CN112207331B - Dynamic integrated modeling method for milling process of single-shaft feeding system - Google Patents

Dynamic integrated modeling method for milling process of single-shaft feeding system Download PDF

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CN112207331B
CN112207331B CN202010956712.6A CN202010956712A CN112207331B CN 112207331 B CN112207331 B CN 112207331B CN 202010956712 A CN202010956712 A CN 202010956712A CN 112207331 B CN112207331 B CN 112207331B
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CN112207331A (en
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张星
张伟
赵钊
赵万华
尹佳
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Xian Jiaotong University
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B23—MACHINE TOOLS; METAL-WORKING NOT OTHERWISE PROVIDED FOR
    • B23C—MILLING
    • B23C1/00—Milling machines not designed for particular work or special operations
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B23—MACHINE TOOLS; METAL-WORKING NOT OTHERWISE PROVIDED FOR
    • B23Q—DETAILS, COMPONENTS, OR ACCESSORIES FOR MACHINE TOOLS, e.g. ARRANGEMENTS FOR COPYING OR CONTROLLING; MACHINE TOOLS IN GENERAL CHARACTERISED BY THE CONSTRUCTION OF PARTICULAR DETAILS OR COMPONENTS; COMBINATIONS OR ASSOCIATIONS OF METAL-WORKING MACHINES, NOT DIRECTED TO A PARTICULAR RESULT
    • B23Q5/00—Driving or feeding mechanisms; Control arrangements therefor
    • B23Q5/22—Feeding members carrying tools or work
    • B23Q5/34—Feeding other members supporting tools or work, e.g. saddles, tool-slides, through mechanical transmission

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Abstract

A dynamics integration modeling method for a milling process of a single-shaft feeding system comprises the steps of carrying out equivalent modeling on a control system module, carrying out equivalent modeling on a PWM (pulse width modulation) and inverter module, carrying out equivalent modeling on a servo motor module, carrying out dynamics equivalent modeling on a mechanical system module, carrying out equivalent modeling on a milling module, carrying out model integration, and carrying out discretization on an integrated model; the invention realizes the physical simulation of the whole process from the instruction input of the ball screw feeding system to the displacement output of the workbench; meanwhile, the simulation calculation of the milling force, the vibration state of the cutter and the machined surface of the part is realized; and establishing an integrated model of the single-shaft ball screw feeding system and the milling process, and realizing the coupling analysis of the interaction between the single-shaft ball screw feeding system and the milling process.

Description

Dynamic integrated modeling method for milling process of single-shaft feeding system
Technical Field
The invention belongs to the technical field of motion control of numerical control machines, and particularly relates to a dynamic integrated modeling method for a milling process of a single-shaft feeding system.
Background
The performance and the precision of the numerical control machine tool as an industrial master machine have a crucial influence on the development of the whole industry. In the milling process of the part, feeding shafts of the machine tool provide feeding motion for the tool and the part, and the coordination of the feeding shafts generates space relative motion between the tool and the part; meanwhile, the cutter removes redundant materials on the part blank through a cutting process, so that the blank is machined into a designed shape and size, namely, a designed final part. In this process, the performance and accuracy of the machine itself, as well as the cutting process, become two major factors affecting the quality of the final part.
The ideal feed motion and cutting process are necessary conditions for processing qualified parts, but there are many influencing factors in the actual processing process, which influence the processing process, thereby reducing the profile accuracy, surface quality and the like of the final parts. From the perspective of a machine tool, due to factors such as non-ideal characteristics of an electrical link in a system, flexibility of a mechanical link and the like, actual displacement output of the system usually deviates from an instruction, so that deviation is generated between the relative position of a cutter and a part; from the viewpoint of the cutting process, the machining process is affected by the vibration, deformation and other factors of the tool. In order to improve the machining precision and quality of parts, a great deal of theoretical modeling and analysis work is carried out by numerous scholars around a machine tool and a cutting process respectively, and remarkable results are obtained.
In fact, however, the feed motion of the machine tool and the cutting process are an integral body, and a complex force-displacement coupling action process exists between the feed motion and the cutting process: due to the influence of non-ideal factors, the instantaneous engagement state of the cutter-part is changed by the speed and displacement fluctuation of the feed motion, so that the cutting process is influenced; on the other hand, the high-frequency fluctuating cutting force and vibration generated in the cutting process also become interference sources of the feeding system, and significantly affect the performance of the feeding system. With the development of production technology, the requirements on the processing quality of parts are higher and higher, and the influence caused by the coupling effect between the machine tool and the cutting process is not negligible.
At present, most theoretical modeling methods are used for modeling a machine tool and a cutting process independently or simplifying one aspect of the machine tool and the cutting process to a greater extent, and the modeling method is difficult to carry out deep analysis research on the coupling effect from the perspective of a system, so that an integrated modeling and analyzing method for a machine tool feeding system and a milling process is lacked.
Disclosure of Invention
In order to overcome the defects of the prior art, the invention aims to provide a dynamic integrated modeling method for a milling process of a single-shaft feeding system, which is used for carrying out detailed modeling and solving on each link in a ball screw feeding system and realizing the physical simulation of the whole process from instruction input to displacement output of a workbench of the ball screw feeding system; modeling the engagement process of the cutter and the part in the milling process, and realizing the simulation calculation of the milling force, the vibration state of the cutter and the machined surface of the part; and integrating a feeding system model and a cutting process model, establishing an integrated model of a single-shaft ball screw feeding system and a milling process, and realizing the coupling analysis of interaction between the feeding system model and the milling process model.
In order to achieve the purpose, the invention adopts the technical scheme that:
a dynamic integrated modeling method for a milling process of a single-shaft feeding system comprises the following steps:
step 1) equivalent modeling of a control system module:
the control system module adopts a closed-loop control mode comprising a position loop, a speed loop and a current loop, the position loop controller adopts a proportional controller, and the speed loop controller, the Q-axis controller and the D-axis controller all adopt proportional-integral controllers; respectively calculating input and output values of a position loop, a speed loop and a current loop by taking a displacement instruction output by a numerical control system as input;
step 2) equivalent modeling of PWM and inverter modules:
based on SVPWM working principle, according to voltage vector amplitude modeling and phase output by a control system module, determining on-off states and on-off time of 6 Insulated Gate Bipolar Transistors (IGBT) and three-phase voltage values applied to a motor stator winding;
step 3) equivalent modeling of the servo motor module:
according to the coupling relation between the electricity, the magnetism, the force and the motion in the motor, three-phase currents of an electricity physical quantity stator, three-phase counter electromotive force of the stator and currents of a D shaft and a Q shaft are respectively measured; physical quantities of the magnetic field include rotor magnetomotive force, stator magnetomotive force and synthetic magnetomotive force; the mechanical physical quantity motor outputs torque and a torque angle; calculating the actual displacement, speed, acceleration, motor rotor displacement and speed of the kinematic physical quantity workbench;
step 4) dynamic equivalent modeling of a mechanical system module:
respectively carrying out equivalence on a motor rotor 21, a workbench 22, a sliding block 23, a guide rail 24, a bearing 25, a nut 26, a lead screw 27 and a coupler 28 which are contained in a mechanical structure of the ball screw feeding system, obtaining a dynamic equation of a mechanical system module by a Newton-Euler equation, further converting the dynamic equation into a state space representation form, and representing the friction force of the mechanical system by using a Stribeck model;
step 5) equivalent modeling of the milling module:
regarding the part as a rigid body, regarding the cutter as a flexible body, establishing a two-degree-of-freedom dynamic model, and calculating the vibration state of the cutter; the actual displacement of the workbench and the vibration of the cutter are considered, and the actual engagement state of the cutter and the part is solved; calculating to obtain instantaneous cutting force according to the instantaneous engagement state of the cutter and the part;
step 6) integration of models:
the output of the control system module is used as the input of the PWM and inverter module, the output of the PWM and inverter module is used as the input of the servo motor module, D, Q shaft current output by the servo motor module is fed back to the control system module, the output motor torque is used as the input of the mechanical system module, the mechanical system module outputs the motion state physical quantity of each motion part, the displacement of the workbench and the speed of the motor rotor are fed back to the control system module, the electrical angle of the motor rotor is fed back to the control system module and the servo motor module, and the displacement of the workbench is output to the milling module; feeding back the cutting force obtained by the calculation of the milling module to the mechanical system module to act on the workbench; the control system module, the PWM and inverter module, the servo motor module, the mechanical system module and the milling module are integrated into a coupled integrated model;
step 7) discretizing the integrated model:
and dispersing continuous time into time steps at equal intervals, replacing continuous variables related to time in the integrated model by values corresponding to the current simulation time, converting all differential equations in the integrated model into differential equations, and converting the continuous form of a state space into a discrete form.
