The gravity balance mechanism plays a vital role in maintaining the equilibrium for robots and assistive devices. The purpose of this paper was to optimize the geometry of a planar spring, which is an essential element of the gravity balance mechanism. To implement the optimization process, a hybrid method is proposed by combining the finite element method, the deep feedforward neural network, and the water cycle algorithm. Firstly, datasets are collected using the finite element method with a full experiment design. Secondly, the output datasets are normalized to eliminate the effects of the difference of units. Thirdly, the deep feedforward neural network is then employed to build the approximate models for the strain energy, deformation, and stress of the planar spring. Finally, the water cycle algorithm is used to optimize the dimensions of the planar spring. The results found that the optimal geometries of the spring include the length of 45 mm, the thickness of 1.029 mm, the width of 9 mm, and the radius of 0.3 mm. Besides, the predicted results determined that the strain energy, the deformation, and the stress are 0.01123 mJ, 33.666 mm, and 79.050 MPa, respectively. The errors between the predicted result and the verifying results for the strain energy, the deformation, and the stress are about 1.87%, 1.69%, and 3.06%, respectively.
A device is balanced when it can maintain equilibrium in any configuration or position without the need for external forces or actuators [
Previously, a gravity balance mechanism with adjusted loads was designed by the combination of a compliant spring and a torsion spring [
Although scientists have had great success in studying many of the different types of mechanisms, a large amount of energy is still required during the adjustment process. Therefore, the present study proposes a new gravity balance mechanism according to the principle of compliant mechanisms. Compliant mechanism is selected to create the planar spring (PS) because it is a monolithic mechanism, which offers less lubricant, no friction, and minimal cost of manufacturing [
In order to serve for a practical application, the PS should concern a full set of performances, including deformation, strain energy, and stress. However, these desirable properties have mutual contradictions, and these properties are very sensitive to PS geometrical dimensions. Therefore, optimization of the PS is necessary to balance its properties. In this study, an optimization process is performed to maximize or minimize one or more properties of the PS. It is noted that the proposed PS is designed based on the principle of compliant mechanisms. It is therefore difficult to build precisely mathematical models that show the relationship between deformation, stress, and strain energy with its geometrical factors. Therefore, in this study, a hybrid method of finite element method (FEM), deep feedforward neural network (DFNN), and water cycle algorithm (WCA) is aimed to build surrogate models and optimize the geometry of the PS.
Nowadays, the FEM is a widely used method to solve complex arithmetic problems [
In summary, the present paper is aimed to optimize the geometry factors of a planar spring, which is used for the gravity compensation mechanism. To perform the optimization process, the FEM method is used to simulate and collect data. Next, regression models are built by the DFNN. Before using the DFNN, the structure of the DFNN is selected by optimizing the controllable parameters of the DFNN by using the Taguchi method. Finally, the WCA is applied to optimize the geometry of the planar spring.
Figure
Structure of gravity balance mechanism.
Then, the moment balance equation of the mechanical system is expressed as follows:
In the proposed design, the bar is made of aluminum alloy with
To ensure an adjustable stiffness in the range of 0.325 N/mm to 1.468 N/mm, deformation of the spring is required from 0 to 30 degrees, and the planar spring is designed with 31 component springs that are connected in series, as shown in Figure
Structure of planar spring.
Structure of component spring.
In order to offer an efficient work for the gravity balance mechanism, the performances of the PS should be enhanced because the efficiency of the overall mechanism is highly dependent on the properties of the PS. Meanwhile, the performances of the PS are very sensitive to the geometrical dimensions of the leaf springs, as given in Figures The generated deformation has to be large enough so that the bar rotates at an angle from 0 to 30 degrees, and creates a moment for balancing. The generated stress must be less than the yield strength of the material. In this study, a maximum load of 3 kg and a stiffness of planar spring of 1.436 N/mm are utilized. The strain energy is as large as possible so that the PS can do highly efficient work. The fatigue should also be considered. It is remarked that the performances of the PS often contradict each other. Therefore, the question arises as to how to balance these properties. So, the WCA optimization algorithm is applied to solve the optimization with multiple constraints.
Based on the proposed design in Figures
As stated in the formulation of optimization problem above, the PS should posses a large strain energy to store and release a good elastic deformation. Therefore, the strain energy
In order to ensure the gravity balance mechanism safely, the PS must create a large enough deformation so that the mechanism can work in the range of 0–30 degrees. In addition, the deformation of the PS should be sufficient to create a moment that is equal to the moment generated by the mass. In addition, the stress of the PS should be less than the yield strength when the mechanism is operating at maximum load to ensure that the spring works safely. Hence, the deformation
The optimization problem for the PS is briefly stated as follows:
Find:
Subject to constraints,
As discussed in Section
Flowchart of the hybrid approach.
