1 Introduction Non-circular CNC turning is an effective method for achieving high efficiency, high flexibility, and high precision machining of non-circular section parts. The core of non-circular CNC turning is the high-frequency response high-precision linear servo unit that drives the tool to perform radial reciprocating motion. It is a typical position servo system. Because the actuator in the linear servo unit always has a certain response speed, the tool will inevitably produce an offset relative to the target position, that is, the actual trajectory of the tool exhibits a superposition of low-frequency ideal vibration and high-frequency noise vibration. The high-frequency noise vibration reflects the stability of the non-circular turning process, and its amplitude directly affects the dimensional accuracy of the cross-sectional profile of non-circular parts. To improve the machining accuracy of non-circular section parts is to improve the non-circular turning stability and reduce the high-frequency noise amplitude. The concept of Variable Spindle Speed ​​Machining (VSM) was first proposed by Professor Stoferle, T. of Germany in 1972 and used to improve the stability of turning operations. In variable speed machining, the spindle speed is continuously and periodically changed at a certain frequency and amplitude at a basic speed. The research on variable speed machining is mainly conducted in two aspects: mechanism and application. In applied research, variable speed machining was applied to turning, grinding and milling, respectively, and the selection and optimization of variable speed cutting parameters were discussed through experimental means. In the aspect of mechanism research, the reasons and conditions for suppressing vibration of variable speed machining and improving processing stability are emphatically discussed from a theoretical perspective. By establishing the mathematical modeling of the machining process, the correctness of the theory is verified by simulations and experiments. The purpose of introducing variable speed machining into non-circular turning to form variable speed non-circular turning is to provide a new way to improve the stability of non-circular turning to meet the ever-increasing profile accuracy requirements of non-circular parts. This article focuses on the system architecture for variable speed non-circular turning. Spindle transmission characteristics description and implementation, linear servo unit design and variable speed machining to improve the stability of the mechanism and other key technologies were studied. 2 variable speed non-circular turning system structure 
Fig. 1 Non-circular turning system of variable speed The non-circular turning of the structure is based on the non-circular CNC turning system and increases the continuous variable speed drive function of the spindle. Its system structure is shown as in Fig. 1. The variable speed non-circular turning system consists of a machine body, a general-purpose digital servo unit, a linear servo unit, a spindle drive unit, and a control computer. The machine tool adopts a general horizontal lathe structure, but the structure of the bed and the spindle is specially designed to ensure that the machine tool has a very good structural rigidity and spindle rotation accuracy. The universal CNC servo unit is driven by the AC servo motor and equipped with a multi-axis motion control board based on DSP. Under the control of the numerical control software, the CNC movement of the lathe along the X direction and the Z direction is realized. The linear servo unit uses a high-frequency linear motor as the driving element, a linear motor is mounted on the horizontal carriage of the lathe, and the cutter is mounted on the output shaft of the motor. The linear servo unit drives the tool to achieve precise reciprocating motion of the tool in the X direction to machine the non-circular cross-sectional profile of the part. At the other end of the output shaft of the linear motor, a grating sensor is provided to feedback the motion position of the linear motor in real time, and the closed loop follow-up control of the linear servo unit is realized under the support of a dedicated controller. The spindle drive unit drives the spindle motor to rotate as required by the control software and the spindle variable speed drive in accordance with the shifting frequency and amplitude of the spindle requested by the shift processing. The spindle encoder at the rear end of the spindle motor provides the spindle rotation signal to the linear motor dedicated controller and the multi-axis motion controller in real time to coordinate the actions between the various parts of the system. The variable-speed non-circular turning system realizes accurate and coordinated movement between the tool and the workpiece under the control of the computer, and processes non-circular parts with different cross-sectional shapes and different precision requirements.
