Low-Carbon Optimization Design of Grinding Machine Spindle Based on Improved Whale Algorithm

Abstract

To achieve a fundamental reduction in the carbon emissions associated with grinding machines, it is imperative to systematically explore low-carbon considerations in the design phase. The spindle is a significant contributor to carbon emissions in grinding machines, and an effective approach for reducing carbon emissions is the structural optimization of the spindle. Most of the current optimization methods aim at improving processability without considering the reduction of carbon emissions. In this context, the present study addresses the issue of carbon emissions within the spindle design phase. Initially, the determination of the spindle’s carbon emissions function and the selection of the optimization objective were undertaken. The structural factors that have a significant influence on the optimization objective were identified as optimization variables. Subsequently, the optimization objective function was established through the application of the fitting method. Finally, the proposed model was refined through the utilization of an enhanced whale algorithm. The findings indicate an 8.22% reduction in carbon emissions associated with the spindle, accompanied by marginal enhancements in both static and dynamic spindle performance. The concluding section of this paper deliberates on the impact of structural parameters on the specified objectives, thereby providing insights for the optimal design of the spindle.

1. Introduction

In the relentless growth of the industrial sector, the need for power resources has steadily risen, representing around 30% of the aggregate power demand, with its corresponding carbon emissions making up approximately 40% of the total emissions [1]. The carbon dioxide produced by the consumption of machine tool equipment energy and raw materials occupies the leading position in the carbon dioxide emission of the equipment manufacturing industry. As high-precision machining machine tools, grinding machine tools account for about 42% of cutting machine tools in China, and a large amount of carbon dioxide is produced during their production, manufacturing, and operation. Hence, the conservation of energy and reduction of emissions in grinding machine tools hold paramount importance for the equipment manufacturing industry. Therefore, the conservation of energy in grinding machines has emerged as a pivotal concern in sustainable manufacturing.

Assessing and quantifying carbon emissions in the components of grinding machines presents a considerable challenge, given the intricate nature of machine elements, their intricate structures, and the interconnected nature of carbon emissions. Traditional approaches to optimizing machine tool structure focus on enhancing their static and dynamic performance by manipulating parameters such as materials, structure, processes, and motion. However, the intricate correlations and potential conflicts between these parameters and carbon emissions make it challenging to concurrently optimize the low-carbon performance and the static and dynamic aspects of grinding machine tool structures. This complexity underscores the need for innovative and integrated approaches to effectively address these challenges and find a balanced solution.

By studying the existing literature, the majority of research on reducing the carbon emissions from machine tools reduces the parameters used in the processing stage by optimizing them [2,3,4,5,6]. However, optimizing process parameters has a constrained impact on reducing carbon emissions, and the machine tool itself cannot curtail its carbon emissions level. The spindle in the grinder, being a constituent of the machining process, also constitutes a significant source of energy consumption for the machine. Approximately 15% of the energy consumption is attributed to it [7]. A spindle system with low-carbon emissions is therefore necessary, but little research has been conducted on how to design a spindle with low-carbon emissions.

Study of the existing literature reveals that there are two aspects of current research. First, the analysis of energy consumption on the one hand. Second, the optimization of design on the other. For the first research aspect, energy consumption was reduced by analyzing energy consumption. For example, there has been a proposal for predicting the spindle system’s energy consumption during machining [8]. A study of spindle systems found that by adjusting motor parameters, 10% of the energy consumption could be reduced [9]. In response to these studies, other scholars began to analyze spindle systems mathematically. Using an empirical spindle system as an energy model, Albertelli et al. [10] examined the energy savings potential of the spindle system. Similarly, the power usage of the CNC lathe spindle and efforts to minimize energy consumption were explored [11]. A study by Ben et al. [12] examined the impact of fluctuations in cutting force on the power consumption of the spindle system during machining. The influence of wear and the installation of components on energy loss was also investigated. Hu et al. [13] modeled the machining sequence of a part and observed variations in machine power consumption with different sequence selections. Hu et al. [14] introduced a multi-objective synergistic operation model to oversee power usage and diminish carbon emissions. Avram et al. [15] simulated the power consumption of machine tool spindles under transient and steady-state conditions. Numerous studies have explored process parameters and the characterization of the milling process to steer low-carbon production and enhance energy efficiency [16,17,18,19]. These articles scrutinize energy consumption models of machine tools or spindles from various perspectives, signifying their importance in advancing theories related to the energy efficiency of spindle systems. However, in the investigation of spindle system energy consumption, the focus of most scholars is on the motor, with little attention given to the mechanical components. It is not easy to express intuitively how the spindle consumes energy. Exploring its energy consumption mechanism holds considerable importance for both conceptual and applied aspects in enhancing the machining efficiency of grinding machines and even higher-end machine tools.

