CN103425811B - The equivalent inertia of NC machine tool feed system and the discrimination method of equivalent damping - Google Patents

Abstract

The present invention relates to a kind of NC machine tool feed system equivalent inertia and the discrimination method of equivalent damping, the steps include: set up the feed system second-order dynamic model simplified and be calculated its difference equation；By G code displacement signal, NC machine tool feed system is continuously and effectively encouraged, and draw current signal and the motor speed signal data of feed servo motor simultaneously；By current signal and the data of tach signal, after bringing described difference equation into, draw equivalent inertia and the equivalent damping of NC machine tool feed system from described difference equation according to method of least square；Present invention mainly solves Identifying Dynamical Parameters problem during Digit Control Machine Tool dynamic modeling, this method can go out inertia and the damping of system at lathe closed loop states accurate recognition, realize the accurate modeling of Servo System of Machine Tools, provide technical foundation for realizing numerical control machine high accurate and high efficiency machining control.

Description

The equivalent inertia of NC machine tool feed system and the discrimination method of equivalent damping

Technical field

The present invention relates to a kind of NC machine tool feed system equivalent inertia and the discrimination method of equivalent damping, in particular with G code displacement signal, NC machine tool feed system is continuously and effectively encouraged, using current of electric data and motor speed data as the discrimination method of parameter.

Background technology

At a high speed, accurate, be efficiently the direction of Modern NC Machine Tool development, but the dynamic characteristic of feed system has important impact to precision and the efficiency of high-speed machine tool.For the high-grade commercial digital control system of full closed loop control, suitable feed system controls parameter and depends on Mechanic system parameter.Inappropriate control parameter not only can make feed system bigger tracking error occur, there is also the vibration of mechanical system, cause the degradation of crudy even to cause device damage, thus affect crudy and working (machining) efficiency time serious.Based on this, obtain the dynamic parameter of the such as feed system such as equivalent inertia and equivalent damping, it is achieved the modeling of feed system, carry out the Research on Dynamic Characteristic of feed system, it is achieved high-speed machine tool high accuracy controls, be to realize high speed, accurate, the premise of highly-efficient processing.It addition, the Digit Control Machine Tool in Sheng Chaning, it is achieved Open-loop Identification is the most difficult, and best bet is closed-loop identification, uses suitable pumping signal to be by the key of closed-loop identification, it is desirable to pumping signal can the excitation of continuous and effective to the system of institute's identification.Being limited by Digit Control Machine Tool opening, to the input signal of numerical control machine tool numerical control system kinematic axis in addition to position signalling, other signal is the most extremely difficult.

Summary of the invention

According to this situation, the present invention proposes a kind of to NC machine tool feed system equivalent inertia with the discrimination method of equivalent damping, this method solve lathe in production and be difficult to a difficult problem for open loop closed loop, equivalent inertia and equivalent damping can be gone out accordingly with accurate recognition, the method proposed is Machine Tool Feeding System dynamic modeling accurately, and the working (machining) efficiency and the crudy that improve Digit Control Machine Tool are significant.

A kind of NC machine tool feed system equivalent inertia and the discrimination method of equivalent damping, it is characterised in that have:

1st step, sets up the feed system second-order dynamic model simplified and is calculated its difference equation；

Second step, is continuously and effectively encouraged NC machine tool feed system by G code displacement signal, and obtains current signal and the motor speed signal data of feed servo motor simultaneously；

Third step, by current signal and the data of tach signal, after bringing difference equation into, draws equivalent inertia and the equivalent damping of NC machine tool feed system according to method of least square from difference equation.

Above in the 1st step, feed system second-order dynamic model and difference equation are drawn by procedure below:

First, with motor shaft as object of study, theoretical according to relevant control, the establishment differential equation:

${\mathrm{iK}}_t=J_e\stackrel{\·}{\mathrm{\ω}}+B_e\mathrm{\ω}+T_e---\left(1\right)$

Wherein, K_tFor motor torque constant, J_eFor equivalent inertia, B_mFor equivalent damping, ω is motor angular velocity, and i is current of electric

Then, the differential equation (1) is rewritten into equation (2):

$K_t(i-T_e/K_t)=J_e\stackrel{\·}{\mathrm{\ω}}+B_e\mathrm{\ω}---\left(2\right)$

Then, by both sides Laplace transformation computing described equation (2) is deformed into equation (3):

