APPROXIMATE COEFFICIENT ENERGY BETWEEN
METHOD
OF D E T E R M I N I N G
OF R E S T I T U T I O N ,
TRANSFER
FORCE,
IN FREE N O R M A L
THE
TIME,
AND
COLLISIONS
BODIES
B. N.
Stikhanovskli
UDC 531.66
In determining the time, coefficient of restitution, and force, we shall include the cases of longitudinal i m pacts of rods or between rods and plates. The i m p a c t is supposed to be elastic (without a p p r e c i a b l e residual deformations) and free: constraints are imposed only by rests with r i g i d i t y comparable with the r i g i d i t y of the colliding bodies. Friction and resistance (due to air, lubricants, guides for the rods, etc.) are several orders of magnitude less than the forces of impact; therefore, they can be n e g l e c t e d during the collision and the latter can be regarded as free. Finally, if the long rod is at rest and the t i m e of i m p a c t is less than the t i m e for passage of the wave over twice the length o f this rod, then this type of i m p a c t is also similar to a free collision. A l l the relations, some of which were derived in D], are simplified in such a way that the errors are m i n i m a l and at the same t i m e there are no a p p r e c i a b l e departures from experiment or from the theories of Saint-Venant, Hertz, and Sears (within the ranges where these are a p p l i c a b l e ) . The m a i n problem in deriving the formulas is to e l u c i d a t e the effect of each parameter on the overall result without appreciable discrepancies with e x p e r i m e n t for limiting or transitional variation of the parameters. Duration of i m p a c t . Bodies of Like Material. We shall denote the ratio of the t i m e r of i m p a c t to the t i m e (the t i m e for passage of a wave twice along the short rod of length l l ) by n i, where i = 0, 1, 2, 3; a t is the v e l o c i t y of propagation of the wave along the rod, so that 2lt/ai
z :
n~
2ll
(1)
al
a) To find n i for rods of equal cross section, If the second rod is semiinfinite (I3 = ~) [1],
n = - -
l
[I
+3.6(I
--
So /a, ho, l. 1 J'
(2)
for steel,
n =
1.3 + 4 . 5 " [ ~ ' } ~ \ v0 /
So ltR'
(2')
where p I is Poisson's ratio; v0 is the absolute v e l o c i t y of r e l a t i v e approach of the bodies i m m e d i a t e l y after the c o l lision; S Ois the cross-sectional area of the rod with smaller area (S 1 = So i f S I - Sz, or S2 = S0 if S2 -< S1, where S 1 and S2 are the cross-sectional areas of the rods, Fig. lb); ] 1 and | z are the lengths of the rods, where lz ~ I t or Z = l ~ / l I ~- 1; and R = RIR~](R1 + R2) is the reduced radius o f the colliding ends of the bodies, where R1 and R2 are the radii of curvature of the ends of the bodies at the point of impact; ff the configurations of the ends are complex, the determining factors are m a i n l y the values of R1 and Rz at the points of contact. Thus the reduced radius (Fig. l c ) is R = RIRz/(R1 + R2) and not R' = RI'R~'/(R1' +R2').
Novosibirsk F l e c t r o t e c h n i c a l Institute. Translated from F i z i k o - T e k h n i c h e s k i e Problemy Razrabotki Poleznykh Iskopaemykh, No. 1, pp. 70-83, January-February, 1971. Original article submitted April 25, 1969. @1971 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15.00.
