Methods of optimisation in machine design

Introduction

Machine building industry has always been known for fast-pace development, and today this trend is being sustained. The industry is facing constant changes while improving methods of design, manufacturing and production management. One of the key drivers for industry development is implementation and development of mathematical modelling methods in production, which date back to ancient centuries. Today, numerical modelling and optimal design methods are becoming more influential in product development [1].

Most enterprises apply mathematical models of production management based on the studies by Adamiecki, Gantt, Taylor, Gastev and other scholars. Development of computer systems helped considerably popularise analytical design, management, and production. Production automation, monitoring systems, electronic document flow, and digital twins are the tools for enterprise digitalisation that allow to manage the entire economic and production system of an enterprise.

In Russia, one of the leading experts in mathematical methods of optimisation of enterprise’s economic efficiency and production is Khakimzyan Amirovich Faskhiyev, Professor, Doctor of Engineering Sciences. His works are related to the enhancement of economic efficiency and competitiveness of an enterprise and its products. His works are valuable both in terms of theory and applications in practice. Most of his works have been developed and now are applied at the premises of the KamAZ vehicle production in the city of Naberezhnye Chelny [2].

Development of optimisation methods

For now, optimal design technologies, to one degree or another, are integral part of any production process, from selection of the best suitable design of a tin can with regard to its material, shape, wall thickness, weight and other characteristics with a single criterion of optimisation, i.e. the minimisation of costs, to optimisation of the entire structure of a guided missile in order to obtain the optimum of the multi-criteria problem where the criterion of optimality is the maximum flight range with the maximum payload.

There are various methods for solving multicriteria problems. Essentially, solving such problems mostly requires a set of Pareto optimal solutions. In this context, the key problem is to choose and justify a particular solution.

Based on the example of a multi-purpose machine for service electric transport, we will analyse the process of multi-criteria optimisation of the electric transport propulsion plant.

The multi-purpose machine for service electric transport (MMSET) has been developed by the Federal State Budget Educational Institution of Higher Education (FGBOU VO) Kalashnikov Izhevsk State Technical University for Sarapul Electric Generator Plant, JSC, following the Decree of the Government of the Russian Federation No. 218 under the high-tech product development project [1].

The MMSET is a modular transport and service vehicle with a wide range of applications. With its modular design, the MMSET allows to perform various types of operations such as bulk and loose cargo transportation, use with a refrigerator plant for perishable products transportation with application of various types of attachable equipment mounted on the front and rear parts of the vehicle [3].

The development of propulsion plant (PP) optimisation methods is scaled and requires the optimality criteria to be defined as the PP elements (local criteria), which can be changed to obtain the required parameters of the machine as well as the optimality criteria of the machine as a whole (global criteria) depending on local ones [1].

As any other project, the propulsion plant development project has a structural hierarchy and can be viewed as a set of structural elements that form the product. By changing the scale of parametric base formation, the following can be considered:
– microlevel, change of materials and technology for manufacturing base members, fasteners and other parts;
– common level, control of characteristics of assembly units, assemblies and units, selection from a variety of standard fasteners and base parts (shafts, gears, etc.);
– macrolevel, creating a general concept of the product from a variety of standard assemblies.

For machine design, it is reasonable to use assemblies the production of which has already been mastered. Therefore, the PP elements will represent a set of local discrete characteristics allowing for reduced costs of machine design and faster launch of full-scale production.

The design layout of the electric vehicle propulsion plant under consideration comprises four assemblies as shown in Figure 1. The layout of assemblies is shown in Figure 2. These assemblies are decisive for all the design characteristics [1].

Moreover, we can single out the mutual arrangement of the elements as the fifth adjustable element. In addition to the variants described above, electric motors can be installed directly in the electric vehicle’s wheels or in the drive axle, but, since the vehicle is to be designed based on the assemblies manufactured at the enterprise, selection of the electric motor for the multi-purpose machine for service electric transport is limited to a series of traction electric motors made by Sarapul Electric Generator Plant, JSC [1].

The next step in design is to identify the relationship between optimality criteria and characteristics of assemblies.

Optimality criteria found feasible in the phase of the machine’s exploratory design and design specification development define the optimisation targets in all subsequent phases of machine development. Criteria forming a set of partial optimality criteria are different in measure and normally have complex interrelations while often being contradictory, i.e. improving one criterion always results in degradation of another. This causes the major difficulty to encounter when solving optimisation problems [5].

Considering the above, the optimisation problem is presented in the following mathematical form [5]:

where {O} - set of Pareto optimal solutions, u_n – n-th optimality criterion, Pm – set of parameters of assembly m; then

where p_ml - parameter l of assembly m.

