Topology optimization of compliant circular-path mechanisms. based on an aggregated linear system and singular value.

Transcription

1 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2000; 00:1 6 [Version: 2002/09/18 v2.02] Topology optimization of compliant circular-path mechanisms based on an aggregated linear system and singular value decomposition A. Takezawa 1,, and M. Kitamura 1 1 Division of Mechanical System and Applied Mechanics, Faculty of Engineering, Hiroshima University, Kagamiyama, Higashi-Hiroshima, Hiroshima , Japan SUMMARY This paper proposes a topology optimization method designing compliant circular path mechanisms, or compliant mechanisms having a set of output displacement vectors with a constant norm, which is induced by a given set of input forces. To perform the optimization, a simple linear system composed of an input force vector, an output displacement vector and a matrix connecting them is constructed in the context of a discretized linear elasticity problem using FEM. By adding constraints that the dimensions of the input and the output vectors are equal and the Euclidian norms of the all local input force vectors are constant, from the singular value decomposition of the matrix connecting the input force vector and the output displacement vector, the optimization problem which specifies and equalizes the norms of all output vectors is formulated as a minimization problem of the weighted summation of the condition number of the matrix and the least square error of the second singular value and the specified value. This methodology is implemented as a topology optimization problem Correspondence to: Hiroshima University, Kagamiyama, Higashi-Hiroshima, Hiroshima , Japan akihiro@hiroshima-u.ac.jp Received Copyright c 2000 John Wiley & Sons, Ltd. Revised

2 2 A. TAKEZAWA AND M. KITAMURA using SIMP method, sensitivity analysis and MMA. The numerical examples illustrate mechanically reasonable compliant circular path mechanisms and other mechanisms having multiple outputs with a constant norm. Copyright c 2000 John Wiley & Sons, Ltd. key words: compliant mechanisms; path generation; topology optimization; singular value decomposition; finite element method; sensitivity analysis 1. INTRODUCTION Compliant mechanisms have a tremendous amount of potential in various engineering fields. They transfer or transform input motions to output motions by whole body flexibility, doing away with the need for mechanical joints. Such devices have advantages in reduction of the total number of mechanical parts, and consequent minimization of noise, size and backlash. These advantages are particularly beneficial in the fabrication of micro-mechanisms. Various compliant mechanisms having equivalent functionality to conventional rigid systems are under development. Howell [1] lists examples of a crimping mechanism, an overrunning clutch, a slider mechanism, and a bi-stable mechanism. Other published examples are a reciprocalcurve mechanism [2], a Dwell mechanism [3], a parallel-guiding mechanism [4], a crank rocker mechanism [5], and a variable stroke mechanism [6]. The fundamental mechanical aspects and design theories for compliant mechanisms are sufficient mature to appear in textbooks [1, 7]. Topology optimization methods can greatly assist in the development of this type of device because they allow an objective fundamental structural optimization coping with changes in topology. The technique is even suitable for difficult design problems involving the optimization of complex mechanisms. The basic method targets maximization of the output displacement

3 OPTIMIZATION OF COMPLIANT CIRCULAR PATH MECHANISMS 3 for a given input load [8, 9, 10], these optimization methods have been extended to various specialized function devices. For example, multiple input-output mechanisms [11], optimization based on geometrical non-linear analysis applied to snap-through mechanisms [12, 13], path generation based on geometrical non-linear analysis [14, 15, 16, 17], optimization of mechanism dynamic responses [18, 19], and optimization of multiphysics actuators [20, 21, 22] have been reported. In this paper, we particularly focus on the optimization of mechanism paths. Initial strategies for path optimization used in [14, 15, 16, 17] are based on the minimization of errors between actual displacement and target displacement at evaluation points forming the path. In this paper, we consider 2D compliant mechanisms exhibiting circular input, and/or output motions. Figure 1 shows examples of these mechanisms implemented using conventional rigidlinks. Such mechanisms have many applications. The Circular-circular mechanism is used in the transmission of rotational motion. Reciprocal-circular motion is fundamental mechanism of reciprocating engines [23]. Circular-reciprocal motion can be used to transform rotation into the action of a manipulator. Compliant mechanisms are particularly appropriate for use in micro-scale applications. However, such complicated mechanisms are difficult to generate by current topology optimization techniques as they generate optimal paths based on evaluation points. Since many evaluation points are required to generate a circular path, the optimization problem can be a very large one, with consequent difficulties in obtaining convergence.