Figure 2 Spindle shift curve 3 Spindle shift characteristics Description and realization of the mathematical model and characteristic parameters of the spindle shift quantitative description In the non-circular shifting process, the spindle speed fluctuates periodically at a basic rotational speed, as shown in Figure 2. Spindle shift characteristics are related to the type and characteristic parameters of the periodic fluctuation function. The angular velocity of the spindle's periodic change can be expressed as: w(t)=w0+Afunction(2pft), (1) where w0 is the basic angular velocity of the spindle, A is the amplitude of the spindle angular velocity change, and f is the frequency of the angular velocity change. ,function is any periodic function such as: sine wave function, triangle wave function, square wave function and so on. The dimensionless expression of this speed change is: w(t)=w0(1+RVAfunction(RVFw0t), (2) where RVA=A/w0 indicates the relative magnitude of the speed change, and RVF=2pf/w0 indicates the speed change. According to the above equation, RVA and RVF quantitatively describe the amplitude and frequency of the spindle speed with respect to the basic rotation speed, which is a characteristic parameter of the spindle speed change.. Spindle Speed ​​Change Implementation As can be seen from the foregoing, the spindle drive excitation function is a cycle. The function, in order to change the spindle speed, but in order to get the controllable spindle motor variable speed performance, should analyze and determine the type and characteristic parameters of the excitation function.There are three kinds of speed trajectory excitation functions with simple contour: sine wave, triangle wave and square wave. If the motor drive excitation function uses a sine wave, then both the angular acceleration and the angular thrust change of the motor rotation are sine or cosine functions that continuously change: when the excitation function adopts a triangular wave, the angular acceleration of the motor rotation is a square wave variation, but the angle The thrust changes to infinity: if the excitation function uses a square wave, then the angular acceleration and angle thrust variation of the motor rotation are infinite. In terms of movement and response, if the angular acceleration of the system is large, then the power and current of the motor driving the main shaft are large: if the angular force of the system changes greatly, then the momentary change of the motor current is large, therefore, from the perspective of obtaining the controllable shift performance of the spindle motor, The sine wave has better tracking performance than other waves and is more suitable as a motor drive excitation function when the spindle is shifted.On the other hand, the allowable RVF and RYA parameter variation ranges of the spindle transmission system are the spindle motor power, the spindle motion unit inertia, The cutting load, spindle servo tracking performance and other factors related to the actual RVA and RVF can only be changed within the allowable parameters.4 Linear servo unit design The linear servo unit in the variable speed non-circular turning system is a closed loop follow-up control system. According to the trajectory given by the program, the tool is driven to perform radial reciprocating precision tracking motion, so that the cross-sectional contour shapes of different non-circular parts can be processed quickly and flexibly to meet the accuracy requirements.
Fig. 3 Linear servo controller's nonlinear tracking controller The structure controller is designed to make the linear servo unit have high tracking accuracy, and it can resist the influence of outside interference and internal parameter changes. This paper designs a linear servo using nonlinear control theory. The nonlinear tracking controller gate in the unit, also called the auto-disturbance rejection controller, is shown in Figure 3. Non-linear tracking controllers are developed based on the transformation of traditional PIDs. The tracking differentiator after the input signal is a nonlinear dynamic link, which extracts the differential signal reasonably and avoids the distortion of the differential signal due to the non-differentiation of the input signal. The weighted sum of the original PIDs is changed to the nonlinear combination summation. A non-linear extended state observer is designed to estimate and control the state variables of the control process in real time to improve the robustness of the system. The controller algorithm is controlled by a linear motor, which can be approximated as a second-order system. The state space model is:
(3) In the formula, 
Let the input signal be v, the output signal be y, and the control quantity be u. Then the specific algorithm of the controller shown in FIG. 3 is as follows: Arranging the transition process, extracting the differential signal in the tracking differentiator, when inputting a signal v, it will generate Two output signals v1 and v2, where v1 tracks the input signal v, v2 is the derivative of v1. The discrete equation for the tracking differentiator is: {v1(k+1)=v1(k)+hv2(k), v2(k+1)=v2(k)+hfst(v1(k)-v(k), V2(k),r,h) (4) where h is the step size and it has a filtering effect on the noise. According to the sampling time, it is determined as follows: the speed factor, which determines the tracking speed. function. The fst expression is: 
(5) Estimated state and total disturbance Since the controlled object is a second-order system, the extended state observer will generate three state signals Z1, Z2, Z3. The discrete equations are: 
(6) where b01, b02, b03, a1, and a2 are adjustment parameters, and the non-linear function fal is: 
(7) The control quantity formation control quantity is calculated by the following formula: 
(8) The algorithm of this controller requires only the ideal input data v(t) and the actual output data y(t) of the linear motor.
