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Impact vibration of a bar
Int. J. mech. Sci., Vol. 19, pp. 471-481.
Pergamon Press 1977. Printed in Great Britain
IMPACT
VIBRATION
OF
A BAR
U . B JORKENSTAM Department of Mechanical Engineering, Link6ping University, S-581 83 Link6ping, Sweden
(Received 14 January 1977; in revised form 23 April 1977) Summary--This paper treats the motion and dynamic stability of a bar having constant stiffness and mass distribution when subjected to a periodic driving force, a distributed and linear damping force and impact forces due to its motion and operation (such as would occur on, e.g. the hammer in a rock drilling machine). Comparisons have been made between the characteristics of a bar operated by either pneumatic or hydraulic means.
NOTATION Young's modulus cross-sectional area mass per unit of length length spatial coordinate displacement time force per unit of length load damping factor wave propagation velocity dimensionless spatial coordinate dimensionless displacement dimensionless time dimensionless load dimensionless damping factor dimensionless wave propagation velocity transformed dimensionless displacement A~, B~ constants of integration ¢k phase angle R coefficient of restitution K load factor E A M L X U T F P D C x u t p d s
1. I N T R O D U C T I O N
IMPACT vibration of systems having one or two degrees of freedom has been investigated in a large number of articles. The results have been applied to mainly acceleration dampers ~-4 and vibrating hammers. 6-9 Only in one case has a continuous system been studied) All these systems are loaded by a periodic driving force. However, only the first two terms of its Fourier expansion are considered, i.e. the constant term and the first harmonic term. The motion of the systems has mostly been determined by the method suggested by Warburton) The steady-state solution is sought by replacing the initial value problem approach by a boundary value problem approach, i.e. a boundary value problem is formulated such that the general solution of the equations of motion satisfies conditions at the beginning and at the end of each period between impacts. A relation between the period of the driving force and the impact period has to be assumed. Due to the non-linearity of the problem, it is theoretically possible to have more than one steady-state solution. But not all of the solutions can exist for longer times. Marsi and Caughey 4 have derived a method of finding the asymptotically stable regions of the solution. Several Soviet authors have dealt with symmetrical and asymmetrical two body systems moving subjected to periodic collisions, t°-~5 Kobrinskii has written a textb o o k 16 o n the subject. There the method of analysing the stability of periodic states is based on the application of finite differences. MS Vol. 19, No. 8---C
471
472
u. BJORKENSTAM
This paper treats the motion and dynamic stability of a periodically impacting bar (see Fig. 1). The bar is assumed to have constant stiffness and mass distribution and is loaded by a periodic force and a distributed linear viscous damping force. The methods derived by Warburton, Marsi and Caughey and Kobrinskii are generalized to hold for a continuous system and a general periodic force. There are several technical applications of the model system studied. Here it is used in order to determine the motion and stability of the hammer in a pneumatic and in a hydraulic driven rock drilling machine.
r~P(T)
///////// FIG. 1. The studied model system. 2. MOTION OF THE SYSTEM Assuming the same impact period as force period possible steady-states of motion for the axially loaded bar in Fig. 1 will be derived in this section. The partial differential equation describing the longitudinal motion U ( X , T ) of a thin rod is
EA
02U(X, T) oEu(x, T) OX 2 + F(X, T) = M OT 2
(la)
Here the distribution of mass M and stiffness E A are assumed to be constants of the spatial coordinate X, while the distributed damping force is assumed to be proportional to the velocity of the longitudinal motion, hence F(X, T)= -DaU(X,
T)/aT.