57
a
vo --+
,_
zc
d
_,
t_
I~;k:llill/lililii/iil/ll/IJ
7,t
_
u
1
e
."
i
Zl
ze
i I_
S2
V
_j ~ae
f ze ma > mt
Fig. 1 The physical significance of the p a r a m e t e r n is as follows. If the part of the long rod lz which influences the contact of the two bodies during the collision is c a l l e d l c, then n = I c / l t and n is found from (2). The quantity I c Is the part of the length of the semltnfinite rod I z = "* which during the collision exerts Impulsive pressure on the short rod l t. The length I c is equal to h a l f of the length of the pulse propagated in rod 19. Comequently, i f | z > l c (l > n), then
(3)
/~1 - - ?/; (see Fig. l a ) ; and ff
lt ~ L. < lc (l .<. l < rt), (see Fig. ld), then n (n-/ $ l ---~ /~1
--
(4)
l ) + 2:t
n+l
In particular, ff | = 1 (l t = |z), it follows from (4) that /~l ~ no "-"
n(n--l)
+2
(4')
b) T o find n i for bodies with different cross-sectional areas. If S2 ~
S 1 and
l ( n ( oo :">~S
S2s~ ~ / 1 ) ,
(see Fig. l e ) , then, t/t ~
r$2 - - tZl
(S-- 1}(l--no} . ~
3
S
andif]
>n(Z~
,
(5)
]/:--no
- Ic ),then 2 (S - - I) (n -- I) /$2 ~
~
-
-
3
(5')
--
S~'"o (n + I)
If $2 < $I (0 -< S < I) and m I m mz, where m I and m 2 are the masses of the bodies (if both bodies are of the same material, m = m z / m x ~ SIII/S2Z2 = Sl -< i), (see Fig. ib), then, n t = n8 - " In'o,
58
(6)
where i f l - <
l < n n2=
nl--
(l -- S) (l -- no)
(7)
k .o and if ! ~ n
' 2(1 -- S ) ( n - - 1) n2 - - n -31/ n--~ (n + 1)
(7')
If Sz < S, (0 -< S < 1) and m ~ SI > 1 (see Fig. If), then ,
n2
(Sl-- l ) ( l - - S ) ( l - - n S [(---l)z+ 11S"
n ~ = n 3 = ~ +
a)
(8)
If the second rod is at rest and n i m l , the collision is like a free i m p a c t . Coefficient of Restitution. Bodies of Like M a t e r i a l [1]. a) Rods of equal cross section. For elastic i m p a c t (i.e., i m p a c t without a p p r e c i a b l e residual deformations) of bodies with lengths l t and 1c, the coefficient of restitution is constant regardless of the reduced radius R [1]: 1 -- 3 ~ ;
/~c ~
(9)
for steel, k c ~ 0.8. For a collision of I t and lz where l z ~ l c (or n ~ l ) (see Fig. l a ) , the coefficient of restitution k, is found with the aid of the law o f conservation of momentum:
(l ~l) n k c + n - - I
k I --=
'
l(n+
(10)
1)
When l , - < l~ < I c (or n > I -> I) (see Fig. ld), ks =
(l 4- 1) lkc--F (n - - l) (n + l + 1)
(11)
n (n + 1)
b) Rods of different cross sections. IfS,~
Sz o r s = s
z/s,
>-1 (see Fig. l e ) , t h e n
1
k =
(12)
1 - - (1 - - k,)---~-.
If S z > S z or S = S2/S , < I (see Figs. Ib, If), then
k =
1 - - (1
k,)S
--
(l-
[
1) (1 -
(l--t-1) 1=- W
S)
(+;q]
(13)
where
t2 ~ 1 ;
l=t?
