Investigating the Accuracy of Beam-based Finite Element ...suresh/Theses/LimayeMastersReport.pdf ·...

Investigating the Accuracy

of

Beam-based Finite Element Modeling

in the Analysis of Compliant Mechanisms

A project report

submitted in partial fulfillment of the

requirements for the degree of

Master of Engineering in

Mechanical engineering

by

Padmanabh Limaye

Under the guidance of

Prof. G. K. Ananthasuresh

Department of Mechanical Engineering

Indian Institute of Science

June 2009

i

Acknowledgement

I am grateful to my advisor Dr. G. K. Ananthsuresh for his precious guidance, who first drew my attention to the project problem. The novel idea of a kit for compliant mechanism was his brainchild. I am also thankful to G. Ramu and A. Ravikumar for working on fabrication of prototypes. My lab mates Meenakshi Sundaram, Narayan Reddy, Sudarshan Hegade gave me innumerable suggestions to make my work fruitful.

All my M.E. classmates and friends at IISc supported me every moment to make ‘Life in IISc’ worth saying so. The journey was always topsy-turvy for me, but because of friends like R. Ganesh, Darshan Pahinkar, Atul Verma, Sandeep Soni, Hari Thakur, Mallikarjuna Rao, Ashvinikumar Patil, Vishal Dixit, Abhra Chatterji it became memorable. However the list is endless.

I take this opportunity to salute all the professors in my department for their valuable talks in classrooms as well as outside, who helped me to grow richer not only in knowledge but also in wisdom. I owe a lot to the institute. It is a beautiful evidence of ‘Unity in Diversity’ of India.

ii

Abstract This project is concerned with an inconsistency between the modeling of large-displacement compliant mechanisms using beam/frame finite element (FE) framework and their fabricated prototypes. The source of the problem is the modeling of the connections where two or more beams of varying widths intersect. We show that the fabricated device is generally much stiffer than the beam FE model used in the analysis and the design optimization because of the connections. We also argue that using continuum finite elements such as plane stress/strain is not an attractive option because another problem that manifests in the form of point flexures in the topology optimization of compliant mechanisms. In this work, we present the evidence of the discrepancy with several examples. In order to resolve the problem, we modify beam FE analysis by adding, at every beam connection, a lumped element that is obtained from continuum finite element analysis. This approach, although being somewhat unorthodox, reduces the error significantly.

We also present a novel idea of building compliant mechanisms with two basic building blocks wherein the aforementioned problem of connections is conveniently circumvented. This leads to developing a compliant mechanism kit similar to the kits available for rigid-body mechanisms. This kit helps the user either to quickly design a compliant mechanism intuitively or to comprehend the result obtained from synthesis process. We have also developed a gradient-based topology optimization procedure that will generate only those designs that can be assembled by hand using the building blocks of the kit. We present several designs and assembled prototypes obtained in this manner. Thus, the kit and the accompanying analysis and optimal synthesis procedure comprise a self-contained educational as well as a research and pragmatic toolset for planar compliant mechanisms.

iii

Contents

1.

1

1. Introduction and related work

The idea of compliant mechanisms originates from the widely used rigid-body mechanisms wherein we merely replace hinges with flexible joints. These are now called lumped compliant mechanisms. Fig. 1a shows one of the earliest lumped compliant mechanisms reported in the literature [1]. Fig. 1b shows another compliant mechanism of this kind used currently in commercial XY stages [2]. On the other hand, there are no such flexible joints in distributed compliant mechanisms in which the elastic flexibility is distributed throughout the structure in the form of slender, narrow, or thin members of different widths. The focus of research today is mostly on distributed compliant mechanisms, which has the advantage of giving, in general, large displacements before permissible stress levels are reached. In planar compliant mechanisms of this kind, invariably slender beams appear whether they are intuitively conceived [3-4] (see Figs. 2a-b), designed using the pseudo rigid-body model-based approach [5] (see Figs. 3a-b), or obtained using topology optimization [6-7] (see Fig. 4a-b). Thus, most compliant mechanisms studied today inherently consist of beam-like members connected to each other and to possibly some relatively rigid segments.

(a) (b)

Figure 1 (a) One of the earliest lumped compliant mechanisms [1] (b) Lumped compliant mechanism currently used in commercial XY stages [2]. Some of the flexural hinges are encircled

There are two options for modeling compliant mechanisms using the finite element analysis (FEA). The first is to use beam/frame finite elements and the second is to use continuum finite elements such as plane stress/strain. Beam element modeling is computationally less expensive than the plane stress/strain case. However, accuracy may be compromised if the beam-like slender segments modeled using beam elements do not obey the assumptions made in the beam theory. In this work, we bring to the fore another source of inconsistency in using beam finite elements for modeling compliant mechanisms. This is concerned with the modeling of the connections where two or more beams of varying widths intersect. This is explained in the next

2

section. Modeling compliant mechanisms using continuum finite elements has its own problems. A very fine mesh is needed for good accuracy and sometimes there may be an issue of deciding between plane stress or plane strain elements, especially at the filleted connections [8].

.

(a)

(b)

Figure 2 (a) Intuitively conceived compliant mechanism used in flexure cam-clamp [3] (b) Another intuitively conceived compliant mechanism used for converting reciprocating motion into enclosing curved path [4]

.

