Optimization Design and Experimental Evaluation of a Low-Cost Linear Motor with Compliant Bearings for Precision Engineering Applications

Abstract

This paper presents the structural optimization and experimental evaluation of a novel linear motor for applications in precision engineering industry. The output linear motion from the motor is supposed to be highly accurate since it is supported by a couple of compliant bearings. The stroke of the proposed motor is determined by the largest displacement of compliant bearings in elastic region. Therefore, the beam-type flexure, which is able to produce large elastic deformation, is suitable for the compliant bearing design. In addition, the bearing design is optimized using the response surface methodology (RSM) in ANSYS to achieve a compliant structure with high deformation capability in desired directions. The optimized bearings will then be assembled with the voice coil actuator (VCA) to create a complete 1-DOF linear motor. Several experimental evaluations are carried out to verify the actual performance of the motor. The experimental results suggest that the proposed linear motor is able to produce precise linear motion with high accuracy and high repeatability. Furthermore, the millimeter-scale stroke of the motor is large enough to use in a variety of devices in precision engineering industry.

INTRODUCTION

For every precise positioning system, two key factors that reduce the precision of desired motions are friction and inertia. Popular actuators and mechanisms, such as pneumatic systems, servo motors with screw shaft, slider-crank linkages, have unavoidable inertia, especially when working in high-speed or in microscale/ nanoscale range. Meanwhile, the friction between mechanical components cannot be estimated accurately after a period of used time due to various reasons, such as lubrication, wear of material caused by dry friction and many other unpredictable errors in manufacturing and assembly processes. To overcome such limitations, compliant mechanisms can be considered as a potential solution for applications with relative-small working range, high precision, good repeatability, lubrication- and maintenance-free.

In this work, the elastic deformation of a thin beam-type compliant mechanism is employed to create desired output motions of compliant bearings. As a result, the compliant bearings can eliminate the inertia by spring force and produce precise displacement based on Hooke’s law. In addition, by combining a pair of compliant bearings with a voice coil actuator (VCA), a novel linear motor actuated by electromagnetic force can be archived to perform frictionless and precise motion.

Referring to previous literatures, the compliant mechanism can be designed based on the combination between rigid and deformable beam elements 1 , 2 , 3 . To improve the maximum deformation of the bearings while the original structure 4 , 5 is remained, the dimensions of each beam are set as design variables for static structural analysis 6 , 7 in ANSYS. By varying the values of these variables according to the constraints for beam dimensions, the output displacement and von Mises stress data can be collected to build the response surfaces 8 , 9 . From these response surfaces, the optimized design that can produce large elastic deformation along the desired direction and has large stiffness in non-actuating directions can be achieved. The mentioned literatures presented a variety of synthesis and optimization methods for compliant mechanism, but most previous work only provided some individual compliant mechanism designs as case studies for demonstrating the methods and have not considered their industrial applications. In this work, a novel design of compliant bearing is optimized based on the existing methods. Its design is specific for a linear motor, which is developed for precise motion systems.

From the optimized compliant bearing, a voice coil module 10 is then designed to fit the dimensions and provide the required force for the deformation of compliant bearings. To simplify the control system and reduce the burden of signal linearization like previous actuators 11 , 12 , the testing module is aimed to have an open-loop control system with a linear transfer function that is developed based on the relationships between applied DC voltage, electromagnetic force and translational displacement. Typically, a VCA needs a translational joint to produce the desired linear motion. In this work, the compliant bearings and voice coil module are designed to fit each other to create a unique linear motor with simple structure, but is capable of producing high accurate linear motion by exploiting the frictionless property of compliant mechanism. A prototype of the motor is then built and experimentally tested to demonstrate its effectiveness.

The remaining of the paper is organized as follow: design of the linear motor, optimization process, experiment setup, optimization result, experiment result and conclusion.

