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Modelling of an Innovative Technology for Pavement Milling
7th 7th IFAC IFAC Symposium Symposium on on Mechatronic Mechatronic Systems Systems 7th IFAC Symposium on MechatronicUniversity, Systems UK September 5-8, 2016. Loughborough 7th IFAC Symposium on MechatronicUniversity, Systems UK September 5-8, 2016. Loughborough September 5-8, 2016. Loughborough Available University,online UK September 5-8, 2016. Loughborough University, UK at www.sciencedirect.com
ScienceDirect IFAC-PapersOnLine 49-21 (2016) 591–597
Modelling Modelling Modelling Modelling
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of an an Innovative Innovative Technology Technology of of an Technology ofPavement an Innovative Innovative Technology Milling Pavement Milling Pavement Milling Pavement Milling
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∗ ∗ J. Blum; Prof. Dr. R. Anderegg J. Blum; Prof. Dr. R. Anderegg ∗ ∗ J. Blum; Prof. Dr. R. Anderegg J. Blum; Prof. Dr. R. Anderegg ∗ Automation, SWITZERLAND Automation, FHNW, FHNW, 5210 5210 Windisch; Windisch; SWITZERLAND Windisch; Automation, [email protected]; [email protected]). Automation, FHNW, FHNW, 5210 5210 Windisch; SWITZERLAND SWITZERLAND [email protected]; [email protected]). [email protected]; [email protected]; [email protected]). [email protected]).
Abstract: A A new, new, dynamical dynamical approach approach for for increasing increasing the the effectiveness effectiveness of of the the pavement pavement milling milling Abstract: Abstract: A dynamical approach for the effectiveness of pavement milling process is shown in this paper. The technology is derived aa dynamical model of aa two-mass Abstract: A new, new, dynamical approach for increasing increasing thefrom effectiveness of the the pavement milling process is shown in this paper. The technology is derived from dynamical model of two-mass process is shown in paper. The technology is derived from dynamical model aa two-mass oscillator a simple the pavement material. are combined simulate process is and shown in this thismodel paper.of derived Both from aamodels dynamical model of of to two-mass oscillator and a simple simple model ofThe thetechnology pavement is material. Both models are combined combined to simulate oscillator and a model of the pavement material. Both models are to simulate the milling operation. Different excitation methods are used and combined to find the optimal oscillator and a simple model of the pavement material. Both models are combined to simulate the milling operation. Different excitation methods are used and combined to find the optimal the milling operation. Different excitation methods are used and to optimal operation of the model by taking the effectiveness The simulation results are the millingpoint operation. Different excitation methods are into usedaccount. and combined combined to find find the the optimal operation point of the model by taking the effectiveness into account. The simulation results are operation point of the model by taking the effectiveness into account. The simulation results are validated on a test stand with suitable proximity settings. With the model being validated, it operation point of the model by taking the effectiveness into account. The simulation results are validated on a test stand with suitable proximity settings. With the model being validated, it validated on a test stand with suitable proximity settings. With the model being validated, it can be scaled to a more complex model that also takes the geometry into account. The benefit validated on a test stand with suitable proximity settings. With the model being validated, it can be scaled to a more complex model that also takes the geometry into account. The benefit can be scaled to complex model that also takes the into account. The benefit of more effective milling process results for example reduction of machine size with can scaled to a a more more complex that takes in theaa geometry geometry account. The benefit of a a be more effective milling processmodel results foralso example in reduction into of the the machine size with of a more effective milling process results for example in a reduction of the machine size maintaining the same work output or in less fuel consumption of existing machines. of a more effective milling process results forfuel example in a reduction of the machine size with with maintaining the same work output or in less consumption of existing machines. maintaining the same work output or in less fuel consumption of existing machines. maintaining same workFederation output or less fuelControl) consumption machines. © 2016, IFACthe (International of in Automatic Hosting of byexisting Elsevier Ltd. All rights reserved. Keywords: Pavement Milling; Dynamical Non-linear Modelling; Material Modelling; Keywords: Pavement Milling; Dynamical Non-linear Modelling; Material Modelling; Keywords: Pavement Milling; Dynamical Non-linear Modelling; Material Modelling; Effectiveness; Excitation Models; Oscillator; Industry Application; Signal Processing; Model Keywords: Pavement Milling; Dynamical Non-linear Material Effectiveness; Excitation Models; Oscillator; IndustryModelling; Application; SignalModelling; Processing; Model Model Effectiveness; Excitation Models; Oscillator; Industry Application; Signal Processing; Scaling Effectiveness; Excitation Models; Oscillator; Industry Application; Signal Processing; Model Scaling Scaling Scaling 2. THE DYNAMICAL MILLING 1. 2. DESCRIPTION DESCRIPTION OF OF THE THE DYNAMICAL DYNAMICAL MILLING MILLING 1. INTRODUCTION INTRODUCTION 2. 1. 2. DESCRIPTION DESCRIPTION OF OFPROCESS THE DYNAMICAL MILLING 1. INTRODUCTION INTRODUCTION PROCESS PROCESS The PROCESS The state state of of the the art art process process in in pavement pavement milling milling nowadays nowadays The state of the art process in pavement milling nowadays is the cold milling method. It was introduced after the The state of milling the art process inItpavement milling nowadays is the cold method. was introduced after the The starting point of the investigation is the of The starting starting point point of of the the investigation investigation is is the the modelling modelling of is the cold method. It after the The hot milling process, which a of modelling of is cold milling milling It was was introduced introduced after hotthe milling process,method. which needed needed a preheating preheating of the the The the dynamic process. The milling situation is simplified. starting point of the investigation is the modelling of the dynamic process. The milling situation is simplified. hot milling process, which needed a preheating of the pavement surface for easier removal of the material. Due to hot milling process, which needed a preheating of the the dynamic process. The milling situation is simplified. pavement surface for easier removal of the material. Due to the Instead of a rotating drill bit on a rotary grinder, dynamic process. The milling situation is simplified. Instead of a rotating drill bit on a rotary grinder, a a pavement surface for removal the material. Due the poor poor overall overall energy effectiveness of the whole process process pavement surfaceenergy for easier easier removal of ofof thethe material. Due to to Instead of rotating drill bit on rotary grinder, aa the effectiveness whole translational bit that attached aa spring and Instead of aa drill rotating drillis bit on aa with rotary grinder, translational drill bit that is attached with spring and the poor overall energy effectiveness of the whole process (heating of the pavement!), it was not sustainable. New the poor of overall energy effectiveness of the whole process drill bit that attached with a chassis spring and (heating the pavement!), it was not not sustainable. New translational damper system to a chassis modelled. The translational drill that is isis spring acts and damper system system to bit chassis isattached modelled.with Thea chassis chassis acts (heating of the it sustainable. New forms rotary grinders drill bits were to aaa that chassis is modelled. The acts (heating the pavement!), pavement!), it was was New damper forms of of of rotary grinders and and drill not bitssustainable. were developed developed as the heavy weight is being driven by a motor. The damper system to chassis is modelled. The chassis acts as the heavy weight that is being driven by a motor. The forms of rotary grinders and drill bits were developed to the of Soon forms of rotary grindersperformance and drill bits weretools. developed the weight that driven by aa motor. The to increase increase the cutting cutting performance of the the tools. Soon as spring and between the chassis and the as the heavy heavy weight system that is is being being driven motor. spring and damper damper system between the by chassis and The the of to the cutting performance Soon the increase tools were were powerful enough to remove remove thetools. pavement between the chassis to increase thepowerful cutting enough performance of the thethe tools. Soon spring and damper system and the the tools to pavement drill bit bitand represent thesystem flexibility in the the the system and and the drill drill spring damper between chassis the drill represent the flexibility in system and the the tools were powerful enough to remove the pavement without the need to preheat the surface. This optimization the toolsthe were powerful enough to remove pavement drill bit the in the and the without need to preheat preheat the surface. surface. Thisthe optimization bit acts as the cutting tool while geometry. drill bit represent represent the flexibility flexibility inneglecting the system systemits and the drill drill bit acts as the cutting tool while neglecting its geometry. without the need to the This optimization of bits respect wear reduction acts as cutting tool neglecting its without the need to preheat theto Thisthe optimization of the the drill drill bits