3

(a)

(b)

Figure 3 (a) Ortho-planar compliant mechanism designed using pseudo-rigid body modeling [5] (b) Compliant gripper designed using pseudo rigid modeling [5]

Hand grips

Output

Input

4

(a)

(b)

Figure 4 (a) and (b) Compliant mechanisms synthesized by topology optimization [6]

The choice of the finite element also matters in the design of compliant mechanisms. This is particularly true in the case of topology optimization. Continuum elements invariably lead to designs that have two elements diagonally connected at a single node suggesting that there should be a point flexure there. This is shown in Fig. 5. The reasons for their appearance are discussed in the literature and some remedies are suggested [7, 9-12]. However, an easily implemental and robust formulation remains elusive. In topology optimization of compliant mechanisms, if we use beam/frame elements instead of continuum elements it does not eliminate

5

this problem completely, but alleviates it to some extent [1, 7]. This is because beam segments inherently have distributed compliance. So, what might have appeared as a point flexure will emerge as a short beam at approximately the same location .See Figs. 6a-b which are the topology solutions obtained for the same specifications using beam and continuum elements [7]. Since designs obtained using point flexures are not amenable for realizing practically useful compliant designs, there tends to be a preference towards using beam/frame elements in topology optimization. However, as noted above, there is a significant source of error in using beam/frame elements in the analysis of compliant mechanisms. This is the motivation for the current work as explained next.

Figure 5 Compliant mechanism designed using continuum elements. Point flexures are encircled [7]

2. Motivation

In the topology optimization of compliant mechanisms using beam/frame elements, the widths or thickness of their rectangular cross-section are taken as the design variables. These variables are varied from their lower limits (set to a very low value but not zero so as to avoid singularities) and upper limits (decided by practical considerations and to be within beam theory assumptions) to fulfill an objective. Usually, the objective is to maximize the ratio of mutual strain energy (MSE) and the strain energy (SE). MSE is computed as the product of strain caused by the applied load and the stress caused by the unit load applied at the output point in the desired direction. MSE is numerically equal to the output displacement. SE is the usual strain energy under the applied load. These two energies are computed using the beam-based FE model. If the FE model is not accurate, so are the optimal designs obtained in topology optimization.

6

As we will demonstrate shortly the beam FE models will be less stiff than the actual prototypes made. Clearly, it will not be a problem in design of stiff structures as fabricated structure is going to be stiffer which always lands on the safer side. But in case of compliant mechanisms, it would mislead the objective function optimized in the algorithm; hence the structure obtained could be away from being optimal. In order to eliminate this problem it becomes necessary to correct the stiffness matrix in beam FEM. We will next explain how beam-intersections lead to more stiffness in the actual prototypes than in the finite element model.

(a)

(b)

Figure 6 Compliant mechanisms designed using (a) continuum elements where point flexures are encircled and (b) beam elements for the same specifications [7]. The encircled beam connections are manifestations of hinges

7

A problem with the modeling of the connections at which two or more beams meet was first noticed in the design of micro-mechanical cycle doubler–a contact-aided mechanism [4] (see Fig. 2b). The response of the prototype was observed to be much different from its beam finite element model. The reason was attributed to the increase in the joint stiffness due to the extra material at the beam-connections when their continuum form is taken in the practical prototype (see Figs.7a-b). Additionally, the effective deformable lengths of the beam segments reduce. The bending stiffness of a beam is inversely proportional to the cube of its length. So a reduction in deformable length of a beam by 25% causes stiffness to increase by about 130%. To counter these two difficulties, the authors of [4] suggested an ad hoc remedy of using additional short beams near the connections to create the extra joint stiffness. They considered the centroid of the fillet portion and drew a line parallel to the fillet’s tangent (see Fig. 8). The intersecting point divides original beam into two portions. The intersecting line is taken as another beam element.

(a)

(b)

Figure 7 (a) Actual beam connection in the fabricated prototype (b) Ideal connection in the beam element model

L2

L1- Deformable length

Extra material L1

L2

Actual deformable length

Fillet (extra material)

Beam Overlapping

8

Thus, an attempt was made to take into account the additional stiffness using additional beam elements. While it worked in that specific application to compare the experimental and simulation results, it is not generally validated. This motivated us to undertake a formal analysis of this problem and suggest a general solution.

Figure 8 Revised beam model for taking joint stiffening into account as per [4]

3. Problem Statement and Scope of the Project

The problem addressed in this work can be stated as follows.

“Demonstrate the stiffness discrepancy between beam FE models and fabricated prototypes, explain the reason for this, and then suggest remedies so as to make compliant mechanism topology design solutions readily usable for manufacturing.”

The work presented here has the following parts.

1. Quantitative comparison of the deformations and stress distributions of compliant mechanisms with linear and geometrically nonlinear beam and plane-stress FE models.

This demonstrates the problem and provides clear motivation for undertaking the work.

2. A general remedy for resolving the enhanced stiffness of the fabricate prototype though FE modeling that is applicable to simulation as well as optimization.

The remedy lies in the development of an automatic construction of connector’s geometry and computation of the stiffness matrix of the super-element. This paves for the topology optimization using this model. Topology optimization results are presented to show the efficacy of the new approach.