q = - - -
t,
Determination of Coefficient of Restitution and T i m e of Impact for Bodies Made o f Different Materials. As the first body we shall take the one for which the time of passage of the wave along it is less than for the second body. Let its length be l p its cross-sectional area S a and so f o r t h Let us reduce the m a t e r i a l of the second body to that
59
of the f i r s t - i.e., let us alter the dimensions of the second body in such a way that the coefficient of restitution, the t i m e of impact, and the force of i m p a c t shall be the same as for the bodies of different m a t e r i a h . This can be done ff we retain the r i g i d i t y of the ends and m a i n bulk of the rod and r e p l a c e the m a t e r i a l of the second body by the m a t e r i a l o f the first. With this substitution of m a t e r i a l , instead of length l s, cross-sectional area S2, and reduced radius R, we shall have
1'2
lo at at
l'=
i'~ 11
p____~_~,S '
S'2 = S., a, a, Pi
--
S~
(14)
$1
R'=R
where 51 = ( 1 - p i~)/Et , Pt is the density o f the m a t e r i a l , and i = 1, 2. Consequently, to d e t e r m i n e k and t for collisions between bodies m~de of different materials, we can use (1)-(13), but with I', s', and R' replacing | , S, and P4 h e r e k = 1-31~ lz, and Pz' = Pl. Impulse and Force of I m p a c t . T h e shape of the pulse will be quite close to r e a l i t y if we express the force of i m p a c t as a function of t i m e in the following form. For S ~ I, 1
Q= A,c[, forS<
_( n#)C];
(15)
1
b
Q-A,bc[1 _(+)c],
(165
where A and A I are constants determined by (22) and (23), C =
nt-- I
(17)
nl where the n i are found from (35-(8), t is the dimensionless t i m e which varies between 0 and ni; I
b --
(n--l)(t
+
o-I)(l-S)
,
(18)
nlo
10 - - S (l o - - 1)
where Z0 = lz/II <- n i and 10 = n i when l z / l l ~ n i, n is found from (2), and S = S2/S 1. The error in determining the force as a function of t i m e Is much lower than when the pulse shape is rectangular, triangular, or sinusoidal. Before finding A andA,, we must c a l c u l a t e the area of the force pulse during an i m p a c t . For the coefficient of restitution given by (95-(14), from the usual formulas of mechanics we find the v e l o c i t y of the centers o f mass of the bodies after the collision [3]:
u,=vl--(l+k)
"'
(v,--v,);
mz -1- m2
u2=v~+(l+k)
m,
(v,--v2),
m t -~ m 2 where v t and v2 are the velocities of the bodies before the collision; the positive direction of v e l o c i t i e s coincides with V,; and the pulse area is P
60
=
m (va --
ul)
--
(1 4- kSmlm2vo , ml + m..
(195
where v 0 is the absolute velocity of r e l a t i v e approach of the bodies before collision (v0 P0 =
=
[ v 1- vzl ), or
(1 -4- k ) m , l+m
(19'5
where m--
- m~ mI
P ml~)0
Starting with (15) and(16), we find the pulse area: For S m 1, nl
AR~ + c
Po : S Q dt =
(20}
(1 + c ) [ c 0 + c ) + 1 1 '
0
for S < 1, nt
A1 b n~ + bc
Po= i Q dt =
(21)
(1 + be) [ c ( l + be) + b]
0
Comparing (19'5 with (205 and (215. we get: For S > 1, A :
(1 + k ) ( l
+e) m[c(1 +c)+l]
(1 + m) n~ +1
,
(22)
for s < 1,
A~ :
[C (1 -t- bc) + b].
(1 q- k) {1 -I- bc) m
(23)
Solving dQ [dt = O, we find the dimensionless t i m e t m a x at which Q = Qmax: For S m 1, C2
"~c
tmax = nl k e-~& l ] ;
(24)
( o r S < 1, C
(25) Putting (24) and (25) into (15) and (16), respectively, we obtain the m a x i m u m dimensionless force: For S =" 1, C
A tinsx
(265
l-bc~ ' for S < 1, cO
Q~a~ = ~At tma x
L+
(27)
c ~-
Multiplying the dimensionless force and t i m e by the appropriate factors, we obtain the force of i m p a c t and the t i m e at Q = Qmax:
61
N--
Q ml fYoal ,"
N=ax = Qm~,
ml voa,
(28)
,"
2 l,
Xmax .-- tmax 2 I.___.LI, al