$W\left(s\right)=\frac{b}{a}\×\frac{a}{s+a}[I\left(s\right)-T_e\left(s\right)/K_t]---\left(3\right)$

Wherein, W, I are the Laplace transformation form of ω, i respectively, and a=B_e/J_e, b=K_t/J_e

Then, with T_sFor the sampling period, to equation (3) discretization, draw difference equation (4):

$\mathrm{\ω}(k+1)=e^{-a{T}_s}\mathrm{\ω}\left(k\right)+\frac{b}{a}(1-e^{-a{T}_s})[i\left(k\right)-T_e\left(k\right)/K_t]---\left(4\right)$

Then, the T of moment of friction will be derived from_e(k)/K_tIt is equivalent to disturb electric current, it is assumed that be d with ω interference electric current in the same direction_f ⁺, reverse interference electric current is d_f ^-, both can be seen as the function of ω, now T_eCan be write as:

$T_e\left(\mathrm{\ω}\left(k\right)\right)=P_V\left(\mathrm{\ω}\left(k\right)\right)\·{d_f}^{+}+N_V\left(\mathrm{\ω}\left(k\right)\right)\·{d_f}^{-}$

Wherein, P_V(ω(k))、N_V(ω (k)) is used to describe d respectively_f ⁺、d_f ^-The function in direction, is defined by the formula:

$P_V\left(\mathrm{\&omega;}\left(k\right)\right)=\left\{\begin{array}{cc}0& \mathrm{\&omega;}\left(k\right)<\mathrm{\&Omega;}\\ 1& \mathrm{\&omega;}\left(k\right)\&GreaterEqual;\mathrm{\&Omega;}\end{array}\right.,N_V\left(\mathrm{\&omega;}\left(k\right)\right)=\left\{\begin{array}{cc}0& \mathrm{\&omega;}\left(k\right)>-\mathrm{\&Omega;}\\ -1& \mathrm{\&omega;}\left(k\right)\&le;-\mathrm{\&Omega;}\end{array}\right.$

Wherein, Ω is the threshold values judging velocity attitude, then draw difference equation (5) according to difference equation (4)

${\mathrm{\&omega;}}_m(k+1)=[\mathrm{\&omega;}\left(k\right)i\left(k\right)-P_V\left(\mathrm{\&omega;}\left(k\right)\right)-N_V\left(\mathrm{\&omega;}\left(k\right)\right)]$

${[e^{-a{T}_s}b(1-e^{-a{T}_s})/\mathrm{ab}(1-e^{-a{T}_s}){d_f}^{+}/\mathrm{ab}(1-e^{-a{T}_s})d_f-/a]}^T---\left(5\right)$

In above-mentioned second step, continuously and effectively being encouraged NC machine tool feed system by G code displacement signal is to realize according to procedure below:

First, required pumping signal u is made up of the second displacement curve of the constant duration of m section difference acceleration, and m < 10, every section of conic section acceleration respectively:

a₀K,a₁K,...,a_iK,...,a_mK

Wherein K is scale factor constant

Then, it is stipulated that formed whole displacement excitation curve, and n by the n little straightway of section constant duration > > m.

Wherein, every little straightway time interval is T_s, it is the minimum interval of curve discrete, takes machine tool position controller and control the integral multiple in cycle.

Then, the speed command v of the little straightway of kth section is obtained_kWith displacement commands y_k:

$v_k=\underset{i=0}{\overset{k}{\mathrm{\&Sigma;}}}{a}_{\mathrm{ij}}{T}_s$

$u_k=\underset{i=0}{\overset{k}{\mathrm{\&Sigma;}}}{v}_i{T}_s$

Wherein, described a_ijRefer to jth section conic section section at i-th section of little straightway.

