The design, development and testing of a prototype boring dynamometer

Int. J. Mach. Tool Des. Res. Vol. 2, pp. 241-266. Pergamon Press 1962. Printed in Great Britain

THE DESIGN, DEVELOPMENT AND TESTING OF A PROTOTYPE BORING DYNAMOMETER

R. A. HALLAM and R. S. ALLSOPP*

I N T R O D U C T I O N

THE DYNAMOMETER described in this paper was designed to fulfil a requirement for an instrument capable of measuring the three components of the cutting force when boring under steady state conditions.

A research programme concerned with the behaviour of cantilever type boring bars was in progress, and it was found that information was needed to relate the effects of depth of cut, feed rate, cutting speed and tool nose radius to each of the cutting force components.

FIG. 1. Photograph of dynamometer.

Searches through published information had yielded little data relevant to the range of cutting conditions and type of tool used in the programme, whilst existing three co-ordinate dynamometers did not appear to be suitable for force measurement whilst boring.

The dynamometer illustrated in Fig. 1 was, therefore, constructed to enable the three force components to be measured when machining bores having diameters from 2.5 in.

* Production Engineering Research Association, Melton Mowbray, Leicestershire. 241

242 R.A. HALLAM and R. S. ,A~LLSOPP

upwards. The instrument was found to be reliable and accurate in performance, and it was rigid enough to prevent the occurrence of unstable chatter vibrations.

A typical selection of the cutting force results obtained when machining cast iron are included in the paper in graphical form. From these results, empirical formulae have been derived using standard law fitting techniques. From these formulae, nomograms have been constructed which indicate the approximate values of cutting force expected for any given cutting condition within the investigated range.

As an Appendix to the paper, an analysis is included which was based on the simple assumption that cutting force is proportional to the cross-sectional area of the uncut chip. This analysis was carried out at an early stage of the investigation in an attempt to predict the type of relationship likely to exist between the cutting forces and the cutting parameters for the particular tool shape used.

DESIGN OF DYNAMOMETER

The basic requirements of a dynamometer for measuring cutting forces over the range of light to medium depths of cut are that it should be sensitive to small changes in cutting force, yet rigid enough to prevent the occurrence of chatter vibration. Adequate rigidity will also ensure that the cutting forces do not deflect the cutting tool enough to significantly alter the cutting angles. There should be little interaction between the three component measuring systems, and the dynamometer should be free from hysteresis to prevent phase shift in readings during uneven cuts. It is preferable that the instrument should be linear and also generally robust for normal workshop usage.

Electrical resistance strain gauges were chosen as transducer elements, as previous experience had shown them to be suitable for the measurement of both static and dynamic loads and deflections. When correctly affixed, they will provide good linearity and negligible hysteresis.

The dynamometer consists essentially of a cranked beam mounted on a rigid shank which is then secured to the toolpost of a lathe. Although fundamentally an instrument for measuring the three components of force when boring, it may also be used for measuring the cutting forces when turning by inverting the dynamometer and cutting tool. The three principal force components are designated as the Tangential force, the Radial force and the Feed force.

The dynamometer was designed to measure a maximum component of force of up to 300 lbf in each of the three directions; the maximum stress in the strain-gauged sections being well within the elastic limit for the steel used in the construction of the dynamometer.

Figure 2 is a drawing of the dynamometer positioned for boring. The design is such that the tool tip may be set on the centre of the co-ordinate axes with the aid of a simple tool-setting bar.

The cranked beam is a hollow cylinder which has a substantial wall thickness at the base to ensure adequate rigidity in bending and thus prevent the occurrence of chatter vibrations under normal cutting conditions. Immediately behind the toolpost, which is welded to the cranked beam, the wall thickness is reduced to increase the strain sensitivity for feed force measurement. However, this reduced portion is sufficiently far removed from the base so as not to materially decrease the bending rigidity which could lead to chatter vibration.

The Design, Development and Testing of a Prototype Boring Dynamometer 243

Force measuring elements

The measuring elements of the dynamometer consist of twelve 600 .(2 strain gauges: two mounted at T1 and T2 for measuring the Tangential force component; two mounted at R1 and Rz for measuring the Radial force component; and eight mounted at F1 to F8 for measuring the Feed force component. The arrangement of the strain gauges is designed to minimize incorrect readings caused by spurious couples, and also to provide a measure of temperature correction.

- ' Z L ! - '

/ F ' ~ T VIEW B /

l ~ m ~,o~ tot T,e4 ¢km'ekq Germ ~ Vll lq Dyncmlx~llcr for Turnin G TeSts

5

L./

~,~ PLA~ Toot ~ I . i~ d td F . . . .

SECTION S-B

Ra41~ Cvttlq F'orc~

, ~ * f W ~ r .oqa Nr I ~ m w ~ l Fs, F Fled Culllq Farce I

TIVE 0

TANGENTIAL AND N ~ I ~ GAUGES

CIRCUIT OIAGRAMS F(~I STRAIN GAUGES ( ~ 1 )

FIG. 2. Three co-ordinate boring dynamometer.