4 4 A. TAKEZAWA AND M. KITAMURA (a) A circular-circular mechanism (b) A reciprocal-circular or circularreciprocal mechanism Figure 1. Examples of mechanisms characterized by circular motion To generate compliant mechanisms that include circular motion, the analysis is conducted using a different methodology to the conventional evaluation point based path generation technique. In a 2D plane, a circular shape can be represented as a set of vectors with a constant norm. That is, a circular path can be traced if all output displacements at a specified point of the compliant mechanism have the same norms for a set of given input forces. To achieve this characteristic, we must optimize the norms of the output vectors induced by given multiple input vectors. We have previously encountered a similar optimization problem, considering norms of local input and output vectors for the design of load cells [24] and maximizing stiffness for the worst load case [25]. We considered a simple linear system composed of local input and output vectors and constructed a matrix connecting these vectors. This equation represents the direct relationship between the set of local inputs and outputs. Since the performance of the structure is independent from input vectors and concentrated in the matrix connecting the vectors, it can be evaluated by a spectrum analysis of the matrix. The technique of reformulating large systems into small systems is called the aggregation approach, and in

5 OPTIMIZATION OF COMPLIANT CIRCULAR PATH MECHANISMS 5 the field of control research, it is used for reducing the dimensions of control systems or state equations, and extracting the elements required for incomplete state feedback control (see e.g. [26, 27]). In this paper, we propose a new optimization method for 2D compliant circular path mechanisms, or compliant mechanisms having a set of output displacement vectors with a constant norm, which is induced by a set of given input forces. A 2D linear elastic problem is first considered and the problem is discretized using the finite element method. An aggregated linear system composed of the input force vector and the output displacement vector is defined and a matrix connecting them is constructed based on the discretized linear elastic system. Adding constraints that the dimensions of the input force vectors and the output displacement vectors are equal and the Euclidian norms of the all local input force vectors are constant, from the singular value decomposition of the matrix connecting the input force vector and the output displacement vector, the optimization problem which specifies and equalizes the norms of all output vectors is formulated as the minimization problem of the weighted summation of the condition number and the least square error of the second singular value and the target value of the matrix. The proposed compliant mechanism optimization method is implemented by the commonly used solid isotropic material with penalization (SIMP) method of topology optimization. Updating the density function of the SIMP method is performed based on sensitivity analysis and the method of moving asymptotes (MMA) [28], this optimizer approach has numerous benefits in various optimization problems by virtue of its combination with topology optimization. The numerical examples provided illustrate mechanically reasonable compliant circular path mechanism structures and demonstrate an extension of our method for other mechanisms.

6 6 A. TAKEZAWA AND M. KITAMURA 2. FORMULATION 2.1. Linear elasticity problem In this research, we consider an optimization method of compliant mechanisms in the context of 2D linear elasticity problems. Let Ω be the domain that varies during the optimization process, which is made of a homogenized isotropic material with an elastic tensor A. We let f and g be the given body force and surface force vector function respectively. The following variational form state equation is then formulated for state variable u: Ae(u) : e(v)dx = f vdx + g vds Ω Ω Ω N (1) for u V, v V, f L 2 (Ω 2 ) 2, g L 2 ( Ω 2 N ) 2 where: V = {v H 1 (Ω) 2 v = 0 on Ω D }, (2) e(u) = 1 2 ( u + ( u) T ), (3) where v is the test function, e is the strain tensor and V is a Sobolev space. We construct our method based on matrix spectrum methods. The linear system is first discretized by the finite element method. The discretized equilibrium equation in Eq.(1) is formulated as follows: KU = F (4) where K is the stiffness matrix, U and F are the discretized displacement and force vectors Definition of compliant circular path mechanisms In this paper, we propose a new optimization method for circular path compliant mechanisms. If all the local displacement vectors at a specified output point of a compliant mechanism,

7 OPTIMIZATION OF COMPLIANT CIRCULAR PATH MECHANISMS 7 which is induced by a given set of input forces have the same norms, the mechanism can be said to have a circular path as shown in Fig.2. Thus, we first define circular path compliant mechanisms as mechanisms which have the set of output displacement vectors with a constant norm, which is induced by the set of given input forces. S et o f o u t p u t dis p l a c em w it h t h e s a m e n o r m en t s S et o f in p u t f o r c es? Fixed Figure 2. An outline of a compliant mechanism with a circular path 2.3. Derivation of the aggregated system To achieve the characteristic, we evaluate the direct relationship between the input force and output displacement vectors. We use a technique from the control research field called aggregation - extracting a small system from a large system (see e.g. [26, 27]). We set the input force vector (local force vector) F l with m dimension and the output displacement vector (local displacement vector) U l with 2 dimensions corresponding to x and y components. Letting the degrees of freedom of the global linear system be n, F l and U l are formulated as follows: F = H f F l, F l = H f T F (5) U l = H u T U (6)