where a l , 11, and m t are the v e l o c i t y of propagation o f the longitudinal wave and the length and mass of the first (short) body. If the colliding bodies are of different materials, we must reduce the m a t e r i a l of the second body to the f i r s t For i d e a l l y flat ends and i d e a l rod m a t e r i a l (the Saint-Venant c a s e ) a n d S >- 1, we get n = 1, and from (175, (22), and (20), c = 0, and A = (1 + k)m/(1 + m) = P0. We can easily show that when c = 0 from (24) we have Tma x = n i = 1 and from (26) Qmax = A = P0, or from (28), N'-"
Nmax____ {I + k5 m m l v o a l 2 (1 + m) l,
(29)
If we substitute Satnt-Venant's value (for S ~ 15 into (29), i.e., k = 2S('1 + m ) / ( 1 + S)m - 1 [4], r e m e m b e r i n g that m l = | l S l P l and a ~ = E l / p l , we get N :
Nmax :
Sl S~ v,~ El ($1 + $2) al
(29'5
Expression (29') is the well-known relation of the o n e - d i m e n s i o n a l theory o f Saint-Venent, which is an e x a m p l e we have derived as a particular case of ('285. Before i m p a c t the bodies do not possess a store of internal energy which might influence their p o s t - i m p a c t v e l o c i t i e s at fl~e t i m e of collision (contactS; for example, the bodies do not vibrate e l a s t i c a l l y before i m p a c t . From the law of conservation of energy it follows that T O + T ~ T 1 + T2, where T o = mlv12/2, T = mzv2Z/2, and T l = mlulg/2, T2 = m~u22/2 are the kinetic energies before and after i m p a c t . The sign of the inequality shows that after i m p a c t part of the kinetic energy is expended on exciting the bodies (it reappears in the form of e l a s t l c vibrations, plastic deformation, and otherlosses). The t o t a l energy of the bodies after i m p a c t can be divided into k i n e t i c energy of translational motion with the velocities o f the centers o f mass (m I and m2) and into internal energy of the bodies (Eint 1, Eint2 ) relative to moving coordinates linked to the centers of mass m I and m z, i.e., moving after i m p a c t with v e l o c i t i e s u I and u2: To +
T = TI +
T2 + E i n t 1+ E i n t ~
(30)
or
AE---- T,--
T+
Ein tzAE
I--
Ti--
To 4- E i n t l ,
where AE and AE 1 are the energies transmitted to ~he second and first bodies. In the formulas for the energy in the i m p a c t of bodies we usually n e g l e c t t h e energies Eint = Ein t ~ + Eint2. regarding them as lost, i.e., we c a l c u l a t e only AT = T 2 - T, where AT ts the k i n e t i c energy transmitted to mz. An exception is the case of calculations of the pulse energy in one or a few reflections of longitudinal waves. In g e n e r a l there is no functional relation between the total energy AE, the kinetic energy AT, the internal energy Ein t transmitted to mz, and the energy AT 1 transmitted from m~ to m t, where the t i m e of passage along the first body is shorter than that for the second. In contrast to [21 here we shall also derive formulas for AT t, the energy transfer efficiency 0int (internal), and other relations. Each of these quantities has a p e r f e c t l y definite meaning. For e x ample, in vibropercussive, percussive, and rotary-percussive machines with r e l a t i v e l y good contact with the solid m e d i u m , we must take account of the t o t a l energy AE and correspondingly the efficiency O -- AE/T0; for instance, in p n e u m a t i c drills in forging and stamping we must know AEin t, which is here usefully expended, and the kinetic energy after i m p a c t is harmful and m e r e l y shakes the equipment and foundation, and therefore the energy transfer efficiency is Oint = AEint/(T + To); and finally, in percussive machines with trammission of energy through a series o f links or by i m p a c t of a bit on a soft m e d i u m , only AT is usefully expended, and 77k = AT/T~
62
Determination of Energy Transfer and Energ]/Transfer Efficiency, As the first body we take the one in which the t i m e of passage of the wave along it is less than for the second body. Here the internal energy of vibrations after impact Eint 1 for the short body is negligible (an exception is the case of plates and casings, but then Eintz is n e g l i gible); therefore (30) is rewritten as T O+
T =
T1 -+- T2 @ E i n t ~ Ta +
7"2 + E i n t z .