With the tool in the position shown, the Tangential force exerts a bending moment giving rise to equal and opposite tensile and compressive stresses on gauges T1 and 7"2 respectively. Since T1 and 7"2 are both active gauges in two separate arms of a common bridge circuit (see wiring diagram in Fig. 2), the net effect of the Feed force on these gauges is zero. The Radial force has no effect on gauges T1 and T2 since they lie on the neutral axis.

Similar remarks apply to the Radial force measuring gauges R1 and Rz. The Feed force measuring gauges F1 to #'8 are positioned at the reduced portion shown

in section A-A, with a more complicated pattern of arrangement designed to produce an approximate self compensating measuring element for dealing with the interference of the

244 R .A . HALLAM and R. S. ALLSOPP

other two forces. The interference is caused by small bending moments induced by the Tangential and Radial forces, and in consequence the gauges have been positioned so that these bending moments tend to be self-cancelling. Referring to section A-A and the wiring diagram, it can be seen that the active gauges are positioned in one arm of the bridge net- work, and the temperature compensating gauges are appropriately positioned in the opposing arm of the bridge. The electrical resistance of the four 600 ~2 gauges, in the ;eries-parallel arrangement as shown, is equivalent to the resistance of one 600 ~2 gauge. However, by connecting F1 in series with F4, any positive increment in F1 due to the Tan- gential force will be compensated by an equal decrease of resistance in F4. Similar remarks apply to F2 and Fa. This arrangement compensates for the effect of the Radial force in a similar manner and also provides a measure of temperature compensation within the active arm of the bridge. During calibration of the dynamometer, however, it was found that the bending moment due to the Tangential force was causing a small residual un- balance of the Feed strain readings during cutting tests.

The temperature compensating gauges, Fs, F6, /'7 and Fs, which have their elements

DYNAMOMETER TERMINAL BOX

J PEN J RECORDER I

STRAIN I MEASURING BRI DGE

I FIG. 3. Block diagram of recording instruments.

positioned circumferentially, i.e. at right angles to the axis of the dynamometer, are virtually free from principal bending strains. Although the wiring of the gauges was designed for self-cancelling temperature compensation, cutting tests revealed that the arrangement was not completely effective in compensating for heat generated at the tool during cutting.

The three strain gauge systems were connected to commercial carrier wave bridges and the outputs fed to d.c. amplifiers and pen recorders.

Figure 3 shows a block diagram of this arrangement as used for both calibration and cutting tests.

Calibration of dynamometer

The dynamometer was calibrated using direct loading with dead weights in each of the three co-ordinate directions. In order to ensure that the points of application of the calibrat- ing forces were the same as those of the cutting forces, a dummy tool shank was used. This was clamped in the tool holder and the load applied to the dummy shank so that the lines of action and points of application were the same as those OCCUlTing when cutting.

The results of the main calibration tests are shown in Figs. 4 and 5 expressed as the


strain recorded* on the carrier wave bridge plotted against the applied load for each direction of force. It can be seen from these graphs that the linearity and sensitivity of each channel is good, with little hysteresis present.

During calibration the deflection produced at the tool tip by a load of 300 lbf applied to the dummy shank was measured in the three principal planes in turn. From these de-

3S,

30

2S

~C) 20 l = z ~ < - x IS

IO

ONr O

// I t

i i ] 'IAL

SO 100 I SO 200 FORCE Ib

FIG. 4. Calibration graph for radial and tangential forces.

flections "3" the equivalent stiffness values "K " have been evaluated as follows:

250 275

Deflection 6 Stiffness Component (in.) (K lbf/in.)

Tangential 0-0041 0'731 × 105 Radial 0.0048 0"625 × 105 Feed 0.0030 1'0 x 105

It has been calculated that the change in angular orientation of the cutting edge will be less than 0.1 o for a 300 lbf load in any direction; this is small compared with the accuracy of the cutting tool angles which were held within -q-0.25 °.

Checks for interaction between the three measuring systems showed this was negligible with the exception of the Feed strain produced by Tangential loading. With the dynamometer positioned for boring, it was found that the Feed strain produced by the Tangential force adds to that produced by the Feed force; whilst in the turning position, the effect of

* Bridge readings not corrected to absolute values of strain.

246 R.A. HALLAM and R. S. ALLSOPP

3 . 0

2.S

2.0

I_o x

z I'S

O~ I-- ~n

1.0

O.S

0

i ,~.4

' 'rrr" I

S j S S e

1 SJ : ~ l s i

I s S S I = i

S o i s S

S

s s S 4 1 1

!

0 50 I00 ISO FORCE Ib

FIG. 5. Calibration graph for feed force.

200

!

0 x

F- i n

0 uJ uJ u .

