Study on the Optimization Model of a Flexible Transmission
A belt drive and a chain drive are the main types of flexible power transmission. In the traditional belt and chain drive design process, engineers need to do a lot of rework to get a design. To solve this problem, taking the Shell Eco-Marathon vehicle as an example, the traditional design and optimization design of the transmission system are carried out. In the optimization design, component optimization and overall optimization design model of the belt and chain drive are first established. Second, the charts in the design manual are converted into formulas by using MATLAB. Finally, an optimization design model is established in Microsoft Excel, and the Excel Solver tool is used to find the optimal design result. The design method proposed in this paper can effectively determine the optimal design of transmission system and provides a new method for the processing of such problems.
Motors are used to provide rotational power, but they typically operate at too high a speed and too low a torque. Therefore, speed reduction between the motor and the driven machine is always needed. Belts and chains represent the major types of flexible power transmission elements. The high speed of the motor makes a belt drive ideal for the first stage of speed reduction. At low speed and high torque conditions, chain drives become desirable. Figure 1 shows a typical application of belt and chain drives, where belts and chains are each used to their best advantage.
It is obvious that the design process of chain drives and belt drives is very similar. Whether designing chain drives or belt drives, many parameters need to be selected from tables or drawings in catalogs. This process always requires a large amount of time, and the designers are not able to evaluate all the solutions. The traditional method has the disadvantage of a low calculation precision. The traditional method is always a feasible solution but not the best solution. Optimal design results can be obtained by using modeling and software tools. This method can shorten the design time and find the best solution.
Optimization has evolved from a methodology of academic interest and has continued to make a significant impact in industry . Based on complex algorithm, Spitas, V. and Spitas, C.  proposed a nonstandard optimal alternative involute gear design, which achieved the same performance as involute gears by satisfying the optimization design with objective function and active constraints. With the development of artificial intelligence in recent years, there have been certain intelligent optimization algorithms, such as genetic algorithms [3–5], ant colony algorithms [6, 7], neural network algorithms [8–10], and particle swarm optimization (PSO) algorithms . Padmanabhan et al.  optimized the bevel gear pair by using a nonlinear programming optimizing software LINGO and metaheuristics. The efficiency of the proposed algorithms is validated through gear design problems and the comparative results are studied . Dolen et al.  used genetic algorithm to study the optimal design of four-stage gear train and proposed five different genetic coding schemes to solve the most critical constraints. Spitas et al.  introduced the concept of nondimensional gear teeth to be used in gear stress minimization problems. This proposed method significantly reduces the calculation time by essentially reducing the total number of design variables . Haj-Fraj et al.  proposed an automatic transmission shift optimization method based on the validation mechanics model of power assembly. For the nonlinear programming problem, the sequential quadratic programming algorithm was used to solve it .
Optimization has been applied in the power transmission system. A multiobjective mathematical model based on the minimum volume of the wheel and the force on the drive shaft of a belt drive was established based on an improved genetic algorithm . Deng et al.  proposed a multiobjective evolutionary algorithm (MOEA) based on nondominated sorting, which can analyze the influence of transmission ratio matching on the fuel economy and emissions. Sun  analyzed a tractor gearbox transmission system and a push spam transmission control system based on determining the parameters of the transmission system. Wang  used theory of the solution of inventive problems (TRIZ) to configure the parameters of belt drive and chain drive, which improved the performance of the whole combined drive. The optimal mathematical model is derived by the method of the internal penalty function, and the design variables, target function, and constraint conditions are determined to obtain the best solution . Based on minimization of the size of the belt drive, Hong et al.  determined the optimum radius of the belt roller by analyzing the design cord and relative data of the ordinary V-belt drive. Using the V-belt’s minimum volume optimization method to establish a target function, V-belt’s optimization design mathematical model was achieved, which is based on the genetic algorithm (GA) principle and uses the GA toolbox of MATLAB . Combining both optimized design ideas and MATLAB and regarding the minimal gross weight of a V-belt pulley as the optimization goal, the modeling and optimization of a multirow V-belt drive is carried out using the exterior penalty method . Therefore, it is necessary to optimize the overall volume of the transmission system.
Many optimization tools have been applied in design optimization. The MATLAB genetic algorithm toolbox is used in the optimization of key parameters . The mathematical models are constructed for a planetary gear reducer in a shield tunneling machine (STM) with 4 objective functions, which are solved by means of MATLAB programming language compiling by using the reliably gray PSO (basic particle swarm optimization) method . Through practical application in Excel of the mechanical system design, Song and Zhang  enumerated the count of applications in Excel for use in the machine design process. By means of the instances in machine design, Excel is used to improve the design velocity and quality in machine design . Therefore, it is necessary to make full use of the superior functions of each software to optimize the design.