(lb)
For convenience dimensionless quantities are introduced as x = XIL u = UIL t=
TO
(2)
p = p[(ML2fl) d = DI(MII) s = C/(Lfl),
where L is the length of the bar and l~ is the frequency of the first harmonic of the periodic driving force while C = ~ / ( E A I M ) is the longitudinal wave propagation velocity. The equation of motion (1) then transforms into
Impact vibration of a bar
02u(x, t) + dOU(X' t) 0-~ Ot
473
s2 02u(x, t) O. --------T-Ox =
(3)
A steady-state solution will be attempted assuming that the period of impacting equals that of the first harmonic. The time the bar is in contact with the surface is assumed to be much less than the periodic time of the first harmonic of the excitation. With the dimensionless time t counted from one impact at steady-state motion the boundary conditions are, for 0 < t < 2~r,
Ou(x, t)] OX
= I ~=o
(4)
Ou(x, t) = O. OX x=l Because of the nature of the boundary conditions, a solution will be attempted by means of a finite cosine transform defined by l
ti(n, t) = fo u(x, t) cos nlrx dx.
(5)
A transformation of (3) together with the boundary conditions (4) lead to
02a(x, t) Ot 2
+
d
Oa(n, t) + (smr)Za(n, t) Ot
= [Ou(x, t) ]~ I _ - - - ~ x cos nlrx + ncru(x, t) sin ncrx o= p(t).
(6)
The periodic load function may be represented by the real component of the Fourier series
p(t) = f~ Pk e ik"+~),
(7)
k=O
where the complex coefficients Pk contain the information regarding the phase angles of the various harmonics while ~ is the phase lag between the period of the force and the impacting period. The general solution of the ordinary differential equation (6) then is the real component of oo
ti(n, t) = e-dt/2(An e-at+ B. e +ia~)+ ~'~ H,~pk e ik(t+¢')
(8)
k=0
where
Hnk = 1/[(sn~r) 2 + ikd - k 2] A = ~/[(smr) 2 - d2/4].
(9)
According to the method suggested by Warburton 3 the initial value problem approach is now replaced by a boundary value problem approach. This is done by studying the steady-state motion of the system at the beginning and at the end of a period between impacts. If the dynamic stiffness of the surface in Fig. 1 is larger than the dynamic stiffness of the bar then the boundary conditions are
u(x, 0 +) = u(x, 27r-)
Ou(X,ot t) t=o÷ = - R Ou(x,ot t) t=z,,-
(10)
474
U. BJORKENSTAM
where 0 ~< R ~< 1. The time of impact will be equal to 2 L / C , which is the time it takes for a w a v e to travel twice the length of the bar. Since the duration of impact is assumed to be much less than the periodic time of the excitation, the dimensionless stiffness s 2= [C[(L~)] 2 must be much larger than 1/w 2. If, e.g. the surface in Fig. 1 is rigid then R is equal to 1. Assuming on the other hand that the surface is another infinitely long bar with the same modulus of elasticity and the same density as the impacting bar but with a different area As then R = (As - A)/(As + A). Hence, As + A gives R = 0 while As -> A gives R = 1. As has to be larger than A since the dynamic stiffness of the surface is to be larger than that of the bar. The t r a n s f o r m e d boundary conditions are ti(n, 0 +) = ti(n, 2w-) O/~(n, t) = - R Oa(n, t) Ot t:0 + Ot z:2,~-
(11)
where n = 1, 2, 3 . . . . . F r o m equations (8) and (11) it follows that A0 is an arbitrary constant and that Bo = (2 ~rpo/ d)l(1 - e -2,~a) A. = (1 - e-"d+~2"a)Q. B. = - ( 1 - e-"a-i2"A)Q.
(12)
where
Q, = (1 + R ) ~, H,,kpkik eiU'/{(d/2 + ih)(1 + R e-"a-i2~a)(1 - e -~a+i2~A) k=0 -
(d/2 - iA)(1 + R e-~d+iz'~a)(1 -- e-~a-i2~a)}.
(13)
The boundary conditions for n = 0 also give the relation f r o m which the unknown phase angle ~ m a y be determined (1 + R ) ~ Hokpkik eik* = 2wp0(1 + R e-2"d)/(l -- e-2"a).