RESEARCH METHOD

Design of the Linear Motor

The linear motor consists of two main modules: a pair of compliant bearings and a voice coil actuator (illustrated in Figure 1 ) with the targeted boundary dimensions being limited to 100mm x 100mm x 180mm to fit popular precise motion systems. The rotor shaft of the VCA, which is assembled with the central flanges (moving bodies) of two bearings, will generate electromagnetic force when a DC voltage value is applied and will make the compliant mechanisms deform. The stator of the VCA is mounted to the fixed bodies of compliant bearings as shown in Figure 2 . The actuated position of the rotor shaft is located at the midpoint between bearing centers so that each compliant mechanism can be acted by equal force and producing identical displacement.

Figure 1 . The kinematic diagram of linear motor.

Based on previous works ⁴ , the initial model of the compliant bearing is designed and calculated by applying the serial-parallel relationships between beam elements ( Figure 2 ). In particular, the bearing consists of four symmetrical legs about its center. Each leg is constructed by five beam-type flexures connected serially. In this design, all of the beams have the same thickness b and width h , only the lengths L of them are changed to well fulfill the given design space. For the bearing material, aluminum alloy 6061 is selected due to its popularity in the compliant mechanism. The detailed mechanical properties of this material are given in Table 1 .

Figure 2 . Original design for one leg of the compliant bearing.

Table 1 Mechanical properties of aluminum alloy 6061

Based on the Hooke’s law in equation (1), to control the displacement x of compliant bearings, the total stiffness k along z axis of beam elements will be determined by using 6x6 stiffness matrix of a typical fixed-end beam (2) for each beam element above and then synthesizing these matrices together following relationships in Figure 3 .

F = k.x (1)

Where: F : Applied force (elastic force) (N)

k : Stiffness of element (Nm)

x : Displacement of element (m)

Figure 3 . Construction of the linear motor (a) and its stiffness modelling (b).

Where: E : Young’s modulus (MPa)

G : Shear modulus (MPa)

A : Cross-sectional area (m ² )

I _y : Inertia moment about y axis (m ⁴ )

I _z : Inertia moment about z axis (m ⁴ )

J : Torsion constant (m ⁴ )

To verify the calculated result, static structural analysis is also performed in Ansys software and the final result of maximum displacements for two methods is shown in Table 2 .

Table 2 Result of the initial motor with 2 compliant bearings

From the required force to archive the maximum displacement of 2 compliant bearings and the boundary dimensions of the motor, a moving-coil VCA was designed with two components: stator and rotor ( Figure 4 ). The strongest output force that this VCA can generate is estimated by the Lorentz force formula in equation (3) with the maximum voltage source 12V and the working current cannot exceed 10A.

Figure 4 . The VCA stator (a) and rotor (b).

F = i _coil l _pass B _gap (3)

F: Electromagnetic force (N)

i _coil : Current through the rotor coil (A)

l _pass : Length of the wire which flux lines pass (m)

B _gap : Magnetic flux density (T)

While i _coil and l _pass can be determined from the number of wiring turns, the magnetic flux density of the VCA is calculated by both analytical method via equivalent closed magnetic circuit model and finite element method with magnetostatic simulation5. It is noted that the stator consists of the permanent magnet and a cover while the rotor is a coil as shown in Figure 4 . From the calculation, the voice coil actuator can have maximum 3.2N linear force. This value will not satisfy the maximum force from Table 2 with nearly 6.4N. Therefore, an optimization is conducted to not only reduce the required force but also increase the maximum displacement.

Structural Optimization

The aim of this optimization is to improve the maximum elastic displacement of the compliant bearings while decreasing the required force from 6.4N to 3.2N or lower. This section demonstrates the general model of optimization in ANSYS with a defined objective, constrains and then analyses the result data to determine the most effective structure for the compliant bearing prototype.

As the compliant bearing is constructed by 4-legged configuration, the structural optimization is conducted for one leg only. For each beam element, there are three dimensions that can be set as design variables for the optimization: Thickness b _i , width h _i and length L _i of the beam i as shown in Figure 5 a where i = 1, 2, 3, 4. From the formula of stiffness of a cantilever beam under bending (illustrated in Figure 5 b) written in equation (4), it is obvious that the minimum thickness will provide the lowest stiffness, which also means the beam can perform a high bending flexibility about this direction effectively. Due to the complexity of using an ultra-thin metal plate for the accurate fabrication and assembly, the beam thicknesses in this work are kept as constant of 0.5mm to create the high flexibility of the compliant bearing and ensure its stable structure during the manufacturing and assembly processes.

