Active Vibration Control of Boring Bar Vibrations
Active Vibration Control of Boring Bar Vibrations L. Andr´en, L. H˚ akansson and I. Claesson Department of Signal Processing, Blekinge Institute of Technology 372 25 Ronneby Sweden July 13, 2004 Abstract The boring operation is a cumbersome manufacturing process plagued by noise and vibration-related problems. A deep internal boring operation in a workpiece is a classic example of chatter-prone machining. The manufacturing industry today is facing tougher tolerances of product surfaces and a desire to process hard-to-cut materials; vibrations must thus be kept to a minimum. An increase in productivity is also interesting from a manufacturing point of view. Penetrating deep and narrow cavities require that the dimensions of the boring bar are long and slender. As a result, the boring bar is inclined to vibrate due to the limited dynamic stiffness. Vibration affects the surface finish, leads to severe noise in the workshop and may also reduce tool life. This report presents an active control solution based on a standard boring bar with an embedded piezo ceramic actuator; this is placed in the area of the peak modal strain energy of the boring bar bending mode to be controlled. An accelerometer is also included in the design; this is mounted as close as possible to the cutting tool. Embedding the electronic parts not only protects them from the harsh environment in a lathe but also enable the design to be used on a general lathe as long as the mounting arrangements are relatively similar. Three different algorithms have been tested in the control system. Since the excitation source of the original vibrations, i.e. the chip formation process cannot be observed directly, the algorithms must be constructed on the basis of a feedback approach. Experimental results from boring operations show that the vibration level can be reduced by 40 dB at the resonance frequency of a fundamental boring bar bending mode; several of its harmonics can also be reduced significantly.
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Introduction
The lathe is a very useful and versatile machine in the workshop, and it is able to perform a wide range of machining operations. A boring operation is a metal cutting operation that bores deep, precise holes in the workpiece. A boring bar is characterized by great length in comparison to its diameter. The boring bar is clamped at one end to a tool post or a revolver and has a cutting tool attached at the free end. The cutting tool is used to perform metal cutting in a bore or cavity of the workpiece. Since a boring bar is usually long and slender, it is inclined to vibrate. Deep internal boring of a workpiece is a classic example of chatter-prone machining. Performing metal cutting under vibrating conditions will yield unsatisfactory results in terms of the surface finish of a workpiece, tool life and undesirable noise levels. The boring bar vibration was investigated in [1] and [2]. The vibration may be characterized by a stochastic process with time varying statistical properties and with non-linear characteristics. [1]. In internal turning operations, the boring bar motion usually consists of components in both the cutting speed direction and the cutting depth direction [1, 2]. However, the motion of a boring bar during a continuous machining operation is generally greatest in the cutting speed direction; and is related to one of the bar’s two fundamental bending modes [1, 2]. A frequent result of this resonant motion is extremely high boring-bar vibration levels [1]. A typical boring operation is illustrated in Fig. 1. A conventional countermeasure to the vibrations is to equip boring bars with a tuned vibration absorber. The absorber is tuned to the frequency range of the fundamental bending mode of the boring bar by adjusting the weight of the reactive mass and the stiffness and damping properties of the elastic element. This will reduce the vibration level during the cutting operation. Active vibration absorbers based on inertial-mass actuators have also been investigated [3]. Active and passive vibration absorbers can provide some relief and are most effective when placed near the tool-end of the bar [7]. Active vibration control of machine-tool vibration, however, comprises a number of different methods for the introduction of a control force to the boring bar. In [8] the approach was to use an active clamping house, i.e. to let the clamping of the boring bar be the secondary vibrating source; the results were good. A similar approach was proposed as early as 1975 in [9]. The method uses a pivoting boring bar and an electro-hydraulic servo system as an actuator; the results are promising. This report presents a different method for the introduction of secondary vibration in boring bars. Here the actuator is mounted in a milled space in a longitudinal direction below the centerline of the boring bar. When the actuator applies a load on the boring bar in its longitudinal direction due to the expansion of the actuator, the boring bar will bend and stretch. By introducing secondary anti-vibrations via the actuator applied bending moment on the boring bar, the original vibrations
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141
The cutting speed direction
The cutting depth direction Boring bar
The feed direction
Workpiece
The cutting tool
Figure 1: A typical boring operation. from the cutting process can be reduced [5, 6, 13]. A challenge is to incorporate electronic devices into the harsh environment of a lathe. An active vibration control application includes actuators and sensors in conjunction with a control system. The actuator and accelerometer must be protected from the metal chips and cutting fluids. One of the goals was to make the active control system applicable to a general lathe. Embedding the active parts, i.e. the actuator and accelerometer, will not only protect them from the surrounding environment but will also allow the design to be used on a general lathe provided that the mounting arrangement is relatively similar. Due to the recent development of piezo ceramic actuators, the technique can be embedded into a boring bar. Milling a space in the boring bar reduces the bending stiffness; with piezo ceramic actuator technology, however, the space can be kept small and the bending stiffness reduction is moderate. The motion of the boring bar usually has components in both the cutting speed and the cutting depth directions. However, the boring bar vibration is to a great extent dominated by the motion in the cutting speed direction [1, 2]. In [5, 6] active vibration control using one actuator in the cutting speed direction indicates that the use of one actuator is a satisfactory