3. A novel building-block based ‘compliant mechanism kit’ concept that circumvents the problem of stiffness discrepancy between the model and the prototype.

5

Additional beam elements introduced at joint (viz. 2, 3, and 4)

Centroid of the fillet area

1

2

4

3

9

This part includes the development of the building blocks and their prototyping. Using this kit, a number of compliant mechanisms can now be quickly assembled by hand. A topology optimization that is attuned to this kit is also developed. Examples are presented to show that whatever design this topology optimization generates can be prototyped with the kit.

4. Comparison between Beam and Continuum FE models

The commercial software COMSOL Multiphysics is used for continuum modeling which would exactly simulate the real structure. Five different mechanisms are studied and comparison is made between COMSOL Multiphysics model and beam model for Linear as well as Nonlinear cases. Realistic force values are taken. Material used is Polypropylene whose Young’s Modulus is taken as E= 2000 MPa. Thickness is taken as 5 mm. Strain energy is taken as the norm for indicating extent of deformation. The deformed configurations and stress distributions are observed to find out where the difference lies.

We have used beam elements which consider bending deformation as well as axial deformation, commonly known as frame elements. The beam FEA is based on Euler-Bernoulli beam theory for bending deformation. It assumes that, plain cross sections remain plain and normal to the axis of beam after the deformation. For a beam orientated along X-axis, there is

only one non-zero stress component present, which is axial normal stress ( x ). Transverse shear

stresses ( xy ) and transverse normal stresses ( y ) are zero. In general, only one principal stress

is non-zero and it is independent of the orientation of the beam. For a beam orientated along X-axis the tensile stress state is shown by Mohr’s circle in Fig.9a. Fig. 9b shows the tensile stress

state when xy is present along with x . If 1 and 2 are the principal stresses such that 1 2 ,

then we can write,

2

21 2 2

x y x y

xy

(1)

2

22 2 2

x y x y

xy

(2)

In absence of y ,

10

2

2

21

1 12

1 12

xy

x

xy

x

(3)

If there is deviation from the assumptions made in beam theory, it gives rise to xy . The

magnitude of xy relative to x can be computed form the ratio 2

1

for a beam oriented in any

direction, using Eq. 3. Thus, magnitude of ratio of principal stresses can be taken as a norm for indicating deviation in stress state. However this norm should not be taken strictly at the

locations where x tends to zero. In Fig. 10, 2

1

is plotted against 2

xy

x

for a practical range of

values.

Figs. 11a-b show the deformation of inverter mechanism obtained by linear beam FEA and continuum analysis using COMSOL Multiphysics respectively. Fig. 11b also plots magnitude of ratio of principal stresses obtained by continuum analysis. We notice that in the region of beam connection, the stress state is much different than that in the assumptions. There are few beams in which considerable shear stress is present. Typically fillet radius is indicated by radius of tool used to manufacture prototype. Larger fillet radius leads to reduction in length of beams and also adds extra material, which makes actual joint much stiffer than that in FE beam

Figure 9 Mohr’s circle representing (a) ideal stress state for a beam oriented along X-axis (b)stress state in presence of transverse shear stress for a beam oriented along X-axis.

σ

τ

σ1 = σxσ2 = σy= 0 σ1 σ

τ

σ2

(σy=0, ‐τxy)

(σx, τxy)

(a) (b)

11

model. It results in more discrepancy in strain energy of beam and continuum model. The joints at the corners where the mechanism is fixed to the ground are also sources of error. To constrain all degrees of freedom at one end of a beam, that end is fixed to the ground in continuum model. But in this case as two beams are meeting at a joint, and we want to constraint all degrees of freedom of both the beams, relatively more portion gets fixed to the ground in continuum model than is intended. This gives additional stiffness to that joint. Fig. 12 plots deformation at output point against load for linear and nonlinear cases.

Figs. 13a-b show the deformation of pliers mechanism obtained by linear beam FEA and continuum analysis using COMSOL Multiphysics respectively. Fig.13b also plots magnitude of ratio of principal stresses obtained by continuum analysis, which is considerable not only in beam joint region but also in beams. There are several locations where beams of different widths are connected together. This causes decrease in deformable lengths of thinner beams significantly. Such joints give more stiffness to actual joints. Fig. 14 plots deformation against load for linear and nonlinear cases.

Figs. 15a-b show the deformation of another mechanism obtained by linear beam FEA

and by continuum analysis using COMSOL Multiphysics respectively. Fig. 15b plots magnitude

of ratio of principal stresses obtained by continuum analysis, which is significant even near the

beam edges at certain locations. Fig. 16 plots deformation against load for linear and nonlinear

cases. At the point of application of load several thin beams are connected to thicker beam. It

makes significant reduction in their deformable lengths making joint stiffer. The difference is

clearly seen in deformation figures.

Figure 10 Ratio of principal stresses plotted against magnitude of transverse shear stress relativeto axial stress for practical range of values

2

1

2xy

x

12

We analyzed another mechanism whose deformation is shown in Fig. 17a obtained using

linear beam FEA and in Fig. 17b obtained using linear continuum analysis. In this case the

discrepancy is smaller as most of the beam members are slender. Still we notice presence of

shear stress from the plot of magnitude of ratio of principal stresses obtained by continuum

analysis. Fig. 18 plots deformation against load for linear and nonlinear cases

In Figs. 19a-b deformation of gripper mechanism is shown obtained using linear beam

FEA and the continuum model respectively. A number of beam members are shorter in lengths.