(14)
k=0
If a value of ~ can be found which satisfies equation (14), then a corresponding one-impact-per-force-period steady-state solution also exists. Knowing the constants of integration the dimensionless displacement is obtained by taking the inverse transformation of (8)
u(x,
t) = a(0, t) + 2 ~ a(n, t) cos n,rx.
(15)
n=l
H e r e a(0, t) is a rigid body motion while the other terms describe the d e f o r m a t i o n of the bar. It follows f r o m (9) and (13) that for an undamped system resonance will occur when s=k/(nzr)
n--1,2,3 .....
(16)
3. STABILITY OF THE MOTION A possible state of motion was determined in the last section. H e r e the stability of that motion will be investigated. The method of analysing the stability of the periodic impact vibrations used here is based on the regularities of the variations in small
475
Impact vibration of a bar
perturbations building up over a number of successive intervals. If the magnitude of the small perturbations remain finite, the corresponding unperturbed motion will be considered stable; if it is discovered that the small perturbations have a tendency to increase infinitely the corresponding unperturbed motion will be regarded as unstable. Following one of the successive impacts, let a small arbitrary disturbance be introduced into the periodic motion, such that the displacement at the next impact differ from the steady-state value u(x, t) = F(x, t, An, Bn, ok). The displacement for the perturbed motion for the ith interval after the initial perturbation can be written u(x, th = F(x, t, An.t, Bn.~, qbt), where An.t= An + AAn.I and Bn.i = Bn + ABn,t are new constants of integration and ~b~= 4~ + A~bt is a new phase lag. If AAn.i, ABn,t and A~t tend to zero when l tends to infinity the disturbed motion will return to its steady-state form and is stable. As before, the dimensionless time is measured from the moment of the last impact. The accumulated effect of the perturbations is given by the perturbations AAn,t and ABn,t while the cumulative perturbation of the phase of the ith impact period is Aq~t = 6~bl + 8~b2+ • • • + &kt. All the quantities AAn.t, ABn.~, Ac~t and &kt are assumed to be small. Hence, terms of the second order of magnitude and smaller, e.g. AAn.t6fft+~, will be neglected in the stability analyses. The condition that the surface is impenetrable gives u(1,0+)~ = u(1, 0 + ),
(17)
while the condition of continuity of two adjacent intervals gives u(x, 0+)l+l = U(X, ( 2 w + 84't+l)-)l
dU(X, t)t+l = - R Ou(x, t)t Ot t=o÷ Ot
(18) t =(2'tr +84,1+1)-"
From equations (17) and (15) it follows that 00
AA0,, + ABo,, + Atb, ~ Hokpkik e 'k* + 2 ~ ( - 1)n k=l
n=l
(AAn,t + ABn,t + A~b~~ Hnkpkik e 'k*) = 0.
(19)
k=l
The transformed initial values (18) give for n = 0 e -2~rd
AAo,t+l - AA0,t +
AB0./+t
ABo.t e -2~rd
-
+
21rpo 1 -- ~'~,~d ~ - d
(A4.+l - A,bt) = 0
(20)
and
R
AB0,t+l + R a/J0.t - ~ e -2*rd- 2¢rp0 1 - (1 + R)Ackt+l(1/d ) ~
e -2~rd e -2¢d (A~I+I
- A~/)
(21)
HokPk(ik) 2 e ik* = O.
k=l
For n > 0 it is found that AAn,I+1
- - -Z-L- a- l n , I e
-¢td-i2~rA. - r ABn .t+t- ABn.t e -'a-i2"~
=
0
(22)
and AAn,l+l + AAn,t R
e -*rd-12*rA
--
(ABn.~+l + ABn.I R e-"d+i2"~) _~ 0.