with with respect tosurface. wear and and the reduction bit bit acts as the the cutting tool1 while while neglecting its geometry. geometry. the reduction of the drill bits with respect to wear and On the right side of figure there is the pavement material cutting forces has made a giant leap in pavement of the drill bits with respect to wear and the reduction On the right side of figure 1 there is the pavement material of the cutting forces has made a giant leap in in pavement pavement On the right side of figure 1 there is the pavement material of the forces made leap model. The overall behaviour of the material is modelled milling and mining mining applications. Modern cold milling On the right side of figure 1 there is the pavement material of the cutting cutting forces has has made aa giant giant leap cold in pavement model. The overall behaviour of the material is modelled milling and applications. Modern milling model. The overall behaviour of is milling and mining applications. Modern cold milling with massless spring and system has machines come in all different sizes. They are overall behaviour of the the material material is modelled modelled milling mining Modern coldavailable milling model. with aa The massless spring and damper damper system that that has the the machinesand come in all allapplications. different sizes. sizes. They are are available machines come in different They available with a massless spring and damper system that has the special ability of being able to break when the reaction from small surface milling machines, which weigh 4 tons, with a massless spring and damper system that has the machines come in all different sizes. They are available special ability of being able to to break break when when the the reaction reaction from small surface milling machines, machines, which which weigh weigh 44 tons, tons, special ability being from small surface force reaches reaches defined limit. When this happens happens the to ton which are for the ability aaof of defined being able able to When break when the reaction from surface milling milling which 4 tons, special force limit. this the to 40 40 small ton machines machines which machines, are designed designed for weigh the complete complete force reaches a defined limit. this the to 40 ton machines which are designed for the complete material model is moved forward by aa prefixed increment. removal of highway pavements. Despite all the progress force reaches aisdefined limit. When When this happens happens the to 40 tonofmachines which are designed for the complete material model moved forward by prefixed increment. removal highway pavements. Despite all the progress material model forward by prefixed removal ofbeen highway pavements. Despite all the that has made in this field, the core has material model is is moved moved forward by aarepresent prefixed increment. increment. removal highway all principle the progress progress that has has ofbeen been made pavements. in this this field, field,Despite the core principle has The combination of these two parts the simple that made in the core principle has The combination of these two parts represent the simple simple always stayed the same. All machines use a rotary grinder that has been made in this field, the core principle has The combination of these two parts represent the always stayed the same. All machines use a rotary grinder The model of the milling process. A schematic of the model is combination of these two parts represent the simple model of the milling process. A schematic of the model is always stayed the same. All machines use a rotary grinder that statically the chassis the machine. always the mounted same. Allto machines use aof rotary grinder model of the milling process. A schematic of the model is that is is stayed statically mounted to the chassis of the machine. shown in fig. 1. model of the milling process. A schematic of the model is mounted to the chassis of the machine. shown in fig. 1. that is statically The more material that has to be milled, the heavier that statically mounted to the of the The is more material that has has to chassis be milled, themachine. heavier shown shown in in fig. fig. 1. 1. The more that be the heavier the that designed the they The more material material that has to to and be milled, milled, thefuel heavier the machines machines that are are designed and the more more fuel they the machines that are designed and the more fuel they consume. the machines that are designed and the more fuel they consume. consume. consume. The paper can enable new The new new technology technology in in this this paper paper can can enable enable aa new new The new technology in this aahigher design generation of milling machines that has a The new technology in this paper can enable new design generation of milling machines that has a higher design generation of machines that aa higher performance to ratio. key for design generation of milling milling machines that has has higher performance to weight weight ratio. The The key element element for achieving achieving performance to weight ratio. The key element for achieving this goal lies in the potential energy of dynamic vibrations. performance to the weight ratio. The keyofelement forvibrations. achieving this goal lies in potential energy dynamic this goal lies in potential energy of vibrations. The paper show possibility of to this liesaims in the theto of dynamic dynamic The goal paper aims topotential show a a energy possibility of how howvibrations. to apply apply possibility The paper aims to show a of how to apply vibrational techniques to a two mass oscillator in way The paper aims to show a possibility of how to vibrational techniques to a two mass oscillator in aaapply way vibrational techniques to oscillator aa way that the the mean mean forces between between themass masses can be bein reduced vibrational techniques to aa two two mass oscillator inreduced way Fig. 1. Model scheme of the milling process that forces the masses can Fig. 1. Model scheme of the milling process that the mean between the masses can be while maintaining the same work with that mean forces forces theoutput. massesE.g. canmilling be reduced reduced whilethe maintaining the between same work work output. E.g. milling with Fig. Fig. 1. 1. Model Model scheme scheme of of the the milling milling process process while maintaining the same output. E.g. milling with the same speed while the mean occurring forces decrease. while maintaining the same work output. E.g. milling with the same speed while the mean occurring forces decrease. decrease. the same speed while the mean occurring forces Lower mean forces imply and the same speed whileper thedefinition mean occurring forces work decrease. Lower mean forces per definition imply lower lower work and The The variables variables and and parameters parameters used used in in this this model model can can be be Lower mean definition therefore less forces energyper consumption. seen variables in table table 1. 1.and Lower mean forces per definition imply imply lower lower work work and and The The variables and parameters parameters used used in in this this model model can can be be therefore less energy consumption. seen in therefore less energy consumption. seen in table 1. therefore less energy consumption. seen in table 1. Copyright 2016 591 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016, 2016 IFAC IFAC 591 Hosting by Elsevier Ltd. All rights reserved. Copyright 2016 responsibility IFAC 591Control. Peer review© of International Federation of Automatic Copyright ©under 2016 IFAC 591 10.1016/j.ifacol.2016.10.665
2016 IFAC MECHATRONICS 592 J. Blum September 5-8, 2016. Loughborough University, UK et al. / IFAC-PapersOnLine 49-21 (2016) 591–597
Table 1. Model variables and parameters Name xs xm xb xi δ ms mm km cm kb cb kbr Fa vc
Unit [m] [m] [m] [m] [m] [kg] [kg] [N/m] [N s/m] [N/m] [N s/m] [N/m] [N ] [m/s]
break it is necessary to define a maximum stress force of the material. The maximum material force Fb,max is assumed to be proportional to the cutting depth δ as in Araujo et al. (2009) and Araujo et al. (2010) or Vlasblom (2007). It follows:
Description Position of the chassis Position of the drill bit Position of the material element Increment of the material or chip size Single cutting depth Mass of the chassis Mass of the drill bit Spring constant of the drill bit Damping constant of the drill bit Spring constant of the material Damping constant of the material Breaking constant of the material Driving force of the chassis Set point for cutting speed
Fb,max = kbr · δ
The two mass oscillator is made of the chassis with position xs and mass ms and a single drill bit with position xm and mass mm . The two masses are connected with a spring with constant km and a damper with constant cm . To move the chassis forward, a force Fa is applied to it. This force is regulated by a PI Controller that keeps the chassis at a designated cutting speed vc . The differential equation for the chassis is in eq. (1) and for the drill bit in eq. (2). Similar dynamical equations are also used in Balachandran (2001), Balachandran and Zhao (2000), Zhao and Balachandran (2001), Araujo et al. (2009), Araujo et al. (2010), Sutherland and DeVor (1988) and Altinta¸s and Budak (1995). ms · x ¨s = Fa − Fm
(6)
The maximum material force therefore depends on the cutting depth. To keep the model simple the cutting depth is assumed to be constant over time. When the ground force Fb reaches the maximum force Fb,max the material element moves by the increment xi . This is described in eq. (7). Another approach is to take the cutting kinematics into account as in section 5. xb + xi , if Fb > kbr · δ xb = (7) xb , else The breaking constant kbr is not to be confused with the spring constant kb . The difference can be seen in the stress-strain diagram for brittle materials (compare fig. 2). Vlasblom (2007) also uses brittle failure for the modelling of rock.
