CN103823787A - Multi-turning-tool parallel turning stability judgment method based on differential quadrature method - Google Patents

Abstract

The invention provides a multi-turning-tool parallel turning stability judgment method based on a differential quadrature method. The multi-turning-tool parallel turning stability judgment method comprises the steps that dynamics modeling is carried out on a multi-turning-tool parallel turning system, and a multi-delay second order differential equation is built; a normalized state space equation is built and obtained; the second kind of Chebyshev points is adopted as discrete points in adjacent unit intervals (0, 1) and (-1, 0); speed items are expressed by displacement items at the discrete points based on a Lagrange's interpolation function by using the differential quadrature method; the intervals where the discrete points of delay items are located are judged, and the delay items are expressed by the second kind of Chebyshev points of the intervals where the discrete points of the delay items are located; state transfer matrixes between the adjacent unit intervals are constructed, and the stability of an original system is judged based on the Floquet theory. Compared with traditional single-turning-tool turning, the dynamic characteristic of the multi-turning-tool parallel turning system is analyzed by adopting the differential quadrature method, optimized cutting parameters are obtained, and machining efficiency is greatly improved.

Description

The parallel turning stability decision method of many lathe tools based on differential quadrature method

Technical field

The present invention relates to a kind of new method that the parallel turning stability of many lathe tools is judged, be differential quadrature method (Differential Quadrature Method), be specially and utilize differential quadrature method choose reasonable cutting parameter to make the parallel turning of many lathe tools avoid the impact of Regenerative Chatter to carry out the processing of high-level efficiency high-quality.

Background technology

In field of machining, turning is one of the most common and conventional processing mode.Traditional bicycle cutter working research is very ripe, and many processing modes parallel lathe tool turning are the new ideas that academia just proposes in recent years, and the walk abreast research of turning stability of especially many lathe tools, still in the starting stage.In theory, the working (machining) efficiency of the parallel turning of many lathe tools will be far away higher than traditional turning, but due to the processing mechanism of the parallel turning of many lathe tools than traditional turning complexity many.The principal element that affects the parallel turning crudy of many lathe tools is processing stability, therefore study the kinetics mechanism of the parallel turning of many lathe tools, choose reasonable machined parameters, can effectively avoid the generation of Regenerative Chatter, thereby realize the even running of process, in guaranteeing crudy, realize the efficient turning of workpiece.

Document 1 " E.Budak; E.Ozturk; Dynamics and stability of parallel turning operations.CIRP Annals-Manufacturing Technology60 (2011) 383-386. " utilizes frequency method to carry out stability analysis to parallel turning, and in addition experimental verification.Many lathe tools in document are fixed on different knife rests, do not have coupling effect between cutter.Its step is as follows:

(1) set up the kinetic model of the parallel turning of many lathe tools;

(2) obtain stability boundaris place kinetics equation;

(3) utilize search procedure to obtain the parallel stabilized zone that is machined in cutting parameter space of many lathe tools;

(4) carry out time-domain-simulation checking and cutting experiment checking.

Document 2 " E.Ozturk, E.Budak, Modeling dynamics of parallel turning operations.Proceedings of4 ^thcIRP International Conference on High Performance Cutting, 2010. " situation of the parallel turning of lathe tool more than two kinds has been discussed: many lathe tool is installed on different knife rests; Many lathe tool is installed on same knife rest.The analysis calculation method of both of these case is similar, identical with the step of document 1.

Frequency domain method step is succinct, and computing velocity is fast, but is unfavorable for considering various complex working conditions.Time-domain-simulation method visual result, but computing velocity is slow, can only obtain the processing stability under single cutting parameter combination condition, is unfavorable for drawing the stability collection of illustrative plates in machined parameters space.Compared with said method, the differential quadrature method that Bellman etc. proposed in nineteen seventies has that computing velocity is fast, result precision advantages of higher, has been widely used in each field of engineering technology.The kinetics equation of the parallel turning of many lathe tools is Differential Equations with Delays, and differential quadrature method is done to suitable popularization and judged to have important productive value and meaning for the processing stability of the parallel turning of many lathe tools.

Summary of the invention

For defect of the prior art, the object of the invention is the dynamics according to the parallel turning of many lathe tools, a kind of new method of stable cutting process sex determination is provided, it is differential quadrature method, for the selection of the parallel turning machined parameters of many lathe tools provides effective foundation, under guaranteeing without the prerequisite of Regenerative Chatter high-quality processing, obtain the high working (machining) efficiency of trying one's best, produce good economic benefit.

The invention provides the parallel turning stability decision method of a kind of many lathe tools based on differential quadrature method, comprise the steps:

The parallel turning system of processing of many lathe tools is carried out to Dynamic Modeling, set up multiple time delay second order differential equation.Carry out state space variation, set up state space equation, be normalized afterwards, obtain the standard equation after normalization;

Upper take the second kind of Chebyshev points (Chebyshev-Gauss-Lobatto Points) as discrete point in adjacent unit interval [0,1] and [1,0], by of equal value standard equation discrete be one group of algebraic equation;

Utilize differential quadrature method, based on Lagrange interpolation function, represent speed term with the displacement item at discrete point place; Judge time lag item discrete point interval of living in, based on Lagrange interpolation function, represent time lag item with the second kind of Chebyshev points between location;

Construct the state-transition matrix between adjacent two unit intervals [0,1] and [1,0], according to the theoretical stability of judging original system of Floquet; If the mould of all eigenwerts of state-transition matrix is all less than 1, system is stable; If the mould of arbitrary eigenwert of state-transition matrix is greater than 1, system is unsettled; Can obtain thus the stability spectrogram of the parallel turning system of many lathe tools in cutting parameter space.

Particularly, according to the parallel turning stability decision method of the many lathe tools based on differential quadrature method provided by the invention, comprise the steps:

Step 1: set up the parallel turning kinetics equation of many lathe tools, the parallel turning kinetics equation of many lathe tools is arranged, obtain multiple time delay second order differential equation;

Step 2: described multiple time delay second order differential equation is carried out to state space conversion, obtain state space equation;

Step 3: described state space equation is normalized, obtains the state space equation of canonical form;

Step 4: the state space equation of canonical form is carried out to period discrete, its equivalence is converted into one group of algebraic equation as state space equation expression formula;

Step 5: based on Lagrange interpolation function, for discrete point, utilize differential quadrature method with the second kind of Chebyshev points on unit interval [0,1], represent the derivative term in state space equation expression formula with displacement item;

Step 6: based on Lagrange interpolation function, the time lag item in state space equation expression formula being carried out to place interval judgement, if the discrete point of time lag item belongs to interval [0,1], is that discrete point represents time lag item with the second kind of Chebyshev points on [0,1]; If the discrete point of time lag item belongs to interval [1,0], is that discrete point represents time lag item with the second kind of Chebyshev points on [1,0];

Step 7: the state-transition matrix between structure adjacent cells interval [0,1] and [1,0], according to the theoretical stability of judging original system of Floquet.