This paper presents the general design of belt drives, chain drives, and the design optimization process in Excel. Section 2 introduces the general design process of belt drive and chain drive and gives the design results of the transmission system applied in Shell Eco-Marathon; Section 3 introduces the construction of the optimization model and the formulation of the tables; Section 4 introduces the construction of the model which combines the belt drive design with the chain drive design; Section 5 introduces the design process of the optimal scheme of the transmission system model; Section 6 compares the general design and the optimization design, including the processes and results, and provides conclusions.
2. General Design Process
The Shell Environmental Marathon requires cars to drive as far as possible with 1 L of gasoline. Under such conditions, the weight of the whole vehicle could be reduced to achieve its goal. According to the rules of the race, the material of the car cannot be changed. The best option is to minimize the space volume of the transmission system so that the vehicle can become lighter and drive farther.
2.1. Belt Drive Principle and the General Design Process
Belt drives are always used as the first transmission. As shown in Figure 2, the belt transmits power by frictional contact between the belt and the driving and driven pulley.
In the general design process, engineers need to follow the process defined in the product catalog. In the process of belt transmission design, it is necessary to know the transmission power, speed of the motor, and output speed (or the drive ratio). The design flowchart of belt drive is shown in Figure 3.
There is a belt drive application in the general design process for the Shell Eco-Marathon vehicle.
Objective: Design the V-belt drive
The power transmitted: =4.123 kW
The vehicle’s motor: Zongshen ZL70
The input rotation speed:
The results of belt drive design are shown in Table 1.
In the process of belt transmission designed by common design method, many parameters need to be queried repeatedly for manuals and charts, which consumes a lot of design time. As far as the belt drive itself is concerned, the final design result cannot guarantee the optimal solution which takes up the smallest space volume.
2.2. Chain Drive Principle and the General Design Process
The chain transmits power between two rotating shafts by meshing with toothed sprockets. Chain drives are usually manufactured using high-strength steel, and for this reason, they are capable of transmitting high torque [28, 29]. The most common type is a roller chain, which is used for high-power transmissions and delivery equipment.
Chain design is based on ensuring that the power transmission capacity is within the limits for three modes of failure: fatigue, impact, and galling .
The chain drive is a transmission device that is widely applied in mechanical equipment. Traditionally, to obtain satisfactory design data for the chain drive, designers must try again and again. In the design of chain drive, it is necessary to know the transmission power, the input speed, and the output speed. The flowchart of chain drive design is shown in Figure 4.
Now, we will show the chain general design process by applying it for the Shell Eco-Marathon vehicle:
Objective: Design the roller chain drive
The power transmitted: P=4.123 kW
The vehicle’s motor: Zongshen ZL70
The transmission system consists of two stages: the belt drive and the chain drive. The belt drive is the first transmission and its drive ratio is =1.5. The input rotation speed is and the output rotation speed is .
The results of chain drive design are shown in Table 2.
In the process of designing chain drive by common design method, it also needs to spend a lot of time in searching manuals to obtain parameters. Unlike the single design of chain drive, this design is a transmission system composed of belt drive and chain drive, and the overall volume of the transmission system is the smallest. This requires that the interaction between the parameters of belt drive and chain drive should be considered as a whole. Obviously, it is difficult to design chain drive separately to take into account the design parameters of belt drive.
3. Optimization Model Construction and Table Formulation
In the traditional design method described above, there is no simple quantitative model that can determine the volumes of overall design of the belt and chain drive to determine whether the final design result is optimal. In the optimization design, the goal is to take up as little space as possible. Therefore, for the overall design of the traditional system (such as a belt drive), the volume of the space occupied by the belt drive is now represented by the volume of the cuboid box that can be mounted on the drive system. The model construction of the belt drive is shown in Figure 5.
After the optimization model is established, many programming platforms and certain integration tools can be used to solve the optimization problem, such as the MATLAB optimization toolbox and Excel Solver. In the general design process, it is necessary to obtain data from the catalog tables. However, for the optimization design, this paper transforms the chart into a formula in order to improve the design speed and quality.
3.1. Overall Objective Function Modeling Process
For the belt drives, the cuboid box model can be seen in Figure 3. The width of the box is the same as the pulleys’ width:The height of the box is the same as the diameter of the large pulley (), so the volume of the belt drive isIn the same way, we can calculate the volume of the chain drives:The volume of the transmission system iswhere is the center distance between the two pulleys, is the small pulley diameter, is the large pulley diameter, is the number of V-belts, is the distance between the two belt grooves, is the distance from the belt groove to the edge, is the center distance between the two sprockets, is the number of teeth of the large sprockets, and is the number of teeth of the small sprockets.