0-1

0"05

0 0 SO

I FeedStrclinProduced by. [ J Tangential Loading- Boring / J Position. J~. j To be Subtracted from J Recorded Feed S t r a i n ~

by Tangential Loading-J Turning Position. J To be Addcd to / Recorded Feed StrainJ

I00 150 TANGENTIAL FORCE, Ib

I I

200 250

FiG. 6. Correction graph for interaction effects.


the Tangential force on the Feed strain is in the reverse direction. The correction graph for the interaction effect of the Tangential force for the two cases is shown in Fig. 6.

Vibration characteristics

The vibration response characteristics of the dynamometer were obtained by applying a constant exciting force over the frequency range 20-1400 c/s. Figure 7 shows the response curves (vibration amplitude versus frequency) determined for the three co-ordinate directions in turn. The dynamometer was rigidly bolted to a reinforced test bed and a vibration

0.0008

0 " 0 0 0 6

0 - 0 0 0 4

0 0 0 0 2

0 0

0 "0010

O<300e

0 .0006

0 .0004

0 .0002

0 0

TANGkNTIAL 'DIRECTIbN

II

, ,

20O 4OO 6OO 8 0 0 FREQUENCY, cls

c

z lal

U

i o I

1 9 0 0 1,200 1,400

nAOl~L O~RECTION

I i a I 200 4 0 0 6 0 0 8 0 0 kO00 1,200 1,400

FREQUENCY, c/s

° o 2o0 ,~o ~ Boo ~ooo ~oo i~oo FREQIJENCY, cls

FIG. 7. Response curves for the co-ordinate directions of the dynamometer•

force applied to the dummy tool shank. This force was held constant at 1.2 lbf vector and the resulting response of the dynamometer measured with an electrodynamic pick-up in contact with the dynamometer toolpost.

The curves show that the principal resonant frequency in each of the co-ordinate directions is above 800 c/s. This frequency was considered high enough to prevent any interference to steady state force measurement due to the dynamometer vibrating.

P E R F O R M A N C E OF D Y N A M O M E T E R

No evidence of chatter was observed under any of the wide range of conditions used in the tests, indicating that the rigidity of the dynamometer was adequate.

248 R . A . HALLAM and R. S. ALLSOPP

Initially, force readings were taken from all three channels simultaneously by means of three complete channels of recording equipment. It was, however, found more convenient to determine each co-ordinate force in turn using only one carrier wave bridge and one pen recorder for all tests. The cutting tools were kept in freshly ground condition to ensure that cutting conditions were essentially constant for repeat tests.

Irk

r

r^--k;-otion of Force Fluctuation due Formation and Spurious Electricel

Freshly_ Ground Tool

-Noise Level Caused by Spurious Electrical Effects with Amplifiers

-7- - - - r . ~ - Operating on Extren~ High Gain Position

o) TYPICAL FEED FORCE TRACE

Mea

c . . . . Ch"*"~*ion due to Chip

_ly Ground Tool.

b) TYPICAL TANGENTIAL FORCE TRACE

_ reshly Ground Tool

~..~-~F-~C--Force Fluctuotion due to Chip

//jj7 i l l ~ l / l ~ ~ "No Spurious Noix LGvel. ~Cut Engaged

¢~ TYPICAL P, AD~AL FORCE TRACE

FIG. 8. Typical pen record traces of force measurement---cast iron workpiece.

Before each test, a short trial cut was taken to enable the amplifier to be adjusted to give just less than full scale pen deflection for the maximum force. The pen recorder was then calibrated by "off-setting" the strain gauge bridge by known amounts corresponding to known forces, and then taking short trace records to obtain a complete calibration.

Force fluctuations

Cast iron. During cutting tests, irregular fluctuations of the force traces were observed as shown in Fig. 8. These fluctuations were found to increase in intensity as the tool showed


signs of wear. They were apparently due to the nature of the chip formation which occurs when machining cast iron, although visual inspection of the machine surface showed no signs of chatter marking.

A high amplification had to be used in the case of the Feed force measuring elements, which resulted in some spurious noise content from the instruments being present even when not cutting.

Steel. The effect of force fluctuation due to chip formation when machining steel is clearly demonstrated in a typical Radial force trace shown in Fig. 9. With an increase in

• IP ,,,,,I-I-I sec Chip Forming Chip Forming Chip being Chip being

Continuous Scroll Continuous Scroll Broken Broken

+ f - * - ),

No Spu~

Fro. 9. Pen record trace of radial force when machining steel.

Force Fluctuation due to Chip Formation

/

ious Noise Level

depth of cut from 0.08 in. to 0.100in. , the chip formation changed from a continuous curling chip to a series of short broken chips. By observing the change in chip formation when cutting, and at the same time watching the recorded trace, it was noticed that the force fluctuations were faithfully reproduced on the pen trace.

In all cutting tests where the form of the recorded trace indicated that force fluctuations were present, the measured deflection caused by the cutting force was taken as the mean path of the pen trace.