Fig. 4 Experimental results of the tracking performance of the linear servo unit Simulation experiment According to the above algorithm, the controlled object is tracked experimentally, as shown in Fig.4. The target value of motion of the linear servo unit in Fig. 4 is the discrete value of formula (2). The values ​​of the parameters are as follows: basic angular velocity w0=251 (rad/s), RVA=0.2, RVF=0.1, (ab)/2= 100 (μm). At this time, the basic rotation speed of the spindle corresponding to piston NC turning is 1200r/min, speed change is ±240r/min, and the ellipticity is 0.4mm. It can be seen from Fig. 4 that the linear servo unit based on nonlinear control can track the input signal well when the speed is not round-turning, and the maximum tracking error is ±4 μm. 5 Variable speed machining improves non-circular turning stability The linear motor that drives the tool in non-circular turning systems is a spring damper system. In the turning process, the electromagnetic force output by the linear motor overcomes the spring force, the damping force and the cutting force to drive the tool to vibrate in the X direction. The actual trajectory of the tool can be considered as the superposition of low-frequency ideal vibration and high-frequency noise vibration. Low-frequency ideal vibration is a kind of forced vibration caused by electromagnetic force. Its motion trajectory is determined by the theoretical cross-sectional contour shape of non-circular parts: High-frequency noise vibration is a kind of self-excited vibration generated during processing, and its amplitude indicates the Cross-sectional dimension accuracy of the part. To improve the stability of non-circular turning is to reduce high-frequency noise and vibration. The self-excited vibration of the tool relative to the workpiece in the non-circular turning process is caused by the dynamic change of the electromagnetic force and the cutting force generated by the linear motor, and is the result of the interaction between the linear servo unit driving the tool and the cutting process. Since the self-excited vibration is maintained by the vibration system itself, the system not only absorbs energy but also consumes energy during each cycle of the vibration. Comparing the energy absorption and consumption of the system during vibration, it can be judged whether the vibration is enhanced or attenuated. In the following, the energy method is used to analyze the cutting force and electromagnetic force of the non-circular turning process on the tool vibration system to determine the energy change of the system. The most widely used non-circular turning application is the machining of elliptical cross-section pistons. Since the difference between the piston length and the half shaft diameter is small relative to the half shaft diameter, if the displacement of the tool at the long axis is set to zero, the ideal vibration of the tool R(t) can be expressed as: R(t) = ab (1-cos( 2wmt)) 2 (9) Let the tool noise vibration be: x(r) = Xcos(wt+Ø), then the cutting force F(t) and the electromagnetic force FE(t) are performed in n vibration cycles. for:
(10) where F(t) = Kcb(x(t)-X(tt) + h0), Kc is the static cutting force coefficient, b is the cutting width, and X(t) and X(tt) are Two rings of noise vibration. FE(t) = 2Kx(t) + F(t) + MR(t). The work done by the regenerative force and the electromagnetic force in each vibration cycle is: 
(11) The difference between the current waveform of tool noise vibration and the previous waveform can be expressed as: Ø=2pMm+em, where Mm= 0,1,2L, 0≤em≤2p. After derivation is available: 