(31)
T 4- Eint-= A T + E i n t == To - - T,,
(32)
The total energy transferred from m t to ms is AE =
T~-
where AT is the kinetic energy transferred from m t to ms and Ein t is the internal energy in the second body. Correspondingly, for transfer of energy from ms to m 1, A E, ~
A T, =
-- A E =
Tt - - To.
(33)
For normal impact v~, vz, ut, and us are directed along the same straight line; when v~ > vz the velocity v2 m a y be either positive or negative, i.e., v = v ~ / v I < 1; when vz > vl, we have v I -- 0 and v2 > 0, i.e., v = v s / v t > 1. If we know analytical methods of determining the value of the coefficient of restitution, or if k is found (for the given geometry, material, and absolute velocity of relative approach) by experiment and analysis, then it is easy to use well-known formulas of mechanics to find the velocities of the bodies after impact, and thus AE, AT, and Eint. Let us use the formulas derived in [2] to find these quantities. The total energy transmitted from m l to mz is -~E~-T0/I-t
[1 L
m m+l
(1 + k ) ( l
-- v)]'~" J'
(34)
the kinetic energy transmitted from m l to ms Is AT=
T,) m v" {[l -t-
1 ~
k
(1
- v
(1 4- m) v
>J } "--I
,
(3s)
where 9 /TIlU ~ To
~ "2
/7l. 2 9 '
/7/.
-m I
9 ,
~e? 7.3
"-"
* ,
',7 t
the internal energy of m~ after impact is m I m~
Eint=
_X E - - • T - -
(36)
2(mr + m:) ( I - - k : ) (v~ - - v.,) '~.
Formula (36) is exactly the same as the formula expressing the vahae of the lost kinetic energy; it can be derived with the aid of Czrnot's theorem. However, the kinetic energy is not always irretrievably lost, but is often useful in transfers of internal and total energy Eint and ~E = AT + Eint, where it may be the largest term. The total energy transferred from m2 to m 1 is equal to the transferred kinetic energy, because we can n e glect the energy Eint i of the mass m 1. For the transfer of energy from the second body with mass m~ to the first body with mass m 1 we can make use of formula (35), reassigning the indexes of 311 the masses and velocities. The axis coinciding with v2 is taken as the positive direction; m 0 = m l [ m 2, v0 = vz/v2. All the subsequent formulas for AT, the transfer efficiency, and the bounds and values of the energy transfer m a x i m a are also valid for AT 1 if i n stead of m and v we read m0 and v0. In this article, for the transfer of energy from ms to m 1 we give all the formulas with the same notation, m = m z / m z, v = vz/v 1, regardless of whether energy is transferred to mz or m 1. The energy transferred from mz to m I ts
m~ +
--
--
v
(37)
mo
63
TABLE 1 T y p e of
Necessary condftfom for energy transfer
energy
m
v
Given
0+~) O-v)-2
Given
m (! + k)
Given
2 .
O
m(1 - - k ) + 2
l>v>--
l--k
--~o
l+k
Given
or
0~m
1--k
l>v>
Given
Given
Given
l+k
k<
v < I
m (1 + v) + 2 m ( 1 - - v)
l+k
l>v>
Given
"Jm + l - - k
Given O~m< AT
(1--k)tl--v)--2 2v
l+k l--k
0>v>
Given
or 0
1>o>/0
Given
Given
v < 1
Given v > Given
k>
--
1 + v(I +2m | --V
1 or
m(I
--k)
Given
4- 2
m(1 +k) 0 ~
Ar~
.m>
m ..< oo
2 (I + k) (1 -- v ) - - 2
Given
v > 1 or 1--k --oo-~ v < 14-k Given
Given
or
fd ~
v > 1 1 --
Given
l>k>
m(1 + v ) + m(1 - - v )
m
oo>~m>~O
Eint
Given
Given
64
Given
co > v >I - - ~
Given
Given
Given.
l > k > 0
TABLE 1 (Cont'd) Ranges o f e x i s t e n c e o f maximum oo > ~ m > . . O
1
Optimal values of parameters
O-.