Heating effect When measuring the Tangential and Radial forces no effects due to temperature varia-

tion were observed at any stage during the tests. However, when measuring the Feed force it was found that the zero force level drifted

if a lengthy cut was taken (Fig. 10). This was thought to be caused by the heat generated

" l '#[- - -~ I iec D r i f t due to hea t t r a n s f e r

C u t E n g a g e d /

Fro. 10. Pen record trace of feed force showing drift due to heat transfer--cast iron workpiece.


at the tool tip being conducted to the gauges. This effect was checked by directly heating the tool, when it was observed that the zero level of the trace drifted in a similar manner to that which occurred whilst cutting.

Although the force level on engagement of the cut apparently differs from the value when the cut is taken off, this is due to non-linearity of the pen when nearing the edge of the chart. Calibration ~showed both actual measured force levels to be equal.

. • . , Note: Increase in Force Fluctuation I SeC ~ / d u g to Worn Tool

c ~ , , ' g ¼a t ~ . - ~ - ~ ! i tt...u.:~2 ¢ I! o~,,.g.g,d Jl T " • " ' '

~-~ T i - i - ' • r ~ . . . . .

CUt Engaged Cut Engaged

FEED FORCE TRACE

lue

b) TANGENTIAL FORCE TRACE

Note: Increase in Force Fluctuation clue to Worn Tool

c) RADIAL FORCE TRACE

FIG. 11. Pen record traces of force measurement showing effect of tool wear---cast iron workpiece.

This drift was eliminated by inserting strips of insulating material between the tool and the adjacent parts of the dynamometer, which restricted heat flow to the Feed force gauges. However, for the main tests this procedure was not necessary as the cuts were limited to short duration in time, during which heat transfer was negligible.

Tool wear

Tool wear was found to have a considerable effect on force values as shown in the traces given in Fig. 11. To overcome this, tools were maintained in a freshly ground condition


and used for only a limited number of tests before regrinding. In this way, repeatability was preserved.

These traces for Tangential, Radial and Feed force tests were taken firstly with a slightly worn tool and secondly with a freshly ground tool. In each case the cutting force was greater with the worn tool--this was particularly noticeable in the case of the Radial force. No systematic tests were conducted to investigate the rate of increase of cutting force with tool wear, as this was beyond the scope of the investigation.

C U T T I N G FORCE TESTS

The main tests were carried out on a selection of common grades of cast iron principally to determine the effect on the three cutting forces of variations in the cutting parameters of depth of cut, feed rate, cutting speed and tool nose radius.

FRONT CUTTING EDGE / / l ~ EA ,'CEANGLE-- ///

x / / I PLA, ANtE PLAN TRAIL ANGLE " ~ ' ~ APPRO

SIDE CUTTING EDGE~ CLEARANCE ANGLE/`

FIG. 12. British standard single point cutting tool terminology.

Both boring and turning tests were carried out on four specimens of cast iron, and inspection of individual results obtained with tools of 0.030 in. nose radius showed little scatter and difference in force level for any given set of cutting conditions. The results of both boring and turning tests on all four specimens, have, therefore, been plotted collectively. Different symbols are used to denote force values obtained from individual specimens, and to distinguish between boring and turning tests. Metallurgical analyses of the four cast irons showed them to be representative of common grades used in general engineering.

Standard carbide tipped tools of ~ in. square shank were used throughout the tests and, as noted earlier, were maintained in the freshly ground condition. Both boring and turning tools were ground to the angles shown below to within -t-¼ ° (see Fig. 12). 4

252 R . A . HALLAM an d R. S. ALLSOPP

Maximum rake plan angle = 90 ° Maximum rake angle ~ 5 ° Plan approach angle = 0 ° Plan trail angle ~-- 10 ° Front cutting edge clearance angle ---- 7° Side cutting edge clearance angle = 7° Secondary front cutting edge clearance angle ---- 20 °

(Tool suitably backed-off to clear workpiece for small diameters o f bore)

s° I . 0 " lOOin . DEPTH OF CUT

nc

o ix.+.~ ~ . ~ ~ ! 7'~" ~-"

z

u ~ O " '

s o I s o 2 5 0 350

U ~, CUTTING SPEED, f t / m i n

4 X Q

5 0 --~ 0 . 0 5 0 in. DEPTH OF ~ CUT

U o r

,9 z $ U

,2 : ,, O

I] x

SO ISO 2 5 0 3 5 0

t.r, CUTTING SPEED, f t / m i n

SO - O.OISin. DEPTH oF CUT

u n, o

L9 Z

5 u

o L

i

i

i 25: ~ - - - -

. i l ~ t ~ - - . "="

o i i

5 0 150 2 5 0 3 5 0

Lr, CUTTING SPEED, f t / m i n

FIG. 13. Radial force versus cutting speed (Tool nose radius = 0.030 in. Feed rate = 0- 006 in/rev).

Tools o f 0 .010in . , 0 .020 in., 0 .030in . and 0.040 in. nose radius were used in the tests, and inspection o f results showed the Tangential and the Feed forces to be substantially independent o f tool nose radius over the investigated range. Tangential and Feed force results obtained with tools o f all values o f nose radius have, therefore, been plotted collectively. In the case o f the Radial force, however, it was found that an increase o f tool


nose radius produced an increase of force for any given conditions. Results were, therefore, plotted separately for each value of tool nose radius tested.

