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Design of a spatial compliant translational joint
Mechanism and Machine Theory ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Design of a spatial compliant translational joint Tai-Shen Yang a, Po-Jen Shih b, Jyh-Jone Lee a,n a b
Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan Department of Civil and Environmental Engineering, National University of Kaohsiung, Kaohsiung, Taiwan
a r t i c l e i n f o
abstract
Article history: Received 12 August 2015 Received in revised form 15 August 2016 Accepted 20 August 2016
This paper presents the design of a novel compliant translational joint with large stiffness ratio and small axis drift. The characteristics of compliant translational joints based on a leaf-spring type joint are investigated. Several types of constructions are analyzed. A novel three-dimensional compliant translational joint is proposed and a design analysis is conducted based on the parametric models. Subsequently, a design optimization is employed with the aid of finite element method. Then, a prototype of the optimized design is fabricated by 3D printing, and tests of specified properties are conducted. Comparisons of the experimental results and simulation results show that this new joint satisfies the design requirements. & 2016 International Federation for the Promotion of Mechanism and Machine Science Published by Elsevier Ltd. All rights reserved.
Keywords: Compliant translational joint Three-dimensional Axis drift Stiffness ratio Optimization
1. Introduction Conventional joints formed by rigid links, such as the revolute pair, the prismatic pair, and ball-and-sockets, are commonly used to connect mechanical parts. However, such joints may contain clearance between the pairing elements and lead to inaccurate positioning during motion. Furthermore, the relative motions between the pairing elements generate friction that results in wear and reliability issues. To overcome these drawbacks, design of the mechanism with compliant joints is proposed. Such mechanism usually consists of a series of rigid links connected by compliant elements and is designed to produce a prescribed motion when a force is applied. Since the mechanism is typically made from a monolith, there exits no clearance between rigid links. Thus, the use of compliant joints can eliminate the presence of friction, backlash, and wear in a mechanism. Many types of mechanisms with compliant joints have been developed over the past decades. Smith [1,2] classified compliant joints in a mechanism into two types, namely, the notch-type joint and leaf-spring type joint. Jones [3,4] was the first researcher to utilize leaf springs to make a four-bar mechanism for translational motion. Plainevaux [5] derived the equations of nonlinear deformation for the cantilever of a leaf spring. Using these equations, the translational motion of the leaf-spring joint subject to dynamic loadings can be predicted. Paros and Weisbord [6] proposed a notch-type compliant joint, which later became a popular configuration used in the compliant mechanisms. Ragulskis et al. [7] applied static finite element analysis to notch-type joints. Their analysis results state that the range of motion of the notch-type joint is limited because of high stress concentrations occurring around the notch area. Over the past few years, due to the advent of new technologies in optical measurement and/or precision machinery, compliant mechanisms with high precision are in demand for research institutes and industries. Goldfarb and Speich [8] n
Corresponding author. E-mail address: [email protected] (J.-J. Lee).
http://dx.doi.org/10.1016/j.mechmachtheory.2016.08.007 0094-114X/& 2016 International Federation for the Promotion of Mechanism and Machine Science Published by Elsevier Ltd. All rights reserved.
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proposed a revolute flexure joint in which the concept of stiffness ratio was established. The stiffness ratio is a useful index when comparing various compliant joints. Later, Awtar and Slocum [9] derived the equations for beam-based compliant structures where non-linearities arising from the force equilibrium conditions in a beam is approximated with moderate complexity. The authors mentioned that any undesirable motion in response to a primary motion is considered an error motion. The error motion is classified as a parasitic error if it occurs along the direction of constraint while it is considered a cross-axis error if it occurs along the direction of motion. Trease et al. [10] noted the necessity for the capacity of large displacement of compliant joints, and proposed compliant joints with large displacement, with three-dimensionally presented configurations. It was claimed that a better off-axis stiffness and lower axis drift (deviation from the line of motion) for a three-dimensional configuration over a planar configuration have been achieved. Gillespie et al. [11] studied a compliant haptic device using three-dimensional compliant rotational joints. Other than these, applications of compliant mechanisms may be found in the fields of MEMS [12–15], biological inspired mimetic [16], and precision engineering [17–20]. It is worth nothing that, except for [10] and [11], most literature that addresses the compliant mechanism design focuses on planar configurations. Since spatial joints are essential to the configuration of spatial mechanisms, it is necessary to explore more types of spatial joints for possible applications in spatial compliant mechanisms. Thus, the purpose of this paper is to study the characteristics of a few types of three-dimensional compliant translational joints and to propose a new one for design. In this work, the conceptual design of the spatial compliant translational joint is discussed. A new type of joint is then presented. The design parameters of the joint are identified and a design analysis is conducted to help identify which parameters are crucial for design. A design optimization is then performed based on these target parameters. Moreover, a prototype of the optimized design is fabricated and experiments for the stiffness are conducted. Finally, comparisons of the simulation and experimental results are performed to verify the design concept.
2. Conceptual design 2.1. Conventional flexure mechanisms with translational motion Many flexure mechanisms that move linearly are built upon a four-link-like primitive construction where the rigid links are connected by leaf springs or a notched joint. The relative motion of the two rigid links, the frame and the moving part, arises from the deflection in the leaf spring or notched portion of the structure, as shown in Fig. 1. For the analysis of the flexure mechanism comprising notched joints, the pseudo-rigid-body model provided in [18] can be used to approximate the flexure mechanism as a traditional bar-connecting linkage and to obtain the motion of the moving part. However, this method cannot be applied to analyze a mechanism consisting of the leaf-spring elements whose non-linearity arises from the conditions when the loads and displacements in one direction may influence the stiffness properties of other directions. The force-displacement relations of the primitive flexure mechanism have been researched using both linear and nonlinear approaches [9]. One significant issue regarding the precision of the flexure mechanism is the off-axis displacement revealed in the primitive construction. To eliminate the axis drift in the primitive construction, Jones [3] proposed different types of structures consisting of series-connected or parallel-connected simple primitives. These series- or parallel-connected structures mainly use the characteristics of symmetry to compensate for the undesirable off-axis deformation in the mechanism. To realize the advantages and drawbacks, the characteristics of flexure mechanisms that are constructed in series connection and in parallel connection will be investigated. 2.2. Flexure mechanisms in series connection In a series connection, the flexure mechanism may comprise two or more simple translational primitives. The primitives may be linked in a line as shown in Fig. 2a or folded beneath a coupler link within the mother primitive as shown in Fig. 2b.
Fig. 1. Flexure joints (a) leaf-spring type (b) notch type.
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Fig. 2. Compliant translation flexures in series connection type (a) in-line type, (b–d) folded type.
