Simulation of a modified rotary timber machiningprocess to improve surface form

N. Brown *, R.M. Parkin, M.R. Jackson

Mechatronics Research Group, Systems Engineering Research Centre, Department of Mechanical

Engineering, Faculty of Engineering, Loughborough University, Loughborough,

Leicestershire LE11 3TU, UK

Received 15 May 2000; accepted 19 October 2000

Abstract

Rotary machining is extensively used for planing and moulding operations within the

woodworking industry. Although the surface form produced by this machining method is

acceptable, the rotary machining action produces cutter marks on the wood surface so that

further finishing operations, such as sanding, are often required to generate a product of

acceptable standard. It has been theorised that the surface finish of planed and moulded

timber products may be improved by oscillation of the cutter block in either a vertical or

horizontal plane. This paper describes the use of a graphical simulation based on Simulink�

software to predict surface finish, and the use of computer simulation to model cutter-block

oscillation. The result is a complete tool for effective design and optimisation of a hydraulic

oscillation system, in combination with a surface form generator in order to improve surface

form. � 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Mechatronic; Simulation; Woodworking; Simulink

1. Introduction

The rotary wood machining process is similar in nature to the upcut milling ofmetals. There are, however, marked differences between the two applications, theprimary one being cutting speed, which lies in the range 30–80 m s�1, compared withtypically 0.5–1.5 m s�1 for the milling of metals [1]. The feed speed in wood ma-chining is correspondingly high, ranging from 0.08–1.6 m s�1. Fig. 1 shows the

Mechatronics 12 (2002) 489–502

*Corresponding author. Tel.: +44-1509-263-171; fax: +44-1509-223-934.

E-mail address: n.brown@lboro.ac.uk (N. Brown).

0957-4158/02/$ - see front matter � 2002 Elsevier Science Ltd. All rights reserved.

PII: S0957-4158 (00 )00069-6

general layout of a two-head (top and bottom) planing and moulding machine.Timber lengths are initially fed into the machine by hand. Feed is then taken up bypowered rollers, which may be tyred with a soft material in order not to damage theproduct. Machining is carried out by knives mounted on cutter blocks. Cutter blocksmay be removed from machine spindles for regrinding and balancing.

NomenclatureA single acting ram area, m2

B general bulk modulus, N m2

fd feedrate, m s�1

h height of cutter wave, mkLE leakage flow, m3 s�1

M ram and slideway mass, kgl frictionN number of cutter-head knives producing surface waven cutter-head rotational speed, rev min�1

P knife marking pitch, mp supply pressure, N m�2

q fluid flow, m3 s�1

rct cutter-tip radius, mV trapped oil volume, m3

x ram displacement, m

Fig. 1. Two-head planing/moulding machine arrangement.

490 N. Brown et al. / Mechatronics 12 (2002) 489–502

Planed and moulded surfaces appear, when viewed closely, as a series of waveswhose peaks are perpendicular to the passage of the product through the machine[2]. Machining errors may exist which affect surface finish, these being surfacewaviness and error of form [1]. Waviness is the component of texture upon which theundulating knife traces (roughness) are superimposed, and can result from cutter-head imbalance, the flexure of the machine frame, and machine vibrations.

Cutter-head imbalance is the principal cause of surface waviness in planed andmoulded timber components [2]. Cutter-wave height and pitch are given by Eqs. (1)and (2), assuming a common cutting circle.

Height ðhÞ ¼ rct � ðr2ct � P 2=4Þ0:5; ð1Þ

where

Pitch ðP Þ ¼ fd � 103=nN2p: ð2ÞFixed-knife planing has been shown to produce an excellent surface finish wherebynone of the defects produced by rotary machining is present. The high feed forcesrequired to draw the timber across a stationary knife, and feed methods used (suchas toothed wheels), however, may mark the product, and the length of fixed-knifemachines may prohibit their use [3]. Only a small amount of material may be re-moved using a fixed knife, around 0.2 mm; hence, fixed-knife machining is notsuitable for moulding.

2. Novel cutter-block motion

It was intended that, through simulation and eventual testing, the effectiveness ofa modified form of cutter-block motion could be verified. Cutting-force control hasbeen used to improve waviness in metal machining [4], but a new approach wasproposed to affect surface form to a greater degree. The idea behind cutter-blockoscillation is that, when each knife contacts the wood surface, the cutter is advancedin a horizontal plane in order to produce a trough-like cutter mark instead of theconventional scallop-shaped cutter mark [5]. This is followed by a retraction of thecutter head prior to the next knife contacting the workpiece. This results in a re-duction in cutter-wave height to the extent that the finished surface bears no visiblecutter marks to the naked eye. Figs. 2 and 3 show how a cutter block with twodegrees of freedom may reduce cutter-wave height. The cutter advances producing aclean cut, cutter-motion detail being shown in Fig. 2. Displacements have been ex-aggerated for clarity, and only one (of four) cutter knives on each cutter block isshown, also for clarity. The numbered cutter blocks refer to the same cutter head,shown in different parts of the oscillation cycle. The knife contacts the surface at thebeginning of the cycle, as shown by the cutter numbered 1. As the cycle progresses,the knife is advanced across the timber surface, as shown by cutters 2, 3 and 4. Theknife moves further away from maximum depth of cut by position 5. The cutter isthen retracted. Referring to Fig. 3, where displacements have been exaggeratedfurther to show this part of the cycle more clearly, the cutter begins the retraction

