- Email: [email protected]
Efficiency of percussive drilling with extension rods
/,~t J. R,)(k ~,tech
tim
5, i & Ge,,mech
4h~'tr Vol 24. No a., pp 2[3 222. 198-
0147<-91)~2'," S 3 0 0 - l ) 0 0 Cop?right ~ !:~,~7 Pergamon Journals Ltd
Printed in G~'at Brlt,xin ~.[1 right. :eser',ed
Efficiency of Percussive Drilling with Extension Rods B. L U N D B E R G +
Extension drilling of rock is simulated numericalh' using a I'ictor desk-top computer. The hammer, adapter and extension rods are tmilbrm amt hare the same characteristic impedances. The same is true jbr the joints and the bit. Two efficiencies are determined as ~mctions of six dimensionless parameters which represent the number of joints, the joint-to-hammer length ratio, the joint-torod characteristic impedance ratio, the initial gap between bit and rock and the bit~rock interaction (two parameters). The first efficiency is defined as the ratio of the work performed on the rock to the impact kinetic energy of the hammer. The second is based on the sum of the work performed on the rock and the rebound khwtic energy of the hammer, which is assumed to he fidl.v retrieved. Both efficiencies generally decrease with the number of joints and the joint-to-hammer length ratio. Exceptionally, however, the efficiencies may increase slight O' with the number of joints. The difference between the second and first efficiency, which is due to hammer rebound, generally increases when the rock becomes increasingly hard. This dependence on rock hardness is strongest when the number of joints and the joint-to-hammer length ratio are low. In the absence of joints the efficiencies have an oscillatory dependence on a small initialgap. When one or several joints are ,tsed, howet'er, the efficiencies essentialO" decrease with the initial gap.
INTRODUCTION Extension rods are used in percussive top hammer drilling of holes in rock to greater depth than the length of a single rod. Commonly, the extension rods are threaded at their ends and are screwed into cylindrical coupling sleeves in firm end-to-end contact with each other. The input end of the first rod is connected to the adapter with the same kind of joint, and the output end of the last rod is similarly screwed into the drill bit. When the hammer of the rock drill hits the adapter, elastic stress waves are generated in the hammer and in the drill rod system. Initially these waves are compressive. When the waves arrive at sections with changes in cross-sectional area or, more generally, in characteristic impedance they are partially reflected and partially transmitted. Thus. reflections occur, for example, at the free end of the hammer and at the adapter/joint, the rod/joint, the rod bit and bit/rock interfaces. Due to the reflection of compressive waves at sections with reduction in characteristic impedance, tensile waves are generated. As a result tensile stresses may occur in certain parts of the rock drill. Often the hammer separates from the adapter and rebounds after a time which is much shorter than the period of percussion. This rebound may occur as a result
+ Department of Mechanical Engineering, T e c h n o l o g y . S-951 87 Lule~,. S w e d e n Rxl~ls :a~ >
Lulefi
University of
213
of reflected compressive waves from the adapter and the first joint only. Then, the rebound is independent of the number of joints and also of the conditions at the bit/rock interface. However, it may also depend on subsequent joints and on the situation at the bit/rock interface. When the hammer separates from the adapter it generally performs a complex motion under the influence of trapped waves. These waves are finally attenuated, and thereafter the hammer leaves the adapter with a constant velocity. As the m o m e n t u m of the hammer is conserved in the absence of external forces, this rebound velocity is equal to the ratio of hammer m o m e n t u m at the time of separation to hammer mass. With a suitable design of the percussive drill, a certain fraction of the kinetic rebound energy can be retrieved. The leading stress wave which reaches the drill bit is generally compressive. It has been modified by the joints on its way to the bit. Therefore it is longer and smoother and has lower amplitude than it would have had otherwise. Also it carries less energy, as a main effect of the joints is to disperse the stress wave energy into the various parts of the rock drill. Dissipation of heat due to friction and slip within the joints is another contribution to the reduction of the energy carried by the leading stress wave. During the reflection of the leading compressive stress wave at the drill bit, the cutters are displaced towards the
214
LUNDBERG:
EFFICIENCY OF PERCUSSIVE DRILLING
rock. After contact has been established they generally which is adequate provided that the incident stress wave penetrate into and crush the rock. Thus, work is per- is much longer than a joint. Paper [8] deals with the formed. The leading compressive wave is followed by transmission of a stress wave through an arbitrary waves which generally have lower amplitudes and which number of identical joints. The results show that the may be tensile as well as compressive. These waves. efficiency of energy transmission at any' joint is higher which result, for example, from multiple reflections than that at its predecessor. Thus, the transmitted wave between the joints, sometimes contribute to the work of is successively modified by the joints in a way which crushing the rock. Finally, and generally before the next favours further energy transmission. The paper [9] deals impact, the waves in the drill rod system are attenuated with the problem of optimizing the shape of the incident mainly due to conversion of stress wave energy into wave such that, for a given single joint, the efficiency of work, dissipation in the joints and material damping. At energy transmission is maximized. Another aspect of this stage, there may remain some kinetic energy associ- optimization is considered by Gupta [10], who deals with ated with motion of the drill rod system away from the the problem of optimizing the characteristic impedance rock. Ideally, the thrust should bring the drill bit back distribution of a single joint with given mass and length in contact with, or at least close to, the rock before the such that, for a given incident wave, the efficiency of energy transmission is maximized. next impact occurs. Two definitions of the efficiency of the drilling process Conversion of stress wave energy into work, finally, are natural. First, the efficiency can be defined as the has been dealt with by a number of authors, for example ratio of the work to the hammer impact kinetic energy. Simon [11], Hustrulid and Fairhurst [12] and Lundberg This efficiency ~/is useful if the hammer rebound kinetic [13-15]. Their studies show that the efficiency of energy energy is not retrieved. Secondly, the efficiency can be transfer to the rock may be as high as 80-90%. In the based on the sum of the work and the rebound kinetic paper [16], the efficiency ~/ of the complete process of energy. This efficiency ~/÷ is adequate if the rock drill percussive drilling is determined for a commercial perdesign is such that the kinetic rebound energy is com- cussive drill, namely an Atlas Copco COP 1038 HD. pletely retrieved. It follows that r/+ exceeds q by e-', The present study concerns the complete process of where e is the coefficient of restitution of the hammer. extension drilling of rock. The efficiencies q and r/+ are It is evident that the following elementary processes determined for a variable number of joints Nj, for are important ingredients in extension drilling of rock: different joint-to-hammer length ratios ).j and for (i) impact between hammer and adapter; (ii) trans- different conditions at the bit/rock interface. Thus, the mission of a stress wave through a joint; and (iii) influence of the dimensionless initial gap 6 [16] and the conversion of stress wave energy into work. In the dimensionless loading parameter fl [13-16] is studied in literature it has been more common to treat these detail. The elastic "swell" model mentioned above is elementary processes separately than to consider the used for the joints which means that the effects of complete process of extension drilling. However, there is friction and slip are neglected. The numerical results are a complex interaction between these elementary pro- obtained with the aid of simulation programs written in cesses and therefore the complete drilling process is not ~.Turbo Pascal for the Victor desk-top computer. These programs are extended versions of the Apple Pascal generally related to its parts in a simple way. Impact between a hammer and an adapter, or a programs DTHD-SIM, DTHD-EFF, HD,SIM and similar device, has been treated in numerous papers. H D - E F F presented in [16]. Commonly I-D models have been used in conjuction with analytical [1,2], graphical [3, 4] or numerical [5, 6] MODEL AND PARAMETERS methods of analysis. The papers [1, 3-6] treat the direct The simulation model for the percussive drilling problem of determining the stress wave produced by process is illustrated in Figs 1 and 2. All quantities a given hammer, whereas the paper [2] concerns the shown are dimensional. Primes (') are used for some corresponding inverse problem. ` of these quantities in order to distinguish them from Transmission of stress waves through joints has been corresponding dimensionless ones given less attention in the literature. Takaoka et al. [7] Figure 1 illustrates the 1-D model of the percussive published experimental results (in Japanese) which were drill to be studied. Figure la shows a uniform rod partly reproduced and further discussed by Fairhurst [1]. configuration and Fig. lb shows an extension rod It was found that the relative energy loss in the trans--configuration with Nj joints. In what follows, Nj = 0 will mission process was 7.5-18.5% for a single joint. This refer to the uniform rod configuration and Nj = 1.2,. loss is due to reflection and dissipation in different to extension rod configurations. proportions for various types of joints investigated; The hammer and the adapter have the same constant Early theoretical studies were made by Fischer [4] who characteristic impedance Zk as the rods, and the bit has used a 1-D model and treated a joint as an elastic the same constant characteristic impedance Z~ as the "swell", that is, as a local increase in characteristic joints. The length of the hammer ts Lh and that of a joint impedance. Thus, friction and slip, which give rise to is L~. The length of an extension rod, exluding the end dissipation, were not considered. The same model was parts which are screwed into either a joint or the bit, is used by, for example, Lundberg [8] and Lundberg et al. 10L~. The length of the adapter external to the first joint [9]. In their papers, use is also made of a rigid joint model
LUNDBERG:
EFFICIENCY OF PERCUSSIVE DRILLING
2[5
e,/
t
t
i
,¢
'2Lj =
eV
(b)
I
Fig, 1. Percussive drill model. (a) Uniform rod configuration. (b) Extension rod configuration.
is Lj. The length of the bit is (1/2)Lj. The length of the integral rod external to the bit is 12Lj. This implies that the total length L' of the rock drill (between the extreme ends of the hammer and the bit) is the same for Nj = 0 and Nj = 1. The material is the same throughout the system and is characterized by the Young's modulus E and the elastic wave speed c. Thus, denoting the common crosssectional area of the hammer, adapter and rods by A, we have Z~, = AE/c. The impact velocity of the hammer is V. The drill rod system is initially quiescent. Also, the hammer and the drill rod system are initially free from stresses. The bit/rock interaction is represented in Fig. 2 by a relation between the force F' acting on the rock and the history of penetration u'. This relation is characterized by the initial gap between bit and rock u~ and by the penetration resistance (slopes) k ' and k;. The system is characterized by the six dimensionless parameters N,,
2 , = Lj/L~,
= u~iu**,
Z j = Zj/Z'R
# = k',./,-**,
7 = k'/k;
where [16] ,,** = (2L~/c)V,
k** = AE/2L'H.