For the construction linked in a line, the flexure mechanism tends to quickly become bulky once the number of blocks increases. For a folded construction, the size of the mechanism is confined within the mother primitive while the motion of the moving part (Fig. 2b–d) will be accordingly confined due to the limited space within the mother primitive. To realize the influence of the number of blocks on the motion of the mechanism, a model of the mechanism with serial connection is built where the range of motion and off-axis displacement for the folded type of construction are investigated using finite element analysis software ANSYS. In the simulation, the large deflection mode is used and the meshing element type Solid_92 is applied. Fig. 3 shows a model of a serially folded connection for simulation where t, L, W, S, and A are, respectively, the thickness, length, width, span of beams, and the width of the rigid link, and where (f, p) are the applied forces. Fig. 4 shows the results of simulation for the motion of the target platform (innermost moving block) versus the number of blocks in the series. From the simulation results, the following observations can be attained. First, the range of motion of the target platform is proportional to the number of serially connected blocks (Fig. 4a). However, the maximum available stroke is still limited by the geometry of the mechanism. Second, the off-axis displacement in the x direction (axis drift) increases with the number of series connections (Fig. 4b). The last feature evidently does not favor the design of a precision flexure mechanism. To conclude, designing the compliant mechanism in series construction may increase the range of motion, but does not support the attainment of precision of the mechanism. This conclusion can also be observed
Fig. 3. Simulation model of series connection in folded manner.
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Fig. 4. Simulation results of deformation versus the numbers of blocks (a) Y displacement (b) X displacement (t ¼0.5 mm, L ¼27.4 mm, W¼ 8 mm, S ¼20 mm, A ¼2 mm, f¼ p ¼1 N, E¼ 2480 MPa, and Poisson's ratio¼ 0.39).
from the existing designs since most of designs do not employ serial configurations [12–14]. 2.3. Flexure mechanisms in parallel connection Jones and Young [4] showed that spring blocks connected in parallel deliver more accurate motion than those connected in series construction. This is because a flexure mechanism in parallel connection may compensate for the axis drift which appears in series connection. Nonetheless, the compliance of a flexure in parallel connection would be decreased due to the increase of the parallel structures. Fig. 5a shows a simple planar parallel construction obtained by reflecting a portion about the axis of motion, while Fig. 5b shows a spatial parallel construction by revolving the plane structure 90° about the axis of motion. Accordingly, the axial stiffness of the configuration is proportional to the number of parallel connections. In addition, the flexure shown by Fig. 5b can prevail over the one shown by Fig. 5a if a lateral force in x direction is applied. 2.4. Combination of series and parallel connections A flexure mechanism formed by the combination of the series and the parallel connections may possess advantages over a single type of construction. There are various types of compound constructions that can be created from individual primitives. Fig. 6a shows a configuration constructed by connecting two blocks in parallel, each with a series connection (Fig. 2a). Similar to the series connection, the flexure mechanism quickly becomes bulky when the number of blocks is increased. Fig. 6b shows a second construction which is also formed from two blocks with folded series connections (Fig. 2b) connected in parallel whose fixed frame is assigned at the outer link. Likewise, constructions can be formed by rearranging the positions of the moving link and the fixed frame as shown in Fig. 6c and d corresponding to Fig. 2c and d. In Fig. 6c, the moving platform is located at the outer side such that it penetrates through the fixed frame. In Fig. 6d, the leaf springs supporting the moving part and the frame are spaced at intervals, resulting in a cross-type structure. Now a three-dimensional structure can be designed based on the planar construction of Fig. 6 by revolving the construction along the axis of motion. As a result, the spatial configuration may have two serial connections and various parallel connections. Note that a tradeoff between the compliance of the flexure and the number of parallel connections could be made to select the best alternative since the more number of parallel connections the flexure has, the stiffer the construction becomes. To evaluate the performance of the four types of constructions in Fig. 6, the lateral stiffness and stiffness ratios are preliminarily investigated via FEA software. As shown in Fig. 7a–d, a three-dimensional construction with three parallel connections is respectively built for each configuration in Fig. 6. The following data are used for nonlinear simulation. Assume the beam spans are of equal distance S (S ¼17 mm) (see Fig. 7a) and the thickness of the spring beam (T) is 0.5 mm,
Frame
Moving
(a)
(b)
Fig. 5. Parallel joint connections: (a) planar parallel connection (b) spatial parallel connection.
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Fig. 6. Constructions with combined series and parallel connections.
the length (L) is 28 mm, and the width (W) is 9 mm. The Young's modulus is 2480 MPa and the Poisson's ratio is 0.39. Four different loading cases are further assumed; Case 1: a 2-Newton (N) force is applied at one end of the moving part along the axis of the motion; Case 2: a 2 N force is applied at the direction perpendicular to the axis of the motion (i.e. off-axis load); Case 3: the 2 N forces are simultaneously applied along the axis of the motion and perpendicular to the axis of the motion; and Case 4: same condition as Case 3 except the magnitude of force is 4 N. The transverse distance d between the beam and the place of applied force is 10 mm for each configuration. Table 1 shows the simulation result. From the result, it can be seen that for the axial loading Case 1, the four types of models have nearly the same axial displacement. For the off-axis loading Case 2, the off-axis deflection in Type 2 is the smallest among the four configurations; therefore, Type 2 has the largest stiffness ratio (axial deflection/off-axial deflection) and Type 1 is the second largest. For the simultaneous loading Cases 3 and 4, the stiffness ratios decrease rapidly in all four models, especially Type 2. Consequently, the stiffness ratio of Type 2 remains the best and Type 4 becomes the second best. It is worth noting that the lateral stiffness and hence the stiffness ratio depends on the span of beams that connect the moving part. Although Type 2 has the largest stiffness ratio at the dimension assumed, it has smaller available stroke of the moving block and is less easy to place the applying force since the moving block is located inside the structure. Therefore, it is determined in this work that Type 4 shown in Fig. 7d will be chosen for further design analysis. Fig. 8a shows the solid model of the joint.
3. Parametric analysis and optimization 3.1. Parametric analysis From the above discussion, the configuration of compound series and parallel connections could provide a higher stiffness ratio than a configuration with a single connection. The stiffness of the compound structure depends on the geometry and material parameters, such as the number of beams, beam spans, length, width, and thickness as were also mentioned by [10–14]. Therefore, analysis of the influence of the design parameters on the stiffness of the structure is an important procedure for sizing new mechanisms. In addition, the result may be useful for design optimization, which will be discussed later. In this section, the Taguchi method [21] of the design experiment is employed to investigate the influence of the design parameters on the performance of the translational flexure mechanism. As shown in Fig. 8b and c, the geometric Please cite this article as: T.-S. Yang, et al., Design of a spatial compliant translational joint, Mechanism and Machine Theory (2016), http://dx.doi.org/10.1016/j.mechmachtheory.2016.08.007i
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Fig. 7. Three-dimensional constructions for the figures in Fig. 6(a) Type 1 (b) Type 2 (c)Type 3 (d) Type 4.
