Statics Analysis and Optimization Design for a Fixed-Guided Beam Flexure

The ratchet mechanism has been used to ensure moving in one direction of rotation, i.e. either clockwise or counter-clockwise. This mechanism is designed based on xed-guided beam exures to reduce friction, and improves accuracy compared to the traditional mechanism. This paper presents a static analysis and parameter optimization for the xed-guided beam exures via using the pseudo-rigid-body model and a fmincon algorithm. The Finite Element Method (FEM) of the xed-guided beam also has been used to verify the maximum stress and the x−direction displacement. Modi ed pseudorigid-body model (M-PRBM) is also applied to signi cantly enhance the accuracy of the maximum stress value. The results show that the averaged errors of maximum stress between MPRBM and FEM are 3.48% for aluminum, and less than 10.9% for titanium, carbon steel, and alloy steel. From the obtained results, the MPRBM is good for prototype design and fabrication of ratchet mechanism in the future.


Introduction
Compliant mechanism (CM) [1] has received much attention in the last two decades because of their advantages such as free friction, no lubricant, reduced maintenance, increased precision, and lightweight compared with a conventionally kinematic mechanism. Ratchet mechanism is a common application of CM. This mechanism allows continuous linear or rotary motion in only one direction while preventing motion in the opposite direction. Ratchet mechanisms are widely used in machinery and tools such as a safety and accuracy device. This mechanism uses two xedguided compliant beams to store energy when it is subjected to force [2]. The xed-guided compliant beams also used in the gripper mechanism [3]. Two common approaches to design and analysis for this mechanism includes pseudo-rigidbody model (PRBM) [4]- [6] and nite element analysis (FEA) [7,8]. But, the error of the maximum stress value between PRBM and FEA is so large. To increase accuracy, a modied pseudorigid-body model (M-PRBM) approach is applied for xed-guided bistable compliant mechanisms [9]. Kennedy et al. [10] designed a ratchet mechanism by using two xed-pinned compliant beams and two xed-guided compliant beams. Although the ratchet mechanism was well de-signed; however, this study has not optimized the parameters of xed-guided beam exures to enhance its performances yet. Recent studies have conducted an optimization design for several compliant mechanisms in various applications, e.g. solar panel [11], a micro-displacement sensor for compliant micro-gripper [12]- [14], or compliant joint for camera positioning system of nanoindentation tester [15,16]. However, an optimization design for the ratchet mechanism has been poorly studied.
In this paper, we introduce an overview of the structural design process of xed-guided beam exures. Unlike previous studies, we optimize geometric parameters to improve the performances of the exures. Then, we have conducted eects of various materials and dimensions on the performances through numerical analysis. The goal of this article is to modeling static performances of the ratchet mechanism. And then, this study conducts an optimization design to enhance its performances. Aluminum material is recommended for this mechanism because of a good ratio of seven between the yield strength and Young's modulus. Other materials are further investigated as a potential of CM to expand the working ability of this mechanism. The maximum stress and the x−direction displacement are calculated by PRBM and nite element method (FEM). After that, M-PRBM of xed-guided mechanisms is proposed to increase maximum stress accuracy.

Selection of conguration
In this paper, the ratchet mechanism is designed by using beam exures. The structure of beam exures is designed with three parts as follows: i) The base is xed to the ground through-bolted joints of two holes with 6 mm in diameter. ii) The body consists of xed-guided beams arranged perpendicular to the base to support the top beam and store energy when the top beam is subjected to force in x−direction. Afterward, it moves to its original position when the top beam is no longer subject to force. iii) The top part is a horizontal beam that is responsible for ensuring the movement of the ratchet.
Three dierent congurations, which have various numbers of vertical beams, are adopted to numerically analyze the horizontal (x−direction) displacement and stress, seen in Figs. 1(a), 1(b) [10], and 1(c) [17]. These congurations may be used to design compliant with a high-precisions mechanism for safety and the arm device. By using ANSYS 18.1 software, the simulation results found that the displacement of one beam is highest, followed by two beams, and the displacement of three beams is lowest, as shown in Fig. 2, respectively.
Another important aspect of the ratchet mechanism is how to improve its positioning precision. This can be achieved by decreasing an undesired motion, called as parasitic error. In the ratchet mechanism, the y-direction displacement is the so-called parasitical displacement [1]. A similar simulation is conducted for three cases in Fig. 2. As depicted in Fig. 3, the results indicated that the parasitic displacement of the rst conguration in Fig. 3(a) is largest, followed by the second conguration, and the last one has the smallest value of parasitic error. Besides, if this error is large, then the ratchet mechanism could not work eciently. The congurations in Figs. 1(a) and 1(c) ensure their working requirement. So, these mechanisms meet the designed requirements. The conguration in Fig. 1(c) has the working space, and its manufacturing process has a higher cost than that in Fig. 1(b). However, these factors are not important, and so the design of that in Fig. 1(b) should be used for the ratchet mechanism. The description of its features is presented below.