8 8 A. TAKEZAWA AND M. KITAMURA where H f is the n m matrix connecting the local and global force vectors and H u is the n 2 matrix connecting the local and global displacement vectors. The reason for retaining both F = H f F l and F l = H T f F in Eq.(5) is that the elements of the vector F are all zero except for the elements corresponding to F l. Using Eq.(4), Eq.(5) and Eq.(6), U l is also formulated as follows: U l = H u T U = H u T K 1 F (7) = H u T K 1 H f F l = CF l where: C = H u T K 1 H f (8) Where C is the 2 m matrix. Eq.(7) is an aggregated system of the discretized equilibrium equation in Eq.(4), which directly shows the relationship between F l and U l. We discuss the evaluation method for the relationship based on the singular value decomposition for the matrix C in the next subsection Relationship between norms of vectors and singular values of the matrix C Here we evaluate the norms of the output displacement vectors U l based on Eq.(7). Adding a constraint that the Euclidian norms of the all local input force vectors, F l, are constant, the relationship among F l, U l and the singular values of the matrix C follows(see e.g. Chapter 7 in [29]):

9 OPTIMIZATION OF COMPLIANT CIRCULAR PATH MECHANISMS 9 max F l U l F l = σ max (9) min F l U l F l = σ min if rank(c) = m 0 if rank(c) < m (10) where U l = CF l (11) Where σ max and σ min are the maximum and minimum singular values of C. That is, the ratio of the norms of F l and U l can be evaluated using the singular value of the matrix C. Following the constraint that F l is constant, the above relationship directly shows the range of the norms of the output displacement vectors U l. Here, we consider the case that the rank of C is equal to the dimension of the input force vector, that is, rank(c) = m in Eq.(10). Since we are considering 2D compliant circular path mechanisms in this paper, the dimension of the output displacement vectors is 2. Therefore, to satisfy rank(c) = m, C must be a 2 2 matrix and have full rank and the dimension of the input force vectors must be 2 (m = 2). In that case, the matrix C has only two singular values and the first singular value σ 1 and the second singular value σ 2 correspond to σ max and σ min respectively and the minimum value of U l / F l equals σ 2. The relationship between the set of F l with the unit norm and U l (= CF l ) can be seen in Fig.3. The set of F l with a constant norm clearly represents the unit circle. The circle is mapped to another space by the matrix C as an ellipsoid. The lengths of the minor and the major axis of the ellipse correspond to the second and first singular values. From the figure, we can see that a circular path can be achieved if the first and second singular values of the matrix C are equal. If the rank of

10 10 A. TAKEZAWA AND M. KITAMURA C is not equal to the dimension of the input force vector, the minimum value of U l / F l becomes 0 and the relationship shown in Fig.3 cannot follow. We have introduced the conditions for the optimization of compliant circular path mechanisms which can be summarized as follows: 1. F l is fixed for all F l. 2. F l has 2 dimensions. 3. C (2 2 matrix) has full rank. 4. The first and second singular values of C are equal. (The condition number of C is 1.) Conditions 1 and 2 must be included in problem definition. We must optimize structures for conditions 3 and 4. It should be noted that compliant circular path mechanisms can be achieved in other ways besides the example case satisfying the above conditions: these are particular conditions related to the given example. However, these numerical examples demonstrate the utility of our methods for design of general useful mechanisms. Figure 3. Illustration of mapping from F l to U l by the matrix C

11 OPTIMIZATION OF COMPLIANT CIRCULAR PATH MECHANISMS Objective function From the summarized conditions for compliant circular path mechanisms discussed in the last subsection, we formulate the objective function. First we consider condition 3, in which the 2 2 matrix C has full rank. Since the rank of a matrix equals the number of positive singular values of the matrix, the second singular value must be positive. We specify the second singular value σ 2 to be a positive value σ 0 by the following objective function which is the least square error between these values: minimize (σ 2 σ 0 ) 2, σ 0 > 0 (12) After condition 3 is satisfied, condition 4 must be satisfied to fix U l for any F l with a constant given norm. The ratio between the maximum and minimum singular values is called the condition number and is commonly used in matrix analyses. When the condition number is equal to the minimum value 1, condition 4 is satisfied. Thus, we formulate another objective function as follows: minimize σ 1 σ 2 (13) When conditions 3 and 4 are simultaneously satisfied, the first and second singular values correspond to the radius of the circular path. Thus, the objective function in Eq.(12) also specifies the radius of the circular path. Finally, using the weighting coefficient method for the two criteria in Eq.(12) and Eq.(13), the objective function is formulated as follows: where w c and w s are weighting coefficients. minimize w c σ 1 σ 2 + w s (σ2 σ 0 ) 2 (14)