parameters consist of the beam span S1, the beam length L, the width W, the thickness t, and the displacement of the flexure mechanism S2. Note that since S2 is the displacement of the translational flexure joint, it can be removed from the control factors. Here, a three-level L9 (34) orthogonal array is established for the analysis as shown in Table 2. The stiffness ratio of each set is calculated by finite element analysis. From the factor responses shown in Fig. 9, it can be seen that increasing the beam length (L) or decreasing the beam thickness (T) tends to increase the stiffness ratio. To determine the confidence in the influence of the parameters, analysis of the variance is performed. Detailed formula and numerical data are listed in Appendices A and B. As shown in Table 3 where the F-factor denotes the significant level of influence of the parameters on the stiffness ratio the results indicate that the beam thickness T is a relatively major influencing parameter, while the beam length L has only a minor effect. Please cite this article as: T.-S. Yang, et al., Design of a spatial compliant translational joint, Mechanism and Machine Theory (2016), http://dx.doi.org/10.1016/j.mechmachtheory.2016.08.007i
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Table 1 Comparison of Stiffness ratios for the configurations in Fig. 7. Configuration
Type 1
Type 2
Type 3
Type 4
Loading case 1
Axial disp. (mm)
4.832
4.831
4.833
4.91
Loading case 2
Off-axis disp. (mm) Stiffness ratio
0.1102 43.8
0.0149 324.2
0.1562 30.94
0.1156 42.47
Loading case 3
Axial disp. (mm) Off-axial disp. (mm) Stiffness ratio
4.921 0.2228 22.08
4.857 0.0557 87.1
4.862 0.1956 24.86
4.934 0.1487 33.18
Loading case 4
Axial disp. (mm) Off-axial disp. (mm) Stiffness ratio
9.879 0.9372 10.54
9.476 0.2028 46.72
9.558 0.5393 17.72
9.646 0.4026 23.96
W
T
S1
L
S2
(a)
(b)
(c)
Fig. 8. (a) A solid model for the spatial compliant translational joint, (b) and (c) geometric parameters of the spatial compliant translation joint.
Table 2 The L9 orthogonal arrays of design parameters. Exp.#
1 2 3 4 5 6 7 8 9
Orthogonal array (L9 array)
Design variable (mm)
Var. 1
Var. 2
Var. 3
Var. 4
W
L
T
S1
1 1 1 2 2 2 3 3 3
1 2 3 1 2 3 1 2 3
1 2 3 2 3 1 3 1 2
1 2 3 3 1 2 2 3 1
6 6 6 8 8 8 10 10 10
20.25 27 33.75 20.25 27 33.75 20.25 27 33.75
0.5 0.8 1.1 0.8 1.1 0.5 1.1 0.5 0.8
22.5 30 37.5 37.5 22.5 30 30 37.5 22.5
3.2. Design optimization After the parametric analysis, the design process can be extended as an optimization problem to better design the flexure mechanism. 3.2.1. Design variable In this work, since the beam length has the least effect, the design parameters are determined as: (1) beam width W, (2) beam thickness T, (3) distance between two beams S2, and (4) distance between beams that support same platform link S1. Please cite this article as: T.-S. Yang, et al., Design of a spatial compliant translational joint, Mechanism and Machine Theory (2016), http://dx.doi.org/10.1016/j.mechmachtheory.2016.08.007i
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Fig. 9. Simulation results for the stiffness ratio versus the control variables.
Table 3 F-factors for the design parameters.
F-factor
W
L
T
S1
0.84
0.53
2.23
0.73
3.2.2. Constraints A challenging task of the design of flexure mechanisms is to keep the joint working in the linear area of the configuration so that it can avoid the complicated behavior of the mechanics due to geometric and/or material nonlinearity. Therefore, it is desirable that the variation of the axial stiffness be limited to a certain range so that the behavior of the flexure is within the linear property of the geometry. These yield the following constraints for the stiffness: 3.2.2.1. Axial stiffness. When the axial and lateral loads are simultaneously applied, the axial stiffness may vary depending on the condition of the loads. It is anticipated that this change be small. Let Kij represent the axial stiffness, as shown in Fig. 10, in which the subscript i denotes the axial load of i N and the subscript j denotes the off-axis load of j N. For instance, K10 is the axial stiffness where the axial load is 1 N and the off-axis load is 0. The rate of change of the stiffness ratio is then limited by an amount a as:
( K10 − K20 )/K10 ( K12 − K22 )/K12 ( K20 − K22 )/K20
(1)
(2)
< a.
(3)
Further, the stiffness ratio has a lower limit as follows:
( Knon− axial/Kaxial )
>b
(4)
in which the axial and non-axial stiffnes are determined at the maximum loading, namely, both axial load and non-axial load are 2 N. For the prototype design in this paper, the rate of change of the axial stiffness is limited by a ¼5%, and the minimum stiffness ratio is given as b¼30.
Fig. 10. Working area of the axial stiffness.
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Fig. 11. (a) Rotational deformation of the spatial compliant translation platform, (b) enlargement of (a).
3.2.2.2. The rotational deformation. The rotational deformation must be small and hence confined by a value α. Let the cross section of the center rod be shown in Fig. 11(a). A point A on the edge of the rod (Fig. 11b) can be used to calculate the torsional deformation as
θ = sin−1
(
)
xT2 + yT2 /r < a
(5)
in which xT and yT are the x- and y-displacements of A, r is the distance between A and the center of the cross section, and assume that the center of the cross section remain stationary during motion. For our prototype of the joint, the limit of the rotational deformation is given by α ¼0.1°. 3.2.2.3. Maximum stress. Regarding the properties of the material, the maximum material yield strength becomes the stress limitation of the platform.
σmax < σallowable
(6)
For our prototype of the joint, the material ABS is used. In addition, taking into account some safety factor for the design, the maximum allowable strength is assumed to be sallowable ¼22.5 MPa. 3.2.2.4. Geometric constraints. The geometric constraint of the cross-type platform reveals that S1 should be larger than S2, and S2 should be larger than the desired range of motion δ .
(7)
S1 > S2 > δ The ratio of the movable displacement S2 over the total length of structure can be stated as
( S2/TL) > β
(8)
in which TL denotes the total length of the structure and equals 2S1–S2. The desired range of motion for the moving platform can be assumed as a specification of the design. In this work, the desired range of motion is specified as δ ¼20 mm, while the length ratio is given by β ¼20%. 3.2.3. Objective function Many objective functions can be established for the optimization problem such as maximizing the stiffness ratio, maximizing the range of motion or minimizing the axis drift of the joint. In particular, the axis drift due to elastic deformation directly reveals the accuracy of the platform. Therefore, axis drift is used as the objective function in this work. The accuracy of the compliant platform can be evaluated according to the axis drift of a point with respect to the actual axial displacement. This can be formulated as
Foptimization = min
(
Δxa2 + Δya2 /Δz a
)
(9)
in which (Δxa , Δya ) are respectively the x component and y component of changes of displacement of a point on the joint center, and Δza is the actual axial displacement when subject to a loading condition. In this work, both axial load and nonaxial load are 2 N for the loading condition. In summary, the optimization problem can be formulated as Minimize
Δxa2 + Δya2 /Δza .