N. Brown et al. / Mechatronics 12 (2002) 489–502 491

cycle with a knife cut having just been completed (position 6). Retraction is rapid,whilst the cutter block continues to rotate, as shown by cutter-block positions 7, 8and 9. The beginning of the next actuation cycle begins with the following cutterknife beginning to contact the timber surface, as shown in position 10.

Achieving a perfectly flat surface would only be possible if cutter-block rotationcould be halted, and inertia due to cutter-block mass in the range 20–80 kg at ro-tational speeds in the range 62–1800 rad s�1 would make this impracticable, thoughit was envisaged that a considerable reduction in cutter-wave height could beachieved through cutter-block oscillation.

2.1. Actuator performance requirements

The stroke of the actuator system is determined by the cutter-wave pitch. In or-der to remove or reduce the effects of cutter marks, the actuator stroke must be equalto or greater than surface-wave pitch, as shown in Eq. (2). For average-quality

Fig. 3. Cutter retract detail.

Fig. 2. Cutter advance detail.

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material, cutter-wave pitch figures in excess of 2.5 mm are commonplace, though forhigh-quality products, wave pitch should be 1.0 mm or less [1].

It can be seen through application of Eqs. (1) and (2) that, when assuming atypical knife tip radius of 70 mm, wave pitches of 2.5 and 1.0 mm will possess heightsof 3.6 and 1.8 lm, respectively. High frequencies of actuation are required; for ex-ample a four-knife cutter rotating at 6000 rev min�1, or 628 rad s�1 requires fouroscillations of the cutter head per revolution, or an actuator capable of operating at400 Hz.

2.1.1. Actuator for cutter-block oscillationHydraulics was seen as the best choice of horizontal actuator. Pneumatic actua-

tors would not offer significant actuation forces and a purely mechanical systemwould not allow as much experimentation with differing actuation waveforms [3]. Inthe hydraulic system, hydraulic cylinders would be controlled by valves capable ofoperating at high frequencies, such as servo valves [6]. Therefore, a hydraulic systemfor the proposed prototype running speed of 314 rad s�1, using a two-knife cutterblock, would require good response up to at least 250 Hz. While the majority ofcutter blocks in industry use four or more knives, a single-knife cutter was chosen forinitial design in order to simplify the design process; nonetheless, the principle re-mains the same.

Where high operating frequencies are required, the trade-off between flow rateand speed of response has placed large constraints on the hydraulic servo designer. A500 Hz hydraulic actuator has been developed [7], though the maximum amplitudeachieved is only 0.2 mm, which is stated to limit system performance. Allan [8] de-scribed a system with a greater amplitude of �12.5 mm, though output frequency islimited to 20 Hz. Although high actuation forces are achieved, this is achieved byusing multiple servo valves in parallel. Such a solution should be avoided for cutter-block oscillation if possible, in order to keep manufacturing costs down.

3. Surface form simulation

3.1. System simulation

Simulink�-based cutter-block oscillation and hydraulic servo models were origi-nally developed as stand-alone blocks, such that individual models were kept simplefor early simulations. Once these blocks had been tested, there remained the simu-lation of the overall system, which consists of five subsystem simulations connectedin parallel. The amount of cutter-block oscillation used was varied throughout themodel, to produce a block of five results in each case. Frequency of oscillation signalwith respect to cutter-block rotary speed may also be varied. The top-level model ofthe overall system simulation is shown in Fig. 4. The design of the simulation hadto be left open to various actuator waveforms in order to study the effects ofcutter-block oscillation; a different approach was called for, therefore, to using afixed mathematical model as proposed by Mezzelkote and Thangaraj [9]. Other

N. Brown et al. / Mechatronics 12 (2002) 489–502 493

simulations, notably those of Ismail et al. [10] and Lin and Liu [11], which are themost applicable to this research, produced simulations whereby factors such assurface energy and tool wear could be considered, which would be potentially usefulfor timber machining. Surface form was the paramount requirement, though, and alocus-based surface-form generator as proposed by Maycock [2] was deemed anappropriate method.

