A chatter stability prediction method for turning slender shaft with tool holder

Abstract

The present invention discloses a method for predicting the chatter stability of a slender shaft with a tool holder during turning, comprising the following steps: S1. Calculating the frequency response function at the cutting point of the slender shaft with a tool holder; S2. Identifying the modal parameters of the orthogonal polynomials of modal parameters to obtain the modal parameters at the cutting point of the slender shaft without a tool holder; S3. Coupling the modal parameters of the slender shaft with the tool holder to obtain the frequency response function at the cutting point of the slender shaft with the tool holder; S4. Constructing a turning chatter stability prediction model to achieve the prediction of the stability lobe diagram of the turning of the slender shaft part with the tool holder. The present invention is based on the analytical expression of the frequency response function at the cutting point of the fixed beam and the overall estimation algorithm of the orthogonal polynomials of modal parameters, combined with the modal coupling of the fixed beam and the tool holder, to obtain the frequency response function of the turning workpiece system of the slender shaft part with the tool holder, and on this basis, the turning stability analysis is also used.

Description

Technical Field

[0001] The invention relates to the technical field of turning processing, and more specifically to a method for predicting vibration stability in turning processing of a slender shaft with a tool holder. Background Art

[0002] Aviation slender shaft parts are one of the important parts in aircraft engines. They are mainly used to transmit transmission torque and support transmission parts. They are usually made of difficult-to-process materials such as titanium alloys and nickel-based alloys. Their aspect ratio is often above 20, and their rigidity is extremely poor. Their turning process is prone to processing deformation and cutting chatter, which not only leads to low processing efficiency of such parts, but also makes it difficult to ensure the dimensional accuracy of the parts. Cutting chatter refers to a self-excited vibration phenomenon that occurs during the cutting process. In mild cases, it leaves vibration marks on the surface of the parts, causing the processed parts to be out of tolerance. In severe cases, it damages the tool and workpiece, resulting in the scrapping of the processed parts.

[0003] The Chinese invention patent with publication number CN104484506B discloses a turning machining chatter prediction method based on reliability lobe diagram, including: obtaining the vibration acceleration of the turning tool under modal force hammer excitation; obtaining the displacement frequency response function of the lathe tool tip point; analyzing and obtaining the main modal parameters; obtaining the mean and standard deviation of each main modal parameter; obtaining the mean of the standard deviation of the lathe spindle speed; obtaining the mean and standard deviation of the chatter frequency, and generating a chatter stability lobe diagram; forming the chatter stability lobe diagram into a two-dimensional plane (Ω, b), calculating the chatter reliability of each node in the gridded two-dimensional plane; drawing the reliability lobe diagram of turning machining; performing turning machining chatter prediction; at different spindle speeds, machining is performed according to the cutting width below the contour line in the reliability lobe diagram. The above technical solution obtains the displacement frequency response function of the lathe tool tip point through modal force hammer excitation, and the operation is cumbersome. Summary of the invention

[0004] The purpose of the present invention is to address the deficiencies in the prior art and provide a method for predicting the vibration stability of a slender shaft turning process with a tool holder. The method is based on the analytical expression of the frequency response function at the cutting point of the clamped beam and the overall estimation algorithm of the orthogonal polynomials of the modal parameters, combined with the modal coupling of the clamped beam and the tool holder, to obtain the frequency response function of the workpiece system during the turning process of a slender shaft part with a tool holder, and on this basis, the turning process stability analysis is also used.

[0005] The purpose of the present invention is achieved through the following technical solutions:

[0006] A method for predicting chatter stability during turning of a slender shaft with a tool holder comprises the following steps:

[0007] S1. Use the vibration equation to obtain the frequency response function of the cutting point of the slender shaft without a tool holder;

[0008] S2. The modal parameters at the cutting point of the slender shaft without a tool holder are obtained by using the orthogonal polynomial modal parameter overall estimation method;

[0009] S3. Coupling the modal parameters of the slender shaft portion with the tool holder portion to obtain the modal parameters of the cutting point of the slender shaft with the tool holder;

[0010] S4. Construct a turning machining chatter stability model to predict the chatter stability lobe diagram of turning a slender shaft with a tool holder.