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subject to
( K10 − K20 )/K10 < 5% ( K12 − K22 )/K12 < 5% ( K20 − K22 )/K20 < 5% ( Knon− axial/Kaxial ) > 30 sin−1
(
(10) (11) (12) (13)
)
xT2 + yT2 /r < 0.1°
(14) (15) (16) (17)
σmax < 22.5 MPa S1 − S2 > 0 S2 > 20 mm S2/TL > 20%
(18)
3.3. Optimization result The optimization of the spatial compliant translational joint was calculated using the optimization module of ANSYS software where the mesh element Solid_92 is used, the large deflection mode is on, and the subproblem approximation method is applied. The Young's modulus of the material used in the simulation is 1240 MPa and the Poisson's ratio is 0.39. The results of the optimization are listed in Table 4.
4. Experiment 4.1. Experimental prototype To cope with the practical accuracy of the prototype, the parameters obtained from Table 4 are approximated as W¼9.9 mm, T ¼0.5 mm, S1 ¼32.6 mm and S2 ¼23.6 mm for simulation. The joint is then fabricated by 3D printing. The prototype is shown in Fig. 12. 4.2. Experiment setup To investigate the stiffness of the spatial compliant translational joint, the axial displacement and off-axis displacement are measured. A cable-pulley-and-weight system is set up to apply the loadings to both the longitudinal and the transverse direction of axis of motion, as shown in Fig. 13. A laser displacement sensor [22] (Fig. 13a) with accuracy up to 2 μm is then placed along the longitudinal axis and used to detect the axial deformation. As shown in Fig. 14, a charge-coupled device (CCD) camera is placed to face the Y–Z plane and used to detect the off-axis deformation. Through the CCD camera, the picture captured after the image process can provide accuracy up to 6.8 μm/pixel. For the experiment, each case of the loading test is repeated ten times and the average result for each test is recorded. 4.3. Experiment results 4.3.1. Axial stiffness The numerical simulations and experimental tests were conducted. Fig. 15 illustrates one of the results of the simulation and Fig. 16 shows the comparisons of the stiffness ratios. The axial displacement ranges from 4.17–11.921 mm while the lateral displacement ranges from 0.087–0.433 mm by experiment [23]. It can be seen that the discrepancies between the simulations and the experiments are within 10%. It also can be seen that the coupling effect is not obvious when the axial load is relatively small; therefore, the stiffness ratios attained at the small loading were larger than those attained at the larger loading. Table 4 Optimization results of the spatial compliant translational joint. Design variable
State variable
Width W (mm) 9.92
Thickness T (mm) 0.50
Beam span S1 (mm) 32.57
Stroke S2 (mm) 23.63
Linearity 1 Eq. (1) 2.27%
Linearity 2 Eq. (2) 2.31%
State variable Linearity 3 Eq. (3) 0.613%
Stiffness ratio 34.36
Twist angle (deg) 1.098E-5
smax (MPa) 21.94
Stroke/overall length 26.8%
Foptimization 0.0291
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Frame
Frame
Moving part
Moving part
Leaf spring
Fig. 12. Prototype of the spatial compliant translational joint manufactured by 3D printing.
Fig. 13. Schematic of experimental setup (a) A diagram of the experiment for axial stiffness (b) diagram of the experiment for rotational stiffness.
Fig. 14. Experimental setup.
4.3.2. Rotational stiffness For the rotational stiffness, a 200 g weight is placed at each side of pulley in Fig. 13b. The average value of the tests is 3.61 105 N mm with standard deviation 1.23 105 N mm and the simulation for this item is 3.94 105 N mm. The difference of the rotational angles between simulation and experiment can be observed. In summary, the difference between the simulation and experimental results may be attributable to the reasons such as the accuracy of the CCD camera used to capture the displacement, the quality of the printing material and the difference between manufactured data and simulation ones. It can be noted that some of the dimensions of the prototype made by the 3D printing are not uniform, especially at the beam thickness, which is sensitive to the performance of the joint.
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Fig. 15. An FEA simulation result for the axial displacements of the design (unit: mm).
Fig. 16. Stiffness ratios for various loading conditions.
5. Conclusions A three-dimensional compliant translational joint was proposed. This joint combines the advantage of the series and the parallel connections of beam-type joints, and it possesses a high stiffness ratio and low axis drift. Benchmarking indices, including a large range of motion, low off-axis drift, and high stiffness ratio, were employed in the design process and realized in this design. An optimized set of dimensions for the joint based on the minimal value of the ratio of off-axis drift and actual axial displacement was allocated to fabricate a practical model. The performance of the mockup was measured and compared with the simulations. It was shown that the relative difference between experiment stiffness ratios and simulated data were within satisfaction.
Appendix A. Formula used for Table 3 [21] Sum of squares between the Groups, SSB k
SSB =
ni
k
∑ ∑ ( Xi ⋅ − X )2 = ∑ i=1 j=1
i=1
Xi2⋅ X2 − ni n
(A.1)
Sum of squares within the groups, SSE Please cite this article as: T.-S. Yang, et al., Design of a spatial compliant translational joint, Mechanism and Machine Theory (2016), http://dx.doi.org/10.1016/j.mechmachtheory.2016.08.007i
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SSE =
13
ni
∑ ∑ ( Xij − Xi ⋅)2
(A.2)
i=1 j=1
where ni is the number of observations in one group, k is the number of groups, and n is the total number of observations. F-factor
⎛ SSB ⎞ ⎛ SSE ⎞ ⎟/ ⎜ F=⎜ ⎟ ⎝ k − 1 ⎠ ⎝ ni − k ⎠
(A.3)
Appendix B. Variance analysis of the design parameters
W
level-1 level-2 level-3 level-1 level-2 level-3 level-1 level-2 level-3 level-1 level-2 level-3
L
T
S1
13.654 9.672 18.652 9.677 15.977 16.323 19.645 15.107 7.225 18.781 10.306 14.326
SSB SSE
121.478 434.155
MSB MSE
60.739 72.359
F-factor
0.839407
SSB SSE
83.977 471.655
MSB MSE
41.989 78.609
F-factor
0.534144
SSB SSE
236.960 318.673
MSB MSE
118.480 53.112
F-factor
2.230746
SSB SSE
109.902 448.605
MSB MSE
54.951 74.768
F-factor
0.734961
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Design of a spatial compliant translational joint Tai-Shen Yang a, Po-Jen Shih b, Jyh-Jone Lee a,n a b
Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan Department of Civil and Environmental Engineering, National University of Kaohsiung, Kaohsiung, Taiwan
a r t i c l e i n f o
abstract
Article history: Received 12 August 2015 Received in revised form 15 August 2016 Accepted 20 August 2016
This paper presents the design of a novel compliant translational joint with large stiffness ratio and small axis drift. The characteristics of compliant translational joints based on a leaf-spring type joint are investigated. Several types of constructions are analyzed. A novel three-dimensional compliant translational joint is proposed and a design analysis is conducted based on the parametric models. Subsequently, a design optimization is employed with the aid of finite element method. Then, a prototype of the optimized design is fabricated by 3D printing, and tests of specified properties are conducted. Comparisons of the experimental results and simulation results show that this new joint satisfies the design requirements. & 2016 International Federation for the Promotion of Mechanism and Machine Science Published by Elsevier Ltd. All rights reserved.