(23)
476
U. BJORKENSTAM
In deriving equations (22) and (23) approximations have been made based on the assumption that s and hence also A are much larger than unity. Equations (19)--(23) constitute a system of linear homogeneous finite difference equations in the unknowns AAn,~, ABn,~and A4~. The solution is sought in the following form AAn,t = an~ :t ABn,I = / 3 n ~ l
A,bt = v~ :t.
(24)
The system of algebraic equations obtained when substituting (24) into (19)-(23) has non-trivial solutions an, /3n and 3' only if the determinant of the system is equal to zero. The system determinant can be decomposed into a product of infinitely many smaller determinants ~
~
~:~(S0t + 2 ~ (-1)nSn ~) n=l e_2,rd (~1+1 _ ~l)2,ffp0 L_1 ~ _ 2~d J
~l+l__~l ~l+l_~le-2,r d 0
R e -2,~a ~/+1 + ~tR e-2#d _(~1+1_ ~l)21rp0__ ~ 1-~
X fi ~+l--~l e-~'d-i27rA ~l+l--~le-~rd+i2~A
~t(1 + R)S02
I
,.=l ~+l + fl R e-,.d-12~.~, _(~H + ~lR e-.,d÷i2.~,) = o where
(25)
o~ Sn I = ~ , Hnkpkik e ik¢'
k=l
0o Sn2 = (l/d) ~ Hnkpk(ik) 2 e ik'~. k=l
(26)
The solution of the problem of the stability of the periodic motion is reduced to a calculation of the quantity ~. If for all roots ~i of equation (25) 161 < 1 then the motion is stable. The roots are found by studying the characteristic equations obtained when each of the determinants in (25) are put equal to zero. The characteristic equations will, after elemination of roots equal to zero, be of the form (27)
an2~ 2 + ant~ + ano = O.
According to Schur's theorem, the necessary and sufficient conditions that both the roots of the second order equations lie inside the unit circle are given by the inequalities lanola,2[ < 1
[anl/(ano+ a.2)l < 1.
(28)
The first determinant gives a02 = 1 +(1 +R)oao~ = - 1 - R 2 e -2,~d
e1- -2~
2;rd
d
®
(1 + R) 2 ~ Hok(pdpo)(ik) z e i~ k=l
-(1 - R e-2'~a)(1 + R)tr aoo = R e-2~d[R -- (1 + R)o']
(29)
Impact vibration of a bar
477
where o- = (l[Tr) ~ ~ (--1)nHnk(Pk/po)ik e 'k~ n~l k=l
(30)
while the remaining determinants give an2 = 1
a~l = -(1 - R) e-"a cos 2IrA ano =
-R
(31)
e -2~rd.
Hence, the necessary and sufficient conditions which give stable motion are
IR
e -2,dR
-
(1 + R)o"
1 - e -2~d 1 + - -
2~'d
IR e-2~dl <
l+R2e-2'~a+(1-Re-2"a)(l+R)cr
r
<1
1
(1 - R) e -'~d cos
I
0o (1 + R) 2 ~ Hok(pk/po)(ik) 2 e ik~ k=l
1 - R e-2"a
I
<1.
(32)
The first, third and fourth inequalities always hold for d > 0 or R < 1, while the second inequality gives the condition from which the stability of the motion, found in Section 2, may be determined. 4. E X A M P L E S
The general equations derived in Sections 2 and 3 will here be used in two simple examples taken from percussive rock drilling. The length of the hammer in an ordinary rock drilling machine is 6-8 times the diameter and can be represented by the model system in Fig. 1. Rock drilling machines are usually either pneumatically or hydraulically driven. The motion and stability of these two kinds of hammers will be compared. ptt)
Po )t
FIG. 2. L o a d function for the p n e u m a t i c h a m m e r (
experimental curve, - - - -
approximate
curve).