On the flip side of the research, the optimized design endeavors to enhance the performance of the equipment or components, addressing aspects such as deformation, frequency, and vibration. Li and colleagues proposed a structural design approach for welding sliding beams in hydraulic presses. Utilizing the topological configuration of machine tool load and mass distribution constraints, this method executed a low-carbon design of the machine tool [20,21]. Chen selected the grinding machine bed, spindle, and wheel seat as the optimization object, and took the performance of the grinding machine as the goal, and used the structural optimization method, the sensitivity analysis method, and the parametric experimental design and analysis method to realize the optimal design of the grinding machine [22]. Liu et al. [23] used grey relational analysis and analytic hierarchy processes to optimize the crossbeams of gantry machines. According to the results, the beam’s natural frequencies were improved in the first four frequencies. A topology and dimension optimization approach was used by Chen et al. [24] to enhance rib design in machining centers, taking static and dynamic considerations into consideration. In order to enhance the dynamic performance of the machine, Law et al. [25] optimized the column topology. In a study by Zhao et al. [26], their dynamic sensitivity method utilized the error propagation model to improve the sensitivity and static rigidity of the machine tool.

From the above research, it is evident that the majority of studies concentrate on the characteristics of carbon emissions, offering a reference and foundation for diminishing carbon emissions at the design stage. However, current efforts seldom delve into an in-depth examination of the low-carbon design method for the spindle. The primary objectives of optimization design research are often directed towards enhancing the static and dynamic processing capabilities of the equipment. The carbon emissions level has progressively evolved into a pivotal indicator for grinding machines. Nevertheless, there is a scarcity of studies that prioritize carbon emissions as the focal point of optimal design.

Table 1. Literature review summary table.

References	Energy Consumption	Static Characteristic	Dynamic Characteristic	Optimization Method
Reference [9]	10% reduction	Not considered	Not considered	Optimize motor parameters
Reference [10]	7% reduction	Not considered	Not considered	Optimized spindle drive mode
Reference [11]	10.6% reduction	Not considered	Not considered	Optimize processing time
Reference [21]	17.6% reduction	The maximum deformation is reduced by 0.8%	Not considered	Optimize topology
Reference [23]	Not considered	Not considered	The first four natural frequencies increased by 17%	Optimize structure

shows the optimization methods used and the optimization objectives considered in five recent pieces of literature.

Based on insights from the literature mentioned previously, we explore new low-carbon design approaches. The low-carbon spindle design methodology introduced in this paper has important theoretical value and practical significance to break through the cooperative optimization problem of carbon emissions and static and dynamic coupling and ensure the low-carbon and efficient operation of the grinding machine. Firstly, the carbon emissions target of this paper was clarified and the mathematical function of spindle carbon emissions was established. In the second phase, the optimization objective was established, and design variables within the structural parameters were carefully chosen. Thirdly, the response surface method was used to derive the fitting function of the optimization target, and the optimization target was transformed into a comprehensive model. Finally, based on the improvement of the whale algorithm, the integrated model was solved using the algorithm.

The subsequent sections of the paper are structured as follows: Section 2 establishes the carbon emissions function and analyzes the static and dynamic performance. Section 3 delineates the design parameters, the all-encompassing optimization goal, and the optimization framework. Section 4 describes the improvement process of the whale algorithm as well as the optimization results and analysis of the model. And in Section 5, the paper presents its conclusions and outlines potential avenues for future research.

2. Spindle Carbon Emissions and Performance Analysis

2.1. Carbon Emissions Function

Building on the force analysis of the grinding machine spindle, the carbon emissions function of the spindle was formulated. There are four stages of grinding processing, which are the following: starting stage, no-load stage, grinding stage, and stopping stage. Among them, the no-load stage and the grinding stage last longer and are also the most energy-consuming stages. The spindle’s input power (P) is predominantly consumed in two facets during the operational process: (1) the power of the grinding wheel to grind the material, which is called grinding power (P_c); and (2) the power of the spindle rotating without processing, which is called no-load power (P_u).

Depending on the processing parameters and configuration, the P_c is directly used for grinding material. Inertia and friction are overcome during rotation by the power P_u of the spindle, including losses caused by friction and changes in speed. Equations (1)–(4) illustrate how this varies with spindle mass, shape, and speed.

$$P=P_c+P_u$$

(1)

$$P_c=\frac{C_p}{1000}{Z^{\prime }}^{0.7}B$$

(2)

$$Z^{\prime }=\frac{1000v_w{f}_a{f}_r}{B}$$

(3)

$$P_u=J_m\omega \frac{d\omega }{dt}+P_f$$

(4)

where C_p is a coefficient related to the workpiece and wheel material, Z′ is the amount of metal removed per unit wheel width, B is the wheel width, $v_w$ is the workpiece speed, f_a is the axial feed, f_r is the radial feed, J_m signifies the equivalent moment of inertia for the spindle, $\omega$ represents the angular velocity, and P_f signifies the power consumed as a result of bearing friction.