3.2. Formulation with the Belt Length Correction Factor
In the ContiTech catalog of V-belts, there is a table of CONTI FO-Z XPZ belt length correction factors, , as shown in Table 3.
The table can be transformed into a formula by MATLAB. The curve in Figure 6 represents the data points and regression curve. The fitting error of the formula is given in Figure 7. From Figures 6 and 7, the formula suits very well with the table. Thus, the functional relationship between and can be described with the following formula:
3.3. Formulation of the Single V-Belt Power Rating Chart
The power rating is a function of , transmission ratio , and small pulley speed and its value can be looked up in a catalog, as shown in Table 4. To make it easier to optimize, it is necessary to find a formula to describe the rated power rating values. As the value of is known, a formula is needed to describe the power rating as a function of and .
First step: linearly interpolate to obtain the value of when .
Second step: process the data in MATLAB.
The power rating of a single V-belt is a function of the belt speed v, small pulley diameter , and transmission ratio :where is the arc of the contact factor and is the belt length correction factor.
In the ContiTech catalog, , so If the speed of the small pulley is known, the formula can be transformed intoThe coefficients of the formulas above can be obtained by MATLAB. The formulas are as follows: The residual case order plot of the formula is given in Figure 8. According to the concentration trend and the concentration degree and shape of the residual distribution in the figure, it can be seen that this formula is fully in accordance with the original data in Table 4.
For the belt optimization only, the transmission ratio can be assumed to be =1.5, then
4. Optimum Model of Belt Drive and Chain Drive
In order to get the global optimal solution, it is necessary to consider the belt drive and chain drive as a whole. In this section, the whole structure of belt drive and chain drive is designed as an optimization model. The model is more complex with more variables. Excel solver can be used to solve optimization problems. The flowchart of the optimal design is shown in Figure 9.
The optimal design process is as follows.
Objective: Design the belt drive and roller chain drive with a minimum volume of the sprockets and pulleys
The power transmitted: =4.123 kW
The vehicle’s motor: Zongshen ZL70
The transmission system consists of two stages: the belt drive and the chain drive. The input rotation speed is and the output rotation speed is .
4.1. Objective Function
The aim of the design optimization is to obtain the minimum value of the objective function. To travel as far as possible within the context of a limited energy supply for the Shell Eco-Marathon vehicle, it is important to reduce the mass (or the volume) of the vehicle. For the belt and chain design optimization, the volume of belt and chain drives transmission system can be chosen as the objective function.
The center distance of the pulleys: The center distance of the sprockets:
4.2. Design Variables
From the formula of the volume, we find that there are six variables:
The diameter of the small pulleys , the diameter of the large pulley , the center distance of the pulleys , the number of belts , the number of teeth on the drive sprocket , the number of teeth on the driven sprockets , and the center distance of the sprockets . However, the value of the belt length and the length of the chain are discrete. We can use the belt length and the chain length to replace the variables and to make it easy to obtain the optimization solution. Thus, we can obtain the following design variables:
4.3. Optimization Constraints
 The belt safety factor should be greater than 1.1, so
 The speed of the belt should not exceed 50 m/s, with , so
 The number of belts should not exceed 10, so
 The center distance:
 For the “CONTI FO-Z XPZ” belt, , , so
 The number of teeth on the drive sprocket should be at least 19, and the number of teeth on the driven sprocket should be less than 114.
 The sum of the teeth should not be less than 50:
 From the safety factor table in the catalog, the safety factor should be greater than 8:
The chain drive ratio:
The chain velocity:
The load in the chain due to the power transmitted:
The load in the chain due to the centripetal acceleration:
The chain axial breaking force:
The chain safety factor There is also a constraint that the center distance of sprockets should be in the range of 30-50 pitches. The constraints can be described as follows:
4.4. Obtain the Optimal Solution
In order to minimize the volume of the objective function, optimization software can be used to obtain the optimal solution. For the optimal design of two-stage drive of belt drive and chain drive, the established model is shown in Table 5.
In the model established in Table 5, from top to bottom are design conditions, design variables, design process of belt drive and chain drive, constraints, and objective function results. Among them, the diameter and length of the small pulley can only be selected from the table, and the “belt drive design process” is used to store these data. Taking the small pulley diameter as an example, the VLOOKUP function is invoked in the cell of the alternative value, and the standard value is obtained by invoking the data in the table.
The main parameters of the design are calculated in the design process of belt drive and chain drive. The constraints of belt drive and chain drive are added to the “constraints”. Finally, the objective function is established. When the design variables are input into the corresponding values, the function values of the objective function are automatically calculated here. The right side of the overall table is the look-up table data.
The model can be used as a design tool for two-stage drive of belt drive and chain drive, which can greatly reduce the repetitive workload of designers.