Preferably, in described step 1, described many lathe tools turning kinetics equation that walks abreast, as shown in Equation (1):

$\begin{array}{c}\left[\begin{array}{c}{\stackrel{\&CenterDot;\&CenterDot;}{z}}_1\left(t\right)\\ {\stackrel{\&CenterDot;\&CenterDot;}{z}}_2\left(t\right)\end{array}\right]+\left[\begin{array}{cc}2{\mathrm{\&zeta;}}_1{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="HDA0000468378720000012.png"\; state="\{\{state\}\}">n</figure-callout>}1}& 0\\ 0& 2{\mathrm{\&zeta;}}_2{\mathrm{\&omega;}}_{n2}\end{array}\right]\left[\begin{array}{c}{\stackrel{\&CenterDot;}{z}}_1\left(t\right)\\ {\stackrel{\&CenterDot;}{z}}_2\left(t\right)\end{array}\right]+\left[\begin{array}{cc}{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="HDA0000468378720000012.png"\; state="\{\{state\}\}">n</figure-callout>}1}^2& 0\\ 0& {\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA0000468378720000012.png"\; state="\{\{state\}\}">n</figure-callout>}2}^2\end{array}\right]\left[\begin{array}{c}{z}_1\left(t\right)\\ z_2\left(t\right)\end{array}\right]=\\ K_f\left[\begin{array}{c}\frac{{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="HDA0000468378720000012.png"\; state="\{\{state\}\}">n</figure-callout>}1}^2}{k_1}{a}_1(-z_1\left(t\right)+z_2(t-\frac{\mathrm{\&tau;}}{2}))\\ \frac{{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA0000468378720000012.png"\; state="\{\{state\}\}">n</figure-callout>}2}^2}{k_2}{a}_1(-z_2\left(t\right)+z_1(t-\frac{\mathrm{\&tau;}}{2}))+\frac{{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA0000468378720000012.png"\; state="\{\{state\}\}">n</figure-callout>}2}^2}{k_2}(a_2-a_1)(-z_2\left(t\right)+z_2(t-\mathrm{\&tau;}))\end{array}\right]\end{array}---\left(1\right)$

Wherein, subscript 1 represents lathe tool A, and subscript 2 represents lathe tool B, and z (t) represents dynamic response displacement, ω _nrepresent inherent circular frequency, K _fthe Cutting Force Coefficient that represents direction of feed, τ represents workpiece swing circle, and ζ represents relative damping ratio, and k represents stiffness coefficient, a ₁represent the cutting depth of lathe tool A, a ₂represent the cutting depth of lathe tool B, represent dynamics acceleration responsive,

represent dynamics speed responsive, t represents the time;

Described multiple time delay second order differential equation, as shown in Equation (2):

$\begin{array}{c}\left[\begin{array}{c}{\stackrel{\&CenterDot;\&CenterDot;}{z}}_1\left(t\right)\\ {\stackrel{\&CenterDot;\&CenterDot;}{z}}_2\left(t\right)\end{array}\right]+\left[\begin{array}{cc}2{\mathrm{\&zeta;}}_1{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="HDA0000468378720000012.png"\; state="\{\{state\}\}">n</figure-callout>}1}& 0\\ 0& 2{\mathrm{\&zeta;}}_2{\mathrm{\&omega;}}_{n2}\end{array}\right]\left[\begin{array}{c}{\stackrel{\&CenterDot;}{z}}_1\left(t\right)\\ {\stackrel{\&CenterDot;}{z}}_2\left(t\right)\end{array}\right]+\left[\begin{array}{cc}{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="HDA0000468378720000012.png"\; state="\{\{state\}\}">n</figure-callout>}1}^2+K_f\frac{{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="HDA0000468378720000012.png"\; state="\{\{state\}\}">n</figure-callout>}1}^2}{k_1}{a}_1& 0\\ 0& {\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA0000468378720000012.png"\; state="\{\{state\}\}">n</figure-callout>}2}^2+K_f\frac{{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA0000468378720000012.png"\; state="\{\{state\}\}">n</figure-callout>}2}^2}{k_2}{a}_2\end{array}\right]\left[\begin{array}{c}{z}_1\left(t\right)\\ z_2\left(t\right)\end{array}\right]=\\ \left[\begin{array}{cc}0& K_f\frac{{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="HDA0000468378720000012.png"\; state="\{\{state\}\}">n</figure-callout>}1}^2}{k_1}{a}_1\\ K_f\frac{{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA0000468378720000012.png"\; state="\{\{state\}\}">n</figure-callout>}2}^2}{k_2}{a}_1& 0\end{array}\right]\left[\begin{array}{c}{z}_1(t-\frac{\mathrm{\&tau;}}{2})\\ z_2(t-\frac{\mathrm{\&tau;}}{2})\end{array}\right]+\left[\begin{array}{cc}0& 0\\ 0& K_f\frac{{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA0000468378720000012.png"\; state="\{\{state\}\}">n</figure-callout>}2}^2}{k_2}(a_2-a_1)\end{array}\right]\left[\begin{array}{c}{z}_1(t-\mathrm{\&tau;})\\ z_2(t-\mathrm{\&tau;})\end{array}\right]\end{array}---\left(2\right).$

Preferably, in step 2, described state space equation, as shown in Equation (3):

Order $Z\left(t\right)=\left[\begin{array}{c}{z}_1\left(t\right)\\ z_2\left(t\right)\\ {\stackrel{\&CenterDot;}{z}}_1\left(t\right)\\ {\stackrel{\&CenterDot;}{z}}_2\left(t\right)\end{array}\right],$ Formula (3) becomes:

$\stackrel{\&CenterDot;}{Z}\left(t\right)=\mathrm{AZ}\left(t\right)+B_{1}Z(t-\frac{\mathrm{\&tau;}}{2})+B_{2}Z(t-\mathrm{\&tau;})---\left(4\right)$

Wherein, expression state speed, A represents state displacement item matrix of coefficients, B ₁represent time lag state displacement item matrix of coefficients, B ₂represent time lag state displacement item matrix of coefficients.

Preferably, described step 3, is specially:

Make t=ξ τ, $\frac{\mathrm{dZ}\left(t\right)}{\mathrm{dt}}=\frac{\mathrm{dZ}\left(\mathrm{\&xi;\&tau;}\right)}{\mathrm{d\&xi;}}\&CenterDot;\frac{\mathrm{d\&xi;}}{\mathrm{dt}}=\frac{1}{\mathrm{\&tau;}}\frac{\mathrm{dZ}\left(\mathrm{\&xi;\&tau;}\right)}{\mathrm{d\&xi;}};$

State space equation after normalization becomes canonical form, as shown in Equation (5):

$\stackrel{\&CenterDot;}{Z}\left(\mathrm{\&xi;}\right)=\mathrm{\&tau;AZ}\left(\mathrm{\&xi;}\right)+\mathrm{\&tau;}{B}_{1}Z(\mathrm{\&xi;}-0.5)+\mathrm{\&tau;}{B}_{2}Z(\mathrm{\&xi;}-1)---\left(5\right).$

Preferably, described step 4, is specially:

In interval, get n+1 Equations of The Second Kind Chebyshev discrete point ξ _i, as shown in Equation (6):

${\mathrm{\&xi;}}_i=\frac{1}{2}[1-\cos \left(\frac{\mathrm{i\&pi;}}{n}\right)],i=0,\&CenterDot;\&CenterDot;\&CenterDot;,n---\left(6\right)$

On each discrete point, need to meet equation (5), that is:

$\stackrel{\&CenterDot;}{Z}\left({\mathrm{\&xi;}}_i\right)=\mathrm{\&tau;AZ}\left({\mathrm{\&xi;}}_i\right)+\mathrm{\&tau;}{B}_{1}Z({\mathrm{\&xi;}}_i-0.5)+\mathrm{\&tau;}{B}_{2}Z({\mathrm{\&xi;}}_i-1)---\left(7\right)$

On [0,1] interval,

$\left[\begin{array}{c}\stackrel{\&CenterDot;}{Z}\left({\mathrm{\&xi;}}_0\right)\\ \stackrel{\&CenterDot;}{Z}\left({\mathrm{\&xi;}}_1\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ \stackrel{\&CenterDot;}{Z}\left({\mathrm{\&xi;}}_n\right)\end{array}\right]=\mathrm{\&tau;}\&CenterDot;I_{n+1}\&CircleTimes;A\left[\begin{array}{c}Z\left({\mathrm{\&xi;}}_0\right)\\ Z\left({\mathrm{\&xi;}}_1\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ Z\left({\mathrm{\&xi;}}_n\right)\end{array}\right]+\mathrm{\&tau;}\&CenterDot;I_{n+1}\&CircleTimes;B_1\left[\begin{array}{c}Z({\mathrm{\&xi;}}_0-0.5)\\ Z({\mathrm{\&xi;}}_1-0.5)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ Z({\mathrm{\&xi;}}_n-0.5)\end{array}\right]+\mathrm{\&tau;}\&CenterDot;I_{n+1}\&CircleTimes;B_2\left[\begin{array}{c}Z({\mathrm{\&xi;}}_0-1)\\ Z({\mathrm{\&xi;}}_1-1)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ Z({\mathrm{\&xi;}}_n-1)\end{array}\right]---\left(8\right)$

Wherein,

represent Kronecker product,

i _n+1represent (n+1) dimension square formation.