[0011] Furthermore, step S1 includes the following steps:

[0012] S11. Simplify the turning process of the slender shaft into a fixed beam structure with the left end clamped and the right end supported by the top;

[0013] S12. Use the vibration equation of the clamped beam to obtain the frequency response function of different cutting point positions.

[0014] Furthermore, the frequency response function at the cutting point in step S1 is:

[0015]

[0016] Where h and x are the positions of the excitation point and the response point, respectively, ωr is the rth order natural frequency of the beam, ar=Aρl, A is the cross-sectional area of ​​the beam, l is the length, and ρ is the mass density; its characteristic function is:

[0017] φ r =coshλ r x-cosλ r x-sinhλ r x+sinλ r x

[0018] Among them, λ r With ω r The relationship between can be expressed as:

[0019] λ r =[(ω r Aρ) / (EI(1+ηi))] 0.25

[0020] Where E is the elastic modulus of the workpiece, η is the damping coefficient that is independent of frequency, and I is the section moment of inertia of the beam, which can be expressed as:

[0021]

[0022] Where D is the outer diameter of the slender shaft, and d is the inner diameter of the slender shaft.

[0023] Furthermore, the specific steps of step S2 are as follows: using an orthogonal polynomial modal parameter overall estimation method to obtain the modal parameters at the cutting position of the clamped beam.

[0024] Furthermore, the frequency response function of the spring-damped inertial system with N degrees of freedom in step S2 can be expressed in the form of an orthogonal polynomial as follows:

[0025]

[0026] Among them, p k ,q k are Forsythe orthogonal polynomials for the numerator and denominator, respectively, and b k 、c k is the number to be solved, ω is the frequency variable, and j is the imaginary number symbol;

[0027] The measured value of the frequency response function is The error between the mathematical model and the measured value is:

[0028]

[0029] Among them, s is the fitting frequency point;

[0030] Using the weighted error e s , so that the error is b k 、c k Linearization:

[0031]

[0032] Written in vector form:

[0033] e=(e(ω1),e(ω2),…,e(ω L ))'=Pb-Qc-d

[0034] Where:

[0035]

[0036]

[0037]

[0038] Let E = e H e, by the least squares method:

[0039]

[0040]

[0041] Changing the above formula into matrix form, we have:

[0042]

[0043] Where: E1 = real (P H P), E3 = real (Q H Q), E2 = -real (P H Q), g1 = real (P H d) g2 = -real(Q H d) When p k ,q k When they are all Forsythe polynomials, the above formula can be rewritten as:

[0044]

[0045] Where I is the identity matrix, the above formula can be rewritten as:

[0046]

[0047]

[0048] After solving, we get the coefficient vectors b and c.

[0049] Furthermore, the Forsythe orthogonal polynomial derivation process is as follows:

[0050] p i (ω s ) = j i s i (ω s )

[0051] Where:

[0052] R0(ω s )=1

[0053] R1(ω s )=ω s

[0054]

[0055]

[0056]

[0057] R i (ω s )=ω i S i-1 (ω s )-V i-1 S i-2 (ω s )

[0058] Ri (ω s )=ω i S i-1 (ω s )-V i-1 S i-2 (ω s )

[0059] For the Forsythe polynomial in the numerator, let ρ 2 s=1, and the Forsythe polynomial in the denominator is

[0060] There is the following conversion relationship between the Forsythe orthogonal polynomial and the power polynomial:

[0061]

[0062] Where:

[0063]

[0064] A 21 =0

[0065]

[0066] A ik =(A i-1,k-1 +V i A i-2,k ) / D(i=3,4,…;k=1,2,…,i)

[0067] The corresponding numerator or denominator power polynomial coefficient γ can be expressed as:

[0068] γ=cA

[0069] In the formula, c is the original orthogonal polynomial vector coefficient, and A is the transformation matrix;

[0070] After obtaining the rational fraction whose numerator and denominator are both power polynomials, the residue and poles are obtained by calling the Residue function in Matlab software, and then the natural frequency and damping ratio corresponding to the poles are obtained by the following calculation formula;

[0071] ω n =|s r |

[0072] ζ=-real(s r / ω n )

[0073]

[0074] Among them, s r is the rth extreme point.