It makes Euler-Bernoulli beam assumption to fail in significant portion, as it is noticeable from

the magnitude of ratio of principal stresses obtained by continuum analysis. It exceeds 0.5 even

near the beam edges indicating considerable amount of shear stress. Hence the inconsistency is

observed between beam and COMSOL Multiphysics model. Fig. 20 plots deformation against

load for linear and nonlinear cases

13

F

F

(a)

(b)

Figure 11 An Inverter mechanism analyzed by (a) Linear beam FE and (b) COMSOL Multiphysicscontinuum model. The deformation is shown for F = 20 N

14

Table 1: Comparison Table

Linear model SE in N.mm (Discrepancy in

brackets)

Geometric Nonlinear SE in N.mm (Discrepancy in

brackets) COMSOL Multiphysics 5.07 6.65 Beam FE analysis 8.24 (63 %) 13.9 (109 %)

Figure 12 Load vs. deformation plots

15

F

F

(a)

(b)

Figure 13 Pliers mechanism analyzed by (a) Linear beam FE and (b) COMSOL model .The deformation is shown for F = 50 N

16



brackets)



Figure 14. Load vs. deformation plots

17

Figure 15 Inverter mechanism analyzed by (a) Linear beam FE and (b) COMSOL model .The deformation is shown for F = 10 N

F

F

(a)

(b)

18



brackets)


brackets) COMSOL Multiphysics 3.76 33.8 Beam FE analysis 4.70 (25 %) 35.7 (5.6 %)


19

Figure 17 Another mechanism analyzed by (a) Linear beam FE and (b) COMSOL model .The deformation is shown for F = 10 N

F 2

F

(a)

(b)

20



brackets)


brackets) COMSOL Multiphysics 68.45 152.4 Beam FE analysis 74.70 (9 %) 186.1 (22.1 %)


21

Figure 19 A gripper mechanism analyzed by (a) Linear beam FE for symmetric half and (b) COMSOL model. The deformation is shown for F = 200 N

F

F 2

(b)

F

(a)

22



brackets)




23

5. Modifying FE model by introducing additional elements at the joints

As we noticed in the examples, the deformation near a beam connection does not follow Euler Bernoulli beam assumptions. In order to eliminate the discrepancy it is necessary estimate the deformation at the beam connection. Therefore the use of continuum model is inevitable. We can separate the region which includes beam connection and filleted portion of the beams. This way every joint can be meshed using continuum elements which are assembled with beam elements. But this makes FE assembly very large and for solving it we have to use numerical techniques such as static condensation. Instead, we shall introduce a 2-D continuum element at every joint region the geometry of which depends upon widths of the connecting beams and fillet radius. The geometry can be intricate in general. This element should have compatibility with beam elements, which means that its every node must have rotational degree of freedom in addition to two translational degrees of freedom. Introducing rotational degrees of freedom in 2-D continuum element is discussed in the literature [13, 14]. But it does not resolve the problem as the deformation pattern of this region is very complex and thus cannot be represented by simple shape functions. As a remedy over this, we mesh the joint-element by ordinary 2-D continuum elements. Hereafter this joint-element is called as super-element. The nodes of the super-element take form of corresponding geometry edges. The stiffness matrix is obtained by influence coefficient method. The principle is that, in Finite Element framework, if we impose zero specified displacement at all degrees of freedom except at a particular degree of freedom where unit displacement is specified, the reaction forces computed at all degrees of freedom represent the column of stiffness matrix corresponding to that particular degree of freedom.

Consider the following example where, an arbitrary joint is considered and using influence coefficient method the stiffness matrix is computed. For simplicity a standard 8-beam connection is considered where all beams have nonzero widths as shown in Fig. 21a.

A MATLAB code is written, for computing geometry as a function of beam widths and fillet radii. The beam widths are taken as 1 mm, 2 mm, 3 mm, 2 mm, 2 mm, 3 mm, 2 mm and 1 mm. The fillet radius is 0.5 mm. The dotted boundary portions are going to be nodes for the super-element. The triangular meshing is done by a code in MATLAB [15], which is shown in Fig. 21b. Fig. 21c, d and e show the perturbation on node number 2 in , , and u v degrees of

freedom respectively. The perturbation in degree of freedom is imposed by considering small rotation of that edge about its mid-point and actually specifying u and v displacements. Consider 0 0( , )X Y as the coordinates of the midpoint of the edge and ( , )X Y represents any point

on the edge, then and u v displacements are given by

0 0- - and -u Y Y v X X (4)

24

Figure 21 Geometry of the super-element shown with node numbers. The dotted edges represent nodes (b) Triangular element mesh generated (c)(d) and (e) Specified boundary displacments on node-2 for Ux, Uy and Uθ degrees of freedon respectively

2

1

3

4

5

6

7

8

(e)

2

1

3

4

5

6

7

8

Fillet radius

(a) (b)

2

1

3

4

5

6

7

8

(c)

2

1

3

4

5

6

7

8

(d)

25

For each perturbation linear FE analysis is done and reactions are computed at every edge viz. , and x yF F F . Again F is physically a moment of force, which is calculated as

0 0- - ( - )x yF F Y Y F X X (5)

This force vector computed in this manner forms the corresponding column in the super-element stiffness matrix. This way the stiffness matrix is obtained for super-element which has rotational degree of freedom too at every node.