(1)
The driving force for the chassis is the applied force Fa (eq. (3)) from the controller and the opposing force Fm comes from the drill bit (eq. (4)). mm · x ¨m = Fm − Fb
(2)
Equation 2 is the one for the drill bit. Its driving force is the drill bit force Fm (eq. (4)) and the opposing force is the ground force Fb (eq. (5)). The applied force Fa from the PI Controller is modelled as in eq. (3). Fa = Kp · (vc − x˙ s ) + Ki · (vc − x˙ s )dt (3) The force Fm between the two masses is modelled as in eq. (4). Fm = cm · (x˙ s − x˙ m ) + km · (xs − xm )
(4)
To describe the ground reaction force Fb a non-linearity has to be introduced. The ground force can only occur while there is contact. Additionally, the contact force can not become negative. Therefore eq. (5): cb · x˙ m + kb · (xm − xb ), if xm > xb and eq. > 0 Fb = (5) 0, else The modelling of the non-linear contact is the same as a jumping ball in Nollau (2009). Next to the two parameters kb and cb , the behaviour of the material is described with three additional parameters. These are the breaking constant kbr , the cutting depth δ and the material increment xi . To allow the material to 592
Fig. 2. Stress-strain diagram for the modelling of brittle material with two coefficients kbr and kb The spring constant kb describes how fast the breaking force Fb,max is reached and the breaking constant kbr describes the limit of Fb or in other words, the value of Fb,max There are two non-linearities which have different behaviours. The first non-linearity comes from the contact condition and the second one from the breaking of the material. The described system can now be excited with different methods. 3. EXCITATION OF THE MODEL The excitation of the model influences the systems behaviour. Since it contains two non-linearities, the systems response is not linear and can drift into chaos. To investigate the systems behaviour, two different excitations are being tested. These are: • Normal excitation and • Waypoint excitation
To be able to compare the different simulation results, it is necessary to introduce an effort coefficient for each parameter setting. The effort coefficient η is calculated
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from a representative force F divided by the maximum force Fb,max which is necessary to break the material. η=
Fx Fb,max
(8)
With this definition, an η of 1 indicates full effort and an η of 0 indicates no effort. The effort coefficient η can be calculated from several properties Fx of the steady-state reaction forces of the simulation. The most interesting aspects considering the effectiveness of the process are: • • • •
Maximum drill bit force: max(Fm ) → Fx , Mean drill bit force: F m → Fx Minimum drill bit force: min(Fm ) → Fx , and Mean driving force F a → Fx
To get the corresponding effort coefficient ηx the property Fx has to be divided by Fb,max . 3.1 Normal excitation
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The dynamics of this normal excitation is dominated by the cutting speed divided by the chip thickness xi of the material (compare fig. 3). The resulting frequency is the so called natural frequency fnat : vc fnat = (9) xi This natural frequency is typically much higher than the resonance frequency of the system because of small chip thicknesses and is thereby damped very well by the systems frequency response. In the fig. 3 simulation however, to do the plausibility test of the model, the chip thickness chosen has to be much bigger. The resulting natural frequency fnat is 2.5Hz. Note that the effort coefficient ηFa is only 0.45. The coefficient is low, because there is not much material to cut (xi = 0.5m). If the chip thickness is decreased to a more practical observable value, the systems behaviour changes. In the next example fig. 4 we can observe a higher natural frequency (fnat = 40Hz) as well as an increased controller effort ηFa = 0.93.
The normal excitation is the simple milling operation. The chassis is moved by the driving force Fa of the controller towards the material and so does the drill bit. When the drill bit makes contact with the material, the ground force Fb occurs. This force is transferred to the drill bit and therefore to the chassis. The effort Fa from the controller has to increase in order to keep the chassis speed vc at the set point. When the ground force reaches the breaking force Fb,max the material breaks. This behaviour can be shown in the simulation data in fig. 3.
Fig. 4. Simulation of normal cutting operation with small chip thickness The above simulation also shows that the drill bit position xm (blue line) is always larger than the material position xb (red line). This means that there is contact at all time. There is no lift off of the material. As long as this state occurs, the effort η for cutting will always be close to 1. The plot in figure 5 shows the effort coefficient η for the normal excitation mode in function of the cutting speed vc .
Fig. 3. Simulation of normal cutting operation with big chip thickness to check the models behaviour 593
It can be seen that if vc is lower than 0.5m/s, no continuous cutting is occurring. Whereas above 0.5m/s, the effort η is close to 1. This can also be seen from the differential equations (1) and (2).
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Fig. 5. Effort coefficient η in function of the cutting speed vc 3.2 Waypoint excitation The waypoint excitation is defined as an additional, vibrational force on the drill bit that generates additional contact pressure. To simulate this scenario, eq. (2) has to be modified as follows: ¨m = Fm − Fb + (Aw · sin(2 · π · fw · t)) mm · x
(10)
Where Aw and fw denote the amplitude, resp. the frequency of the superimposed force. To analyse the behaviour of the waypoint excited system, the frequency fw is varied while the cutting speed vc and the amplitude Aw of the Excitation are kept constant. The result is the following plot (fig. 6) that shows a significant drop of the effort in the region of 10Hz excitation frequency.
Fig. 7. Simulation result for the waypoint excitation with fw = 10Hz The effort coefficient η drops to 0.48. This mean that only half the force is needed to maintain the cutting speed vc of 2m/s. This is where the energy saving can take place. 3.3 Conclusion of the simulation model for different excitations The simulation model behaves as expected. As soon as the drill bit makes contact with the material, the ground force rises and is transferred to the chassis. To maintain the set point of the cutting speed, the controller has to produce more effort. This effort can be used to measure the effectiveness of the current parameter setting. The normal excitation (classical milling process) shows no increase of effectiveness with higher cutting speeds. But the behaviour of the system is as expected.