Keywords: Compliant translational joint Three-dimensional Axis drift Stiffness ratio Optimization
1. Introduction Conventional joints formed by rigid links, such as the revolute pair, the prismatic pair, and ball-and-sockets, are commonly used to connect mechanical parts. However, such joints may contain clearance between the pairing elements and lead to inaccurate positioning during motion. Furthermore, the relative motions between the pairing elements generate friction that results in wear and reliability issues. To overcome these drawbacks, design of the mechanism with compliant joints is proposed. Such mechanism usually consists of a series of rigid links connected by compliant elements and is designed to produce a prescribed motion when a force is applied. Since the mechanism is typically made from a monolith, there exits no clearance between rigid links. Thus, the use of compliant joints can eliminate the presence of friction, backlash, and wear in a mechanism. Many types of mechanisms with compliant joints have been developed over the past decades. Smith [1,2] classified compliant joints in a mechanism into two types, namely, the notch-type joint and leaf-spring type joint. Jones [3,4] was the first researcher to utilize leaf springs to make a four-bar mechanism for translational motion. Plainevaux [5] derived the equations of nonlinear deformation for the cantilever of a leaf spring. Using these equations, the translational motion of the leaf-spring joint subject to dynamic loadings can be predicted. Paros and Weisbord [6] proposed a notch-type compliant joint, which later became a popular configuration used in the compliant mechanisms. Ragulskis et al. [7] applied static finite element analysis to notch-type joints. Their analysis results state that the range of motion of the notch-type joint is limited because of high stress concentrations occurring around the notch area. Over the past few years, due to the advent of new technologies in optical measurement and/or precision machinery, compliant mechanisms with high precision are in demand for research institutes and industries. Goldfarb and Speich [8] n
Corresponding author. E-mail address: [email protected] (J.-J. Lee).
http://dx.doi.org/10.1016/j.mechmachtheory.2016.08.007 0094-114X/& 2016 International Federation for the Promotion of Mechanism and Machine Science Published by Elsevier Ltd. All rights reserved.
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proposed a revolute flexure joint in which the concept of stiffness ratio was established. The stiffness ratio is a useful index when comparing various compliant joints. Later, Awtar and Slocum [9] derived the equations for beam-based compliant structures where non-linearities arising from the force equilibrium conditions in a beam is approximated with moderate complexity. The authors mentioned that any undesirable motion in response to a primary motion is considered an error motion. The error motion is classified as a parasitic error if it occurs along the direction of constraint while it is considered a cross-axis error if it occurs along the direction of motion. Trease et al. [10] noted the necessity for the capacity of large displacement of compliant joints, and proposed compliant joints with large displacement, with three-dimensionally presented configurations. It was claimed that a better off-axis stiffness and lower axis drift (deviation from the line of motion) for a three-dimensional configuration over a planar configuration have been achieved. Gillespie et al. [11] studied a compliant haptic device using three-dimensional compliant rotational joints. Other than these, applications of compliant mechanisms may be found in the fields of MEMS [12–15], biological inspired mimetic [16], and precision engineering [17–20]. It is worth nothing that, except for [10] and [11], most literature that addresses the compliant mechanism design focuses on planar configurations. Since spatial joints are essential to the configuration of spatial mechanisms, it is necessary to explore more types of spatial joints for possible applications in spatial compliant mechanisms. Thus, the purpose of this paper is to study the characteristics of a few types of three-dimensional compliant translational joints and to propose a new one for design. In this work, the conceptual design of the spatial compliant translational joint is discussed. A new type of joint is then presented. The design parameters of the joint are identified and a design analysis is conducted to help identify which parameters are crucial for design. A design optimization is then performed based on these target parameters. Moreover, a prototype of the optimized design is fabricated and experiments for the stiffness are conducted. Finally, comparisons of the simulation and experimental results are performed to verify the design concept.
2. Conceptual design 2.1. Conventional flexure mechanisms with translational motion Many flexure mechanisms that move linearly are built upon a four-link-like primitive construction where the rigid links are connected by leaf springs or a notched joint. The relative motion of the two rigid links, the frame and the moving part, arises from the deflection in the leaf spring or notched portion of the structure, as shown in Fig. 1. For the analysis of the flexure mechanism comprising notched joints, the pseudo-rigid-body model provided in [18] can be used to approximate the flexure mechanism as a traditional bar-connecting linkage and to obtain the motion of the moving part. However, this method cannot be applied to analyze a mechanism consisting of the leaf-spring elements whose non-linearity arises from the conditions when the loads and displacements in one direction may influence the stiffness properties of other directions. The force-displacement relations of the primitive flexure mechanism have been researched using both linear and nonlinear approaches [9]. One significant issue regarding the precision of the flexure mechanism is the off-axis displacement revealed in the primitive construction. To eliminate the axis drift in the primitive construction, Jones [3] proposed different types of structures consisting of series-connected or parallel-connected simple primitives. These series- or parallel-connected structures mainly use the characteristics of symmetry to compensate for the undesirable off-axis deformation in the mechanism. To realize the advantages and drawbacks, the characteristics of flexure mechanisms that are constructed in series connection and in parallel connection will be investigated. 2.2. Flexure mechanisms in series connection In a series connection, the flexure mechanism may comprise two or more simple translational primitives. The primitives may be linked in a line as shown in Fig. 2a or folded beneath a coupler link within the mother primitive as shown in Fig. 2b.
Fig. 1. Flexure joints (a) leaf-spring type (b) notch type.
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Fig. 2. Compliant translation flexures in series connection type (a) in-line type, (b–d) folded type.