3.1.1. SubsystemsThe hydraulic subsystem is shown in Fig. 5. The kernel is a state-space imple-

mentation of a transfer-function-based model which is described in full in AppendixA. The servo-valve model is derived from experimental data produced by Moog [12].

In order to allow for drift within the hydraulic system and save on setup timewhen altering actuation waveforms, a phase-locked loop has been devised (Fig. 8)which alters the control-signal lag in order to compensate for any phase discrep-ancies between the control signal and the hydraulic servo output. The phase detectoris based on two zero-crossing detectors sensing the phase of input and output signals.Following these functions are flip-flops which trigger only on the positive-going cyclefor each signal. The switching arrangement which then follows captures time fromthe simulations internal clock; crossing-point times are subtracted to find the phaselag. This is then ported to a variable transport delay back in the next level up (thehydraulic subsystem) in order to provide phase compensation. The cutter-block

Fig. 4. Parallel implementation of simulation of actuation system and timber surface.

494 N. Brown et al. / Mechatronics 12 (2002) 489–502

vertical locus is basically a cosine function followed by a cut-off which restricts thevertical part of the cutter-block waveform roughly to the region of interest, i.e., thetimber surface. Fig. 6(a) shows the subsystem for generating 5� of cutter-block os-cillation phase lag. This subsystem was used in determining the effect of cutter-blockoscillation phase lag upon the timber surface. The base delay is settable from input 1,which then rises in increments set on input 2. The output is multiplexed for furtherprocessing. Input 3 carries the cutter-block oscillation waveform from the hydraulic

Fig. 6. (a) Subsystem for generating differing degrees of cutter-block oscillation phase lag. (b) Subsystem

for generating varying amounts of cutter-block oscillation amplitude.

Fig. 5. Hydraulic subsystem.

N. Brown et al. / Mechatronics 12 (2002) 489–502 495

servo model. Fig. 6(b) shows the subsystem for generating 5� of cutter-block oscil-lation amplitude. This subsystem was used in determining the effect of differingamounts of cutter-block oscillation upon the timber surface. The base amplitude issettable from input 1, which then rises in increments set on input 2. The output ismultiplexed for further processing. Input 3 carries the cutter-block oscillationwaveform from the hydraulic servo model.

3.1.2. Secondary componentsReferring to Fig. 7, a number of simulation blocks exist which were developed in

order to improve the efficiency of the simulation. The five-way two-pole relay sub-system enables switching between multiplexed lines. The problem of switchingmultiplexed lines can often crop up during modelling within environments likeSimulink�. The solution is often untidy and makes for clutter in the model, makingdebugging difficult. The solution proposed here neatly encapsulates a multiwayswitching system in a single block, comprising five two-way switches to control dataflow of the multiplexed lines, embedded in a subfunction.

After adding feedrate to the multiplexed cutter-block oscillation lines, the resultsare ported back to the Matlab� environment for examination. To this end, ademultiplexer is used to provide separate arrays, offering simplified data handlingthrough providing five separate 10 000-element arrays, whereas a multiplexed outputwould produce a single 5� 10000 contiguous block of data. Such a large 2-D arraywould be far more likely to cause runtime operating system errors than individual1-D arrays.

The simulation results are based upon the number of samples taken, sincea variable-time-step algorithm within the simulation code means that, were the

Fig. 7. Phase detection and lag determination for PLL.

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simulation results plotted using a system based on time, arrays of variable lengthwould be produced. A sample counter still allows a variable time step to be used,enhancing simulation performance over a fixed-time-step simulation. The simulationis made to stop when the limit of 10 000 samples has been reached.

4. Results

Fig. 8 shows the cutter loci and resultant timber surface generated using theconventional cutting method (indicated with cross points), and a modified methodusing a cutter-block oscillation amplitude equivalent to 25% of the timber feed percutter revolution, which is indicated with a solid line.

Fig. 9 shows the effect of two different amplitudes of square-wave cutter-blockoscillation, along with a conventional surface wave (indicated by ‘+’) for compari-son. As can be seen from the image, a cutter block oscillation of 25% of the feed perrevolution brings about a small reduction in cutter-wave height. A cutter-oscillationamplitude of 100% of the feed per revolution (i.e., 2 mm for a 2-mm-per-revolutionfeedrate) produces a flat surface, since the loci overlap.

Fig. 10 shows the effect of introducing the hydraulic servo simulation into theoverall machining model. As can be seen, small variances in phase response for thehydraulic servo produce a marked effect on the periodicity of the cutter-tip locus

Fig. 8. Ideal surface produced with sine-wave oscillation.

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(indicated with a solid line), but cutter-wave height reduction is still very large. Onceagain, the conventional cutter path is illustrated with a ‘+’ for reference.