The parameters ).j and Zj represent the length and the characteristic impedance, respectively, of each of the N~ joints. These parameters also represent the length of the adapter and the characteristic impedance of the bit, respectively. As the material is the same throughout the system, Zj can also be considered to represent the cross-sectional area of the joints and the bit. 6 represents the initial gap between bit and rock, and fl and 7represent the bit/rock interaction. After sufficiently long time, the work W ' = ~F'du" performed on the rock will remain constant. Also, the hammer will rebound with velocity eV, where e is the coefficient of restitution, and there will be no reestablished contact between hammer and adapter (up to this time there may have been a number of separations
and re-established contacts). Then the kinetic energy associated with rebound is e:Wf<, where W~ is the kinetic impact energy of the hammer. The efficiency of the drilling process is defined as ,7 = w ' / w ' ~
or ,7 + = ( w ' +
e-'W;:),.'w~.
From these definitions there follows the relation r/+ = r/ + e: already mentioned. In [17], the slightly more general efficiency r/ + 0e 2 was introduced, where 0 is the fraction of the kinetic rebound energy which can be retrieved. The efficiencies r/ and q* are functions of the six dimensionless parameters .A,~e {0, 1. . . . I', 21E [0, CC), Zje [1, oo), 6 • [0, ~), fl • [0, m) and 7 • [0, 1]. The meaning of given values of Nj. ).j, Zj and ",' is readily understood. The significance of prescribed values of and fl is less easily assessed but was thoroughly discussed in [16]. For a semi-infinite system (Nj--, :z) which is perfectly uniform (Zj = 1) 5 = 1 represents an initial gap which is closed exactly at the moment when the reflection of the incident stress wave is completed. Thus, the gap is closed but no work is performed. For the same semi-infinite
F'
J
Fig. 2. Bit/rock interaction model.
216
LUNDBERG:
E F F I C I E N C Y OF PERCUSSIVE D R I L L I N G
and uniform system but in initial contact with the rock (~ = 0) the efficiency rl is high when /~ is of the order of unity (rlm~,= 0.815(1 - 7 ) for /~ = 1.26) but is low (r/<< 1) when /~ << 1 or /~ >>1. In this sense fl << 1, /~ - 1 and /~ >>1 represent soft, medium hard and hard rock, respectively. The numerical procedures used are described in [16]. The rock drill is subdivided into L segments, each with length d = L'/L and constant characteristic impedance. Correspondingly there are the relations Lh = LHd for the hammer and L5 = L j d for a joint, where LH and L j a r e the number of segments used to represent the hammer and a joint, respectively. The length of a time step is the transit time h = d/c for a stress wave through a segment. Numerical approximations are used for determination of the moments of contact re-establishment between hammer and adapter or integral rod, and also for the bit/rock interaction. Otherwise exact relations of the I-D theory of elastic stress waves are used. The accuracy of simulation increases with the number of segments used to represent a given system. In [16] it was found that good accuracy is obtained with LH = 5 or even less.
SIMULATION AND RESULTS
Variation of parameters The joint-to-rod impedance ratio Zj and the unloading parameter 7 are kept constant at the representative values Zj = 2 and 7 = 0.1 throughout the simulations. Thus, there remains to study the dependence of the efficiencies r/and r/+ on the number of joints Nj, the joint-to-hammer length ratio ).j, the initial gap 6 and the loading parameter/L For these parameters, the values N, e {0, 1,2, 5,9}, ).j ~ {0.2, 0.5, 1}, 6 ~ {0, 0.1,0.2 . . . . I} and/~ e {0.1, 0.2, 0.5 . . . . ,10} are chosen. In a real drilling process the initial gap 6 and the loading parameter/~ vary in irregular ways which can hardly be predicted or accurately controlled. Therefore it is of interest to determine average values of q and ~ + when one of these parameters varies within its full range given above while the other parameters remain constant. The average values are defined by
Y = .f[ y(x) dx/(b - a) where y = 1/ or r/+, x = 6 or ln(/~), a = xmi,, b = Xm~, and where linear interpolation is used to evaluate the integral from the computed points (x, y).
Number of segments and duration of simulation
-
The number of segments L which can be used to represent the rock drill between the extreme ends of~da¢ hammer and the bit is limited due to the fact that there is only space for 100 segments on the computer display. As each extension rod is represented on the display by only one segment at each of its ends and by one empty segment space between (cf. Fig. 1b), the largest possible number of segments is not fixed but depends on the parameters Nj and 2j. From the computational point of
Table 1. Number of segments used to represent the hammer, L H, a joint, Lj, and the entire rock drill. L, vs number of joints Nj and joint-to-hammer length ratio ,:.j, Nj
~.j
LH
Lj
L
0 0 0 1 1 I 2 2 2 5 5 5 9 9 9
0.2 0.5 1 0.2 0.5 i 0.2 0.5 I 0.2 0.5 1 0.2 0.5 1
60 24 12 60 24 12 50 20 10 30 12 6 20 8 4
12 12 12 12 12 12 10 10 10 6 6 6 4 4 4
70 42 30 210 174 162 285 255 245 369 351 345 422 410 406
view all rod segments are equivalent, however, independently of their representation o n the display. The number of segments used to represent the hammer, LH, a joint, Lj, and the entire rock drill, L, are given in Table 1 for the various combinations of Nj and 2 j . It can be seen that LH varies from 4 to 60, Lj from 4 tO t2 and L from 30 to 422. Each simulation should have sufficient duration to allow the work W' and the coefficient of restitution e to assume their final values. Then correct efficiencies ~/and + are obtained. In this study, a duration of simulation of six stress wave transits through the entire system is considered adequate. This corresponds to the dimensional time 6L'/c or to 6L time steps (cf. Table 1). It should be pointed out, however, that this duration may not be strictly sufficient in every simulation, In exceptional situations, therefore, one or both of the efficieneies may be somewhat underestimated.
Results of simulation The results of the simulation are shown in Figs 3-9. Results for ~/and r/- are represented with open and filled symbols, respectively. Squares, circles and triangles are used for ).j = 0.2, 0.5 and 1, respectively. First it is assumed that there is initial contact between bit and rock, that is, 6 = 0. Figure 3 shows the dependence of the efficiency 17 on fl, N j and ).j for Zj = 2, 6 = 0 and "; = 0.1. Generally q first increases and then decreases with /~ such that ~/,.~ is assumed for/~ e {0.5, 1, 2} (medium hard rock). When Nj and ).j assume larger values (larger number of joints which are longer relative to the hammer), r / ~ tends to be assumed for lower values of ~ (softer rock). Also, r/ generally decreases with both Nj and 2j. The decrease with Nj is commonly faster for low Nj than for high. Notice that exceptionally (;tj = 1,/~ = 0.2, Fig. 3c) r/may increase slightly when Nj is increased from one value (Nj = 5) to another (Nj = 9). Figure 4 shows the dependence of the average efficiency r7 on Nj and ~.j for Zj = 2, 6 = 0 and 7 = 0.1. Each point in Fig. 4 is obtained from a curve in Fig. 3. The following observations correspond to some of those
LUNDBERG:
EFFICIENCY
made directly from Fig. 3: (i) ~ decreases with N;, and this decrease is generally faster for low Nj than for high; (ii) r7 decreases with ,:.j for :Vj = 1, 2 . . . . . 9. The results plotted in Fig. 4 are gixen also in Table 2. Figure 5 shows the dependence of the efficiency r/+ on '8, Nj and ,;.j for Z j = 2 , ~ = 0 and 7 = 0 . 1 . From a comparison with Fig. 3, there follows that the difference between r/- and r/generally increases (hammer rebound becomes more important) when '8 increases (harder rock). This dependence on ,8 is stronger when -\:3 and ,:.j assume lower values (fewer joints which are shorter relative to the hammer).
OF PERCUSSIVE
DRILLING
217
{C; 1.0X =I
zj=2 -o x "0.1
~7 0 5
(a) 1I 0 ~-
X • 0.2
i I
Z*2
i
~'0
i
7" ,0.1
L
N:O /
0 [ 0, I
I
I
I
0,2
0.5
1
__1_ ~
I ~
I0
# Fig. 3. Efficienc? q vs loading parameter fl and n u m b e r of joints ,¥j for j o i n t - t o - h a m m e r length ratio (a) ,:.j = 0.2, (b)),j = 0.5 and (c) ,,;.j = 1. Z;-" (5=0and-=0.1
-~ 0 . 5 ]
[
X
I
i
O[ 0.1
I
I
I
0.2
O. 5
1
2
1
]
5
10
Figure 6 shows the dependence of the average efficiency 0 - on Nj and ).j for Zj = 2, 6 = 0 and 7 = 0.1. Each point in Fig. 6 is obtained from a curve in Fig. 5. F r o m a comparison with Fig. 4 it can be assessed that r7+ and r7 depend on :\j and 2j similarly. However, r~+ decreases with 2; more slowly than r7. The results plotted in Fig. 6 are given also in Table 3. 1.O~ I I
&i2
(b) 1.0
I ( [0.1 , 10]
X • 0.5
X "0.1
Zj,2 ,0 X " 0.1
j=O,2
0.5
'"7
I
0
0.!