12 12 A. TAKEZAWA AND M. KITAMURA 2.6. Analysis of phase difference Since only the norm of the displacement is considered in the above objective function, the phase change from the input force vector to the output displacement vector cannot be specified. However, the phase difference can be calculated analytically. The singular value decomposition of the matrix C is as follows: C = V ΣW T (15) Where V and W are 2 2 orthogonal matrices and Σ is the diagonal matrix having σ 1 and σ 2 as its elements. Thus, the transformation from F l into U l by C are divided into following three steps: 1, Rotation of coordinate system by W T (F l and contraction of F l = W T F l ). 2, Expansion along x and y directions by Σ (F l = ΣF l ). 3, Rotation of coordinate system by V (U l = V F l ). Where F l and F l indicate intermediate matrices during the above transformation and rotation of coordinate system includes transformation from the left-handed coordinate system into the right-handed one. Figure 4 shows an illustration [ ] of the above transformation when C = which is decomposed as V ΣW T = [ ] [ 1 2 ] ] [ ] 3 2 [ ] = [ ] cos π 6 sin π 6. V and W T represent π/ [ cos π 3 sin π 3 sin π 3 cos π 3 sin π 6 cos π 6 and π/6 rotations of the coordinate system. In Fig.4, α is the initial phase of F l and β represents phase change from F l into F l upon the expansion and contraction of the coordinate system by Σ.

13 OPTIMIZATION OF COMPLIANT CIRCULAR PATH MECHANISMS 13!" # Figure 4. Detailed illustration of mapping from F l to U l by the matrix C When the matrix C has the same singular values as the ideal optima of the proposed methodology, the phase change β by the matrix Σ does not yield since the space is expanded or contracted along x and y directions with the same rate. Then, the phase change is calculated as the sum of the two rotations by the matrices V and W T Topology optimization We use a topology optimization method as an optimizer of the linear elastic domain Ω because the method can perform the most fundamental optimization of the domain including the shape and the number of holes. The fundamental idea of topology optimization is to introduce a fixed, extended design domain D that includes, a priori the optimal shape Ω and use of the following

14 14 A. TAKEZAWA AND M. KITAMURA characteristic function: 1 if x Ω χ(x) = 0 if x D \ Ω (16) Using this function, the original design problem of Ω is replaced by a material distribution problem incorporating an elasticity tensor, χa, in the extended design domain D, where A is the elasticity tensor of the original material of Ω. Unfortunately, the optimization problem does not have any optimal solutions in L (D; {0, 1})[30]. Thus some regularization techniques must be introduced to obtain optimal solutions. A homogenization method is used to perform the relaxation of the solution space [30, 31]. In this way, the original material distribution optimization problem with respect to the characteristic function is replaced by an optimization problem of the composite consisting of the original material and a very weak material imitating voidage with respect to the density function. This density function represents the volume fraction of the original material and can be regarded as a weak limit of the characteristic function. In the optimization problem, the relationship between the material properties of the composite and the density function must be defined. Several approaches have been proposed, for example, a simple microstructure can be defined and used to calculate material properties using an asymptotic method [31, 32]. An alternative approach is to set a completely artificial material property [33, 34, 35], this is known as the solid isotropic material with a penalization (SIMP) method, and is the most popular method in topology optimization. We use the SIMP method in this research. In this method, the material property is defined using the following simple equation with the penalized material density: A = ρ p A (0 ρ 1) (17)

15 OPTIMIZATION OF COMPLIANT CIRCULAR PATH MECHANISMS 15 where A is the material property of the composite, ρ is the density function representing the volume fraction of the original material and p is a positive penalization parameter. This method has the advantage of controlling the non-linearity between the material property of the composite and the density function. This has the significant benefit of avoiding the gray domain between material and voidage. Finally, adding a volume constraint and introducing the vector ρ of the discretized density function, the topology optimization problem of the discretized linear elastic domain composed of N finite elements for compliant circular-path mechanism is formulated as follows: minimize ρ w c σ 1 σ 2 + w s (σ2 σ 0 ) 2 (18) where C = H T u K 1 H f (19) F = H f F l (20) U = H u U l (21) 0 ρ i 1, for i = 1,..., N (22) V olume(ρ) V U (23) where V olume( ) denotes the function calculating the volume of the domain, and V U is the upper limit of the volume. 3. NUMERICAL IMPLEMENTATION 3.1. Algorithm The optimization procedure is as follows:

16 16 A. TAKEZAWA AND M. KITAMURA 1. Set an initial shape. 2. Iterate the following procedure until convergence: (a) Calculate the matrix C by the finite element method. (b) Calculate the singular values and condition number of C, objective function and the total volume. (c) Calculate the sensitivities of the objective function and the total volume. (d) Based on the sensitivities, update the design variables using the method of moving asymptotes (MMA) [28] Computation of the matrix C Here, we consider the computation of C in Eq.(8). Since the equation contains the inverse matrix of the stiffness matrix, the calculation requires a significant amount of computation. We reformulate the equation by introducing the adjoint variable matrix Z u and Z f which are n 2 matrices (n:degrees of freedom of the global system) as follows: C = H u T K 1 H f = Z u T H f (24) = Z u T KZ f where: KZ u = H u (25) KZ f = H f (26) Note that, we do not need to introduce the adjoint variable Z f only to calculate C, it is also used in sensitivity analysis described in the next subsection.

17 OPTIMIZATION OF COMPLIANT CIRCULAR PATH MECHANISMS 17 In our method, C is 2 2 matrix and the computational cost for calculating its singular values is almost ignorable. The main computational costs for the adjoint variable matrix Z u and Z f are reasonable, being is equal to four times of the cost for solving the ordinal equilibrium equation in Eq.(4). Since the stiffness matrix is constant in these four linear systems, the computational cost can be reduced further if we use additional techniques such as first performing the LU decomposition of K and repeating only forward and backward substitutions to solve these linear systems Sensitivity analysis As shown above and because we use MMA to update design variables, first order sensitivities of the objective function are required. The derivative of the objective function in Eq.(14) with respect to i-th discretized density function ρ i is derived as follows: ( ) f 1 σ1 σ 2 σ 2 σ 0 σ = w c σ ρ i σ 2 2 σ 1 + w s 2 (27) 2 ρ i ρ i (σ2 σ 0 ) 2 ρ i Clearly, we must calculate the derivative of singular value. We use following relationship between the singular value and the eigenvalue (see e.g. Chapter 7 in [29]). σ j (C) = λ j (C T C) (28) Let σ j ( ) and λ j ( ) denote the j-th singular value and j-the eigenvalue of the matrix respectively. That is, the sensitivity of the j-th singular value with respect to the density function ρ i is obtained as follows using the sensitivity of the j-th eigenvalue: ( σ j (C) 1 λ j C T C ) = (29) ρ i λj (C T C) ρ i

18 18 A. TAKEZAWA AND M. KITAMURA The sensitivity of λ j ( C T C ) with respect to ρ i is obtained as follows [36]: λ j ( C T C ) ρ i where φ j is normalized j-th eigenvector. T = φ ( C T C ) j ρ i ( T C T = φ j φ j ρ i C + C T C ρ i ) φ j (30) If the above j-th eigenvalue is a repeated eigenvalue, this sensitivity cannot be used, since repeated eigenvalues have only directional derivatives. In this case, the sensitivities are obtained as results from the following eigenvalue problem [36]. Ma = λ ( j C T C ) a, M R s s, a R s (31) ρ i where ( T C T M kl = φ k ρ i C + C T C ρ i ) φ l, k, l = 1,..., s (32) where s is the number of repeated eigenvalues, M is the s s matrix whose components are represented by Eq.(32), and a is the s dimensional eigenvector representing the derivative directions. We must calculate C/ ρ i here. The derivatives of both sides of Eq.(7) are first calculated as follows: C ρ i F l = U l ρ i = H u T U ρ i (33) The derivative of the global displacement vector U can be obtained as follows from Eq.(4), U 1 K = K U (34) ρ i ρ i

19 OPTIMIZATION OF COMPLIANT CIRCULAR PATH MECHANISMS 19 Substituting Eq.(34) into Eq.(33) and using Eq.(5), C F l = H T 1 K u K U ρ i ρ i = H T 1 K u K K 1 F ρ i = H T 1 K u K K 1 H f F l ρ i (35) Substituting the adjoint variable matrices in Eq.(25) and Eq.(26), the following equation is finally obtained. C T K F l = Z u Z f F l (36) ρ i ρ i That is, C T K = Z u Z f (37) ρ i ρ i 3.4. Filtering method We used the SIMP method as our topology optimization method. In the 2D problem, the SIMP method can encounter a numerical instability known as checkerboard patterns [37, 38]. One way to prevent this problem is to introduce so-called filtering techniques (see [39] and references therein). We use the projection method [40] as a filter. This method sets the design variables in addition to the density function and projects the design variable onto the density function using a projection function. By adjusting the effective range and shape of the function, we avoid the checkerboard problem. We consider a weighted average function using a simple linear projection function as follows: µ(d(x p )) = Ω p dw(x x p )dx Ω p w(x x p )dx (38)