The spindle of the grinder is solid and stepped. J_m is shown in Equation (5).

$$J_m=\frac{1}{2}\rho \pi {\displaystyle \sum l_i{R}_i^4}$$

(5)

where $\rho$ is the density, l_i represents the length of the i-th order shaft, and R is the i-th order shaft outer diameter.

P_f denotes the loss of power attributable to friction during spindle rotation, defined as illustrated in Equations (6) and (7).

$$P_f=M_f\omega$$

(6)

$$M_f=\frac{1}{2}upR$$

(7)

where M_f denotes frictional torque of the bearing, $u$ represents the coefficient of friction, p is the bearing load, and R denotes the shaft diameter.

According to Equations (1)–(7), Equation (8) describes P_u in detail.

$$P_u=\frac{1}{2}\rho \pi \omega \frac{d\omega }{dt}{\displaystyle \sum l_i{R}_i^4}+\frac{1}{2}upR\omega$$

(8)

Derived from the preceding analysis, the energy consumption function is depicted in Equation (9).

$$E=E_c+E_u={\displaystyle {\int }_0^{t}Pdt}$$

(9)

In this context, E_c represents the material removal energy consumption, E_u denotes the no-load energy consumption, and t signifies the spindle running time.

Carbon emissions from the main axes can be calculated from the energy consumption with the following formula:

$$C=E\cdot f$$

(10)

where C is the spindle’s carbon emissions and $f$ is the electrical energy carbon emissions factor.

To streamline the calculation of carbon emissions and assess the environmental implications of electric energy consumption, this study employs the carbon emissions factor associated with electric energy. The baseline emission factors for China’s regional power grid, as determined by the Ministry of Ecology and Environment, Department of Climate Change, are presented in

Table 2. Carbon emissions factors of electric energy across the country.

District	Chinese North	Chinese Northeast	Chinese East	Chinese Central	Chinese Northwest	Chinese South
Carbon emissions factor ($\mathrm{K}\mathrm{g}\mathrm{C}{\mathrm{O}}_2/\mathrm{K}\mathrm{W}\cdot \mathrm{h}$)	0.9419	1.0826	0.792	0.8587	0.8922	0.804

[27].

2.2. Static and Dynamic Performance Analysis

2.2.1. Static Performance

Grinding accuracy and performance are directly affected by the spindle, which is the key component of grinders. Therefore, this section selects and analyzes the key static and dynamic aspects.

As a result of the spindle’s static performance, the grinder’s ability to withstand static external loads is directly affected. This metric is used to judge the static performance of the spindle because it measures its resistance to deformation under static loads [28]. And the static stiffness is calculated as shown in Equation (11).

$$K=\frac{F}{\delta }$$

(11)

In this equation, K represents stiffness, F denotes the load, and $\delta$ signifies the deformation.

Therefore, deformation is identified as the primary factor influencing the spindle’s stiffness, making it the chosen key indicator for the static performance of the spindle.

2.2.2. Dynamic Performance

The dynamic performance of a spindle is determined by the characteristics of the spindle when it is under dynamic loads. During grinding, the force generated by the grinding wheel induces spindle vibrations, affecting both machining accuracy and surface quality. This means it should possess excellent static characteristics and, equally importantly, superior dynamic characteristics.

The vibration equation of the spindle can be computed as presented in Equation (12).

$$K\cdot X+R\cdot \stackrel{\cdot }{X}+M\cdot \stackrel{\cdot \cdot }{X}=F$$

(12)

where, K, R, and M are the stiffness, damping, and mass coefficients, respectively. $X$, $\dot{X}$, and $\ddot{X}$ are the displacement, velocity, and acceleration quantities. F is the external load.

Modal and natural frequencies are the primary metrics for gauging the dynamic performance of the system. The first natural frequency is considered the principal vibration mode and the key indicator of dynamic performance for the spindle.

For the free vibration state, external excitation can be ignored. At this point, the first-order natural frequency can be obtained as follows:

$$f_1=\frac{1}{2\pi }\sqrt{\frac{K}{M}}$$

(13)

where f₁ is the first-order intrinsic frequency.

If the natural frequency of the system is low, it can easily resonate with external excitation [29].

Spindle optimization aims to improve the span and deformation of the spindle using the primary performance metrics discussed above. Spindle performance can both be improved statically and dynamically by minimizing deformation and improving natural frequency.