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solution. When an active boring bar is used, a suitable control algorithm is needed. The first requirement with respect to the algorithm is that it should be able to handle the non-stationary environments which a boring process gives rise to. Since it is not possible to distinguish between the the original boring bar vibrations and the secondary anti-vibrations, the algorithm must be based on a feedback approach. It may also be advantageous to consider the forward path, i.e. the signal transfer from the adaptive filter to the error sensor. A simple proportional or P controller, which is widely used in control theory, may help satisfy the first two requirements, i.e. handle non-stationary environments and based on a feedback approach. Where variations in the forward path are considered, more advanced controllers may be preferred. An algorithm that meets all requirements is the feedback filtered-x lms algorithm. This algorithm has proved successful in both the active control of tool vibration in external longitudinal turning and the active control of boring bar vibration [5, 6, 13]. Both the feedback filtered-x LMS algorithm and the Internal Model Control (IMC) controller based on an adaptive control FIR filter and governed by the Filtered-x LMS algorithm have been tested. Internal model control causes a feedback controller to work as a feedforward equivalent, provided the estimate of the forward path matches the actual forward path. This report deals with active vibration control in boring operations using three different control strategies and three different active boring bar designs.
2 2.1
Materials and Methods Experimental Setup
All the experiments have been carried out on a MAZAK 250 Quickturn lathe, see Fig. 2; this has 18.5 kW spindle power, a maximum machining diameter of 300 mm and 1007 mm between the centers. In order to save material, the cutting operations were performed as external turning operations, although a boring bar was used as a tool holder, see Fig. 3. 2.1.1
Measurement Equipment and Setup
Three different control algorithms were used in the active control measurements: an ordinary PID controller, a feedback filtered-x LMS algorithm, and an Internal Model Control (IMC) controller based on an adaptive control FIR filter governed by the Filtered-x LMS algorithm. A block diagram of the experimental setup for the active vibration control in boring operations can be seen in Fig. 4. A signal conditioning unit is also included in the digital controller in order to be able to adjust the level of the input and output of the DSP. When using the
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Figure 2: The lathe in which all the experiments have been carried out. filtered-x LMS algorithms, the transfer function of the signal chain D/A converter, amplifier, structural transfer path from the actuator to the accelerometer and an A/D converter must be estimated in an initial phase prior the active vibration control. The experimental setup for investigating the forward path is illustrated in the block diagram in Fig. 5. The accelerometer in the cutting speed direction was used when estimating the forward path. The following equipment was used in the measurements. • 2 Br¨ uel & Kjær 4374 accelerometers. • 1 Br¨ uel & Kjær NEXUS 2 channel conditioning amplifier 2692. • TEAC RD-200T DAT recorder. • A custom-designed amplifier designed for capacitive loads. • Texas Instruments DSP TMS320C32. • Active boring bars with an embedded piezo ceramic actuator, see section 2.2.
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Figure 3: The turning operation used in the experiments. 2.1.2
Cutting Data and Machining Parameters
The workpiece material in the cutting experiments was chromium molybdenum nickel steel. The diameter of the workpiece was large (< 200mm) to ensure that the workpiece vibrations were negligible. The workpiece material SS 254103, chromium molybdenum nickel steel, is a quenched and tempered steel. This material excites the machine-tool system with a narrow bandwidth in the cutting operation [27]. It facilitates the introduction of major narrow-banded tool vibration in a turning operation, resulting in a deterioration in surface finish and severe acoustic noise levels [1, 27]. The cutting tools used were standard 55◦ diagonal inserts. These have a tool geometry designated by the ISO code DNMG 150618-SL and with chip breaker geometry for medium roughing. The carbide grade was TN7015. The cutting data was selected in order to produce significant tool vibrations. These resulted in an observable deterioration in the workpiece surface as well as severe acoustic noise. After a preliminary set of trials, a suitable combination of cutting data and tool geometry was selected, see Table 1. Cutting data set No. 1 was selected for the production of significant tool vibrations for evaluating the three different control algorithms used in active control of boring bar vibration. Cutting
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Active Vibration Control of Boring Bar Vibrations
HP signal analyzer ch1
ch2
DAT recorder ch1 ch2 ch3 Amplifier A = 10
Digital Controller
NEXUS Charge Amplifier
Accelerometer in the cutting depth direction bk4374
Piezo actuator
Boring Bar NEXUS Charge Amplifier
Accelerometer in the cutting speed direction bk4374
Figure 4: A block diagram describing the experimental setup for active vibration control. Cutting data Geometry Cutting speed set v (m/min) No. 1 DNMG 150608-SL 80 No. 2 DNMG 150608-SL 100-150
depth of cut Feed a (mm) s (mm/rev) 1.0 0.2-0.3 0.5-1.5 0.2
Table 1: Cutting data and tool geometry
data set No. 1 was also selected to facilitate investigation of the influence of the actuator location in the boring bar on the vibration control performance in the metal cutting process. To vary the tool vibration level in a controlled manner, a low cutting speed was selected, i.e. just beyond the limit of build up of edge effects. The initial feed rate was selected in accordance with the lower chip-breaking limit of the insert. In the cutting experiments the cutting depth was increased to the limit of the control of the active boring bar. The cutting depth was subsequently slightly reduced to the maximum machining depth where maintaining control was possible; the feed rate was then gradually increased to the limit of active control. By using cutting data set No. 2, it was occasionally possible to perform the boring operation without any large vibrations. Under such circumstances, it was possible to record data that enabled online estimation of the forward path during continuous turning.