(12) It can be known from equation (12) that at constant speed non-circular turning, when em=0 and em=p, the system is in critical stability state; when 0 When the system is unstable, when em=3p/2, the ∆W reaches a positive maximum, and the system stability is the worst. This shows that non-circular turning and ordinary round turning have the same regenerative vibration characteristics: the tool noise vibration, ie, the resonating frequency, increases continuously with the increase of the workpiece speed, but changes in a segmental sawtooth shape; on the stability map, it corresponds to each The Mm value has a lug-like curve. When the speed of the spindle changes during the non-circular speed change, the cutting process alternates between the stable area and the non-steady area. This cutting condition disturbs the continuous execution of the regenerative chatter. On the other hand, in the non-stable zone, the tool noise vibration is an inverter excitation response whose amplitude is smaller than the response of fixed frequency excitation at constant speed turning, which makes variable-speed machining can effectively reduce the cutting tool during non-circular turning. Noise vibration, thereby improving non-circular turning stability. 6 Conclusions Non-circular CNC turning is an effective method for machining non-circular cross-section parts. In order to improve the stability of non-circular turning, this paper introduces variable speed machining to non-circular turning, and studies the key technologies of variable speed non-circular turning. The following conclusions are drawn: The RVA and RVF parameters can quantitatively describe the amplitude and frequency of the mainshaft shift, and the sine-shaped periodic function is an easy-to-implement spindle variable-speed drive excitation function. The linear servo unit based on nonlinear tracking control has good tracking and robustness and can meet the precision requirements of non-circular turning. The noise vibration during variable-speed non-circular turning is a kind of self-excited vibration, and it has regenerative vibration characteristics. Variable speed machining improves the non-circular turning stability mechanism based on variable speed spindle speed disturbances and reduces the original continuous regenerative chatter.
Fig. 1 Non-circular turning system of variable speed The non-circular turning of the structure is based on the non-circular CNC turning system and increases the continuous variable speed drive function of the spindle. Its system structure is shown as in Fig. 1. The variable speed non-circular turning system consists of a machine body, a general-purpose digital servo unit, a linear servo unit, a spindle drive unit, and a control computer. The machine tool adopts a general horizontal lathe structure, but the structure of the bed and the spindle is specially designed to ensure that the machine tool has a very good structural rigidity and spindle rotation accuracy. The universal CNC servo unit is driven by the AC servo motor and equipped with a multi-axis motion control board based on DSP. Under the control of the numerical control software, the CNC movement of the lathe along the X direction and the Z direction is realized. The linear servo unit uses a high-frequency linear motor as the driving element, a linear motor is mounted on the horizontal carriage of the lathe, and the cutter is mounted on the output shaft of the motor. The linear servo unit drives the tool to achieve precise reciprocating motion of the tool in the X direction to machine the non-circular cross-sectional profile of the part. At the other end of the output shaft of the linear motor, a grating sensor is provided to feedback the motion position of the linear motor in real time, and the closed loop follow-up control of the linear servo unit is realized under the support of a dedicated controller. The spindle drive unit drives the spindle motor to rotate as required by the control software and the spindle variable speed drive in accordance with the shifting frequency and amplitude of the spindle requested by the shift processing. The spindle encoder at the rear end of the spindle motor provides the spindle rotation signal to the linear motor dedicated controller and the multi-axis motion controller in real time to coordinate the actions between the various parts of the system. The variable-speed non-circular turning system realizes accurate and coordinated movement between the tool and the workpiece under the control of the computer, and processes non-circular parts with different cross-sectional shapes and different precision requirements.