The invention has the beneficial effects that:
according to the invention, through detailed modeling of each electrical and mechanical link and the milling process in the ball screw feeding system, an integrated model of the ball screw feeding system and the milling process comprising a control system, a servo amplifier, a PWM module, a servo motor, a mechanical system and the milling process is constructed, the whole-process physical simulation from displacement instruction input to final part machining surface output in the milling process is realized, and the calculation prediction of each electrical, magnetic, force, motion and other physical quantities in the machine tool and the milling process is realized, so that the simulation prediction and the system coupling analysis of the machine tool running state and the actual cutting state are realized. Compared with the existing modeling method, the modeling method has the advantages that the machine tool and the milling process can be regarded as a whole, systematic coupling analysis can be carried out, the force-displacement interaction and influence between the machine tool and the milling process in the milling process can be further analyzed, and the generation, transmission, coupling and influence mechanisms of various errors in the part milling process can be further comprehensively and deeply researched.
Drawings
Fig. 1 is a schematic diagram of the milling process of the structure of the single-shaft ball screw feeding system.
Fig. 2 is a schematic diagram of an integrated model of a single-shaft ball screw feeding system and a milling process.
Fig. 3 is a schematic diagram of an inverter.
Fig. 4 is a schematic structural diagram of a three-phase ac permanent magnet synchronous motor.
Fig. 5 is a model of a three-phase ac permanent magnet synchronous motor.
Fig. 6 is a mechanical structure diagram of the ball screw feeding system.
Fig. 7 is an equivalent dynamic model of the ball screw feed system.
Fig. 8 is a milling process dynamics model.
Fig. 9 is a schematic view of a geometric model of the tool.
Fig. 10 is a schematic view showing a state of instantaneous tool-part engagement.
Fig. 11 is a flow chart of model solving integrating the uniaxial ball screw feeding system and the milling process.
FIG. 12 shows simulation results of the following error of the stage.
Fig. 13 shows the result of the table feed speed simulation.
Fig. 14 is a three-phase current simulation result of the servo motor.
Fig. 15 is a simulation result of three-phase voltages applied to three-phase windings of the servo motor.
Fig. 16 shows a three-phase back electromotive force simulation result of the servo motor.
Fig. 17 shows a simulation result of the output torque of the servo motor.
Fig. 18 shows the results of friction simulation of a mechanical system.
Fig. 19 shows a stator synthetic magnetomotive force simulation result of the servo motor.
Fig. 20 shows the results of the milling force simulation in three directions.
Fig. 21 is a simulation result of the vibration state of the tool.
Fig. 22 shows the results of surface simulation after the part machining.
Detailed Description
The invention will be further described by way of example with reference to the accompanying drawings.
As shown in fig. 1, the single-shaft ball screw feeding system consists of a numerical control system, a servo driving system and a ball screw mechanical transmission system; the numerical control system has the main functions of decoding an NC code program through a decoder, performing interpolation calculation through an interpolator, generating a displacement and speed instruction sequence and sending the displacement and speed instruction sequence to the servo driving system; the servo driving system has the main functions that an input command sequence passes through a position controller, a speed controller, a current controller, a PWM and an inverter module to generate driving voltage; under the action of driving voltage, a permanent magnet motor in the ball screw mechanical transmission system generates torque through electromechanical energy conversion, drives a coupler and a screw to rotate, and further converts the rotary motion into the translational motion of the workbench through a screw-nut pair; in the process, the current controller detects and feeds back the driving current to realize current closed-loop control; the motor encoder detects and feeds back the actual rotating speed of the motor rotor, and transmits the actual rotating speed to the speed controller to realize speed closed-loop control; and a grating ruler on the workbench detects and feeds back the actual displacement of the workbench, and conveys the actual displacement to a position controller to realize position closed-loop control. The displacement of the table produces a relative movement between the tool and the part, the tool removing the excess material of the blank by means of a milling process, resulting in a final part, in which process cutting forces are generated, accompanied by vibrations of the tool. The cutting force is transmitted to the worktable, and excitation and disturbance are generated on the worktable, and the movement and control of the machine tool are influenced.
Referring to fig. 2, the dynamic integrated model of the milling process of the single-shaft feeding system constructed by the invention comprises a control system module, a PWM and inverter module, a servo motor module, a mechanical system module and a milling module.
A dynamic integrated modeling method for a milling process of a single-shaft feeding system comprises the following steps:
step 1) equivalent modeling of a control system module:
the control system module adopts a closed loop control mode comprising a position loop, a speed loop and a current loop, wherein the position loop controller adopts a proportional controller, and the gain of the proportional controller is KpDenotes that t is time, xr(t) is a displacement command, theta, output by the numerical control systemwta(t) when the actual displacement output of the workbench is converted into the rotating angle corresponding to the motor rotor end, the output of the position ring controller is
ωr(t)＝Kp(xr(t)η-θwta(t)) (1)
In the formula, eta is the transmission ratio of the nut-screw pair, omegar(t) is a speed command value;
the speed loop controller adopts a proportional-integral controller, and the gain of the proportional-integral controller is KvExpressed by the time constant TvThat means, the output of the speed loop controller is:
Figure BDA0002678879720000051
in the formula, ωra(t) is the actual angular velocity value of the rotor of the electric machine, ir(t) is a current command value;
the output of the speed loop controller is used as the reference input of the Q-axis current loop, and the Q-axis and D-axis current loop controllers are proportional-integral controllers with gains of KiqAnd KidThe time constants are represented by TiqAnd TidIndicates that the output of the Q-axis current loop controller is
Figure BDA0002678879720000052
In the formula，iqa(t) is the actual value of the motor Q-axis current, Vq(t) is a Q-axis voltage command value;
the D-axis uses zero input, so the output of the D-axis current loop controller is:
Figure BDA0002678879720000053
in the formula ida(t) is the actual value of the D-axis current of the motor, Vd(t) is a Q-axis voltage command value;
output value V of current loop controllerq(t) and Vd(t) all need to pass through an amplitude limiting link, and the limited amplitude is determined by the performance of an actual control servo system; the voltage values after the amplitude limiting links are respectively set as Vqr(t) and Vdr(t) a voltage value expressed in a two-phase rotating coordinate system, and subjected to coordinate conversion as follows
Figure BDA0002678879720000054
Obtaining a voltage value V under a two-phase static coordinate systemα(t) and Vβ(t) in the formula, θre(t) is the actual electrical angle value of the motor; further expressing the voltage value under the two-phase static coordinate system by using a polar coordinate to obtain the final output value of the control system module as
Figure BDA0002678879720000055
Step 2) equivalent modeling of PWM and inverter modules:
the PWM and inverter module mainly carries out modeling representation on a driver in an actual servo system and has the function of obtaining a three-phase voltage value V actually applied to a servo motor according to an output value of the control system moduleA(t)、VB(t) and VC(t), the function is realized by adopting the current mature SVPWM algorithm;