The mechanical design process is performed by the following steps: Step 1: identify the problem When the working load is changed, the balance condition will be broken. Besides, when the load is adjusted, the gravity balance mechanism needs to be changed to maintain the equilibrium condition. One of the popular methods for adjusting the gravity balance mechanism is an adjustment of the stiffness of the spring. In this study, the proposed planar spring is designed to adjust easily the number of active leaf springs. So, the stiffness of PS can be adjusted without extra energy. To meet work requirements of the gravity balance mechanism, the design and optimization of the PS is essential. Step 2: original design Based on the requirements of the problem, the original gravity balance mechanism is designed, as shown in Figure Step 3: define design variables, objective functions, and constraints The PS is designed to meet technical requirements. A deformation must be large enough, the stress must be less than the yield strength, and the strain energy is large. The objective function, constraint, and design variable are presented in Section
The purpose of this stage is to create datasets to build the approximate models, and the datasets were generated by the numerical simulation in ANSYS 18.2 software. The sequence of steps is as follows: Step 4: experiment design Experimental design is the statistical technique that is widely used in product development. To accurately evaluate the effect of each design variable on the performance of a product, the full factor experiment is used. Step 5: simulate and collect data Simulation is a technique of predicting the behavior of the structure. It is applied in engineering to reduce the costs of experimentation. A 3D model is built, and the boundary conditions and load are set up. The simulation process is performed to collect data. Step 6: normalize data The properties of a planar spring have different units; to avoid the effect of the different units on the optimal result, the output response should be normalized. This normalization only changes the value of the response, but does not change the nature of the data. Moreover, this process can evaluate fairly. In this study, the properties of PS need to be standardized to unify the units. The standardization process is performed by the following formula: where
To approximate the relationship between the geometry sizes and the output characteristics of the PS, the DFNN is applied to create regression models. The working diagram of the DFNN is shown in Figure
Structure of deep feedforward neural network.
The net input of node
The approximate models are greatly dependent on the structure of the DFNN and the actual data. To get the exact approximate models, the structure of the DFNN is optimized in this study. The optimization process for the structure of the DFNN is performed as follows: Step 7: define the objective function of the DFNN structure. The accuracy of the approximate model is usually assessed through coefficient of determination ( The MSE and where Step 8: select the input parameters of the DFNN structure A basic structure of the DFNN includes the number of hidden layers, the number of nodes in each hidden layer, training function, activation function, bias coefficient, ratio of division of data for training, validation, and testing. However, in this study, the number of hidden layers, the number of nodes in each hidden layer, the transfer function and the ratio of division of data are chosen as controllable variables. Normally, researchers usually use training functions: trainlm, trainbr, and trainscg. However, trainbr only uses 2 datasets: training and validating while the other transfer functions use 3 datasets: training, validating, and testing. Therefore, to evaluate equity, this study proposes to choose 3 training functions: trainlb, traincgs, and trainscg. The number of nodes in hidden layers can be selected differently based on the position of the hidden layer and the number of nodes of the input layer. According to Chen [ where According to Seo et al. [ Step 9: collect data and optimize the structure of the DFNN Taguchi is often used to improve a product’s quality [ Nominal is the best: The smaller is better: The bigger is better: where The Taguchi method used orthogonal arrays to reduce the number of experiments required but ensured that design variables are evaluated independently. The purpose of this step is to find the structure of the DFNN that best matches the existing dataset by minimizing MSE by using equation ( Step 10: evaluate the value of MSE After optimizing the structure of the DFNN, the value of MSE has to be evaluated to find the most suitable structure for the dataset. If the value of MSE is not satisfied, it means that the most suitable structure has not been found, then go back to step 8. If the value of MSE is satisfied, then go to the next step.
The WCA is an optimal algorithm inspired by the water cycle. It was developed by Eskandar [ Step 11: choose the initial parameters of the WCA Initial rain drops ( where Step 12: generate random initial population Initial population is generated by random variables as follows: where Step 13: calculate the value of the initial raindrop In order to select raindrops representing the sea and rivers, the value of the initial raindrop needs to be calculated and they are calculated according to the following formula: where Step 14: determine the intensity of flow for rivers and sea The raindrops flow to the rivers or sea depending on the intensity of the flow. The intensity of flow for rivers and sea are calculated by the following formula: where Step 15: the streams flow to the rivers During the streams’ flow to rivers, the positions of streams are continuously updated. The new position for the stream is determined as follows: where rand is a random number that is chosen between 0 and 1, Step 16: the rivers flow to the sea Similar to a stream, when rivers flow to the sea, the position of the river is always updated, and the new position of the river is determined as follows: where rand is a random number that is chosen between 0 and 1, Step 17: exchange positions of the river with a stream To offer the best solution, the new cost of a stream is updated. If the solution given by a stream is better than its connecting river, the positions of the river and stream are exchanged Step 18: exchange positions of the sea with a river Similar to streams, if the solution of the river is better than the sea, the positions of the sea and the river will be swapped Step 19: check the evaporation condition Evaporation is one factor that prevents the algorithm from being optimized locally. As seen in nature, water from ponds, lakes, rivers, streams, seas, and the ground evaporates into clouds. Clouds fly up high then condense in the cold air to form water particles. Water particles fall down and form rain. Raindrops will form new streams that flow into rivers and the sea. In the WCA, the condition evaporation is determined as follows: where When the distance from the river to sea is less than Step 20: reduce the value of In the WCA, the value of Step 21: check the convergence criteria Like other optimization algorithms, the stop condition of the WCA algorithm can be based on the maximum number of iterations, the CPU time, or the error between the last two results less than a certain tolerance value. If the stopping condition is satisfied, the optimization will be finished, otherwise return to step 5.
Flowchart of the WCA.
To perform simulation and collect data, firstly, the experimental design was constructed by a full experimental design. The four design variables of PS include
Design variables with three levels.
Variables | Level 1 | Level 2 | Level 3 |
---|---|---|---|
1.0 | 1.2 | 1.4 | |
45 | 50 | 55 | |
9 | 11.5 | 14 | |
0.1 | 0.2 | 0.3 |
Mechanical properties of the material.
Density | Yield strength (MPa) | Poisson’s ratio | Young modulus (MPa) |
---|---|---|---|
2810 kg/m3 | 503 | 0.33 | 71700 |
In this study, a nonlinear FEM is applied for the simulation process. The simulation process is set up as follows: the boundary condition and loads are given as shown in Figure
Meshing model and boundary conditions.