For the construction linked in a line, the flexure mechanism tends to quickly become bulky once the number of blocks increases. For a folded construction, the size of the mechanism is confined within the mother primitive while the motion of the moving part (Fig. 2b–d) will be accordingly confined due to the limited space within the mother primitive. To realize the influence of the number of blocks on the motion of the mechanism, a model of the mechanism with serial connection is built where the range of motion and off-axis displacement for the folded type of construction are investigated using finite element analysis software ANSYS. In the simulation, the large deflection mode is used and the meshing element type Solid_92 is applied. Fig. 3 shows a model of a serially folded connection for simulation where t, L, W, S, and A are, respectively, the thickness, length, width, span of beams, and the width of the rigid link, and where (f, p) are the applied forces. Fig. 4 shows the results of simulation for the motion of the target platform (innermost moving block) versus the number of blocks in the series. From the simulation results, the following observations can be attained. First, the range of motion of the target platform is proportional to the number of serially connected blocks (Fig. 4a). However, the maximum available stroke is still limited by the geometry of the mechanism. Second, the off-axis displacement in the x direction (axis drift) increases with the number of series connections (Fig. 4b). The last feature evidently does not favor the design of a precision flexure mechanism. To conclude, designing the compliant mechanism in series construction may increase the range of motion, but does not support the attainment of precision of the mechanism. This conclusion can also be observed
Fig. 3. Simulation model of series connection in folded manner.
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Fig. 4. Simulation results of deformation versus the numbers of blocks (a) Y displacement (b) X displacement (t ¼0.5 mm, L ¼27.4 mm, W¼ 8 mm, S ¼20 mm, A ¼2 mm, f¼ p ¼1 N, E¼ 2480 MPa, and Poisson's ratio¼ 0.39).
from the existing designs since most of designs do not employ serial configurations [12–14]. 2.3. Flexure mechanisms in parallel connection Jones and Young [4] showed that spring blocks connected in parallel deliver more accurate motion than those connected in series construction. This is because a flexure mechanism in parallel connection may compensate for the axis drift which appears in series connection. Nonetheless, the compliance of a flexure in parallel connection would be decreased due to the increase of the parallel structures. Fig. 5a shows a simple planar parallel construction obtained by reflecting a portion about the axis of motion, while Fig. 5b shows a spatial parallel construction by revolving the plane structure 90° about the axis of motion. Accordingly, the axial stiffness of the configuration is proportional to the number of parallel connections. In addition, the flexure shown by Fig. 5b can prevail over the one shown by Fig. 5a if a lateral force in x direction is applied. 2.4. Combination of series and parallel connections A flexure mechanism formed by the combination of the series and the parallel connections may possess advantages over a single type of construction. There are various types of compound constructions that can be created from individual primitives. Fig. 6a shows a configuration constructed by connecting two blocks in parallel, each with a series connection (Fig. 2a). Similar to the series connection, the flexure mechanism quickly becomes bulky when the number of blocks is increased. Fig. 6b shows a second construction which is also formed from two blocks with folded series connections (Fig. 2b) connected in parallel whose fixed frame is assigned at the outer link. Likewise, constructions can be formed by rearranging the positions of the moving link and the fixed frame as shown in Fig. 6c and d corresponding to Fig. 2c and d. In Fig. 6c, the moving platform is located at the outer side such that it penetrates through the fixed frame. In Fig. 6d, the leaf springs supporting the moving part and the frame are spaced at intervals, resulting in a cross-type structure. Now a three-dimensional structure can be designed based on the planar construction of Fig. 6 by revolving the construction along the axis of motion. As a result, the spatial configuration may have two serial connections and various parallel connections. Note that a tradeoff between the compliance of the flexure and the number of parallel connections could be made to select the best alternative since the more number of parallel connections the flexure has, the stiffer the construction becomes. To evaluate the performance of the four types of constructions in Fig. 6, the lateral stiffness and stiffness ratios are preliminarily investigated via FEA software. As shown in Fig. 7a–d, a three-dimensional construction with three parallel connections is respectively built for each configuration in Fig. 6. The following data are used for nonlinear simulation. Assume the beam spans are of equal distance S (S ¼17 mm) (see Fig. 7a) and the thickness of the spring beam (T) is 0.5 mm,
Frame
Moving
(a)
(b)
Fig. 5. Parallel joint connections: (a) planar parallel connection (b) spatial parallel connection.
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Fig. 6. Constructions with combined series and parallel connections.
the length (L) is 28 mm, and the width (W) is 9 mm. The Young's modulus is 2480 MPa and the Poisson's ratio is 0.39. Four different loading cases are further assumed; Case 1: a 2-Newton (N) force is applied at one end of the moving part along the axis of the motion; Case 2: a 2 N force is applied at the direction perpendicular to the axis of the motion (i.e. off-axis load); Case 3: the 2 N forces are simultaneously applied along the axis of the motion and perpendicular to the axis of the motion; and Case 4: same condition as Case 3 except the magnitude of force is 4 N. The transverse distance d between the beam and the place of applied force is 10 mm for each configuration. Table 1 shows the simulation result. From the result, it can be seen that for the axial loading Case 1, the four types of models have nearly the same axial displacement. For the off-axis loading Case 2, the off-axis deflection in Type 2 is the smallest among the four configurations; therefore, Type 2 has the largest stiffness ratio (axial deflection/off-axial deflection) and Type 1 is the second largest. For the simultaneous loading Cases 3 and 4, the stiffness ratios decrease rapidly in all four models, especially Type 2. Consequently, the stiffness ratio of Type 2 remains the best and Type 4 becomes the second best. It is worth noting that the lateral stiffness and hence the stiffness ratio depends on the span of beams that connect the moving part. Although Type 2 has the largest stiffness ratio at the dimension assumed, it has smaller available stroke of the moving block and is less easy to place the applying force since the moving block is located inside the structure. Therefore, it is determined in this work that Type 4 shown in Fig. 7d will be chosen for further design analysis. Fig. 8a shows the solid model of the joint.
3. Parametric analysis and optimization 3.1. Parametric analysis From the above discussion, the configuration of compound series and parallel connections could provide a higher stiffness ratio than a configuration with a single connection. The stiffness of the compound structure depends on the geometry and material parameters, such as the number of beams, beam spans, length, width, and thickness as were also mentioned by [10–14]. Therefore, analysis of the influence of the design parameters on the stiffness of the structure is an important procedure for sizing new mechanisms. In addition, the result may be useful for design optimization, which will be discussed later. In this section, the Taguchi method [21] of the design experiment is employed to investigate the influence of the design parameters on the performance of the translational flexure mechanism. As shown in Fig. 8b and c, the geometric Please cite this article as: T.-S. Yang, et al., Design of a spatial compliant translational joint, Mechanism and Machine Theory (2016), http://dx.doi.org/10.1016/j.mechmachtheory.2016.08.007i
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Fig. 7. Three-dimensional constructions for the figures in Fig. 6(a) Type 1 (b) Type 2 (c)Type 3 (d) Type 4.