[0075] Further, step S3 includes the following steps:

[0076] S31. Simplify the tool holder into a spring-damped inertial system and establish a coupling model between the clamped beam and the tool holder;

[0077] S32. Derive the calculation formula of the frequency response function of the cutting point in turning of a slender shaft with a tool holder.

[0078] Furthermore, the specific process of step S32 is as follows:

[0079] The dynamic differential equation is established as follows:

[0080]

[0081] After Laplace transform, the transfer function at the tool tip is:

[0082]

[0083] The corresponding frequency response function is:

[0084]

[0085] Further, after the turning process in step S42 undergoes k complete vibration cycles, 2πf c T=2kπ+ε

[0086] Among them, ε=3π+2Ψ, and the spindle speed can be obtained from n=60 / T:

[0087]

[0088] By setting the real part of the characteristic equation to zero, the critical axial cutting depth can be obtained as:

[0089]

[0090] Compared with the prior art, the present invention has the following beneficial effects:

[0091] The present invention simplifies the turning system of the slender shaft part without a tool holder into a fixed-support beam structure, and directly uses the vibration equation of the beam to obtain the frequency response function of the cutting point position, which can avoid the cumbersome method of obtaining the frequency response function through a modal hammer experiment; an orthogonal polynomial modal parameter overall identification method is used to obtain the modal parameters of the cutting point position of the slender shaft part without a tool holder, which has fast identification speed, high accuracy and easy operation.

[0092] The present invention can conveniently obtain the frequency response function of the cutting point position of the slender shaft part with the tool holder by coupling the part without the tool holder and the two parts with the tool holder, and accurately consider the influence of the tool holder on the turning stability. BRIEF DESCRIPTION OF THE DRAWINGS

[0093] Figure 1 This is a simplified model diagram for turning of slender shaft parts without tool holder;

[0094] Figure 2 It is the flow chart of overall identification of modal parameters orthogonal polynomials;

[0095] Figure 3 Comparison diagram of the fitting curve and the measured curve;

[0096] Figure 4 Simplify the dynamic model of the turning system for slender shaft parts with a tool holder;

[0097] Figure 5 This is the diagram of the chatter mechanism in turning of slender shaft parts;

[0098] Figure 6 The simulation flow chart of turning stability of slender shaft parts with tool holder;

[0099] Figure 7 Simulate the stability of turning of slender shaft parts with tool holders. DETAILED DESCRIPTION

[0100] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0101] Example 1

[0102] This embodiment provides a method for predicting the chatter stability of a slender shaft with a tool holder during turning, which specifically includes the following steps:

[0103] S1. Use vibration theory to obtain the frequency response function of the cutting point of the slender shaft part without tool holder;

[0104] S2. The modal parameters at the cutting point of the slender shaft part without tool holder are obtained by using the orthogonal polynomial modal parameter overall estimation method;

[0105] S3. Coupling the modal parameters of the slender shaft portion and the tool holder portion to obtain the modal parameters of the cutting point of the slender shaft part with the tool holder;

[0106] S4. Construct a turning machining chatter stability model to predict the chatter stability lobe diagram of turning a slender shaft with a tool holder.

[0107] Example 2

[0108] This embodiment refers to Embodiment 1, and provides a method for predicting the chatter stability of a slender shaft turning process with a tool holder, which specifically includes the following steps:

[0109] S1. Using vibration theory to obtain the frequency response function of the cutting point of the slender shaft part without a tool holder, including the following steps:

[0110] S11. Simplify the turning process of the slender shaft into a fixed beam structure with the left end clamped and the right end supported by the top;

[0111] S12. Use the vibration equation of the clamped beam to obtain the frequency response function of different cutting point positions.

