CN101934487A - Determination Method of Chatter Stability Limit Diagram for Ball Screw Grinding - Google Patents

Abstract

The invention relates to a method for determining the chatter stability limit diagram of ball screw grinding. The third is to determine the grinding process parameters to ensure the stability of the grinding process according to the grinding chatter stability limit diagram. The invention mainly solves the problem of suppressing chatter phenomenon in the process of grinding the ball screw by the CNC thread grinder, and accurately identifies the parameters of the grinding dynamic model through the proposed experimental design method, and determines the stability of the grinding chatter accordingly The limit diagram provides the grinding process parameters to ensure the stability of the grinding process, so as to effectively solve the problem of chatter suppression in the grinding process. According to the curve, the grinding process parameters of grinding wheel spindle speed n and grinding depth h to avoid chatter phenomenon can be determined, which provides a basis for improving the grinding quality and grinding efficiency of the ball screw.

Description

Determination Method of Chatter Stability Limit Diagram for Ball Screw Grinding

technical field

The invention relates to a method for analyzing the chatter stability of ball screw rod grinding, in particular to a method for accurately judging the chatter stability of ball screw rod grinding by using a limit diagram.

Background technique

Grinding is the last process in the processing of ball screw. The flutter phenomenon caused by unstable grinding during the grinding process is one of the important factors affecting the surface quality of the workpiece. Generally, there are two ways to suppress the grinding chatter phenomenon. One is to measure the signals (such as acoustic emission signals, etc.) when chatter occurs and when no chatter occurs during the grinding process of the ball screw. The pattern recognition theory establishes the discriminant formula of the grinding chatter phenomenon, and controls the grinding process parameters according to the discriminant formula to avoid the chatter phenomenon; the other is to establish a grinding dynamic model by measuring the dynamic characteristics of the machine tool , and according to the theory of stability criterion in cybernetics, the stability region of the grinding system is analyzed, and the appropriate grinding process parameters are selected according to the stability region to avoid the chatter phenomenon. Of the above two methods, the former is only suitable for one type of ball screw grinding. When the grinding conditions change (such as grinding wheel replacement, workpiece replacement), it is necessary to re-study the discriminant of the grinding chatter phenomenon. The latter has the problem that the grinding dynamics model is difficult to accurately determine its model parameters. This makes the suppression of grinding chatter a major problem in precision grinding. According to this situation, the present invention proposes an experimental design method for accurately determining the parameters of the grinding dynamics model. This method solves the technical problem that the parameters of the grinding dynamics model are difficult to accurately determine. sex limit chart. The proposed method is of great significance to improve the surface processing quality and processing efficiency of ball screw.

Contents of the invention

The present invention provides a method for determining the chatter stability limit diagram of ball screw grinding, which can optimize the grinding process parameters of the ball screw, suppress the grinding chatter phenomenon, and improve the grinding efficiency.

In order to achieve the above object, the technical solution of the present invention is: a method for determining the chatter stability limit diagram of ball screw grinding, comprising:

1. Establish the dynamic model of the grinding process through the experimental method;

2. Determine the grinding chatter stability limit diagram according to the stability criterion in cybernetics;

3. According to the grinding chatter stability limit diagram, determine the grinding process parameters to ensure the stability of the grinding process.

The specific method of establishing the dynamic model of grinding process by experimental method:

During the grinding process of the ball screw, the top of the tailstock of the machine tool supports one end of the ball screw, and the top of the headstock supports and fixes the other end of the ball screw, and the motor drives the ball screw to rotate to adjust the grinding angle of the grinding wheel of the machine tool for grinding. The differential equation of motion of the grinding dynamics model is shown in formula (1):

$\mathrm{<span\; class="notranslate">\; m</span>}\stackrel{<span\; class="notranslate">\; \&Center\; Dot;</span><span\; class="notranslate">\; \&Center\; Dot;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; c</span>}\stackrel{<span\; class="notranslate">\; \&CenterDot;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; wxya</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$

In the formula: m is the mass of the grinding wheel, kg; c is the damping of the grinding wheel, N/m; k is the stiffness of the grinding wheel, N/m; X(t) is the amplitude of the grinding wheel at time t, m; F(t) is the grinding force, N .