Fig. 6. Effort coefficient η in function of the frequency fw with waypoint excitation In order for this to work, the stiffness between the chassis and the drill bit has to be small to allow the additional movement the drill bit is making. The next plot (fig. 7) shows the named situation at 10Hz. It can be seen that the drill bit is oscillating which results in a periodically lift off of the drill bit. When the drill bit makes contact again, several cuts occur shortly behind each other. In this phase the kinematic energy of the drill bit is used to cut the material. 594
The waypoint excitation shows a great increase of effectiveness between an excitation frequency fw of 5..25Hz. For this to work the drill bit has to be able to move relatively independently from the chassis. A higher frequency is not necessarily better because the effect is damped by the systems damping at higher frequencies. 4. VALIDATION OF THE MODEL The validation of the model is an important step to undertake before advancing to the scaling.. 4.1 Experimental Set-up The experimental setup takes place on a test stand, designed for optimizing milling processes in stone-like materials. A single drill bit is equipped with strain gauges and
2016 IFAC MECHATRONICS September 5-8, 2016. Loughborough University, J. Blum UK et al. / IFAC-PapersOnLine 49-21 (2016) 591–597
an accelerometer. The strain gauges measure the contact force Fm of the drill bit, while the accelerometer measures the accelerations of the drill bit in all three principal axes. Because it is not practical to cut in a straight line, the movement of the drill bit was chosen to be a helix. The drill bit faces constant cutting conditions during its movement on a semicircle, while it does not make any contact with the material on the other side. This setup ensures perfectly reproducible conditions for each cut (see fig. 8).
window is calculated. This step is shown in figure 9.A and B. (2) The resulting trend (9.B) of the mean is continuously monitored for a local maximum. This is shown in 9.C: The derivation of the mean-trend is the argument of the signum-function. The resulting signum-funcion is derived again. If the derived signum-function is negative, a local maximum has been found. (3) The found maximum in the last step is added to the list of maximums. (4) The representative value (red circle in fig. 9.D) for the set of cuts is the mean of all found maximums.
Fig. 8. Overview of the experimental setup The tested parameters can be seen in the next table 2. Table 2. Experimental parameters Parameter Material Cutting depth (mm) Cutting speed (m/s) Excitation Waypoint excitation freq. (Hz)
Values C30/37 3 2 none 10
5 2.5 waypoint
7 3
With each parameter setting, twenty cuts were undertaken. The data of the strain gauges show the contact and no-contact phases of the helix cutting movement (compare figure 9.A). 4.2 Classification of the Experiments To classify a set of cuts with constant parameters, it is necessary to reduce the data to a simple, representing value. In this case, a multi-level averaging of the data with local maximum value detection is used. The algorithm works as follows: (1) A rectangle window is moved over the raw data, at every position, the mean value of all data in the 595
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Fig. 9. Signalprocessing of the strain gauge signal
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The rating coefficient converges with the number of found maximums. This can also be shown in figure 9.D. The rating coefficient can be interpreted as the mean reaction force for a specific parameter set and is therefore a good representation of a set of cuts. 4.3 Experimental Results With the classification algorithm from section 4.2 the overall result can be seen in figure 10. Fig. 11. Overview of the milling situation of a single disk in this case are simple if the movement is not disturbed by the cutting forces. For simplicity reasons the cutting forces are neglected to calculate the trajectories. The trajectory of one drill bit is the superposition of a rotational and a translational movement. This can be calculated for each drill bit with an angular offset. The total cutting depth is the sum of the single cutting depths. δ=
N 2 · r · π · vf sin(Θi ) · M at(Θi ) N · vc i=1
(11)
with Θi = Fig. 10. Experimental results The following statements can be made from the measurements: (1) The mean reaction force of the drill bit (Fm ) increases with increasing cutting depth δ. This is the validation of equation (6). (2) The mean reaction force of the drill bit (Fm ) decreases, if the system is excited. This is the validation of the model behaviour that shows the reduction of the effort coefficient. For these reasons, the simple model for the drill bit material interaction is validated and can be used for further investigations. 5. SCALABILITY OF THE MODEL With the model being validated, the next step can be undertaken: The scaling. A model with only one drill bit is not suitable to describe the overall milling process. The next step consists of taking the geometry of a rotary grinder into account to calculate the variation of the cutting depth δ over time. Figure 11 gives an overview of the milling situation. The single disk model of a rotary grinder introduces four more parameters. These are the number of drill bits in the trace N , the radius r, the feed rate vf of the grinder and the cutting height h. The cutting speed vc is the same as in the last sections. 5.1 Trajectories of the drill bits To describe the cutting depth δ over time, it is necessary to know the trajectories of the drill bits. The trajectories 596
and
vc 2·π ·t+ (i − 1) r N
h 1, if 0 < Θi < acos 1 − M at(Θi ) = r 0, else
(12)
(13)
Equation (11) consists of a constant term that describes the cutting depth in the horizontal plane with the current parameter setting. The sine function is the correction term for the i-th drill bit angle. The M at(Θi )-function is a simple contact condition. It becomes 1 if the i-th drill bit is in the range (angle) of the material. Otherwise there is no contact and the cutting depth δ is 0. B. Balachandran Balachandran (2001) uses a similar model for the description of a metal milling process in his paper. The modelling of this disk has an influence on the cutting depth δ which means that the maximum force Fb,max changes in time, depending on the parameters r, vf , N , vc and h. A typical course of this function (eq. (11)) can be seen in the plot (fig. 12). One can see in the course of the function when two drill bits are cutting simultaneously and when only one drill bit is cutting. The Fourier-transform of this signal looks as in fig. 12 below. This course of δ can now be implemented in the simulation model to investigate the drill disks behaviour. 6. CONCLUSION The simple model from section 2 is sufficient for the modelling of the main effects of the milling process of a single drill bit with constant cutting depth. The excitation of the system in a vibrational way results in a lower driving
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Balachandran, B. and Zhao, M. (2000). A mechanics based model for study of dynamics of milling operations. Meccanica, 35(2), 89–109. Nollau, R. (2009). Modellierung und Simulation technischer Systeme: eine praxisnahe Einf¨ uhrung. Springer Science & Business Media. Sutherland, J.W. and DeVor, R. (1988). A dynamic model of the cutting force system in the end milling process. Diss. Abstr. Int., 49(1), 293. Vlasblom, W. (2007). Cutting of rock. Lecture Book Dredging Processes. Zhao, M. and Balachandran, B. (2001). Dynamics and stability of milling process. International Journal of Solids and Structures, 38(10), 2233–2248.