[0112] Under the chuck-center clamping condition, the turning process of slender shaft parts without tool holder can be simplified as follows: Figure 1 For the clamped beam structure shown in the figure, according to the vibration theory of beams, the frequency response function at the cutting point can be expressed as:

[0113]

[0114] Where h and x are the positions of the excitation point and the response point, respectively, and ω r is the rth natural frequency of the beam, a r =Aρl, A is the cross-sectional area of ​​the beam, l is the length of the beam, ρ is the mass density, and its characteristic function can be expressed as:

[0115] φ r =coshλ r x-cosλ r x-sinhλ r x+sinλ r x (2)

[0116] In the formula, λ r lSee Table 1;

[0117] Table 1 λ of clamped beam r List

[0118]

[0119] λ r With ω r The relationship between can be expressed as:

[0120] λ r =[(ω r Aρ) / (EI(1+ηi))] 0.25 (3)

[0121] Where E is the elastic modulus of the workpiece, η is the damping coefficient that is independent of frequency, and I is the section moment of inertia of the beam, which can be expressed as:

[0122]

[0123] Where D is the outer diameter of the slender shaft, and d is the inner diameter of the slender shaft;

[0124] S2. The modal parameters at the cutting point of the slender shaft part without tool holder are obtained by using the orthogonal polynomial modal parameter overall estimation method, as follows: The frequency response function of the spring-damped inertial system with N degrees of freedom can be expressed in the form of orthogonal polynomials:

[0125]

[0126] In the formula, p k ,q k are Forsythe orthogonal polynomials for the numerator and denominator, respectively, and b k 、c k is the number to be solved, ω is the frequency variable, and j is the imaginary number symbol. Assume that the measured value of the frequency response function is The error between the mathematical model and the measured value is:

[0127]

[0128] Where s is the fitting frequency point.

[0129] Using the weighted error e s , so that the error is b k 、c k Linearization:

[0130]

[0131] Written in vector form:

[0132] e=(e(ω1),e(ω2),…,e(ω L ))'=Pb-Qc-d (8)

[0133] Where:

[0134]

[0135]

[0136]

[0137] Let E = e H e, by the least squares method:

[0138]

[0139] Changing the above formula into matrix form, we have:

[0140]

[0141] Where: E1 = real (P H P), E3 = real (Q H Q), E2 = -real (P H Q), g1 = real (P H d) g2 = -real(Q H d) When p k ,q k When they are all Forsythe polynomials, the above formula can be rewritten as:

[0142]

[0143] Where I is the identity matrix, the above formula can be rewritten as:

[0144]

[0145] Solving the above equations, we can obtain the coefficient vectors b and c.

[0146] The derivation process of Forsythe orthogonal polynomial is as follows:

[0147] p i (ω s ) = j i s i (ω s ) (13)

[0148] Where:

[0149] R0(ω s )=1

[0150] R1(ω s )=ω s

[0151]

[0152]

[0153]

[0154] R i (ω s )=ω i S i-1 (ω s )-V i-1 S i-2 (ω s )

[0155] R i (ω s )=ω i Si-1 (ω s )-V i-1 S i-2 (ω s )

[0156] For the Forsythe polynomial in the numerator, let ρ 2 s=1, and the Forsythe polynomial in the denominator is

[0157] There is the following conversion relationship between Forsythe orthogonal polynomials and power polynomials:

[0158]

[0159] Where:

[0160]

[0161] A 21 =0

[0162]

[0163] A ik =(A i-1,k-1 +V i A i-2,k ) / D(i=3,4,…;k=1,2,…,i)

[0164] The corresponding numerator or denominator power polynomial coefficient γ can be expressed as:

[0165] γ=cA (15)

[0166] Where c is the original orthogonal polynomial vector coefficient and A is the transformation matrix.