In formula (1), the grinding force F(t) represented by the material removal rate can be expressed by formula (2):

F(t)＝-k _m hba(t) (2)

in:

a(t)=[bX(t-T)-X(t)]

In the formula: k _m is the grinding force coefficient of the grinding wheel, N/m ³ ; h is the grinding depth of the grinding wheel, m; b is the grinding contact width, m; a(t) is the vibration pattern on the workpiece surface, m; b is the grinding wheel The overlap coefficient of the front and rear two revolutions; T is the rotation period of the grinding wheel,

Through Laplace transform, formula (1) is expressed as formula (3) in the frequency domain:

(ms ² +cs+k)X(s)=F(s) (3)

In the formula: s is the Laplace operator, dimensionless.

Substituting the expression (2) of the grinding force F(t) into the formula (3) after Laplace transformation, taking X(s) as the output of the chatter system and a ₀ (s) as the input, the grinding system can be obtained The transfer function of , as shown in formula (4):

$\frac{\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}\mathrm{<span\; class="notranslate">\; h</span>}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; k</span>}}\mathrm{<span\; class="notranslate">\; W</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; be</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; s\; T</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}\mathrm{<span\; class="notranslate">\; h</span>}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; k</span>}}\mathrm{<span\; class="notranslate">\; wxya</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>$

in:

$\mathrm{<span\; class="notranslate">\; W</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\frac{{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&zeta;</span>}}{{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}$

Where: W(s) is the dynamic compliance of the grinding wheel system, m/N; ω _n is the fixed angular frequency of the grinding wheel system, rad/s; ζ is the damping ratio of the grinding wheel system, dimensionless.

According to the stability criterion in cybernetics, the specific method to determine the grinding chatter stability limit diagram is as follows:

According to the transfer function formula (4) of the grinding wheel chatter system, and according to the stability criterion in cybernetics, the grinding chatter stability limit diagram can be determined, and the denominator of the transfer function of formula (4) is set to zero. The chatter condition occurs in the grinding wheel system, that is,

$\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; be</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; s\; T</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}\mathrm{<span\; class="notranslate">\; hb</span>}\frac{\mathrm{<span\; class="notranslate">\; W</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$

和表征磨削工艺参数T、h、k′_m的传递函数，得到稳定极限条件，如式(6)所示：According to the first discriminant method of system stability Lyapunov, the characteristic root s=jω is substituted into formula (5) to represent the dynamic characteristics of workpiece and grinding wheel structure

and the transfer function that characterizes the grinding process parameters T, h, _k′m, the stability limit condition is obtained, as shown in formula (6):

$\frac{\mathrm{<span\; class="notranslate">\; W</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j\&omega;</span>}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; be</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; j\&omega;T</span>}}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 6</span>}<span\; class="notranslate">\; )</span>$

Where: ω is the chatter angular frequency of the grinding wheel, rad/s.

Let λ=ω/ω _n , from formula (6), it can be deduced that the grinding depth h and the grinding wheel spindle speed n to ensure the stability of the grinding process are respectively

$\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; b</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&omega;T</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; [</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&zeta;\&lambda;</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ]</span>}{{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; b</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&omega;T</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; b</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 7</span>}<span\; class="notranslate">\; )</span>$

$\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 60</span>}\mathrm{<span\; class="notranslate">\; \&omega;</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; arcsin</span>}\frac{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&zeta;\&lambda;</span>}}{\mathrm{<span\; class="notranslate">\; b</span>}\sqrt{{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&zeta;\&lambda;</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; arctan</span>}\frac{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&zeta;\&lambda;</span>}}{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}<span\; class="notranslate">\; )</span>$

According to the constant parameter grinding wheel system damping ratio ζ, grinding wheel system fixed angular frequency ω _n , grinding wheel grinding force coefficient k _m , grinding wheel stiffness k, overlap coefficient b, and the stability of grinding chatter is determined by formula (7) and formula (8) According to the grinding process parameters determined in the stable region of the grinding chatter stability limit diagram, the occurrence of grinding chatter can be avoided. Among them, the hammering test is performed on the grinding wheel frame of the machine tool to obtain the frequency response function of the grinding wheel frame, and the natural frequency ω _n and damping coefficient ζ in the above formula can be determined through modal analysis. The method proposed by the present invention is mainly to determine the values of the parameters _km , k, and b through experiments, and accurately determine the grinding chatter stability limit diagram.

