11038 FACTA UNIVERSITATIS Series: Mechanical Engineering https://doi.org/10.22190/FUME220822044P © 2020 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND Original scientific paper GUIDING ACCURACY OF THE WATT COMPLIANT COGNATE MECHANISMS Nenad T. Pavlović, Dušan Stojiljković University of Niš, Faculty of Mechanical Engineering, Serbia Abstract. In kinematics, cognate linkages are those that ensure the same input-output relationship or coupler curve geometry, while being dimensionally dissimilar. Compliant mechanisms gain some or all of their mobility from the relative flexibility of their joints rather than from the rigid-body joints only, which is the case with classical mechanisms. In this paper the guiding accuracy of the compliant cognate Watt four-bar linkages has been presented. The compliant cognate mechanisms have been developed as the counterparts of the rigid-body Watt mechanism, where the coupler point can be guided on an approximate rectilinear path. The guiding accuracy of the "coupler" point on the path segment (minimal deviation between exact rectilinear and realized path) has been calculated for the new-designed compliant mechanisms. Key Words: Compliant Mechanisms, Four-bar Linkages, Cognate Linkages, Watt Mechanism, Coupler Point Guiding 1. INTRODUCTION Compliant mechanisms gain some or all of their mobility from the relative flexibility of their joints rather than from the rigid-body joints only, which is the case with classical mechanisms. There are many advantages of using the compliant joints in the mechanism structure: a mechanism can be built in one piece, the weight can be reduced and wear, clearance, friction, noise and need for lubrication can be eliminated. Therefore, they are suitable for application in micromachining. On the other hand, because of the elastic deformation, the motion range of these flexible segments is restricted [1]. There are many papers considering the structure and function of the compliant joints and compliant mechanisms. A number of authors have established basic nomenclature and classification for the components of compliant mechanisms [2, 3, 4]. A method to aid the design of a class of compliant mechanisms wherein the flexible sections (flexural pivots) are small in length compared to the relatively rigid sections has been also Received August 22, 2022 / Accepted October 26, 2022 Corresponding author: Nenad T. Pavlović University of Niš, Faculty of Mechanical Engineering, Aleksandra Medvedeva 14, 18106 Niš, Serbia E-mail: nenad.t.pavlovic@masfak.ni.ac.rs 2 N.T. PAVLOVIĆ, D. STOJILJKOVIĆ suggested [2, 5]. Ananthasuresh and Kota [6] have presented a formal structural optimization technique called the homogenization method in order to design flexible structures (compliant mechanisms). Böttcher et al. [7] have introduced new ideas of technically realizable joints from nature and their integration into elastically movable structures for motion tasks in positioning and manipulating engineering. The influence of the geometry as well as the material