Evaluating the quality of the meshing.
In this study, strain energy is considered an objective function; therefore, strain energy is a factor that needs data collection. In addition, deformation and stress are considered constraints so they also need to be collected as data. The results of data collection are shown in Table
Experimental design and simulation data.
No. | Deformation (mm) | Stress (MPa) | Strain energy (mJ) | ||||
---|---|---|---|---|---|---|---|
1 | 1 | 45 | 9 | 0.1 | 37.648 | 143.88 | 0.01136 |
2 | 1 | 45 | 9 | 0.2 | 37.359 | 126.56 | 0.011931 |
3 | 1 | 45 | 9 | 0.3 | 37.181 | 91.524 | 0.01208 |
4 | 1 | 45 | 11.5 | 0.1 | 29.158 | 110.5 | 0.0075272 |
5 | 1 | 45 | 11.5 | 0.2 | 28.933 | 97.469 | 0.0073316 |
6 | 1 | 45 | 11.5 | 0.3 | 28.794 | 70.283 | 0.0074207 |
7 | 1 | 45 | 14 | 0.1 | 23.743 | 89.134 | 0.004959 |
8 | 1 | 45 | 14 | 0.2 | 23.559 | 78.896 | 0.0052279 |
9 | 1 | 45 | 14 | 0.3 | 23.445 | 56.8 | 0.0052896 |
10 | 1 | 50 | 9 | 0.1 | 52.955 | 135.96 | 0.015283 |
11 | 1 | 50 | 9 | 0.2 | 52.482 | 183.47 | 0.015191 |
12 | 1 | 50 | 9 | 0.3 | 52.271 | 103.04 | 0.01534 |
13 | 1 | 50 | 11.5 | 0.1 | 41.035 | 105.25 | 0.010162 |
14 | 1 | 50 | 11.5 | 0.2 | 40.666 | 141.99 | 0.01011 |
15 | 1 | 50 | 11.5 | 0.3 | 40.502 | 79.32 | 0.010206 |
16 | 1 | 50 | 14 | 0.1 | 33.423 | 85.454 | 0.0067236 |
17 | 1 | 50 | 14 | 0.2 | 33.121 | 115.14 | 0.0066931 |
18 | 1 | 50 | 14 | 0.3 | 32.987 | 64.151 | 0.0067545 |
19 | 1 | 55 | 9 | 0.1 | 72.04 | 175.37 | 0.018836 |
20 | 1 | 55 | 9 | 0.2 | 71.362 | 203.9 | 0.018837 |
21 | 1 | 55 | 9 | 0.3 | 71.012 | 114.56 | 0.019014 |
22 | 1 | 55 | 11.5 | 0.1 | 55.856 | 135.11 | 0.012552 |
23 | 1 | 55 | 11.5 | 0.2 | 55.326 | 158.02 | 0.012564 |
24 | 1 | 55 | 11.5 | 0.3 | 55.053 | 88.312 | 0.012679 |
25 | 1 | 55 | 14 | 0.1 | 45.508 | 109.33 | 0.008325 |
26 | 1 | 55 | 14 | 0.2 | 45.075 | 128.3 | 0.0083395 |
27 | 1 | 55 | 14 | 0.3 | 44.852 | 71.513 | 0.008414 |
28 | 1.2 | 45 | 9 | 0.1 | 22.22 | 96.941 | 0.0065149 |
29 | 1.2 | 45 | 9 | 0.2 | 22.13 | 66.215 | 0.0063295 |
30 | 1.2 | 45 | 9 | 0.3 | 21.933 | 65.468 | 0.0062087 |
31 | 1.2 | 45 | 11.5 | 0.1 | 17.205 | 74.773 | 0.0040134 |
32 | 1.2 | 45 | 11.5 | 0.2 | 17.135 | 51.129 | 0.003901 |
33 | 1.2 | 45 | 11.5 | 0.3 | 16.981 | 50.323 | 0.003827 |
34 | 1.2 | 45 | 14 | 0.1 | 14.007 | 60.582 | 0.0028689 |
35 | 1.2 | 45 | 14 | 0.2 | 13.95 | 41.457 | 0.0027898 |
36 | 1.2 | 45 | 14 | 0.3 | 13.824 | 40.669 | 0.0027374 |
37 | 1.2 | 50 | 9 | 0.1 | 31.178 | 108.37 | 0.0082735 |
38 | 1.2 | 50 | 9 | 0.2 | 31.011 | 90.501 | 0.0080522 |
39 | 1.2 | 50 | 9 | 0.3 | 30.784 | 73.985 | 0.0079012 |
40 | 1.2 | 50 | 11.5 | 0.1 | 24.156 | 83.785 | 0.0055193 |
41 | 1.2 | 50 | 11.5 | 0.2 | 24.025 | 69.716 | 0.0053739 |
42 | 1.2 | 50 | 11.5 | 0.3 | 23.848 | 57.009 | 0.005274 |
43 | 1.2 | 50 | 14 | 0.1 | 19.671 | 67.908 | 0.0036635 |
44 | 1.2 | 50 | 14 | 0.2 | 19.565 | 56.337 | 0.003569 |
45 | 1.2 | 50 | 14 | 0.3 | 19.419 | 46.096 | 0.0035029 |
46 | 1.2 | 55 | 9 | 0.1 | 42.206 | 125.93 | 0.010211 |
47 | 1.2 | 55 | 9 | 0.2 | 42.035 | 82.775 | 0.010003 |
48 | 1.2 | 55 | 9 | 0.3 | 41.738 | 81.78 | 0.0098 |
49 | 1.2 | 55 | 11.5 | 0.1 | 32.717 | 97.869 | 0.0068261 |