Fig. 12. Cutting depth of a drill disk with 6 drill bits and its spectrum effort of the system by the same mean cutting speed and cutting depth. The experimental setup is able to confirm the simulation results by measuring the current drill bit force with strain gauges. The model can be scaled to a single rotary disk by adjusting the cutting depth δ with the geometrical information of the system. The reduced driving force for the same work output can enable a new generation of milling machines that have a higher performace to weight ratio or that consume less fuel for the milling process itself. ACKNOWLEDGEMENTS The author thanks the supporting industry partner (confidential) who enabled the possibility for the experiments. Thank you to my supporting advisor: Prof. Dr. Roland Anderegg for guiding me through the project. REFERENCES Altinta¸s, Y. and Budak, E. (1995). Analytical prediction of stability lobes in milling. CIRP Annals-Manufacturing Technology, 44(1), 357–362. Araujo, A.C., Pacheco, P., and Savi, M.A. (2009). Dynamical analysis of an end milling process. In 20th International Congress of Mechanical Engineering. Araujo, A.C., Savi, M.A., and Pacheco, P. (2010). Experimental and numerical dynamical analysis of an end milling process. In VI Congresso Nacional de engenharia Mecˆ anica, Campina Grande, PB, Brazil. Balachandran, B. (2001). Nonlinear dynamics of milling processes. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 359(1781), 793–819. 597
ScienceDirect IFAC-PapersOnLine 49-21 (2016) 591–597
Modelling Modelling Modelling Modelling
∗ ∗ ∗ ∗ ∗
Institute for Institute for Institute for (e-mail: Institute for (e-mail: (e-mail: (e-mail:
of an an Innovative Innovative Technology Technology of of an Technology ofPavement an Innovative Innovative Technology Milling Pavement Milling Pavement Milling Pavement Milling
for for for for
∗ ∗ J. Blum; Prof. Dr. R. Anderegg J. Blum; Prof. Dr. R. Anderegg ∗ ∗ J. Blum; Prof. Dr. R. Anderegg J. Blum; Prof. Dr. R. Anderegg ∗ Automation, SWITZERLAND Automation, FHNW, FHNW, 5210 5210 Windisch; Windisch; SWITZERLAND Windisch; Automation, [email protected]; [email protected]). Automation, FHNW, FHNW, 5210 5210 Windisch; SWITZERLAND SWITZERLAND [email protected]; [email protected]). [email protected]; [email protected]; [email protected]). [email protected]).
Abstract: A A new, new, dynamical dynamical approach approach for for increasing increasing the the effectiveness effectiveness of of the the pavement pavement milling milling Abstract: Abstract: A dynamical approach for the effectiveness of pavement milling process is shown in this paper. The technology is derived aa dynamical model of aa two-mass Abstract: A new, new, dynamical approach for increasing increasing thefrom effectiveness of the the pavement milling process is shown in this paper. The technology is derived from dynamical model of two-mass process is shown in paper. The technology is derived from dynamical model aa two-mass oscillator a simple the pavement material. are combined simulate process is and shown in this thismodel paper.of derived Both from aamodels dynamical model of of to two-mass oscillator and a simple simple model ofThe thetechnology pavement is material. Both models are combined combined to simulate oscillator and a model of the pavement material. Both models are to simulate the milling operation. Different excitation methods are used and combined to find the optimal oscillator and a simple model of the pavement material. Both models are combined to simulate the milling operation. Different excitation methods are used and combined to find the optimal the milling operation. Different excitation methods are used and to optimal operation of the model by taking the effectiveness The simulation results are the millingpoint operation. Different excitation methods are into usedaccount. and combined combined to find find the the optimal operation point of the model by taking the effectiveness into account. The simulation results are operation point of the model by taking the effectiveness into account. The simulation results are validated on a test stand with suitable proximity settings. With the model being validated, it operation point of the model by taking the effectiveness into account. The simulation results are validated on a test stand with suitable proximity settings. With the model being validated, it validated on a test stand with suitable proximity settings. With the model being validated, it can be scaled to a more complex model that also takes the geometry into account. The benefit validated on a test stand with suitable proximity settings. With the model being validated, it can be scaled to a more complex model that also takes the geometry into account. The benefit can be scaled to complex model that also takes the into account. The benefit of more effective milling process results for example reduction of machine size with can scaled to a a more more complex that takes in theaa geometry geometry account. The benefit of a a be more effective milling processmodel results foralso example in reduction into of the the machine size with of a more effective milling process results for example in a reduction of the machine size maintaining the same work output or in less fuel consumption of existing machines. of a more effective milling process results forfuel example in a reduction of the machine size with with maintaining the same work output or in less consumption of existing machines. maintaining the same work output or in less fuel consumption of existing machines. maintaining same workFederation output or less fuelControl) consumption machines. © 2016, IFACthe (International of in Automatic Hosting of byexisting Elsevier Ltd. All rights reserved. Keywords: Pavement Milling; Dynamical Non-linear Modelling; Material Modelling; Keywords: Pavement Milling; Dynamical Non-linear Modelling; Material Modelling; Keywords: Pavement Milling; Dynamical Non-linear Modelling; Material Modelling; Effectiveness; Excitation Models; Oscillator; Industry Application; Signal Processing; Model Keywords: Pavement Milling; Dynamical Non-linear Material Effectiveness; Excitation Models; Oscillator; IndustryModelling; Application; SignalModelling; Processing; Model Model Effectiveness; Excitation Models; Oscillator; Industry Application; Signal Processing; Scaling Effectiveness; Excitation Models; Oscillator; Industry Application; Signal Processing; Model Scaling Scaling Scaling 2. THE DYNAMICAL MILLING 1. 2. DESCRIPTION DESCRIPTION OF OF THE THE DYNAMICAL DYNAMICAL MILLING MILLING 1. INTRODUCTION INTRODUCTION 2. 1. 2. DESCRIPTION DESCRIPTION OF OFPROCESS THE DYNAMICAL MILLING 1. INTRODUCTION INTRODUCTION PROCESS PROCESS The PROCESS The state state of of the the art art process process in in pavement pavement milling milling nowadays nowadays The state of the art process in pavement milling nowadays is the cold milling method. It was introduced after the The state of milling the art process inItpavement milling nowadays is the cold method. was introduced after the The starting point of the investigation is the of The starting starting point point of of the the investigation investigation is is the the modelling modelling of is the cold method. It after the The hot milling process, which a of modelling of is cold milling milling It was was introduced introduced after hotthe milling process,method. which needed needed a preheating preheating of the the The the dynamic process. The milling situation is simplified. starting point of the investigation is the modelling of the dynamic process. The milling situation is simplified. hot milling process, which needed a preheating of the pavement surface for easier removal of the material. Due to hot milling process, which needed a preheating of the the dynamic process. The milling situation is simplified. pavement surface for easier removal of the material. Due to the Instead of a rotating drill bit on a rotary grinder, dynamic process. The milling situation is simplified. Instead of a rotating drill bit on a rotary grinder, a a pavement surface for removal the material. Due the poor poor overall overall energy effectiveness of the whole process process pavement surfaceenergy for easier easier removal of ofof thethe material. Due to to Instead of rotating drill bit on rotary grinder, aa the effectiveness whole translational bit that attached aa spring and Instead of aa drill rotating drillis bit on aa with rotary grinder, translational drill bit that is attached with spring and the poor overall energy effectiveness of the whole process (heating of the pavement!), it was not sustainable. New the poor of overall energy effectiveness of the whole process drill bit that attached with a chassis spring and (heating the pavement!), it was not not sustainable. New translational damper system to a chassis modelled. The translational drill that is isis spring acts and damper system system to bit chassis isattached modelled.with Thea chassis chassis acts (heating of the it sustainable. New forms rotary grinders drill bits were to aaa that chassis is modelled. The acts (heating the pavement!), pavement!), it was was New damper forms of of of rotary grinders and and drill not bitssustainable. were developed developed as the heavy weight is being driven by a motor. The damper system to chassis is modelled. The chassis acts as the heavy weight that is being driven by a motor. The forms of rotary grinders and drill bits were developed to the of Soon forms of rotary grindersperformance and drill bits weretools. developed the weight that driven by aa motor. The to increase increase the cutting cutting performance of the the tools. Soon as spring and between the chassis and the as the heavy heavy weight system that is is being being driven motor. spring and damper damper system between the by chassis and The the of to the cutting performance Soon the increase tools were were powerful enough to remove remove thetools. pavement between the chassis to increase thepowerful cutting enough performance of the thethe tools. Soon spring and damper system and the the tools to pavement drill bit bitand represent thesystem flexibility in the the the system and and the drill drill spring damper between chassis the drill represent the flexibility in system and the the tools were powerful enough to remove the pavement without the need to preheat the surface. This optimization the toolsthe were powerful enough to remove pavement drill bit the in the