Preferably, described step 5, is specially:

Utilize differential quadrature method, by the speed term of displacement item representation formula (8) left end; In order to express easily, note

for

f (ξ _i) expression mode bit transposition scalar statement symbol;

First use (n+1) individual point (ξ ₀, f (ξ ₀)), (ξ ₁, f (ξ ₁)) ..., (ξ _n, f (ξ _n)) carry out Lagrange's interpolation, result as shown in Equation (9):

$f\left(\mathrm{\&xi;}\right)=\underset{j=0}{\overset{n}{\mathrm{\&Sigma;}}}{l}_j\left(\mathrm{\&xi;}\right)f\left({\mathrm{\&xi;}}_j\right)---\left(9\right)$

Wherein, Interpolation-Radix-Function is:

$l_i\left(\mathrm{\&xi;}\right)=\underset{\underset{k\&NotEqual;i}{k=0}}{\overset{n}{\mathrm{\&Pi;}}}\frac{(\mathrm{\&xi;}-{\mathrm{\&xi;}}_k)}{({\mathrm{\&xi;}}_i-{\mathrm{\&xi;}}_k)}---\left(10\right)$

To Lagrange interpolation function differentiate, then by the each time point substitution of formula (8) left end speed term, result as shown in Equation (11):

Wherein, the expression formula of H is as shown in Equation (12):

$H_{\mathrm{ij}}=\frac{{\mathrm{dl}}_j\left(\mathrm{\&xi;}\right)}{\mathrm{d\&xi;}}{|}_{\mathrm{\&xi;}={\mathrm{\&xi;}}_i}=\left\{\begin{array}{c}\frac{\underset{\underset{k\&NotEqual;i,j}{k=0}}{\overset{n}{\mathrm{\&Pi;}}}({\mathrm{\&xi;}}_i-{\mathrm{\&xi;}}_k)}{\underset{\underset{k\&NotEqual;j}{k=0}}{\overset{n}{\mathrm{\&Pi;}}}({\mathrm{\&xi;}}_j-{\mathrm{\&xi;}}_k)},j\&NotEqual;i\\ \underset{\underset{k\&NotEqual;i}{k=0}}{\overset{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="HDA0000468378720000012.png"\; state="\{\{state\}\}">n</figure-callout>}}{\mathrm{\&Sigma;}}}\frac{1}{{\mathrm{\&xi;}}_j-{\mathrm{\&xi;}}_k},j=i\end{array}\right.---\left(12\right).$

Preferably, described step 6, is specially:

Time lag displacement item Z (ξ-0.5) represents with the second kind of Chebyshev points on [1,0] and [0,1] interval; Find i to make ξ _i-0.5 < 0 and ξ _i+1-0.5 > 0 (i=0,1 ..., n); Be that interpolation point obtains Lagrange interpolation function with the second kind of Chebyshev points on [1,0] and [0,1] interval, then by the time point substitution of time lag displacement item, obtain:

Wherein,

$T_{\mathrm{ij}}^d=\left\{\begin{array}{c}\frac{\underset{\underset{k\&NotEqual;j}{k=0}}{\overset{n}{\mathrm{\&Pi;}}}[({\mathrm{\&xi;}}_i-0.5)-({\mathrm{\&xi;}}_k-1)]}{\underset{\underset{k\&NotEqual;j}{k=0}}{\overset{n}{\mathrm{\&Pi;}}}[({\mathrm{\&xi;}}_j-1)-({\mathrm{\&xi;}}_k-1)]},({\mathrm{\&xi;}}_i-0.5)\&NotElement;\{{\mathrm{\&xi;}}_k-1|k=\mathrm{0,1},\&CenterDot;\&CenterDot;\&CenterDot;,n\}\\ 0,({\mathrm{\&xi;}}_i-0.5)\&Element;\{{\mathrm{\&xi;}}_k-1|k=\mathrm{0,1},\&CenterDot;\&CenterDot;\&CenterDot;,j-1,j+1,\&CenterDot;\&CenterDot;\&CenterDot;,n\}\\ 1,{\mathrm{\&xi;}}_i-0.5={\mathrm{\&xi;}}_j-1\end{array}\right.---\left(14\right)$

Wherein,

$T_{\mathrm{ij}}^p=\left\{\begin{array}{c}\frac{\underset{\underset{k\&NotEqual;j}{k=0}}{\overset{n}{\mathrm{\&Pi;}}}\left[({\mathrm{\&xi;}}_i-0.5){-\mathrm{\&xi;}}_k\right]}{\underset{\underset{k\&NotEqual;j}{k=0}}{\overset{n}{\mathrm{\&Pi;}}}({\mathrm{\&xi;}}_j-{\mathrm{\&xi;}}_k)},({\mathrm{\&xi;}}_i-0.5)\&NotElement;\{{\mathrm{\&xi;}}_k-1|k=\mathrm{0,1},\&CenterDot;\&CenterDot;\&CenterDot;,n\}\\ 0,({\mathrm{\&xi;}}_i-0.5_1)\&Element;\left\{{\mathrm{\&xi;}}_k\right|k=\mathrm{0,1},\&CenterDot;\&CenterDot;\&CenterDot;,j-1,j+1,\&CenterDot;\&CenterDot;\&CenterDot;,n\}\\ 1,{\mathrm{\&xi;}}_i-0.5={\mathrm{\&xi;}}_j\end{array}\right.---\left(16\right)$

Discrete on interval [1,0] of time lag item Z (ξ-1), as shown in Equation (17):

Can obtain T=I _n+1,

Wherein, T represents

Preferably, described step 7, is specially:

Structure Floquet transfer matrix:

Matrix H, T ^d, T ^p, T disappears after the first row and is designated as respectively

Wherein, T ^drepresent

t ^prepresent

Comprehensive above various, can show that the state transitions pass between adjacent two intervals [1,0] and [0,1] is:

$P\left[\begin{array}{c}Z\left({\mathrm{\&xi;}}_0\right)\\ Z\left({\mathrm{\&xi;}}_1\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ Z\left({\mathrm{\&xi;}}_n\right)\end{array}\right]=Q\left[\begin{array}{c}Z({\mathrm{\&xi;}}_0-1)\\ Z({\mathrm{\&xi;}}_1-1)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ Z({\mathrm{\&xi;}}_n-1)\end{array}\right]---\left(18\right)$

Wherein,

Floquet state-transition matrix Φ is as shown in formula:

Wherein, represent Penrose-Moor generalized inverse;

According to Floquet theory, if the mould of all eigenwerts of Φ is all less than 1, system is stable; If the mould of arbitrary eigenwert is greater than 1 in Φ, system is unsettled.