referring to FIG. 3, the inverter operates the sourceSimply know, by controlling T1-T6The on-off states of 6 Insulated Gate Bipolar Transistors (IGBTs) in total convert the DC voltage to AC voltage, where T2、T4、T6Are always respectively connected with T1、T3、T5Is in the opposite on-off state, so that only T needs to be obtained1、T3、T5The on-off state is only required; according to the SVPWM algorithm principle, three on-off states, the on-off state and the on-off time of each IGBT and three-phase voltage V applied to a motor winding are contained in one PWM switching period timeA(t)、VB(t) and VC(t) is dependent on the magnitude V of the voltage vectorm(t) and its phase θV(t), which is specifically related as follows:
2.1) when
Figure BDA0002678879720000056
The method comprises the following steps:
the first state:
on-off state: t is1Is turned on, T3、T5Disconnecting;
duration:
Figure BDA0002678879720000061
three-phase voltage:
Figure BDA0002678879720000062
wherein, TsFor PWM switching period, UdcIs a direct current voltage;
and a second state:
on-off state: t is1、T3Is turned on, T5Disconnecting;
duration:
Figure BDA0002678879720000063
three-phase voltage:
Figure BDA0002678879720000064
and a third state:
on-off state: t is1、T3、T5All are disconnected;
duration: t is t3＝Ts-t1-t2；
Three-phase voltage: vA(t)＝Vb(t)＝Vc(t)＝0；
2.2) when
Figure BDA0002678879720000065
The method comprises the following steps:
the first state:
on-off state: t is3Is turned on, T1、T5Disconnecting;
duration:
Figure BDA0002678879720000066
three-phase voltage:
Figure BDA0002678879720000067
and a second state:
on-off state: t is1、T3Is turned on, T5Disconnecting;
duration:
Figure BDA0002678879720000068
three-phase voltage:
Figure BDA0002678879720000069
and a third state:
on-off state: t is1、T3、T5All are disconnected;
duration: t is t3＝Ts-t1-t2；
Three-phase voltage: vA(t)＝Vb(t)＝Vc(t)＝0；
2.3) when
Figure BDA00026788797200000610
The method comprises the following steps:
the first state:
on-off state: t is3Is turned on, T1、T5Disconnecting;
duration:
Figure BDA00026788797200000611
three-phase voltage:
Figure BDA00026788797200000612
and a second state:
on-off state: t is3、T5Is turned on, T1Disconnecting;
duration:
Figure BDA0002678879720000071
three-phase voltage:
Figure BDA0002678879720000072
and a third state:
on-off state: t is1、T3、T5All are disconnected;
duration: t is t3＝Ts-t1-t2；
Three-phase voltage: vA(t)＝Vb(t)＝Vc(t)＝0；
2.4) when
Figure BDA0002678879720000073
The method comprises the following steps:
the first state:
on-off state: t is5Is turned on, T1、T3Disconnecting;
duration:
Figure BDA0002678879720000074
three-phase voltage:
Figure BDA0002678879720000075
and a second state:
on-off state: t is3、T5Is turned on, T1Disconnecting;
duration:
Figure BDA0002678879720000076
three-phase voltage:
Figure BDA0002678879720000077
and a third state:
on-off state: t is1、T3、T5All are disconnected;
duration: t is t3＝Ts-t1-t2；
Three-phase voltage: vA(t)＝Vb(t)＝Vc(t)＝0；
2.5) when
Figure BDA0002678879720000078
The method comprises the following steps:
the first state:
on-off state: t is5Is turned on, T1、T3Disconnecting;
duration:
Figure BDA0002678879720000079
three-phase voltage:
Figure BDA00026788797200000710
and a second state:
on-off state: t is1、T5Is turned on, T3Disconnecting;
duration:
Figure BDA00026788797200000711
three-phase voltage:
Figure BDA00026788797200000712
and a third state:
on-off state: t is1、T3、T5All are disconnected;
duration: t is t3＝Ts-t1-t2；
Three-phase voltage: vA(t)＝Vb(t)＝Vc(t)＝0；
2.6) when
Figure BDA0002678879720000081
The method comprises the following steps:
the first state:
on-off state: t is1Is turned on, T3、T5Disconnecting;
duration:
Figure BDA0002678879720000082
three-phase voltage:
Figure BDA0002678879720000083
and a second state:
on-off state: t is1、T5Is turned on, T3Disconnecting;
duration:
Figure BDA0002678879720000084
three-phase voltage:
Figure BDA0002678879720000085
and a third state:
on-off state: t is1、T3、T5All are disconnected;
duration: t is t3＝Ts-t1-t2；
Three-phase voltage: vA(t)＝Vb(t)＝Vc(t)＝0；
The on-off time of each IGBT and the three-phase voltage applied to the motor can be obtained by the formula; the on-off sequence of the intelligent high-voltage power supply can be further optimized by adopting a mature seven-segment SVPWM algorithm to obtain better performance;
step 3) equivalent modeling of the servo motor module:
the modeling is carried out on the three-phase alternating current permanent magnet synchronous motor, the internal structure of the three-phase alternating current permanent magnet synchronous motor is shown in figure 4, and the three-phase alternating current permanent magnet synchronous motor mainly comprises a stator iron core 11, a stator winding 12, a permanent magnet 13, a rotor iron core 14 and the like;
as shown in FIG. 5, a local coordinate system X of the motor is establishedmYmZmThe origin is located at the center of the rotating shaft of the servo motor, ZmThe axis of the shaft coinciding with the axis of the rotor, XmThe axis direction of the shaft is consistent with that of the A phase winding of the motor stator, and Y ismThe axis being perpendicular to XmAxis and ZmA shaft;
the rotor permanent magnet generates magnetic flux with the density of
Figure BDA0002678879720000086
In the formula, BrThe residual magnetic flux density of the permanent magnet, g is the length of the air gap, h is the thickness of the permanent magnet, murmFor the relative permeability of the permanent magnet, the air gap flux of each pole of the motor rotor is:
Figure BDA0002678879720000087
in the formula, DrDenotes the diameter of the rotor, L is the axial length of the rotor core, PnThe number of magnetic pole pairs of the motor rotor is counted;
the magnetomotive force of each pole of the rotor of the motor is
Figure BDA0002678879720000088
In the formula, mu0Is air permeability, AmThe air gap area corresponding to each pole of the permanent magnet; the total magnetic potential of the magnetic circuit is as follows:
Fr＝2Fp (10)
for the stator, the phase current is obtained by the following formula
Figure BDA0002678879720000091
In the formula, ea(t)、eb(t)、ec(t) are respectively stator three-phase winding counter electromotive force, and R is stator phase resistance;
the magnetomotive force generated by each pole of each phase winding is as follows:
Figure BDA0002678879720000092
in the formula, kwIs a winding coefficient for considering the winding distribution effect; t isphThe number of turns of each phase winding; respectively make each phase magnetomotive force along XmAxis and YmPerforming axial projection to obtain the magnetic potential of each pole of the stator synthetic magnetic field in XmAxis, YmThe components of the axis are:
Figure BDA0002678879720000093
determining stator resultant flux edge XmAxis, YmThe components of the axis are:
Figure BDA0002678879720000094
the rotor flux is respectively directed along XmAxis and YmThe axis is projected to obtain the components:
Figure BDA0002678879720000095
so as to obtain the components of the total magnetic flux of the rotor and the stator on the x axis and the y axis respectively as follows:
Figure BDA0002678879720000096
the amplitude of the total flux vector inside the motor and its phase are
Figure BDA0002678879720000097
Here, θΦsr(t) has a value range of [ - π, π]The specific quadrant thereof is according to phisrx(t) and Φsry(t) judging the symbol;
further, the total magnetic flux at the three-phase winding of the stator is obtained as follows:
Figure BDA0002678879720000101
the back emf of the stator windings is then:
Figure BDA0002678879720000102
according to the formula (13), the stator synthetic magnetomotive force is calculated as follows:
Figure BDA0002678879720000103
the total resultant magnetomotive force of the stator and rotor is then calculated by:
Figure BDA0002678879720000104
wherein, deltasr(t) represents the included angle between the stator magnetomotive force axis and the rotor magnetomotive force axis and
Figure BDA0002678879720000105
fix (·) denotes rounding to zero;
the motor output torque is obtained according to the following formula:
Figure BDA0002678879720000106
for d and q axis currents of the motor, the d and q axis currents are obtained by carrying out Park conversion on three phase currents of the motor, namely
Figure BDA0002678879720000107
The mechanical angular velocity of the motor rotor is calculated by the following mechanical motion equation:
Figure BDA0002678879720000108
step 4) dynamic equivalent modeling of a mechanical system module:
as shown in fig. 6, the structure considered by the dynamic model of the ball screw feeding system mainly comprises a motor rotor 21, a workbench 22, a slide block 23, a guide rail 24, a bearing 25, a nut 26, a screw 27 and a coupling 28; the equivalent dynamic model is shown in fig. 7, in which the motor rotor 21 is equivalent to the moment of inertia Jr(ii) a The working table 22 is equivalent to a mass mwt(ii) a The screw 27 is equally divided into three sections which are respectively equivalent to three masses ms1、ms2、ms3And three moments of inertia Js1、Js2、Js3And respectively pulling and pressing the rigidity k by a screw rodssL、kssRAnd screw torsional stiffness kθsL、kθsRConnecting; the coupling 28 is equivalent to a torsion spring unit with a torsional stiffness kθcRepresents; the nut 26 and the likeEffective as tension-compression springs, with stiffness ksnRepresents; the bearings 25 at the two ends of the screw rod are respectively equivalent to have the rigidity of ksbLAnd ksbRThe tension and compression spring; the damping element considered comprises essentially the sliding damping c between the guide rail 24 and the slide 23wtDamping of the translation of the nut 26 csnAnd a rotational damping cθnDamping of translation c of bearing 25sbL、csbRAnd a rotational damping cθbL、cθbRTranslation damping c of the screw 27ssL、cssRAnd a rotational damping cθsL、cθsRRotational damping of the coupling 28 cθcAnd motor rotor bearing damping cθr；
From the Newton-Euler equation, the kinetic equation of the mechanical system can be obtained as
Figure BDA0002678879720000111
Wherein M is the mass matrix of the mechanical system, and
Figure BDA0002678879720000112
k is the stiffness matrix of the mechanical system,
Figure BDA0002678879720000113
c is the damping matrix of the mechanical system,
Figure BDA0002678879720000114
q (t) is the generalized coordinate vector of the mechanical system,
q(t)＝[xwt(t),xs1(t),xs2(t),xs3(t),θs1(t),θs2(t),θs3(t),θr(t)]T (29)
wherein x iswtIs the displacement of the table, xs1、xs2、xs3Translational displacement, theta, of three lead screws, respectivelys1、θs2、θs3The torsion angles, theta, of three lead screws respectivelyrThe torsion angle of the motor rotor;
f (t) is a generalized force vector,
F(t)＝[Ff(t)+Fx(t) 0 0 0 0 0 0 Te(t)]T (30)
wherein, Fx(t) milling force in X-axis direction, Ff(t) is the mechanical system friction, calculated using the Stribeck model as follows:
Figure BDA0002678879720000121
wherein v iswt(t) Table speed, Fc、Fs、VsCan be identified through experiments, delta is 2;
further expressing the mechanical system dynamic equation in a state space form, and selecting a state variable x1(t)＝q(t)，
Figure BDA0002678879720000122
The state space expression of the mechanical system is as follows:
Figure BDA0002678879720000123
wherein, the state vector:
Figure BDA0002678879720000124
system matrix:
Figure BDA0002678879720000125
constant coefficient matrix:
Figure BDA0002678879720000126
outputting a matrix: c ═ I O]And inputting a vector: u (t) f (t), y (t) is output toAn amount;
step 5) equivalent modeling of the milling module:
(1) dynamic modeling of the milling process:
as shown in FIG. 8, a tool local coordinate system O is establishedtXtYtZtOrigin of coordinate system OtAt the center of the bottom of the tool in the initial state, and the coordinate axis XtSame direction of feed of the machine tool, ZtIn the direction of the tool axis, YtPerpendicular to XtOtZtAnd (4) a plane. The part is regarded as a rigid body, the cutter is simplified into a two-degree-of-freedom dynamic system, and the vibration directions of the two-degree-of-freedom dynamic system are respectively equal to XtAxis, YtThe axial directions are the same;
setting a certain time edge XtAxis, YtThe milling forces generated in the axial direction are respectively Fx(t) and Fy(t), the vibration state of the tool can be obtained by the following equation:
Figure BDA0002678879720000127
in the formula, mxt、mytRespectively being a tool in XtAxis, YtMass in the axial direction, cxt、cytAre each XtAxis, YtDamping coefficient in axial direction, kxt、kytAre each XtAxis, YtThe rigidity in the axial direction and the above physical quantities can be obtained by performing a modal test experiment on the cutter. x is the number oft，ytRespectively the center position of the cutter is at XtAxis, YtCoordinates of the axis;
(2) and (3) simulation calculation of the instantaneous engagement state of the cutter and the part and the machined surface of the part:
as shown in FIG. 9, when the helical end mill has Z teeth, beta helix angle, and R radius, the tooth spacing angle phi is equal to that of a milling cutter with uniformly distributed cutter teethpCan be expressed as
Figure BDA0002678879720000128
Cutting edges of the cutter are divided into M cutting microelements with the thicknesses of dz along the axial direction, and when the initial moment is set, the instantaneous radial contact angle of the cutting microelements at the end part of the 1 st cutter tooth is 0 degrees, and then when a certain moment t is obtained, the instantaneous radial contact angle of the microelements of the ith cutting edge on the jth cutter tooth is as follows:
Figure BDA0002678879720000131
in the formula, S is the rotating speed of the main shaft;
for the machine tool with the worktable moving in a feeding way and the cutter being static, at a certain moment t, the displacement x of the worktable can be obtained by the step 4)wt(t) the instantaneous vibration position of the tool is (x) from the equation (33)t(t),yt(t)), it can be seen that at time t, the actual position of the center of the tool in the part coordinate system is the actual position of the center of the tool in consideration of the actual displacement of the table and the tool vibration
Figure BDA0002678879720000132
The coordinate S of the first cutting edge infinitesimal on the jth tooth at the time t can be further determinedjl(t) is
Figure BDA0002678879720000133
The upper formula is a space trajectory curved surface expression swept by each cutting edge of the spiral milling cutter in the milling process, and an envelope surface formed by the space trajectory curved surface expression is a processed surface of the part;
as shown in FIG. 10, the actual undeformed chip thickness h takes into account the actual displacement of the table and the tool vibrationjl(t) is the current cutting edge infinitesimal position Sjl(t) the distance between the intersection point of the line passing through the point and the axis of the tool and the path of the previous cutting edge;
(3) calculating the milling force:
the milling force calculation adopts a common instantaneous rigid force calculation model,
Figure BDA0002678879720000134
in the formula: dFt,jl，dFr,jl，dFa,jlRespectively representing tangential, radial and axial cutting force infinitesimal; ds represents the cutting edge infinitesimal length; ktc，Krc，KacRepresenting the tangential, radial and axial cutting force coefficients, respectively; kte，Kre，KaeRespectively representing tangential, radial and axial cutting edge force coefficients, wherein the cutting force coefficient and the cutting edge force coefficient can be obtained through experimental identification;
considering that the cutting edge infinitesimal height is small, ds ≈ dz, the tangential, radial and axial cutting force infinitesimal on the l-th cutting edge infinitesimal of thickness dz acting on tooth j can be expressed as:
Figure BDA0002678879720000135
in the formula: gjl(φjl(t)) is a unit step function for indicating whether the current cutting edge infinitesimal participates in cutting, which is defined as:
Figure BDA0002678879720000136
in the formula: phi is ast，φex-representing the cut-in angle and the cut-out angle, respectively;
for backmilling, the calculated expressions for the entry angle and exit angle are:
Figure BDA0002678879720000141
in the formula, aeIndicates the cutting width;
for down-cut, the calculated expressions for the cut-in angle and cut-out angle are:
Figure BDA0002678879720000142
through coordinate transformation, the following cutting force components acting on three axes in the rectangular coordinate system can be obtained:
Figure BDA0002678879720000143
by integrating in the axial direction and summing each tooth, the instantaneous cutting forces acting on the entire milling cutter in the feed, normal and axial directions can be found to be:
Figure BDA0002678879720000144
step 6) integration of models:
on the basis of the control system module, the PWM and inverter module, the servo motor module, the mechanical system module and the milling module which are constructed in the steps 1) to 5), the five modules are coupled and integrated to obtain an integrated model of the single-shaft ball screw feeding system and the milling process shown in the figure 2: will control the output V of the system modulem(t) and θV(t) as input to the PWM and inverter modules; combining PWM with output V of inverter moduleA(t)、VB(t)、VC(t) as an input to the servo motor module; output i of servo motor moduleqa(t)、ida(T) feedback to the control system module, which outputs Te(t) as an input to a mechanical system module; the mechanical system module outputs the motion state physical quantity of each motion part, feeds back the displacement of the workbench and the speed of the motor rotor to the control system module, and feeds back the electrical angle of the motor rotor to the control system module and the servo motor module; the displacement of the worktable is input into the milling module, and the milling force is fed back to the mechanical system module to act on the worktable, thereby forming a single-shaft ball screw feedingAn integrated model of the system and the milling process; it can be seen from the modeling process of each module and the integration process that the established integration model comprises the calculation of multiple physical quantities such as electricity, magnetism, force, motion and the like, and the physical quantities are mutually influenced, so that the system physical simulation model has strong coupling characteristics;