Figure 5 . Design variables of each beam (a), bending cantilever beam model (b).

k _cantilever = 3EI/l ³ = Ebh ³ /4l ³ (4)

Take advantage of the symmetrical design of the compliant bearing, seven dimensions of a leg are analysed as input variables as illustrated in Figure 6 a. By this way, instead of setting ten variables for the lengths and widths of 5 beams, the number of design variables is minimized to 7 variables.

After defining the input variables for constraining, the output variables for the objective of optimization are also considered. There will be two parameters that are set as target variables: the center displacement (VCA rotor shaft position) of the compliant mechanism under the acting of 1N force (also known as the linear compliance of the bearing under an actuating force from the VCA) and the maximum von Mises stress in the bearing structure.

The objective of this optimization is to maximize the displacement of the compliant bearing under desired actuating force while its original structure is remained, and the von Mises stress is constrained to be lower than the yield strength of material as shown in Table 1 .

Figure 6 . Optimization variables (a) Considered variables for each optimization case (b).

The result for displacement and von Mises stress variables with initial beam dimensions in Figure 2 will be calculated first in ANSYS and connected to 3 single-objective optimization (Pareto optimality) setups with different constraining cases, i.e., the input variables will change beam widths only, the input variables will change beam lengths only, the input variables will change both beam widths and lengths ( Figure 6 b). The constraints for input variables will be defined based on the boundary dimensions of the motor model to avoid the penetration between beam elements. Each optimization setup will provide an output displacement - stress data and this result will be used to build response surfaces. By using the genetic algorithm (GA) with defined constraints and a single objective to maximize the output displacement along Z axis, the optimized dimensions of beam elements can be determined from ANSYS software.

There is also another method to archive the optimized dimensions, which is identifying the sweet spot of response surfaces by analysing the data from static structural analysis. Because the target is to maximize the displacement and minimize the stress, the ratio t between these two parameters can be determined for each dimensions combination in the result data following equation (5).

t = c _j /σmax _j (5)

t: Ratio between max displacement and max stress

C _j : Maximum displacement under 1N force (mm)

σmax _j : Maximum von Mises stress (MPa)

The combination with highest t will be the best result for optimization due to both high displacement and low stress. In reverse, the model with highest t or sweet spots of response surfaces is not always the finest design to build the prototype. Too low compliance can cause unstable motion when applying and dependent displacements along remaining DOFs. Thus, beside considering the sweet spots, the points which locate close to these spots must also need to be noticed.

Results

Optimization result

From the optimization process, response surface results for 3 cases with the most influential design variables can be obtained and presented in Figure 7 , Figure 8 and 9. Overall, 3 cases have quite similar optimized results for the new dimensions of beam elements and the best combination of each case is shown in Table 3 . By using this data, each combination will be calculated by the stiffness matrix method from equation (2) and also performed in simulation again to determine which is the best one for building the prototype of the linear motor later.

Figure 7 . Response surface of displacement (a) and maximum von Mises stress (b) in beam widths only optimization.

Figure 8 . Response surface of displacement (a) and maximum von Mises stress (b) in beam lengths only optimization.

Figure 9 . Response surface of displacement (a) and von Mises stress (b) in both beam widths and lengths optimization.

Table 3 Finest combinations of each optimization cases

By using above optimized dimensions, the new compliant bearing with ununiform beam elements is redesign in Figure 10 and analysed by finite element method in Figure 11 .

Figure 10 . The optimized model of compliant bearing.

The new model shown better deforming distribution especially for beams 1 and 3. The maximum displacement for the optimized motor is shown in Table 4 . The motor stroke is determined by the ratio between the yield strength of aluminum and maximum von Mises stress from Figure 11 b, this ratio is then multiplied with the highest displacement in Figure 11 a.

Table 4 Result of optimized motor with 2 compliant bearings

Figure 11 . Simulation of the motor with two bearings for displacement (a) and von Mises stress (b) acted by 1N force.