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HP signal analyzer ch1
ch2
DAT recorder ch1
Amplifier A = 10
ch2
Signal Source
Piezo actuator (double P804.10)
Boring Bar
Accelerometer bk 4374 s/n 2243930 0.1418 pC/ms-2
NEXUS Charge Amplifier 1mV/ms-2 0.142 pC/ms-2
Figure 5: The block diagram describing the experimental setup for both offline and online estimation of the forward path. No cutting fluid was applied during machining.
2.2
Active Boring Bar
A boring bar is usually long and slender in order to facilitate metal cutting in the bore of a workpiece. The boring bar used in the experiments was based on standard WIDAX S40T PDUNR15 boring bars, see Fig. 6. The diameter of the boring bar was 40 mm and the length 300 mm; 100 mm is required for the clamping. The ovarhang part thus constitutes 200 mm.
Figure 6: A CAD model of the standard boring bar WIDAX S40T PDUNR15.
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To perform active vibration control, an actuator and accelerometer must be applied to the vibrating object. The environment in which active control of boring bar vibration is designed to operate is harsh. The actuator and accelerometer must be protected from cutting fluids and metal chips resulting from the cutting operation. One possibility is to embed and seal the electronic parts into the boring bar. Accelerometers are usually so small that incorporating them into the design will have negligible effect on the bending stiffness of the boring bar. The accelerometer was mounted 25 mm from the tool tip and senses the vibrations in the cutting speed direction. The actuator, on the other hand, must be sufficiently large to produce adequate secondary vibrational forces to enable a sufficient increase of the dynamic bending stiffness. There are several ways of mounting the actuator in the boring bar. Three different mounting locations for the actuator have been tested in real-life cutting experiments. The difference between the active boring bars is
Cutting depth direction
α
15 mm
18.5 mm
m 40
Cutting speed direction
m
15 mm
Actuator
Figure 7: Boring bar cross section with embedded actuator and α is the actuator offset angle. the actuator offset angle, i.e. the angle between the cutting speed direction and the boring bar cross section radius which intersects the actuator center towards the reversed cutting depth direction. The actuator offset angle is illustrated in Fig. 7 and is denoted α. In all three cases, the actuator was mounted in a longitudinal direction below the centerline of the boring bar and adjacent to the clamping. When the actuator expands in length, it applies a load to the boring bar in its longitudinal direction; as a result, the boring bar will bend and stretch. Secondary anti-vibrations may thus be introduced by the actuator applied bending moment in order to reduce the original boring bar vibration, excited by the chip formation process during continuous machining. A schematic figure of the active boring bar control system is shown in Fig. 8 The first active boring bar design was based on an embedded actuator with 0◦ actuator offset angle. The second active boring bar design was based on an embedded actuator with
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Cutting speed direction Primary excitation introduced by the material deformation process
Workpiece Boring Bar Embedded actuator
Feedback controller
Secondary excitation via active actuator
W
Figure 8: A schematic figure of the active boring bar control system. 15◦ actuator offset angle. The third active boring bar design was based on an embedded actuator with ◦ 30 actuator offset angle.
2.3
Control Algorithms
Control is the process of causing a system variable to conform to some desired value known as a reference value [17]. A special class of control theory is feedback control, which is used in the active vibration control application discussed here. A block diagram of the elementary parts of feedback control is provided in Fig. 9. Feedback control can be found in a wide range of products ranging from simple heating systems to very complex processes. The central component is the plant, which must be controlled in some way. An output sensor senses a variable that is designed to imitate a reference signal. The controller uses the information from the reference signal and the output of the plant to produce a control signal to the actuator. The actuator is the physical part of the control system that has direct control authority on the plant. A simple analogy is a heating system. Here the reference is the desired temperature in the room, and a radiator is used as the
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149
Disturbance
Reference sensor
Σ
Output Controller
Actuator
Plant
Output sensor
Figure 9: Block diagram of an elementary feedback control system.