Here the geometry computed is inadequate to implement it in general where only few of the connecting beams are present and in the topology optimization framework where some of the beams disappear as the optimization progresses. For that, consider three-beam joint where beam widths are 0 mm, 0 mm, 2 mm, 0 mm, 2 mm, 2 mm, 0 mm and 0 mm fillet radius being 2 mm. Fig. 22a shows this joint as it appears in the prototype and Fig. 22b shows the basic geometry computed which is different than that in prototype. To resolve this situation, the geometry computation is extended to identify beams of zero width. The nodes corresponding to these beams are represented by small edges formed over the external fillet of small radius (0.2mm) (see Fig. 22c). The code also introduces compulsory fictitious widths to take care of geometries where none or only one beam of non-zero width is present. For the nodes which are going to be fixed to the ground special care is taken as shown in Fig. 22d. Although the beams connected at node number 3 and 5 are absent, since we have to restrict all planar degrees of freedom at those nodes, compulsory width is employed. This width equals to width of the beam connected at opposite end of the super-element.

In the present work, we compute stiffness matrices for two types of joints used in ground structure, 8-beam connections and 4-beam connections. Here, beam widths and fillet radii are the variables but the angles at which they meet are standard and fixed. This scheme is extendable to any arbitrary joint where the number beams, their widths, inclination angles and fillet radius, all are variables.

Fig. 23a shows the deformation of the inverter mechanism earlier shown in Fig. 11 analyzed by linear FEA after implementing super-elements. The discrepancy in the strain energy between COMSOL Multiphysics model and beam FE model is reduced from 63% to 12%. Boundaries of each super-element are drawn in undeformed configuration. Deformed configuration shows only beams. Fig. 23a shows another mechanism for which the discrepancy is reduced from 20% to 11%.

The mechanisms are also analyzed as large deformation problems. Nonlinearity due to large strain is not considered here because it rarely exists in compliant mechanism. Thus considering large displacement and rotations is adequate. Here we use co-rotational beam elements. The load is applied in steps. At every step the equilibrium equation is solved using the Newton-Raphson method. It requires computing the tangent stiffness matrix and the internal

26

Figure 22 (a) A beam joint as it appears in the prototype (b) Geometry for the same joint generated without identifying zero-width beams (c) Geometry calculation extended accordingly so that it replicates the joint in the prototype (d) Compulsory widths provided for the nodes that are fixed to the ground as node number 3 and 5 take the widths of beams connected at node 7 and 1 respectively

(a)

(c)

2

1

3 4

5

6 7

8

(b)

3

4

5

6

2

1

7

8

(d)

27

force, which is implemented in [16] for beam elements. It is necessary to separate rigid displacements from the total displacements. Fig. 23 shows the large displacement of a joint which includes rigid rotation , where the local co-ordinate system (L.C.S) makes an angle with the global coordinate system (G.C.S). For simplicity only two nodes are considered,

which have coordinates 1 1,X Y and 2 2,X Y , with respect to origin O . LU and gU are

displacements in L.C.S. and G.C.S. respectively.

1 1 1 2 2 2

1 1 1 2 2 2

L

g

T

x y x y L

T

x y x y g

U U U U U U U

U U U U U U U

(6)

The mapping between L.C.S and G.C.S is given by transformation,

int int

L g

L g

U U

F F

(7)

Where transformation matrix is given by

cos sin 0 0 0 0

sin cos 0 0 0 0

0 0 1 0 0 0

0 0 0 cos sin 0

0 0 0 sin cos 0

0 0 0 0 0 1

(8)

1 1( , )X Y

2 2( , )X Y

O G.C.S

L.C.S

1x gU

1y gU

1gU

1x LU

1y LU

1LU

Figure 23 Solid lines show undeformed joint geometry and dotted lines show deformed configuration. Mapping is obtained between L.C.S and G.C.S.

28

LU is obtained from gU by subtracting displacements due to rigid rotation of and mapping them

to L.C.S.by transformation matrix .

( )L gU U (9)

1

1

2

2

cos 1 sin 0 0 0 0

sin cos 1 0 0 0 0

10 0 0 0 0

0 0 0 cos 1 sin 0

0 0 0 sin cos 1 0

10 0 0 0 0

X

Y

X

Y

(10)

intL LF KU (11)

From Eq. 4-8 we can write

int Tg gF K U (12)

Hence the tangent stiffness matrix is obtained by

intgtgt

gg

FK

U

(13)

tgt TgK K (14)

A test problem is analyzed as shown in Fig. 24, where two beam elements of lengths 44.5 mm, width 1mm and thickness of 2mm, are connected together with a super-element. Young’s modulus is 210 GPa. The stiffness of the joint is kept very high so as to make it perfectly rigid. It finds application in rigid connector compliant mechanisms, discussed in next section. We notice that the joint perfectly undergoes large rotation.

Thus mechanism shown in Fig. 25a is analyzed considering geometric nonlinearity, for which the discrepancy in strain energy is reduced from 109% to 24% by use of super-elements. Fig. 25b shows the comparison between load-deformation curves. Similar analysis is done for the second mechanism for which discrepancy is reduced from 66% to 47%. The deformation is shown in Fig. 26a and load vs. deformation curves are shown in Figure 26b.