Maximum energy tramfer efficiency
v - -m k- - --I m(l +k)
1
--
k
"-2" >~ l + k or
m U
1 ~ k > ~
1
--m
v
k -- v (l + k)
>/--1
1
>m>/
1 --' 2v
or
1-+-urn
k -
m(l
- - v)
m~l
1
m
2m
m<2k--1
or
L~k>
l+k
m~-I
m (i + k)
m-k 2m+ i--k
2m+1
--k
l~-k
3+k
1 -k
k > --
1+ l--v
(1 + k ) ( 1 ~ v ) + 2v (1 + k ) ( l -- v ) - - 2 v
V w h e n v < O;
1
-~- [(1 + k) (1 --v) + 2v]~
3v-- 1 k > whenl > v >0 l--v
oo >~ m > ~ O
1--k - l+k
v<
O
or
~>/
mO --k)
I
3+k
~ ) ~ ) 0
U=
m~
l+k
+ 2
m(l ~k)
1 --km
(l--v)
(1 +
m--
1
1
k) + 2
(1 - - v) (1 + k ) - -
2
4v-'-'~[(1 + k) (v - - I) + 21a
(1 - - k~)
U
v
+2
(1 -- v p (1 + v)~
1
O
1 ~
k2
1
m~O
m m~O
oo >/ ~ >_. ~ oo
k=O
m (1 - - v p
(1 + m ) ( l
+ m y ~-)
65
II _ I -"t'~.o....-I 4>10~ ~/-.I/e
0.8L
-J
.'%,
~
I
4Y747
-2
I%o 0
--1
+f
T.~:--~ , .-~_ l\
0 t
Z
~Y
/
to
It]
tO
5
l
m
-g
L
O.d
" o
/
/i
>~F,q-
/
~71
-- I
#
0
O.8 "Ko l ~
_t tO
IS
_t 5
t
.,r...I
5
1o
f
m
7 0,8
e
o.6 o,4
/
0,2
0
0.2
0,4
o,6
o.8
x
0
o, /i 0.2
0.4
o.a
O,d
x
Fig. 2 In transfer of energy from m2 and m I part of the p r e i m p a c t k i n e t i c energy of ms goes to increase the k i n e t i c e n e r g y of m l, and part to increase its own i n t e r n a l energy Ein t. Consequently, for this e n e r g y transfer, the k i n e t i c e n e r g y lost by the second body as a result of the collision has the form AT 1 + Fin t. T h e transfer e f f i c i e n c y for transfer of the total energy from m I to m2 is AE
[
~-1-
~-
t
m ,,,+~ ( l + k ~ ( l - v ) ~ ; ].
(38)
that for transfer of k i n e t i c e n e r g y from m I to m2 is
.
.
~qk
.
.
.
my
To
2
I+
1 -4- k
(1
--
V
--
l
.
(39)
(1 4- m ) . v
T h e transfer e f f i c i e n c y for i n t e r n a l energy is Eint
66
_
m, m 2 ( l -- k - ~ ) ( v ~ - v~?-
=
m (1 -- k 2) (I - - v)* (1 + m) (1 § mv~)
(40)
Tlint 0.~ ~ , ( ~.~', W"
~
K.+-o.4 = ~' ~
/
~.-
/
2 m=0.25
_.-r :~, b2 I'-%,
----... =1
,K----'O
_
,.T x=O
~,~=~ u=_ll
----
4
5
6
7
8
>,"-\
~
I
f
..n ..- ~ ' ~
",',,
~ i-"I-.__I.,_,~:o.~. '~ I
o~
-6
I
.'r~_~
_~ -2
-4
0
.4
2
0---
m~
6
~
1,0 c
~ %,%
0.8
~.~
o.8 0.4
"'~-'o
0.2
m=2
0
0.2
\,C. \
0.4
0.6
0.,$
K
Fig. 3 The efficiency for transfer of energy from the second
~1 :
T
- -m- -y- ~
{[1
bodywith mass mz to the first with mass m t is m +
i
§
--
1}
"
41,
If m = 0 , 9
n, = --:-(1 + k)(v~2
1).