The customer’s requirements are given in Table 1.

Based on constraints, local optimality criteria regarding the MMSET can be formulated. The maximum efficiency of such criteria will be achieved by optimising the design of the propulsion plant.

During optimisation, the remaining requirements shall be taken into account as input constraints.

Further, it is necessary to determine how input parameters affect the state of the system, using the empirical method – by changing parts, assemblies and operation control algorithms, and then to identify the relationship through the correlation-regression analysis.

In this case, it is necessary to calculate the coefficient of pair correlation between any possible pairs of parameters. The coefficient characterises the relationship between two random values. If one parameter is designated y1, and another parameter is designated y2, the number of experiments (measurements) is N, i.e. the current number of experiments is u = 1, 2, …, N, then the coefficient of pair correlation r is calculated by the following formula:

where  and   - arithmetic mean values [5].

The values of coefficient of pair correlation r_у1, y₂ may lie in the range of –1 to +1. If an increase in the value of one parameter causes an increase in the value of another parameter, the coefficient sign will be positive; otherwise, if the parameter value is decreased, the sign will be negative. The closer the calculated value r_у1, y2 to unity, the higher the dependence of the value of one parameter on another. In this case, it is sufficient to consider only one of the parameters in studies (experiments) [5].

To check the significance of the coefficient of pair correlation, it shall be compared with a tabular (critical) value of r. Quantiles of Student’s t-distribution are given in the table of critical values of the Student’s correlation coefficients [2].

To use the table, the quantity of degrees of freedom f = N – 2 shall be known and a certain level of the significance, for instance, equal to α = 0.05, shall be selected. This value of the significance level is called the 5%-risk level corresponding to the probability of right answer when testing our hypothesis P = 1 – α = 0.95, or 95 %. This means there is only a 5 % chance of making an error during the hypothesis testing [2].

Below we will look into the method based on the analysis of mathematical interrelations of physical formulas.

The mathematical formulation of criteria is defined as follows.

The first criterion is u₁. The operating time t in the driving cycle can be mathematically described as the percentage of the battery load mode and is presented as the following formula:

where n – number of load modes, i – load mode number; t – total changeover time, (h); w_i – power consumption of the i-th load mode in percent of maximum load mode, (%); τ_i – share of operating time in the i-th mode, (%); E – battery energy, (kW∙h); W – electric motor power consumption at maximum load (kW).

The second, third and fifth criteria u₂, u₃ and u₅, maximum speed of empty vehicle V_п (km/h), maximum speed of loaded vehicle V_з (km/h) and maximum climbing angle α (deg) are derived from the power balance equation [7]:

where N_eη_тр – drive wheel power with regard to transmission efficiency η_тр (kW);
N_f – power consumed for wheel rolling resistance to be calculated by the following formula [8]:

where V_а – driving speed (m/s); P_f – wheel rolling resistance force (kN); α – climbing angle (deg); G_a – vehicle weight (N); f_a – wheel rolling resistance coefficient depending on propulsion plant and road surfacing.

N_i – climbing resistance power equal to [8]:

where P_i – climbing resistance force (kN). N_в – air resistance power excluding wind, to be calculated by formula [8]:

where P_в – air resistance force (kN); К₀ – air resistance coefficient (N×s² /м⁴ ); F_a – vehicle front area (m2 ).

N_j – acceleration resistance power, to be calculated by formula [8]:

where P_j – acceleration resistance force (kN); δ_a – rotational inertia coefficient; g – gravity acceleration (~9.8 m/s² ); j_a – vehicle acceleration, to be calculated by formula [8]:

where D – vehicle dynamic factor (kN/kg); Ѱ – road resistance coefficient; δ_a – rotational inertia coefficient.

With consideration given to values of power consumed to overcome all types of resistance, the power balance equation is as follows:

The speed can be derived from the equation:

This expression is difficult to analyse, so let us consider the criteria as systems of equations. After conversion, we obtain the following systems of equations for determining the maximum speed:

The system of equations for the maximum climbing angle derived from the power balance:

The drawbar force formula is known [8]:

where P_кр – drawbar force (N); М_д – engine torque (Nm); i_с – transmission gear ratio; η_с – transmission efficiency r_к – wheel radius (m).

The local optimum can be achieved at the maximum value of P_кр. Thus, in a general form, the mathematical description of the MMSET PP optimisation problem will be represented as follows:

To continue problem solving, the parameters need to be introduced into the relative coordinate system while the criterion shall be normalised to unified U(u). There are various methods of normalisation. As an example, we will consider the method of logarithmic normalisation, the advantage of which is the transition from absolute to relative increments of parameters.