In above-mentioned third step, by current signal and the data of tach signal, after bringing difference equation into, draw equivalent inertia and the equivalent damping of NC machine tool feed system according to method of least square from difference equation, realized by procedure below:

First against difference equation (5) definition following equalities:

Y⁰=[ω(2)ω(3)...ω(N)]^T

${\mathrm{\&Phi;}}^0=\left[\begin{array}{cccc}\mathrm{\&omega;}\left(1\right)& i\left(1\right)& -P_V\left(\mathrm{\&omega;}\left(1\right)\right)& -N_V\left(\mathrm{\&omega;}\left(1\right)\right)\\ \mathrm{\&omega;}\left(2\right)& i\left(2\right)& -P_V\left(\mathrm{\&omega;}\left(2\right)\right)& -N_V\left(\mathrm{\&omega;}\left(2\right)\right)\\ ...& ...& ...& ...\\ \mathrm{\&omega;}(N-1)& i(N-1)& -P_V\left(\mathrm{\&omega;}(N-1)\right)& -N_V\left(\mathrm{\&omega;}(N-1)\right)\end{array}\right]$

${\mathrm{\&theta;}}^0={[e^{-a{T}_s}b(1-e^{-a{T}_s})/\mathrm{ab}(1-e^{-a{T}_s}){d_f}^{+}/\mathrm{ab}(1-e^{-a{T}_s}){d_f}^{-}/a]}^T$

Then, according to least squares identification theoretical method, θ is drawn⁰Optimal estimation vector:

$\widehat{{\mathrm{\&theta;}}^0}={\left({\left({\mathrm{\&Phi;}}^0\right)}^T{\mathrm{\&Phi;}}^0\right)}^{-1}{\left({\mathrm{\&Phi;}}^0\right)}^T{Y}^0---\left(6\right)$

Wherein, ω (2), ω (3) ... ω (N) is the 2nd, 3 ... the motor shaft angular velocity in N number of sampling period,

I (1), i (2) ... i (N-1) is the 1st, 2 respectively ... the current of electric in N-1 sampling period,

Then, definition is respectively intended to describe d_f ⁺、d_f ^-The function P in direction_V(ω(k))、N_V(ω (k)):

$P_V\left(\mathrm{\&omega;}\left(k\right)\right)=\left\{\begin{array}{cc}0& \mathrm{\&omega;}\left(k\right)<\mathrm{\&Omega;}\\ 1& \mathrm{\&omega;}\left(k\right)\&GreaterEqual;\mathrm{\&Omega;}\end{array}\right.,N_V\left(\mathrm{\&omega;}\left(k\right)\right)=\left\{\begin{array}{cc}0& \mathrm{\&omega;}\left(k\right)>-\mathrm{\&Omega;}\\ -1& \mathrm{\&omega;}\left(k\right)\&le;-\mathrm{\&Omega;}\end{array}\right.$

Wherein, Ω is the threshold values of the axle zero angular velocity defining servomotor.

Thus, NC machine tool feed system equivalent inertia is drawn:

J_e=K_t/ b,

Thus, equivalent damping is drawn:

B_e=aJ_e

Wherein, $a=-1n\left(\widehat{{\mathrm{\&theta;}}^0}\left(1\right)\right)/T_s,b=a\widehat{{\mathrm{\&theta;}}^0}\left(2\right)/(1-e^{-a{T}_s})$

Represent respectivelyThe 1st of vector, 2 row components

When carrying out identification test, first taking less scale factor K, be then gradually increased, stop experiment when the result of identification tends towards stability, the experimental result value after taking finally is as identification result.

Invention effect and effect

The present invention is by utilizing G code can effectively encourage the dynamic continuance of feed system, and then the discrimination method proposed, and accurate recognition goes out equivalent inertia and equivalent damping, it is thus possible to overcome the Digit Control Machine Tool in production to be difficult to Open-loop Identification and a difficult problem for effectively excitation；

Provide simple and practical discrimination method accurately and obtain equivalent inertia and equivalent damping, it is thus possible to provide technical foundation for the modeling of NC machine tool feed system dynamical system accurately；

Foundation can be provided for improving the solution of Digit Control Machine Tool crudy in high-speed, high precision is processed and working (machining) efficiency problem.

Accompanying drawing illustrates:

Fig. 1 is the identification principle figure in embodiment

Fig. 2 is the electric current of the identification in embodiment, tach signal data

Fig. 3 is the NC machine tool feed system kinetic model of the simplification in embodiment

A () is time, accelerating curve

B () is time, rate curve

C () is time, displacement curve

Fig. 4 is the result empirical curve example of the identification in embodiment

A () is the identification result of equivalent inertia

B () is the identification result of equivalent damping

Detailed description of the invention:

The NC machine tool feed system equivalent inertia provided in present embodiment and equivalent damping carry out the experimental technique of identification and comprise the following steps:

1st step: set up the feed system second-order dynamic model simplified and be calculated its difference equation.