The sine-wave-oscillated hydraulic servo system model produces a cutter-waveshape as shown in Fig. 11. The effects of square-wave harmonics no longer trigger offhigh-order resonances within the hydraulic servo, and phase response is cleaner. Thesolid line denotes the modified cutter locus; in this case, oscillation amplitude is equalto 100% of the feed per revolution. Cutter-wave height reduction is better than 85%.

5. Conclusions

Simulation of the new machining system strongly suggests that the use of themodified machining process produces a flatter timber surface whilst using the sametooling as a conventional machining process. A 180 mm diameter cutter block,when oscillated with a sine-wave forcing function, would produce cutter marksmuch smaller than those produced by conventional tooling, and would be of thesame size as those made by a cutter block of over 600 mm diameter. The net effect,therefore, when using sine-wave oscillation, is to utilise the benefits of using a verylarge cutter block to produce a reduced wave height, whilst actually using astandard cutter block, with a massively lower balance requirement. Simulatingthe system allows the design of a hydraulically oscillated cutter block where

Fig. 9. Ideal surface produced with square-wave oscillation.

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construction of a mechanical prototype is unfeasible at the design stage due toeconomic and safety requirements, and makes evaluation of the performance ofnovel cutter-block motion possible. The ‘virtual cutter block’ thus produced pro-duces a flatter surface with no exaggerated imbalance problems. The use of square-wave oscillation produces a flatter surface still, whereby it is foreseen that cuttermarks may be almost invisible to the naked eye. This could remove the require-ment for sanding, in which case a whole machining process would be eliminated intimber component production.

Appendix A. Ram and slideway transfer function derivation

In order to derive the ram–slideway transfer function, a single acting hydraulicram is considered. The piston cross-sectional area is taken as A, the input fluid flowto the ram is given by qðtÞ, pðtÞ is the fluid pressure and the ram position is given byxðtÞ.

In order to provide a comprehensive model, it is necessary to consider the massmoved, (i.e., ram, slideway and cutter block), any leakage which may exist betweenram and cylinder wall, effects of viscous friction, and the compressibility of the fluid,i.e., the fluid bulk modulus.

Fig. 10. Surface wave produced with square-wave oscillation and hydraulics.

N. Brown et al. / Mechatronics 12 (2002) 489–502 499

The governing equation for a simple ram is given by

qðtÞ ¼ AdxðtÞdt

: ðA:1Þ

Taking into account leakage, where a leakage flow of kLEpðtÞ exists, the right-handside of the flow continuity equation then becomes

AdxðtÞdt

þ kLEpðtÞ: ðA:2Þ

Calculation of pðtÞ due to inertia is achievable through applying Newton’s secondlaw, i.e., force ¼ mass� acceleration.

pðtÞA ¼ Md2xðtÞdt2

: ðA:3Þ

The ram will be subject to a viscous load, whereby the coefficient of viscous friction laffects the force available from the ram. Static friction is not included in the modelsince simulations show that its effect is negligible with respect to cutter-block systeminertia. The pressure pfðtÞ required to overcome this frictional loss is, therefore,

pfðtÞ ¼ ldxðtÞdt

: ðA:4Þ

It is then necessary to consider the effects of fluid compressibility. Bulk modulus maybe defined as change in fluid pressure divided by change in volume per unit volume,i.e.,

Fig. 11. Surface wave produced with sine-wave oscillation and hydraulics.

500 N. Brown et al. / Mechatronics 12 (2002) 489–502

B ¼ DpDv

: ðA:5Þ

Therefore,

dvdt

¼ VBdpdt

: ðA:6Þ

The complete flow continuity equation is, therefore,

qin ¼ qvel þ qleakage þ qcompressibility: ðA:7Þ

Therefore,

qðtÞ ¼ AdxðtÞdt

þ kLEpðtÞ þVBdpðtÞdt

� �: ðA:8Þ

Viscous friction and inertia are used to calculate the pressure required to move theram:

pðtÞA� ldxðtÞdt

¼ Md2xðtÞdt2

: ðA:9Þ

Therefore,

qðtÞ ¼ AdxðtÞdt

þ kLEMAd2xðtÞdt2

�þ lAdxðtÞdt

�

þ VB

MAd3xðtÞdt3

�þ lAd2xðtÞdt2

�: ðA:10Þ

Therefore,

qðtÞ ¼ VMBA

d3xðtÞdt3

þ kLEMA

�þ lV

BA

�d2xðtÞdt2

þ kLElA

�þ A

�dxðtÞdt

: ðA:11Þ

Taking Laplace transforms for zero initial conditions gives the ram–slidewaytransfer function:

X ðsÞQðsÞ ¼

1

ðMV =BAÞs3 þ ðkLEM=Aþ lV =BAÞs2 þ ðkLEl=Aþ AÞs : ðA:12Þ

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