Preferably, also comprise the steps:

Step 8: and draw out the stability collection of illustrative plates of system in Delay Parameters space.

Compared with prior art, the present invention has following beneficial effect:

The present invention, compared with traditional bicycle cutter turning processing, adopts differential quadrature method to analyze the parallel turning system dynamics of many lathe tools, obtains the cutting parameter after optimizing, and has greatly improved working (machining) efficiency.

Accompanying drawing explanation

By reading the detailed description of non-limiting example being done with reference to the following drawings, it is more obvious that other features, objects and advantages of the present invention will become:

Fig. 1 is the parallel turning system schematic diagram of many lathe tools, and two lathe tools are arranged on different knife rests.

Fig. 2 is the parallel lathe tool cutting depth parameter space (a that are machined in of many lathe tools ₁, a ₂) stability collection of illustrative plates, wherein black part is divided expression unstable region.

In Fig. 1, each as follows for the represented implication of formula:

A represents lathe tool A, and B represents lathe tool B, C _wrepresent the ratio of damping of workpiece, K _wrepresent the stiffness coefficient of workpiece, C ₁represent the ratio of damping of lathe tool A, K ₁represent the stiffness coefficient of lathe tool A, C ₂represent the ratio of damping of lathe tool B, K ₂represent the stiffness coefficient of lathe tool B, a ₁represent the cutting depth of lathe tool A, a ₂represent the cutting depth of lathe tool B, z ₁represent the dynamic response displacement of lathe tool A, z ₂represent the dynamic response displacement of lathe tool B, Z represents lathe tool flexibility direction.

In Fig. 2, a ₁represent the cutting depth of lathe tool A, a ₂represent the cutting depth of lathe tool B.

Embodiment

Below in conjunction with specific embodiment, the present invention is described in detail.Following examples will contribute to those skilled in the art further to understand the present invention, but not limit in any form the present invention.It should be pointed out that to those skilled in the art, without departing from the inventive concept of the premise, can also make some distortion and improvement.These all belong to protection scope of the present invention.

Many lathe tools based on differential quadrature method provided by the invention turning stability decision method that walks abreast, comprises the following steps:

(1) set up the kinetics equation that the parallel turning of many lathe tools is processed, i.e. multiple time delay second order differential equation.Determine the correlation parameter in equation according to vibration test/two modal testing results.

(2) differential equation is carried out to state space conversion, obtain state space equation.

(3) state space equation is normalized, is turned to the canonical form of being convenient to Differential Quadrature Method.

(4) based on differential quadrature method, use the discrete point on adjacent cells interval [0,1] or [1,0] to represent derivative term and the time lag item in state space equation, multiple time delay equation equivalence is converted into one group of Algebraic Equation set that only contains cutting parameter.

(5) judge many lathe tools Tutrning Process stability that walks abreast based on Floquet theorem, and draw stability collection of illustrative plates in cutting parameter space.

More specifically, below in conjunction with concrete processing instance explanation specific embodiment of the invention scheme, instance parameter draws experiment 2. in document 2 and adopts the turning that walks abreast of two lathe tools, and lathe tool is installed on different knife rests, there is no each other vibration coupling, its schematic diagram as shown in Figure 1.The less lathe tool of note cutting-in is lathe tool A, and the lathe tool that cutting-in is larger is lathe tool B, and the diameter d of processing work is d=35mm, and speed of mainshaft Ω is Ω=1800rpm, direction of feed Cutting Force Coefficient K _ffor K _f=872MPa, modal test parameter is as shown in table 1:

The parallel cutting of lathe tool more than table 1 example modal test parameter

Mode Shape	f n(Hz)	ζ(%)	k(N/m)
				1	2238.9	3.23	4.769*10 7
2	2372.3	4.51	1.166*10 8

Wherein, f _nrepresent natural frequency, ζ represents relative damping ratio, and k represents stiffness coefficient.

Step (1), sets up the parallel turning kinetics equation of many lathe tools, as shown in Equation (1):

$\begin{array}{c}\left[\begin{array}{c}{\stackrel{\&CenterDot;\&CenterDot;}{z}}_1\left(t\right)\\ {\stackrel{\&CenterDot;\&CenterDot;}{z}}_2\left(t\right)\end{array}\right]+\left[\begin{array}{cc}2{\mathrm{\&zeta;}}_1{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="HDA0000468378720000012.png"\; state="\{\{state\}\}">n</figure-callout>}1}& 0\\ 0& 2{\mathrm{\&zeta;}}_2{\mathrm{\&omega;}}_{n2}\end{array}\right]\left[\begin{array}{c}{\stackrel{\&CenterDot;}{z}}_1\left(t\right)\\ {\stackrel{\&CenterDot;}{z}}_2\left(t\right)\end{array}\right]+\left[\begin{array}{cc}{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="HDA0000468378720000012.png"\; state="\{\{state\}\}">n</figure-callout>}1}^2& 0\\ 0& {\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA0000468378720000012.png"\; state="\{\{state\}\}">n</figure-callout>}2}^2\end{array}\right]\left[\begin{array}{c}{z}_1\left(t\right)\\ z_2\left(t\right)\end{array}\right]=\\ K_f\left[\begin{array}{c}\frac{{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="HDA0000468378720000012.png"\; state="\{\{state\}\}">n</figure-callout>}1}^2}{k_1}{a}_1(-z_1\left(t\right)+z_2(t-\frac{\mathrm{\&tau;}}{2}))\\ \frac{{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA0000468378720000012.png"\; state="\{\{state\}\}">n</figure-callout>}2}^2}{k_2}{a}_1(-z_2\left(t\right)+z_1(t-\frac{\mathrm{\&tau;}}{2}))+\frac{{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA0000468378720000012.png"\; state="\{\{state\}\}">n</figure-callout>}2}^2}{k_2}(a_2-a_1)(-z_2\left(t\right)+z_2(t-\mathrm{\&tau;}))\end{array}\right]\end{array}---\left(1\right)$

Wherein, subscript 1 represents lathe tool A, and subscript 2 represents lathe tool B, and z (t) represents dynamic response displacement, ω _nrepresent inherent circular frequency, K _fthe Cutting Force Coefficient that represents direction of feed, τ represents workpiece swing circle, and ζ represents relative damping ratio, and k represents stiffness coefficient, a ₁represent the cutting depth of lathe tool A, a ₂represent the cutting depth of lathe tool B,

represent acceleration,

expression speed, t represents the time.

Formula (1) is arranged, obtains multiple time delay second order differential equation as shown in Equation (2):