Fig. 1 is the identification principle figure in embodiment, as shown in Figure 1:

First, with motor shaft as object of study, theoretical according to relevant control, the establishment differential equation:

${\mathrm{iK}}_t=J_e\stackrel{\&CenterDot;}{\mathrm{\&omega;}}+B_e\mathrm{\&omega;}+T_e---\left(1\right)$

Wherein, K_tFor motor torque constant, J_eFor equivalent inertia, B_mFor equivalent damping, ω is motor angular velocity, and i is current of electric

Fig. 2 is the electric current of the identification in embodiment, tach signal data.As in figure 2 it is shown, when determining the axis servomotor of identification, for the electric current i required for the identification of J, B and the tach signal ω of motor, the external interface provided by servo-driver is obtained.

Then, the differential equation (1) is rewritten into equation (2):

$K_t(i-T_e/K_t)=J_e\stackrel{\&CenterDot;}{\mathrm{\&omega;}}+B_e\mathrm{\&omega;}---\left(2\right)$

Then, by both sides Laplace transformation computing described equation (2) is deformed into equation (3):

$W\left(s\right)=\frac{b}{a}\&times;\frac{a}{s+a}[I\left(s\right)-T_e\left(s\right)/K_t]---\left(3\right)$

Wherein, W, I are the Laplace transformation form of ω, i respectively, and a=B_e/J_e, b=K_t/J_e

Then, with T_sFor the sampling period, to equation (3) discretization, draw difference equation (4):

$\mathrm{\&omega;}(k+1)=e^{-a{T}_s}\mathrm{\&omega;}\left(k\right)+\frac{b}{a}(1-e^{-a{T}_s})[i\left(k\right)-T_e\left(k\right)/K_t]---\left(4\right)$

Then, the T of moment of friction will be derived from_e(k)/K_tIt is equivalent to disturb electric current, it is assumed that be d with ω interference electric current in the same direction_f ⁺, reverse interference electric current is d_f ^-, both can be seen as the function of ω, now T_eCan be write as:

$T_e\left(\mathrm{\&omega;}\left(k\right)\right)=P_V\left(\mathrm{\&omega;}\left(k\right)\right)\&CenterDot;{d_f}^{+}+N_V\left(\mathrm{\&omega;}\left(k\right)\right)\&CenterDot;{d_f}^{-}$

Wherein, P_V(ω(k))、N_V(ω (k)) is used to describe d respectively_f ⁺、d_f ^-The function in direction, is defined by the formula:

$P_V\left(\mathrm{\&omega;}\left(k\right)\right)=\left\{\begin{array}{cc}0& \mathrm{\&omega;}\left(k\right)<\mathrm{\&Omega;}\\ 1& \mathrm{\&omega;}\left(k\right)\&GreaterEqual;\mathrm{\&Omega;}\end{array}\right.,N_V\left(\mathrm{\&omega;}\left(k\right)\right)=\left\{\begin{array}{cc}0& \mathrm{\&omega;}\left(k\right)>-\mathrm{\&Omega;}\\ -1& \mathrm{\&omega;}\left(k\right)\&le;-\mathrm{\&Omega;}\end{array}\right.$

Wherein, Ω is the threshold values judging velocity attitude, then draw difference equation (5) according to difference equation (4)

${\mathrm{\&omega;}}_m(k+1)=[\mathrm{\&omega;}\left(k\right)i\left(k\right)-P_V\left(\mathrm{\&omega;}\left(k\right)\right)-N_V\left(\mathrm{\&omega;}\left(k\right)\right)]$

${[e^{-a{T}_s}b(1-e^{-a{T}_s})/\mathrm{ab}(1-e^{-a{T}_s}){d_f}^{+}/\mathrm{ab}(1-e^{-a{T}_s})d_f-/a]}^T---\left(5\right)$

NC machine tool feed system equivalent inertia involved by the present embodiment and the second step of the discrimination method of equivalent damping: continuously and effectively being encouraged NC machine tool feed system by G code displacement signal is to realize according to procedure below:

Input signal is input to Digit Control Machine Tool positioner.