step 7), solving process of the integrated model:
in order to facilitate the solution simulation of the integrated model, the integrated model is converted into a time domain discrete model; in the discrete time domain, the continuous time t is dispersed into N time steps with equal intervals, wherein the time step is t/N, and then the time t is any timeiAnd transforming the integrated model as follows:
7.1) discretization of control system modules:
from equation (1), the output of the position loop controller is:
ωr(ti)＝Kp(xr(ti)η-θwta(ti-1)) (46)
the expression of the output of the speed loop controller in the discrete time domain is:
Figure BDA0002678879720000151
wherein ir0(ti)＝ir0(ti-1)+(ωr(ti)-ωra(ti-1))Δt；
Likewise, the expressions for the outputs of the Q-axis and D-axis current loop controllers in the discrete time domain may be converted to the following forms, respectively:
Figure BDA0002678879720000152
Figure BDA0002678879720000153
formulas (5) and (6) can be converted to the following formulas, respectively:
Figure BDA0002678879720000154
Figure BDA0002678879720000155
7.2) discretization of a servo motor module:
when the established three-phase alternating current permanent magnet synchronous motor model is converted into a time domain discrete model for simulation, an abnormal divergence phenomenon exists, and in order to solve the problem, the following calculation method is adopted:
back electromotive force of the stator winding is
Figure BDA0002678879720000156
The total flux change at the three-phase winding of the stator in the delta t time is
Figure BDA0002678879720000157
Therefore, the total magnetic flux at the three-phase winding of the stator at the current moment can be obtained as follows:
Figure BDA0002678879720000158
in order to avoid overlarge calculation error caused by undersize of the magnetic flux of the three-phase winding at a certain moment, the following three conditions are respectively calculated according to the size of the three-phase magnetic flux:
11 when max [ phi ]a(ti),Φb(ti),Φc(ti)}＝Φa(ti) The method comprises the following steps:
Figure BDA0002678879720000161
22 when max [ phi ]a(ti),Φb(ti),Φc(ti)}＝Φb(ti) The method comprises the following steps:
Figure BDA0002678879720000162
33 when max { phi [ ]a(ti),Φb(ti),Φc(ti)}＝Φc(ti) The method comprises the following steps:
Figure BDA0002678879720000163
then the total flux vector magnitude can be found to be:
Figure BDA0002678879720000164
it is in Xm、YmThe components of the axis are:
Figure BDA0002678879720000165
rotor flux at Xm、YmThe component of the axis can be calculated by:
Figure BDA0002678879720000166
thereby obtaining the stator magnetic flux at Xm、YmThe components of the axis are:
Figure BDA0002678879720000167
further obtaining stator magnetomotive force Xm、YmThe components of the axis are:
Figure BDA0002678879720000168
the stator three-phase magnetomotive force can be calculated according to the following formula:
Figure BDA0002678879720000169
wherein, Fso(ti) For introduced virtual items, Fso(ti)≡0；
The three-phase current of the motor stator can be directly obtained through magnetomotive force:
Figure BDA0002678879720000171
the amplitude and the phase of the stator synthetic magnetomotive force are as follows:
Figure BDA0002678879720000172
the included angle between the stator magnetomotive force axis and the rotor magnetomotive force axis is as follows:
Figure BDA0002678879720000173
the total resultant magnetomotive force of the stator and rotor can be obtained by the following formula
Figure BDA0002678879720000174
The motor output torque can be obtained according to the following formula
Figure BDA0002678879720000175
7.3) discretization of mechanical system modules:
converting a mechanical system state space equation represented by the formula into a discrete form:
Figure BDA0002678879720000176
wherein G ═ eAΔt,
Figure BDA0002678879720000177
7.4) discretization of the milling module:
7.4.1) discrete solution of the milling process dynamics model:
the milling process kinetic equation in the form of a differential equation obtained in step 5 is converted into a differential form, and it can be seen from equation (33) that t is the time tiAt any moment, the tool edge XtAxis and YtThe instantaneous acceleration of the shaft can be found by the following equation,
Figure BDA0002678879720000178
further, it is possible to obtain an instantaneous speed of
Figure BDA0002678879720000179
Finally, its vibrational displacement can be obtained by:
Figure BDA00026788797200001710
7.4.2) search solution of undeformed chip thickness:
in the time domain discrete solving process, the space track swept by each cutting edge of the helical milling cutter obtained by the formula (37) is a series of space discrete points, and for convenience of calculation, the thickness of the undeformed chip in the step 5) is used as the infinitesimal instantaneous radial contact angle phi between the last cutting edge track and the current cutting edgejl(ti) Close to the same position, infinitesimal distance from the current cutting edgeThe distance between the closest points is replaced. Since the cutting edge trajectory is a discrete point, the instantaneous radial contact angle on the upper cutting edge trajectory is
Figure BDA0002678879720000181
Point of (2)
Figure BDA0002678879720000182
A search is made in the vicinity to obtain the closest point and further the undeformed chip thickness, i.e.
Figure BDA0002678879720000183
Wherein, TωThe time of one rotation of the main shaft is K is a given integer and is used for designating a search interval;
for the calculation of the milling force, only the continuous time t in the corresponding expression is used as the discrete time tiReplacing;
the simulation flow of the integrated equivalent model is shown in fig. 11, and in order to further show the effect of the present invention, the simulation results of an embodiment are shown as follows, the simulation parameters of each control system module, PWM and inverter module, servo motor module and mechanical system module are respectively shown in tables 1-4, the main geometric parameters and milling process parameters of the tool are shown in tables 5-6, and the simulation parameter table of the control system module in table 1 is shown in table 6
Figure BDA0002678879720000184
TABLE 2 simulation parameter table for PWM and inverter modules
Figure BDA0002678879720000185
TABLE 3 simulation parameter table for servo motor module
Figure BDA0002678879720000186
TABLE 4 simulation parameter table for mechanical system module
Figure BDA0002678879720000187
Figure BDA0002678879720000191
TABLE 5 main geometrical parameters of the tool
Figure BDA0002678879720000192
TABLE 6 milling Process parameters
Figure BDA0002678879720000193
Referring to fig. 12-22, the present invention obtains the simulation results of the main physical quantities of each machine tool and the milling process in the milling process by using a linear motion command as an input, wherein fig. 12 is the following error of the worktable; FIG. 13 is the actual speed of the table; FIG. 14 is a three-phase current of the servo motor; FIG. 15 is a three-phase voltage applied to a three-phase winding of a servo motor; FIG. 16 is a three-phase back EMF of the servo motor; FIG. 17 shows the output torque of the servo motor; FIG. 18 is a friction force of a mechanical system; FIG. 19 is a stator resultant magnetomotive force of the servo motor; FIG. 20 shows milling forces in three directions; FIG. 21 shows the vibration state of the tool; FIG. 22 is a surface of the part after machining; from the simulation results, it can be seen that due to the disturbance effect in the cutting process, high-frequency fluctuation occurs in the three-phase current, the torque, the displacement, the speed and the like of the servo motor. Indicating that the cutting process has a significant effect on the operation of the machine tool.
The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention, and various modifications and changes may be made by those skilled in the art. Any modification, equivalent replacement, and improvement made according to the structure, shape, principle of the present invention shall be included in the protection scope of the present invention.