actuator. The plant would be the room and the output of the plant is the actual temperature of the room. The temperature of the room is easily controlled by switching the radiator on and off. Feedback control is not a new science. In 1788 James Watt invented the centrifugal governor; this was the first feedback device to attract the attention of the entire engineering community and be accepted internationally [23]. The earliest feedback device known, however, can be found among the works of Ktesibios, Philon and Heron from the Hellenistic period ca 300 B.C. For further reading on the history of feedback control, see [23]. The choice of control algorithms must be based on the application. The boring operation is a process which has non-stationary stochastic properties [1]; the algorithm must be able to handle variations in the plant being controlled. Another Forward path
Amplifier
Actuator
Structural transfer path
Figure 10: The physical components of the forward path in the boring bar vibration control system.
important factor which has strong effect the choice of control algorithm is that the excitation source, the chip formation process, cannot be observed directly. The accelerometer senses both the vibrations resulting from the cutting process and those induced by the actuator. The algorithm must thus based on a feedback approach. The forward path, which is always present in an active vibration control
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application [19] must also be considered. In the particular control problem under discussion, the forward path basically consists of an amplifier, actuator and the structural transfer path in the boring bar, see Fig. 10. Strictly speakin, the forward path also includes D/A and A/D converters as well as an accelerometer. Three controller algorithms suitable for evaluating purposes with respect to the application under discussion are the simple PID controller, the more advanced feedback filtered-x LMS algorithm and the Internal Model Control, IMC, controller based on an adaptive control FIR filter governed by the Filtered-x LMS algorithm. In the filtered-x LMS algorithm and the IMC-based controller the estimate of the forward path was a 40 coefficient FIR filter. Both the adaptive controllers used an adaptiver FIR filter with 35 weights . 2.3.1
PID Controller
The proportional integral derivative PID controller is well-known and is, for instance, widely accepted in the processing industry [24]. Originally, it was implemented using analog technology [24]. However, today almost all controllers are implemented in computers [24]. The digital PID controller can be viewed as an approximation of its analog counterpart. In a boring operation, the plant or forward path may be observed by an accelerometer mounted on the boring bar. Since the goal is to reduce vibration, acceleration should be as small as possible. The acceleration signal is denoted e(t); the control signal to the actuator is denoted y(t). The PID controller in continuous time t can be written as [24] de(t) 1 t (1) e(τ )dτ + Td y(t) = K e(t) + Ti 0 dt where K is the gain of the controller, Ti is the integration time and Td is the derivative time. The proportional part of the controller sets the constant gain. The integral part in conjunction with a proportional part improves steady state properties; when combined with derivative control, it also improves the transient properties [17]. A discretized version of the analog PID controller can be approximated as [24] y(n) = P (n) + I(n) + D(n)
(2)
where P (n) = Ke(n − 1) K e(n − 1) Fs Ti Fs KTd N Fs Td D(n − 1) − e(n − 1) − e(n − 2) D(n) = Fs Td + N Fs Td + N I(n) = I(n − 1) +
(3) (4) (5)
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where Fs is the sampling frequency and N is a high frequency gain limitation of the derivative part. e(n) is the resulting error; in active vibration control in a boring operation it is also the accelerometer signal. Fig. 11 shows a block diagram of a d(n) K
y(n)
Forward path C
yC(n)
Σ e(n)
x(n) = e(n-1)
Unit delay z-1
Figure 11: Block diagram of a feedback control system based on a digital P controller. feedback control system with a digital P controller. Here the box with the unit delay z −1 between e(n) and e(n − 1) at the input to the controller indicated that the subject is a digital controller in a feedback control system. 2.3.2
LMS Algorithm
The least mean square LMS algorithm was developed by Widrow and Hoff in 1960. The least square approach provides a powerful approach to digital filtering in situations where a fixed, finite length filter is applicable. This approach has been widely used in many areas [20]. It is an important member of the family of stochastic gradient algorithms [18]. The LMS algorithm is very simple and has therefore been made the standard against which other adaptive filtering algorithms are benchmarked [18]. In the active control of vibration application discussed here, an LMS algorithm was used when estimating the forward path of the system. A forward path is always present in active control applications [19]; if the filtered-x LMS algorithm is used as a controller, an estimate of the forward path is also needed. A block diagram of the forward path estimation using an LMS algorithm is shown in Fig. 12. The task of the LMS algorithm is to imitate a desired signal d(n) by letting an input signal x(n) pass an adaptive filter wn (k) to produce an output y(n). The algorithm adjusts the weights wn (k) so that the error signal e(n) is minimized in the mean
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Noise generator
x(n)
Forward path
d(n)
Adaptive filter w(n)
y(n)
+ − Σ e(n)
LMS
Figure 12: Block diagram of the forward path estimation using an LMS algorithm.