I
I
5
I
I
!
I
I
10
%
i
O
I
i
1
02
0 .5
I
I
I
I
1
2
5
10
#
Fig. 4. Average efficiency ~ with respect to ln(fl) (or ~ ~[0. I, I] vs n u m b e r of joints ,Vj and j o i n t - t o - h a m m e r length ratio ,;.;. Zj = 2, 6 = 0 and 7 = 0 . l.
218
LUNDBERG:
E F F I C I E N C Y O F PERCUSSIVE D R I L L I N G
Table 2. Average efficiency # with respect to lnffl) for fl~[0.1, 1] vs number of joints Nj and joint-tohammer length ratio 21. Zj = 2, 5 = 0 and 7 = 0.1.
(b] 1.0FI
~,j " 0 . 5
L zj-2 B-O
# -Vj
;.j = 0.2
,;,j = 0.5
,;.j = 1
0 1 2 5 9
0.629 0.599 0.565 0.504 0.469
0.647 0.539 0.475 0.360 0.288
0.632 0.468 0.356 0.232 0.188
!
r? ÷
Finally the influence of initial gap 6 between bit and rock is studied. Figures 7 and 8 show the dependence of the efficiencies 17 and 17+, respectively, on 5 and Nj for 2j = 0.5, Zj = 2, fl = 1 and 7 = 0.1. For N j = 0 t7 and 17+ oscillate with 6, but for Nj > 0 they essentially decrease with 6. Figure 9 shows the dependence of the average efficiencies ~ and ~ + on Nj for the same values of 2 j , Z j , fl and 7. Each point on the curve for ~ is obtained from a curve in Fig. 7. Similarly, each point on the curve for + is obtained from a curve in Fig. 8. The results plotted in Fig. 9 are given also in Table 4.
7 ,0.1
0.5
0.1
I
I
1
1
I
0.5
1
2
5
10
I 2
I 5
I 10
(c)
1.0
DISCUSSION
I
0.2
Xj=I
A useful background for interpretation of the results obtained from the simulations is provided by previous results [8, 13] for the three elementary processes mentioned in the Introduction: (i) impact between hammer and adapter; (ii) transmission of a stress wave through a joint; and (iii) conversion of stress wave energy into work. Impact efficiency r/i, transmission efficiency 17Tand conversion efficiency 17o respectively, and also some other results for these elementary processes are put together in the Appendix.
r'0.1
Nj,0
/
•
1
r/+ 0.5
(a) 1.0 Xj=0.2
Zj,Z F 8-o
Nj,0
.//m
OL
0.1
I 0.2
I 0,,5
I 1
Fig. 5. E~ciency ~/+ vs loading parameter fl and number of joints Nj for j o i n t - t o - h a ~ length ratio (a) ,;.j = 0.2; (b) 2j = 015 and (C) 2j = 1. Z~= 2, ~ =0 and 7 =0.1.
'r/+ 0.5
0.1
0.2
0.5
1
B
2
5
10
It can be established from Fig. 3 t h a t the efficiency 17 of the complete drilling process depends on the loading parameter fl in a similar way as the conversion efficiency 17c. Thus, both 17and 17c assume their maxima for values of fl of the order of unity (medium hard rock). Also, both efficiencies are low when fl is small (soft rock) or large (hard rock). In the former case, the leading incident compressive wave is reflected mainly as a tensile wave at the bit/rock interface (nearly free end); in the latter case,
LUNDBERG:
EFFICIENCY OF PERCUSSIVE DRILLING
219
1.o Z'2
X j-0.5
BE Co.I,~O~
B
zj,2 i
]v+:c
,o
,F , 0 . 1
iI
~'o~-
~
--------------____~
I
'
I
I
t
;
o.5
I
I
5
:9
~+
10
0
~ "-''~ -
i
i
i
i 0'5
1.0
%
Fig. 6. Average efficiency ~- with respect to In(fl) for fl~ [0. l, I] vs number of joints ~ and joint-to-hammer length ratio ).j. Zj = 2, d = 0 and 7 = 0. I.
Fig. 8. Efficiency r/+ vs initial gap 6 and number of joints +vj. ,;.j = 0.5, Z j - _ . fl = l and ;' =0.1.
Table 3. Average efficiency~i- with respect to Ln (fl) For fl ~ [0.1, [] vs number of joints Nj and joint-tohammer length ratio ,:.j. Zj =-~,~ ~5= 0 and 7 = 0. l.
it is reflected mainly as a compressive wave (nearly fixed end). As the loading parameter fl can be expressed fl = 2 L ' H k ' / A E the intermediate " m a t c h e d " conditions can be obtained by choosing a suitable value for k'. The main effect of" the joints is to decrease the efficiency ~1- As can be seen in Figs 3 and 4 this decrease in efficiency is more important for large values o f the j o i n t - t o - h a m m e r length ratio ,:.j than for low values. This dependence on 2; corresponds to that of the transmission efficiency ~/T, which is a decreasing function of 2j. Thus, a small value o f the parameter ,;-s = L~,'L;{ is favourable. It can be realized by using a short joint or a long h a m m e r or both. A n o t h e r effect of the joints, which is demonstrated by Fig. 3, is the removal o f the m a x i m u m for ~1 to lower values o f ft. With the aid of the Appendix this can be explained as follows. In the absence o f joints the conversion efficiency tic assumes its m a x i m u m value when fl = 2 L ' n k ' / A E assumes a value of the order o f unity. 2L~ is the length of the incident stress wave. If joints are presents, however, the incident wave at the bit will be longer than 2L~. It will also have a different shape, which is less important. Therefore, with joints present, a lower value of fl will correspond to an incident stress wave at the bit with such a length that the conversion efficiency Y/c is maximum. Therefore the m a x i m u m of the efficiency ~/ is also removed to lower values of ft. It is also found from Figs 3 and 4 that the decrease in efficiency r/ with the number of joints Xj is generally faster for low/Vj than for high. This is in correspondence with previous results [8] for the efficiency of energy transmission through a chain o f rigid joints. At first it might appear strange that there exist conditions which make the efficiency ~1 increase with the
3,'j 0 l 2 5 9
,~j = 02 0.779 0.705 0.604 0.512 0.475
,;.; = 0.5 0.765 0,574 0.509 0.394 0.323
,:.j = I 0.740 0.579 0.467 0.343 0.299
1.0'j l i
X "0,5 Zu'2
,8=I ii
7" =0.1
l
9
J i 0
l
L
I
L 0.5
1
I
I
I
l 1.0
Fig. ?. Efficiency r/ ',s initial gap 6 and number of joints Xj. ,:.j = 0.5. Z j - - 2. fl = l and 7 = 0 . 1 .
220
LUNDBERG:
1.0
E F F I C I E N C Y OF PERCUSSIVE D R I L L I N G
ks, 0.5 2'.,2 a ( [o,q
3"I X "0.1
O.
77
I
I
I
I
I
5
I
I
I
t ..... 1
10
Fig. 9. Average efficiencies ~ and r7+ with respect to 6 for 6 ~[0, 1] vs number of joints Nj. 2j = 0.5, Z 1 = 2, fl = 1 and 7 = 0.1.
Table 4. Averageefficiencies~ and r1+ with respect to 6 for 6 e [0, 1] vs number of joints Nj.
/.j=0.5,
Zj =2,
fl--" 1
"/=0.1.
0 1 2 5 9
0.759 0.551 0,430 0.264 0.236
0.766 0.586 0.464 0.299 0.271
and
discussion above, it is clear that the influence of the joints is less important when the parameters Nj and 2j assume lower values. Therefore it is also understood that the difference between the efficiencies *l- and rl depends more on fl when Nj and :.j are low; a long chain of long joints effectively acts as a shield between the hammer and the bit which makes the hammer rebound less dependent on the conditions at the bit'rock interface. It is not difficult to understand why an initial gap 6 between bit and rock may increase or decrease the efficiencies r/and r/* in the absence of joints, whereas it mainly has a negative effect in the presence of joints. The reason is as follows. As long as there is a gap between bit and rock the incident compressive stress wave is completely reflected, mainly as a tensile wave. The shape of the reflected wave differs somewhat from the shape of the incident wave due to the geometry of the bit. In the absence of joints the leading compressive part of the reflected wave is trapped in the hammer and causes a rebound, but the remaining part is reflected again and returns intact, mainly as a compressive wave, at the bit where it may either further reduce the gap or contribute to the work. In the presence of joints, the corresponding doubly reflected wave is largely dispersed within the rods because of the reflections at the joints. Therefore its leading part has less ability to produce work when it arrives at the bit. It is noticed that ~ = r/l(r/T)'~ r/C is a simple-minded approximation of the efficiency r/. The impact efficiency is rh = 1 while the efficiencies r/T and ~/c are given in Tables A1 and A2, respectively. The approximation neglects the following effects: (i) the influence on the length and shape of the transmitted wave by each joint; (ii) the multiple reflections of stress waves within the adapter and the extension rods. The comparison of r/ and ~ in Table 5 indicates the importance of the neglected effects under various conditions. For the lowest joint-to-hammer length ratio (2j = 0.2) and medium hard (fl = 1) or hard (/3 = 10) rock the approximation is reasonably accurate (r7 ~ 0.8 r/) and therefore the ignored effects have little importance. For the highest joint-to-hammer length ratio (,;.j= 1) and soft rock (/3 =0.1), however, the approximation ~ leads to an underestimation of r/ by more than a factor of 10 (r7 ~ 0.09 r/). Consequently the ignored effects are important in this case. These observations can be well understood from arguments given in the discussion above.