and the without need to preheat preheat the surface. surface. Thisthe optimization bit acts as the cutting tool while geometry. drill bit represent represent the flexibility flexibility inneglecting the system systemits and the drill drill bit acts as the cutting tool while neglecting its geometry. without the need to the This optimization of bits respect wear reduction acts as cutting tool neglecting its without the need to preheat theto Thisthe optimization of the the drill drill bits with with respect tosurface. wear and and the reduction bit bit acts as the the cutting tool1 while while neglecting its geometry. geometry. the reduction of the drill bits with respect to wear and On the right side of figure there is the pavement material cutting forces has made a giant leap in pavement of the drill bits with respect to wear and the reduction On the right side of figure 1 there is the pavement material of the cutting forces has made a giant leap in in pavement pavement On the right side of figure 1 there is the pavement material of the forces made leap model. The overall behaviour of the material is modelled milling and mining mining applications. Modern cold milling On the right side of figure 1 there is the pavement material of the cutting cutting forces has has made aa giant giant leap cold in pavement model. The overall behaviour of the material is modelled milling and applications. Modern milling model. The overall behaviour of is milling and mining applications. Modern cold milling with massless spring and system has machines come in all different sizes. They are overall behaviour of the the material material is modelled modelled milling mining Modern coldavailable milling model. with aa The massless spring and damper damper system that that has the the machinesand come in all allapplications. different sizes. sizes. They are are available machines come in different They available with a massless spring and damper system that has the special ability of being able to break when the reaction from small surface milling machines, which weigh 4 tons, with a massless spring and damper system that has the machines come in all different sizes. They are available special ability of being able to to break break when when the the reaction reaction from small surface milling machines, machines, which which weigh weigh 44 tons, tons, special ability being from small surface force reaches reaches defined limit. When this happens happens the to ton which are for the ability aaof of defined being able able to When break when the reaction from surface milling milling which 4 tons, special force limit. this the to 40 40 small ton machines machines which machines, are designed designed for weigh the complete complete force reaches a defined limit. this the to 40 ton machines which are designed for the complete material model is moved forward by aa prefixed increment. removal of highway pavements. Despite all the progress force reaches aisdefined limit. When When this happens happens the to 40 tonofmachines which are designed for the complete material model moved forward by prefixed increment. removal highway pavements. Despite all the progress material model forward by prefixed removal ofbeen highway pavements. Despite all the that has made in this field, the core has material model is is moved moved forward by aarepresent prefixed increment. increment. removal highway all principle the progress progress that has has ofbeen been made pavements. in this this field, field,Despite the core principle has The combination of these two parts the simple that made in the core principle has The combination of these two parts represent the simple simple always stayed the same. All machines use a rotary grinder that has been made in this field, the core principle has The combination of these two parts represent the always stayed the same. All machines use a rotary grinder The model of the milling process. A schematic of the model is combination of these two parts represent the simple model of the milling process. A schematic of the model is always stayed the same. All machines use a rotary grinder that statically the chassis the machine. always the mounted same. Allto machines use aof rotary grinder model of the milling process. A schematic of the model is that is is stayed statically mounted to the chassis of the machine. shown in fig. 1. model of the milling process. A schematic of the model is mounted to the chassis of the machine. shown in fig. 1. that is statically The more material that has to be milled, the heavier that statically mounted to the of the The is more material that has has to chassis be milled, themachine. heavier shown shown in in fig. fig. 1. 1. The more that be the heavier the that designed the they The more material material that has to to and be milled, milled, thefuel heavier the machines machines that are are designed and the more more fuel they the machines that are designed and the more fuel they consume. the machines that are designed and the more fuel they consume. consume. consume. The paper can enable new The new new technology technology in in this this paper paper can can enable enable aa new new The new technology in this aahigher design generation of milling machines that has a The new technology in this paper can enable new design generation of milling machines that has a higher design generation of machines that aa higher performance to ratio. key for design generation of milling milling machines that has has higher performance to weight weight ratio. The The key element element for achieving achieving performance to weight ratio. The key element for achieving this goal lies in the potential energy of dynamic vibrations. performance to the weight ratio. The keyofelement forvibrations. achieving this goal lies in potential energy dynamic this goal lies in potential energy of vibrations. The paper show possibility of to this liesaims in the theto of dynamic dynamic The goal paper aims topotential show a a energy possibility of how howvibrations. to apply apply possibility The paper aims to show a of how to apply vibrational techniques to a two mass oscillator in way The paper aims to show a possibility of how to vibrational techniques to a two mass oscillator in aaapply way vibrational techniques to oscillator aa way that the the mean mean forces between between themass masses can be bein reduced vibrational techniques to aa two two mass oscillator inreduced way Fig. 1. Model scheme of the milling process that forces the masses can Fig. 1. Model scheme of the milling process that the mean between the masses can be while maintaining the same work with that mean forces forces theoutput. massesE.g. canmilling be reduced reduced whilethe maintaining the between same work work output. E.g. milling with Fig. Fig. 1. 1. Model Model scheme scheme of of the the milling milling process process while maintaining the same output. E.g. milling with the same speed while the mean occurring forces decrease. while maintaining the same work output. E.g. milling with the same speed while the mean occurring forces decrease. decrease. the same speed while the mean occurring forces Lower mean forces imply and the same speed whileper thedefinition mean occurring forces work decrease. Lower mean forces per definition imply lower lower work and The The variables variables and and parameters parameters used used in in this this model model can can be be Lower mean definition therefore less forces energyper consumption. seen variables in table table 1. 1.and Lower mean forces per definition imply imply lower lower work work and and The The variables and parameters parameters used used in in this this model model can can be be therefore less energy consumption. seen in therefore less energy consumption. seen in table 1. therefore less energy consumption. seen in table 1. Copyright 2016 591 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016, 2016 IFAC IFAC 591 Hosting by Elsevier Ltd. All rights reserved. Copyright 2016 responsibility IFAC 591Control. Peer review© of International Federation of Automatic Copyright ©under 2016 IFAC 591 10.1016/j.ifacol.2016.10.665
2016 IFAC MECHATRONICS 592 J. Blum September 5-8, 2016. Loughborough University, UK et al. / IFAC-PapersOnLine 49-21 (2016) 591–597
Table 1. Model variables and parameters Name xs xm xb xi δ ms mm km cm kb cb kbr Fa vc
Unit [m] [m] [m] [m] [m] [kg] [kg] [N/m] [N s/m] [N/m] [N s/m] [N/m] [N ] [m/s]
break it is necessary to define a maximum stress force of the material. The maximum material force Fb,max is assumed to be proportional to the cutting depth δ as in Araujo et al. (2009) and Araujo et al. (2010) or Vlasblom (2007). It follows:
Description Position of the chassis Position of the drill bit Position of the material element Increment of the material or chip size Single cutting depth Mass of the chassis Mass of the drill bit Spring constant of the drill bit Damping constant of the drill bit Spring constant of the material Damping constant of the material Breaking constant of the material Driving force of the chassis Set point for cutting speed
Fb,max = kbr · δ
The two mass oscillator is made of the chassis with position xs and mass ms and a single drill bit with position xm and mass mm . The two masses are connected with a spring with constant km and a damper with constant cm . To move the chassis forward, a force Fa is applied to it. This force is regulated by a PI Controller that keeps the chassis at a designated cutting speed vc . The differential equation for the chassis is in eq. (1) and for the drill bit in eq. (2). Similar dynamical equations are also used in Balachandran (2001), Balachandran and Zhao (2000), Zhao and Balachandran (2001), Araujo et al. (2009), Araujo et al. (2010), Sutherland and DeVor (1988) and Altinta¸s and Budak (1995). ms · x ¨s = Fa − Fm
(6)
The maximum material force therefore depends on the cutting depth. To keep the model simple the cutting depth is assumed to be constant over time. When the ground force Fb reaches the maximum force Fb,max the material element moves by the increment xi . This is described in eq. (7). Another approach is to take the cutting kinematics into account as in section 5. xb + xi , if Fb > kbr · δ xb = (7) xb , else The breaking constant kbr is not to be confused with the spring constant kb . The difference can be seen in the stress-strain diagram for brittle materials (compare fig. 2). Vlasblom (2007) also uses brittle failure for the modelling of rock.