Mechanism design
A mechanism, as shown in Fig. 4, consists of two vertical parallel xed-guided beams (i.e. the beam ends link to a xed base, and the others connect to the horizontal top beam). The dimensions of the xed base and the horizontal top beam are the constant values given in Tab. 1. They are referenced in [10] with a scale to match the mechanism size. Since the horizon- tal top beam is exerted the x−direction force F , and it will be displaced in the x−direction. The two vertical xed-guided beams will be bent, and they store elastic energy. Thanks to this energy, the top beam is back the original location when the force F is released. The process will be repeated with the constant value of the force F .
The two xed-guided beams not only must be designed to ensure the domestic machinability but also depend on the requirement of the compliant mechanism [1]. Therefore, the beam height, h, should be varied between 0.4 mm and 0.6 mm. The fabrication method for the prototype of mechanism can be used such as wire electric discharge matching. The two xed-guided beams not only must be designed to ensure the domestic machinability but also depend on the requirement of the compliant mechanism. Domestic machining by the EDM method is 0.4 Tab According to the theory of elastic deformation [18] based on the Kirchho hypothesis, the thin plate requires the ratio between the width and thickness must be greater than 30.
The material used in the whole mechanism is AL 7075-T6 [15]. It has a good ratio of seven be-  This value is ideal for compliant mechanisms [1]. The materials with the highest strengthto-modulus ratio will allow a larger deection before failure. Good materials have a high ratio. The parameters of this material are given in Tab. 2.
The structure of the whole mechanism should be a block as seen in Fig. 4, (i.e. there are no joints between the beams). The dimensions of two xed-guided beams are designed by applying the pseudo-rigid-body model (PRBM) theory [1], and are optimized by using the Optimization tool of the Matlab software [19]. These will be presented in the next section.

Pseudo-rigid-body model
In this section, the xed-guided mechanism is analyzed though PRBM, as shown in Fig. 5. This method has been proposed by Howell [1], and has been applied to other mechanisms [4,9]. This model is particularly useful for the large deection analysis. Two xed-guided beams used in this mechanism have no original curvature, and the moment is zero at the mid-point of the beam. The design parameters of the two beams are given in Tab. 3. The values of K Θ , γΘ max are referenced in [1]. The safety factor, SF = 1.5 [20] for well-known materials, under reasonably constant environmental conditions, subjected to loads and stresses that can be determined readily. The symbols indicated the width, height, and length of the xed-guided beam are given as w, h and L, respectively. Those dimensions need to be designed beams.
The x-direction displacement of the beam is given by Howell [1] as The y−direction displacement of the beam is addressed as where, The spring constant of the beam is computed by where, the moment of inertia of the beam Using the virtual work principle, force F causes the deformation of the beam to be measured by an equation Maximum stress at the end of the beam is where, c is distance from neutral shaft to outer surface of beam Permissible stress is calculated by From Eqs. (4)-(6) equations, the length of beam exure is computed as or, From (1) and (11) equations. The x−direction displacement of the beam where, p is displacement coecient in From Eqs. (3), (5), and (7), the stress is found as Maximum stress at the end of the beam where, q is coecient of maximum stress

Optimization design
The optimal design [19] is always mentioned in structural design problems to ensure the minimum use of materials to minimize the production cost and maximize the performance of the mechanism. Design variables need to be optimized including the width, height, and length of the xed-guided beam. To achieve these optimal values, the problem is set up as follows: -Design variables: The length of the xedguided beam is calculated using equation (11).
-Design vector [X]: -Objective function: The objective function is set based on the work requirement of mechanism. The horizontal x−direction displacement reaches the desired value. Therefore, based on equation (12), the objective function in this problem is given -Equality constraint: The ratio between displacements in the y and the x−directions -Inequality constraints: To ensure the working ability of the mechanism, the maximum stress of the beam σ max will be not exceed the permissible stress [σ u ]. Combining equations (9) and (15), we obtain or, Besides, in the mechanism design, the ratio between the width and height is given or, The displacement of the y-direction of the beam is the parasitical displacement. Therefore, it needs to have a constraint not to aect the performance of the function of the mechanism or, where, k is displacement coecient in y−direction Finally, we use the Matlab R2016b software to optimize the objective function with the "fmincon" function (i.e. objective function is now −f (x)). The optimal results are obtained: So, the optimal sizes are used to design the mechanism as w = 15 mm; h = 0.5 mm; L = 136 mm.