50 | 1.2 | 55 | 11.5 | 0.2 | 32.584 | 64.17 | 0.0066893 |
51 | 1.2 | 55 | 11.5 | 0.3 | 32.353 | 63.11 | 0.0065549 |
52 | 1.2 | 55 | 14 | 0.1 | 26.652 | 79.533 | 0.0045419 |
53 | 1.2 | 55 | 14 | 0.2 | 26.543 | 52.124 | 0.004453 |
54 | 1.2 | 55 | 14 | 0.3 | 26.354 | 51.099 | 0.0043642 |
55 | 1.4 | 45 | 9 | 0.1 | 14.31 | 70.515 | 0.0028812 |
56 | 1.4 | 45 | 9 | 0.2 | 14.21 | 71.532 | 0.0023315 |
57 | 1.4 | 45 | 9 | 0.3 | 14.106 | 66.273 | 0.0021754 |
58 | 1.4 | 45 | 11.5 | 0.1 | 11.078 | 54.279 | 0.0017708 |
59 | 1.4 | 45 | 11.5 | 0.2 | 11 | 55.096 | 0.0014267 |
60 | 1.4 | 45 | 11.5 | 0.3 | 10.919 | 51.038 | 0.0013319 |
61 | 1.4 | 45 | 14 | 0.1 | 9.0172 | 43.868 | 0.0012658 |
62 | 1.4 | 45 | 14 | 0.2 | 8.9536 | 44.557 | 0.0010189 |
63 | 1.4 | 45 | 14 | 0.3 | 8.8876 | 41.251 | 0.00095013 |
64 | 1.4 | 50 | 9 | 0.1 | 20.005 | 79.204 | 0.0036771 |
65 | 1.4 | 50 | 9 | 0.2 | 19.885 | 73.889 | 0.002852 |
66 | 1.4 | 50 | 9 | 0.3 | 19.747 | 74.814 | 0.0027744 |
67 | 1.4 | 50 | 11.5 | 0.1 | 15.496 | 61.129 | 0.0024474 |
68 | 1.4 | 50 | 11.5 | 0.2 | 15.403 | 57.082 | 0.0018876 |
69 | 1.4 | 50 | 11.5 | 0.3 | 15.295 | 57.768 | 0.0018403 |
70 | 1.4 | 50 | 14 | 0.1 | 12.618 | 49.426 | 0.0016241 |
71 | 1.4 | 50 | 14 | 0.2 | 12.541 | 46.18 | 0.0012489 |
72 | 1.4 | 50 | 14 | 0.3 | 12.453 | 46.705 | 0.0012188 |
73 | 1.4 | 55 | 9 | 0.1 | 27.052 | 88.311 | 0.0038068 |
74 | 1.4 | 55 | 9 | 0.2 | 26.893 | 75.666 | 0.0035977 |
75 | 1.4 | 55 | 9 | 0.3 | 26.726 | 81.191 | 0.0034145 |
76 | 1.4 | 55 | 11.5 | 0.1 | 20.967 | 68.245 | 0.0025272 |
77 | 1.4 | 55 | 11.5 | 0.2 | 20.843 | 58.445 | 0.002389 |
78 | 1.4 | 55 | 11.5 | 0.3 | 20.713 | 62.752 | 0.0022659 |
79 | 1.4 | 55 | 14 | 0.1 | 17.078 | 55.247 | 0.0016786 |
80 | 1.4 | 55 | 14 | 0.2 | 16.976 | 47.321 | 0.0015868 |
81 | 1.4 | 55 | 14 | 0.3 | 16.87 | 50.783 | 0.0015041 |
The properties of the PS have different units. Therefore, in this study, datasets are normalized to eliminate the units of measurement for data. This normalization makes the evaluation process fair. Data are normalized using (
Data normalization results.
No. | |||||||
---|---|---|---|---|---|---|---|
1 | 1 | 45 | 9 | 0.1 | 0.455413 | 0.6323 | 0.576281 |
2 | 1 | 45 | 9 | 0.2 | 0.450836 | 0.526193 | 0.607891 |
3 | 1 | 45 | 9 | 0.3 | 0.448018 | 0.311552 | 0.61614 |
4 | 1 | 45 | 11.5 | 0.1 | 0.320976 | 0.427805 | 0.364101 |
5 | 1 | 45 | 11.5 | 0.2 | 0.317413 | 0.347973 | 0.353273 |
6 | 1 | 45 | 11.5 | 0.3 | 0.315212 | 0.181424 | 0.358205 |
7 | 1 | 45 | 14 | 0.1 | 0.235231 | 0.296911 | 0.221928 |
8 | 1 | 45 | 14 | 0.2 | 0.232317 | 0.23419 | 0.236814 |
9 | 1 | 45 | 14 | 0.3 | 0.230512 | 0.098823 | 0.240229 |
10 | 1 | 50 | 9 | 0.1 | 0.697795 | 0.58378 | 0.793455 |
11 | 1 | 50 | 9 | 0.2 | 0.690305 | 0.87484 | 0.788362 |
12 | 1 | 50 | 9 | 0.3 | 0.686964 | 0.382103 | 0.796611 |
13 | 1 | 50 | 11.5 | 0.1 | 0.509045 | 0.395642 | 0.509961 |
14 | 1 | 50 | 11.5 | 0.2 | 0.503202 | 0.620722 | 0.507082 |
15 | 1 | 50 | 11.5 | 0.3 | 0.500605 | 0.236787 | 0.512397 |
16 | 1 | 50 | 14 | 0.1 | 0.388511 | 0.274366 | 0.319614 |
17 | 1 | 50 | 14 | 0.2 | 0.383729 | 0.456231 | 0.317926 |
18 | 1 | 50 | 14 | 0.3 | 0.381607 | 0.143857 | 0.321325 |
19 | 1 | 55 | 9 | 0.1 | 1 | 0.825217 | 0.990146 |