These examples show that modifying FE analysis using super-element can alleviate the problem of joint stiffening to a considerable extent. In addition to beam connections, in some portion of the beams the Euler-Bernoulli beam assumption does not hold good as discussed in the earlier section illustrated with examples. We have used plane stress assumption for computing super-element stiffness matrix, which may not be a good assumption to work with.

29

Due to these reasons some discrepancy is still present even on implementing the super-element at the joints.

Table 6: Comparison Table for the mechanism shown in Fig. 25


brackets)


brackets) COMSOL Multiphysics 5.07 6.65 Beam FE analysis 8.24 (63 %) 13.9 (109 %) Beam FE analysis with super-element at joint

5.68 (12%) 8.24 (24 %)

Table 7: Comparison Table for the mechanism shown in Fig. 26


brackets)


brackets) COMSOL Multiphysics 147.9 147.9 Beam FE analysis 178.4 (20 %) 245.8 (66 %) Beam FE analysis with super-element at joint

164.2 (11 %) 217.3 (47 %)

F

Figure 24 Test problem of the rigid connector for large deformation problem. F is 14.14 N

30

F(a)

(b)

Figure 25 (a) Linear FE analysis of inverter mechanism after implementing super-elements (b) Load-deformation curves show comparison between beam FEA ,COMSOL Multiphysics analysis and beam FEA with super-elements for Linear and Geometric Nonlinear cases

31

Uout

(a)

(b)

F

Figure 26 (a) Linear FE analysis of pliers mechanism after implementing super-elements (b) Load-deformation curves show comparison between beam FEA, COMSOL Multiphysics analysis and beam FEA with super-elements for Linear and Geometric Nonlinear cases

32

Obtaining optimal topologies with modified FE analysis

Gradient-based topology optimization method requires sensitivities of the objective function with respect to the design variables. Since super-element stiffness matrix as a function of connecting beam widths is not available explicitly, direct sensitivity computation is not practicable with this method. Therefore, it is not considered in the current work. Nevertheless, super-element stiffness matrix can be obtained as a function of the design variables, by interpolation. Here, the matrix is computed for all each possible combination of discrete values of variables. The data obtained can be used in two ways.

1. To find polynomial interpolation function in the number of design variables and degree indicated by a number of data points in each variable. It gives a continuous function; therefore, expression for the gradient can be obtained. It requires huge amount of computation as interpolation has to be done in 9-D space, where 9 is the number of design variables. Once it is done, exact FE analysis along with gradient calculation is possible.

2. During optimization routine, simple linear interpolation can be done between two neighboring data points in each variable. It gives an approximate linear function in number of design variables. The drawback of this method is that gradients computed will be constant and piecewise continuous.

Because both of these approaches are computationally in efficient, optimization is not implemented considering correction in FE model. Instead, another practical approach is pursued for which topology optimization is also implemented. This is discussed in the next section.

6. Design of a Rigid connector compliant mechanism and the development of a kit

As discussed in the earlier sections, the discrepancy with beam FE model is due to inaccurate estimation of the deformation in the beam joint region. So, we circumvent this problem by using rigid connectors in both FE model and prototype. This gives an idea of assembling the prototypes from two types of building blocks, which are deformable beams of standard lengths and rigid connectors. Similar to the kits available for rigid link mechanisms, a kit is developed for compliant mechanisms.

Figs. 27a-b show an example of a compliant mechanism assembled using the parts in the kit. Both, the connectors and beams are made of spring steel and manufactured by wire-cut electro discharge machining. The 8-beam connector is shown in Fig. 27c. It is designed so that the end of a beam can be snap-fitted into the slot. The notch provided in between two slots allows deformation of the slot while inserting a beam. It restricts translational and rotational displacement of that end of beam with respect to the connector. We have used beams of two

33

standard lengths that are found in the square ground structure. The radius of connectors is 5.5 mm and horizontal and vertical spacing is 50 mm. All beams are of standard width, which is 0.2 mm. It is proposed to use a breadboard having square pits located equispaced as the connectors in ground structure and special connectors having square cross section extrusion. This enables the user to fix the mechanism to ground by press-fitting this connector into the pit, which restricts its all planar degrees of freedom.

Figure 27 (a) A compliant mechanism assembled from the kit (b) The deformed configuration (c) The connector made of spring steel

(a) (b)

(c)

5.5 mm

34

The motivation behind making the kit is to create a platform where users can quickly realize a compliant mechanism using their creativity and mechanics intuition. The prototypes can be easily hand-assembled from the available parts as it is done in Lego toys. It is also possible to verify or modify mechanisms synthesized systematically as in topology optimization. Conventionally a compliant mechanism prototype is fabricated from a single block of material by machining, where it is possible to produce different beams of different width. However a prototype assembled from the kit, where all beams are of same width, also functions in a similar manner since topology remains the same. It is noticed that the topology is of utmost importance among the factors such as size and shape which decide the behavior of the mechanism. This makes using the kit worthwhile and it may lead the way for a real compliant mechanism for practical use.