$\begin{array}{c}\left[\begin{array}{c}{\stackrel{\&CenterDot;\&CenterDot;}{z}}_1\left(t\right)\\ {\stackrel{\&CenterDot;\&CenterDot;}{z}}_2\left(t\right)\end{array}\right]+\left[\begin{array}{cc}2{\mathrm{\&zeta;}}_1{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="HDA0000468378720000012.png"\; state="\{\{state\}\}">n</figure-callout>}1}& 0\\ 0& 2{\mathrm{\&zeta;}}_2{\mathrm{\&omega;}}_{n2}\end{array}\right]\left[\begin{array}{c}{\stackrel{\&CenterDot;}{z}}_1\left(t\right)\\ {\stackrel{\&CenterDot;}{z}}_2\left(t\right)\end{array}\right]+\left[\begin{array}{cc}{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="HDA0000468378720000012.png"\; state="\{\{state\}\}">n</figure-callout>}1}^2+K_f\frac{{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="HDA0000468378720000012.png"\; state="\{\{state\}\}">n</figure-callout>}1}^2}{k_1}{a}_1& 0\\ 0& {\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA0000468378720000012.png"\; state="\{\{state\}\}">n</figure-callout>}2}^2+K_f\frac{{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA0000468378720000012.png"\; state="\{\{state\}\}">n</figure-callout>}2}^2}{k_2}{a}_2\end{array}\right]\left[\begin{array}{c}{z}_1\left(t\right)\\ z_2\left(t\right)\end{array}\right]=\\ \left[\begin{array}{cc}0& K_f\frac{{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="HDA0000468378720000012.png"\; state="\{\{state\}\}">n</figure-callout>}1}^2}{k_1}{a}_1\\ K_f\frac{{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA0000468378720000012.png"\; state="\{\{state\}\}">n</figure-callout>}2}^2}{k_2}{a}_1& 0\end{array}\right]\left[\begin{array}{c}{z}_1(t-\frac{\mathrm{\&tau;}}{2})\\ z_2(t-\frac{\mathrm{\&tau;}}{2})\end{array}\right]+\left[\begin{array}{cc}0& 0\\ 0& K_f\frac{{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA0000468378720000012.png"\; state="\{\{state\}\}">n</figure-callout>}2}^2}{k_2}(a_2-a_1)\end{array}\right]\left[\begin{array}{c}{z}_1(t-\mathrm{\&tau;})\\ z_2(t-\mathrm{\&tau;})\end{array}\right]\end{array}---\left(2\right)$

Step (2), carries out state space conversion to formula (2), obtains state space equation as shown in Equation (3):

Order $Z\left(t\right)=\left[\begin{array}{c}{z}_1\left(t\right)\\ z_2\left(t\right)\\ {\stackrel{\&CenterDot;}{z}}_1\left(t\right)\\ {\stackrel{\&CenterDot;}{z}}_2\left(t\right)\end{array}\right],$ Formula (3) becomes:

$\stackrel{\&CenterDot;}{Z}\left(t\right)=\mathrm{AZ}\left(t\right)+B_{1}Z(t-\frac{\mathrm{\&tau;}}{2})+B_{2}Z(t-\mathrm{\&tau;})---\left(4\right)$

Wherein, expression state speed term, A represents state displacement item matrix of coefficients, B ₁represent time lag state displacement item matrix of coefficients, B ₂represent time lag state displacement item matrix of coefficients.

Step (3), is normalized state space equation.Might as well make t=ξ τ,

state space equation after normalization becomes canonical form, as shown in Equation (5):

$\stackrel{\&CenterDot;}{Z}\left(\mathrm{\&xi;}\right)=\mathrm{\&tau;AZ}\left(\mathrm{\&xi;}\right)+\mathrm{\&tau;}{B}_{1}Z(\mathrm{\&xi;}-0.5)+\mathrm{\&tau;}{B}_{2}Z(\mathrm{\&xi;}-1)---\left(5\right)$

Step (4), carries out period discrete to formula (5), and its equivalence is converted into one group of algebraic equation.In interval, get n+1 Equations of The Second Kind Chebyshev discrete point ξ _i, as shown in Equation (6):

${\mathrm{\&xi;}}_i=\frac{1}{2}[1-\cos \left(\frac{\mathrm{i\&pi;}}{n}\right)],i=0,\&CenterDot;\&CenterDot;\&CenterDot;,n---\left(6\right)$

On each discrete point, need to meet equation (5), that is:

$\stackrel{\&CenterDot;}{Z}\left({\mathrm{\&xi;}}_i\right)=\mathrm{\&tau;AZ}\left({\mathrm{\&xi;}}_i\right)+\mathrm{\&tau;}{B}_{1}Z({\mathrm{\&xi;}}_i-0.5)+\mathrm{\&tau;}{B}_{2}Z({\mathrm{\&xi;}}_i-1)---\left(7\right)$

On [0,1] interval,

$\left[\begin{array}{c}\stackrel{\&CenterDot;}{Z}\left({\mathrm{\&xi;}}_0\right)\\ \stackrel{\&CenterDot;}{Z}\left({\mathrm{\&xi;}}_1\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ \stackrel{\&CenterDot;}{Z}\left({\mathrm{\&xi;}}_n\right)\end{array}\right]=\mathrm{\&tau;}\&CenterDot;I_{n+1}\&CircleTimes;A\left[\begin{array}{c}Z\left({\mathrm{\&xi;}}_0\right)\\ Z\left({\mathrm{\&xi;}}_1\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ Z\left({\mathrm{\&xi;}}_n\right)\end{array}\right]+\mathrm{\&tau;}\&CenterDot;I_{n+1}\&CircleTimes;B_1\left[\begin{array}{c}Z({\mathrm{\&xi;}}_0-0.5)\\ Z({\mathrm{\&xi;}}_1-0.5)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ Z({\mathrm{\&xi;}}_n-0.5)\end{array}\right]+\mathrm{\&tau;}\&CenterDot;I_{n+1}\&CircleTimes;B_2\left[\begin{array}{c}Z({\mathrm{\&xi;}}_0-1)\\ Z({\mathrm{\&xi;}}_1-1)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ Z({\mathrm{\&xi;}}_n-1)\end{array}\right]---\left(8\right)$

Wherein,

represent Kronecker product,

i _n+1represent (n+1) dimension square formation.

Step (5), differential quadrature method

Utilize differential quadrature method, by the speed term of displacement item representation formula (8) left end.In order to express easily, note

for

f (ξ _i) expression mode bit transposition scalar statement symbol.First use (n+1) individual point (ξ ₀, f (ξ ₀)), (ξ ₁, f (ξ ₁)) ..., (ξ _n, f (ξ _n)) carry out Lagrange (Lagrange) interpolation, result as shown in Equation (9):

$f\left(\mathrm{\&xi;}\right)=\underset{j=0}{\overset{n}{\mathrm{\&Sigma;}}}{l}_j\left(\mathrm{\&xi;}\right)f\left({\mathrm{\&xi;}}_j\right)---\left(9\right)$

Wherein, Interpolation-Radix-Function is:

$l_i\left(\mathrm{\&xi;}\right)=\underset{\underset{k\&NotEqual;i}{k=0}}{\overset{n}{\mathrm{\&Pi;}}}\frac{(\mathrm{\&xi;}-{\mathrm{\&xi;}}_k)}{({\mathrm{\&xi;}}_i-{\mathrm{\&xi;}}_k)}---\left(10\right)$

To Lagrange interpolation function differentiate, then by the each time point substitution of formula (8) left end speed term, result as shown in Equation (11):

Wherein, the expression formula of H is as shown in Equation (12):

$H_{\mathrm{ij}}=\frac{{\mathrm{dl}}_j\left(\mathrm{\&xi;}\right)}{\mathrm{d\&xi;}}{|}_{\mathrm{\&xi;}={\mathrm{\&xi;}}_i}=\left\{\begin{array}{c}\frac{\underset{\underset{k\&NotEqual;i,j}{k=0}}{\overset{n}{\mathrm{\&Pi;}}}({\mathrm{\&xi;}}_i-{\mathrm{\&xi;}}_k)}{\underset{\underset{k\&NotEqual;j}{k=0}}{\overset{n}{\mathrm{\&Pi;}}}({\mathrm{\&xi;}}_j-{\mathrm{\&xi;}}_k)},j\&NotEqual;i\\ \underset{\underset{k\&NotEqual;i}{k=0}}{\overset{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="HDA0000468378720000012.png"\; state="\{\{state\}\}">n</figure-callout>}}{\mathrm{\&Sigma;}}}\frac{1}{{\mathrm{\&xi;}}_j-{\mathrm{\&xi;}}_k},j=i\end{array}\right.---\left(12\right)$