Finite element method
FEM is used to verify the deviations of the PRBM presented in the previous section. This method uses the constant parameters in Tab. 1 with the material used is AL 7075-T6. The AN-SYS 18.1 software used in this method helps to visually illustrate the movement of the device after applying the force F in the x−direction.
Meshing method: In FEM, the mesh is very important that aects the simulation results. In FEM we use three common types of mesh in Fig.  6 such as meshing automatic method, renement coupled with an automatic method, and sizing method. Automatic meshing reduces time but the results of the mesh may not converge, resulting in less accurate calculations. For better calculation results, renement mesh can be used. However, in some cases such as a simple structure or a dicult mesh to split, the use of renement mesh is not necessary. Sizing mesh based on size to mesh should take time to nd optimal results. To assess the quality of the mesh, skewness standards are used.
The optimal size values are simulated with three dierent mesh types. The simulation results of displacements in the x−direction and maximum stresses will be compared with those of PRBM, seen in Tab. 4. Besides, skewness standard, number of elements and number of nodes are used to assess the quality of the meshes. With the results shown in Tab. 4, the sizing mesh is not used in simulation because Skewness value has pool and error of displacements in the x−direction and maximum stresses are large. The displacements error of automatic mesh and renement mesh are small so FEM is suitable for calculation with this value. Opposite, the maximum stress errors of above two methods are large. Among the above meshes, the Skewness standard of automatic mesh has the best value. So, the automatic mesh is only the best t in this design. In addition, the errors between the PRBM and FEM in case of stress is very large. Therefore, the PRBM is almost suitable to model the displacement but this method is failed to analyze the stress.
According to the simulation results in Fig.  3(b), the value of parasitical displacement is very small. So, it will not aect the operation of the mechanism, and is neglected.