square sense. The error signal can be written as e(n) = d(n) − y(n) = d(n) −
L−1
wn (l)x(n − l),
(6)
l=0
where the adaptive FIR filter has L coefficients. The weights are updated on average in the negative direction of the gradient. The gradient estimate in the LMS algorithm consists of the derivatives of the square error signal with respect to each of the weights of the adaptive filter
∂e2 (n) = ∂wn (k)
∂ d(n) −
L−1
2 x(n − l)wn (l)
l=0
∂wn (k)
= −2x(n − k)e(n)
(7)
for k = 0, 1, . . . , L − 1. The weights can now be updated in the negative direction of the gradient estimate as wn+1 (0) wn (0) x(n) wn+1 (1) wn (1) x(n − 1) (8) = + µ e(n), .. .. .. . . . wn+1 (L − 1) wn (L − 1) x(n − L + 1) To enable convergence in the mean square of the LMS algorithm the step size µ is usually selected according to the inequality [28] 0
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Introduction
The lathe is a very useful and versatile machine in the workshop, and it is able to perform a wide range of machining operations. A boring operation is a metal cutting operation that bores deep, precise holes in the workpiece. A boring bar is characterized by great length in comparison to its diameter. The boring bar is clamped at one end to a tool post or a revolver and has a cutting tool attached at the free end. The cutting tool is used to perform metal cutting in a bore or cavity of the workpiece. Since a boring bar is usually long and slender, it is inclined to vibrate. Deep internal boring of a workpiece is a classic example of chatter-prone machining. Performing metal cutting under vibrating conditions will yield unsatisfactory results in terms of the surface finish of a workpiece, tool life and undesirable noise levels. The boring bar vibration was investigated in [1] and [2]. The vibration may be characterized by a stochastic process with time varying statistical properties and with non-linear characteristics. [1]. In internal turning operations, the boring bar motion usually consists of components in both the cutting speed direction and the cutting depth direction [1, 2]. However, the motion of a boring bar during a continuous machining operation is generally greatest in the cutting speed direction; and is related to one of the bar’s two fundamental bending modes [1, 2]. A frequent result of this resonant motion is extremely high boring-bar vibration levels [1]. A typical boring operation is illustrated in Fig. 1. A conventional countermeasure to the vibrations is to equip boring bars with a tuned vibration absorber. The absorber is tuned to the frequency range of the fundamental bending mode of the boring bar by adjusting the weight of the reactive mass and the stiffness and damping properties of the elastic element. This will reduce the vibration level during the cutting operation. Active vibration absorbers based on inertial-mass actuators have also been investigated [3]. Active and passive vibration absorbers can provide some relief and are most effective when placed near the tool-end of the bar [7]. Active vibration control of machine-tool vibration, however, comprises a number of different methods for the introduction of a control force to the boring bar. In [8] the approach was to use an active clamping house, i.e. to let the clamping of the boring bar be the secondary vibrating source; the results were good. A similar approach was proposed as early as 1975 in [9]. The method uses a pivoting boring bar and an electro-hydraulic servo system as an actuator; the results are promising. This report presents a different method for the introduction of secondary vibration in boring bars. Here the actuator is mounted in a milled space in a longitudinal direction below the centerline of the boring bar. When the actuator applies a load on the boring bar in its longitudinal direction due to the expansion of the actuator, the boring bar will bend and stretch. By introducing secondary anti-vibrations via the actuator applied bending moment on the boring bar, the original vibrations
Active Vibration Control of Boring Bar Vibrations
141
The cutting speed direction
The cutting depth direction Boring bar
The feed direction
Workpiece
The cutting tool
Figure 1: A typical boring operation. from the cutting process can be reduced [5, 6, 13]. A challenge is to incorporate electronic devices into the harsh environment of a lathe. An active vibration control application includes actuators and sensors in conjunction with a control system. The actuator and accelerometer must be protected from the metal chips and cutting fluids. One of the goals was to make the active control system applicable to a general lathe. Embedding the active parts, i.e. the actuator and accelerometer, will not only protect them from the surrounding environment but will also allow the design to be used on a general lathe provided that the mounting arrangement is relatively similar. Due to the recent development of piezo ceramic actuators, the technique can be embedded into a boring bar. Milling a space in the boring bar reduces the bending stiffness; with piezo ceramic actuator technology, however, the space can be kept small and the bending stiffness reduction is moderate. The motion of the boring bar usually has components in both the cutting speed and the cutting depth directions. However, the boring bar vibration is to a great extent dominated by the motion in the cutting speed direction [1, 2]. In [5, 6] active vibration control using one actuator in the cutting speed direction indicates that the use of one actuator is a satisfactory
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solution. When an active boring bar is used, a suitable control algorithm is needed. The first requirement with respect to the algorithm is that it should be able to handle the non-stationary environments which a boring process gives rise to. Since it is not possible to distinguish between the the original boring bar vibrations and the secondary anti-vibrations, the algorithm must be based on a feedback approach. It may also be advantageous to consider the forward path, i.e. the signal transfer from the adaptive filter to the error sensor. A simple proportional or P controller, which is widely used in control theory, may help satisfy the first two requirements, i.e. handle non-stationary environments and based on a feedback approach. Where variations in the forward path are considered, more advanced controllers may be preferred. An algorithm that meets all requirements is the feedback filtered-x lms algorithm. This algorithm has proved successful in both the active control of tool vibration in external longitudinal turning and the active control of boring bar vibration [5, 6, 13]. Both the feedback filtered-x LMS algorithm and the Internal Model Control (IMC) controller based on an adaptive control FIR filter and governed by the Filtered-x LMS algorithm have been tested. Internal model control causes a feedback controller to work as a feedforward equivalent, provided the estimate of the forward path matches the actual forward path. This report deals with active vibration control in boring operations using three different control strategies and three different active boring bar designs.