20 | 1 | 55 | 9 | 0.2 | 0.989264 | 1 | 0.990201 |
21 | 1 | 55 | 9 | 0.3 | 0.983722 | 0.452677 | 1 |
22 | 1 | 55 | 11.5 | 0.1 | 0.743731 | 0.578573 | 0.642269 |
23 | 1 | 55 | 11.5 | 0.2 | 0.735339 | 0.718926 | 0.642934 |
24 | 1 | 55 | 11.5 | 0.3 | 0.731016 | 0.291875 | 0.6493 |
25 | 1 | 55 | 14 | 0.1 | 0.579873 | 0.420637 | 0.408266 |
26 | 1 | 55 | 14 | 0.2 | 0.573017 | 0.536853 | 0.409069 |
27 | 1 | 55 | 14 | 0.3 | 0.569486 | 0.188959 | 0.413193 |
28 | 1.2 | 45 | 9 | 0.1 | 0.211115 | 0.344738 | 0.308061 |
29 | 1.2 | 45 | 9 | 0.2 | 0.20969 | 0.156502 | 0.297797 |
30 | 1.2 | 45 | 9 | 0.3 | 0.20657 | 0.151926 | 0.29111 |
31 | 1.2 | 45 | 11.5 | 0.1 | 0.131704 | 0.208931 | 0.16958 |
32 | 1.2 | 45 | 11.5 | 0.2 | 0.130595 | 0.064081 | 0.163358 |
33 | 1.2 | 45 | 11.5 | 0.3 | 0.128157 | 0.059143 | 0.159261 |
34 | 1.2 | 45 | 14 | 0.1 | 0.081064 | 0.121993 | 0.106221 |
35 | 1.2 | 45 | 14 | 0.2 | 0.080162 | 0.004828 | 0.101843 |
36 | 1.2 | 45 | 14 | 0.3 | 0.078166 | 0 | 0.098942 |
37 | 1.2 | 50 | 9 | 0.1 | 0.352962 | 0.414756 | 0.405415 |
38 | 1.2 | 50 | 9 | 0.2 | 0.350318 | 0.305285 | 0.393164 |
39 | 1.2 | 50 | 9 | 0.3 | 0.346723 | 0.204103 | 0.384805 |
40 | 1.2 | 50 | 11.5 | 0.1 | 0.241771 | 0.264141 | 0.252945 |
41 | 1.2 | 50 | 11.5 | 0.2 | 0.239696 | 0.17795 | 0.244896 |
42 | 1.2 | 50 | 11.5 | 0.3 | 0.236894 | 0.100104 | 0.239366 |
43 | 1.2 | 50 | 14 | 0.1 | 0.170752 | 0.166874 | 0.15021 |
44 | 1.2 | 50 | 14 | 0.2 | 0.169074 | 0.095987 | 0.144978 |
45 | 1.2 | 50 | 14 | 0.3 | 0.166762 | 0.033247 | 0.141319 |
46 | 1.2 | 55 | 9 | 0.1 | 0.527587 | 0.522333 | 0.512674 |
47 | 1.2 | 55 | 9 | 0.2 | 0.524879 | 0.257953 | 0.501159 |
48 | 1.2 | 55 | 9 | 0.3 | 0.520177 | 0.251858 | 0.489921 |
49 | 1.2 | 55 | 11.5 | 0.1 | 0.377332 | 0.350424 | 0.325289 |
50 | 1.2 | 55 | 11.5 | 0.2 | 0.375226 | 0.143974 | 0.317715 |
51 | 1.2 | 55 | 11.5 | 0.3 | 0.371568 | 0.13748 | 0.310275 |
52 | 1.2 | 55 | 14 | 0.1 | 0.281294 | 0.238092 | 0.198837 |
53 | 1.2 | 55 | 14 | 0.2 | 0.279568 | 0.070177 | 0.193916 |
54 | 1.2 | 55 | 14 | 0.3 | 0.276575 | 0.063897 | 0.189 |
55 | 1.4 | 45 | 9 | 0.1 | 0.085862 | 0.182845 | 0.106902 |
56 | 1.4 | 45 | 9 | 0.2 | 0.084279 | 0.189076 | 0.076471 |
57 | 1.4 | 45 | 9 | 0.3 | 0.082632 | 0.156857 | 0.06783 |
58 | 1.4 | 45 | 11.5 | 0.1 | 0.034684 | 0.083379 | 0.045432 |
59 | 1.4 | 45 | 11.5 | 0.2 | 0.033449 | 0.088384 | 0.026382 |
60 | 1.4 | 45 | 11.5 | 0.3 | 0.032167 | 0.063523 | 0.021134 |
61 | 1.4 | 45 | 14 | 0.1 | 0.002052 | 0.019598 | 0.017475 |
62 | 1.4 | 45 | 14 | 0.2 | 0.001045 | 0.023819 | 0.003807 |
63 | 1.4 | 45 | 14 | 0.3 | 0 | 0.003565 | 0 |
64 | 1.4 | 50 | 9 | 0.1 | 0.176041 | 0.236076 | 0.150963 |
65 | 1.4 | 50 | 9 | 0.2 | 0.174141 | 0.203515 | 0.105286 |
66 | 1.4 | 50 | 9 | 0.3 | 0.171955 | 0.209182 | 0.10099 |
67 | 1.4 | 50 | 11.5 | 0.1 | 0.104642 | 0.125344 | 0.082888 |
68 | 1.4 | 50 | 11.5 | 0.2 | 0.103169 | 0.100551 | 0.051898 |
69 | 1.4 | 50 | 11.5 | 0.3 | 0.101459 | 0.104753 | 0.049279 |
70 | 1.4 | 50 | 14 | 0.1 | 0.05907 | 0.053648 | 0.03731 |
71 | 1.4 | 50 | 14 | 0.2 | 0.057851 | 0.033762 | 0.01654 |
72 | 1.4 | 50 | 14 | 0.3 | 0.056457 | 0.036978 | 0.014873 |
73 | 1.4 | 55 | 9 | 0.1 | 0.287628 | 0.291869 | 0.158143 |