5. The Optimal Solution in the Optimal Model
5.1. Optimal Solution Obtained by “GRG Nonlinear”
Considering that the variables in the optimization model are discrete variables, this paper adopts the “GRG Nonlinear ” method to solve this problem.
On the basis of the optimization model established in Table 5, open the Excel Solver tool, the cell corresponding to the objective function is selected as the optimization objective, the cell corresponding to the “minimum” design variable is selected as the variable cell, and integer constraints are added. Constraints less than or equal to zero are added to all constraints cells in the constraints section.
Several initial values are randomly input into the design variables and solved by optimization tools. The optimal solution is the design variables corresponding to the minimum value of the objective function.
Compared with the solutions with different starting points, the following solution is the best, and the value of the objective function is 1312888 mm3.
5.2. Calculate the Other Parameters of the Belt Drives and Chain Drives
The belt drive:
The diameter of the large pulley:
 The center distance:
 The center adjustments x and y:
The arc of contact around the small pulley:
 The belt speed:
 The belt flex frequency:
 The effective pull:
 The axle load:
The axle load for two-pulley drives can be calculated fromThe dynamic wave loading needed for dimensional purposes is derived fromThe belt tension factor k1 is a function of under specific service conditions. Obtain the value of k1 that we need in MATLAB:
k1=[1.70 1.73 1.76 1.79 1.83 1.87 1.91 1.95 2.00 2.05 2.11 2.17 2.24 2.31 2.39 2.48 2.58 2.69 2.82];
From Table 15, on page 40 of the belt catalog, we can obtain the centrifugal force factor =0.072. So
 Control of the belt tension:
The calculated static belt tension:The static belt tension from Table 59, on page 78:
The test force from Table 7, on page 42 of the belt catalog:
The read-off value from Table 7, on page 42 of the belt catalog isThe free span length:
With this deflection, the belt tension complies with the rated value calculated.
The chain drive:
The center distance of the sprockets isCheck the size constraints:
Calculate the length and width of the transmission system and check if they meet the constraints.
From the Table 6, it can be seen that the solution meets the size requirements.
5.3. Design Result
Using the CONTI FO-Z heavy-duty cogged raw edge V-belt Section XPZ and Renold roller chain (12.7 mm Pitch), here are the values of the main design parameters:
The belt number: 2. The small pulley diameter: 71 mm. The belt length: 590 mm. The number of teeth on the drive sprocket: 19. The number of teeth on the driven sprocket: 38. The chain length: 89 (pitches).
In contrast to the traditional design results (as shown in Table 7), the overall drive system volume is reduced by 15.95% compared to the conventional designs. After comparison, the combination of the optimization model and software tools is faster, and the optimization effect is better.
In order to test the reliability of the optimization results, considering that all variables are discrete variables, the exhaustive method is used to solve the optimization problem in MATLAB. The final result is the same as the solution in this paper, but the solution time in MATLAB is up to 3 hours, while the solution in this paper is only a few minutes.
Taking the Shell Eco-Marathon vehicle transmission system design as an example, this paper analyzes the traditional design process of the belt and chain drive based on product design manuals and proposes an optimization design scheme. First, an optimization design model is built, including component optimization and overall optimization design model of a belt and chain drive. Second, MATLAB is used to formulate the parameters of the chart. Finally, the optimization design model is established in Microsoft Excel, and the Excel Solver tool is used to find the optimization design result.
In the Shell Eco-Marathon vehicle transmission system design example, compared with the traditional design results, the optimization design volume of the parts is reduced by 15.38%, and the optimization design volume of the total transmission system is reduced by 15.95%. The result of this optimization makes the vehicle lighter and allows it to travel farther in the case where the material is unchanged. Compared to the traditional design, the optimization solution can efficiently obtain the optimal solution by minimizing the size of the Shell Eco-Marathon vehicle drive system.
When Excel Solver is used to solve the optimization design model, if the Excel Solver has problems of “providing only local optimization” or “error using evolution method” due to the complicated engineering problems, we can use different “GRG nonlinear” methods to obtain the best solution in less time, for example, the overall optimization design of the transmission system presented in this paper. In the Similarity Study of establishing objective function by using minimum volume optimization method, compared with the model based on genetic algorithm, the optimal solution is faster and the model is more universal.
For discrete design variables, this paper uses an exhaustive solution method. This method can quickly find the optimal solution when there are a few variables, but when there are a large number of variables in the problem, the solution takes a long time. Therefore, to address optimization problems, we should choose the appropriate optimization tools and methods according to the problem.
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
This work was supported by National Fundamental Research Funds for the Central Universities of China (Grant no. 2014YJ02), National Undergraduate Innovation and Entrepreneurship Training Program (Grant no. 201611413086), and Beijing College Students Innovation Training Project (Grant no. K201504024).
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