Figure 13 illustrates the variation of Radial force with cutting speed--these results being obtained with tools of 0.030 in. nose radius. It will be seen that the effect of cutting speed, whilst noticeable at the lower end of the scale, does not appear to be significant within the normal working range 225 to 350 ft/min. Similar results were obtained for the Tangential and the Feed forces. In the derivation of empirical formulae relating the forces to the cutting conditions, minor effects due to speed variation in the range given have therefore been ignored.

Using standard law fitting techniques, formulae were derived from the results relating the three forces to feed rate, depth of cut, and in the case of the Radial force the effect of the tool nose radius.

The formulae derived for the three forces are as follows:

Tangential force

pt = 6.04 x 104f °'72 (d q- 0.004)

Radial force

(i) when d ~< R pr = 3'42 X 104.] .o.9 d °'7 (0-7 q- R)

(ii) when d > R pr ----- 3"42 x 104f °'9 [(0"7 q- R) R °'7 q- 0"234 (d - R)]

Feed force

(i) when d ~< 0.030 in. pf = 1.30 x 104f °'~s d 1"2

(ii) when d > 0.030 in. pf---- 3.45 x 10sf °'38 [d °'7 - 0.031]

where pt = Tangential force (lbf) Pr = Radial force (lbf) Pl = Feed force (lbf)

f = feed rate (in./rev) d = depth of cut (in.) R = tool nose radius (in.)

Curves have been calculated from these empirical formulae, and are drawn in on the graphs shown in Figs. 14-17, which illustrate typical cutting force results.

In Fig. 14 is shown the variation of Tangential force with depth of cut. The top graph shows the individual experimental values whilst the lower graph shows the arithmetic mean points of these values used in the derivation of the empirical formulae. SatisfactotT correlation exists between experimental results and the forces calculated from the formulae-- the latter being represented by the actual curve drawn in on the graph.


2OO

_.e d taJ u e~ o

o IOO Z I,- I--

U

n

I J ~ * I _

, I ~ . ~ l l

I "l*~, *t~ I

O 0 0"020 0 "040 0 -060 0 -080 0.100

d, DEPTH OF CUT, in.

2 o o A R I T H M E T I C , MEAN POINTS OF EXPERIMENTAL VALUES

0 0

,ff

o tL

o I 0 0 Z

U

0-o20 0 -040 o-06o o . 0 0 0 0.10o d, DEPTH 0~. CUT, in.

FIG. 14. Tangential force versus depth of cut (Cutting speed = 250 ft/min. Feed rate ::~ 0"006 in./rev).

Figure 15 illustrates the variation of Radial force with depth of cut, using four values of tool nose radius. It can be seen that although an increase of tool nose radius results in an increased Radial force, the characteristic form of the force--depth of cut relationship remains unchanged. In all cases the Radial force rises sharply--though at a decreasing rate--until the depth of cut is equal to the tool nose radius. For these conditions the radiused portion of the tool only is cutting, making the effective angle of approach of the tool large. Further increase of depth of cut until the tool nose radius is submerged reduces the resultant approach angle, thereby increasing the Feed force in relation to the Radial force. Once the radius is completely submerged, i.e. the depth of cut exceeds the tool nose radius, no increase should take place, since the portion of chip cut by the curved section of the tool is not increased when using tools of zero plan approach angle. However, a slight increase in the Radial force does in fact occur, and could be due to the combination of frictional effects and the direction of the chip flow over the top surface of the tool. As before, empirical formulae have been obtained relating cutting force in this case to depth of cut, feed rate and tool nose radius; and good correlation is obtained.

The Design, Deve lopment and Test ing o f a Prototype Boring D y n a m o m e t e r

2 5 - ....

I ~ ~ . . . . i ~ - - - - - - ~

I I I SO

2 5

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o o z_ s o I . - I . -

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. 25

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I , _ . . . . . ~ _ ~ - - t 1 - , . - - " - " ~

• 4 I I I

0 0.020 0-040 0-060 0-080 0.100

d,DEPTH OF CUT, in. FIG. 15. Radial force vs. depth o f cut (Cut t ing speed = 250 ft /min.

Feed rate = 0-006 in./rev).

7 5 ~ I 0-1 O0 in, DEPTH OF CUT I

,, ( x " )

u \ ( . o ) ~¢,,- : tz 5 0

" I z (~*) J m l I

~ - I (~o) , , ~ J ' - j * " o 2 s - _ . / ,

I I

J , t " , - I , / / ( ~ ' ~ ) i " O.OI5in.DEPTH OF CUT

o ~ , , " ' / I J 0 0 . 0 0 5 0 . 0 1 0 0.015

f, FEED RATE, in./rev

FxG. 16. Radia l force vs. feed rate (Tool nose radius ~ 0 .030 in. Cut t ing speed = 250 ft/min).