74 | 1.4 | 55 | 9 | 0.2 | 0.28511 | 0.214402 | 0.146567 |
75 | 1.4 | 55 | 9 | 0.3 | 0.282466 | 0.248249 | 0.136425 |
76 | 1.4 | 55 | 11.5 | 0.1 | 0.191274 | 0.168938 | 0.087305 |
77 | 1.4 | 55 | 11.5 | 0.2 | 0.18931 | 0.108901 | 0.079655 |
78 | 1.4 | 55 | 11.5 | 0.3 | 0.187252 | 0.135287 | 0.07284 |
79 | 1.4 | 55 | 14 | 0.1 | 0.129693 | 0.089309 | 0.040327 |
80 | 1.4 | 55 | 14 | 0.2 | 0.128077 | 0.040752 | 0.035245 |
81 | 1.4 | 55 | 14 | 0.3 | 0.126399 | 0.061961 | 0.030667 |
As presented in Section
DFNN parameters with 3 levels.
Variable | Level 1 | Level 2 | Level 3 |
---|---|---|---|
Training function | Trainlm | Traincgb | Trainscg |
Number of hidden layers | 2 | 3 | 4 |
Number of nodes | 7 | 9 | 11 |
Divide data | 60 : 20 : 20 | 70 : 15 : 15 | 80 : 10 : 10 |
Experimental design using
No. | Training function | Number of hidden layers | Number of nodes | Divide data |
---|---|---|---|---|
1 | trainlm | 2 | 7 | 60 : 20 : 20 |
2 | trainlm | 3 | 9 | 70 : 15 : 15 |
3 | trainlm | 4 | 11 | 80 : 10 : 10 |
4 | traincgb | 2 | 9 | 80 : 10 : 10 |
5 | traincgb | 3 | 11 | 60 : 20 : 20 |
6 | traincgb | 4 | 7 | 70 : 15 : 15 |
7 | trainscg | 2 | 11 | 70 : 15 : 15 |
8 | trainscg | 3 | 7 | 80 : 10 : 10 |
9 | trainscg | 4 | 9 | 60 : 20 : 20 |
As shown in Section
MSE results for deformation.
No. | MSE | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
1 | 0.0003 | 0.0000 | 0.0002 | 0.0006 | 0.0009 | 0.0002 | 0.0001 | 0.0001 | 0.0004 | 0.0002 |
2 | 0.0031 | 0.0072 | 0.0001 | 0.0004 | 0.0000 | 0.0007 | 0.0019 | 0.0048 | 0.0086 | 0.0006 |
3 | 0.0166 | 0.0029 | 0.0006 | 0.0008 | 0.0039 | 0.0058 | 0.0040 | 0.0026 | 0.0057 | 0.0013 |
4 | 0.0108 | 0.0036 | 0.0030 | 0.0010 | 0.0038 | 0.0150 | 0.0005 | 0.0131 | 0.0036 | 0.0110 |
5 | 0.0043 | 0.0038 | 0.0047 | 0.0058 | 0.0014 | 0.0079 | 0.0059 | 0.0158 | 0.0039 | 0.0061 |
6 | 0.0027 | 0.0110 | 0.0164 | 0.0069 | 0.0012 | 0.0093 | 0.0033 | 0.0064 | 0.0312 | 0.0104 |
7 | 0.0088 | 0.0018 | 0.0015 | 0.0032 | 0.0075 | 0.0045 | 0.0037 | 0.0061 | 0.0025 | 0.0057 |
8 | 0.0130 | 0.0239 | 0.0017 | 0.0141 | 0.0276 | 0.0028 | 0.0113 | 0.0034 | 0.0031 | 0.0106 |
9 | 0.0103 | 0.0121 | 0.0146 | 0.0134 | 0.0222 | 0.0109 | 0.0091 | 0.0068 | 0.0205 | 0.0075 |
MSE results for stress.
No. | MSE | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
1 | 0.0051 | 0.0023 | 0.0181 | 0.0022 | 0.0116 | 0.0063 | 0.0035 | 0.0023 | 0.0053 | 0.0039 |
2 | 0.0165 | 0.0046 | 0.0100 | 0.0075 | 0.0143 | 0.0084 | 0.0140 | 0.0056 | 0.0034 | 0.0032 |
3 | 0.0091 | 0.0103 | 0.0091 | 0.0009 | 0.0171 | 0.0078 | 0.0032 | 0.0104 | 0.0302 | 0.0036 |
4 | 0.0090 | 0.0076 | 0.0069 | 0.0150 | 0.0068 | 0.0087 | 0.0157 | 0.0043 | 0.0088 | 0.0086 |
5 | 0.0066 | 0.0129 | 0.0104 | 0.0082 | 0.0066 | 0.0155 | 0.0079 | 0.0075 | 0.0073 | 0.0090 |
6 | 0.0144 | 0.0106 | 0.0109 | 0.0127 | 0.0166 | 0.0088 | 0.0153 | 0.0132 | 0.0114 | 0.0179 |
7 | 0.0015 | 0.0115 | 0.0115 | 0.0024 | 0.0086 | 0.0118 | 0.0078 | 0.0049 | 0.0096 | 0.0146 |
8 | 0.0542 | 0.0154 | 0.0151 | 0.0037 | 0.0125 | 0.0138 | 0.0375 | 0.0162 | 0.0303 | 0.0137 |
9 | 0.0160 | 0.0124 | 0.0110 | 0.0356 | 0.0048 | 0.0106 | 0.0173 | 0.0270 | 0.0100 | 0.0283 |
MSE results for strain energy.