Fig. 3 (a) is time, accelerating curve.As shown in Fig. 3 (a)

Required pumping signal u is made up of the second displacement curve of the constant duration of m section difference acceleration, and m < 10, every section of conic section acceleration respectively:

a₀K,_a1K,...,a_iK,...,a_mK

Wherein K is scale factor constant

Then, it is stipulated that formed whole displacement excitation curve, and n by the n little straightway of section constant duration > > m

Wherein, every little straightway time interval is T_s, it is the minimum interval of curve discrete, takes machine tool position controller and control the integral multiple in cycle

Then, the speed command v of the little straightway of kth section is obtained_kWith displacement commands y_k:

$v_k=\underset{i=0}{\overset{k}{\mathrm{\&Sigma;}}}{a}_{\mathrm{ij}}{T}_s$

$u_k=\underset{i=0}{\overset{k}{\mathrm{\&Sigma;}}}{v}_i{T}_s$

Wherein, described a_ijRefer to jth section conic section section at i-th section of little straightway

Third step: by current signal and the data of tach signal, after bringing difference equation into, is drawn equivalent inertia and the equivalent damping of NC machine tool feed system, is realized by procedure below according to method of least square from difference equation

First against difference equation (5) definition following equalities:

Y⁰=[ω(2)ω(3)...ω(N)]^T

${\mathrm{\&Phi;}}^0=\left[\begin{array}{cccc}\mathrm{\&omega;}\left(1\right)& i\left(1\right)& -P_V\left(\mathrm{\&omega;}\left(1\right)\right)& -N_V\left(\mathrm{\&omega;}\left(1\right)\right)\\ \mathrm{\&omega;}\left(2\right)& i\left(2\right)& -P_V\left(\mathrm{\&omega;}\left(2\right)\right)& -N_V\left(\mathrm{\&omega;}\left(2\right)\right)\\ ...& ...& ...& ...\\ \mathrm{\&omega;}(N-1)& i(N-1)& -P_V\left(\mathrm{\&omega;}(N-1)\right)& -N_V\left(\mathrm{\&omega;}(N-1)\right)\end{array}\right]$

${\mathrm{\&theta;}}^0={[e^{-a{T}_s}b(1-e^{-a{T}_s})/\mathrm{ab}(1-e^{-a{T}_s}){d_f}^{+}/\mathrm{ab}(1-e^{-a{T}_s}){d_f}^{-}/a]}^T$

Then, according to least squares identification theoretical method, θ is drawn⁰Optimal estimation vector:

$\widehat{{\mathrm{\&theta;}}^0}={\left({\left({\mathrm{\&Phi;}}^0\right)}^T{\mathrm{\&Phi;}}^0\right)}^{-1}{\left({\mathrm{\&Phi;}}^0\right)}^T{Y}^0---\left(6\right)$

Wherein, ω (2), ω (3) ... ω (N) is the 2nd, 3 ... the motor shaft angular velocity in N number of sampling period,

I (1), i (2) ... i (N-1) is the 1st, 2 respectively ... the current of electric in N-1 sampling period,

Then, definition is respectively intended to describe d_f ⁺、d_f ^-The function P in direction_V(ω(k))、N_V(ω (k)):

$P_V\left(\mathrm{\&omega;}\left(k\right)\right)=\left\{\begin{array}{cc}0& \mathrm{\&omega;}\left(k\right)<\mathrm{\&Omega;}\\ 1& \mathrm{\&omega;}\left(k\right)\&GreaterEqual;\mathrm{\&Omega;}\end{array}\right.,N_V\left(\mathrm{\&omega;}\left(k\right)\right)=\left\{\begin{array}{cc}0& \mathrm{\&omega;}\left(k\right)>-\mathrm{\&Omega;}\\ -1& \mathrm{\&omega;}\left(k\right)\&le;-\mathrm{\&Omega;}\end{array}\right.$

Wherein, Ω is the threshold values of the axle zero angular velocity defining servomotor

Thus, NC machine tool feed system equivalent inertia is drawn

J_e=K_t/ b,

Thus, equivalent damping is drawn

B_e=aJ_e

Wherein, $a=-1n\left(\widehat{{\mathrm{\&theta;}}^0}\left(1\right)\right)/T_s,b=a\widehat{{\mathrm{\&theta;}}^0}\left(2\right)/(1-e^{-a{T}_s})$

Wherein, Represent respectivelyThe 1st of vector, 2 row components,

Fig. 4 (a) is the result of the equivalent inertia identification in embodiment；

Fig. 4 (b) is the result of the equivalent damping identification in embodiment.