3. Low-Carbon Design Approach for Spindles

Initially, the structural parameters and objectives were chosen in Section 3.1, and the variables were identified through analysis. Then, the central composite experiment method was selected to create the fitting function of the target. Finally, the model was formulated in Section 3.2.

3.1. Structural and Objective Selection

3.1.1. Structural Parameters

The spindle is a solid shaft with various technological features, including chamfering. To simplify simulation modeling and calculations, the spindle’s structure was streamlined, and certain minor technological features were disregarded, as depicted in Figure 1. These simplifications did not compromise the spindle’s performance.

Certain spindle structures are designated for the installation of other components, necessitating the preservation of their parameters throughout the optimization process. On the basis of this analysis, some structural parameters were put forward to optimize the main shaft, including the outer diameter (D₁, D₂, D₃, D₄, D₅, D₆) and length (l₁, l₂, l₃, l₄, l₅, l₆) of the stepped shaft.

3.1.2. Design Variables

In Section 3.1.1, 12 structural parameters were selected initially, but optimizing multiple parameters simultaneously is a challenging and time-consuming task.

Various structural parameters exert distinct effects on the optimization objectives, and certain parameters with minimal impact can be disregarded to streamline the modeling and calculation process. In this part of the study, a simulation experiment was devised, the spindle’s simulation data were acquired, and the sensitivity of the preselected structural parameters was computed. After analysis, the primary design variables were identified.

3.1.3. Determination of Design Variables

Each input variable must be quantitatively analyzed to determine its impact on the output variable. Therefore, structural parameters with a significant influence on the optimization objectives were identified through sensitivity analysis [30].

A parameter’s sensitivity is determined by the difference in the indicator function when the parameter is changed. In a mathematical sense, sensitivity is the degree to which each parameter influences the final result by taking the partial derivative of a function. It is usually expressed by Formula (14) [31].

$$S_{{\chi }_i}=\frac{\partial O({\chi }_i)}{\partial {\chi }_i}$$

(14)

where $S_{{\chi }_i}$ represents the parameter sensitivity. $O\left(x_i\right)$ is the indicator function of the spindle. ${\chi }_i$ denotes the structural parameter. Parameter sensitivity has a greater impact on the indicator when its absolute value is larger.

In this paper, simulation data were obtained using Workbench 16.0 software, parameter correlation modules were selected, and the linear correlation between parameters was evaluated using the Spearman coefficient. A sensitivity matrix was generated to prove the correlation between the input and output parameters, and the sensitivity of the output to the input parameters. In

Table 3. Sensitivity results.

Objectives	D1	l1	D2	l2	D3	l3	D4	l4	D5	l5	D6	l6
Carbon emissions	0.146	0.289	0.017	0.287	0.9	0.048	0.321	0.103	0.224	0.092	0.209	0.357
Maximum deformation	0.52	0.65	0.66	0.297	0.278	0.257	0.227	0.245	0.199	0.185	0.129	0.104
First-order frequency	−0.327	−0.965	−0.512	−0.167	−0.451	−0.283	−0.069	−0.504	−0.169	−0.598	−0.421	−0.037

, each structural parameter is assessed for its sensitivity to the index. Figure 2 illustrates how the results are depicted visually as a histogram to illustrate the level of sensitivity.

Based on the outcomes of the sensitivity analysis, it was observed that D₃ had a significant effect on the carbon emissions, the maximum deformation was strongly influenced by D₁, l₁, and D₂, and the first-order intrinsic frequency was strongly influenced by l₁. l₁, D₂, and D₃ were the structural parameters that had a significant effect on the optimization objective. Therefore, these variables were identified as the design variables.

3.2. Integrated Optimization Objectives

Given the multivariable and multi-objective nature of spindle optimization, this section employed the response surface method as an effective fitting technique to establish the optimization objectives. The fundamental concept behind the response surface method involves employing polynomial functions to interpolate sample points within the design space. This method establishes an approximate mathematical model linking multiple variables with response values, facilitating the prediction of responses for non-tested points. It is a mathematical modeling approach aimed at addressing multi-parameter problems by identifying optimal structural parameters through the analysis of regression equations.

3.2.1. Establishment of Test Methods

In this study, a central composite experimental design was employed to generate experimental design points. This experimental design method offers data sample points for constructing the response surface mathematical model. It boasts advantages such as simplicity in design, high efficiency, and robust predictability. Taking the values of the variables, the tests were carried out to read the carbon emissions C, the maximum deformation U, and the first-order intrinsic frequency E. The findings are presented in

Table 4. Results of experimental design.