number of joints Nj. With the aid of the Appendix, however, this phenomenon can be explained. Notice in Fig. 3c that it occurs for the relatively low value 0.2 of the loading parameter/3. Therefore, when the number of joints is only Nj = 5 the leading incident wave at the bit is relatively short and consequently the conversion efficiency r/c is relatively low. When the number of joints increases to Nj = 9 less energy is carried to the bit by the leading wave. However, this wave is longer than with N~ = 5 and therefore the conversion efficiency r/c is increased. In the example of Fig. 3c, which appears to be exceptional, the increased number of joints leads to an increased conversion efficiency r/c which more than .. Table 5. Comparison of efficiency r/and approximation ~ for different compensates the decreased supply of stress wave energy values of joint-to-hammer length ratio .,;.j and loading parameter /L to the bit. Therefore the efficiency r/increases with t-he N ~ = 9 , Z ~ = 2 , 6 = 0 and 7 =0.1. number of joints Nj. The observed fact that the difference between theo0.2 0.1 0.437 0,152 0.35 efficiencies r/+ and r/ increases with the loading par0.2 I 0.578 0.487 0.84 0.2 10 0.177 0.146 0.82 ameter/3 is due to the increased compressive part in the 0.5 0.1 0.280 0.053 0.19 stress wave which is reflected at the drill bit. This 0.5 t 0.370 0.253 0.68 compressive part of the reflected stress wave is reflected 0.5 10 0.105 0.079 0.75 l 0. I 0.209 0.019 0.09 and dispersed by each joint. Finally, parts of it may be 1 1 0.222 0.087 0.39 trapped by the hammer which will then get a backward 1 I0 0.042 0.045 1.07 momentum and separate from the adapter. From the
LUNDBERG:
E F F I C I E N C Y OF PERCUSSIVE D R I L L I N G
221
As heating of joints is an important problem in percussive drilling of rock with extension rods, efforts should be made in future work to incorporate friction and slip with the joint model.
In the elementary process of transmission s h o ~ n in Fig. Alb. a transmitted stress wave is generated. Generally this stress wave is longer, has lower amplitude and carries less energy than the incident stress wave. The efficiency of energy transmission tin brief "'transmission efficienc}"), defined as the ratio of transmitted to incident energy, is [8]
Ackno~dedgements--This work is part of a project supported by the Swedish Board of Technical Development (STU). The author is in debt to STU for this support, and to Mr Arne Nilsson lbr excellent programming.
tit = I - 2p:(l - p : " ) (t - [ ? ) m
Receiced 5 February 1986; recLved 26 Norember 1986.
REFERENCES 1. Fairhurst C. Wave mechanics of percussive drilling. Mme Quarry Engng 27, 122 130 (1961). 2. Lundberg B. and Lesser M. On impactor synthesis. J. Sound Vibr. 58(1), 5-[4 (1978). 3. Fischer H. C. On longitudinal impact I. Appl. Scient. Res. A8, 105-139 (1959). 4. Fischer H. C. On longitudinal impact II. Appl. Scient. Res. A8, 278-308 (1959). 5. Durra P. K. The determination of stress waveforms produced b? percussive driIl pistons of various geometrical designs. Int. J. Rock Mech. Min. Sci. 5, 501 518 (1968). 6. Lundberg B. Microcomputer simulation of longitundinat impact between nonuniform elastic rods. I J M E E 9, 301 315 (198l). 7. Takaoka S., Hayamizu H. and Misawa S. On the reflection of elastic waves at a rod joint. J. 3tin. Metall. Inst. Japan 74, 7 I2 (t958). 8. Lundberg B. Energy transfer in percussive rock destruction--Ill. Stress wave transmission through joints. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 10, 421 435 (1973). 9. Lundberg B.. Gupta R. and Andersson L.-E. Optimum transmission of elastic waves through joints. Ware Motion 1, 193-200 (1979). 10. Gupta R. Optimum design of wave transmitting joints. Wave Motion 4, 75--83 (1982). 11. Simon R. Transfer of the stress wave energy in the drill steel of a percussive drill to the rock. Mr. J. Rock Mech. Atin. Sci. I, 397"-t,1 l (1964). 12. Hustrulid W. A. and Eairhurst C. A. A theoretical and experimental study of the percussive drilling of rock. Int. J. Rock Mech. AIin. Sci. 8,311 333; 335-356 (1971): 9 , 4 1 7 - 4 2 9 : 4 3 1 4 4 9 ( 1 9 7 2 ) . 13. Lundberg B. Energy transfer in percussive rock destruction--l. Comparison of percussive methods. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 10, 381--399 (I973). 14. Lundberg B. Energy transfer in percussive rock destruction--ll. Supplement on hammer drilling, lnt. J. Rock Mech. Min. Sci. & Geomech. Astr. 10, 401-419 (1973). 15. Lundberg B. Microcomputer simulation of stress wave enegy transfer to rock in percussive drilling. Mr. J. rock Mech. Min. Sci. & Geomech. Abstr. 19, 229-239 (1982). 16. Lundberg B. Microcomputer simulation of percussive drilling. 1hr. J. Rock Mech. Min. Sci. & Geomech. Abstr. 22, 237-249 (1985). 17. Lundberg B. and Karlsson L. G. Influence of geometrical design on the efficiency of a simple down-the-hole percussive drill. Int. J. Rock Mech. Mm. Sci. & Geomech. Abstr. 23, 281-287 (1986).
APPENDIX
where p = (Zj - 1)(Zj ~ 1) and m = I ).j = l. 2 . . . . TEe result for 2j = 1 is valid also for 21 > I. From this result, there tbllows that th decreases with both 2j~ [ . . . . 12, I~,• and Zj s[1, ~ ) . Also, 2 j ~ 0 or Zj ~ l implies rh ~ 1. Thus, the transmission efficiency tit is high if the joint is short relative to the incident ~ave (or conversely, if the incident wave is long relative to the joint), or if the characteristic impedance (cross-sectional area) of the joint is dose to that of the rods. Some numerical results for the transmission efficiency r/T are given in Table A1. In the elementary process of conversion of stress wa~e energy, into work shown in Fig. Ale. a reflected stress ,save is generated. The efficiency of energy transfer to the rock (in brief "'conversion efficiency"), defined as the ratio of work to the energ? or the incident wave, is generally a function t/c of Zj, 2j, ~, fi and 7- For Zj = I and (5 = 0 this function takes the simple form [13] P1c=2(1-7)(I -e
From this result, it follows that r/c increases ~ith fi it: the inter~al (0, 1.26) and decreases with fl in the interval ( 1.26, z I. Also, r/c ~ 0 as f l ~ 0 or fl~-z_, and r/cm,~=0.815(l-;.') for [~=1.26. As fl = 2L'Hk'/AE ~ (2L;0k' it also follows that high con;e:sion efficiency (a)
z~
z;
V
L
L' H
J
2C H
v
i
(b)
I
z~ I
L
J
2C~
--
i
1c1
Efficiencies o1" Elementar)" Processes
t
Three elementary processes which are important ingredients in extension drilling of rock are illustrated in Fig. A1. They are: (i) impact between hammer and adapter: (ii) transmission of a stress wave through a joint and (iii) conversion of stress wave energy into w o r k . The drill rods are assumed to be reflection-free at the dashed crosssections. In the elementary process of impact shown in Fig. Ala, a compressive stress wave is generated in the adapter. It is a rectangular pulse with length 2L~ and force amplitude (I/2)Z~, V. The ener_v carried b ; this wave is equal to the impact kinetic energy. Therefore the efficiencY, of conversion of impact kinetic energy into stress wave energy (in brief "'impact efficienc?") is
t
th - I
'~):fi
I
z'~
zCH --i
Fig. A1. Elementary processes in extension drilling of rock. (a) Impact between hammer and adapter. (b) Transmission of a stress v, ave through a joint. (c) Conversion of stress v, ave energ? into work.