Time lag displacement item Z (ξ-0.5) will represent with the second kind of Chebyshev points on [1,0] and [0,1] interval.Can find i to make ξ _i-0.5 < 0 and ξ _i+1-0.5 > 0 (i=0,1 ..., n); Be that interpolation point obtains Lagrange interpolation function with the second kind of Chebyshev points on [1,0] and [0,1] interval, then by the time point substitution of time lag displacement item, can obtain:

Wherein,

$T_{\mathrm{ij}}^d=\left\{\begin{array}{c}\frac{\underset{\underset{k\&NotEqual;j}{k=0}}{\overset{n}{\mathrm{\&Pi;}}}[({\mathrm{\&xi;}}_i-0.5)-({\mathrm{\&xi;}}_k-1)]}{\underset{\underset{k\&NotEqual;j}{k=0}}{\overset{n}{\mathrm{\&Pi;}}}[({\mathrm{\&xi;}}_j-1)-({\mathrm{\&xi;}}_k-1)]},({\mathrm{\&xi;}}_i-0.5)\&NotElement;\{{\mathrm{\&xi;}}_k-1|k=\mathrm{0,1},\&CenterDot;\&CenterDot;\&CenterDot;,n\}\\ 0,({\mathrm{\&xi;}}_i-0.5)\&Element;\{{\mathrm{\&xi;}}_k-1|k=\mathrm{0,1},\&CenterDot;\&CenterDot;\&CenterDot;,j-1,j+1,\&CenterDot;\&CenterDot;\&CenterDot;,n\}\\ 1,{\mathrm{\&xi;}}_i-0.5={\mathrm{\&xi;}}_j-1\end{array}\right.---\left(14\right)$

Wherein,

$T_{\mathrm{ij}}^p=\left\{\begin{array}{c}\frac{\underset{\underset{k\&NotEqual;j}{k=0}}{\overset{n}{\mathrm{\&Pi;}}}\left[({\mathrm{\&xi;}}_i-0.5){-\mathrm{\&xi;}}_k\right]}{\underset{\underset{k\&NotEqual;j}{k=0}}{\overset{n}{\mathrm{\&Pi;}}}({\mathrm{\&xi;}}_j-{\mathrm{\&xi;}}_k)},({\mathrm{\&xi;}}_i-0.5)\&NotElement;\{{\mathrm{\&xi;}}_k-1|k=\mathrm{0,1},\&CenterDot;\&CenterDot;\&CenterDot;,n\}\\ 0,({\mathrm{\&xi;}}_i-0.5_1)\&Element;\left\{{\mathrm{\&xi;}}_k\right|k=\mathrm{0,1},\&CenterDot;\&CenterDot;\&CenterDot;,j-1,j+1,\&CenterDot;\&CenterDot;\&CenterDot;,n\}\\ 1,{\mathrm{\&xi;}}_i-0.5={\mathrm{\&xi;}}_j\end{array}\right.---\left(16\right)$

Discrete very easy on interval [1,0] of time lag item Z (ξ-1), as shown in Equation (17):

Yi Zhi, T=I _n+1,

Wherein, T represents

Step (6), structure Floquet transfer matrix

Matrix H, T ^d, T ^p, T disappears after the first row and is designated as respectively

Wherein, T ^drepresent

t ^prepresent

Comprehensive above various, can show that the state transitions pass between adjacent two intervals [1,0] and [0,1] is:

$P\left[\begin{array}{c}Z\left({\mathrm{\&xi;}}_0\right)\\ Z\left({\mathrm{\&xi;}}_1\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ Z\left({\mathrm{\&xi;}}_n\right)\end{array}\right]=Q\left[\begin{array}{c}Z({\mathrm{\&xi;}}_0-1)\\ Z({\mathrm{\&xi;}}_1-1)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ Z({\mathrm{\&xi;}}_n-1)\end{array}\right]---\left(18\right)$

Wherein,

Floquet state-transition matrix Φ is as shown in formula:

Wherein,

represent Penrose-Moor generalized inverse;

According to Floquet theory, if the mould of all eigenwerts of Φ is all less than 1, system is stable; If the mould of arbitrary eigenwert is greater than 1 in Φ, system is unsettled.

At the cutting depth parameter space (a of lathe tool A and lathe tool B ₁, a ₂), the stability collection of illustrative plates of the parallel turning of many lathe tools is as shown in Figure 2.

According to document 1, if keep other experiment conditions constant, only adopt lathe tool A to carry out traditional turning, the Limit cutting depth of stability processing is 3.4mm, only adopts lathe tool B to carry out traditional turning, the Limit cutting depth of stability processing is 12.6mm.With reference to the accompanying drawings 2, after adopting the parallel turning of many lathe tools, the Limit cutting depth of stability processing obtains greatly and improves, if adopt (a ₁, a ₂)=(8,26) cutting-in parameter combinations carries out without Regenerative Chatter stability processing, and the cutting-in of lathe tool A is brought up to 8mm by 3.4mm, has improved 135.3%, and the cutting-in of lathe tool B is brought up to 26mm by 12.6mm, has improved 106.3%.As can be seen here, compared with traditional bicycle cutter cutting, the parallel turning of many lathe tools has greatly improved working (machining) efficiency, and the raising of working (machining) efficiency is not only embodied in many in lathe tool turning simultaneously, is also embodied in the parallel turning processing of many lathe tools and has greatly improved the stabilized zone that traditional bicycle cutter turning is processed.

Above specific embodiments of the invention are described.It will be appreciated that, the present invention is not limited to above-mentioned specific implementations, and those skilled in the art can make various distortion or modification within the scope of the claims, and this does not affect flesh and blood of the present invention.

Claims (9)

1. the parallel turning stability decision method of the many lathe tools based on differential quadrature method, is characterized in that, comprises the steps:

Step 1: set up the parallel turning kinetics equation of many lathe tools, the parallel turning kinetics equation of many lathe tools is arranged, obtain multiple time delay second order differential equation;

Step 2: described multiple time delay second order differential equation is carried out to state space conversion, obtain state space equation;

Step 3: described state space equation is normalized, obtains the state space equation of canonical form;

Step 4: the state space equation of canonical form is carried out to period discrete, its equivalence is converted into one group of algebraic equation as state space equation expression formula;

Step 5: based on Lagrange interpolation function, for discrete point, utilize differential quadrature method with the second kind of Chebyshev points on unit interval [0,1], represent the derivative term in state space equation expression formula with displacement item;

Step 6: based on Lagrange interpolation function, the time lag item in state space equation expression formula being carried out to place interval judgement, if the discrete point of time lag item belongs to interval [0,1], is that discrete point represents time lag item with the second kind of Chebyshev points on [0,1]; If the discrete point of time lag item belongs to interval [1,0], is that discrete point represents time lag item with the second kind of Chebyshev points on [1,0];

Step 7: the state-transition matrix between structure adjacent cells interval [0,1] and [1,0], according to the theoretical stability of judging original system of Floquet.