Serial Number	${\mathit{l}}_1$/mm	${\mathit{D}}_2$/mm	${\mathit{D}}_3$/mm	$\mathit{C}$/Kg	$\mathit{U}$/mm	$\mathit{H}$/Hz
1	50.00	65.00	70.00	0.129	6.489 × 10−3	1120.09
2	50.00	65.00	63.00	0.127	6.493 × 10−3	1119.82
3	50.00	65.00	77.00	0.131	6.489 × 10−3	1120.55
4	50.00	58.50	70.00	0.117	6.072 × 10−3	1120.38
5	50.00	71.50	70.00	0.141	6.778 × 10−3	1119.97
6	45.00	65.00	70.00	0.129	6.327 × 10−3	1120.55
7	55.00	65.00	70.00	0.129	6.659 × 10−3	1120.69
8	45.93	59.72	64.31	0.118	6.032 × 10−3	1120.47
9	45.93	59.72	75.69	0.121	6.029 × 10−3	1120.82
10	45.93	70.28	64.31	0.137	6.579 × 10−3	1120.50
11	45.93	70.28	75.69	0.141	6.575 × 10−3	1120.53
12	54.07	59.72	64.31	0.118	6.279 × 10−3	1120.50
13	54.07	59.72	75.69	0.121	6.277 × 10−3	1121.33
14	54.07	70.28	64.31	0.137	6.859 × 10−3	1120.64
15	54.07	70.28	75.69	0.141	6.848 × 10−3	1120.24

.

3.2.2. Response Surface Modeling

The mathematical models of the response surface include the linear function model, second-order model and third-order model. The relationship between carbon emissions, static and dynamic performance, and each dimension variable is complex, so the second-order response surface model was used to express the relationship between carbon emissions, static and dynamic performance, and dimension variable [32]. Expression 15 is as follows:

$$y={\alpha }_0+{\alpha }_1{x}_1+{\alpha }_2{x}_2+{\alpha }_3{x}_3+{\alpha }_{12}{x}_1{x}_2+{\alpha }_{13}{x}_1{x}_3+{\alpha }_{23}{x}_2{x}_3+{\alpha }_{11}{x}_1^2+{\alpha }_{22}{x}_2^2+{\alpha }_{33}{x}_3^2$$

(15)

where y denotes the response variable, x₁, x₂, and x₃ denote the two independent variables, and ${\alpha }_0$, ${\alpha }_1$, ${\alpha }_2$, ${\alpha }_3$, ${\alpha }_{12}$, ${\alpha }_{13}$, ${\alpha }_{23}$, ${\alpha }_{11}$, ${\alpha }_{22}$, and ${\alpha }_{33}$ represent the regression coefficients of the model.

3.2.3. Establishment of Response Surface Model

The feasibility of fitting carbon emissions, maximal form variables, and first-order frequencies was analyzed using Design-Expert 10.0.1 software with a central composite design. p-value is a statistical concept that indicates the probability of an observed statistic or a more extreme situation occurring if the original hypothesis holds. Typically, when the p-value falls below a predetermined level of significance, commonly set at 0.05, the original hypothesis is rejected, indicating the significance of the parameter in question. The p-value is calculated using the Formulas (16) and (17) as follows:

$$F=\frac{MSR/dfr}{MSE/dfe}$$

(16)

$$P=1-P\left(F_{\mathrm{value}}\le F\right)$$

(17)

where MSR represents the regression sum of squares and dfr represents the regression degrees of freedom; MSE represents the residual sum of squares and dfe represents the residual degrees of freedom. $P(F_{\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}}\le F)$ is the cumulative distribution function value at a given point in the F-distribution.

The statistical software Design-Expert 10.0.1 was used to fit the selected response surface model and experimental data.

• Carbon emissions regression equation.

The linear regression equation of carbon emissions is as follows:

$$\begin{array}{c}C=0.021391-6.42403\times {10}^{-5}A+1.7522\times {10}^{-3}B-5.2666\times {10}^{-4}C-5.81846\times {10}^{-7}AB-5.40287\times \\ {10}^{-7}AC+4.1561\times {10}^{-7}BC+1.42111\times {10}^{-6}{A}^2+8.40891\times {10}^{-7}\times B^2+5.82709\times {10}^{-6}\times C^2\end{array}$$

(18)

where A represents l₁ (mm), B represents D₂ (mm), and C represents D₃ (mm). The parameters in Equations (19) and (20) are the same.

Their linear fit data are shown in

Table 5. Carbon emissions linear fit data.

Carbon Emissions	$\mathit{M}\mathit{S}\mathit{R}$	$\mathit{d}\mathit{f}\mathit{r}$	$\mathit{M}\mathit{S}\mathit{E}$	$\mathit{d}\mathit{f}\mathit{e}$
$C$	1.097 × 10−3	9	4.74 × 10−9	5

.