255

256 R. A. HALLAM and R. S. ALLSOPP

125 I I ,~ O. I O0 in. DEPTH OF CUlT ~,&)

e I ,oo . . . . . . V - ~ , ' 4 - - ~ ; - -,~

j o ~ y ~ I I i^i / ~ i •

w 7 5 - - - - o

I g i l

I / o.om Ln.DEPTH OF CUT 2 s I - I . . . . . ~ - -

I I I I I I ~,, ( *~ , , o ~ = . . . ( ~ ) ' - ' - - ' , ~ II , L , ~ r ~ " ~ : (,=' •

0 0 , 0 0 5 0 " 0 1 0 0"015 f, FEED RATE,in/rcv

FIG. 17. Feed force vs. feed rate (Cutting speed = 250 ft/min).

Figure 16 illustrates the variation of Radial force with feed rate when using tools of 0.030 in. nose radius. As for the two other components of force, the variation with feed rate is to a power less than unity, in this case 0.9.

The variation of the Feed force component with feed rate is shown in Fig. 17. Satis- factory correlation again exists between the experimental points and the empirical formulae

1= ..$

1 ~o.o,#-. = oJ --... O "-

TOOL ANGLES

Max. R a k . Plan A n g l . Max. Rake Ar~l . Plan Approach A n g l . Plan Trail Angl. Front Curt|r ig Edg. Clearanc¢ Angf¢ = 7 ° Side C.E.CJI,. = 7 ° Secondary Front C.E.C,A. - 2 0 °

.~ .n P r = 3 ' 4 2 x IO4f O '9 d O'7 ( O ' 7 + R )

O.

. # °'O/o

o O.o~ °

= so oj : ,o: O.~oJ . . . . . . . . . . . . . . ~--_of \

O ~ - - - _ lb . .

°"< Z ~-.. oa

.eo

: 4 o = °

i,o

- O

FIG. 18. Nomogram for determining radial force component (d < R). Material--cast iron. (Cutting speed 225-400 ft/min).


relating Feed force to depth of cut and feed rate. No effect of tool nose radius on Feed force was apparent over the investigated range.

Similar correlation of results was achieved for the other force-cutting parameter relationships which are not shown graphically.

Nomograms were constructed from the empirical formulae for all three components of the cutting force. Two examples are presented in Figs. 18 and 19 which represent the Radial force equations for the two ranges of depth of cut less than and greater than the tool nose radius respectively.

TOOL ANGI. ES

Max.Rake Plan Angle = 9 0 ° Max.Rake Angle S ° Plan Approach Angle = 0 ° Plan Tool Angle = I 0 ° Front Cutting Edge Clearance

Af~gl¢ = 7 Side C.E.C.A. = 7 ° Secondary Front C.E.C.A. : 2 0 °

o.oo, q :~o.2ool . ~--

o o,oq , o,oo "~ o.o=o -I o . - oJ %oj

d > R P r - 3 ' 4 2 x 104 f 0 " 9 [ ( 0 . 7 + R ) R 0"7 + 0 . , 3 4 ~d-R~) ]

o.% i-o , ' / ] ,,,9 %z /

. . . . . . . . . . . . . . . O o : / . . . . . t O y /

-40 o~ ~ . / ,o

-,,o =~

MATERIAL - CAST IRON CUTTING SPEED - 22S to 4 O O f t l m i n

- BO

- I 0 0

FI~. 19. Nomogram for determining radial force component (d > R). Material---cast iron. (Cutting speed 225-400 ft/min.)

Acknowledgements--The authors wish to thank the Director and Council of the Production Engineering Research Association for permission to use the information presented in this paper.

APPENDIX

A Simplified Analysis of Cutting Force Components

During an early stage of the investigation, a simplified analysis of the cutting force component distribution was attempted. Such an analysis, it was thought, would serve a dual purpose inasmuch as it would provide a possible independent check on the general shape of the experimental curves relating cutting forces with the variables, and would throw some light on the type of relationships to be expected between the variables.

To meet the conditions of the simple assumption, no account was taken of the advanced metal cutting theories, and the only basic assumption made was that the cutting force is proportional to the cross sectional area of the uncut chip.

Subsequent work showed that the formulae so derived did not provide a fit over all the variables, and it was decided that the empirical formulae only would be used for final correlation of results.

258 R .A . HALLAM and R. S. ALLSOPP

However, the analysis is presented in this paper since it is felt that subsequent development, based on this approach, will be of value to other research workers engaged in the study of dynamometry and cutting forces.

Derivation of cutting Jorce component relationships

(a) Depth of cut <~ tool nose radius. The chip cross-section shown in Fig. 20 has been sub-divided by a mesh consisting of the following two families of curves:

, J_ (,-v) d

.YYI / I / ,, .wv~/ / / iY- i ' -_

/ / .¥%IX/ / / . ~ " ~-"

a j l t n / / v J / " / I ' I ' . ~ I ' ~ -

\ f

TOOL TIP

C

I1

I

R i O'

FIG. 20. Croas-section of chip removed during a boring operation for depth of cut < tool nose radius.