2. the parallel turning stability decision method of the many lathe tools based on differential quadrature method according to claim 1, is characterized in that, in described step 1, and described many lathe tools turning kinetics equation that walks abreast, as shown in Equation (1):

$\begin{array}{c}\left[\begin{array}{c}{\stackrel{\&CenterDot;\&CenterDot;}{z}}_1\left(t\right)\\ {\stackrel{\&CenterDot;\&CenterDot;}{z}}_2\left(t\right)\end{array}\right]+\left[\begin{array}{cc}2{\mathrm{\&zeta;}}_1{\mathrm{\&omega;}}_{n1}& 0\\ 0& 2{\mathrm{\&zeta;}}_2{\mathrm{\&omega;}}_{n2}\end{array}\right]\left[\begin{array}{c}{\stackrel{\&CenterDot;}{z}}_1\left(t\right)\\ {\stackrel{\&CenterDot;}{z}}_2\left(t\right)\end{array}\right]+\left[\begin{array}{cc}{\mathrm{\&omega;}}_{n1}^2& 0\\ 0& {\mathrm{\&omega;}}_{n2}^2\end{array}\right]\left[\begin{array}{c}{z}_1\left(t\right)\\ z_2\left(t\right)\end{array}\right]=\\ K_f\left[\begin{array}{c}\frac{{\mathrm{\&omega;}}_{n1}^2}{k_1}{a}_1(-z_1\left(t\right)+z_2(t-\frac{\mathrm{\&tau;}}{2}))\\ \frac{{\mathrm{\&omega;}}_{n2}^2}{k_2}{a}_1(-z_2\left(t\right)+z_1(t-\frac{\mathrm{\&tau;}}{2}))+\frac{{\mathrm{\&omega;}}_{n2}^2}{k_2}(a_2-a_1)(-z_2\left(t\right)+z_2(t-\mathrm{\&tau;}))\end{array}\right]\end{array}---\left(1\right)$

Wherein, subscript 1 represents lathe tool A, and subscript 2 represents lathe tool B, and z (t) represents dynamic response displacement, ω _nrepresent inherent circular frequency, K _fthe Cutting Force Coefficient that represents direction of feed, τ represents workpiece swing circle, and ζ represents relative damping ratio, and k represents stiffness coefficient, a ₁represent the cutting depth of lathe tool A, a ₂represent the cutting depth of lathe tool B,

represent dynamics acceleration responsive,

represent dynamics speed responsive, t represents the time;

Described multiple time delay second order differential equation, as shown in Equation (2):

$\begin{array}{c}\left[\begin{array}{c}{\stackrel{\&CenterDot;\&CenterDot;}{z}}_1\left(t\right)\\ {\stackrel{\&CenterDot;\&CenterDot;}{z}}_2\left(t\right)\end{array}\right]+\left[\begin{array}{cc}2{\mathrm{\&zeta;}}_1{\mathrm{\&omega;}}_{n1}& 0\\ 0& 2{\mathrm{\&zeta;}}_2{\mathrm{\&omega;}}_{n2}\end{array}\right]\left[\begin{array}{c}{\stackrel{\&CenterDot;}{z}}_1\left(t\right)\\ {\stackrel{\&CenterDot;}{z}}_2\left(t\right)\end{array}\right]+\left[\begin{array}{cc}{\mathrm{\&omega;}}_{n1}^2+K_f\frac{{\mathrm{\&omega;}}_{n1}^2}{k_1}{a}_1& 0\\ 0& {\mathrm{\&omega;}}_{n2}^2+K_f\frac{{\mathrm{\&omega;}}_{n2}^2}{k_2}{a}_2\end{array}\right]\left[\begin{array}{c}{z}_1\left(t\right)\\ z_2\left(t\right)\end{array}\right]=\\ \left[\begin{array}{cc}0& K_f\frac{{\mathrm{\&omega;}}_{n1}^2}{k_1}{a}_1\\ K_f\frac{{\mathrm{\&omega;}}_{n2}^2}{k_2}{a}_1& 0\end{array}\right]\left[\begin{array}{c}{z}_1(t-\frac{\mathrm{\&tau;}}{2})\\ z_2(t-\frac{\mathrm{\&tau;}}{2})\end{array}\right]+\left[\begin{array}{cc}0& 0\\ 0& K_f\frac{{\mathrm{\&omega;}}_{n2}^2}{k_2}(a_2-a_1)\end{array}\right]\left[\begin{array}{c}{z}_1(t-\mathrm{\&tau;})\\ z_2(t-\mathrm{\&tau;})\end{array}\right]\end{array}---\left(2\right).$

3. the parallel turning stability decision method of the many lathe tools based on differential quadrature method according to claim 2, is characterized in that, in step 2, and described state space equation, as shown in Equation (3):

Order $Z\left(t\right)=\left[\begin{array}{c}{z}_1\left(t\right)\\ z_2\left(t\right)\\ {\stackrel{\&CenterDot;}{z}}_1\left(t\right)\\ {\stackrel{\&CenterDot;}{z}}_2\left(t\right)\end{array}\right],$ Formula (3) becomes:

$\stackrel{\&CenterDot;}{Z}\left(t\right)=\mathrm{AZ}\left(t\right)+B_{1}Z(t-\frac{\mathrm{\&tau;}}{2})+B_{2}Z(t-\mathrm{\&tau;})---\left(4\right)$

Wherein,

expression state speed, A represents state displacement item matrix of coefficients, B ₁represent time lag state displacement item matrix of coefficients, B ₂represent time lag state displacement item matrix of coefficients.

4. the parallel turning stability decision method of the many lathe tools based on differential quadrature method according to claim 3, is characterized in that, described step 3, is specially:

Make t=ξ τ, $\frac{\mathrm{dZ}\left(t\right)}{\mathrm{dt}}=\frac{\mathrm{dZ}\left(\mathrm{\&xi;\&tau;}\right)}{\mathrm{d\&xi;}}\&CenterDot;\frac{\mathrm{d\&xi;}}{\mathrm{dt}}=\frac{1}{\mathrm{\&tau;}}\frac{\mathrm{dZ}\left(\mathrm{\&xi;\&tau;}\right)}{\mathrm{d\&xi;}};$

State space equation after normalization becomes canonical form, as shown in Equation (5):

$\stackrel{\&CenterDot;}{Z}\left(\mathrm{\&xi;}\right)=\mathrm{\&tau;AZ}\left(\mathrm{\&xi;}\right)+\mathrm{\&tau;}{B}_{1}Z(\mathrm{\&xi;}-0.5)+\mathrm{\&tau;}{B}_{2}Z(\mathrm{\&xi;}-1)---\left(5\right).$

5. the parallel turning stability decision method of the many lathe tools based on differential quadrature method according to claim 4, is characterized in that, described step 4, is specially:

In interval, get n+1 Equations of The Second Kind Chebyshev discrete point ξ _i, as shown in Equation (6):

${\mathrm{\&xi;}}_i=\frac{1}{2}[1-\cos \left(\frac{\mathrm{i\&pi;}}{n}\right)],i=0,\&CenterDot;\&CenterDot;\&CenterDot;,n---\left(6\right)$

On each discrete point, need to meet equation (5), that is:

$\stackrel{\&CenterDot;}{Z}\left({\mathrm{\&xi;}}_i\right)=\mathrm{\&tau;AZ}\left({\mathrm{\&xi;}}_i\right)+\mathrm{\&tau;}{B}_{1}Z({\mathrm{\&xi;}}_i-0.5)+\mathrm{\&tau;}{B}_{2}Z({\mathrm{\&xi;}}_i-1)---\left(7\right)$

On [0,1] interval,

$\left[\begin{array}{c}\stackrel{\&CenterDot;}{Z}\left({\mathrm{\&xi;}}_0\right)\\ \stackrel{\&CenterDot;}{Z}\left({\mathrm{\&xi;}}_1\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ \stackrel{\&CenterDot;}{Z}\left({\mathrm{\&xi;}}_n\right)\end{array}\right]=\mathrm{\&tau;}\&CenterDot;I_{n+1}\&CircleTimes;A\left[\begin{array}{c}Z\left({\mathrm{\&xi;}}_0\right)\\ Z\left({\mathrm{\&xi;}}_1\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ Z\left({\mathrm{\&xi;}}_n\right)\end{array}\right]+\mathrm{\&tau;}\&CenterDot;I_{n+1}\&CircleTimes;B_1\left[\begin{array}{c}Z({\mathrm{\&xi;}}_0-0.5)\\ Z({\mathrm{\&xi;}}_1-0.5)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ Z({\mathrm{\&xi;}}_n-0.5)\end{array}\right]+\mathrm{\&tau;}\&CenterDot;I_{n+1}\&CircleTimes;B_2\left[\begin{array}{c}Z({\mathrm{\&xi;}}_0-1)\\ Z({\mathrm{\&xi;}}_1-1)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ Z({\mathrm{\&xi;}}_n-1)\end{array}\right]---\left(8\right)$

Wherein,

represent Kronecker product,

i _n+1represent (n+1) dimension square formation.