[0167] After obtaining the rational fraction whose numerator and denominator are both power polynomials, the residue and poles are obtained by calling the Residue function in Matlab software, and then the natural frequency and damping ratio corresponding to the poles are obtained by the following calculation formula.

[0168]

[0169] Among them, s r is the rth extreme point.

[0170] Based on the above theoretical analysis, Figure 2 The flowchart shown in the figure can be used to realize the overall identification of modal parameter orthogonal polynomials using MATLAB software. Figure 3 shown.

[0171] S3. Coupling the modal parameters of the slender shaft part and the tool holder part to obtain the modal parameters at the cutting point of the slender shaft part with the tool holder, including the following steps:

[0172] S31. The dynamic model of the workpiece system for turning a slender shaft part with a tool holder can be simplified as follows: Figure 4 As shown in the figure, m, c, k are mass, damping and stiffness respectively, and the subscripts 1 and 2 represent the workpiece and the tool rest respectively.

[0173] S32.According to Figure 4 , the dynamic differential equation can be established as follows:

[0174]

[0175] After Laplace transform, the transfer function at the tool tip is:

[0176]

[0177] The corresponding frequency response function is:

[0178]

[0179] S4. Construct a turning vibration stability model to predict the vibration stability lobe diagram of turning a slender shaft with a tool holder, including the following steps:

[0180] S41. When turning slender shaft parts, they are only weakly rigid in the direction perpendicular to the axis. The vibration generation mechanism is as follows: Figure 5 As shown, its dynamic differential equation can be expressed as:

[0181]

[0182] In the formula, m is the equivalent mass, c is the equivalent damping, k is the equivalent stiffness, K f is the cutting force coefficient in the feed direction, a p The amount of knife damage on the back.

[0183] S42. Perform Laplace transformation on the above formula to obtain the dynamic cutting thickness:

[0184] h(s)=h0+(e -st -1)K f a p h(s)Φ(s) (21)

[0185] The ratio of dynamic and static cutting thickness is:

[0186]

[0187] The stability of the closed-loop transfer function depends on the roots of its characteristic equation, namely:

[0188] 1-(e -sT -1)K f a p Φ(s)=0 (23)

[0189] Let the root of the characteristic equation be s = σ + iω c , only when σ=0(s=iω c ), the system is in a critical stable state, and the workpiece vibrates at a frequency ω c and constant amplitude oscillation. At this point, the characteristic equation of the system becomes:

[0190] 1+(1-e -ωiT )K f a plim Φ(iω c )=0 (24)

[0191] In the formula, a plim is the maximum axial cutting depth without chatter. The frequency response function is decomposed into the form of real and imaginary parts, that is, Ф(jω c )=R+iI, and applying Euler's formula, the characteristic equation becomes:

[0192] 1+K f a plim [R(1-cosω c T)-Isinω c T]+iK f a plim [I(1-cosω c T)+Rsinω c T]=0

[0193] Let the imaginary part be zero, we have:

[0194]

[0195] Where Ψ is the phase shift of the transfer function, which can be further transformed into:

[0196]

[0197] During turning, when the spindle speed is n and the chatter frequency is f c (Hz), the number of vibration lines generated by one rotation of the spindle is:

[0198]

[0199] Where k is the integer part of the vibration fringes, ε is the phase difference between the inner and outer vibration fringes, and ε / 2π is the fractional part of the vibration fringes.

[0200] Example 3

[0201] This embodiment refers to Embodiment 2 and provides a method for predicting the chatter stability of a slender shaft with a tool holder during turning. The method is different from Embodiment 2 in that it further includes the following steps:

[0202] The turning process undergoes k complete vibration cycles, then:

[0203] 2πf c T=2kπ+ε (28) In the formula, ε=3π+2Ψ, and the spindle speed can be obtained from n=60 / T:

[0204]

[0205] By setting the real part of the characteristic equation to zero, the critical axial cutting depth can be obtained as:

[0206]

[0207] Based on the above theoretical foundation, according to Figure 6 The simulation flow chart shown in the figure is used to develop Figure 7 The simulation interface shown can realize the prediction of the stability lobe diagram for turning parts with slender shafts with tool holders, and the simulation results can be saved in the form of graphics and data tables.