No. | MSE | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
1 | 0.0061 | 0.0001 | 0.0004 | 0.0003 | 0.0004 | 0.0001 | 0.0003 | 0.0037 | 0.0014 | 0.0016 |
2 | 0.0222 | 0.0037 | 0.0086 | 0.0007 | 0.0032 | 0.0024 | 0.0035 | 0.0164 | 0.0082 | 0.0008 |
3 | 0.0013 | 0.0011 | 0.0203 | 0.0011 | 0.0175 | 0.0019 | 0.0296 | 0.0005 | 0.0010 | 0.0034 |
4 | 0.0152 | 0.0060 | 0.0007 | 0.0093 | 0.0016 | 0.0072 | 0.0178 | 0.0049 | 0.0054 | 0.0036 |
5 | 0.0045 | 0.0150 | 0.0039 | 0.0067 | 0.0071 | 0.0058 | 0.0058 | 0.0018 | 0.0078 | 0.0051 |
6 | 0.0048 | 0.0043 | 0.0049 | 0.0058 | 0.0075 | 0.0103 | 0.0037 | 0.0107 | 0.0093 | 0.0043 |
7 | 0.0119 | 0.0118 | 0.0027 | 0.0143 | 0.0046 | 0.0025 | 0.0086 | 0.0039 | 0.0051 | 0.0067 |
8 | 0.0062 | 0.0047 | 0.0015 | 0.0063 | 0.0015 | 0.0020 | 0.0423 | 0.0073 | 0.0061 | 0.0273 |
9 | 0.0043 | 0.0034 | 0.0355 | 0.0061 | 0.0158 | 0.0357 | 0.0091 | 0.0047 | 0.0200 | 0.0104 |
Based on MSE results obtained for deformation, stress, and strain energy, the
The S/N analysis results for deformations (Figure
Results analysis S/N for deformation.
Results of rank for deformation.
Level | Training function | Number of hidden layers | Number of nodes | Divide data |
---|---|---|---|---|
1 | 40.84 | 51.82 | 47.60 | 49.46 |
2 | 53.31 | 42.68 | 42.27 | 43.84 |
3 | 40.07 | 39.72 | 44.34 | 40.91 |
Delta | 13.24 | 12.10 | 5.33 | 8.54 |
Rank | 1 | 2 | 4 | 3 |
For the stress dataset, Figure
Results analysis S/N for stress.
Results of rank for stress.
Level | Training function | Number of hidden layers | Number of nodes | Divide data |
---|---|---|---|---|
1 | 39.34 | 41.01 | 37.16 | 38.90 |
2 | 40.03 | 37.44 | 38.16 | 39.37 |
3 | 35.52 | 36.44 | 39.57 | 36.62 |
Delta | 4.51 | 4.58 | 2.41 | 2.75 |
Rank | 2 | 1 | 4 | 3 |
Figure
Results analysis S/N for strain energy.
Results of rank for strain energy.
Level | Training function | Number of hidden layers | Number of nodes | Divide data |
---|---|---|---|---|
1 | 42.32 | 45.08 | 43.74 | 43.34 |
2 | 43.56 | 39.59 | 38.62 | 41.66 |
3 | 37.31 | 38.52 | 40.83 | 38.19 |
Delta | 6.25 | 6.56 | 5.12 | 5.14 |
Rank | 2 | 1 | 4 | 3 |
Table
The optimal structure of DFNN.
Function | Training function | Number of hidden layers | Number of nodes | Divide data |
---|---|---|---|---|
Deformation | trainlm | 2 | 7 | 60 : 20 : 20 |
Stress | trainlm | 2 | 11 | 70 : 15 : 15 |
Strain energy | trainlm | 2 | 7 | 60 : 20 : 20 |
To evaluate the effectiveness of the structure of the DFNN, the appropriateness of the approximate model built by the DFNN and the linear model is compared with each other.
Figure
The good fitness of model (a) for deformation, (b) for stress, and (c) for strain energy.
In addition, to evaluate the accuracy of the proposed model, the value
Compare the value
Response | Model data | Training data | Validating data | Testing data | ||||
---|---|---|---|---|---|---|---|---|
DFNN | Linear | DFNN | Linear | DFNN | Linear | DFNN | Linear | |
Deformation | 0.99 | 0.91 | 0.99 | 0.91 | 1.0 | 0.92 | 0.99 | 0.96 |
Stress | 0.97 | 0.76 | 0.90 | 0.73 | 0.95 | 0.78 | 0.95 | 0.82 |
Strain energy | 0.99 | 0.89 | 0.99 | 0.97 | 1.0 | 0.89 | 0.99 | 0.93 |
Compare the value MSE of the proposed model and the linear model.