Beneficial effects of the present invention:

The invention mainly solves the problem of suppressing chatter phenomenon in the process of grinding the ball screw by the CNC thread grinder, and accurately identifies the parameters of the grinding dynamics model through the proposed experimental design method, and determines the stability of the grinding chatter accordingly The limit diagram provides the grinding process parameters to ensure the stability of the grinding process, so as to effectively solve the problem of chatter suppression in the grinding process. Through this invention, the problem of how to accurately determine the stability limit diagram of grinding chatter is solved. According to this curve, the grinding process parameters n and grinding depth h of the grinding wheel spindle to avoid chatter phenomenon can be determined, in order to improve the grinding quality of ball screw And grinding efficiency provides a basis.

Description of drawings

Fig. 1 is a schematic diagram of the ball screw grinding process;

Figure 2 is a simplified model diagram of the ball screw grinding system;

为磨削稳定域实测点。Fig. 3 is the limit diagram of the chatter stability limit of ball screw grinding, in which: A is the unstable region, B is the stable region, O is the measured point in the grinding unstable region,

It is the measured point in the grinding stable region.

Detailed ways

The present invention will be further described below in conjunction with the accompanying drawings and embodiments.

The method for determining the chatter stability limit diagram of ball screw grinding of the present invention includes the following steps: one is to establish a dynamic model of the grinding process through an experimental method; The chatter stability limit diagram; the third is to determine the grinding process parameters to ensure the stability of the grinding process according to the grinding chatter stability limit diagram.

Concrete method of the present invention:

1. Method of establishing grinding dynamics model

As shown in Figure 1, during the ball screw grinding process, the top end of the tailstock 13 of the machine tool withstands one end of the ball screw 11, the top end of the headstock 10 supports and fixes the other end of the ball screw 11, and the motor drives the ball screw 11 to rotate. Adjust the grinding angle of the grinding wheel 12 of the machine tool for grinding.

Fig. 2 is a schematic diagram of a simplified grinding wheel-ball screw grinding system model, and the differential equation of motion of the grinding dynamics model is shown in formula (1):

$\mathrm{<span\; class="notranslate">\; m</span>}\stackrel{<span\; class="notranslate">\; \&Center\; Dot;</span><span\; class="notranslate">\; \&Center\; Dot;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; c</span>}\stackrel{<span\; class="notranslate">\; \&Center\; Dot;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; wxya</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$

In the formula: m is the mass of the grinding wheel, kg; c is the damping of the grinding wheel, N/m; k is the stiffness of the grinding wheel, N/m; X(t) is the amplitude of the grinding wheel at time t, m; F(t) is the grinding force, N .

In formula (1), the grinding force F(t) represented by the material removal rate can be expressed by formula (2):

F(t)＝-k _m hba(t) (2)

in:

a(t)=[bX(t-T)-X(t)]

In the formula: k _m is the grinding force coefficient of the grinding wheel, N/m ³ ; h is the grinding depth of the grinding wheel, m; b is the grinding contact width, m; a(t) is the vibration pattern on the workpiece surface, m; b is the grinding wheel The overlap coefficient of the front and rear two revolutions; T is the rotation period of the grinding wheel,

Through Laplace transform, formula (1) is expressed as formula (3) in the frequency domain:

(ms ² +cs+k)X(s)=F(s) (3)

In the formula: s is the Laplace operator, dimensionless.