(i) Circles of radius R touching the line MN. (ii) Straight lines parallel to MN. Consider an element of area AA defined by this mesh containing the point P. Then a force Ap = LAA acting in the plane of the cross-section may be ascribed to

the element, where L is a force constant. It is assumed that the element of force Ap acts


in the direction of the inward normal through P. This force may be resolved into perpen= dicular components Apt, Aps, acting in the radial and feed directions respectively.

Integrating, to obtain the total components, pr, acting in the radial direction:

pr = fLda sin 0 (A. 1)

and the total force component in the feed direction is given by:

= fLdA cos 0 (A.2) Pt

Hence, referring to Fig. 20 it can be seen that:

s i n 0 - - R - V, and, c o s 0 _ {V(2R V)} ½ R R

Substituting in equations (1) and (2) respectively we get:

and

(A. la)

0 0

s

2 2 f ( f 3 u)2du 0

L ( {V(2R - V)} ½ dA P! (A. 2a) 2 R

Now the infinitesimal dements of area have the form of parallelograms in which the sides parallel to MN are of Icngth du, the width being equal to d V, and in the limiting case:

dA = du d V (A. 3)

In the following equation the first integral expression gives the total radial force acting in the area NMST (Fig. 20), but since the area NM Y is uncut the radial force required to cut this area must be subtracted from the first integral expression.

By substituting (A. 3) in (A. I) and putting in the appropriate limits it is found that

I a I ~

pr:Lf f ( )dud'-L f f ( X)dudV 0 0 0 0

where the limit B is given by,

and is obtained by considering an element of the area N M Y such as MIIQ of Fig. 21. Therefore,

s !

260

or,

R. A. HALLAM and R. S. ALLSOPP

2 2 [ ( f]2R- u)3]jo'r

2 2 12R

p , - = ~ - (A. 4)

R _1

0

t

f l u

2

- u

FIG. 21. Determination of limits for integral.

Now although equation (A.4) is mathematically correct, it implies that a negative radial cutting force exists for zero depth of cut, which is impossible. However, for practical cutting conditions, the term 1/12 ( f /R) 2 is small and so it may be neglected when bringing the equation into practical form. Hence, re-writing equation (A. 4) with this modification:

pr = ~ -

Considering now the Feed Force (pf).


where

Now:

Let

therefore,

Substituting (A. 3) in (A. 2) it is found, since the same limits apply, that: $ a f ~ Ps:L f f {V(2R-V)}fdudV~ -L f f {V(2RR-V)}½dudV

0 0 0 0

- ~ f - - ] j , as before.

f {V(2R R - V)}½ dV : -- R f (1 -- ZZ) ½ dZ

which is a standard integral integrating to"

-Rf ( 1 - ZZ)½ dZ = -R[sin-lZ+Z(l~ -ZZ) ½]

Thus : s

p, =L f [ - ff {sin -1 (1 - ~) q- [1 - ~] [1 - ( 1 - ~)2]½}]S du- o

s L f [-~{sin -1 (1 -~ ) + [l -~-] [ l - (1 -~]Vi2]½/]'duj 'Jo

O

where/3 has the value as before. After integration and rearrangement, PI is given by,

tRf [cos_l f sin -1 ( 1 - d ) _ [ 1 - d] [ 1 - ( 1 - d ) 2]½-t- ps = ~ - ~ -

2R,4 [1 ~f~2]½ 1 [ 1 f z I _ _

+ 7 t ] \~-h! j - 3 (A.5)

Considering the Tangential force (Pt). The Tangential force is given by an expression analogous to those for pr and p f, but it

will of course be acting perpendicular to the plane of the section. Hence if m is the force constant for the Tangential direction:

p,--mffdudV-mffdudV 0 0 0 0

where fl is given as before.


Now since:

f (! - Z2) ~ dZ = ½ [sin -I Z ÷ Z(I - Z2) ½]

therefore:

f i R 2 ( f ; u ) 2 ] ~ d u = - R 2 [sin -1 ( ~ R ~u) [f-~R~U]. {1 - i ] / ] ' f - u 2 ½

Thus, I

O

which, after integrating and rearranging gives,

R i { 1 " f 2½ Pt m R f [ d - l + ~ s i n - l f R + ~ (A.6)

Approximations: It is considered that equations (4a), (5) and (6) will be closest to the truth when f i r is

fairly small; simplified approximate formulae are therefore of interest. Neglecting (fiR) 3 and higher powers, the equations may be reduced to simpler forms.