When carrying out identification test, first take less scale factor K, be then gradually increased, stop experiment when the result of identification tends towards stability, take final experimental result value as identification result, as shown in Figure 4.

Finally, the present invention is not only only limited to embodiment, based on spirit of the invention, can carry out various change, these not got rid of from the scope of the present invention.

Claims (3)

1. a NC machine tool feed system equivalent inertia and the discrimination method of equivalent damping, it is characterised in that have:

1st step, sets up the feed system second-order dynamic model simplified and is calculated its difference equation；

Second step, by G code displacement signal, NC machine tool feed system is continuously and effectively encouraged, process is: required pumping signal u is made up of the second displacement curve of the constant duration of m section difference acceleration, and m < 10, every section of conic section acceleration respectively:

a₀K,a₁K,···,a_iK,···,a_mK

Wherein K is scale factor constant,

Then, it is stipulated that formed whole displacement excitation curve, and n by the n little straightway of section constant duration > > m,

Wherein, every little straightway time interval is T_s, it is the minimum interval of curve discrete, takes machine tool position controller and control the integral multiple in cycle,

Then, the speed command v of the little straightway of kth section is obtained_kWith displacement commands u_k:

Wherein, described a_ijRefer to jth section conic section section at i-th section of little straightway,

And draw current signal data and the motor speed signal data of feed servo motor simultaneously；

Third step, by described current signal data and described tach signal data, after bringing described difference equation into, draws equivalent inertia and the equivalent damping of NC machine tool feed system according to method of least square from described difference equation.

NC machine tool feed system equivalent inertia the most according to claim 1 and the discrimination method of equivalent damping, it is characterised in that:

In described 1st step, described feed system second-order dynamic model and described difference equation are drawn by procedure below:

First, theoretical according to relevant control, with motor shaft as object of study, the establishment differential equation:

Wherein, K_tFor motor torque constant, J_eFor equivalent inertia, B_eFor equivalent damping, ω is motor angular velocity, and i is current of electric,

Then, the described differential equation is rewritten into equation (2):

Then, by both sides Laplace transformation computing described equation (2) is deformed into equation (3):

Wherein, W, I are the Laplace transformation form of ω, i respectively, and a=B_e/J_e, b=K_t/J_e, s is Laplace transform operator,

Then, with T_sFor the sampling period, to described equation (3) discretization, draw difference equation (4):

Then, the T of moment of friction will be derived from_e(k)/K_tIt is equivalent to disturb electric current, it is assumed that with ω interference electric current be in the same directionReverse interference electric current isBoth can be seen as the function of ω, now T_eCan be write as:

Wherein, P_V(ω(k))、N_V(ω (k)) is used to describe respectivelyThe function in direction, is defined by the formula:

Wherein, Ω is the threshold values judging velocity attitude, then draw difference equation (5) according to described difference equation (4),

。

NC machine tool feed system equivalent inertia the most according to claim 2 and the discrimination method of equivalent damping, it is characterised in that:

In described third step, after bringing described current signal data and described tach signal data into described difference equation, draw described equivalent inertia and equivalent damping according to method of least square from described difference equation, realized by procedure below:

First, for described difference equation (5) definition following equalities:

Y⁰=[ω (2) ω (3) ... ω (N)]^T

Then, according to least squares identification theoretical method, draw described θ⁰Optimal estimation vector:

Wherein, ω (2), ω (3), ω (N) is the 2nd, 3 ... the motor shaft angular velocity in N number of sampling period,

Wherein, i (2), i (3), ω (N-1) are the 1st, 2 respectively ... the current of electric in N-1 sampling period,

Then, definition is respectively intended to describeThe function P in direction_V(ω(k))、N_V(ω (k)):

Wherein, Ω is the threshold values judging velocity attitude,

Thus, NC machine tool feed system equivalent inertia is drawn:

J_e=K_t/ b,

Thus, equivalent damping is drawn:

B_e=aJ_e

Wherein,

Represent respectivelyThe 1st of vector, 2 row components.