Substituting the expression (2) of the grinding force F(t) into the formula (3) after Laplace transformation, taking X(s) as the output of the chatter system and a ₀ (s) as the input, the grinding system can be obtained The transfer function of , as shown in formula (4):

$\frac{\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}\mathrm{<span\; class="notranslate">\; h</span>}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; k</span>}}\mathrm{<span\; class="notranslate">\; W</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; be</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; s\; T</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}\mathrm{<span\; class="notranslate">\; h</span>}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; k</span>}}\mathrm{<span\; class="notranslate">\; wxya</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>$

in:

$\mathrm{<span\; class="notranslate">\; W</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\frac{{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&zeta;</span>}}{{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}$

Where: W(s) is the dynamic compliance of the grinding wheel system, m/N; ω _n is the fixed angular frequency of the grinding wheel system, rad/s; ζ is the damping ratio of the grinding wheel system, dimensionless.

According to the stability criterion in cybernetics, the specific method to determine the grinding chatter stability limit diagram is as follows:

According to the transfer function formula (4) of the grinding wheel chatter system, and according to the stability criterion in cybernetics, the grinding chatter stability limit diagram can be determined, and the denominator of the transfer function in formula (4) is set to zero, and we get The chatter condition occurs in the grinding wheel system, that is,

$\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; be</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; s\; T</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}\mathrm{<span\; class="notranslate">\; hb</span>}\frac{\mathrm{<span\; class="notranslate">\; W</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$

和表征磨削工艺参数T、h、k′_m的传递函数，得到稳定极限条件，如式(6)所示：According to the first discriminant method of system stability Lyapunov, the characteristic root s=jω is substituted into formula (5) to represent the dynamic characteristics of workpiece and grinding wheel structure

and the transfer function that characterizes the grinding process parameters T, h, _k′m, the stability limit condition is obtained, as shown in formula (6):

$\frac{\mathrm{<span\; class="notranslate">\; W</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j\&omega;</span>}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; be</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; j\&omega;T</span>}}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 6</span>}<span\; class="notranslate">\; )</span>$

Where: ω is the chatter angular frequency of the grinding wheel, rad/s.

Let λ=ω/ω _n , from formula (6), it can be deduced that the grinding depth h and the grinding wheel spindle speed n to ensure the stability of the grinding process are respectively

$\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; b</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&omega;T</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; [</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&zeta;\&lambda;</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ]</span>}{{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; b</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&omega;T</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; b</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 7</span>}<span\; class="notranslate">\; )</span>$

$\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 60</span>}\mathrm{<span\; class="notranslate">\; \&omega;</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; arcsin</span>}\frac{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&zeta;\&lambda;</span>}}{\mathrm{<span\; class="notranslate">\; b</span>}\sqrt{{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&zeta;\&lambda;</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; arctan</span>}\frac{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&zeta;\&lambda;</span>}}{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}<span\; class="notranslate">\; )</span>$

According to the constant parameter grinding wheel system damping ratio ζ, grinding wheel system fixed angular frequency ω _n , grinding wheel grinding force coefficient k _m , grinding wheel stiffness k, overlap coefficient b, and the grinding chatter can be determined by formula (7) and formula (8) The stability limit diagram is shown in Figure 3. According to the grinding process parameters determined in the stability region of the grinding chatter stability limit diagram, the occurrence of grinding chatter can be avoided. The hammering test is carried out on the grinding wheel frame of the machine tool by using the hammering method, and the frequency response function of the grinding wheel frame can be obtained. The natural frequency ω _n and the damping coefficient ζ in the above formula can be determined through modal analysis. However, due to the complexity of the grinding process and the measurement and calculation methods of various parameters, constant parameters such as the grinding force coefficient k _m , the grinding wheel stiffness k and the overlap coefficient b cannot be accurately determined, resulting in the inability to accurately determine the grinding chatter stability limit diagram , thus affecting the solution to the problem of grinding chatter.