Claims (1)

1. A dynamic integrated modeling method for a milling process of a single-shaft feeding system is characterized by comprising the following steps:
step 1) equivalent modeling of a control system module:
the control system module adopts a closed-loop control mode comprising a position loop, a speed loop and a current loop, the position loop controller adopts a proportional controller, and the speed loop controller, the Q-axis controller and the D-axis controller all adopt proportional-integral controllers; respectively calculating input and output values of a position loop, a speed loop and a current loop by taking a displacement instruction output by a numerical control system as input;
step 2) equivalent modeling of PWM and inverter modules:
based on SVPWM working principle, according to voltage vector amplitude modeling and phase output by a control system module, determining on-off states and on-off time of 6 Insulated Gate Bipolar Transistors (IGBT) and three-phase voltage values applied to a motor stator winding;
step 3) equivalent modeling of the servo motor module:
according to the coupling relation between the electricity, the magnetism, the force and the motion in the motor, three-phase currents of an electricity physical quantity stator, three-phase counter electromotive force of the stator and currents of a D shaft and a Q shaft are respectively measured; physical quantities of the magnetic field include rotor magnetomotive force, stator magnetomotive force and synthetic magnetomotive force; the mechanical physical quantity motor outputs torque and a torque angle; calculating the actual displacement, speed, acceleration, motor rotor displacement and speed of the kinematic physical quantity workbench;
step 4) dynamic equivalent modeling of a mechanical system module:
respectively carrying out equivalence on a motor rotor 21, a workbench 22, a sliding block 23, a guide rail 24, a bearing 25, a nut 26, a lead screw 27 and a coupler 28 which are contained in a mechanical structure of the ball screw feeding system, obtaining a dynamic equation of the mechanical system by a Newton-Euler equation, further converting the dynamic equation into a state space representation form, and representing the friction force of the mechanical system by using a Stribeck model;
step 5) equivalent modeling of the milling module:
regarding the part as a rigid body, regarding the cutter as a flexible body, establishing a two-degree-of-freedom dynamic model, and calculating the vibration state of the cutter; the actual displacement of the workbench and the vibration of the cutter are considered, and the actual engagement state of the cutter and the part is solved; calculating to obtain instantaneous cutting force according to the instantaneous engagement state of the cutter and the part;
step 6) integration of models:
the output of the control system module is used as the input of the PWM and inverter module, the output of the PWM and inverter module is used as the input of the servo motor module, D, Q shaft current output by the servo motor module is fed back to the control system module, the output motor torque is used as the input of the mechanical system module, the mechanical system module outputs the motion state physical quantity of each motion part, the displacement of the workbench and the speed of the motor rotor are fed back to the control system module, the electrical angle of the motor rotor is fed back to the control system module and the servo motor module, and the displacement of the workbench is output to the milling module; feeding back the cutting force obtained by the calculation of the milling module to the mechanical system module to act on the workbench; the control system module, the PWM and inverter module, the servo motor module, the mechanical system module and the milling module are integrated into a coupled integrated model;
step 7) discretizing the integrated model:
dispersing continuous time into time steps at equal intervals, replacing continuous variables related to time in the integrated model by values corresponding to the current simulation time, converting all differential equations in the integrated model into differential equations, and converting the continuous form of a state space into a discrete form;
the position loop controller of the control system module adopts a proportional controller, and the gain of the proportional controller is KpDenotes that t is time, xr(t) is a displacement command, theta, output by the numerical control systemwta(t) when the actual displacement output of the worktable is converted into the corresponding rotation angle of the motor rotor end, the output of the position ring controllerIs composed of
ωr(t)＝Kp(xr(t)η-θwta(t)) (1)
In the formula, eta is the transmission ratio of the nut-screw pair, omegar(t) is a speed command value;
the speed loop controller adopts a proportional-integral controller, and the gain of the proportional-integral controller is KvExpressed by the time constant TvThat means, the output of the speed loop controller is:
Figure FDA0003141685520000021
in the formula, ωra(t) is the actual angular velocity value of the rotor of the electric machine, ir(t) is a current command value;
the output of the speed loop controller is used as the reference input of the Q-axis current loop, and the Q-axis and D-axis current loop controllers are proportional-integral controllers with gains of KiqAnd KidThe time constants are represented by TiqAnd TidIndicates that the output of the Q-axis current loop controller is
Figure FDA0003141685520000022
In the formula iqa(t) is the actual value of the motor Q-axis current, Vq(t) is a Q-axis voltage command value;
the D-axis uses zero input, so the output of the D-axis current loop controller is:
Figure FDA0003141685520000023
in the formula ida(t) is the actual value of the D-axis current of the motor, Vd(t) is a Q-axis voltage command value;
output value V of current loop controllerq(t) and Vd(t) all need to pass through an amplitude limiting link, and the limited amplitude is controlled by actual controlDetermining the performance of the system; the voltage values after the amplitude limiting links are respectively set as Vqr(t) and Vdr(t) a voltage value expressed in a two-phase rotating coordinate system, and subjected to coordinate conversion as follows
Figure FDA0003141685520000024
Obtaining a voltage value V under a two-phase static coordinate systemα(t) and Vβ(t) in the formula, θre(t) is the actual electrical angle value of the motor; further expressing the voltage value under the two-phase static coordinate system by using a polar coordinate, and obtaining the final output value of the control system module as follows:
Figure FDA0003141685520000025
the PWM and inverter module comprises the following modeling steps:
voltage vector amplitude V output by control system modulem(t) and its phase θV(T) determining the on-off status of 6 Insulated Gate Bipolar Transistors (IGBTs), wherein T2、T4、T6Are always respectively connected with T1、T3、T5The on-off state of (1) is opposite, and the specific relation is as follows:
3.1) when
Figure FDA0003141685520000031
The method comprises the following steps:
the first state:
on-off state: t is1Is turned on, T3、T5Disconnecting;
duration:
Figure FDA0003141685520000032
three-phase voltage:
Figure FDA0003141685520000033
wherein, TsFor PWM switching period, UdcIs a direct current voltage;
and a second state:
on-off state: t is1、T3Is turned on, T5Disconnecting;
duration:
Figure FDA0003141685520000034
three-phase voltage:
Figure FDA0003141685520000035
and a third state:
on-off state: t is1、T3、T5All are disconnected;
duration: t is t3＝Ts-t1-t2；
Three-phase voltage: vA(t)＝Vb(t)＝Vc(t)＝0；
3.2) when
Figure FDA0003141685520000036
The method comprises the following steps:
the first state:
on-off state: t is3Is turned on, T1、T5Disconnecting;
duration:
Figure FDA0003141685520000037
three-phase voltage:
Figure FDA0003141685520000038
and a second state:
on-off state: t is1、T3Is turned on, T5Disconnecting;
duration:
Figure FDA0003141685520000039
three-phase voltage:
Figure FDA00031416855200000310
and a third state:
on-off state: t is1、T3、T5All are disconnected;
duration: t is t3＝Ts-t1-t2；
Three-phase voltage: vA(t)＝Vb(t)＝Vc(t)＝0；
3.3) when
Figure FDA00031416855200000311
The method comprises the following steps:
the first state:
on-off state: t is3Is turned on, T1、T5Disconnecting;
duration:
Figure FDA0003141685520000041
three-phase voltage:
Figure FDA0003141685520000042
and a second state:
on-off state: t is3、T5Is turned on, T1Disconnecting;
duration:
Figure FDA0003141685520000043
three-phase voltage:
Figure FDA0003141685520000044
and a third state:
on-off state:T1、T3、T5all are disconnected;
duration: t is t3＝Ts-t1-t2；
Three-phase voltage: vA(t)＝Vb(t)＝Vc(t)＝0；
3.4) when
Figure FDA0003141685520000045
The method comprises the following steps:
the first state:
on-off state: t is5Is turned on, T1、T3Disconnecting;
duration:
Figure FDA0003141685520000046
three-phase voltage:
Figure FDA0003141685520000047
and a second state:
on-off state: t is3、T5Is turned on, T1Disconnecting;
duration:
Figure FDA0003141685520000048
three-phase voltage:
Figure FDA0003141685520000049
and a third state:
on-off state: t is1、T3、T5All are disconnected;
duration: t is t3＝Ts-t1-t2；
Three-phase voltage: vA(t)＝Vb(t)＝Vc(t)＝0；
3.5) when
Figure FDA00031416855200000410
The method comprises the following steps:
the first state:
on-off state: t is5Is turned on, T1、T3Disconnecting;
duration:
Figure FDA00031416855200000411
three-phase voltage:
Figure FDA00031416855200000412
and a second state:
on-off state: t is1、T5Is turned on, T3Disconnecting;
duration:
Figure FDA00031416855200000413
three-phase voltage:
Figure FDA0003141685520000051
and a third state:
on-off state: t is1、T3、T5All are disconnected;
duration: t is t3＝Ts-t1-t2；
Three-phase voltage: vA(t)＝Vb(t)＝Vc(t)＝0；
3.6) when
Figure FDA0003141685520000052
The method comprises the following steps:
the first state:
on-off state: t is1Is turned on, T3、T5Disconnecting;
duration:
Figure FDA0003141685520000053
three-phase voltage:
Figure FDA0003141685520000054
and a second state:
on-off state: t is1、T5Is turned on, T3Disconnecting;
duration:
Figure FDA0003141685520000055
three-phase voltage:
Figure FDA0003141685520000056
and a third state:
on-off state: t is1、T3、T5All are disconnected;
duration: t is t3＝Ts-t1-t2；
Three-phase voltage: vA(t)＝Vb(t)＝Vc(t)＝0；
The servo motor module comprises the following modeling steps:
establishing a local coordinate system X of the motormYmZmThe origin is located at the center of the rotating shaft of the servo motor, ZmThe axis of the shaft coinciding with the axis of the rotor, XmThe axis direction of the shaft is consistent with that of the A phase winding of the motor stator, and Y ismThe axis being perpendicular to XmAxis and ZmA shaft;
the rotor permanent magnet generates magnetic flux with the density of
Figure FDA0003141685520000057
In the formula, BrThe residual magnetic flux density of the permanent magnet, g is the length of the air gap, h is the thickness of the permanent magnet, murmThe relative permeability of the permanent magnet is that of air at each pole of the rotor of the motorThe gap flux is:
Figure FDA0003141685520000058
in the formula, DrDenotes the diameter of the rotor, L is the axial length of the rotor core, PnThe number of magnetic pole pairs of the motor rotor is counted;
the magnetomotive force of each pole of the rotor of the motor is
Figure FDA0003141685520000059
In the formula, mu0Is air permeability, AmThe air gap area corresponding to each pole of the permanent magnet; the total magnetic potential of the magnetic circuit is as follows:
Fr＝2Fp (10)
for the stator, the phase current is obtained by the following formula
Figure FDA0003141685520000061
In the formula, ea(t)、eb(t)、ec(t) are respectively stator three-phase winding counter electromotive force, and R is stator phase resistance;
the magnetomotive force generated by each pole of each phase winding is as follows:
Figure FDA0003141685520000062
in the formula, kwIs a winding coefficient for considering the winding distribution effect; t isphThe number of turns of each phase winding; respectively make each phase magnetomotive force along XmAxis and YmPerforming axial projection to obtain the magnetic potential of each pole of the stator synthetic magnetic field in XmAxis, YmThe components of the axis are:
Figure FDA0003141685520000063
determining stator resultant flux edge XmAxis, YmThe components of the axis are:
Figure FDA0003141685520000064
the rotor flux is respectively directed along XmAxis and YmThe axis is projected to obtain the components:
Figure FDA0003141685520000065
so as to obtain the components of the total magnetic flux of the rotor and the stator on the x axis and the y axis respectively as follows:
Figure FDA0003141685520000066
the amplitude of the total flux vector inside the motor and the phase thereof are:
Figure FDA0003141685520000067
here, θΦsr(t) has a value range of [ - π, π]The specific quadrant thereof is according to phisrx(t) and Φsry(t) judging the symbol;
further, the total magnetic flux at the three-phase winding of the stator is obtained as follows:
Figure FDA0003141685520000071
the back emf of the stator windings is then:
Figure FDA0003141685520000072
according to the formula (13), the stator synthetic magnetomotive force is calculated as follows:
Figure FDA0003141685520000073
the total resultant magnetomotive force of the stator and rotor is then calculated by:
Figure FDA0003141685520000074
wherein, deltasr(t) represents the included angle between the stator magnetomotive force axis and the rotor magnetomotive force axis and
Figure FDA0003141685520000075
fix (·) denotes rounding to zero;
the motor output torque is obtained according to the following formula:
Figure FDA0003141685520000076
for d and q axis currents of the motor, the d and q axis currents are obtained by carrying out Park conversion on three phase currents of the motor, namely
Figure FDA0003141685520000077
The mechanical angular velocity of the motor rotor is calculated by the following mechanical motion equation:
Figure FDA0003141685520000078
the mechanical system module comprises the following modeling steps:
ball screw inletThe structure considered by the system dynamic model comprises a motor rotor (21), a workbench (22), a slide block (23), a guide rail (24), a bearing (25), a nut (26), a lead screw (27) and a coupling (28); the motor rotor (21) is equivalent to moment of inertia Jr(ii) a The working table (22) is equivalent to a mass block mwt(ii) a Equally dividing the screw rod (27) into three sections which are respectively equivalent to three masses ms1、ms2、ms3And three moments of inertia Js1、Js2、Js3And respectively pulling and pressing the rigidity k by a screw rodssL、kssRAnd screw torsional stiffness kθsL、kθsRConnecting; the coupling (28) is equivalent to a torsion spring unit with a torsional rigidity of kθcRepresents; the nut (26) is equivalent to a tension-compression spring, and the rigidity of the tension-compression spring is ksnRepresents; the bearings (25) at the two ends of the screw rod are respectively equivalent to have the rigidity of ksbLAnd ksbRThe tension and compression spring; the damping element considered includes a sliding damping c between the guide rail (24) and the slide (23)wtDamping of the translation of the nut (26)snAnd a rotational damping cθnThe translational damping c of the bearing (25)sbL、csbRAnd a rotational damping cθbL、cθbRThe translational damping c of the screw (27)ssL、cssRAnd a rotational damping cθsL、cθsRRotational damping of the coupling (28)θcAnd motor rotor bearing damping cθr；
From the Newton-Euler equation, the kinetic equation of the mechanical system is obtained as
Figure FDA0003141685520000081
Wherein M is the mass matrix of the mechanical system, and
Figure FDA0003141685520000082
k is the stiffness matrix of the mechanical system,
Figure FDA0003141685520000083
c is the damping matrix of the mechanical system,
Figure FDA0003141685520000084
q (t) is the generalized coordinate vector of the mechanical system,
q(t)＝[xa(t),xs1(t),xs2(t),xs3(t),θs1(t),θs2(t),θs3(t),θr(t)]T (29)
wherein x iswtIs the displacement of the table, xs1、xs2、xs3Translational displacement, theta, of three lead screws, respectivelys1、θs2、θs3The torsion angles, theta, of three lead screws respectivelyrThe torsion angle of the motor rotor;
f (t) is a generalized force vector,
F(t)＝[Ff(t)+Fx(t) 0 0 0 0 0 0 Te(t)]T (30)
wherein, Fx(t) milling force in X-direction, Ff(t) is the friction force experienced by the mechanical system, calculated using the Stribeck model as follows:
Figure FDA0003141685520000091
wherein v iswt(t) Table speed, Fc、Fs、VsCan be identified through experiments, delta is 2;
further expressing the mechanical system dynamic equation in a state space form, and selecting a state variable x1(t)＝q(t)，
Figure FDA0003141685520000092
Obtaining the state space of the mechanical systemThe expression is as follows:
Figure FDA0003141685520000093
wherein, the state vector:
Figure FDA0003141685520000094
system matrix:
Figure FDA0003141685520000095
constant coefficient matrix:
Figure FDA0003141685520000096
outputting a matrix: c ═ I O]And inputting a vector: u (t) f (t), and y (t) is an output vector;
the equivalent modeling of the milling module comprises the following modeling steps:
(1) dynamic modeling of the milling process:
establishing a local coordinate system O of the tooltXtYtZtOrigin of coordinate system OtAt the center of the bottom of the tool in the initial state, and the coordinate axis XtSame direction of feed of the machine tool, ZtIn the direction of the tool axis, YtPerpendicular to XtOtZtA plane; the part is regarded as a rigid body, the cutter is simplified into a two-degree-of-freedom dynamic system, and the vibration directions of the two-degree-of-freedom dynamic system are respectively equal to XtAxis, YtThe axial directions are the same;
setting a certain time edge XtAxis, YtThe milling forces generated in the axial direction are respectively Fx(t) and Fy(t), the vibration state of the tool is obtained by the following equation:
Figure FDA0003141685520000097
in the formula, mxt、mytRespectively being a tool in XtAxis, YtMass in the axial direction, cxt、cytAre each XtAxis, YtDamping coefficient in axial direction, kxt、kytAre each XtAxis, YtThe rigidity in the axial direction and the physical quantities can be obtained by performing a modal test experiment on the cutter; x is the number oft，ytRespectively the center position of the cutter is at XtAxis, YtCoordinates of the axis;
(2) and (3) simulation calculation of the instantaneous engagement state of the cutter and the part and the machined surface of the part:
the tooth number of the spiral end mill is Z, the spiral angle is beta, the radius is R, and the tooth space angle phi of the milling cutter with evenly distributed cutter teethpExpressed as:
Figure FDA0003141685520000098
cutting edge of the tool is divided into M parts with thickness d along axial directionzAnd when the initial moment is set, the instantaneous radial contact angle of the cutting micro element at the end part of the 1 st cutter tooth is 0 degrees, and then when a certain moment t is obtained, the instantaneous radial contact angle of the micro element of the l-th cutting edge on the jth cutter tooth is:
Figure FDA0003141685520000099
in the formula, S is the rotating speed of the main shaft;
for the machine tool with the worktable moving forward and the cutter being stationary, at a certain moment t, the displacement of the worktable is determined as x in the step 4)wt(t) the instantaneous vibration position of the tool is (x) from the equation (33)t(t),yt(t)), it can be seen that at time t, the actual position of the center of the tool in the part coordinate system, taking into account the actual displacement of the table and the tool vibration, is:
Figure FDA0003141685520000101
further, the coordinate of the first cutting edge infinitesimal on the jth cutter tooth at the time t is obtained as
Figure FDA0003141685520000102
The upper formula is a space trajectory curved surface expression swept by each cutting edge of the spiral milling cutter in the milling process, and an envelope surface formed by the space trajectory curved surface expression is a processed surface of the part;
actual undeformed chip thickness h taking into account the actual displacement of the table and tool vibrationsjl(t) is the current cutting edge infinitesimal position Sjl(t) the distance between the intersection point of the line passing through the point and the axis of the tool and the path of the previous cutting edge;
(3) calculating the milling force:
the milling force calculation adopts a common instantaneous rigid force calculation model,
Figure FDA0003141685520000103
in the formula: dFt,jl，dFr,jl，dFa,jlRespectively representing tangential, radial and axial cutting force infinitesimal; ds represents the cutting edge infinitesimal length; ktc，Krc，KacRepresenting the tangential, radial and axial cutting force coefficients, respectively; kte，Kre，KaeRespectively representing tangential, radial and axial cutting edge force coefficients; the cutting force coefficient and the cutting edge force coefficient are obtained through experimental identification;
considering that the cutting edge infinitesimal height is small, ds ≈ dz, the tangential, radial and axial cutting force infinitesimal on the l-th cutting edge infinitesimal of thickness dz acting on tooth j is expressed as:
Figure FDA0003141685520000104
in the formula：gjl(φjl(t)) is a unit step function for indicating whether the current cutting edge infinitesimal participates in cutting, which is defined as:
Figure FDA0003141685520000105
in the formula: phi is ast，φex-representing the cut-in angle and the cut-out angle, respectively;
for backmilling, the calculated expressions for the entry angle and exit angle are:
Figure FDA0003141685520000106
in the formula, aeIndicates the cutting width;
for down-cut, the calculated expressions for the cut-in angle and cut-out angle are:
Figure FDA0003141685520000107
through coordinate transformation, the cutting force components acting on three axes in the rectangular coordinate system are obtained as follows:
Figure FDA0003141685520000111
by integrating in the axial direction and summing each cutter tooth, the instantaneous cutting forces acting on the entire milling cutter in the feed, normal and axial directions are obtained:
Figure FDA0003141685520000112
the model integration comprises the following steps:
will control the output V of the system modulem(t) and θV(t) preparation ofIs the input of the PWM and inverter module; combining PWM with output V of inverter moduleA(t)、VB(t)、VC(t) as an input to the servo motor module; output i of servo motor moduleqa(t)、ida(T) feedback to the control system module, which outputs Te(t) as an input to a mechanical system module; the mechanical system module outputs the motion state physical quantity of each motion part, feeds back the displacement of the workbench and the speed of the motor rotor to the control system module, and feeds back the electrical angle of the motor rotor to the control system module and the servo motor module; the displacement x of the working platform obtained by the calculation of the mechanical system modulewt(t) input to the milling module and calculating the milling force F obtained by the milling modulex(t) feeding back to the mechanical system module to act on the worktable;
the discretization step comprises the following steps:
in the discrete time domain, the continuous time t is dispersed into N time steps with equal intervals, wherein the time step is t/N, and then the time t is any timeiAnd transforming the electromechanical integration model as follows:
7.1) discretization of control system modules:
the output of the position loop controller is given by equation (1):
ωr(ti)＝Kp(xr(ti)η-θwta(ti-1)) (46)
the expression of the output of the speed loop controller in the discrete time domain is:
Figure FDA0003141685520000113
wherein ir0(ti)＝ir0(ti-1)+(ωr(ti)-ωra(ti-1))Δt；
Likewise, the expressions for the outputs of the Q-axis and D-axis current loop controllers in the discrete time domain are also converted to the following forms, respectively:
Figure FDA0003141685520000114
Figure FDA0003141685520000115
formulas (5) and (6) are converted to the following formulae, respectively:
Figure FDA0003141685520000121
Figure FDA0003141685520000122
7.2) discretization of a servo motor module:
when the established three-phase alternating current permanent magnet synchronous motor model is converted into a time domain discrete model for simulation, the following calculation method is adopted:
back electromotive force of the stator winding is
Figure FDA0003141685520000123
During the time delta t, the total flux variation at the three-phase winding of the stator is as follows:
Figure FDA0003141685520000124
so as to obtain the total magnetic flux at the three-phase winding of the stator at the current moment as follows:
Figure FDA0003141685520000125
according to the magnitude of the three-phase magnetic flux, the following three conditions are calculated respectively:
when ma isx{Φa(ti),Φb(ti),Φc(ti)}＝Φa(ti) The method comprises the following steps:
Figure FDA0003141685520000126
when max { phi [ ]a(ti),Φb(ti),Φc(ti)}＝Φb(ti) The method comprises the following steps:
Figure FDA0003141685520000127
when max { phi [ ]a(ti),Φb(ti),Φc(ti)}＝Φc(ti) The method comprises the following steps:
Figure FDA0003141685520000128
then the total flux vector magnitude is obtained as:
Figure FDA0003141685520000129
it is in Xm、YmThe components of the axis are:
Figure FDA0003141685520000131
rotor flux at Xm、YmThe component of the axis is calculated by:
Figure FDA0003141685520000132
thereby obtaining stator magnetGeneral in Xm、YmThe components of the axis are:
Figure FDA0003141685520000133
further obtaining stator magnetomotive force Xm、YmThe components of the axis are:
Figure FDA0003141685520000134
then the stator three-phase magnetomotive force is calculated according to the following formula:
Figure FDA0003141685520000135
wherein, Fso(ti) For introduced virtual items, Fso(ti)≡0；
The three-phase current of the motor stator is directly obtained through magnetomotive force:
Figure FDA0003141685520000136
the amplitude and the phase of the stator synthetic magnetomotive force are as follows:
Figure FDA0003141685520000137
the included angle between the stator magnetomotive force axis and the rotor magnetomotive force axis is as follows:
Figure FDA0003141685520000138
the total resultant magnetomotive force of the stator and rotor can be obtained by the following formula
Figure FDA0003141685520000139
The motor output torque is obtained according to the following formula
Figure FDA00031416855200001310
7.3) discretization of mechanical system modules:
converting a mechanical system state space equation represented by the formula into a discrete form:
Figure FDA0003141685520000141
wherein G ═ eAΔt,
Figure FDA0003141685520000142
7.4) discretization of the milling module:
7.4.1) discrete solution of the milling process dynamics model:
transforming the milling process kinetic equation in the form of a differential equation obtained in step 5) into a differential form, according to equation (33), at tiAt any moment, the tool edge XtAxis and YtThe instantaneous acceleration of the shaft is determined by
Figure FDA0003141685520000143
Further, it is possible to obtain an instantaneous speed of
Figure FDA0003141685520000144
Finally, its vibrational displacement can be obtained by:
Figure FDA0003141685520000145
7.4.2) search solution of undeformed chip thickness:
in the time domain discrete solving process, the space track swept by each cutting edge of the helical milling cutter obtained by the formula (37) is a series of space discrete points, and the instantaneous radial contact angle phi between the undeformed chip thickness in the step 5) and the current cutting edge infinitesimal on the previous cutting edge track is usedjl(ti) Near the same position, the distance between the current cutting edge infinitesimal closest point is replaced, since the cutting edge trajectory is a discrete point, by the instantaneous radial contact angle on the last cutting edge trajectory being
Figure FDA0003141685520000146
Point of (2)
Figure FDA0003141685520000147
A search is made in the vicinity to obtain the closest point and further the undeformed chip thickness, i.e.
Figure FDA0003141685520000148
Wherein, TωThe time of one rotation of the main shaft is K is a given integer and is used for designating a search interval;
for the calculation of the milling force, only the continuous time t in the corresponding expression is used as the discrete time tiAnd (4) replacing.
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