According to the calculation, the significant difference value of p is less than 0.05 and the significance level is significant, which indicates that the established response model is highly reliable.

By solving the response surface model, the response surface diagram of carbon emissions was obtained, as shown in Figure 3.

2.

Solving the maximum deformation regression equation.

The linear regression equation of maximum deformation is as follows:

$$\begin{array}{c}U=-7.14391\times {10}^{-3}+4.5631\times {10}^{-5}A+2.72339\times {10}^{-4}B+3.18588\times {10}^{-5}C+3.36481\times {10}^{-7}AB-\\ 3.36599\times {10}^{-8}AC-3.54515\times {10}^{-8}BC-3.26635\times {10}^{-7}{A}^2-1.79422\times {10}^{-6}{B}^2-2.01853\times {10}^{-7}{C}^2\end{array}$$

(19)

Their linear fit data are shown in

Table 6. Maximum deformation linear fit data.

Maximum Deformation	$\mathit{M}\mathit{S}\mathit{R}$	$\mathit{d}\mathit{f}\mathit{r}$	$\mathit{M}\mathit{S}\mathit{E}$	$\mathit{d}\mathit{f}\mathit{e}$
$U$	1.084 × 10−6	9	2.672 × 10−10	5

.

According to the calculation, the significant difference value of p is less than 0.05 for the established deformation response surface model; the significance level is significant, which indicates that the established response model is highly reliable.

By solving the response surface model, the response surface diagram with maximum deformation was obtained, as shown in Figure 4.

3.

Solving the first-order natural frequency regression equation.

The first-order natural frequency linear regression equation is as follows:

$$\begin{array}{c}H=1164.67794-2.07421A+0.2001B+0.011864C-4.01473\times {10}^{-3}AB+2.70143\\ \times {10}^{-4}AC-6.44195\times {10}^{-3}BC+0.023288A^2+3.24736\times {10}^{-3}{B}^2+3.0041\times {10}^{-3}{C}^2\end{array}$$

(20)

Their linear fit data are shown in

Table 7. First-order natural frequency linear fit data.

First-Order Natural Frequency	$\mathit{M}\mathit{S}\mathit{R}$	$\mathit{d}\mathit{f}\mathit{r}$	$\mathit{M}\mathit{S}\mathit{E}$	$\mathit{d}\mathit{f}\mathit{e}$
$H$	1.84	9	0.19	5

.

According to the calculation, the significant difference value of p is less than 0.05 for the establishment of the first-order intrinsic frequency response surface model; the significance level is significant, which indicates that the reliability of the established response model is high.

By solving the response surface model, the response surface diagram of the first-order natural frequency was obtained, as shown in Figure 5.

3.2.4. Comprehensive Optimization Modeling

In the actual process, it was essential to maintain the values of structural parameters within a reasonable range, determined by actual processing. Additionally, an optimization model as shown in Formula (21) was established by using design variables, comprehensive objectives, and the above constraints.

$$\begin{array}{c}\begin{array}{l}\min C\\ \min U\\ \max H\end{array}\\ \begin{array}{ll}s.t.& 45<A<55\\ & 60<B<70\\ & 65<C<75\end{array}\end{array}$$

(21)

4. Algorithms and Results of Optimization

4.1. Algorithm for Optimization

Based on the above, the optimal solution formed an optimization problem with multiple objectives and variables. Considering the characteristics of this problem, a multi-objective optimization algorithm based on whale behavior simulation in nature was adopted [33,34]. The algorithm leverages the collective behavior and migratory characteristics observed in whale populations to address the multi-objective optimization challenge. The whale algorithm is a novel swarm intelligence algorithm designed to simulate the bubble net hunting strategy employed by humpback whales. Humpback whales mainly hunt in three ways, namely, encircling predation, bubble net predation, and random predation. The reasons for choosing the whale algorithm in this paper are as follows:

• In comparison to similar classical algorithms, this algorithm has the more important ability to jump out of local optimization.
• The algorithm offers three search methods: surrounding, random, and bubble net, demonstrating robust local search capabilities. Finding discrete and discontinuous optimal solutions is the key to spindle optimization.

4.2. Improvement of the Whale Algorithm

4.2.1. Initialization of the Cat Chaotic Map

Initialization is the first step of a swarm intelligence optimization algorithm, and the computational efficiency and optimization ability change with this step, so its importance cannot be ignored. In this paper, chaotic mapping was employed to replace the random function initialization mode of the traditional whale algorithm. By this method, the rate of convergence and travers ability of the original algorithm could be improved. The chaotic mapping in this paper was formed by choosing the basic cat chaotic map (also known as Arnold chaotic map). Compared with other chaotic maps, the cat chaotic map has the following advantages:

The cat chaotic map is a simple chaotic map with fast calculation speed. This makes cat chaotic mapping have the advantages of high efficiency and real-time performance in practical applications.