222
LUNDBERG:
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for soft rock (k' small) is obtained with a long incident wave (2L~ large). Conversely, high conversion efficiency for hard rock is obtained with a short incident wave. For Zj > I and fi = 0 the dependence of gc on fl is qualitatively the same [15]. Some numerical results for the conversion efficiency g o obtained from simulations with L . = 20, are given in Table A2. Table A1. Transmission efficiency r/r vs joint-to-hammer length ratio (twice the joint-to-incident wave length ratio) 21 • Rectangular incident wave, Zj = 2. ),j 02 0.5 I> 1
r/r 0.955 0.889 0.800
Table A2. Conversion efficiency r/c vs joint-to-hammer length ratio (four times the bit-to-incident wave length ratio) ;.j and loading parameter ft. Rectangular incident wave, Zj = 2. ;.J
3
'c
0.2 0.2 0.2 0.5 0.5 0.5
0. l 1 I0 0.1 1 I0 O.l l 10
0.159 0.737 0.221 0.152 0.730 0.227 0.141 0.647 0.336
1
I 1
tim
5, i & Ge,,mech
4h~'tr Vol 24. No a., pp 2[3 222. 198-
0147<-91)~2'," S 3 0 0 - l ) 0 0 Cop?right ~ !:~,~7 Pergamon Journals Ltd
Printed in G~'at Brlt,xin ~.[1 right. :eser',ed
Efficiency of Percussive Drilling with Extension Rods B. L U N D B E R G +
Extension drilling of rock is simulated numericalh' using a I'ictor desk-top computer. The hammer, adapter and extension rods are tmilbrm amt hare the same characteristic impedances. The same is true jbr the joints and the bit. Two efficiencies are determined as ~mctions of six dimensionless parameters which represent the number of joints, the joint-to-hammer length ratio, the joint-torod characteristic impedance ratio, the initial gap between bit and rock and the bit~rock interaction (two parameters). The first efficiency is defined as the ratio of the work performed on the rock to the impact kinetic energy of the hammer. The second is based on the sum of the work performed on the rock and the rebound khwtic energy of the hammer, which is assumed to he fidl.v retrieved. Both efficiencies generally decrease with the number of joints and the joint-to-hammer length ratio. Exceptionally, however, the efficiencies may increase slight O' with the number of joints. The difference between the second and first efficiency, which is due to hammer rebound, generally increases when the rock becomes increasingly hard. This dependence on rock hardness is strongest when the number of joints and the joint-to-hammer length ratio are low. In the absence of joints the efficiencies have an oscillatory dependence on a small initialgap. When one or several joints are ,tsed, howet'er, the efficiencies essentialO" decrease with the initial gap.
INTRODUCTION Extension rods are used in percussive top hammer drilling of holes in rock to greater depth than the length of a single rod. Commonly, the extension rods are threaded at their ends and are screwed into cylindrical coupling sleeves in firm end-to-end contact with each other. The input end of the first rod is connected to the adapter with the same kind of joint, and the output end of the last rod is similarly screwed into the drill bit. When the hammer of the rock drill hits the adapter, elastic stress waves are generated in the hammer and in the drill rod system. Initially these waves are compressive. When the waves arrive at sections with changes in cross-sectional area or, more generally, in characteristic impedance they are partially reflected and partially transmitted. Thus. reflections occur, for example, at the free end of the hammer and at the adapter/joint, the rod/joint, the rod bit and bit/rock interfaces. Due to the reflection of compressive waves at sections with reduction in characteristic impedance, tensile waves are generated. As a result tensile stresses may occur in certain parts of the rock drill. Often the hammer separates from the adapter and rebounds after a time which is much shorter than the period of percussion. This rebound may occur as a result
+ Department of Mechanical Engineering, T e c h n o l o g y . S-951 87 Lule~,. S w e d e n Rxl~ls :a~ >
Lulefi
University of
213
of reflected compressive waves from the adapter and the first joint only. Then, the rebound is independent of the number of joints and also of the conditions at the bit/rock interface. However, it may also depend on subsequent joints and on the situation at the bit/rock interface. When the hammer separates from the adapter it generally performs a complex motion under the influence of trapped waves. These waves are finally attenuated, and thereafter the hammer leaves the adapter with a constant velocity. As the m o m e n t u m of the hammer is conserved in the absence of external forces, this rebound velocity is equal to the ratio of hammer m o m e n t u m at the time of separation to hammer mass. With a suitable design of the percussive drill, a certain fraction of the kinetic rebound energy can be retrieved. The leading stress wave which reaches the drill bit is generally compressive. It has been modified by the joints on its way to the bit. Therefore it is longer and smoother and has lower amplitude than it would have had otherwise. Also it carries less energy, as a main effect of the joints is to disperse the stress wave energy into the various parts of the rock drill. Dissipation of heat due to friction and slip within the joints is another contribution to the reduction of the energy carried by the leading stress wave. During the reflection of the leading compressive stress wave at the drill bit, the cutters are displaced towards the
214
LUNDBERG:
EFFICIENCY OF PERCUSSIVE DRILLING
rock. After contact has been established they generally which is adequate provided that the incident stress wave penetrate into and crush the rock. Thus, work is per- is much longer than a joint. Paper [8] deals with the formed. The leading compressive wave is followed by transmission of a stress wave through an arbitrary waves which generally have lower amplitudes and which number of identical joints. The results show that the may be tensile as well as compressive. These waves. efficiency of energy transmission at any' joint is higher which result, for example, from multiple reflections than that at its predecessor. Thus, the transmitted wave between the joints, sometimes contribute to the work of is successively modified by the joints in a way which crushing the rock. Finally, and generally before the next favours further energy transmission. The paper [9] deals impact, the waves in the drill rod system are attenuated with the problem of optimizing the shape of the incident mainly due to conversion of stress wave energy into wave such that, for a given single joint, the efficiency of work, dissipation in the joints and material damping. At energy transmission is maximized. Another aspect of this stage, there may remain some kinetic energy associ- optimization is considered by Gupta [10], who deals with ated with motion of the drill rod system away from the the problem of optimizing the characteristic impedance rock. Ideally, the thrust should bring the drill bit back distribution of a single joint with given mass and length in contact with, or at least close to, the rock before the such that, for a given incident wave, the efficiency of energy transmission is maximized. next impact occurs. Two definitions of the efficiency of the drilling process Conversion of stress wave energy into work, finally, are natural. First, the efficiency can be defined as the has been dealt with by a number of authors, for example ratio of the work to the hammer impact kinetic energy. Simon [11], Hustrulid and Fairhurst [12] and Lundberg This efficiency ~/is useful if the hammer rebound kinetic [13-15]. Their studies show that the efficiency of energy energy is not retrieved. Secondly, the efficiency can be transfer to the rock may be as high as 80-90%. In the based on the sum of the work and the rebound kinetic paper [16], the efficiency ~/ of the complete process of energy. This efficiency ~/÷ is adequate if the rock drill percussive drilling is determined for a commercial perdesign is such that the kinetic rebound energy is com- cussive drill, namely an Atlas Copco COP 1038 HD. pletely retrieved. It follows that r/+ exceeds q by e-', The present study concerns the complete process of where e is the coefficient of restitution of the hammer. extension drilling of rock. The efficiencies q and r/+ are It is evident that the following elementary processes determined for a variable number of joints Nj, for are important ingredients in extension drilling of rock: different joint-to-hammer length ratios ).j and for (i) impact between hammer and adapter; (ii) trans- different conditions at the bit/rock interface. Thus, the mission of a stress wave through a joint; and (iii) influence of the dimensionless initial gap 6 [16] and the conversion of stress wave energy into work. In the dimensionless loading parameter fl [13-16] is studied in literature it has been more common to treat these detail. The elastic "swell" model mentioned above is elementary processes separately than to consider the used for the joints which means that the effects of complete process of extension drilling. However, there is friction and slip are neglected. The numerical results are a complex interaction between these elementary pro- obtained with the aid of simulation programs written in cesses and therefore the complete drilling process is not ~.Turbo Pascal for the Victor desk-top computer. These programs are extended versions of the Apple Pascal generally related to its parts in a simple way. Impact between a hammer and an adapter, or a programs DTHD-SIM, DTHD-EFF, HD,SIM and similar device, has been treated in numerous papers. H D - E F F presented in [16]. Commonly I-D models have been used in conjuction with analytical [1,2], graphical [3, 4] or numerical [5, 6] MODEL AND PARAMETERS methods of analysis. The papers [1, 3-6] treat the direct The simulation model for the percussive drilling problem of determining the stress wave produced by process is illustrated in Figs 1 and 2. All quantities a given hammer, whereas the paper [2] concerns the shown are dimensional. Primes (') are used for some corresponding inverse problem. ` of these quantities in order to distinguish them from Transmission of stress waves through joints has been corresponding dimensionless ones given less attention in the literature. Takaoka et al. [7] Figure 1 illustrates the 1-D model of the percussive published experimental results (in Japanese) which were drill to be studied. Figure la shows a uniform rod partly reproduced and further discussed by Fairhurst [1]. configuration and Fig. lb shows an extension rod It was found that the relative energy loss in the trans--configuration with Nj joints. In what follows, Nj = 0 will mission process was 7.5-18.5% for a single joint. This refer to the uniform rod configuration and Nj = 1.2,. loss is due to reflection and dissipation in different to extension rod configurations. proportions for various types of joints investigated; The hammer and the adapter have the same constant Early theoretical studies were made by Fischer [4] who characteristic impedance Zk as the rods, and the bit has used a 1-D model and treated a joint as an elastic the same constant characteristic impedance Z~ as the "swell", that is, as a local increase in characteristic joints. The length of the hammer ts Lh and that of a joint impedance. Thus, friction and slip, which give rise to is L~. The length of an extension rod, exluding the end dissipation, were not considered. The same model was parts which are screwed into either a joint or the bit, is used by, for example, Lundberg [8] and Lundberg et al. 10L~. The length of the adapter external to the first joint [9]. In their papers, use is also made of a rigid joint model
LUNDBERG:
EFFICIENCY OF PERCUSSIVE DRILLING
2[5
e,/
t
t
i
,¢
'2Lj =
eV
(b)
I
Fig, 1. Percussive drill model. (a) Uniform rod configuration. (b) Extension rod configuration.
is Lj. The length of the bit is (1/2)Lj. The length of the integral rod external to the bit is 12Lj. This implies that the total length L' of the rock drill (between the extreme ends of the hammer and the bit) is the same for Nj = 0 and Nj = 1. The material is the same throughout the system and is characterized by the Young's modulus E and the elastic wave speed c. Thus, denoting the common crosssectional area of the hammer, adapter and rods by A, we have Z~, = AE/c. The impact velocity of the hammer is V. The drill rod system is initially quiescent. Also, the hammer and the drill rod system are initially free from stresses. The bit/rock interaction is represented in Fig. 2 by a relation between the force F' acting on the rock and the history of penetration u'. This relation is characterized by the initial gap between bit and rock u~ and by the penetration resistance (slopes) k ' and k;. The system is characterized by the six dimensionless parameters N,,
2 , = Lj/L~,
= u~iu**,
Z j = Zj/Z'R
# = k',./,-**,
7 = k'/k;
where [16] ,,** = (2L~/c)V,
k** = AE/2L'H.