parameters consist of the beam span S1, the beam length L, the width W, the thickness t, and the displacement of the flexure mechanism S2. Note that since S2 is the displacement of the translational flexure joint, it can be removed from the control factors. Here, a three-level L9 (34) orthogonal array is established for the analysis as shown in Table 2. The stiffness ratio of each set is calculated by finite element analysis. From the factor responses shown in Fig. 9, it can be seen that increasing the beam length (L) or decreasing the beam thickness (T) tends to increase the stiffness ratio. To determine the confidence in the influence of the parameters, analysis of the variance is performed. Detailed formula and numerical data are listed in Appendices A and B. As shown in Table 3 where the F-factor denotes the significant level of influence of the parameters on the stiffness ratio the results indicate that the beam thickness T is a relatively major influencing parameter, while the beam length L has only a minor effect. Please cite this article as: T.-S. Yang, et al., Design of a spatial compliant translational joint, Mechanism and Machine Theory (2016), http://dx.doi.org/10.1016/j.mechmachtheory.2016.08.007i
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Table 1 Comparison of Stiffness ratios for the configurations in Fig. 7. Configuration
Type 1
Type 2
Type 3
Type 4
Loading case 1
Axial disp. (mm)
4.832
4.831
4.833
4.91
Loading case 2
Off-axis disp. (mm) Stiffness ratio
0.1102 43.8
0.0149 324.2
0.1562 30.94
0.1156 42.47
Loading case 3
Axial disp. (mm) Off-axial disp. (mm) Stiffness ratio
4.921 0.2228 22.08
4.857 0.0557 87.1
4.862 0.1956 24.86
4.934 0.1487 33.18
Loading case 4
Axial disp. (mm) Off-axial disp. (mm) Stiffness ratio
9.879 0.9372 10.54
9.476 0.2028 46.72
9.558 0.5393 17.72
9.646 0.4026 23.96
W
T
S1
L
S2
(a)
(b)
(c)
Fig. 8. (a) A solid model for the spatial compliant translational joint, (b) and (c) geometric parameters of the spatial compliant translation joint.
Table 2 The L9 orthogonal arrays of design parameters. Exp.#
1 2 3 4 5 6 7 8 9
Orthogonal array (L9 array)
Design variable (mm)
Var. 1
Var. 2
Var. 3
Var. 4
W
L
T
S1
1 1 1 2 2 2 3 3 3
1 2 3 1 2 3 1 2 3
1 2 3 2 3 1 3 1 2
1 2 3 3 1 2 2 3 1
6 6 6 8 8 8 10 10 10
20.25 27 33.75 20.25 27 33.75 20.25 27 33.75
0.5 0.8 1.1 0.8 1.1 0.5 1.1 0.5 0.8
22.5 30 37.5 37.5 22.5 30 30 37.5 22.5
3.2. Design optimization After the parametric analysis, the design process can be extended as an optimization problem to better design the flexure mechanism. 3.2.1. Design variable In this work, since the beam length has the least effect, the design parameters are determined as: (1) beam width W, (2) beam thickness T, (3) distance between two beams S2, and (4) distance between beams that support same platform link S1. Please cite this article as: T.-S. Yang, et al., Design of a spatial compliant translational joint, Mechanism and Machine Theory (2016), http://dx.doi.org/10.1016/j.mechmachtheory.2016.08.007i
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Fig. 9. Simulation results for the stiffness ratio versus the control variables.
Table 3 F-factors for the design parameters.
F-factor
W
L
T
S1
0.84
0.53
2.23
0.73
3.2.2. Constraints A challenging task of the design of flexure mechanisms is to keep the joint working in the linear area of the configuration so that it can avoid the complicated behavior of the mechanics due to geometric and/or material nonlinearity. Therefore, it is desirable that the variation of the axial stiffness be limited to a certain range so that the behavior of the flexure is within the linear property of the geometry. These yield the following constraints for the stiffness: 3.2.2.1. Axial stiffness. When the axial and lateral loads are simultaneously applied, the axial stiffness may vary depending on the condition of the loads. It is anticipated that this change be small. Let Kij represent the axial stiffness, as shown in Fig. 10, in which the subscript i denotes the axial load of i N and the subscript j denotes the off-axis load of j N. For instance, K10 is the axial stiffness where the axial load is 1 N and the off-axis load is 0. The rate of change of the stiffness ratio is then limited by an amount a as:
( K10 − K20 )/K10 ( K12 − K22 )/K12 ( K20 − K22 )/K20
(1)
(2)
< a.
(3)
Further, the stiffness ratio has a lower limit as follows:
( Knon− axial/Kaxial )
>b
(4)
in which the axial and non-axial stiffnes are determined at the maximum loading, namely, both axial load and non-axial load are 2 N. For the prototype design in this paper, the rate of change of the axial stiffness is limited by a ¼5%, and the minimum stiffness ratio is given as b¼30.
Fig. 10. Working area of the axial stiffness.
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Fig. 11. (a) Rotational deformation of the spatial compliant translation platform, (b) enlargement of (a).
3.2.2.2. The rotational deformation. The rotational deformation must be small and hence confined by a value α. Let the cross section of the center rod be shown in Fig. 11(a). A point A on the edge of the rod (Fig. 11b) can be used to calculate the torsional deformation as
θ = sin−1
(
)
xT2 + yT2 /r < a
(5)
in which xT and yT are the x- and y-displacements of A, r is the distance between A and the center of the cross section, and assume that the center of the cross section remain stationary during motion. For our prototype of the joint, the limit of the rotational deformation is given by α ¼0.1°. 3.2.2.3. Maximum stress. Regarding the properties of the material, the maximum material yield strength becomes the stress limitation of the platform.
σmax < σallowable
(6)
For our prototype of the joint, the material ABS is used. In addition, taking into account some safety factor for the design, the maximum allowable strength is assumed to be sallowable ¼22.5 MPa. 3.2.2.4. Geometric constraints. The geometric constraint of the cross-type platform reveals that S1 should be larger than S2, and S2 should be larger than the desired range of motion δ .
(7)
S1 > S2 > δ The ratio of the movable displacement S2 over the total length of structure can be stated as
( S2/TL) > β
(8)
in which TL denotes the total length of the structure and equals 2S1–S2. The desired range of motion for the moving platform can be assumed as a specification of the design. In this work, the desired range of motion is specified as δ ¼20 mm, while the length ratio is given by β ¼20%. 3.2.3. Objective function Many objective functions can be established for the optimization problem such as maximizing the stiffness ratio, maximizing the range of motion or minimizing the axis drift of the joint. In particular, the axis drift due to elastic deformation directly reveals the accuracy of the platform. Therefore, axis drift is used as the objective function in this work. The accuracy of the compliant platform can be evaluated according to the axis drift of a point with respect to the actual axial displacement. This can be formulated as
Foptimization = min
(
Δxa2 + Δya2 /Δz a
)
(9)
in which (Δxa , Δya ) are respectively the x component and y component of changes of displacement of a point on the joint center, and Δza is the actual axial displacement when subject to a loading condition. In this work, both axial load and nonaxial load are 2 N for the loading condition. In summary, the optimization problem can be formulated as Minimize
Δxa2 + Δya2 /Δza .
