physical systems. It has been often used in design and simulation of
planar mechanisms. Adams

Given numerous advances in compliant mechanism theories in the past two decades, there is no design software that is dedicated to design of compliant mechanisms. In this paper, we aim to fill this gap by developing a conceptual design tool that integrates kinetostatic analysis and synthesis theory for design of compliant mechanisms. Kinematics of rigid-body mechanisms will also be a side product of this software since they are considered as a special subset of compliant mechanisms. The software was developed in MATLAB in order to take advantage of the built-in functions, such as nonlinear equation solvers and optimization routines. Also, the software tool has a fully interactive graphical user interface to aid users during the design process.

First section of the paper explains the theory behind the kinematic and static analysis modules of the software. Then, a detailed explanation of the implementation of these theories for the different analysis modules are presented with flowcharts and algorithms. An overview of the graphical user interface is given in the next section. Finally, four case studies are illustrated to demonstrate and verify different analysis modules.

It is well known that analysis of compliant mechanisms involves simultaneously solving a set of kinematic equations coupled with static force equilibrium equations which are called kinetostatic (kinematic and static) equations. In this section, we present the mathematical formulation of kinetostatic equations.

Complex number method is one of the most commonly used methods in kinematic
analysis of rigid mechanisms:

For instance, the mechanism shown in Fig.

Two of the three available independent loops that are required to analyze the mechanism are shown.

Compliant mechanisms have at least one compliant link that can be deformed
under external forces. Pseudo-rigid-body models (PRB) have been widely used
in the analysis and synthesis of compliant mechanisms. In this approach,

The PRB-3R model (left) and the finite segment model (right) for compliant beams.

Recently,

Different PRB models have varying accuracies and thus, the accuracy of static
analysis greatly depends on the PRB models used in the analysis. The accuracy
of pseudo-rigid-body models can be illustrated with a basic example of a
single beam loaded from one end and fixed to wall at the other end. The beam
is discretized using PRB-3R and PRB-FSM (11 segments) methods and compared
with Bernoulli–Euler large deflection equation and GEBT. The first test case
is a large force and moment acting in the same direction and the next case is
an opposing force and moment. Table

Comparison of PRB-3R, PRB-FSM and GEBT methods versus the analytical model in static analysis of a beam.

Typically, PRB models with higher number of segments yield more accurate
results. PRB-FSM model with more than ten segments will be always reasonably
accurate for any compliant mechanism regardless of the loading condition.
However, employing PRB-FSM model can increase the degrees of freedom of a
compliant mechanism beyond a limit where an analysis is not possible within
an acceptable time frame. There exists some PRB models
(

Kinetostatic analysis is defined as determining the deflection of a compliant
mechanism upon a given load or vice versa. Several methods exist for
derivation of kinetostatic equations (coupled kinematic and static
equilibrium). Direct stiffness approach

On the other hand, energy methods

The total work done on a mechanism is the sum of work done by external forces
(forces and moments) and internal forces (springs),

Total work done on the system by external forces (along a path

Potential energy of the system is the negative of the work function; i.e.,

Equation (

Equation (

If the direction of a force is fixed during minimization at
Eq. (

Algorithm for incrementally increasing force

initial guess

power=[0,

force

next configuration

initial guess

In this section, we describe the implementation details of DAS (Design,
Analysis and Synthesis) 2D, a computer-aided design tool for planar compliant
mechanisms. The compliant mechanism design software is implemented in MATLAB
using object oriented programming (Fig.

DAS 2D class structure is shown. The main classes can be categorized under three different groups.

Graph theory is often employed in representing mechanisms

A slider-crank mechanism its the graph theory representations: the classical graph representation (top) and the new graphical representation implemented in DAS-2D.