In this case, the i-th controlled parameter u_i is converted into non-dimensional x_i as follows [7]:

where ϛ_i – coefficient numerically equal to the unit of measurement of parameter u_i . The other two possible methods include application of the methods of statistical analysis by means of arithmetic mean value [7]:

where u_{i ср} – arithmetic mean of all values of u_i by formula:

where u_ij – the j-th controlled parameter u_i ; – n quantity of all values u_ij; and average median value

where   - average median value of sorted sequence u_ij.

Besides, the criteria may have the perpendicular direction of the optimisation vector, for example, an increase in the maximum attainable speed normally brings nearer the optimum derivation while an increase in the acceleration time is a negative result. For this purpose, the following equation can be used [7]:

where u_ia – value of increasing criteria; u_ib – value of decreasing criteria; u_{i max}, u_{i min} – maximum and minimum values of criterion. The direction of criterion optimisation can be determined as follows:

where   - refined value of criterion, [u_i] – admissible value of the i-th criterion.

After the conditions of the optimisation problem are formulated, we may proceed with problem solving. There are various methods of optimum determination. The author has chosen the method of normalisation to the unified criterion. The essence of the methods is to normalise a variety of optimality criteria to one criterion by means of different techniques.

These methods include the method of constrained maximisation, linear convolution, methods of scalarisation, Chebyshev method, as well as methods for determining criteria weight coefficients, comprising the common ones – expert evaluation methods. The essence of expert evaluation methods is to employ human intelligence and their ability to seek and find the solution to ill-defined problems. Some methods are developed based on the theory of expert evaluation. Let us consider two methods.
1. Ranking method. Assume that expert evaluation is conducted by a group of L experts highly qualified in the field where the relevant decision is to be made. The ranking method implies that each expert shall rank partial criteria of the object to be designed in the order of their priority. Number 1 is assigned to the partial criterion of the highest priority, number 2 is assigned to the next partial criterion in the order of priority, and so on. These ranks are converted in such a way that rank 1 is given rating m (number of partial criteria), rank 2 – rating m – 1 and so on to rank m, which is given rating 1. We designate the resulted ratings rik, where i – number of the i-th expert, k – number of the k-th criterion. Expert opinion poll results can be presented in the table form, see Table 3 [2].

Row (L+1) presents sums of ratings given to criteria by experts. In this case, weight coefficients are determined as follows [2]:

2. The scoring method is based on experts evaluation of a partial criterion on the scale. Experts are allowed to evaluate the priority expressed in fractional values or to assign the same value from the selected scale to several criteria. The score given by the i-th expert for k criterion is designated h_ik, then

where h_jk is the sum of the i-th row. r_ik is called the weight calculated for k criterion by the i-th expert [2].

In this respect, taking into account that , we get  

The scoring method is the most labour-consuming method in terms of calculations, but it is more convenient and more precise for expert evaluation.

With respect to expert weigh coefficients and normalisation, the mathematical formulation of the optimisation problem can be represented as follows:

where U – unified normalised value of criterion for the system, i – number of criterion, n – number of variants of assemblies combinations, l₁ – expert weight coefficient of the i-th increasing criterion; l₂ – expert weight coefficient of the i-th decreasing criterion u, [u_i] – admissible value of the i-th criterion, u_ia – value of increasing criteria, u_ib – value of decreasing criteria, u_{i max}, u_{i min} – maximum and minimum values of criterion; u∙_1a u∙_1b – refined values of criterion.

After conversions, we get the final formula represented as follows:

With the maximum value of the normalised criterion determined, the optimality of the propulsion plant design for the service electric transport vehicle is supposed to be achieved.

Conclusion

In the course of the study we developed the methods for obtaining optimal performance characteristics of service electric transport by using the most reasonable (optimal) design of the propulsion plant for arbitrary criteria.

The next step is to form a parametric base using available parameters and to determine the final configuration of the machine. The developed methods can be used not only for electric vehicle development, but also for making the best possible decision in multiple projects, for example:
– design of a hybrid vehicle with a combined propulsion plant [8][9][10];
– design of machine assemblies and units [11];
– optimisation of enterprise capacity utilisation [5];
– integration of methods of lean production and cost reduction in the enterprise [2].

Also, the methods can be applied to solve qualitative problems, such as improvement of the vehicle operation comfort [12].

When applied, such methods allow to accelerate the process of object design, to reduce costs of production launch and to improve the product performance without affecting other characteristics.

Advanced modelling results in creation of a digital twin of the product and production. At this level, the automatic analysis may give not only information about optimal characteristics, but also to build complex prediction models that allow an enterprise to gain a significant advantage in the market.