3. Experimental Design Method for Accurately Determining the Limit Diagram of Grinding Chatter Stability

From the above, the reason why the grinding chatter stability limit diagram cannot be accurately determined is mainly due to the uncertainty of the parameters _km , k, and b. Therefore, how to determine the values of the parameters _km , k, and b through experiments becomes the key to the application of this technology. Assume that the value ranges of k _m , k, and b are k _{m min} ≤ _{k m} ≤ _{km max} , k _min ≤ k ≤ _{k max} , b _min ≤ b ≤ b _max , and randomly generate N groups of data within the value range{ k _mi , k _i , bi _} , i=1, 2, . . . , N. As shown in Figure 3, calculate the maximum variation of the grinding depth h when the grinding wheel spindle speed n is fixed. Assuming that at the grinding wheel spindle speed n _k1 , the variation of the grinding depth h is the largest, it means that the uncertainty of the grinding chatter stability limit diagram is the largest here, and experiments should be carried out at this place to eliminate its uncertainty. It is determined through experiments that (n _k1 , h ₁₀ ) and h(n _k1 , n ₁₁ ) are the points where grinding chatter does not occur and where it occurs, respectively, and the grinding chatter stability limit diagram should be between these two points. According to the experimental results, in randomly generated N sets of data {k _mi , _ki , _bi }, i=1, 2, ..., N, remove n=n _k1 , h ₁₀ ≤h≤h Points other than ₁₁ . Then _, N sets of _data {k _mi , _ki , _bi }, i=1, ₂ , . Calculate the maximum variation of the grinding depth h when the grinding wheel spindle speed n is fixed. Assuming that at the grinding wheel spindle speed _nk2 , the variation of the grinding depth h is the largest, it means that the uncertainty of the grinding chatter stability limit diagram is the largest here, and experiments should be carried out at this place to eliminate its uncertainty. It is determined through experiments that (n _k2 , h ₂₀ ) and h( _nk2 , h ₂₁ ) are the points where grinding chatter does not occur and occurs, respectively, and the grinding chatter stability limit diagram should be between these two points. According to the experimental results, in randomly generated N sets of data {k′ _mi , _ki , bi _} , i=1, 2, ..., N, remove n=n _k2 , h ₂₀ ≤h≤ Points other than h ₂₁ . By analogy in this way, until a satisfying

n=n _k1 , h ₁₀ ≤ h ≤ _{h 11}

n=n _k2 , h ₂₀ ≤ h ≤ _{h 21}

n=n _km , h _m0 ≤ h ≤ _{h m1}

And the variation of the grinding depth h is less than the allowable value {k _mi , _ki , _bi } parameter, and the finally determined grinding chatter stability limit diagram is obtained from this parameter.

Claims (3)