7. FE modeling of Rigid connector compliant mechanisms and design using topology optimization

For modeling a rigid connector compliant mechanism, we model each connector as a separate finite element as done for super-element in the previous section. Each node has three degrees of freedom. The stiffness matrix for this element is obtained by the method of influence coefficients which is discussed. Again two types of connectors, 8-beam and 4-beam connectors are considered. The actual connector is circular in shape, which is modeled by a polygon so that the radius of a circle passing through all nodal points is equal to the radius of the connector. It is necessary to obtain singular mode for small rigid rotations and translations. Stiffness matrix

obtained this way is multiplied by a large number 10n to achieve rigidity of the connector.

To check the validity of this model and decide the value of n , we consider a simple problem where eight beams are connected by a connector and arbitrary load cases are considered. All beams have unit width and thickness and lengths of 50 mm. One such case is shown in Fig. 28. Analysis is done for different values of n .Total strain energy and strain energy in the beams are plotted against n.We notice that from 0n to 3n the difference between the two, which is the strain energy in the connector, reduces to almost zero; further

increase in n can lead to numerical errors. Hence a factor of 310 is chosen to multiply to the stiffness matrix in all the forthcoming examples.

Having the above model in hand and using topology optimization technique, optimal topologies are obtained for given specifications for compliant mechanism. Ratio of mutual strain energy (MSE) and strain energy (SE) is considered as objective function to be maximized with respect to width of the beams as design variable ix . A spring is provided at the output port where

the displacement is to be maximized.

35

0

i

d d

x

i

F

MSEMaximize

SEsubject to KU F

KU

V Vx w or

1

2

Td

T

where

MSE U KU

SE U KU

(15)

Fx

Figure 28 (a) A test problem considered for which deformation is shown for n = 0, Fx = 1 N andFy = 2 N (b) Total strain energy and the strain energy in the beams is plotted against n

Fy

36

Since only one standard width is available in the kit, this is in fact a binary optimization problem. But we can consider it as a continuous optimization problem, provided penalization is done on the design variable. So, following penalization is used where p is the penalty factor and K becomes a function of X

px

X ww

(16)

Some benchmark problems are solved using this formulation. MMA (the Method of Moving Asymptotes) algorithm is used for optimization [17]. In all the following examples w is 0.2 mm. Lower bound on x is 1e-3. Material being spring steel, Young’s modulus of 210e3 MPa is used. Out of plane thickness is 1 mm.

Figs. 29-30 show the results of optimization for 1p and 2p respectively for the

inverter mechanism. Volume constraint of 0.4 is used. Spring value is 1e-10 times E. If volume constraint is changed to 0.8 results obtained are shown in Fig. 31 and Fig. 32 for 1p and

2p respectively. Distribution of beam widths is shown in each adjacent figure. We notice that

widths of almost all members are pushed to either 0 or w , not only in the cases where 2p but

also for 1p . The reason is attributed to the bending stiffness of beams which is proportional to

cube of their widths, which acts as inherent penalization. Similar results are obtained if spring stiffness is changed to 1e-6 times E.

When boundary conditions are changed to give a pliers mechanism, results obtained are shown in Fig. 33, where volume constraint is 0.8, spring value is 1e-6 times E and 2p . Using 1p

very much similar result was obtained. From width distribution we notice that a small number of beams have attained intermediate widths. But it is negligible and the mechanism behaves similarly, when those particular beams are deleted. Fig. 34 shows a gripper mechanism obtained for volume constraint 0.8, output spring of 1e-10 times E and 2p . This work is also extended

for obtaining optimal compliant mechanisms for geometric nonlinear analysis. The optimization problem is stated as

int

0

i

out

in

ext

tgt

x

i

U

U

U

Maximize

subject to F F

K U F

V V

x w or

(17)

37

The sensitivity analysis is done by adjoint variable method discussed in [18]. Again the same penalization is used for design variable ix as in Eq. 16. Fig. 35 shows inverter mechanism

obtained for 1p , volume constraint of 0.8 and spring 1e-10 times E. For different values of

p and spring stiffness, same result is obtained. Gripper mechanism obtained for 2p and spring

value 1e-6 times E, is shown in Fig. 36. Distribution of beam widths is shown in the plot next to it. Reinforcement is provided for the beams along the axis of symmetry to prevent buckling.

FdF

Figure 29 Inverter mechanism shown with deformed configuration for F=100 N. Distribution of widthsplotted (right). Displacement amplification is 1.96. The prototype made using the kit is also shownalong with deformed configuration

38

FdF

Figure 30 Inverter mechanism shown with deformed configuration for F=100 N. Distribution of widths plotted (right). Displacement amplification is 2.19. The prototype made using the kit is also shown along with deformed configuration

39

FdF


40

FdF


41

Fd

F

Figure 33 Pliers mechanism shown with deformed configuration for F=500 N. Distribution of widths plotted below. Displacement amplification is 0.85. The prototype made using the kit is shown along with deformed configuration

42

F

Fd

Figure 34 Inverter mechanism shown with deformed configuration for F=300 N. Distribution of widths plotted below. Displacement amplification is 1.3. The prototype made using the kit is also shown along with deformed configuration

43

Fd F

Figure 35 Symmetric half of inverter mechanism obtained with Geometric nonlinear analysis shown with deformed configuration for F=300N. Distribution of widths plotted below. Displacement amplification is 1.28. The prototype made using the kit is shown along with deformed configuration

44

Figure 36 Symmetric half of gripper mechanism obtained with geometric nonlinear analysis shown with deformed configuration for F=300 N. Distribution of widths plotted below. Displacement amplification is 0.69. The prototype made using the kit is also shown along with deformed configuration.