2 2.1
Materials and Methods Experimental Setup
All the experiments have been carried out on a MAZAK 250 Quickturn lathe, see Fig. 2; this has 18.5 kW spindle power, a maximum machining diameter of 300 mm and 1007 mm between the centers. In order to save material, the cutting operations were performed as external turning operations, although a boring bar was used as a tool holder, see Fig. 3. 2.1.1
Measurement Equipment and Setup
Three different control algorithms were used in the active control measurements: an ordinary PID controller, a feedback filtered-x LMS algorithm, and an Internal Model Control (IMC) controller based on an adaptive control FIR filter governed by the Filtered-x LMS algorithm. A block diagram of the experimental setup for the active vibration control in boring operations can be seen in Fig. 4. A signal conditioning unit is also included in the digital controller in order to be able to adjust the level of the input and output of the DSP. When using the
Active Vibration Control of Boring Bar Vibrations
143
Figure 2: The lathe in which all the experiments have been carried out. filtered-x LMS algorithms, the transfer function of the signal chain D/A converter, amplifier, structural transfer path from the actuator to the accelerometer and an A/D converter must be estimated in an initial phase prior the active vibration control. The experimental setup for investigating the forward path is illustrated in the block diagram in Fig. 5. The accelerometer in the cutting speed direction was used when estimating the forward path. The following equipment was used in the measurements. • 2 Br¨ uel & Kjær 4374 accelerometers. • 1 Br¨ uel & Kjær NEXUS 2 channel conditioning amplifier 2692. • TEAC RD-200T DAT recorder. • A custom-designed amplifier designed for capacitive loads. • Texas Instruments DSP TMS320C32. • Active boring bars with an embedded piezo ceramic actuator, see section 2.2.
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Figure 3: The turning operation used in the experiments. 2.1.2
Cutting Data and Machining Parameters
The workpiece material in the cutting experiments was chromium molybdenum nickel steel. The diameter of the workpiece was large (< 200mm) to ensure that the workpiece vibrations were negligible. The workpiece material SS 254103, chromium molybdenum nickel steel, is a quenched and tempered steel. This material excites the machine-tool system with a narrow bandwidth in the cutting operation [27]. It facilitates the introduction of major narrow-banded tool vibration in a turning operation, resulting in a deterioration in surface finish and severe acoustic noise levels [1, 27]. The cutting tools used were standard 55◦ diagonal inserts. These have a tool geometry designated by the ISO code DNMG 150618-SL and with chip breaker geometry for medium roughing. The carbide grade was TN7015. The cutting data was selected in order to produce significant tool vibrations. These resulted in an observable deterioration in the workpiece surface as well as severe acoustic noise. After a preliminary set of trials, a suitable combination of cutting data and tool geometry was selected, see Table 1. Cutting data set No. 1 was selected for the production of significant tool vibrations for evaluating the three different control algorithms used in active control of boring bar vibration. Cutting
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Active Vibration Control of Boring Bar Vibrations
HP signal analyzer ch1
ch2
DAT recorder ch1 ch2 ch3 Amplifier A = 10
Digital Controller
NEXUS Charge Amplifier
Accelerometer in the cutting depth direction bk4374
Piezo actuator
Boring Bar NEXUS Charge Amplifier
Accelerometer in the cutting speed direction bk4374
Figure 4: A block diagram describing the experimental setup for active vibration control. Cutting data Geometry Cutting speed set v (m/min) No. 1 DNMG 150608-SL 80 No. 2 DNMG 150608-SL 100-150
depth of cut Feed a (mm) s (mm/rev) 1.0 0.2-0.3 0.5-1.5 0.2
Table 1: Cutting data and tool geometry
data set No. 1 was also selected to facilitate investigation of the influence of the actuator location in the boring bar on the vibration control performance in the metal cutting process. To vary the tool vibration level in a controlled manner, a low cutting speed was selected, i.e. just beyond the limit of build up of edge effects. The initial feed rate was selected in accordance with the lower chip-breaking limit of the insert. In the cutting experiments the cutting depth was increased to the limit of the control of the active boring bar. The cutting depth was subsequently slightly reduced to the maximum machining depth where maintaining control was possible; the feed rate was then gradually increased to the limit of active control. By using cutting data set No. 2, it was occasionally possible to perform the boring operation without any large vibrations. Under such circumstances, it was possible to record data that enabled online estimation of the forward path during continuous turning.