Cat chaotic mapping can evenly disperse the points in the input space to the positions in the output space.

It is defined by the following Equation (22).

$$\left[\begin{array}{c}{X}_{n+1}\\ Y_{n+1}\end{array}\right]=\left[\begin{array}{cc}1& a_1\\ b_1& a_1{b}_1+1\end{array}\right]\left[\begin{array}{c}{x}_n\\ y_n\end{array}\right](mod1)$$

(22)

In Formula (22), $a_1$ and $b_1$ are both real numbers, and mod1 is meant to find the fractional part of the real number a1. In particular, it should be noted that the correct cat chaotic sequence is generated when the determinant of the linear transformation matrix in Formula (22) is equal to 1. The cat chaotic map has a simple structure, and the unit chaotic sequence formed in [0, 1] has good ergodic uniformity, and avoids the problem of a small cycle or fixed point, so the cat chaotic map was selected to produce the initial population.

4.2.2. Golden Sine Algorithm

The traditional whale algorithm realizes global exploration through random predation mechanism. The random predation mechanism changes with the change of vector a, and the optimal search is achieved by adjusting the value of a. The global search affects the accuracy of the solution, so this link is particularly important. In 2001, Tanyildizi E. put forward the Golden Sine Algorithm [35] (Golden-SA), which attracted many scholars’ attention because of its advantages such as low implementation difficulty, fast decreasing speed, and few adjustable variables. In this manuscript, the Golden Sine Algorithm was introduced into the random predation mechanism of the traditional whale algorithm, which was helpful to fully explore the global optimal solution. It shortened the space between the whale individual and the best individual, and also helped to weigh the two links of “exploration” and “development”. To enhance solution quality and rate of convergence to a certain extent, the expression for the stochastic predation mechanism can be rewritten as Equation (23):

$$\begin{array}{l}X\left(t+1\right)=X_{rand}\cdot \left|\sin \left(r_1\right)\right|-r_2\cdot \sin \left(r_1\right)A\cdot D\\ D=\left|x_1\cdot C\cdot X_{rand}-x_{2}X\right|\end{array}$$

(23)

where the golden section coefficients are $x_1=-\pi +(1-z)\cdot 2\pi$ and $x_2=-\pi +z\cdot 2\pi$, and where z is $\sqrt{5}-1/2$, r₁ is a randomly generated number within the range of 0 to 2π, and r₂ is a randomly generated number within the range of 0 to π.

Figure 6 shows the flow chart of the improved whale algorithm.

4.3. Analyses and Results of Optimization

The optimization model was addressed using a modified whale algorithm, and the outcomes were rounded to derive the optimal solution. The algorithm was deployed in MATLAB R2018b. Carbon emissions calculations were carried out using MATLAB. ANSYS Workbench was employed for the analysis of the spindle’s maximum deflection and first-order intrinsic frequency. The initial mass of the spindle was 17.7328 kg, and the spindle was made of Cr40 material with a density of 7.9 g/cm³.

Furthermore, to ascertain the effectiveness of the proposed approach, a comparative assessment was conducted against a lightweight optimization methodology. In this alternative optimization approach, the singular objective was the minimization of spindle mass. The optimization variable was defined as the size parameter, with additional consideration given to the constraints imposed by both static and dynamic performance throughout the optimization process.

The Pareto optimal solution set for the spindle parameters could be obtained by optimizing with the improved whale algorithm, as shown in Figure 7. Each point in it represents the values corresponding to the carbon emissions, maximum deformation, and frequency obtained in the range. The interconstraints between carbon emissions and static and dynamic performance may result in the improvement of one target performance, often at the expense of other target performances, and there was no possible solution that could achieve the optimal performance for all targets. Therefore, a compromise result was selected from the solution set as the result of optimization.

The algorithm was run 10 times and the results were averaged. The results of the multi-objective optimization of the spindle parameters are shown in

Table 8. Optimization results.

Design Variables/ Objectives	Before Optimization	Lightweight Optimization	Traditional Whale Algorithm	Improved Whale Algorithm
$l_1$/mm	50	51.4	46.2	44.5
$D_2$/mm	65	61.5	62.5	61
$D_3$/mm	70	67	66.4	65.5
$C$/Kg	0.129	0.1143	0.1232	0.1184
U/mm	0.0065	0.00706	0.00614	0.00601
$H$/Hz	1120.09	1119.64	1120.52	1120.41

. After optimization, the carbon emissions of the spindle were reduced from 0.129 kg to 0.1184 kg, the maximum deformation was reduced from 0.0065 mm to 0.00601 mm, and the first-order intrinsic frequency was increased from 1120.09 Hz to 1120.41 Hz, so the optimization effect was obvious.