20 20 A. TAKEZAWA AND M. KITAMURA where: Ω p (x p ) = {x x x p r min, x D} (39) where Ω p is the circular effective area of the projection function of a design variable d at x p, d is the design variable function, r min is a radius of the effective area of the projection function and w is the following linear weight function: r min x x e if x Ω p w(x x p ) = r min (40) 0 if x D \ Ω p The above functions are calculated numerically by appropriate discretization. In the original paper [40], the discretized design variable function was set to the nodes of the finite element mesh. We discretize the function at the center of the element for simplicity in the sensitivity calculations. 4. NUMERICAL EXAMPLES The following numerical examples are provided to confirm the utility of the proposed method. In all examples, a 1[mm] polyethylene plate with Young s modulus E of 1.4[Gpa] and Poisson s ratio ν of 0.4 is used as the subject material. In addition, a ground spring with a spring constant of [N/m] is set as the input point in each example. To imitate displacement inputs as force inputs, the spring constant is set to a large value relative to the norms of the stiffness matrices of the structures. When the input force vectors have the norms [N], 1[mm] displacements are yielded at the input point. The ground spring with a spring constant of [N/m] is also set at the output point in each example. The penalization parameter p in Eq.(17) is set to 3. The parameter r min for the projection method used in Eq.(39) is set

21 OPTIMIZATION OF COMPLIANT CIRCULAR PATH MECHANISMS 21 to 1.5 times the mesh size. Quadrangular isoparametric elements are used for discretizations by the finite element method. All optimal configurations are plotted as the distribution of a filtered density function of the optimal results A circular input and output example As a benchmark problem of the circular path compliant mechanism optimization, optimization of a compliant mechanism transforming a circular input motion into a circular output motion is performed as illustrated in Fig.5. In an ordinary articulated mechanism, this function is performed by, for example a parallelogram linkage. Our method can generate mechanisms having output displacement vectors with a constant norm, which is induced by the set of input force vectors with a constant norm. By setting the output displacement and input force vectors as 2 dimensional vectors composed of the x and y elements of the displacement or the force at each point, these vectors with fixed norms represent circles. The design domain is a 20[mm] 10[mm] rectangle with a fixed boundary condition on the left hand side. The applied point of an input force set is set at the top of the right side. The output point is set at the center of the bottom. The target value of the second singular value is set to be (= Target ratio of expansion or contraction of displacement/ground spring rate at the input point = 0.43/ ). The volume constraint is set to 30% of the total volume. The domain is discretized with a rectangular mesh. The initial value of the density function is 0.3 in all areas of the domain. The both weighting coefficients w c and w s in Eq.(14) are set to 0.5.

22 22 A. TAKEZAWA AND M. KITAMURA Figure 5. Design domain for a circular input and circular output mechanism Figure 6 shows the optimal configuration obtained after 100 iterations. Figure 7 shows the convergence history of the first and second singular values of the matrix C. Although some oscillations are observed during the optimization process, both singular values finally converge to the target value. Figure 8 shows the deformed shapes of the optimal configuration for some input forces. The input forces are set to be F l = (cos θ, sin θ) T, (θ = π 6 i, i = 0,..., 11). The scale factor of the displacements is 3.0. In these figures, circles with radii 3.0[mm] ( [mm]) and 1.2[mm] ( [mm]) are displayed on the input and output points respectively. The output point is also indicated by an arrow. We can see the input and output points tracked these circles and the optimal configuration worked as the mechanism transforming circular input motion into circular output motion. In typical articulated mechanisms, circular input and output with the same radii can be easily performed by parallelogram linkages. However, to perform the function of such mechanism with different radii between the input and output, additional mechanical devices such as sliders are required. Our compliant mechanism can perform this function with a single component.