type of the compliant joints on guiding accuracy of some point of the compliant mechanisms has been also analyzed [8]. Pavlović and Pavlović [9] have presented the original designs of the compliant parallel-guiding mechanisms. There are many papers that offer information on large motion compliant mechanisms. Mackay et al. [10] have introduced metrics for large-displacement linear- motion compliant mechanisms. Dirksen et al. [11] have presented the topology synthesis of large displacement compliant mechanisms with specific output motion paths. The large-range compliant mechanisms have been also investigated [12–14]. Several papers are dedicated to the consideration of the pantograph apparatus as a basis for developing a compliant mechanism. Merriam et al. [15] have described a statically balanced concept and have demonstrated its optimization, testing, and implementation for a haptic pantograph mechanism. A number of authors have presented how compliant pantograph mechanisms can be used for linear displacement applications [16, 17]. Patil et al. have dealt with a large motion displacement analyzed with an FEM simulation. For the model shown in [18], the pantograph apparatus is not included but a similar double parallel mechanism is shown and used to create a specific desired motion. Stojiljković et al. [19] have presented the designing of the compliant mechanism based on two symmetrical pantograph rigid-body mechanisms in order to achieve large scissors-like motion. The designing and guiding accuracy of the compliant cognate four-bar linkages developed as the counterparts of the rigid-body Roberts–Chebyshev mechanism, have been also presented [20, 21, 22]. The frequent method to synthesize a compliant mechanism is to design it as the counterpart of the rigid-body linkage being able to realize pre-defined function of the compliant mechanism (rigid-body replacement method). This paper deals with the design and guiding accuracy analyze of the compliant cognate four-bar linkages. The compliant cognate mechanisms have been developed as the counterparts of the rigid-body Watt mechanism, where the coupler point can be guided on an approximate rectilinear path. The aim of this paper is to analyze the guiding accuracy of the "coupler" point on the path segment (minimal deviation between exact rectilinear and realized path) for the new- designed compliant mechanisms. 2. WATT RIGID-BODY FOUR-BAR LINKAGE The coupler point C of the Watt four-bar linkage (Fig. 1), located in the middle of the coupler, can be guided on an approximate rectilinear path segment [6]. Guiding Accuracy of the Watt Compliant Cognate Mechanisms 3 Fig. 1Watt four-bar linkage The links lengths of this mechanism as well as the position of the input crank, that enable realizing of horizontal displacement of coupler point C of ΔxC = 5 mm, with the small angular displacement of the input crank (Δ ≤ 5o) and the minimal difference between realized and exact rectilinear path (ΔyC), has been shown as following [6]: 𝑎 = 𝐴0𝐴̅̅ ̅̅ ̅ = 61.406 𝑚𝑚 (1a) 