[0208] Based on the analytical expression of the frequency response function at the cutting point of the fixed beam and the overall estimation algorithm of the orthogonal polynomial of the modal parameters, the present application combines the modal coupling of the fixed beam and the tool holder to obtain the frequency response function of the turning workpiece system of the slender shaft part with the tool holder. On this basis, with the help of the turning stability analysis, the rapid calculation of the stability lobe diagram of the turning of the slender shaft part with the tool holder is realized.

[0209] It should be noted that if the embodiments of the present invention involve directional indications (such as up, down, left, right, front, back, etc.), the directional indications are only used to explain the relative position relationship, movement status, etc. between the components under a certain specific posture (as shown in the accompanying drawings). If the specific posture changes, the directional indication will also change accordingly.

[0210] In addition, if there are descriptions involving "first", "second", etc. in the embodiments of the present invention, the descriptions of "first", "second", etc. are only used for descriptive purposes and cannot be understood as indicating or suggesting their relative importance or implicitly indicating the number of the indicated technical features. Therefore, the features defined as "first" and "second" may explicitly or implicitly include at least one of the features. In addition, the technical solutions between the various embodiments can be combined with each other, but they must be based on the ability of ordinary technicians in the field to implement them. When the combination of technical solutions is contradictory or cannot be implemented, it should be deemed that such a combination of technical solutions does not exist and is not within the scope of protection required by the present invention.

Claims

1. A method for predicting chatter stability in turning of slender shafts with toolholders. It is characterized in that The following steps are involved: S1. Use the vibration equation to obtain the frequency response function of the cutting point of the slender shaft without a tool holder; S11. Simplify the turning process of the slender shaft into a fixed beam structure with the left end clamped and the right end supported by the top; S12. Use the vibration equation of the clamped beam to find the frequency response function at different cutting points. The frequency response function at the cutting point is: Where h and x are the positions of the excitation point and the response point, respectively, and ω r is the rth order natural frequency of the beam, m r =Aρl, A is the cross-sectional area of ​​the beam, l is the length of the beam, and ρ is the mass density; its characteristic function is: f r =coshλ r x-cosλ r x-sinhλ r x+sinλ r x Among them, λ r With ω r The relationship between can be expressed as: l r =[(ω r (Ar) / (EI(1+ηi))] 0.25 Where E is the elastic modulus of the workpiece, η is the damping coefficient that is independent of frequency, and I is the section moment of inertia of the beam, which can be expressed as: Where D is the outer diameter of the slender shaft, and d is the inner diameter of the slender shaft; S2. The modal parameters at the cutting point of the slender shaft without a tool holder are obtained by using the orthogonal polynomial modal parameter overall estimation method; The frequency response function of the spring-damper inertial system with N degrees of freedom can be expressed in the form of orthogonal polynomials as follows: Among them, p k ,q k are Forsythe orthogonal polynomials for the numerator and denominator, respectively, and b k 、c k is the number to be solved, ω is the frequency variable, and j is the imaginary number symbol; The measured value of the frequency response function is The error between the mathematical model and the measured value is: Among them, s is the fitting frequency point; Using the weighted error e s , so that the error is b k 、c k Linearization: Written in vector form: e=(e(ω 1 ),e(ω 2 ),…,e(ω L ))'=Pb-Qc-d Where: Let E = e H e, by the least squares method: Changing the above formula into matrix form, we have: Where: E 1 =real(P H P) and E 3 =real(Q H Q) and E 2 =-real(P H Q), g 1 =real(P H d) g 2 =-real(Q H d); when p k ,q k When they are all Forsythe polynomials, the above formula can be rewritten as: Where I is the identity matrix, the above formula can be rewritten as: After solving, we get coefficient vectors b and c; S3. Coupling the modal parameters of the slender shaft portion with the tool holder portion to obtain the modal parameters of the cutting point of the slender shaft with the tool holder; S31. Simplify the tool holder into a spring-damped inertial system and establish a coupling model between the clamped beam and the tool holder; S32. The calculation formula of the frequency response function of the cutting point in turning of a slender shaft with a tool holder is derived and expressed as: S4. Construct a turning vibration stability model to predict the vibration stability lobe diagram of turning a slender shaft with a tool holder; S41. When turning slender shaft parts, they are only weakly rigid in the direction perpendicular to the axis, and their dynamic differential equation can be expressed as: In the formula, m is the equivalent mass, c is the equivalent damping, k is the equivalent stiffness, K f is the cutting force coefficient in the feed direction, a p To eat knife measure for the back; S42. Perform Laplace transformation on the above formula to obtain the dynamic cutting thickness: h(s)=h 0 +(e -st -1)K f a p h(s)Φ(s) The ratio of dynamic and static cutting thickness is: The stability of the closed-loop transfer function depends on the roots of its characteristic equation, namely: 1-(e -sT -1)K f a p Φ(s)=0 Let the root of the characteristic equation be s = σ + iω c , only when σ=0, s=iω c , the system is in a critical stable state, and the workpiece vibrates at a frequency ω c and constant amplitude oscillation; at this time, the characteristic equation of the system becomes: 1+(1-e -ωiT )K f a plim Φ(iω c )=0 In the formula, a plim is the maximum axial cutting depth without chatter, and the frequency response function is decomposed into the form of real and imaginary parts, that is, Ф(jω c )=R+iI, and applying Euler's formula, the characteristic equation becomes: 1+K f a plim [R(1-cosω c Τ)-Isinω c T]+iK f a plim [I(1-cosω c T)+Rsinω c T]=0 Let the imaginary part be zero, we have: Where Ψ is the phase shift of the transfer function, which can be further transformed into: During turning, when the spindle speed is n and the chatter frequency is f c (Hz), the number of vibration lines generated by one rotation of the spindle is: Where k is the integer part of the vibration fringes, ε is the phase difference between the inner and outer vibration fringes, and ε / 2π is the fractional part of the vibration fringes.