Radial force

pr = ~

which reduces without approximation to:

pr--Lfd 1 - ~ (A.7)

Feed force Equation (A. 5), neglecting higher powers than (fiR) 2, and since f/2R is small, taking

cos -1 (f/2R) = zr/2, becomes,

A further approximation of equation (A.8) is possible, and may be carried out as follows:

Re-writing equation (A. 8) with (d/R) denoted by x and considering only the function of x enclosed in the square brackets we have:

f ( x )=~- - s in 1(1 - x ) --(1 - x ) 1 - ( 1 - - x ) S i (i)

Differentiating and simplifying we obtain:

f'(x) = 2x~(2 - x) -~ (ii)


Expanding (ii) gives

½(x ½ x ~ x~ ) (iii) f '(x) = 2 . 2 4 32 "'"

Integrating term by term, noting that f(0) = 0

. f ( x ) = 4 . 2 ~ x 1 - 3 3 x 2 _ 3 20 x - - ~ "'"

or

3 3 x2 - ) f ( x ) = 1.875x ~ 1 - ~ - 6 x - - 2 2 4 "'" (iv)

We take an approximating function g(x) which has the same form as the first term of (iv), and takes the same value as (i) when x = 1. Noting that f(1) = rr/2 we obtain,

x ~ g(x) =-~

then when x = 0 or x = 1 we have g(x) = f(x). Hence, for this approximation we have the expression for the feed force reduced to:

Ps = 0.79 L (A. 8a)

Tangential force Equation (A. 6) for the Tangential force may be approximated by expansion as follows:

fa R {2~fR q- -k } + 1 { 1 - ~RZ-k }]

= t u R f [ d 2~R2], neglecting higher powers of f /R (A.9)

However, this implies that a negative tangential cutting force exists for zero depth of cut, which is most unlikely in normal cutting operations. Since, in the above approximation, small positive terms have been neglected, and the termf2/24R 2 will be small in comparison with d/R, from practical considerations the following form will be closer to the actual conditions.

pt -= mfd (A. 9a)

By differentiating (A. 7), (A. 8a) and (A. 9a) partially with respect to d, the following relationships are obtained:

0pl _ 1.185 Lf (A. 11) gd

apt _ mf (A. 12) ad


Also, collecting together the approximate formulae for the three forces we have:

pr----Lfd 1 - 2-R (A.7)

Pl = 0.79 LRf (A. 8a)

pt = mfd (A. 9a)

d

I L n _L d- q~oMi~ of Chip Cut "-mporti~ o~ ~,| byTool Nose Radius [ChipCut by x l

. . \ \ \ ~ t ~ a ~ g ~ t \ P ~ r ~ ] o f T o o ,

0 I" -I N

\

\ \ \ \ \ x

TOOL TIP

FIG. 22. Cross-section of chip removed during a boring operation for depth of cut > tool nose radius.

Depth of cut > tool nose radius

For depths of cut in excess of the tool nose radius, using tools with 0 ° plan approach angle the cross-section of the chip removed will be comprised of a portion of rectangular form in addition to the area removed by the radiused portion of the cutting edge of the tool.

The area of this rectangular portion, see Fig. 22, will be equal to f ( d - R), where as before f , d and R are feed rate, depth of cut and tool nose radius respectively.


In the case of the Radial force, there should theoretically be no further increase from the value found for the case d = R, since for all depths of cut greater than R, the actual radiused portion of the tool will still be cutting the same area of chip as when d = R.

In the case of the Feed force, there will be an additional force L l f ( d - R) due to the cutting of the rectangular portion of the chip, where L1 is a force constant and the equation (A. 8a) will become:

p / = 0.79 L R f + Laf (d - R) (A. 13)

since (A. 8a) is valid for the case d = R.

2 0 0

..,,s

O ' I

~ 100 I- F-

J J I I TANGENTIAI.__FORCE t ~ " - - ~ ' '

EXP.ERIMENT .~,'~"L

y THEORY

'~ I ~ , F,d~

1 . ~ r.toav,

0

d I DEPTH OF'CUT

J

0 0'020 0"040 0"060 0"080 0"100 0-100 d, DEPTH OF CUT

I / 1 EXPERm4[NT I " l - - I. . . . . . ' _ ? _ _ _ * . . . . . . .o L, I

L"

o o 8 o o l o

d, OEm OF CUT

FIG. 23. Comparison of theoretical relationship between cutting force and depth of cut with experimental results.

Similar remarks apply to the Tangential force, and the resulting equation for depths of cut greater than the tool nose radius will be:

pt = m f R + m l f ( d - R) (A. 14)

where ml is the force constant under these conditions. Because of the simple assumptions made in this analysis, it does not accurately predict

the observed forms of the three forces as functions of the feed rate. The analysis predicts a linear dependency with feed, whilst the experimental results

show the actual relationship to be as follows:

Tangential force oc f0.72

Radial force ocf0.9

Feed force oc f0.s8

However, the degree of agreement of the relationship between the three forces and depth of cut, shown in Fig. 23, is good. The experimental curves are taken from the main


results, whilst the theoretical curves have been plotted from the equations of this analysis after derivation of cutting force constants. It is considered that the degree of similarity in form for all three forces justifies the method of approach used in this simple analysis.

Documents

The design, development and testing of a prototype boring dynamometer