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HP signal analyzer ch1
ch2
DAT recorder ch1
Amplifier A = 10
ch2
Signal Source
Piezo actuator (double P804.10)
Boring Bar
Accelerometer bk 4374 s/n 2243930 0.1418 pC/ms-2
NEXUS Charge Amplifier 1mV/ms-2 0.142 pC/ms-2
Figure 5: The block diagram describing the experimental setup for both offline and online estimation of the forward path. No cutting fluid was applied during machining.
2.2
Active Boring Bar
A boring bar is usually long and slender in order to facilitate metal cutting in the bore of a workpiece. The boring bar used in the experiments was based on standard WIDAX S40T PDUNR15 boring bars, see Fig. 6. The diameter of the boring bar was 40 mm and the length 300 mm; 100 mm is required for the clamping. The ovarhang part thus constitutes 200 mm.
Figure 6: A CAD model of the standard boring bar WIDAX S40T PDUNR15.
Active Vibration Control of Boring Bar Vibrations
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To perform active vibration control, an actuator and accelerometer must be applied to the vibrating object. The environment in which active control of boring bar vibration is designed to operate is harsh. The actuator and accelerometer must be protected from cutting fluids and metal chips resulting from the cutting operation. One possibility is to embed and seal the electronic parts into the boring bar. Accelerometers are usually so small that incorporating them into the design will have negligible effect on the bending stiffness of the boring bar. The accelerometer was mounted 25 mm from the tool tip and senses the vibrations in the cutting speed direction. The actuator, on the other hand, must be sufficiently large to produce adequate secondary vibrational forces to enable a sufficient increase of the dynamic bending stiffness. There are several ways of mounting the actuator in the boring bar. Three different mounting locations for the actuator have been tested in real-life cutting experiments. The difference between the active boring bars is
Cutting depth direction
α
15 mm
18.5 mm
m 40
Cutting speed direction
m
15 mm
Actuator
Figure 7: Boring bar cross section with embedded actuator and α is the actuator offset angle. the actuator offset angle, i.e. the angle between the cutting speed direction and the boring bar cross section radius which intersects the actuator center towards the reversed cutting depth direction. The actuator offset angle is illustrated in Fig. 7 and is denoted α. In all three cases, the actuator was mounted in a longitudinal direction below the centerline of the boring bar and adjacent to the clamping. When the actuator expands in length, it applies a load to the boring bar in its longitudinal direction; as a result, the boring bar will bend and stretch. Secondary anti-vibrations may thus be introduced by the actuator applied bending moment in order to reduce the original boring bar vibration, excited by the chip formation process during continuous machining. A schematic figure of the active boring bar control system is shown in Fig. 8 The first active boring bar design was based on an embedded actuator with 0◦ actuator offset angle. The second active boring bar design was based on an embedded actuator with
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Cutting speed direction Primary excitation introduced by the material deformation process
Workpiece Boring Bar Embedded actuator
Feedback controller
Secondary excitation via active actuator
W
Figure 8: A schematic figure of the active boring bar control system. 15◦ actuator offset angle. The third active boring bar design was based on an embedded actuator with ◦ 30 actuator offset angle.
2.3
Control Algorithms
Control is the process of causing a system variable to conform to some desired value known as a reference value [17]. A special class of control theory is feedback control, which is used in the active vibration control application discussed here. A block diagram of the elementary parts of feedback control is provided in Fig. 9. Feedback control can be found in a wide range of products ranging from simple heating systems to very complex processes. The central component is the plant, which must be controlled in some way. An output sensor senses a variable that is designed to imitate a reference signal. The controller uses the information from the reference signal and the output of the plant to produce a control signal to the actuator. The actuator is the physical part of the control system that has direct control authority on the plant. A simple analogy is a heating system. Here the reference is the desired temperature in the room, and a radiator is used as the
Active Vibration Control of Boring Bar Vibrations
149
Disturbance
Reference sensor
Σ
Output Controller
Actuator
Plant
Output sensor
Figure 9: Block diagram of an elementary feedback control system.
actuator. The plant would be the room and the output of the plant is the actual temperature of the room. The temperature of the room is easily controlled by switching the radiator on and off. Feedback control is not a new science. In 1788 James Watt invented the centrifugal governor; this was the first feedback device to attract the attention of the entire engineering community and be accepted internationally [23]. The earliest feedback device known, however, can be found among the works of Ktesibios, Philon and Heron from the Hellenistic period ca 300 B.C. For further reading on the history of feedback control, see [23]. The choice of control algorithms must be based on the application. The boring operation is a process which has non-stationary stochastic properties [1]; the algorithm must be able to handle variations in the plant being controlled. Another Forward path
Amplifier
Actuator
Structural transfer path
Figure 10: The physical components of the forward path in the boring bar vibration control system.