By comparing results from Tables 2 and 4, it is seen that the optimized design has its deforming capability improved nearly 55%. Meanwhile, the maximum required force of the optimized design also fits to the output force generated by the proposed VCA module.

Experimental setup

The experiment is conducted to study the linear transfer function with discrete input DC signal and has a preliminary investigation on the operating tolerance of the motor prototype.

To verify the actual performance of the optimized design, a linear motor prototype ( Figure 12 a) was built and an experimental setup was conducted to measure its translational displacement corresponding to a DC current value ( Figure 12 b). In this experiment, the motor prototype is clamped by a fixture. The output shaft then contacts directly with the dial indicator head to measure its translational displacement along the desired DOF. The dial indicator and fixture are both fixed to the ground to avoid external disturbances from the environment. The resolution for devices will be as follows: 0.2A for the DC power supply and 0.01mm for the dial indicator. The maximum current that the power supply can archive is 5A.

The measurement was carried out at room temperature of 20-25ºC and repeated 10 measured times, the DC current increases from 1 to 5A with 0.2A interval. As a result, there are 25 data points for each measuring time.

Figure 12 . Linear motor prototype (a) and experimental setup for displacement measurement (b).

Actual Performance of the Frictionless Linear Motor

Based on the experimental result, the average curve demonstrates the highly linear relationship between input current and output displacement can be archived as shown in Figure 13 . The maximum deviation between the average curve and measuring values is 19µm, while the highest deviation between measuring times is nearly 45µm. These micro-level differences can be explained by the limited resolution of 10µm and 0.2A from the dial indicator and DC power supply respectively. The experimental stroke has come up to 0.45mm at 4A input current due to the limitation of the DC power supply. With more advanced devices, the linear motor is supposed to achieve better performance in terms of resolution, accuracy, stroke and a linear characteristic. Overall, based on the experimental result, this motor model can guarantee the workspace up to more than 10mm (simulation result from Table 4 ). The experiments were repeated 25 times with 10 measured points each time. The measured results were then analyzed and the achieved result showed that the motor is able to perform a positioning tolerance of ±22.5 micrometers.

Figure 13 . The average current-displacement curve with error bars for measured points

DISCUSSION

From the average curve of experimental result, the linear function in equation (6) demonstrating the relationship between the output displacement C (mm) and the input current i _coil (A) can be determined by linearization. This can be also used as the transfer function for the open-loop control system of the linear motor.

C = 0.117 ×i _coil (6)

It is noted that for the measurement in the electrical current range from 0A to 1A, the measured value can be unclear due to the small displacement and the initial reaction force by the dial indicator. Therefore, the measurement is conducted from 1A to 5A.

CONCLUSION

The linear motor presented in this paper has shown the capability to work in micro-scale tolerance while still ensuring high repeatability as demonstrated by the experimental results. With the maximum stroke of more than 10 mm together with the linear characteristic for open-loop control system, this type of motor will be a potential solution for high-precision industrial applications such as actuating systems of compact robotic modules, pick and place units of production lines, control valves in pneumatic systems, or even actuating apparatuses for aerospace engineering due to the maintenance-free characteristic, etc.

For the future, it is necessary to conduct the experiment in this paper again by devices with more accurate resolution and continuous input DC current; so that the tolerance of the motor can be identified precisely and the dynamic operation in a frequency range can be studied. Furthermore, the effect of temperature and errors caused by fabrication and assembly processes will be also considered to investigate.

ACKNOWLEDGMENT

This research is funded by Ho Chi Minh City University of Technology – VNU-HCM, under grant number To-CK-2023-03.

COMPETING INTEREST

The authors declare that they have no competing interests.

AUTHORS CONTRIBUTION

H. N. Le: Conceptualization, Prototype Development, Investigation and Manuscript Preparation. H. T. Nguyen: Investigation, Validation and Manuscript Editing. T. A. Le: Design Optimization and Validation. M. T. Pham: Methodology, Funding Acquisition, Resources, Validation and Manuscript Editing.