(4v)
Formulas (34) and (38) are valid both for elastic and for viscoelastic collisions, if we can neglect the energies of elastic vibrations, plastic deformations, and other modes of energy dissipation in the first body with mass m I (by the first body we mean that for which the time of passage of the wave along it is less than for the second body). For elastic collision (i.e., without residual deformations), the energy of vibrations in the first body is negligible. Formulas (35), (37), and (39)-(41') are valid for elastic and viscoelastic impact, regardless of the different types of energy dissipation during the impact, i.e., their correcmess is determined only by the accuracy with which the coefficient of restitution is found. In particular, for m = co we have "%m=- -
2 v (1 + k) (1 - - v ) .
(39')
The results of analysis of formulas (34)-(41') are listed in Table 1. In percussive machines and impact testers, one of the colliding bodies will be stationary, i.e., v 1 = 0 (v = v2/vt = i oo), or v 2 = 0 (v = vg/v I = 0); therefore, for these commonly-encountered cases we shall write formulas (34)-(41) with the limitations given in Table 1. The energy
67
~
f-i
a
i I l",t, ".
o,4
) I .\.r
"1",~ ]l -8
-~
-4
i#,~ ~
|0,~ I ., ~
--2
0
o7-9.--
2
~
K--f--
,4
6
t d
~
"
o~ :o,6 o,4
t"
fl
/
:
o,2 o,2
0
~4
o,6
o,6
~
Fig. 4 transfer e f f i c i e n c i e s corresponding to the total, k i n e t i c , and i n t e r n a l e n e r g i e s transferred from m I to ms for v = 0 are as follows: (I -- k m ) ~ . %=0=
"~*v=O--
1 --
(I + m F
(42) '
(1 + k)2 m . (1 + m ) 2 '
(43)
m (1 - - k2)
flinty=~ --
1 + m
When v t = 0 (v -- i ~ ) we h a v e energy transfer from the second body to the first. we g e t 1 -- k ~
~illta~ = 0
(44) Putting v = i oo in (40) and (41),
(45)
l+m (1 + k)~ m
(45')
Expressions (43) and (45') are versions o f the w e l l - k n o w n f o r m u l a o f B. N. Bokii [5], which is a p a r t i c u l a r case of formulas (39) or (41). From (43) and (45') we see that the transfer e f f i c i e n c y for k i n e t i c e n e r g y is i n d e p e n d e n t o f which of t h e bodies is stationary before i m p a c t . Figures 2 - 4 are plots of various curves o f 77, Ok' Tint, and ~z as functions of m, v, and k. If t h e v a l u e s found for m, v, or k are a p p r o p r i a t e l y substituted in the i n e q u a l i t i e s in T a b l e 1, we can e a s i l y find t h e i n t e r v a l s in which energy is transferred from m I to m 2 or from m2 to m l .
~8
In designing percussive machines with the aid of the above formulas, we can make a rational choice of the variable coefficient of restitution, the mass ratio, or the velocity ratio. At maximum efficiency we get low values of harmful energy dissipation in the form of vibratiom or compressive shock waves which are propagated to tile m a chine components and the bases and foundations and spread to neighboring structures and their attendant personnel LITERATURE lo 2. 31 4. 5.
CITED
B. N. Sttkhanovskii, In: Mechanics of MachInes [in Russian], Issue 17-18, Nauka, Moscow (1969). B. N. Stikhanovskil, In: Transactions of Intercollegiate Scientific Conference on Electric Percussive Machines [In Russian], Izd. NTO Mashprom, NTO Gornoe t IGD SO AN SSSP,. Novostbirsk (1967). B. M. Yavorskli and A. A. Detlaf, Handbook of Physics [in Russian], Nauka, Moscow (1964). S. A. Zegzhda, Vestn. LGU, No. 1 (1966). B. N. Bokli, Practical Course of MIning Practice [in Russian], Vol. 2, GITI~ Moscow (1981).
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