Response | Model data | Training data | Validating data | Testing data | ||||
---|---|---|---|---|---|---|---|---|
DFNN | Linear | DFNN | Linear | DFNN | Linear | DFNN | Linear | |
Deformation | 2.82 | 0.0050 | 8.02 | 0.0053 | 4.14 | 0.0046 | 5.00 | 0.0014 |
Stress | 0.0015 | 0.0121 | 0.0046 | 0.0099 | 5.98 | 0.0127 | 0.0031 | 0.0112 |
Strain energy | 3.28 | 0.0069 | 9.5 | 0.0065 | 1.3 | 0.0061 | 6.70 | 0.0049 |
As illustrated in Figure 13(a), the MSE values of the training dataset of the deformation continuously decreased at the 9th epoch, but the MSE values of the testing and validating dataset still remained stable. As depicted in Figure
The best validation performance: (a) for deformation, (b) for stress, and (c) for strain energy.
Figure
Error histogram (a) for deformation, (b) for stress, and (c) for strain energy.
Figure
The fitness of model (a) for deformation, (b) for stress, and (c) for strain energy.
After optimizing the structure of the DFNN, an alternative model using the DNFF was established. Then, the WCA algorithm was carried out using Matlab R2018a to optimize the parameters of the planar spring. The parameters of the WCA were initialized as initial rain drops of 50, number of rivers and sea of 4, maximum distance between streams and rivers, between river and sea of 10−5, and the maximum iteration of 2000.
The optimal results were generated as Table
Optimal results.
No. | WCA | ER-WCA | ||
---|---|---|---|---|
Strain energy (mJ) | Time (s) | Strain energy (mJ) | Time (s) | |
1 | 0.011509 | 2.62 | 0.011509 | 3.05 |
2 | 0.011509 | 2.59 | 0.011509 | 3.08 |
3 | 0.010654 | 2.63 | 0.010654 | 2.84 |
4 | 0.010654 | 2.63 | 0.010654 | 2.90 |
5 | 0.011127 | 2.62 | 0.011127 | 2.96 |
6 | 0.011314 | 2.63 | 0.011314 | 3.61 |
7 | 0.011364 | 2.63 | 0.011364 | 3.56 |
8 | 0.011509 | 2.62 | 0.011509 | 3.57 |
9 | 0.010717 | 2.63 | 0.010717 | 3.17 |
10 | 0.011509 | 2.63 | 0.011509 | 3.07 |
11 | 0.010675 | 2.65 | 0.010675 | 3.03 |
12 | 0.011509 | 2.33 | 0.011509 | 2.90 |
13 | 0.011509 | 2.67 | 0.011509 | 2.98 |
14 | 0.011509 | 2.67 | 0.011509 | 3.05 |
15 | 0.011509 | 2.90 | 0.011509 | 3.02 |
16 | 0.010716 | 2.68 | 0.010716 | 3.05 |
17 | 0.011509 | 2.68 | 0.011509 | 2.89 |
18 | 0.011509 | 2.72 | 0.011509 | 2.85 |
19 | 0.011509 | 2.85 | 0.011509 | 2.94 |
20 | 0.011509 | 2.80 | 0.011509 | 2.85 |
21 | 0.011257 | 2.95 | 0.011257 | 2.86 |
22 | 0.010708 | 2.69 | 0.010708 | 2.93 |
23 | 0.010656 | 2.94 | 0.010656 | 3.02 |
24 | 0.011509 | 2.94 | 0.011509 | 3.06 |
25 | 0.011509 | 2.96 | 0.011509 | 3.09 |
26 | 0.010675 | 2.63 | 0.010675 | 3.19 |
27 | 0.011509 | 2.65 | 0.011509 | 3.05 |
28 | 0.011509 | 2.68 | 0.011509 | 3.05 |
29 | 0.011509 | 2.95 | 0.011509 | 3.05 |
30 | 0.010716 | 2.96 | 0.010716 | 2.84 |
Average | 0.01123 | 2717.755 | 0.01119 | 3057.928 |
Standard deviation | 0.000373 | 149.2229 | 0.000568 | 202.8228 |
Optimized results are used to build the 3D model. The model was then used for finite element analysis. Finite element analysis results are presented in Table
Results of evaluation.
Strain energy (mJ) | Deformation (mm) | Stress (MPa) | Life (106 cycle) | |
---|---|---|---|---|
Prediction | 0.01123 | 33.666 | 79.050 | |
FEM | 0.01102 | 34.236 | 81.475 | 299 |
Error (%) | 1.87 | 1.69 | 3.06 |
FEM results for (a) deformation, (b) stress, (c) strain energy, and (d) life.
This paper proposes a method for the development and optimization of a planar spring used for a gravity balance mechanism. The proposed method is a combination of finite FEM, DFNN, and WCA. First, the FEM was used to collect data. Next, the approximated model was constructed using the DFNN, and finally, the geometry of the planar spring was optimized by the WCA. The effectiveness of the proposed method was tested by comparison with the optimal results of the ER-WCA. The comparison results show that the convergence speed and search stability of the proposed algorithm are better than the ER-WCA. The optimal parameters of PS are used to build 3D models. This model is used to be FEM. Comparing the FEM results with the optimal prediction results shows that the errors of energy strain, deformation, and stress are 1.87%, 1.69%, and 3.06%, respectively. This error shows that the proposed solution is highly robust. In addition, the life of the PS was also predicted with 299 million cycles. For future research, prototypes will be manufactured and measured to verify the numerical results. The optimization method will be utilized for other compliant mechanisms.
The data used to support the findings of this study are included within the article.
The authors declare that there are no conflicts of interest regarding the publication of this article.
The authors are thankful for the financial support from the HCMC University of Technology and Education, Vietnam, under Grant No. T2021-01NCS.