Figure 2 Spindle shift curve 3 Spindle shift characteristics Description and realization of the mathematical model and characteristic parameters of the spindle shift quantitative description In the non-circular shifting process, the spindle speed fluctuates periodically at a basic rotational speed, as shown in Figure 2. Spindle shift characteristics are related to the type and characteristic parameters of the periodic fluctuation function. The angular velocity of the spindle's periodic change can be expressed as: w(t)=w0+Afunction(2pft), (1) where w0 is the basic angular velocity of the spindle, A is the amplitude of the spindle angular velocity change, and f is the frequency of the angular velocity change. ,function is any periodic function such as: sine wave function, triangle wave function, square wave function and so on. The dimensionless expression of this speed change is: w(t)=w0(1+RVAfunction(RVFw0t), (2) where RVA=A/w0 indicates the relative magnitude of the speed change, and RVF=2pf/w0 indicates the speed change. According to the above equation, RVA and RVF quantitatively describe the amplitude and frequency of the spindle speed with respect to the basic rotation speed, which is a characteristic parameter of the spindle speed change.. Spindle Speed ​​Change Implementation As can be seen from the foregoing, the spindle drive excitation function is a cycle. The function, in order to change the spindle speed, but in order to get the controllable spindle motor variable speed performance, should analyze and determine the type and characteristic parameters of the excitation function.There are three kinds of speed trajectory excitation functions with simple contour: sine wave, triangle wave and square wave. If the motor drive excitation function uses a sine wave, then both the angular acceleration and the angular thrust change of the motor rotation are sine or cosine functions that continuously change: when the excitation function adopts a triangular wave, the angular acceleration of the motor rotation is a square wave variation, but the angle The thrust changes to infinity: if the excitation function uses a square wave, then the angular acceleration and angle thrust variation of the motor rotation are infinite. In terms of movement and response, if the angular acceleration of the system is large, then the power and current of the motor driving the main shaft are large: if the angular force of the system changes greatly, then the momentary change of the motor current is large, therefore, from the perspective of obtaining the controllable shift performance of the spindle motor, The sine wave has better tracking performance than other waves and is more suitable as a motor drive excitation function when the spindle is shifted.On the other hand, the allowable RVF and RYA parameter variation ranges of the spindle transmission system are the spindle motor power, the spindle motion unit inertia, The cutting load, spindle servo tracking performance and other factors related to the actual RVA and RVF can only be changed within the allowable parameters.4 Linear servo unit design The linear servo unit in the variable speed non-circular turning system is a closed loop follow-up control system. According to the trajectory given by the program, the tool is driven to perform radial reciprocating precision tracking motion, so that the cross-sectional contour shapes of different non-circular parts can be processed quickly and flexibly to meet the accuracy requirements.
Fig. 3 Linear servo controller's nonlinear tracking controller The structure controller is designed to make the linear servo unit have high tracking accuracy, and it can resist the influence of outside interference and internal parameter changes. This paper designs a linear servo using nonlinear control theory. The nonlinear tracking controller gate in the unit, also called the auto-disturbance rejection controller, is shown in Figure 3. Non-linear tracking controllers are developed based on the transformation of traditional PIDs. The tracking differentiator after the input signal is a nonlinear dynamic link, which extracts the differential signal reasonably and avoids the distortion of the differential signal due to the non-differentiation of the input signal. The weighted sum of the original PIDs is changed to the nonlinear combination summation. A non-linear extended state observer is designed to estimate and control the state variables of the control process in real time to improve the robustness of the system. The controller algorithm is controlled by a linear motor, which can be approximated as a second-order system. The state space model is:
(3) In the formula, Let the input signal be v, the output signal be y, and the control quantity be u. Then the specific algorithm of the controller shown in FIG. 3 is as follows: Arranging the transition process, extracting the differential signal in the tracking differentiator, when inputting a signal v, it will generate Two output signals v1 and v2, where v1 tracks the input signal v, v2 is the derivative of v1. The discrete equation for the tracking differentiator is: {v1(k+1)=v1(k)+hv2(k), v2(k+1)=v2(k)+hfst(v1(k)-v(k), V2(k),r,h) (4) where h is the step size and it has a filtering effect on the noise. According to the sampling time, it is determined as follows: the speed factor, which determines the tracking speed. function. The fst expression is: (5) Estimated state and total disturbance Since the controlled object is a second-order system, the extended state observer will generate three state signals Z1, Z2, Z3. The discrete equations are: (6) where b01, b02, b03, a1, and a2 are adjustment parameters, and the non-linear function fal is: (7) The control quantity formation control quantity is calculated by the following formula: (8) The algorithm of this controller requires only the ideal input data v(t) and the actual output data y(t) of the linear motor. Fig. 4 Experimental results of the tracking performance of the linear servo unit Simulation experiment According to the above algorithm, the controlled object is tracked experimentally, as shown in Fig.4. The target value of motion of the linear servo unit in Fig. 4 is the discrete value of formula (2). The values ​​of the parameters are as follows: basic angular velocity w0=251 (rad/s), RVA=0.2, RVF=0.1, (ab)/2= 100 (μm). At this time, the basic rotation speed of the spindle corresponding to piston NC turning is 1200r/min, speed change is ±240r/min, and the ellipticity is 0.4mm. It can be seen from Fig. 4 that the linear servo unit based on nonlinear control can track the input signal well when the speed is not round-turning, and the maximum tracking error is ±4 μm. 5 Variable speed machining improves non-circular turning stability The linear motor that drives the tool in non-circular turning systems is a spring damper system. In the turning process, the electromagnetic force output by the linear motor overcomes the spring force, the damping force and the cutting force to drive the tool to vibrate in the X direction. The actual trajectory of the tool can be considered as the superposition of low-frequency ideal vibration and high-frequency noise vibration. Low-frequency ideal vibration is a kind of forced vibration caused by electromagnetic force. Its motion trajectory is determined by the theoretical cross-sectional contour shape of non-circular parts: High-frequency noise vibration is a kind of self-excited vibration generated during processing, and its amplitude indicates the Cross-sectional dimension accuracy of the part. To improve the stability of non-circular turning is to reduce high-frequency noise and vibration. The self-excited vibration of the tool relative to the workpiece in the non-circular turning process is caused by the dynamic change of the electromagnetic force and the cutting force generated by the linear motor, and is the result of the interaction between the linear servo unit driving the tool and the cutting process. Since the self-excited vibration is maintained by the vibration system itself, the system not only absorbs energy but also consumes energy during each cycle of the vibration. Comparing the energy absorption and consumption of the system during vibration, it can be judged whether the vibration is enhanced or attenuated. In the following, the energy method is used to analyze the cutting force and electromagnetic force of the non-circular turning process on the tool vibration system to determine the energy change of the system. The most widely used non-circular turning application is the machining of elliptical cross-section pistons. Since the difference between the piston length and the half shaft diameter is small relative to the half shaft diameter, if the displacement of the tool at the long axis is set to zero, the ideal vibration of the tool R(t) can be expressed as: R(t) = ab (1-cos( 2wmt)) 2 (9) Let the tool noise vibration be: x(r) = Xcos(wt+Ø), then the cutting force F(t) and the electromagnetic force FE(t) are performed in n vibration cycles. for:
(10) where F(t) = Kcb(x(t)-X(tt) + h0), Kc is the static cutting force coefficient, b is the cutting width, and X(t) and X(tt) are Two rings of noise vibration. FE(t) = 2Kx(t) + F(t) + MR(t). The work done by the regenerative force and the electromagnetic force in each vibration cycle is: (11) The difference between the current waveform of tool noise vibration and the previous waveform can be expressed as: Ø=2pMm+em, where Mm= 0,1,2L, 0≤em≤2p. After derivation is available: (12) It can be known from equation (12) that at constant speed non-circular turning, when em=0 and em=p, the system is in critical stability state; when 0 Our harmonizing lines Force Control System, Harmonizing lines Constant Force Actuator, Harmonizing lines active contact flange have many advantages over other grinders. It can realize flexible grinding, quick repsond to surface changing, and instant adjusting. Traditional mechanic hand lacks flexibility, and it is not easy to adjust, hard to realize mass production.
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