6. the parallel turning stability decision method of the many lathe tools based on differential quadrature method according to claim 5, is characterized in that, described step 5, is specially:

Utilize differential quadrature method, by the speed term of displacement item representation formula (8) left end; In order to express easily, note

for

f (ξ _i) expression mode bit transposition scalar statement symbol;

First use (n+1) individual point (ξ ₀, f (ξ ₀)), (ξ ₁, f (ξ ₁)) ..., (ξ _n, f (ξ _n)) carry out Lagrange's interpolation, result as shown in Equation (9):

$f\left(\mathrm{\&xi;}\right)=\underset{j=0}{\overset{n}{\mathrm{\&Sigma;}}}{l}_j\left(\mathrm{\&xi;}\right)f\left({\mathrm{\&xi;}}_j\right)---\left(9\right)$

Wherein, Interpolation-Radix-Function is:

$l_i\left(\mathrm{\&xi;}\right)=\underset{\underset{k\&NotEqual;i}{k=0}}{\overset{n}{\mathrm{\&Pi;}}}\frac{(\mathrm{\&xi;}-{\mathrm{\&xi;}}_k)}{({\mathrm{\&xi;}}_i-{\mathrm{\&xi;}}_k)}---\left(10\right)$

To Lagrange interpolation function differentiate, then by the each time point substitution of formula (8) left end speed term, result as shown in Equation (11):

Wherein, the expression formula of H is as shown in Equation (12):

$H_{\mathrm{ij}}=\frac{{\mathrm{dl}}_j\left(\mathrm{\&xi;}\right)}{\mathrm{d\&xi;}}{|}_{\mathrm{\&xi;}={\mathrm{\&xi;}}_i}=\left\{\begin{array}{c}\frac{\underset{\underset{k\&NotEqual;i,j}{k=0}}{\overset{n}{\mathrm{\&Pi;}}}({\mathrm{\&xi;}}_i-{\mathrm{\&xi;}}_k)}{\underset{\underset{k\&NotEqual;j}{k=0}}{\overset{n}{\mathrm{\&Pi;}}}({\mathrm{\&xi;}}_j-{\mathrm{\&xi;}}_k)},j\&NotEqual;i\\ \underset{\underset{k\&NotEqual;i}{k=0}}{\overset{n}{\mathrm{\&Sigma;}}}\frac{1}{{\mathrm{\&xi;}}_j-{\mathrm{\&xi;}}_k},j=i\end{array}\right.---\left(12\right).$

7. the parallel turning stability decision method of the many lathe tools based on differential quadrature method according to claim 6, is characterized in that, described step 6, is specially:

Time lag displacement item Z (ξ-0.5) represents with the second kind of Chebyshev points on [1,0] and [0,1] interval; Find i to make ξ _i-0.5 < 0 and ξ _i+1-0.5 > 0 (i=0,1 ..., n); Be that interpolation point obtains Lagrange interpolation function with the second kind of Chebyshev points on [1,0] and [0,1] interval respectively, then by the time point substitution of time lag displacement item, obtain:

Wherein,

$T_{\mathrm{ij}}^d=\left\{\begin{array}{c}\frac{\underset{\underset{k\&NotEqual;j}{k=0}}{\overset{n}{\mathrm{\&Pi;}}}[({\mathrm{\&xi;}}_i-0.5)-({\mathrm{\&xi;}}_k-1)]}{\underset{\underset{k\&NotEqual;j}{k=0}}{\overset{n}{\mathrm{\&Pi;}}}[({\mathrm{\&xi;}}_j-1)-({\mathrm{\&xi;}}_k-1)]},({\mathrm{\&xi;}}_i-0.5)\&NotElement;\{{\mathrm{\&xi;}}_k-1|k=\mathrm{0,1},\&CenterDot;\&CenterDot;\&CenterDot;,n\}\\ 0,({\mathrm{\&xi;}}_i-0.5)\&Element;\{{\mathrm{\&xi;}}_k-1|k=\mathrm{0,1},\&CenterDot;\&CenterDot;\&CenterDot;,j-1,j+1,\&CenterDot;\&CenterDot;\&CenterDot;,n\}\\ 1,{\mathrm{\&xi;}}_i-0.5={\mathrm{\&xi;}}_j-1\end{array}\right.---\left(14\right)$

Wherein,

$T_{\mathrm{ij}}^p=\left\{\begin{array}{c}\frac{\underset{\underset{k\&NotEqual;j}{k=0}}{\overset{n}{\mathrm{\&Pi;}}}\left[({\mathrm{\&xi;}}_i-0.5){-\mathrm{\&xi;}}_k\right]}{\underset{\underset{k\&NotEqual;j}{k=0}}{\overset{n}{\mathrm{\&Pi;}}}({\mathrm{\&xi;}}_j-{\mathrm{\&xi;}}_k)},({\mathrm{\&xi;}}_i-0.5)\&NotElement;\{{\mathrm{\&xi;}}_k-1|k=\mathrm{0,1},\&CenterDot;\&CenterDot;\&CenterDot;,n\}\\ 0,({\mathrm{\&xi;}}_i-0.5_1)\&Element;\left\{{\mathrm{\&xi;}}_k\right|k=\mathrm{0,1},\&CenterDot;\&CenterDot;\&CenterDot;,j-1,j+1,\&CenterDot;\&CenterDot;\&CenterDot;,n\}\\ 1,{\mathrm{\&xi;}}_i-0.5={\mathrm{\&xi;}}_j\end{array}\right.---\left(16\right)$

Discrete on interval [1,0] of time lag item Z (ξ-1), as shown in Equation (17):

Can obtain T=I _n+1,

Wherein, T represents

8. the parallel turning stability decision method of the many lathe tools based on differential quadrature method according to claim 7, is characterized in that, described step 7, is specially:

Structure Floquet transfer matrix:

Matrix H, T ^d, T ^p, T disappears after the first row and is designated as respectively

Wherein, T ^drepresent

t ^prepresent

Comprehensive above various, can show that the state transitions pass between adjacent two intervals [1,0] and [0,1] is:

$P\left[\begin{array}{c}Z\left({\mathrm{\&xi;}}_0\right)\\ Z\left({\mathrm{\&xi;}}_1\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ Z\left({\mathrm{\&xi;}}_n\right)\end{array}\right]=Q\left[\begin{array}{c}Z({\mathrm{\&xi;}}_0-1)\\ Z({\mathrm{\&xi;}}_1-1)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ Z({\mathrm{\&xi;}}_n-1)\end{array}\right]---\left(18\right)$

Wherein,

Floquet state-transition matrix Φ is as shown in formula:

Wherein,

represent Penrose-Moor generalized inverse;

According to Floquet theory, if the mould of all eigenwerts of Φ is all less than 1, system is stable; If the mould of arbitrary eigenwert is greater than 1 in Φ, system is unsettled.

9. the parallel turning stability decision method of the many lathe tools based on differential quadrature method according to claim 1, is characterized in that, also comprises the steps:

Step 8: and draw out the stability collection of illustrative plates of system in Delay Parameters space.

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