23 OPTIMIZATION OF COMPLIANT CIRCULAR PATH MECHANISMS 23 Figure 6. Optimal configuration of the circular input and circular output mechanism 6x10-7 5x10-7 Singular value 4x10-7 3x10-7 2x10-7 1x10-7 First (largest) singular value Second singular value Iteration Figure 7. The convergence history of the singular values

24 24 A. TAKEZAWA AND M. KITAMURA (a) θ = 0 (b) θ = π 6 (c) θ = π 3 (d) θ = π 2 (e) θ = 2π 3 (f) θ = 5π 6 (g) θ = π (h) θ = 7π 6 (i) θ = 4π 3 (j) θ = 3π 2 (k) θ = 5π 3 (l) θ = 11π 6 Figure 8. Deformed shapes for the set of input forces F l = (cos θ, sin θ) T

25 OPTIMIZATION OF COMPLIANT CIRCULAR PATH MECHANISMS 25 The phase difference between the input and output vectors are analyzed. The matrix C corresponding to the optimal configuration is [ ] which is deconposed as V ΣW T = [ ] [ ] [ ] The matrices V and W T are corresponding to and rotations. Thus, analytical phase change is Figure 9 shows the plots of the phases of input and output vector. The results show that the numerical phase difference is mostly corresponding to the analytical phase difference. 5x10-4 U lx Output displacement 0 U ly -5x π/2 π 3π/2 2π Phase of input load Figure 9. Phase difference between input force and output displacement 4.2. Example showing double reciprocal inputs with a circular output In our method, the norms of the input force vector and output vector are specified. The straightforward use of the method is for a circular input and circular output mechanism of the type shown in the first example. However, our method can be extended to other mechanisms with non-circular motion under the constraints of the vectors norms. Here, we optimize a mechanism transforming double reciprocal motions to a circular motion. The design domain is a 10[mm] 20[mm] rectangle as shown in Fig.10. The applied forces are assumed to be a

26 26 A. TAKEZAWA AND M. KITAMURA horizontal force at the top of the right side and a vertical force at the bottom of the right side. The components of local force vector are the x component of the force at the top of the right side and the y component of the force at the bottom of the right side. At the input points, to perform a reciprocal motion, the slide boundary conditions are introduced. The output displacement point is set at the center of the left side. The components of the output displacement vector are x and y components of the local displacement at the output point as the same with the first example. The target value of the second singular value is set to be (= Target ratio of expansion or contraction of displacement/ground spring rate at inputpoint = 0.65/ ). The volume constraint is set to 40% of the total volume. The domain is discretized with a rectangular mesh. The initial value of the density function is 0.4 in all areas of the domain. The weighting coefficients w c and w s in Eq.(14) are set to 0.01 and 0.99 respectively.

27 OPTIMIZATION OF COMPLIANT CIRCULAR PATH MECHANISMS 27 Figure 10. Design domain for double reciprocal inputs with a circular output mechanism Figure 11 shows the optimal configuration obtained after 300 iterations. Figure 12 shows the convergence history of the first and second singular values of the matrix C. Both singular values were finally converged to the target value. Figure 13 shows the deformed shapes of the optimal configuration for the following set of input forces. The input forces with unit norm are set to be F l = (cos θ, sin θ) T, (θ = π 6 i, i = 0,..., 11). The first and second elements of the input force vector are corresponding to the x component of the horizontal force on the top of the right side and the y component of the vertical force on the bottom of the right side. The displacement scale factor is 3.0. In these figures, a circle with radius 1.95[mm] ( [mm]) is displayed on the output point. The output point is also indicated by an arrow. Input forces are shown by dotted arrows at the input points in each figure. We can see the output points tracked the circle although input forces are simple vertical and horizontal linear motion: the

28 28 A. TAKEZAWA AND M. KITAMURA optimal configuration successfully converts the motion. This example shows that a compliant circular motion mechanism can be generated for non-circular inputs by our method. Figure 11. Optimal configurations for a double reciprocal input with circular output mechanism 6x10-7 5x10-7 First (largest) singular value Second singular value Singular value 4x10-7 3x10-7 2x10-7 1x Iteration Figure 12. The convergence history of the singular values

29 OPTIMIZATION OF COMPLIANT CIRCULAR PATH MECHANISMS 29 (a) θ = 0 (b) θ = π 6 (c) θ = π 3 (d) θ = π 2 (e) θ = 2π 3 (f) θ = 5π 6 (g) θ = π (h) θ = 7π 6 (i) θ = 4π 3 (j) θ = 3π 2 (k) θ = 5π 3 (l) θ = 11π 6 Figure 13. Deformed shapes for the set of input forces F l = (cos θ, sin θ) T

