A study on robust design of cruciform torsional beam/cantilever beam

The electro-thermal actuator of a micro-electromechanical system (MEMS) has high power consumption and low response speed. To overcome this shortcoming, a steady state mechanism of cross torsion beam/cantilever beam is proposed in conjunction with the micro-electro-thermal actuator. Through the analysis of the structure and working principle of the cross torsion beam/cantilever beam, as well as the randomness analysis of its design variables and noise factors, the robust design is carried out, and the algorithm program of the stochastic model is written to solve the robust design model. The results show that the robust design method can not only ensure that the cantilever beam will not collapse excessively after the whole mechanism is released from the air but also ensure the robustness of the design solution when the design variables and noise factors change.


Introduction
Currently, most of the research and application of microactuators at home and abroad focus on piezoelectric drives, electrostatic drives, and so on.In contrast, the research on electric heating drivers needs to be more.Compared with electrostatic drive and electromagnetic drive, electric drive has the advantages of large displacement, large output torque, and no interference from an external magnetic field.However, in general, the power consumption of the MEMS electrothermal driver is large when working, and the response speed is slow when responding.To reduce the power consumption of the driver, this paper proposes a steady state mechanism of cross torsion beam/cantilever beam working in conjunction with the micro-electrothermal driver.The mechanism can fully use the driving force at the heating stage of the drive, avoid the slow cooling process, and achieve the purpose of rapid cooling.Therefore, it is expected that the power consumption of the driver can be reduced, and the response time can be shortened, thus improving the response speed [1][2][3] .
Because there are many uncontrollable factors in the actual manufacturing process, such as production environment and manufacturing process errors, even minor errors may greatly impact MEMS devices [4- 5] .Therefore, it is necessary to study the robust design of a cross torsion beam/cantilever beam.The robust design method considers the influence of various variations on the device, improving the quality performance of the cross torsional beam/cantilever beam and further making the device have a certain degree of robustness.This research is significant for exploring MEMS devices in the micro field.

Theoretical analysis of cruciform torsional beam/cantilever beam
The combined action of the cross torsion beam/cantilever beam and the electric heating driver is shown in Figure 1.In the work project, it is required that the torsional beam has a low torsional stiffness and the cantilever beam has a high flexural stiffness to ensure that the process of quick closure and disconnection can be realized when one end is closed.The other end is switched [6][7] .The overall stiffness should be analyzed and improved through robust design to ensure that the cantilever beam will not collapse excessively after the whole mechanism is released from the air.Figure 2 shows a schematic diagram of the geometric dimensions of the cross torsional beam/cantilever beam, where AA' and BB' are the cross-section views of the torsional beam and cantilever beam, respectively.

Figure 2. Geometric dimension diagram of cross torsion beam/cantilever beam mechanism.
A mechanical analysis of the mechanism is conducted, as shown in Figure 3. Figure 3 is the mechanical system model of the cruciform torsional beam/cantilever beam.It can be seen from the figure that when the free end of the cantilever beam is subjected to a load, the two fixed ends of the flexible torsional beam generate a support reaction, a support moment, and a supporting torque, respectively.The whole system is in a state of mechanical equilibrium.Mechanical system model of cross torsion beam/cantilever beam.The mechanical system model in Figure 3, combined with the geometric size of the mechanism in Figure 2, can be obtained according to the equilibrium conditions of mechanics. (2) For point A of the twisted beam, since point A is the fixed end of the twisted beam, the displacement of point A is 0 = A  .Therefore, it can be concluded that: In Formula (4), point A bending moment is the elastic modulus of the material, G is the shear modulus, 1 I and 1 J , respectively, represents the moment of inertia of the torsional beam and the torque constant [8][9][10] , where the and expression is: The expression of r F , r T , r M can be obtained by combining the above Formulas ( 1) -( 6).
Similarly, after the corresponding mechanical analysis of points B and C in Figure 3., the interaction force of each element can be calculated.
For a torsional beam, according to the same static equilibrium conditions, we can get: In the same way, the reaction force of the twisted beam on the cantilever beam can be obtained, and the torsion beam reaction moment is on the cantilever beam.According to Cartesian's second law of material mechanics [6]   , the expression of displacement at point C of the free end of the cantilever beam in the cross-shaped torsional beam/cantilever beam mechanism can be written as:  [7-10]   , according to Hooke's law, we can get: Therefore, the elastic coefficient of the mechanical system of the whole structure, namely the equivalent stiffness, s k is expressed as:

Design variable
After the mechanical system analysis and modeling of the cruciform torsional beam/cantilever beam, it can be known that the main parameters affecting its performance (the overall stiffness) are the length, width, and thickness of the torsional beam and cantilever beam, respectively.Because when w/t=1, 1  =0.141.Therefore, the design variables involved in this paper are the length of the torsional beam, the length, width, and thickness of the cantilever beam, and their respective variations:

Noise factors
The noise factors influencing product quality characteristics in robust design are called stochastic parameters [12]   .Through the theoretical analysis of the established mathematical model, it can be known that the noise factors affecting the cross torsion beam/cantilever beam are as follows: elastic modulus of a cross beam, shear modulus of a cross beam, Poisson's ratio, and allowable bending stress.Therefore, the noise factor is expressed as:

Objective function
In this paper, the elastic coefficient (equivalent stiffness) of the mechanical system of a cross-shaped torsional beam/cantilever beam is taken as the design characteristic of the composite microbeam.To improve its mass performance (the overall equivalent stiffness) to a certain extent and minimize the fluctuation of the influence within a certain range, the desired small characteristics of the mean and variance of the stiffness variation are taken as the quality characteristics of the product [13]  , and the minimum value is solved.It can be seen from the three figures shown in Figure 4 that when other parameters of the mechanism do not change, the width and thickness of the cruciform torsional beam and cantilever beam have an increasing relationship with the elastic coefficient of the mechanism.In contrast, the length of the cruciform torsional beam and cantilever beam has a decreasing relationship with the elastic coefficient of the mechanism.But this increasing or decreasing relationship is not arbitrary.
Because the whole mechanism works, the torsional beam should have a small rigidity, the cantilever beam should have a large flexural rigidity, and the whole mechanism should have a certain overall rigidity.Here, MATLAB software is used to analyze the influence of the corresponding geometric parameters of the design variables on the torsional rigidity of the cantilever beam, the flexural rigidity of the cantilever When the cantilever beam is subjected to external loads, it can be seen from Figure 3 that the maximum displacement will occur at point C of the free end of the cantilever beam.According to the mechanical theory, the displacement at point C of the free end of the cantilever beam is less than the maximum bending and torsional deformation limits [11] .We can get: After mechanical analysis, the stress constraint of the whole mechanism is distributed at positions A and B in Figure 3(a), and the maximum bending stress at point B is not greater than the allowable bending stress.

 
Since point A is the fixed end of the torsional beam, the combined effect of bending moment and torque will be generated during deformation, which can be obtained according to the third strength theory of mechanics of materials.

Mathematical model of robust design
In summary, the following robust design mathematical model based on the stochastic model is established.

Design variables and noise factors
The initial values and upper and lower bounds of the design variables are shown in Table 1.
Table 1.The initial upper and lower values of the design variable.By referring to Chen et al. [14] , it can be found that the probability distribution types and parameters followed by noise factors involved in this paper are shown in Table 2.  6, and compared with the original scheme, the result is shown in Table 3.It can be seen from Table 3 that the overall stiffness of the mechanism after robust design optimization is significantly improved, from 2.569 m N   / in the original scheme to 3.348 m N   / , an increase of about 31%, which ensures the safety and stability performance of the mechanism.

Simulated analysis
Abaqus software is used to conduct the robust cross torsion beam/cantilever beam mechanism design based on a stochastic model.For the simulation analysis of stress and displacement under fixed load, the model after robust design can be seen from Figure 7  .The difference between the theoretical value and the simulation value is 0.14%, and the error rate is less than 5%, which is within the reasonable range.Figure 8 shows that the maximum stress distribution position at the junction of the torsion beam and cantilever beam is 66.49Mpa, which is far less than its allowable valueBending stress of 440 MPa [15] .The above can better verify the accuracy and rationality of robust design., an increase of about 31%.
2) In the process of robust design, the influence of noise factor and the size variation of the mechanism on the whole mechanism has been included, so the solution of robust design can make the mass performance of the cross-shaped torsional beam/cantilever beam mechanism fluctuate less when the size parameter changes, and further improve the manufacturing yield of the cross-shaped microbeam.The results show that it is feasible to apply the robust design method to the cross torsion beam/cantilever beam research.

Figure 1 .
Figure 1.The combined action of cross torsion beam/cantilever beam and electrothermal driver.Figure2shows a schematic diagram of the geometric dimensions of the cross torsional beam/cantilever beam, where AA' and BB' are the cross-section views of the torsional beam and cantilever beam, respectively.

Figure 3 .
Figure 3. Mechanical system model of cross torsion beam/cantilever beam.The mechanical system model in Figure3, combined with the geometric size of the mechanism in Figure2, can be obtained according to the equilibrium conditions of mechanics.

F
is the reaction force of the cantilever beam on the twisted beam, and is the reaction moment of the cantilever beam on the twisted beam.So, we can get:

Figure 4 .
condition Before analyzing the constraint conditions, MATLAB software was used to analyze the influence trend of one parameter change on the elastic coefficient (equivalent stiffness) of the whole mechanism when the length, width, and height of the torsional beam and cantilever beam were fixed in pairs.The influence of the thickness, width, and length of the torsion beam and cantilever beam on the elastic coefficient KS of the system.

Figure 5 .
Figure 5. Influence of length of torsion beam, length, width, and thickness of cantilever beam on KT, KC, and KS.From the four figures shown in Figure5, it can be concluded that when the torsional rigidity of the whole mechanism is small, the flexural rigidity of the cantilever beam is large, and the whole mechanism has a certain equivalent stiffness, the boundary constraint conditions established are: The constraint expression of mechanism performance can be obtained by combining positions A, B and C as follows:

Figure 6 .
Figure 6.Schematic diagram of program results.Table3.Comparison between the robust solution of the original scheme and the stochastic model.Design variable in applying certain load under the action of the maximal displacement of 29.91

Table 2 .
Probability distribution types and parameters followed by noise factors.Our established mathematical model is edited into the main program module, and the values of design variables, noise factors, and other related parameters are written into the corresponding data.We run the corresponding algorithm program in the data file and conduct gradual debugging to obtain a robust design solution based on the random model.The program result is shown in Figure