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subject to
( K10 − K20 )/K10 < 5% ( K12 − K22 )/K12 < 5% ( K20 − K22 )/K20 < 5% ( Knon− axial/Kaxial ) > 30 sin−1
(
(10) (11) (12) (13)
)
xT2 + yT2 /r < 0.1°
(14) (15) (16) (17)
σmax < 22.5 MPa S1 − S2 > 0 S2 > 20 mm S2/TL > 20%
(18)
3.3. Optimization result The optimization of the spatial compliant translational joint was calculated using the optimization module of ANSYS software where the mesh element Solid_92 is used, the large deflection mode is on, and the subproblem approximation method is applied. The Young's modulus of the material used in the simulation is 1240 MPa and the Poisson's ratio is 0.39. The results of the optimization are listed in Table 4.
4. Experiment 4.1. Experimental prototype To cope with the practical accuracy of the prototype, the parameters obtained from Table 4 are approximated as W¼9.9 mm, T ¼0.5 mm, S1 ¼32.6 mm and S2 ¼23.6 mm for simulation. The joint is then fabricated by 3D printing. The prototype is shown in Fig. 12. 4.2. Experiment setup To investigate the stiffness of the spatial compliant translational joint, the axial displacement and off-axis displacement are measured. A cable-pulley-and-weight system is set up to apply the loadings to both the longitudinal and the transverse direction of axis of motion, as shown in Fig. 13. A laser displacement sensor [22] (Fig. 13a) with accuracy up to 2 μm is then placed along the longitudinal axis and used to detect the axial deformation. As shown in Fig. 14, a charge-coupled device (CCD) camera is placed to face the Y–Z plane and used to detect the off-axis deformation. Through the CCD camera, the picture captured after the image process can provide accuracy up to 6.8 μm/pixel. For the experiment, each case of the loading test is repeated ten times and the average result for each test is recorded. 4.3. Experiment results 4.3.1. Axial stiffness The numerical simulations and experimental tests were conducted. Fig. 15 illustrates one of the results of the simulation and Fig. 16 shows the comparisons of the stiffness ratios. The axial displacement ranges from 4.17–11.921 mm while the lateral displacement ranges from 0.087–0.433 mm by experiment [23]. It can be seen that the discrepancies between the simulations and the experiments are within 10%. It also can be seen that the coupling effect is not obvious when the axial load is relatively small; therefore, the stiffness ratios attained at the small loading were larger than those attained at the larger loading. Table 4 Optimization results of the spatial compliant translational joint. Design variable
State variable
Width W (mm) 9.92
Thickness T (mm) 0.50
Beam span S1 (mm) 32.57
Stroke S2 (mm) 23.63
Linearity 1 Eq. (1) 2.27%
Linearity 2 Eq. (2) 2.31%
State variable Linearity 3 Eq. (3) 0.613%
Stiffness ratio 34.36
Twist angle (deg) 1.098E-5
smax (MPa) 21.94
Stroke/overall length 26.8%
Foptimization 0.0291
Please cite this article as: T.-S. Yang, et al., Design of a spatial compliant translational joint, Mechanism and Machine Theory (2016), http://dx.doi.org/10.1016/j.mechmachtheory.2016.08.007i
T.-S. Yang et al. / Mechanism and Machine Theory ∎ (∎∎∎∎) ∎∎∎–∎∎∎
11
Frame
Frame
Moving part
Moving part
Leaf spring
Fig. 12. Prototype of the spatial compliant translational joint manufactured by 3D printing.
Fig. 13. Schematic of experimental setup (a) A diagram of the experiment for axial stiffness (b) diagram of the experiment for rotational stiffness.
Fig. 14. Experimental setup.
4.3.2. Rotational stiffness For the rotational stiffness, a 200 g weight is placed at each side of pulley in Fig. 13b. The average value of the tests is 3.61 105 N mm with standard deviation 1.23 105 N mm and the simulation for this item is 3.94 105 N mm. The difference of the rotational angles between simulation and experiment can be observed. In summary, the difference between the simulation and experimental results may be attributable to the reasons such as the accuracy of the CCD camera used to capture the displacement, the quality of the printing material and the difference between manufactured data and simulation ones. It can be noted that some of the dimensions of the prototype made by the 3D printing are not uniform, especially at the beam thickness, which is sensitive to the performance of the joint.
Please cite this article as: T.-S. Yang, et al., Design of a spatial compliant translational joint, Mechanism and Machine Theory (2016), http://dx.doi.org/10.1016/j.mechmachtheory.2016.08.007i
T.-S. Yang et al. / Mechanism and Machine Theory ∎ (∎∎∎∎) ∎∎∎–∎∎∎
12
Fig. 15. An FEA simulation result for the axial displacements of the design (unit: mm).
Fig. 16. Stiffness ratios for various loading conditions.
5. Conclusions A three-dimensional compliant translational joint was proposed. This joint combines the advantage of the series and the parallel connections of beam-type joints, and it possesses a high stiffness ratio and low axis drift. Benchmarking indices, including a large range of motion, low off-axis drift, and high stiffness ratio, were employed in the design process and realized in this design. An optimized set of dimensions for the joint based on the minimal value of the ratio of off-axis drift and actual axial displacement was allocated to fabricate a practical model. The performance of the mockup was measured and compared with the simulations. It was shown that the relative difference between experiment stiffness ratios and simulated data were within satisfaction.
Appendix A. Formula used for Table 3 [21] Sum of squares between the Groups, SSB k
SSB =
ni
k
∑ ∑ ( Xi ⋅ − X )2 = ∑ i=1 j=1
i=1
Xi2⋅ X2 − ni n
(A.1)
Sum of squares within the groups, SSE Please cite this article as: T.-S. Yang, et al., Design of a spatial compliant translational joint, Mechanism and Machine Theory (2016), http://dx.doi.org/10.1016/j.mechmachtheory.2016.08.007i
T.-S. Yang et al. / Mechanism and Machine Theory ∎ (∎∎∎∎) ∎∎∎–∎∎∎ k
SSE =
13
ni
∑ ∑ ( Xij − Xi ⋅)2
(A.2)
i=1 j=1
where ni is the number of observations in one group, k is the number of groups, and n is the total number of observations. F-factor
⎛ SSB ⎞ ⎛ SSE ⎞ ⎟/ ⎜ F=⎜ ⎟ ⎝ k − 1 ⎠ ⎝ ni − k ⎠
(A.3)
Appendix B. Variance analysis of the design parameters
W
level-1 level-2 level-3 level-1 level-2 level-3 level-1 level-2 level-3 level-1 level-2 level-3
L
T
S1
13.654 9.672 18.652 9.677 15.977 16.323 19.645 15.107 7.225 18.781 10.306 14.326
SSB SSE
121.478 434.155
MSB MSE
60.739 72.359
F-factor
0.839407
SSB SSE
83.977 471.655
MSB MSE
41.989 78.609
F-factor
0.534144
SSB SSE
236.960 318.673
MSB MSE
118.480 53.112
F-factor
2.230746
SSB SSE
109.902 448.605
MSB MSE
54.951 74.768
F-factor
0.734961
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