Fd

F

45

8. Conclusion

In this work, an inconsistency between the modeling of small and large-displacement compliant mechanisms using beam/frame finite element (FE) framework and their fabricated prototypes is shown. The discrepancy is quantified by comparing strain energies of beam/frame element FEA and 2-D plane stress analysis using COMSOL Multiphysics for linear and geometrically nonlinear cases. The prototype proves to be stiffer and the reason is attributed to stiffer beam connections. As a remedy, it is suggested to introduce a super-element at beam connection which is an additional finite element constructed by doing geometry computation and separate 2-D plane stress analysis of the joint region. This approach reduces the aforementioned discrepancy. However, for optimal mechanism synthesis large computation is required.

A concept of a compliant mechanism kit is also presented with its prototypes and optimal design method. It consists of deformable beams and rigid connectors, which can be easily hand-assembled using snap-fits. The prototypes are made in spring steel. It enables the user to quickly realize a compliant mechanism with mechanics intuition or comprehend the result of optimal synthesis. The FE model is described where the inconsistency is eliminated and optimal mechanisms are obtained which can be assembled using the kit. This approach thus circumvents the aforementioned joint stiffening problem and provides a research and pragmatic toolset for synthesizing compliant mechanisms as a parallel to the kits available for rigid link mechanisms.

References

1. Ananthsuresh, G.K., “A new design paradigm for Micro-Electro-Mechanical-Systems & Investigations on the Compliant Mechanism Synthesis, PhD thesis,” Univesity of Michigan, 1994.

2. Dinesh, M., and Ananthasuresh, G.K., “A topology-optimized large-range compliant X-Y micro stage,” National Conference on Machines and Mechanisms-109, 2007.

3. Luzhong, Y., and Ananthasuresh, G.K., Eder, J., “Optimal design of a cam-flexure clamp,” Finite Elements in Analysis and Design, Vol.40, Issue 9-10, 2004, pp. 1157–1173.

4. Mankame, N. “Investigations on contact aided compliant mechanisms, PhD Dissertation,” University of Pennsylvania, 2004.

5. Howell, L., “Compliant Mechanisms,” John Wiley & Sons, Inc., New York, 2001.

6. Saxena, A., and Ananthasuresh, G.K., “On an optimal property of compliant topologies,” Struct. Multidisciplinary Optimization, Vol.19, 2000, pp. 36-49.

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7. Sangmesh, R.D., Sahu, D., Dinesh, M., Ananthasuresh, G.K., “A Comparative study of the formulations and benchmark problems for the topology optimization of Compliant Mechanisms,” Journal of Mechanisms and Robotics, Vol.1, 2008, pp. 011003-8.

8. Zettl, B., Szyszkowski, W., Zhang, W.J. “Accurate low DOF modeling of a planar compliant mechanism with flexure hinges: the equivalent beam methodology,” Precision Engineering, Vol.29, 2005, pp. 237–245.

9. Yoon, G., Kim, Y., Bendsoe, M., & Sigmund, O., “Hinge-free topology optimization with embedded translation-invariant differentiable wavelet shrinkage,” Structural and Multidisciplinary Optimization, Vol.27, 2004, pp. 139–150.

10. Yin, L., and Ananthasuresh, G. K., “Design of distributed compliant mechanisms,”Mechanics Based Design of Structures and Machines, Vol. 31, No. 2, 2003, pp. 151–179.

11. Poulsen, T. A “A new scheme for imposing a minimum length scale in topology optimization,” International Journal for Numerical Methods in Engineering, Vol. 57, 2003, pp. 741–760.

12. Wang, M. Y. “A kinetoelastic approach to continuum compliant mechanism optimization,”ASME 2008 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE New York, USA. Paper No. DETC 2008-49426. 2008.

13. Hughes, T. J. R., Brezzi, F., “On drilling degrees of freedom,” Computer Methods in Applied

Mechanics and Engineering, Vol. 72, 1989, pp. 105-121. 14. Hughes, T. J. R., Masud, A., Harari, I. “Numerical assessment of some membrane elements with

drilling degrees of freedom,” Computers and Structures, Vol. 55, No. 2, 1995, pp. 297-314. 15. Darren Engwirda, “Unstructured Mesh Generation (Mesh2D v2.3)” http://www.mathworks.com,

2008. 16. Sivanagendra, P., “Geometrically Nonlinear Elastic Analysis of Frames with Application to

Vision-Based Force-sensing and Mechanics of Plant Stems, M.E. Thesis,” Indian Institute of Science, 2006.

17. Svanberg, K., “The method of moving asymptotes – a new method for structural optimization,” International Journal for Numerical Methods in Engineering, Vol. 24, 359–373

18. Saxena, A., and Ananthasuresh, G.K., “Topology Synthesis of Compliant Mechanisms for

Nonlinear Force-deflection and Curved Path Specifications,” Journal of Mechanical Design, Vol. 123, 2001, pp. 33-42.

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