As shown in Table 8, after light weight optimization, the carbon emissions of the spindle were reduced by 11%, the maximum deformation displacement was increased by 8.6%, and the first-order natural frequency was reduced by 0.04%. After the optimization of the traditional whale algorithm, the carbon emissions of the spindle were reduced by 4.5%, the maximum deformation displacement was reduced by 5.5%, and the first-order natural frequency was increased by 0.039%. After the optimization of the improved whale algorithm, the main axis carbon emissions was reduced by 8.22%, the maximum deformation displacement was reduced by 7.5%, and the first-order natural frequency was increased by 0.029%. According to the results, although the light weight optimization was better in terms of carbon emissions, the maximum deformation and the first-order natural frequency had a large deterioration. The low-carbon optimization of the main axis could not be done at the expense of static and dynamic performance. In terms of carbon emissions and maximum deformation of the spindle, the improved algorithm obtained better results. In the first-order natural frequency, although the improved algorithm was not as good as the traditional algorithm, there was not much difference. The mechanical characteristics of the spindle had been greatly improved to meet the requirements of structural optimization and performance.

4.4. Design Variables and Their Impact on Goals

To examine the impacts of the three design variables on the three objectives, one design variable was varied while keeping the other design variables constant. Subsequently, a graph illustrating the change in objectives concerning that design variable was plotted.

Figure 8 illustrates the impact of l₁ on the target. An increase in l₁ leads to an increase and then a decrease in carbon emissions. The maximum deformation increases due to the increase of l₁ and the slope of the maximum deformation curve increases slightly. The first-order intrinsic frequency decreases initially and then increases as the parameter l₁ increases.

The impact of D₂ on the target is shown in Figure 9. An increase in D₂ results in an increase in both carbon emissions and maximum deformation, with a slight decrease in the slope of both curves. The first-order intrinsic frequency decreases but at a reduced rate as D₂ increases.

Figure 10 shows the influence of D₃ on the optimization objective. An increase in D₃ corresponds to an increase in both carbon emissions and the first-order intrinsic frequency, and the slopes of both curves increase slightly. As D₃ increases, the maximum deformation initially decreases and then increases after surpassing the inflection point.

From the aforementioned analysis, it can be concluded that spindle carbon emissions generally increase when l₁, D₂, and D₃ increase. The increase in these structural parameters decreases the carbon emissions performance of the spindle. As l₁ and D₂ increase, the maximum deformation increases and the first-order intrinsic frequency decreases. As D₃ increases, the deformation decreases and the first-order intrinsic frequency increases. The increase in l₁ and D₂ leads to a decrease in static and dynamic performance and an increase in energy consumption. The increase in D₃ leads to an increase in static and dynamic performance and an increase in energy consumption. Different structural parameters have different effects on carbon emissions and performance, which indicates that carbon emissions is not the only consideration in the low-carbon optimization of spindles, but static and dynamic performance should also be considered.

5. Conclusions

Given that the spindle serves as the primary source of carbon emissions in the grinding process, optimizing the spindle for low-carbon output emerges as a crucial approach to mitigating carbon emissions from the grinding machine. To facilitate the low-carbon optimization of the spindle, in this study, an optimization design method was introduced that integrates considerations of carbon emissions and performance. Initially, the spindle carbon emissions function was formulated. Subsequently, through an analysis of both the spindle carbon emissions function and performance, the optimization variables and objectives for the low-carbon design of the spindle were determined. Then, the fitting method was chosen to model the function of the optimization target. This method allows the spindle to reduce carbon emissions without sacrificing structural performance. Using the selected spindle as a case study, this paper optimized the structural parameters, resulting in a slight increase in the first-order intrinsic frequency, an 8.22% reduction in carbon emissions, and a decrease in maximum deformation. Through an analysis of the impact of structural parameters on the objectives, it offered insights for guiding spindle low-carbon optimization.

Carbon emissions from grinding machine equipment can be reduced through low-carbon optimization. The paper illustrates the method through an example of the structure of a grinding machine spindle due to the complexity of low-carbon optimization. Since this paper has only validated the method in simulation for the time being, in future work, the optimization method will first be validated in specific experiments. In addition, more complex components and full life-cycle phases in grinding machines will be explored. And this is not the only strategy to reduce carbon emissions in grinding machine equipment, rather, for grinding machines, different strategies should be considered, such as environment, economy, and machining time.