Modied
Pseudo-rigid-body model (M-PRBM) From the FEM results obtained, the averaged errors of displacements in the x−direction are less than 7%. Those of the maximum stresses are greater than 50%. There is a large error of max. Stress between the FEA simulations and the PRBM using for thin bar elastic deformation. This may be explained by the following reasons: 1) the compliant mechanisms are often used in structures that require small displacements and 2) the limitation of the PRBM model, which replaces the links between beams by the torsion springs, should ensure the accuracy of deformation in the x−direction only. In order to obtain the accuracy of the maximum stress value in the calculation, the coecient α is added as, α = σ tinh toan σ mo phong = 320.326 168.75 = 1.8982, Where α is the conversion coecient of stress. Thus, to calculate the maximum stress values when using the PRBM model, we use the following equation: The calculated M-PRBM results after using the coecients α = 1.8982 and their errors with those of FEM ones are presented in Tab. 5.
With the results shown in Tab. 5, the difference in the maximum stress values between the M-PRBM and FEM with the height h = 0.4 mm is less than 5%. That of h = 0.5 mm and 0.6 mm is less than 6% in Fig. 7. Thus, the value of deviation between the PRBM and FEM depends heavily on the height value. This is entirely consistent with the theory of compliant mechanism. In the viewpoint of machinability, 4. Inuence of materials and congurations to maximum stress and In this section, the dierent materials and congurations are examined in order to enhance the applicability of the mechanism. The parameter of four dierent materials is shown in Tab. 6. The good materials must have the ratio between the yield strength and Young's modulus more than 5.  Fig. 8 and Fig. 9. Maximum stress distributions of four dierent materials are plotted against the beam width, seen in Fig. 8. They are calculated by the M-PRBM and FEM. By increasing the beam width, all of the stresses gradually decrease (i.e. the maximum stress is in- .42% for the carbon steel, stainless steel, titanium, and aluminum, respectively. There are large dierences between two methods for the carbon steel, stainless steel, and titanium while there is a good agreement between them for the aluminum. This may be explained by the elastic modulus of aluminum smaller than the other materials. It is noted that the elastic modulus aects the calculation of the maximum stress. When the modulus E value of each material is changed, L value also changes according to equation (11). This results in that of the aluminum obtain the smallest values.
Calculated displacements in the x−direction prole of four dierent materials are drawn against the beam width, seen in Fig. 9. They are computed by the PRBM and FEM methods. By growing the beam width, all of the displacements considerably gradual rise (i.e. the displacements in the x−direction is proportional to the increase of the beam width). The maximum displacements in the x−direction of carbon and stainless steels are 31.77 mm and 31.45 mm, respectively. Those of non-steel such as titanium and aluminum are 24.15 mm and 18.71 mm, respectively. The maximum displacements of the carbon and stainless steels are approximate values, and are higher than those of the titanium and aluminum. The averaged errors between the PRBM and FEM are 11.30%, 11.56%, 2.89% and 1.27% for carbon steel, stainless steel, titanium, and aluminum, respectively. There are large differences between the two aforementioned methods for carbon steel, stainless steel. There are good agreements between them for titanium and aluminum. Similar to the calculated results of the maximum stresses, the elastic modulus of aluminum is smaller than that of the other materials. This leads to the displacement of the aluminum obtains the smallest values.  Fig. 10 and Fig. 11.
The data in Fig. 10 shows that maximum stress distributions of four dierent materials are plotted against the beam width. Similar to the calculated results of the maximum stresses show in Fig. 8. All of the stresses also decrease when the beam width increase. The averaged er-rors of the carbon steel, stainless steel, titanium, and aluminum are 16.44%, 12.02%, 14.02%, and 3.54%, respectively. The maximum stresses of those are 320 N/mm 2 , 317 N/mm 2 , 238 N/mm 2 , and 197 N/mm 2 , respectively. The maximum stresses of the titanium and aluminum lower than of the carbon and stainless steel. The carbon steel, stainless steel, and titanium have large averaged errors while the aluminum is nearly accuracy. This may be explained similarly to the calculated results of the maximum stresses with h =0.4 mm. The data in Fig. 11 shows that calculated displacements in the x−direction prole of four dierent materials are drawn against the beam width. The maximum displacements in the x−direction of titanium and aluminum are 32.94 mm and 26.14 mm, respectively while carbon and stainless steels of this value are 44.4 mm and 43.96 mm, respectively. The averaged errors of the carbon steel, stainless steel, titanium and aluminum between the PRBM and FEM are 14.56%, 15.21%, 4.83%, and 2.96%, respectively. Similarly, as Fig. 9, the displacement of the aluminum obtains the smallest values.  Calculation results for h = 0.6 mm are presented in Tab. 9, and plotted in Fig. 12 and Fig. 13.
Maximum stress results drawn in Fig. 12 can be compared with the data in Fig. 8. The averaged error of titanium is 13.84%, and of the carbon steel, stainless steel, and aluminum are 4.96%, 4.56%, and 4.5%, respectively. There is a large dierence between the PRBM and FEM for the titanium while there is a good agreement between them for the other materials. The maximum stresses of those are 292.5 N/mm 2  of the titanium and aluminum are approximate values and less than the carbon and stainless steels. The averaged errors between the PRBM and FEM are 6.92%, 5.32%, 6.58%, and 2.6% for the carbon steel, stainless steel, titanium, and aluminum, respectively. There are small dierences between two methods for all materials.
Summarily, the errors of maximum stress and x−direction displacements between the M-PRBM and FEM are generated by calculated. The M-PRBM calculates the line while FEM base on the area. Moreover, materials change (elastic modulus E change) greatly affect the accuracy of the maximum stress and the x−direction displacement between M-PRBM and FEM. Therefore, M-PRBM theory is only suitable for materials with elastic modulus similar to aluminum material. Besides, when h = 0.4 mm and 0.5 mm, the error of maximum stress and the x−direction displacement increases as the material has an elastic modulus larger (i.e. the error is proportional to the elastic modulus E). However, when h = 0.6 mm, this error does not change much when materials are changed, except for the average error of the maximum stress for titanium material.

Conclusions
In this paper, static analysis and optimization design are proposed for the xed-guided beam exures. These exures are intended for driving the ratchet mechanism. The PRBM theory for large deformations of xed-guided parallel beams exures are used to calculate the maximum stress and the x−direction displacement. Predicted values from the analytical method are veried by FEM using ANSYS 18.1 software. The FEM simulations are established for three cases of the beams that have the height varying from 0.4 mm to 0.6 mm with four dierent materials. The conguration of the xedguided beam has been optimized for maximum x−direction displacement. Because there is a large error of maximum stress between PRBM and FEM, so M-PRBM is designed to obtain the accuracy of the maximum stress value. The value of y−direction displacement (parasitical displacement) is very small, so it is neglected. Compared with the FEM simulations, the M-PRBM is a better calculation of the maximum stress than the PRBM. The averaged errors between the M-PRBM and the FEM simulation are 3.48% for aluminum, and less than 10.9% for titanium, carbon steel, and alloy steel. The M-PRBM is therefore good for the design and fabrication of the compliant mechanism. Future work, the proposed method is extended for related compliant mechanisms.