30 30 A. TAKEZAWA AND M. KITAMURA The matrix C corresponding to the optimal configuration is [ ] which is deconposed as V ΣW T = [ ] [ ] [ ] Since the matrices V and W T correspond to and 1.67 rotations, the analytical phase change is Figure 14 shows plots of the phases of input and output vectors. The results show that the numerical phase difference is mostly due to the analytical phase difference. 8x10-4 Output displacement 6x10-4 4x10-4 2x x x10-4 U lx U ly -6x x π/2 π 3π/2 2π (=π/2-0.78) Phase of input load Figure 14. Phase difference between input force and output displacement 4.3. Example with circular input and double reciprocal outputs Similarly to the second example, we extend our method for the design of another mechanism including non-circular motion. Here, we assume double reciprocal output motions induced by a circular input. Since the norms of the displacement vectors will be fixed as a specified value, the output points will alternately move in and out. The design domain is a 20[mm] 20[mm] square as shown in Fig.15. The applied forces are assumed to be at the center of the right side. The components of the input force vector are x and y components of the local force vector at the input point. The output displacement points are set at the top and the bottom of the left side. A

31 OPTIMIZATION OF COMPLIANT CIRCULAR PATH MECHANISMS 31 horizontal output motion and a vertical motion are assumed to be on the top and the bottom of the left side respectively. That is, the components of output vector are the x components of the displacement at the bottom of the left side and the y component of the displacement at the top of the left side. At output points, to perform a linear motion, the slide boundary conditions are introduced. The target value of the second singular value is set to be (= Target ratio of expansion or contraction of displacement/ground spring rate at inputpoint = 1.0/ ). The volume constraint is set to 40% of the total volume. The domain is discretized with a rectangular mesh. The initial value of the density function is 0.4 in all areas of the domain. The weighting coefficients w c and w s in Eq.(14) are set to 0.1 and 0.9 respectively. Figure 15. Design domain for a circular input and double reciprocal outputs mechanism Figure 16 shows the optimal configuration obtained after 100 iterations. Figure 17 shows the convergence history of the first and second singular values of the matrix C. Both singular

32 32 A. TAKEZAWA AND M. KITAMURA values finally converge to the target value as in the previous examples. Figure 18 shows the deformed shapes of the optimal configuration for the following set of input forces. The input forces with unit norm are set to be F l = (cos θ, sin θ) T, (θ = π 6 i, i = 0,..., 11). The scale factor of the displacements is 3.0. In these figures, the circle with radius 3.0 ( ) representing the set of the input force vectors is displayed on the input point. Due to the ground spring, the circle can be approximately regarded as the set of input displacements. Output displacements are shown by dotted arrows on output points in each figure. From these figures, we can see the optimal configuration successfully converts the motion. Figure 16. Optimal configurations of the circular input and double reciprocal outputs mechanism

33 OPTIMIZATION OF COMPLIANT CIRCULAR PATH MECHANISMS 33 8x10-7 6x10-7 Singular value 4x10-7 2x10-7 First (largest) singular value Second singular value Iteration Figure 17. The convergence history of the singular values

34 34 A. TAKEZAWA AND M. KITAMURA (a) θ = 0 (b) θ = π 6 (e) θ = 2π 3 (f) θ = 5π 6 (i) θ = 4π 3 (j) θ = 3π 2 (c) θ = π 3 (g) θ = π (k) θ = 5π 3 (d) θ = (h) θ = (l) θ = π 2 7π 6 11π 6 Figure 18. Deformed shapes for the set of input forces Fl = (cos θ, sin θ)t [ ] 7 The matrix C corresponding to the optimal configuration is 5.46 which is [ ] [ ] [ ] 0 decomposed as V ΣW T = Since the matrices V and W T are corresponding to and 0.32 rotations, analytical phase change is Figure 19 shows plots of the input and output phase vectors. The results show that as with the other Copyright c 2000 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2000; 00:1 6

35 OPTIMIZATION OF COMPLIANT CIRCULAR PATH MECHANISMS 35 examples, the numerical phase difference is mostly due to the analytical phase difference. 1.5x10-3 Output displacement 1x10-3 5x x x10-3 U lx U ly -1.5x π/2 π 3π/2 2π Phase of input load Figure 19. Phase difference between input force and output displacement 5. CONCLUSION We have proposed a new optimization method for 2D compliant circular path mechanisms, or compliant mechanisms having a set of output displacement vectors with a constant norm, which is induced by a set of input forces. We achieved the optimization by constructing a simple linear system composed of an input force vector, an output displacement vector and a matrix connecting them. To construct such aggregated system, a linear elasticity problem is first discretized by the finite element method. The aggregated linear system was then defined based on the discretized linear elasticity system. Adding constraints that the dimensions of the input force vectors and the output displacement vectors are equal and the Euclidian norms of the all local input force vectors are constant, from the singular value decomposition of the matrix connecting the input force vector and the output displacement vector, the optimization problem which specifies and equalizes the norms of all output vectors was formulated as a minimization

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