The main challenge of kinematic analysis is finding the independent loops for
formulating the kinematic constraint equations. In the graph theory, the
topology of any mechanisms can be represented with an adjacency matrix and
the number of independent kinematic loops can be calculated with the Euler's
formula. Once the adjacency matrix and the number of independent kinematic
loops are determined, back edges of the graph can be found by a depth first
search algorithm (Algorithm

Depth first search for finding the back edges of a graph.

startNode

currentNode

Parent(currentNode)

Visited(currentNode)

currentNode

currentNode-currentNode.child(i)

Parent(currentNode.child(i))

Visited(currentNode.child(i))

currentNode

The schematic view (left) and flowchart (right) of the static force
analysis problem.

After all independent kinematic loops are found, nonlinear kinematic
equations of the mechanism can be derived. The kinematic equation for the

Following this notation, for a mechanism with

By applying Eq. (

The schematic view (left) and flowchart (right) of the distance
analysis problem.

The schematic view (left) and flowchart (right) of mechanical
advantage analysis problem.

Static force analysis can be described as determining the relationship
between the external loading and the deflection of the compliant mechanism.
Figure

Contrary to the static force analysis, the distance analysis defined as
determining the required load(s) that will result in a prescribed
deformation. As shown in Fig.

After designing a mechanism, the desired motion of a node or a link and unknown loads on the mechanism are defined. Then, the magnitudes of unknown loads are obtained by an optimization process which minimizes the difference between current deformation and the desired deformation.

Mechanical advantage is defined as the ratio of the output force

The energy/torque vs. the input load curve gives the designer a visualized
way to evaluate the quality of a compliant mechanism.
Algorithm

Algorithm for obtaining torque and energy within a specified range

distance=[0,

Link Angle(Slider Position)

minimize total potential energy (Subject to kinematic equations)

calculate Energy(i)

Torque(i)

To expedite the design process, a graphical user interface has been developed
with MATLAB and interfaces for the some of the design and analysis modules
are shown in Fig.

The graphic user interface for the six modules:

In this section, four case studies are presented to demonstrate the capability and the validity of the kinetostatic analysis software described in the previous sections.

To test the static solver, we give an example of a parallelogram flexure
mechanism which consists of two identical compliant beams (length

Static analysis is performed on the mechanism for a large range of load
magnitude (

The two PRB models used (left) and the software output of the deformation of the flexure mechanism (right).

Comparison of the software results with analytical model (BCM) and
ABAQUS software. Magnitudes of

Figure

Compound multibeam parallelogram mechanisms (CMPM) are composed of multiple
parallelogram flexure mechanisms.

A typical compound multibeam parallelogram mechanism loaded at the primary direction.

Three different PRB models (PRB-RPR, PRB-2(PR)R, PRB-FSM with 10 springs;
Table

Comparison of the software results with analytical model

We considered a fully compliant bistable mechanism

The undeflected mechanism (top left) and deflected position (bottom left). The compliant Link 3–4 is fixed to the ground and connected to Link 2–3 via a pin joint. The comparison of the torque-energy plot with Adams software (right). The dotted line is the Adams output and the solid line is the output by the DAS-2D program.

The target vertical displacement of the mechanism is set to

A compliant four bar bistable mechanism (Fig.

Figure

This paper presents DAS-2D, a kinetostatic solver for conceptual
design of planar compliant mechanisms. The recently developed theories in
compliant mechanism research such as pseudo-rigid-body models have been
implemented. Static force analysis is performed by minimizing the total
potential energy of the system. Considering rigid-body mechanisms as a
special case of compliant mechanisms, kinematic analysis routines based on
vector loop closure equations have been developed. Implementation details of
the different modules are presented and they are demonstrated with four
representative case studies. Future work includes the incorporation of
kinematic and static synthesis of complaint mechanisms and modeling compliant
members with analytical models that can provide strain energy calculation for
the Eq. (

List of the PRB Matrix values for the PRB Models used in the paper.
First row of the PRB matrix of a PRB-FSM model with

This material is based upon work supported by National Science Foundation under Grant No:CMMI-1144022 and CMMI-1161841 and the Air Force Office of Scientific Research under contract AFOSR FA9550-12-1-0070. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the funding agencies. Edited by: G. Hao Reviewed by: G. Chen and two anonymous referees