1. A method for determining the chatter stability limit diagram of ball screw grinding, characterized in that: comprising the following steps: (1) Establish grinding process dynamics model by experimental method; (2) Determine the grinding chatter stability limit diagram according to the stability criterion in cybernetics; (3) According to the grinding chatter stability limit diagram, determine the grinding process parameters to ensure the stability of the grinding process. 2. the method for determining the chatter stability limit diagram of ball screw grinding according to claim 1, is characterized in that: the specific method of setting up the dynamic model of grinding process by experimental method is: During the grinding process of the ball screw, the top of the tailstock of the machine tool supports one end of the ball screw, and the top of the headstock supports and fixes the other end of the ball screw, and the motor drives the ball screw to rotate to adjust the grinding angle of the grinding wheel of the machine tool for grinding. The differential equation of motion of the grinding dynamics model is shown in formula (1): $\mathrm{<span\; class="notranslate">\; m</span>}\stackrel{<span\; class="notranslate">\; \&Center\; Dot;</span><span\; class="notranslate">\; \&CenterDot;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; c</span>}\stackrel{<span\; class="notranslate">\; \&CenterDot;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; wxya</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$ In the formula: m is the mass of the grinding wheel, kg; c is the damping of the grinding wheel, N/m; k is the stiffness of the grinding wheel, N/m; X(t) is the amplitude of the grinding wheel at time t, m; F(t) is the grinding force, N , In formula (1), the grinding force F(t) represented by the material removal rate can be expressed by formula (2): F(t)＝-k _m hba(t) (2) in: a(t)=[bX(t-T)-X(t)] In the formula: k _m is the grinding force coefficient of the grinding wheel, N/m ³ ; h is the grinding depth of the grinding wheel, m; b is the grinding contact width, m; a(t) is the vibration pattern on the workpiece surface, m; b is the grinding wheel The overlap coefficient of the front and rear two revolutions; T is the rotation period of the grinding wheel, Through Laplace transform, formula (1) is expressed as formula (3) in the frequency domain: (ms ² +cs+k)X(s)=F(s) (3) In the formula: s is the Laplace operator, dimensionless, Substituting the expression (2) of the grinding force F(t) into the formula (3) after Laplace transformation, taking X(s) as the output of the chatter system and a ₀ (s) as the input, the grinding system’s Transfer function, as shown in formula (4): $\frac{\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}\mathrm{<span\; class="notranslate">\; h</span>}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; k</span>}}\mathrm{<span\; class="notranslate">\; W</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; be</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; s\; T</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}\mathrm{<span\; class="notranslate">\; h</span>}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; k</span>}}\mathrm{<span\; class="notranslate">\; wxya</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>$ in: $\mathrm{<span\; class="notranslate">\; W</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\frac{{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&zeta;</span>}}{{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}$ Where: W(s) is the dynamic compliance of the grinding wheel system, m/N; ω _n is the fixed angular frequency of the grinding wheel system, rad/s; ζ is the damping ratio of the grinding wheel system, dimensionless. 3. The determination method of the ball screw grinding chatter stability limit diagram according to claim 1, characterized in that: according to the stability criterion in cybernetics, the specific method for determining the grinding chatter stability limit diagram is: According to the transfer function formula (4) of the grinding wheel chatter system, and according to the stability criterion in cybernetics, determine the grinding chatter stability limit diagram, let the denominator of the transfer function of formula (4) be zero, and get the grinding wheel A flutter condition occurs in the system, namely $\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; be</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; s\; T</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}\mathrm{<span\; class="notranslate">\; hb</span>}\frac{\mathrm{<span\; class="notranslate">\; W</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$ According to the first discriminant method of system stability Lyapunov, the characteristic root s=jω is substituted into formula (5) to represent the dynamic characteristics of workpiece and grinding wheel structure

and the transfer function that characterizes the grinding process parameters T, h, _k′m, the stability limit condition is obtained, as shown in formula (6): $\frac{\mathrm{<span\; class="notranslate">\; W</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j\&omega;</span>}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; be</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; j\&omega;T</span>}}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 6</span>}<span\; class="notranslate">\; )</span>$ Where: ω is the chatter angular frequency of the grinding wheel, rad/s. Let λ=ω/ω _n , from formula (6), it can be deduced that the grinding depth h and the grinding wheel spindle speed n to ensure the stability of the grinding process are respectively $\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; b</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&omega;T</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; [</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&zeta;\&lambda;</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ]</span>}{{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; b</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&omega;T</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; b</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 7</span>}<span\; class="notranslate">\; )</span>$ $\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 60</span>}\mathrm{<span\; class="notranslate">\; \&omega;</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; arcsin</span>}\frac{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&zeta;\&lambda;</span>}}{\mathrm{<span\; class="notranslate">\; b</span>}\sqrt{{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&zeta;\&lambda;</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; arctan</span>}\frac{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&zeta;\&lambda;</span>}}{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}<span\; class="notranslate">\; )</span>$ According to the constant parameters grinding wheel system damping ratio ζ, grinding wheel system fixed angular frequency ω _n , grinding wheel grinding force coefficient k _m , grinding wheel stiffness k, grinding contact width b, and the grinding chatter is determined by equations (7) and (8). vibration stability limit diagram, according to the grinding process parameters determined in the stability region of the grinding chatter stability limit diagram, to avoid the occurrence of grinding chatter phenomenon, determine the values of parameters _km , k, and b through experiments, and Accurate determination of grinding chatter stability limit diagrams.