(1)
The driving force for the chassis is the applied force Fa (eq. (3)) from the controller and the opposing force Fm comes from the drill bit (eq. (4)). mm · x ¨m = Fm − Fb
(2)
Equation 2 is the one for the drill bit. Its driving force is the drill bit force Fm (eq. (4)) and the opposing force is the ground force Fb (eq. (5)). The applied force Fa from the PI Controller is modelled as in eq. (3). Fa = Kp · (vc − x˙ s ) + Ki · (vc − x˙ s )dt (3) The force Fm between the two masses is modelled as in eq. (4). Fm = cm · (x˙ s − x˙ m ) + km · (xs − xm )
(4)
To describe the ground reaction force Fb a non-linearity has to be introduced. The ground force can only occur while there is contact. Additionally, the contact force can not become negative. Therefore eq. (5): cb · x˙ m + kb · (xm − xb ), if xm > xb and eq. > 0 Fb = (5) 0, else The modelling of the non-linear contact is the same as a jumping ball in Nollau (2009). Next to the two parameters kb and cb , the behaviour of the material is described with three additional parameters. These are the breaking constant kbr , the cutting depth δ and the material increment xi . To allow the material to 592
Fig. 2. Stress-strain diagram for the modelling of brittle material with two coefficients kbr and kb The spring constant kb describes how fast the breaking force Fb,max is reached and the breaking constant kbr describes the limit of Fb or in other words, the value of Fb,max There are two non-linearities which have different behaviours. The first non-linearity comes from the contact condition and the second one from the breaking of the material. The described system can now be excited with different methods. 3. EXCITATION OF THE MODEL The excitation of the model influences the systems behaviour. Since it contains two non-linearities, the systems response is not linear and can drift into chaos. To investigate the systems behaviour, two different excitations are being tested. These are: • Normal excitation and • Waypoint excitation
To be able to compare the different simulation results, it is necessary to introduce an effort coefficient for each parameter setting. The effort coefficient η is calculated
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from a representative force F divided by the maximum force Fb,max which is necessary to break the material. η=
Fx Fb,max
(8)
With this definition, an η of 1 indicates full effort and an η of 0 indicates no effort. The effort coefficient η can be calculated from several properties Fx of the steady-state reaction forces of the simulation. The most interesting aspects considering the effectiveness of the process are: • • • •
Maximum drill bit force: max(Fm ) → Fx , Mean drill bit force: F m → Fx Minimum drill bit force: min(Fm ) → Fx , and Mean driving force F a → Fx
To get the corresponding effort coefficient ηx the property Fx has to be divided by Fb,max . 3.1 Normal excitation
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The dynamics of this normal excitation is dominated by the cutting speed divided by the chip thickness xi of the material (compare fig. 3). The resulting frequency is the so called natural frequency fnat : vc fnat = (9) xi This natural frequency is typically much higher than the resonance frequency of the system because of small chip thicknesses and is thereby damped very well by the systems frequency response. In the fig. 3 simulation however, to do the plausibility test of the model, the chip thickness chosen has to be much bigger. The resulting natural frequency fnat is 2.5Hz. Note that the effort coefficient ηFa is only 0.45. The coefficient is low, because there is not much material to cut (xi = 0.5m). If the chip thickness is decreased to a more practical observable value, the systems behaviour changes. In the next example fig. 4 we can observe a higher natural frequency (fnat = 40Hz) as well as an increased controller effort ηFa = 0.93.
The normal excitation is the simple milling operation. The chassis is moved by the driving force Fa of the controller towards the material and so does the drill bit. When the drill bit makes contact with the material, the ground force Fb occurs. This force is transferred to the drill bit and therefore to the chassis. The effort Fa from the controller has to increase in order to keep the chassis speed vc at the set point. When the ground force reaches the breaking force Fb,max the material breaks. This behaviour can be shown in the simulation data in fig. 3.
Fig. 4. Simulation of normal cutting operation with small chip thickness The above simulation also shows that the drill bit position xm (blue line) is always larger than the material position xb (red line). This means that there is contact at all time. There is no lift off of the material. As long as this state occurs, the effort η for cutting will always be close to 1. The plot in figure 5 shows the effort coefficient η for the normal excitation mode in function of the cutting speed vc .
Fig. 3. Simulation of normal cutting operation with big chip thickness to check the models behaviour 593
It can be seen that if vc is lower than 0.5m/s, no continuous cutting is occurring. Whereas above 0.5m/s, the effort η is close to 1. This can also be seen from the differential equations (1) and (2).
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Fig. 5. Effort coefficient η in function of the cutting speed vc 3.2 Waypoint excitation The waypoint excitation is defined as an additional, vibrational force on the drill bit that generates additional contact pressure. To simulate this scenario, eq. (2) has to be modified as follows: ¨m = Fm − Fb + (Aw · sin(2 · π · fw · t)) mm · x
(10)
Where Aw and fw denote the amplitude, resp. the frequency of the superimposed force. To analyse the behaviour of the waypoint excited system, the frequency fw is varied while the cutting speed vc and the amplitude Aw of the Excitation are kept constant. The result is the following plot (fig. 6) that shows a significant drop of the effort in the region of 10Hz excitation frequency.
Fig. 7. Simulation result for the waypoint excitation with fw = 10Hz The effort coefficient η drops to 0.48. This mean that only half the force is needed to maintain the cutting speed vc of 2m/s. This is where the energy saving can take place. 3.3 Conclusion of the simulation model for different excitations The simulation model behaves as expected. As soon as the drill bit makes contact with the material, the ground force rises and is transferred to the chassis. To maintain the set point of the cutting speed, the controller has to produce more effort. This effort can be used to measure the effectiveness of the current parameter setting. The normal excitation (classical milling process) shows no increase of effectiveness with higher cutting speeds. But the behaviour of the system is as expected.