The parameters ).j and Zj represent the length and the characteristic impedance, respectively, of each of the N~ joints. These parameters also represent the length of the adapter and the characteristic impedance of the bit, respectively. As the material is the same throughout the system, Zj can also be considered to represent the cross-sectional area of the joints and the bit. 6 represents the initial gap between bit and rock, and fl and 7represent the bit/rock interaction. After sufficiently long time, the work W ' = ~F'du" performed on the rock will remain constant. Also, the hammer will rebound with velocity eV, where e is the coefficient of restitution, and there will be no reestablished contact between hammer and adapter (up to this time there may have been a number of separations
and re-established contacts). Then the kinetic energy associated with rebound is e:Wf<, where W~ is the kinetic impact energy of the hammer. The efficiency of the drilling process is defined as ,7 = w ' / w ' ~
or ,7 + = ( w ' +
e-'W;:),.'w~.
From these definitions there follows the relation r/+ = r/ + e: already mentioned. In [17], the slightly more general efficiency r/ + 0e 2 was introduced, where 0 is the fraction of the kinetic rebound energy which can be retrieved. The efficiencies r/ and q* are functions of the six dimensionless parameters .A,~e {0, 1. . . . I', 21E [0, CC), Zje [1, oo), 6 • [0, ~), fl • [0, m) and 7 • [0, 1]. The meaning of given values of Nj. ).j, Zj and ",' is readily understood. The significance of prescribed values of and fl is less easily assessed but was thoroughly discussed in [16]. For a semi-infinite system (Nj--, :z) which is perfectly uniform (Zj = 1) 5 = 1 represents an initial gap which is closed exactly at the moment when the reflection of the incident stress wave is completed. Thus, the gap is closed but no work is performed. For the same semi-infinite
F'
J
Fig. 2. Bit/rock interaction model.
216
LUNDBERG:
E F F I C I E N C Y OF PERCUSSIVE D R I L L I N G
and uniform system but in initial contact with the rock (~ = 0) the efficiency rl is high when /~ is of the order of unity (rlm~,= 0.815(1 - 7 ) for /~ = 1.26) but is low (r/<< 1) when /~ << 1 or /~ >>1. In this sense fl << 1, /~ - 1 and /~ >>1 represent soft, medium hard and hard rock, respectively. The numerical procedures used are described in [16]. The rock drill is subdivided into L segments, each with length d = L'/L and constant characteristic impedance. Correspondingly there are the relations Lh = LHd for the hammer and L5 = L j d for a joint, where LH and L j a r e the number of segments used to represent the hammer and a joint, respectively. The length of a time step is the transit time h = d/c for a stress wave through a segment. Numerical approximations are used for determination of the moments of contact re-establishment between hammer and adapter or integral rod, and also for the bit/rock interaction. Otherwise exact relations of the I-D theory of elastic stress waves are used. The accuracy of simulation increases with the number of segments used to represent a given system. In [16] it was found that good accuracy is obtained with LH = 5 or even less.
SIMULATION AND RESULTS
Variation of parameters The joint-to-rod impedance ratio Zj and the unloading parameter 7 are kept constant at the representative values Zj = 2 and 7 = 0.1 throughout the simulations. Thus, there remains to study the dependence of the efficiencies r/and r/+ on the number of joints Nj, the joint-to-hammer length ratio ).j, the initial gap 6 and the loading parameter/L For these parameters, the values N, e {0, 1,2, 5,9}, ).j ~ {0.2, 0.5, 1}, 6 ~ {0, 0.1,0.2 . . . . I} and/~ e {0.1, 0.2, 0.5 . . . . ,10} are chosen. In a real drilling process the initial gap 6 and the loading parameter/~ vary in irregular ways which can hardly be predicted or accurately controlled. Therefore it is of interest to determine average values of q and ~ + when one of these parameters varies within its full range given above while the other parameters remain constant. The average values are defined by
Y = .f[ y(x) dx/(b - a) where y = 1/ or r/+, x = 6 or ln(/~), a = xmi,, b = Xm~, and where linear interpolation is used to evaluate the integral from the computed points (x, y).
Number of segments and duration of simulation
-
The number of segments L which can be used to represent the rock drill between the extreme ends of~da¢ hammer and the bit is limited due to the fact that there is only space for 100 segments on the computer display. As each extension rod is represented on the display by only one segment at each of its ends and by one empty segment space between (cf. Fig. 1b), the largest possible number of segments is not fixed but depends on the parameters Nj and 2j. From the computational point of
Table 1. Number of segments used to represent the hammer, L H, a joint, Lj, and the entire rock drill. L, vs number of joints Nj and joint-to-hammer length ratio ,:.j, Nj
~.j
LH
Lj
L
0 0 0 1 1 I 2 2 2 5 5 5 9 9 9
0.2 0.5 1 0.2 0.5 i 0.2 0.5 I 0.2 0.5 1 0.2 0.5 1
60 24 12 60 24 12 50 20 10 30 12 6 20 8 4
12 12 12 12 12 12 10 10 10 6 6 6 4 4 4
70 42 30 210 174 162 285 255 245 369 351 345 422 410 406
view all rod segments are equivalent, however, independently of their representation o n the display. The number of segments used to represent the hammer, LH, a joint, Lj, and the entire rock drill, L, are given in Table 1 for the various combinations of Nj and 2 j . It can be seen that LH varies from 4 to 60, Lj from 4 tO t2 and L from 30 to 422. Each simulation should have sufficient duration to allow the work W' and the coefficient of restitution e to assume their final values. Then correct efficiencies ~/and + are obtained. In this study, a duration of simulation of six stress wave transits through the entire system is considered adequate. This corresponds to the dimensional time 6L'/c or to 6L time steps (cf. Table 1). It should be pointed out, however, that this duration may not be strictly sufficient in every simulation, In exceptional situations, therefore, one or both of the efficieneies may be somewhat underestimated.
Results of simulation The results of the simulation are shown in Figs 3-9. Results for ~/and r/- are represented with open and filled symbols, respectively. Squares, circles and triangles are used for ).j = 0.2, 0.5 and 1, respectively. First it is assumed that there is initial contact between bit and rock, that is, 6 = 0. Figure 3 shows the dependence of the efficiency 17 on fl, N j and ).j for Zj = 2, 6 = 0 and "; = 0.1. Generally q first increases and then decreases with /~ such that ~/,.~ is assumed for/~ e {0.5, 1, 2} (medium hard rock). When Nj and ).j assume larger values (larger number of joints which are longer relative to the hammer), r / ~ tends to be assumed for lower values of ~ (softer rock). Also, r/ generally decreases with both Nj and 2j. The decrease with Nj is commonly faster for low Nj than for high. Notice that exceptionally (;tj = 1,/~ = 0.2, Fig. 3c) r/may increase slightly when Nj is increased from one value (Nj = 5) to another (Nj = 9). Figure 4 shows the dependence of the average efficiency r7 on Nj and ~.j for Zj = 2, 6 = 0 and 7 = 0.1. Each point in Fig. 4 is obtained from a curve in Fig. 3. The following observations correspond to some of those
LUNDBERG:
EFFICIENCY
made directly from Fig. 3: (i) ~ decreases with N;, and this decrease is generally faster for low Nj than for high; (ii) r7 decreases with ,:.j for :Vj = 1, 2 . . . . . 9. The results plotted in Fig. 4 are gixen also in Table 2. Figure 5 shows the dependence of the efficiency r/+ on '8, Nj and ,;.j for Z j = 2 , ~ = 0 and 7 = 0 . 1 . From a comparison with Fig. 3, there follows that the difference between r/- and r/generally increases (hammer rebound becomes more important) when '8 increases (harder rock). This dependence on ,8 is stronger when -\:3 and ,:.j assume lower values (fewer joints which are shorter relative to the hammer).
OF PERCUSSIVE
DRILLING
217
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I ~
I0
# Fig. 3. Efficienc? q vs loading parameter fl and n u m b e r of joints ,¥j for j o i n t - t o - h a m m e r length ratio (a) ,:.j = 0.2, (b)),j = 0.5 and (c) ,,;.j = 1. Z;-" (5=0and-=0.1
-~ 0 . 5 ]
[
X
I
i
O[ 0.1
I
I
I
0.2
O. 5
1
2
1
]
5
10
Figure 6 shows the dependence of the average efficiency 0 - on Nj and ).j for Zj = 2, 6 = 0 and 7 = 0.1. Each point in Fig. 6 is obtained from a curve in Fig. 5. F r o m a comparison with Fig. 4 it can be assessed that r7+ and r7 depend on :\j and 2j similarly. However, r~+ decreases with 2; more slowly than r7. The results plotted in Fig. 6 are given also in Table 3. 1.O~ I I
&i2
(b) 1.0
I ( [0.1 , 10]
X • 0.5
X "0.1
Zj,2 ,0 X " 0.1
j=O,2
0.5
'"7
I
0
0.!