𝑏 = 𝐵0𝐵̅̅ ̅̅ ̅ = 𝑎 (1b) c= 𝐴𝐵̅̅ ̅̅ = 1.04𝑎 (1c) d= 𝐴0𝐵0̅̅ ̅̅ ̅̅ ̅ = 2.1𝑎 (1d) 𝐴𝐶̅̅̅̅ = 𝐵𝐶̅̅ ̅̅ = 𝑐 2 (1e) 𝜑 = 313 ÷ 318 𝑜 (1f) 𝛥𝑦𝐶 = 6.4 𝜇𝑚 (1g) The cognate four-bar linkages are those that ensure the same input-output relationship or coupler curve geometry, while being dimensionally dissimilar. The Roberts–Chebyshev theorem states that each coupler curve can be generated by three different four-bar linkages. These four-bar linkages can be constructed using similar triangles and parallelograms. Fig. 2 shows two cognate Watt rigid-body four-bar linkages: A0A1CB1K0 and B0B2CA2K0, introduced in this paper, where coupler point C, located in the end of the coupler, can be also guided on an approximate rectilinear path segment. 4 N.T. PAVLOVIĆ, D. STOJILJKOVIĆ Fig. 2 The cognate Watt four-bar linkages Geometrical relations defining link positions are: 𝐴0𝐴1̅̅ ̅̅ ̅̅ ̅ǁ𝐴𝐶̅̅̅̅ ˄ 𝐴0𝐴̅̅ ̅̅ ̅ǁ𝐴1𝐶̅̅ ̅̅ ̅ (2a) 𝐵0𝐵2̅̅ ̅̅ ̅̅ ̅ǁ𝐵𝐶̅̅ ̅̅ ˄ 𝐵0𝐵̅̅ ̅̅ ̅ǁ𝐵2𝐶̅̅ ̅̅ ̅ (2b) 𝐴𝐶̅̅̅̅ : 𝐶𝐵̅̅ ̅̅ = 𝐴1𝐵1̅̅ ̅̅ ̅̅ : 𝐵1𝐶̅̅ ̅̅ ̅ ⇒ 𝐵1𝐶̅̅ ̅̅ ̅ = 1 2 𝐴1𝐶̅̅ ̅̅ ̅ (2c) 𝐵𝐶̅̅ ̅̅ : 𝐶𝐴̅̅̅̅ = 𝐵2𝐴2̅̅ ̅̅ ̅̅ ̅: 𝐴2𝐶̅̅ ̅̅ ̅ ⇒ 𝐴2𝐶̅̅ ̅̅ ̅ = 1 2 𝐵2𝐶̅̅ ̅̅ ̅ (2d) 𝐴𝐶̅̅̅̅ : 𝐶𝐵̅̅ ̅̅ = 𝐴0𝐾0̅̅ ̅̅ ̅̅ ̅: 𝐾0𝐵0̅̅ ̅̅ ̅̅ ̅ ⇒ 𝐴0𝐾0̅̅ ̅̅ ̅̅ ̅ = 1 2 𝐴0𝐵0̅̅ ̅̅ ̅̅ ̅ (2e) Coordinates of the cognate mechanism joints have been calculated based on plane analytic geometry, respectively:  joint A1 𝑥𝐴1 = 𝑦𝐴0−𝑦𝐶+ 𝑦𝐴0−𝑦𝐴 𝑥𝐴0−𝑥𝐴 𝑥𝐶− 𝑦𝐶−𝑦𝐴 𝑥𝐶−𝑥𝐴 𝑥𝐴0 𝑦𝐴0−𝑦𝐴 𝑥𝐴0−𝑥𝐴 − 𝑦𝐶−𝑦𝐴 𝑥𝐶−𝑥𝐴 (3a) 𝑦𝐴1 = 𝑦𝐶 + 𝑦𝐴0−𝑦𝐴 𝑥𝐴0−𝑥𝐴 (𝑥𝐴1 − 𝑥𝐶)=𝑦𝐴0 + 𝑦𝐶−𝑦𝐴 𝑥𝐶−𝑥𝐴 (𝑥𝐴1 − 𝑥𝐴0) (3b)  joint B1 𝑥𝐵1 = 1 2 (𝑥𝐴1 + 𝑥𝐶) (3c) 𝑦𝐵1 = 1 2 (𝑦𝐴1 + 𝑦𝐶) (3d) Guiding Accuracy of the Watt Compliant Cognate Mechanisms 5  joint B2 𝑥𝐵2 = 𝑦𝐵0−𝑦𝐶+ 𝑦𝐵0−𝑦𝐵 𝑥𝐵0−𝑥𝐵 𝑥𝐶− 𝑦𝐶−𝑦𝐵 𝑥𝐶−𝑥𝐵 𝑥𝐵0 𝑦𝐵0−𝑦𝐵 𝑥𝐵0−𝑥𝐵 − 𝑦𝐶−𝑦𝐵 𝑥𝐶−𝑥𝐵 (3e) 𝑦𝐵2 = 𝑦𝐶 + 𝑦𝐵0−𝑦𝐵 𝑥𝐵0−𝑥𝐵 (𝑥𝐵2 − 𝑥𝐶)=𝑦𝐵0 + 𝑦𝐶−𝑦𝐵 𝑥𝐶−𝑥𝐵 (𝑥𝐵2 − 𝑥𝐵0) (3f)  joint A2 𝑥𝐴2 = 1 2 (𝑥𝐵2 + 𝑥𝐶) (3g) 𝑦𝐴2 = 1 2 (𝑦𝐵2 + 𝑦𝐶) (3h)  joint K0 𝑥𝐾0 = 1 2 (𝑥𝐴0 + 𝑥𝐵0) (3i) 𝑦𝐾0 = 1 2 (𝑦𝐴0 + 𝑦𝐵0) (3j) 3. WATT COMPLIANT FOUR-BAR LINKAGE Fig. 3 shows a notch joint (circular flexure hinge) as a characteristic type of the compliant joint [6]. This compliant joint is fully determined by two parameters: the width of relatively rigid segments wR and the width of relatively elastic segments wE. Fig. 3 A notch compliant joint Based on the rigid-body Watt mechanism, a compliant Watt mechanism with notch joints was developed (Fig. 4a). Compliant joints A0 and A have been oriented in the direction of rigid-body input crank A0A, while compliant joints B0 and B have been oriented in the direction of rigid-body follower B0B. 6 N.T. PAVLOVIĆ, D. STOJILJKOVIĆ a) b) Fig. 4 The basic Watt compliant mechanism with notch joints (circular flexure hinges) in: a) undeformed position, b) deformed position The input force may act in the middle of the "input crank" (Fa), in the middle of the "follower" (Fb) or at the end of "coupler" (Fc). The deformed position of the compliant Watt mechanism is shown in Fig. 4b [6]. The symmetry axes of the compliant notch joints cross each other at the point 1 (Fig. 5) corresponding to revolute joint of the rigid-body