Fig. 6. Effort coefficient η in function of the frequency fw with waypoint excitation In order for this to work, the stiffness between the chassis and the drill bit has to be small to allow the additional movement the drill bit is making. The next plot (fig. 7) shows the named situation at 10Hz. It can be seen that the drill bit is oscillating which results in a periodically lift off of the drill bit. When the drill bit makes contact again, several cuts occur shortly behind each other. In this phase the kinematic energy of the drill bit is used to cut the material. 594
The waypoint excitation shows a great increase of effectiveness between an excitation frequency fw of 5..25Hz. For this to work the drill bit has to be able to move relatively independently from the chassis. A higher frequency is not necessarily better because the effect is damped by the systems damping at higher frequencies. 4. VALIDATION OF THE MODEL The validation of the model is an important step to undertake before advancing to the scaling.. 4.1 Experimental Set-up The experimental setup takes place on a test stand, designed for optimizing milling processes in stone-like materials. A single drill bit is equipped with strain gauges and
2016 IFAC MECHATRONICS September 5-8, 2016. Loughborough University, J. Blum UK et al. / IFAC-PapersOnLine 49-21 (2016) 591–597
an accelerometer. The strain gauges measure the contact force Fm of the drill bit, while the accelerometer measures the accelerations of the drill bit in all three principal axes. Because it is not practical to cut in a straight line, the movement of the drill bit was chosen to be a helix. The drill bit faces constant cutting conditions during its movement on a semicircle, while it does not make any contact with the material on the other side. This setup ensures perfectly reproducible conditions for each cut (see fig. 8).
window is calculated. This step is shown in figure 9.A and B. (2) The resulting trend (9.B) of the mean is continuously monitored for a local maximum. This is shown in 9.C: The derivation of the mean-trend is the argument of the signum-function. The resulting signum-funcion is derived again. If the derived signum-function is negative, a local maximum has been found. (3) The found maximum in the last step is added to the list of maximums. (4) The representative value (red circle in fig. 9.D) for the set of cuts is the mean of all found maximums.
Fig. 8. Overview of the experimental setup The tested parameters can be seen in the next table 2. Table 2. Experimental parameters Parameter Material Cutting depth (mm) Cutting speed (m/s) Excitation Waypoint excitation freq. (Hz)
Values C30/37 3 2 none 10
5 2.5 waypoint
7 3
With each parameter setting, twenty cuts were undertaken. The data of the strain gauges show the contact and no-contact phases of the helix cutting movement (compare figure 9.A). 4.2 Classification of the Experiments To classify a set of cuts with constant parameters, it is necessary to reduce the data to a simple, representing value. In this case, a multi-level averaging of the data with local maximum value detection is used. The algorithm works as follows: (1) A rectangle window is moved over the raw data, at every position, the mean value of all data in the 595
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Fig. 9. Signalprocessing of the strain gauge signal
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The rating coefficient converges with the number of found maximums. This can also be shown in figure 9.D. The rating coefficient can be interpreted as the mean reaction force for a specific parameter set and is therefore a good representation of a set of cuts. 4.3 Experimental Results With the classification algorithm from section 4.2 the overall result can be seen in figure 10. Fig. 11. Overview of the milling situation of a single disk in this case are simple if the movement is not disturbed by the cutting forces. For simplicity reasons the cutting forces are neglected to calculate the trajectories. The trajectory of one drill bit is the superposition of a rotational and a translational movement. This can be calculated for each drill bit with an angular offset. The total cutting depth is the sum of the single cutting depths. δ=
N 2 · r · π · vf sin(Θi ) · M at(Θi ) N · vc i=1
(11)
with Θi = Fig. 10. Experimental results The following statements can be made from the measurements: (1) The mean reaction force of the drill bit (Fm ) increases with increasing cutting depth δ. This is the validation of equation (6). (2) The mean reaction force of the drill bit (Fm ) decreases, if the system is excited. This is the validation of the model behaviour that shows the reduction of the effort coefficient. For these reasons, the simple model for the drill bit material interaction is validated and can be used for further investigations. 5. SCALABILITY OF THE MODEL With the model being validated, the next step can be undertaken: The scaling. A model with only one drill bit is not suitable to describe the overall milling process. The next step consists of taking the geometry of a rotary grinder into account to calculate the variation of the cutting depth δ over time. Figure 11 gives an overview of the milling situation. The single disk model of a rotary grinder introduces four more parameters. These are the number of drill bits in the trace N , the radius r, the feed rate vf of the grinder and the cutting height h. The cutting speed vc is the same as in the last sections. 5.1 Trajectories of the drill bits To describe the cutting depth δ over time, it is necessary to know the trajectories of the drill bits. The trajectories 596
and
vc 2·π ·t+ (i − 1) r N
h 1, if 0 < Θi < acos 1 − M at(Θi ) = r 0, else
(12)
(13)
Equation (11) consists of a constant term that describes the cutting depth in the horizontal plane with the current parameter setting. The sine function is the correction term for the i-th drill bit angle. The M at(Θi )-function is a simple contact condition. It becomes 1 if the i-th drill bit is in the range (angle) of the material. Otherwise there is no contact and the cutting depth δ is 0. B. Balachandran Balachandran (2001) uses a similar model for the description of a metal milling process in his paper. The modelling of this disk has an influence on the cutting depth δ which means that the maximum force Fb,max changes in time, depending on the parameters r, vf , N , vc and h. A typical course of this function (eq. (11)) can be seen in the plot (fig. 12). One can see in the course of the function when two drill bits are cutting simultaneously and when only one drill bit is cutting. The Fourier-transform of this signal looks as in fig. 12 below. This course of δ can now be implemented in the simulation model to investigate the drill disks behaviour. 6. CONCLUSION The simple model from section 2 is sufficient for the modelling of the main effects of the milling process of a single drill bit with constant cutting depth. The excitation of the system in a vibrational way results in a lower driving
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Balachandran, B. and Zhao, M. (2000). A mechanics based model for study of dynamics of milling operations. Meccanica, 35(2), 89–109. Nollau, R. (2009). Modellierung und Simulation technischer Systeme: eine praxisnahe Einf¨ uhrung. Springer Science & Business Media. Sutherland, J.W. and DeVor, R. (1988). A dynamic model of the cutting force system in the end milling process. Diss. Abstr. Int., 49(1), 293. Vlasblom, W. (2007). Cutting of rock. Lecture Book Dredging Processes. Zhao, M. and Balachandran, B. (2001). Dynamics and stability of milling process. International Journal of Solids and Structures, 38(10), 2233–2248.
Fig. 12. Cutting depth of a drill disk with 6 drill bits and its spectrum effort of the system by the same mean cutting speed and cutting depth. The experimental setup is able to confirm the simulation results by measuring the current drill bit force with strain gauges. The model can be scaled to a single rotary disk by adjusting the cutting depth δ with the geometrical information of the system. The reduced driving force for the same work output can enable a new generation of milling machines that have a higher performace to weight ratio or that consume less fuel for the milling process itself. ACKNOWLEDGEMENTS The author thanks the supporting industry partner (confidential) who enabled the possibility for the experiments. Thank you to my supporting advisor: Prof. Dr. Roland Anderegg for guiding me through the project. REFERENCES Altinta¸s, Y. and Budak, E. (1995). Analytical prediction of stability lobes in milling. CIRP Annals-Manufacturing Technology, 44(1), 357–362. Araujo, A.C., Pacheco, P., and Savi, M.A. (2009). Dynamical analysis of an end milling process. In 20th International Congress of Mechanical Engineering. Araujo, A.C., Savi, M.A., and Pacheco, P. (2010). Experimental and numerical dynamical analysis of an end milling process. In VI Congresso Nacional de engenharia Mecˆ anica, Campina Grande, PB, Brazil. Balachandran, B. (2001). Nonlinear dynamics of milling processes. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 359(1781), 793–819. 597