I
I
5
I
I
!
I
I
10
%
i
O
I
i
1
02
0 .5
I
I
I
I
1
2
5
10
#
Fig. 4. Average efficiency ~ with respect to ln(fl) (or ~ ~[0. I, I] vs n u m b e r of joints ,Vj and j o i n t - t o - h a m m e r length ratio ,;.;. Zj = 2, 6 = 0 and 7 = 0 . l.
218
LUNDBERG:
E F F I C I E N C Y O F PERCUSSIVE D R I L L I N G
Table 2. Average efficiency # with respect to lnffl) for fl~[0.1, 1] vs number of joints Nj and joint-tohammer length ratio 21. Zj = 2, 5 = 0 and 7 = 0.1.
(b] 1.0FI
~,j " 0 . 5
L zj-2 B-O
# -Vj
;.j = 0.2
,;,j = 0.5
,;.j = 1
0 1 2 5 9
0.629 0.599 0.565 0.504 0.469
0.647 0.539 0.475 0.360 0.288
0.632 0.468 0.356 0.232 0.188
!
r? ÷
Finally the influence of initial gap 6 between bit and rock is studied. Figures 7 and 8 show the dependence of the efficiencies 17 and 17+, respectively, on 5 and Nj for 2j = 0.5, Zj = 2, fl = 1 and 7 = 0.1. For N j = 0 t7 and 17+ oscillate with 6, but for Nj > 0 they essentially decrease with 6. Figure 9 shows the dependence of the average efficiencies ~ and ~ + on Nj for the same values of 2 j , Z j , fl and 7. Each point on the curve for ~ is obtained from a curve in Fig. 7. Similarly, each point on the curve for + is obtained from a curve in Fig. 8. The results plotted in Fig. 9 are given also in Table 4.
7 ,0.1
0.5
0.1
I
I
1
1
I
0.5
1
2
5
10
I 2
I 5
I 10
(c)
1.0
DISCUSSION
I
0.2
Xj=I
A useful background for interpretation of the results obtained from the simulations is provided by previous results [8, 13] for the three elementary processes mentioned in the Introduction: (i) impact between hammer and adapter; (ii) transmission of a stress wave through a joint; and (iii) conversion of stress wave energy into work. Impact efficiency r/i, transmission efficiency 17Tand conversion efficiency 17o respectively, and also some other results for these elementary processes are put together in the Appendix.
r'0.1
Nj,0
/
•
1
r/+ 0.5
(a) 1.0 Xj=0.2
Zj,Z F 8-o
Nj,0
.//m
OL
0.1
I 0.2
I 0,,5
I 1
Fig. 5. E~ciency ~/+ vs loading parameter fl and number of joints Nj for j o i n t - t o - h a ~ length ratio (a) ,;.j = 0.2; (b) 2j = 015 and (C) 2j = 1. Z~= 2, ~ =0 and 7 =0.1.
'r/+ 0.5
0.1
0.2
0.5
1
B
2
5
10
It can be established from Fig. 3 t h a t the efficiency 17 of the complete drilling process depends on the loading parameter fl in a similar way as the conversion efficiency 17c. Thus, both 17and 17c assume their maxima for values of fl of the order of unity (medium hard rock). Also, both efficiencies are low when fl is small (soft rock) or large (hard rock). In the former case, the leading incident compressive wave is reflected mainly as a tensile wave at the bit/rock interface (nearly free end); in the latter case,
LUNDBERG:
EFFICIENCY OF PERCUSSIVE DRILLING
219
1.o Z'2
X j-0.5
BE Co.I,~O~
B
zj,2 i
]v+:c
,o
,F , 0 . 1
iI
~'o~-
~
--------------____~
I
'
I
I
t
;
o.5
I
I
5
:9
~+
10
0
~ "-''~ -
i
i
i
i 0'5
1.0
%
Fig. 6. Average efficiency ~- with respect to In(fl) for fl~ [0. l, I] vs number of joints ~ and joint-to-hammer length ratio ).j. Zj = 2, d = 0 and 7 = 0. I.
Fig. 8. Efficiency r/+ vs initial gap 6 and number of joints +vj. ,;.j = 0.5, Z j - _ . fl = l and ;' =0.1.
Table 3. Average efficiency~i- with respect to Ln (fl) For fl ~ [0.1, [] vs number of joints Nj and joint-tohammer length ratio ,:.j. Zj =-~,~ ~5= 0 and 7 = 0. l.
it is reflected mainly as a compressive wave (nearly fixed end). As the loading parameter fl can be expressed fl = 2 L ' H k ' / A E the intermediate " m a t c h e d " conditions can be obtained by choosing a suitable value for k'. The main effect of" the joints is to decrease the efficiency ~1- As can be seen in Figs 3 and 4 this decrease in efficiency is more important for large values o f the j o i n t - t o - h a m m e r length ratio ,:.j than for low values. This dependence on 2; corresponds to that of the transmission efficiency ~/T, which is a decreasing function of 2j. Thus, a small value o f the parameter ,;-s = L~,'L;{ is favourable. It can be realized by using a short joint or a long h a m m e r or both. A n o t h e r effect of the joints, which is demonstrated by Fig. 3, is the removal o f the m a x i m u m for ~1 to lower values o f ft. With the aid of the Appendix this can be explained as follows. In the absence o f joints the conversion efficiency tic assumes its m a x i m u m value when fl = 2 L ' n k ' / A E assumes a value of the order o f unity. 2L~ is the length of the incident stress wave. If joints are presents, however, the incident wave at the bit will be longer than 2L~. It will also have a different shape, which is less important. Therefore, with joints present, a lower value of fl will correspond to an incident stress wave at the bit with such a length that the conversion efficiency Y/c is maximum. Therefore the m a x i m u m of the efficiency ~/ is also removed to lower values of ft. It is also found from Figs 3 and 4 that the decrease in efficiency r/ with the number of joints Xj is generally faster for low/Vj than for high. This is in correspondence with previous results [8] for the efficiency of energy transmission through a chain o f rigid joints. At first it might appear strange that there exist conditions which make the efficiency ~1 increase with the
3,'j 0 l 2 5 9
,~j = 02 0.779 0.705 0.604 0.512 0.475
,;.; = 0.5 0.765 0,574 0.509 0.394 0.323
,:.j = I 0.740 0.579 0.467 0.343 0.299
1.0'j l i
X "0,5 Zu'2
,8=I ii
7" =0.1
l
9
J i 0
l
L
I
L 0.5
1
I
I
I
l 1.0
Fig. ?. Efficiency r/ ',s initial gap 6 and number of joints Xj. ,:.j = 0.5. Z j - - 2. fl = l and 7 = 0 . 1 .
220
LUNDBERG:
1.0
E F F I C I E N C Y OF PERCUSSIVE D R I L L I N G
ks, 0.5 2'.,2 a ( [o,q
3"I X "0.1
O.
77
I
I
I
I
I
5
I
I
I
t ..... 1
10
Fig. 9. Average efficiencies ~ and r7+ with respect to 6 for 6 ~[0, 1] vs number of joints Nj. 2j = 0.5, Z 1 = 2, fl = 1 and 7 = 0.1.
Table 4. Averageefficiencies~ and r1+ with respect to 6 for 6 e [0, 1] vs number of joints Nj.
/.j=0.5,
Zj =2,
fl--" 1
"/=0.1.
0 1 2 5 9
0.759 0.551 0,430 0.264 0.236
0.766 0.586 0.464 0.299 0.271
and
discussion above, it is clear that the influence of the joints is less important when the parameters Nj and 2j assume lower values. Therefore it is also understood that the difference between the efficiencies *l- and rl depends more on fl when Nj and :.j are low; a long chain of long joints effectively acts as a shield between the hammer and the bit which makes the hammer rebound less dependent on the conditions at the bit'rock interface. It is not difficult to understand why an initial gap 6 between bit and rock may increase or decrease the efficiencies r/and r/* in the absence of joints, whereas it mainly has a negative effect in the presence of joints. The reason is as follows. As long as there is a gap between bit and rock the incident compressive stress wave is completely reflected, mainly as a tensile wave. The shape of the reflected wave differs somewhat from the shape of the incident wave due to the geometry of the bit. In the absence of joints the leading compressive part of the reflected wave is trapped in the hammer and causes a rebound, but the remaining part is reflected again and returns intact, mainly as a compressive wave, at the bit where it may either further reduce the gap or contribute to the work. In the presence of joints, the corresponding doubly reflected wave is largely dispersed within the rods because of the reflections at the joints. Therefore its leading part has less ability to produce work when it arrives at the bit. It is noticed that ~ = r/l(r/T)'~ r/C is a simple-minded approximation of the efficiency r/. The impact efficiency is rh = 1 while the efficiencies r/T and ~/c are given in Tables A1 and A2, respectively. The approximation neglects the following effects: (i) the influence on the length and shape of the transmitted wave by each joint; (ii) the multiple reflections of stress waves within the adapter and the extension rods. The comparison of r/ and ~ in Table 5 indicates the importance of the neglected effects under various conditions. For the lowest joint-to-hammer length ratio (2j = 0.2) and medium hard (fl = 1) or hard (/3 = 10) rock the approximation is reasonably accurate (r7 ~ 0.8 r/) and therefore the ignored effects have little importance. For the highest joint-to-hammer length ratio (,;.j= 1) and soft rock (/3 =0.1), however, the approximation ~ leads to an underestimation of r/ by more than a factor of 10 (r7 ~ 0.09 r/). Consequently the ignored effects are important in this case. These observations can be well understood from arguments given in the discussion above.