counterpart. Fig. 5 Characteristic key points of a compliant notch joint The other characteristic key points of a compliant notch joint (Fig. 5) can be calculated by using the set of equations: Guiding Accuracy of the Watt Compliant Cognate Mechanisms 7 𝑟2⃗⃗ ⃗= 𝑟1⃗⃗⃗ + 𝑤𝑅 2 𝑒 𝑖(𝜙− 𝜋 2 ) (4a) 𝑟3⃗⃗ ⃗= 𝑟2⃗⃗ ⃗+ 𝑤𝑅−𝑤𝐸 2 𝑒𝑖𝜙 (4b) 𝑟4⃗⃗⃗ = 𝑟3⃗⃗ ⃗+ 𝑤𝑅−𝑤𝐸 2 𝑒𝑖(𝜙−𝜋) (4c) 𝑟5⃗⃗ ⃗= 𝑟4⃗⃗⃗ + 𝑤𝑅−𝑤𝐸 2 𝑒 𝑖(𝜙+ 𝜋 2 ) (4d) 𝑟6⃗⃗ ⃗= 𝑟1⃗⃗⃗ + 𝑤𝑅 2 𝑒 𝑖(𝜙+ 𝜋 2 ) (4e) 𝑟7⃗⃗ ⃗= 𝑟6⃗⃗ ⃗+ 𝑤𝑅−𝑤𝐸 2 𝑒𝑖𝜙 (4f) 𝑟8⃗⃗ ⃗= 𝑟6⃗⃗ ⃗+ 𝑤𝑅−𝑤𝐸 2 𝑒𝑖(𝜙−𝜋) (4g) 𝑟9⃗⃗⃗ = 𝑟6⃗⃗ ⃗+ 𝑤𝑅−𝑤𝐸 2 𝑒 𝑖(𝜙− 𝜋 2 ) (4h) Angle determines the orientation of a compliant joint in reference to the x-axis (Fig. 5). Fig. 6a shows the first cognate compliant Watt mechanism with notch joints, introduced in this paper, that has been developed based on cognate Watt rigid-body four- bar linkage A0A1CB1K0 (Fig. 6b). a) b) Fig. 6 The first Watt cognate compliant mechanism with notch joints (circular flexure hinges) (a) and its rigid-body counterpart A0A1CB1K0 (b)  8 N.T. PAVLOVIĆ, D. STOJILJKOVIĆ Fig. 7a shows the second cognate compliant Watt mechanism with notch joints, introduced in this paper, that has been developed based on the cognate Watt rigid-body four-bar linkage B0B2CA2K0 (Fig. 7b). Fig. 7 The second Watt cognate compliant mechanism with notch joints (circular flexure hinges) (a) and its rigid-body counterpart B0B2CA2K0 (b) The deformed position of the Watt cognate compliant mechanisms with notch joints are shown in Fig. 8. a) b) Fig. 8 The first (a) and the second (b) Watt cognate compliant mechanism with notch joints in deformed position Guiding Accuracy of the Watt Compliant Cognate Mechanisms 9 4. GUIDING ACCURACY OF THE WATT COMPLIANT MECHANISMS The guiding accuracy of "coupler" point C (on the path segment of ΔxC = 5 mm) of the compliant Watt mechanism as well its cognate compliant mechanisms with notch joints were analyzed. The lengths of the mechanism "links" have been defined in equations (1) with the following compliant joint parameters: the width of relatively rigid segments wR = 10 mm and the width of relatively elastic segments wE = 1mm. The position analysis of the compliant mechanisms was performed using the ANSYS Software for material piacryl (modulus of elasticity E = 3700 N/mm2, bending strength σbs = 90 N/mm 2) as well as material thickness of δ = 4 mm. This material is not expensive; it is available on the market as well as suitable for further experimental research of these compliant mechanisms. The calculation was performed for the elements with a rectangular cross-sectional area using PLANE 183 (2-D 8-Node Structural Solid) as a characteristic ANSYS element type (Fig. 9). Fig. 9 ANSYS element type PLANE 183 (2-D 8-Node Structural Solid) Due to the fact that the acting point of the input force does not have to be located on the "input crank" of the compliant mechanism, the guiding accuracies were compared for three different cases of location of input force acting points: in the middle of the "input crank" - Fa, in the middle of the "follower" - Fb and