important factor which has strong effect the choice of control algorithm is that the excitation source, the chip formation process, cannot be observed directly. The accelerometer senses both the vibrations resulting from the cutting process and those induced by the actuator. The algorithm must thus based on a feedback approach. The forward path, which is always present in an active vibration control
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application [19] must also be considered. In the particular control problem under discussion, the forward path basically consists of an amplifier, actuator and the structural transfer path in the boring bar, see Fig. 10. Strictly speakin, the forward path also includes D/A and A/D converters as well as an accelerometer. Three controller algorithms suitable for evaluating purposes with respect to the application under discussion are the simple PID controller, the more advanced feedback filtered-x LMS algorithm and the Internal Model Control, IMC, controller based on an adaptive control FIR filter governed by the Filtered-x LMS algorithm. In the filtered-x LMS algorithm and the IMC-based controller the estimate of the forward path was a 40 coefficient FIR filter. Both the adaptive controllers used an adaptiver FIR filter with 35 weights . 2.3.1
PID Controller
The proportional integral derivative PID controller is well-known and is, for instance, widely accepted in the processing industry [24]. Originally, it was implemented using analog technology [24]. However, today almost all controllers are implemented in computers [24]. The digital PID controller can be viewed as an approximation of its analog counterpart. In a boring operation, the plant or forward path may be observed by an accelerometer mounted on the boring bar. Since the goal is to reduce vibration, acceleration should be as small as possible. The acceleration signal is denoted e(t); the control signal to the actuator is denoted y(t). The PID controller in continuous time t can be written as [24] de(t) 1 t (1) e(τ )dτ + Td y(t) = K e(t) + Ti 0 dt where K is the gain of the controller, Ti is the integration time and Td is the derivative time. The proportional part of the controller sets the constant gain. The integral part in conjunction with a proportional part improves steady state properties; when combined with derivative control, it also improves the transient properties [17]. A discretized version of the analog PID controller can be approximated as [24] y(n) = P (n) + I(n) + D(n)
(2)
where P (n) = Ke(n − 1) K e(n − 1) Fs Ti Fs KTd N Fs Td D(n − 1) − e(n − 1) − e(n − 2) D(n) = Fs Td + N Fs Td + N I(n) = I(n − 1) +
(3) (4) (5)
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Active Vibration Control of Boring Bar Vibrations
where Fs is the sampling frequency and N is a high frequency gain limitation of the derivative part. e(n) is the resulting error; in active vibration control in a boring operation it is also the accelerometer signal. Fig. 11 shows a block diagram of a d(n) K
y(n)
Forward path C
yC(n)
Σ e(n)
x(n) = e(n-1)
Unit delay z-1
Figure 11: Block diagram of a feedback control system based on a digital P controller. feedback control system with a digital P controller. Here the box with the unit delay z −1 between e(n) and e(n − 1) at the input to the controller indicated that the subject is a digital controller in a feedback control system. 2.3.2
LMS Algorithm
The least mean square LMS algorithm was developed by Widrow and Hoff in 1960. The least square approach provides a powerful approach to digital filtering in situations where a fixed, finite length filter is applicable. This approach has been widely used in many areas [20]. It is an important member of the family of stochastic gradient algorithms [18]. The LMS algorithm is very simple and has therefore been made the standard against which other adaptive filtering algorithms are benchmarked [18]. In the active control of vibration application discussed here, an LMS algorithm was used when estimating the forward path of the system. A forward path is always present in active control applications [19]; if the filtered-x LMS algorithm is used as a controller, an estimate of the forward path is also needed. A block diagram of the forward path estimation using an LMS algorithm is shown in Fig. 12. The task of the LMS algorithm is to imitate a desired signal d(n) by letting an input signal x(n) pass an adaptive filter wn (k) to produce an output y(n). The algorithm adjusts the weights wn (k) so that the error signal e(n) is minimized in the mean
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Noise generator
x(n)
Forward path
d(n)
Adaptive filter w(n)
y(n)
+ − Σ e(n)
LMS
Figure 12: Block diagram of the forward path estimation using an LMS algorithm.
square sense. The error signal can be written as e(n) = d(n) − y(n) = d(n) −
L−1
wn (l)x(n − l),
(6)
l=0
where the adaptive FIR filter has L coefficients. The weights are updated on average in the negative direction of the gradient. The gradient estimate in the LMS algorithm consists of the derivatives of the square error signal with respect to each of the weights of the adaptive filter
∂e2 (n) = ∂wn (k)
∂ d(n) −
L−1
2 x(n − l)wn (l)
l=0
∂wn (k)
= −2x(n − k)e(n)
(7)
for k = 0, 1, . . . , L − 1. The weights can now be updated in the negative direction of the gradient estimate as wn+1 (0) wn (0) x(n) wn+1 (1) wn (1) x(n − 1) (8) = + µ e(n), .. .. .. . . . wn+1 (L − 1) wn (L − 1) x(n − L + 1) To enable convergence in the mean square of the LMS algorithm the step size µ is usually selected according to the inequality [28] 0