number of joints Nj. With the aid of the Appendix, however, this phenomenon can be explained. Notice in Fig. 3c that it occurs for the relatively low value 0.2 of the loading parameter/3. Therefore, when the number of joints is only Nj = 5 the leading incident wave at the bit is relatively short and consequently the conversion efficiency r/c is relatively low. When the number of joints increases to Nj = 9 less energy is carried to the bit by the leading wave. However, this wave is longer than with N~ = 5 and therefore the conversion efficiency r/c is increased. In the example of Fig. 3c, which appears to be exceptional, the increased number of joints leads to an increased conversion efficiency r/c which more than .. Table 5. Comparison of efficiency r/and approximation ~ for different compensates the decreased supply of stress wave energy values of joint-to-hammer length ratio .,;.j and loading parameter /L to the bit. Therefore the efficiency r/increases with t-he N ~ = 9 , Z ~ = 2 , 6 = 0 and 7 =0.1. number of joints Nj. The observed fact that the difference between theo0.2 0.1 0.437 0,152 0.35 efficiencies r/+ and r/ increases with the loading par0.2 I 0.578 0.487 0.84 0.2 10 0.177 0.146 0.82 ameter/3 is due to the increased compressive part in the 0.5 0.1 0.280 0.053 0.19 stress wave which is reflected at the drill bit. This 0.5 t 0.370 0.253 0.68 compressive part of the reflected stress wave is reflected 0.5 10 0.105 0.079 0.75 l 0. I 0.209 0.019 0.09 and dispersed by each joint. Finally, parts of it may be 1 1 0.222 0.087 0.39 trapped by the hammer which will then get a backward 1 I0 0.042 0.045 1.07 momentum and separate from the adapter. From the
LUNDBERG:
E F F I C I E N C Y OF PERCUSSIVE D R I L L I N G
221
As heating of joints is an important problem in percussive drilling of rock with extension rods, efforts should be made in future work to incorporate friction and slip with the joint model.
In the elementary process of transmission s h o ~ n in Fig. Alb. a transmitted stress wave is generated. Generally this stress wave is longer, has lower amplitude and carries less energy than the incident stress wave. The efficiency of energy transmission tin brief "'transmission efficienc}"), defined as the ratio of transmitted to incident energy, is [8]
Ackno~dedgements--This work is part of a project supported by the Swedish Board of Technical Development (STU). The author is in debt to STU for this support, and to Mr Arne Nilsson lbr excellent programming.
tit = I - 2p:(l - p : " ) (t - [ ? ) m
Receiced 5 February 1986; recLved 26 Norember 1986.
REFERENCES 1. Fairhurst C. Wave mechanics of percussive drilling. Mme Quarry Engng 27, 122 130 (1961). 2. Lundberg B. and Lesser M. On impactor synthesis. J. Sound Vibr. 58(1), 5-[4 (1978). 3. Fischer H. C. On longitudinal impact I. Appl. Scient. Res. A8, 105-139 (1959). 4. Fischer H. C. On longitudinal impact II. Appl. Scient. Res. A8, 278-308 (1959). 5. Durra P. K. The determination of stress waveforms produced b? percussive driIl pistons of various geometrical designs. Int. J. Rock Mech. Min. Sci. 5, 501 518 (1968). 6. Lundberg B. Microcomputer simulation of longitundinat impact between nonuniform elastic rods. I J M E E 9, 301 315 (198l). 7. Takaoka S., Hayamizu H. and Misawa S. On the reflection of elastic waves at a rod joint. J. 3tin. Metall. Inst. Japan 74, 7 I2 (t958). 8. Lundberg B. Energy transfer in percussive rock destruction--Ill. Stress wave transmission through joints. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 10, 421 435 (1973). 9. Lundberg B.. Gupta R. and Andersson L.-E. Optimum transmission of elastic waves through joints. Ware Motion 1, 193-200 (1979). 10. Gupta R. Optimum design of wave transmitting joints. Wave Motion 4, 75--83 (1982). 11. Simon R. Transfer of the stress wave energy in the drill steel of a percussive drill to the rock. Mr. J. Rock Mech. Atin. Sci. I, 397"-t,1 l (1964). 12. Hustrulid W. A. and Eairhurst C. A. A theoretical and experimental study of the percussive drilling of rock. Int. J. Rock Mech. AIin. Sci. 8,311 333; 335-356 (1971): 9 , 4 1 7 - 4 2 9 : 4 3 1 4 4 9 ( 1 9 7 2 ) . 13. Lundberg B. Energy transfer in percussive rock destruction--l. Comparison of percussive methods. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 10, 381--399 (I973). 14. Lundberg B. Energy transfer in percussive rock destruction--ll. Supplement on hammer drilling, lnt. J. Rock Mech. Min. Sci. & Geomech. Astr. 10, 401-419 (1973). 15. Lundberg B. Microcomputer simulation of stress wave enegy transfer to rock in percussive drilling. Mr. J. rock Mech. Min. Sci. & Geomech. Abstr. 19, 229-239 (1982). 16. Lundberg B. Microcomputer simulation of percussive drilling. 1hr. J. Rock Mech. Min. Sci. & Geomech. Abstr. 22, 237-249 (1985). 17. Lundberg B. and Karlsson L. G. Influence of geometrical design on the efficiency of a simple down-the-hole percussive drill. Int. J. Rock Mech. Mm. Sci. & Geomech. Abstr. 23, 281-287 (1986).
APPENDIX
where p = (Zj - 1)(Zj ~ 1) and m = I ).j = l. 2 . . . . TEe result for 2j = 1 is valid also for 21 > I. From this result, there tbllows that th decreases with both 2j~ [ . . . . 12, I~,• and Zj s[1, ~ ) . Also, 2 j ~ 0 or Zj ~ l implies rh ~ 1. Thus, the transmission efficiency tit is high if the joint is short relative to the incident ~ave (or conversely, if the incident wave is long relative to the joint), or if the characteristic impedance (cross-sectional area) of the joint is dose to that of the rods. Some numerical results for the transmission efficiency r/T are given in Table A1. In the elementary process of conversion of stress wa~e energy, into work shown in Fig. Ale. a reflected stress ,save is generated. The efficiency of energy transfer to the rock (in brief "'conversion efficiency"), defined as the ratio of work to the energ? or the incident wave, is generally a function t/c of Zj, 2j, ~, fi and 7- For Zj = I and (5 = 0 this function takes the simple form [13] P1c=2(1-7)(I -e
From this result, it follows that r/c increases ~ith fi it: the inter~al (0, 1.26) and decreases with fl in the interval ( 1.26, z I. Also, r/c ~ 0 as f l ~ 0 or fl~-z_, and r/cm,~=0.815(l-;.') for [~=1.26. As fl = 2L'Hk'/AE ~ (2L;0k' it also follows that high con;e:sion efficiency (a)
z~
z;
V
L
L' H
J
2C H
v
i
(b)
I
z~ I
L
J
2C~
--
i
1c1
Efficiencies o1" Elementar)" Processes
t
Three elementary processes which are important ingredients in extension drilling of rock are illustrated in Fig. A1. They are: (i) impact between hammer and adapter: (ii) transmission of a stress wave through a joint and (iii) conversion of stress wave energy into w o r k . The drill rods are assumed to be reflection-free at the dashed crosssections. In the elementary process of impact shown in Fig. Ala, a compressive stress wave is generated in the adapter. It is a rectangular pulse with length 2L~ and force amplitude (I/2)Z~, V. The ener_v carried b ; this wave is equal to the impact kinetic energy. Therefore the efficiencY, of conversion of impact kinetic energy into stress wave energy (in brief "'impact efficienc?") is
t
th - I
'~):fi
I
z'~
zCH --i
Fig. A1. Elementary processes in extension drilling of rock. (a) Impact between hammer and adapter. (b) Transmission of a stress v, ave through a joint. (c) Conversion of stress v, ave energ? into work.
222
LUNDBERG:
E F F I C I E N C Y O F PERCUSSIVE D R I L L I N G
for soft rock (k' small) is obtained with a long incident wave (2L~ large). Conversely, high conversion efficiency for hard rock is obtained with a short incident wave. For Zj > I and fi = 0 the dependence of gc on fl is qualitatively the same [15]. Some numerical results for the conversion efficiency g o obtained from simulations with L . = 20, are given in Table A2. Table A1. Transmission efficiency r/r vs joint-to-hammer length ratio (twice the joint-to-incident wave length ratio) 21 • Rectangular incident wave, Zj = 2. ),j 02 0.5 I> 1
r/r 0.955 0.889 0.800
Table A2. Conversion efficiency r/c vs joint-to-hammer length ratio (four times the bit-to-incident wave length ratio) ;.j and loading parameter ft. Rectangular incident wave, Zj = 2. ;.J
3
'c
0.2 0.2 0.2 0.5 0.5 0.5
0. l 1 I0 0.1 1 I0 O.l l 10
0.159 0.737 0.221 0.152 0.730 0.227 0.141 0.647 0.336
1
I 1