at the "coupler" point nearby the joint A or B– Fc (Fig. 4a, 6a, 7a). The results are shown in Table 1. Table 1 Guiding accuracy ΔyC [μm] of the Watt compliant mechanisms on the horizontal displacement of ΔxC = 5 mm Fa Fb Fc 0 2.3 8.8 11.7 1 0.3 6.5 17.8 2 29.2 0.6 1.3 0 - the basic Watt compliant mechanism 1 - the first cognate Watt compliant mechanism 2 - the second cognate Watt compliant mechanism 10 N.T. PAVLOVIĆ, D. STOJILJKOVIĆ 5. CONCLUSION The frequent method to synthesize a compliant mechanism is to design it as the counterpart of the rigid-body linkage being able to realize pre-defined function of the compliant mechanism (rigid-body replacement method). This paper deals with the design and guiding accuracy analyze of the compliant cognate four-bar linkages. The basic compliant mechanism as well as two compliant cognate mechanisms have been developed as the counterparts of the rigid-body Watt mechanism, where the coupler point can be guided on an approximate rectilinear path. The guiding accuracy, that is, the difference between realized and exact rectilinear path, have been analyzed and compared for all above mentioned compliant mechanisms being suitable to realize approximate rectilinear guiding of the coupler point. The basic compliant model with the input force acting in the middle of the "input crank", the first cognate compliant model with the input force acting in the middle of the "input crank" as well as the second cognate compliant model of the Watt mechanism with the input force acting in the middle of the "follower" and in the end of "follower", realize better rectilinear guiding accuracy than the rigid-body counterpart linkage. The differences between realized and exact rectilinear path of these compliant mechanisms rate in range yC = 0.3  2.3 m on the path segment of ΔxC = 5 mm, comparing with the yC = 6.4 m of the rigid-body counterpart linkage. Also, the producing of the rigid-body linkages is not possible without appearing of the tolerance of the links lengths and clearance in the revolute joints, which decrease the guiding accuracy. It means that the Roberts–Chebyshev theorem, concerning a coupler curve being generated by two different four-bar linkages, should be also applied on the compliant mechanism. The best guiding accuracy has been provided by the first cognate Watt compliant mechanism (yC = 0.3 m on the path segment of ΔxC = 5 mm) with the input force acting point located in the middle of the "input crank". Thus the cognate compliant mechanism has realized a smaller deviation between exact rectilinear and realized path than the basic compliant mechanism, and this fact confirms our suggestion that the results offered by the cognate compliant mechanisms should be taken into consideration in the synthesis procedure of the compliant mechanisms. Acknowledgements: This research was financially supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia (Contract No. 451-03-68/2022- 14/ 200109). REFERENCES 1. Zentner, L. 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