2. The method for predicting the chatter stability of a slender shaft turning process with a tool holder according to claim 1, It is characterized in that The Forsythe orthogonal polynomial derivation process is as follows: p i (oh s )=j i s i (oh s ) Where: R 0 (oh s )=1 R 1 (oh s )=ω s S i (oh s )=R i (oh s ) / D i 1 / 2 R i (oh s )=ω i S i-1 (oh s )-V i-1 S i-2 (oh s ) R i (oh s )=ω i S i-1 (oh s )-V i-1 S i-2 (oh s ) For the Forsythe polynomial in the numerator, let The Forsythe polynomial on the denominator is There is the following conversion relationship between the Forsythe orthogonal polynomial and the power polynomial: Where: A 21 ＝0 And ik =(A i-1,k-1 +V i And i-2,k ) / D,i=3,4,...;k=1,2,...,i The corresponding numerator or denominator power polynomial coefficient γ can be expressed as: γ=cA In the formula, c is the original orthogonal polynomial vector coefficient, and A is the transformation matrix; After obtaining the rational fraction whose numerator and denominator are both power polynomials, the residue and poles are obtained by calling the Residue function in Matlab software, and then the natural frequency and damping ratio corresponding to the poles are obtained by the following calculation formula; oh n =|s r | ζ=-real(s r / h n ) Among them, s r is the rth extreme point.

3. The method for predicting the chatter stability of a slender shaft turning process with a tool holder according to claim 1, It is characterized in that After the turning process in step S42 has undergone k complete vibration cycles, 2πf c T=2kπ+ε Among them, ε=3π+2Ψ, and the spindle speed can be obtained from n=60 / T: By setting the real part of the characteristic equation to zero, the critical axial cutting depth can be obtained as:

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