A typical load history for one period of a pneumatically driven hammer is shown in Fig. 2. The experimental curve may be approximated by the cosine function
478
U. BJORKENSTAM
p(t) = Po+ pk cos k(t + d~)
(33)
where in practice k should be equal to 1. However, it is of theoretical interest to also study k larger than 1, i.e. a one impact per the kth force period. The load history for a hydraulic hammer can be represented with good accuracy by the square-wave function in Fig. 3. It is described by the real component of the Fourier series
p ( t ) = q {K-- l +(217r) ~ [sinkKcr + i (coskKcr--1)]e'k(~'÷~)[k}.
(34)
k=l
For an undamped pneumatic hammer it follows, from equations (14) and (33) when d ~ 0, that
(35)
sin k~k = ¢r(kpo/pk)(1 - R)/(1 + R). p(t)
+q
% K~
2~
-q
FIG. 3. Load function for the hydraulic hammer.
This equation gives the possible values of the phase angle ~b, which corresponds to a steady-state motion with one impact per the kth load period. According to (32) and (33) the obtained motion is stable provided the inequality
I1 + (1 + R )2/(1 + R 2+ (1 -
R 2)crp)(pk/pok) cos
k4,l <
1,
(36)
where c~
o'p = (l/Tr) ~ (--1)nHnk(Pk/pO) sin k~b n=l
1 -R
~-~ ~
k2(-1)
~
1 + R ~/7'=1(sncr) 2 - k 2'
(37)
is satisfied. The phase angle ~k, derived from equation (35), is drawn in Fig. 4 as a function of the load factor po/pk for different values of the coefficient of restitution R. Values of ~b corresponding to unstable forms of motion, according to (36), are marked with a dashed line. From equation (35) the upper bound of the load factor po/pk follows as
kpo/Pk <--(l/Tr)(1 + R)/(1 - R)
(38)
while the inequality (36) gives the upper and lower bounds of stable forms of motion
Impact vibration of a bar
479
k¢
~AAX~
3~/4
~r12
t Illl Jlllj/ /I I] / I IIII i, / // IIIIi//
/ /
[ ~}t///. I'//// / _ . / /
~i'-"
!1!!////
/
~
/1
j
l
I"1" / /
Y
/
>
I
2
3
4
5
6
k .o/p 1
FIG. 4. Phase angle 4, as a function of the load factor PolP~for R =0.0,0.1 . . . . . 0.9 ( stable solutions, - - - - unstable solutions). Pneumatic hammer. as
(1 + R ) 2 X/[4(1 + R 2 + (1 -
R2)o~)21k 2 + Ir2(1 -
R2) 2] <
kpopk< ¢tl 11+ RR
(39)
I n Fig. 5 t h e s e b o u n d a r i e s a r e d r a w n f o r a n u n d a m p e d p n e u m a t i c h a m m e r w i t h k = 1. T h e b o u n d a r i e s f o r a s l i g h t l y d a m p e d s y s t e m w i t h t h e r e l a t i v e d a m p i n g 2~rd = 0.03 a r e a l s o s h o w n . S i n c e scr >> 1 a n d t h e l o w e s t r e s o n a n c e f r e q u e n c y is o b t a i n e d , a c c o r d i n g to e q u a t i o n (16), f o r s~r~ 1 n o i n f l u e n c e o f t h e s t i f f n e s s s 2 o f t h e b a r o n t h e s t a b i l i t y o f t h e m o t i o n c a n b e f o u n d w h e n k = 1. F o r k l a r g e r t h a n 1 t h e e f f e c t o f t h e s t i f f n e s s is e s s e n t i a l , as s e e n f r o m Fig. 6 w h e r e k - - ' 1 0 is s t u d i e d f o r d i f f e r e n t v a l u e s o f sir. P o s s i b l e v a l u e s o f t h e p h a s e a n g e l 4, f o r a n u n d a m p e d d y d r a u l i c h a m m e r f o l l o w f r o m e q u a t i o n s (14) a n d (34)
(2/¢r) ~ , [sin krcr sin k~b + ( c o s k r c r - 1) c o s k~b] = (K - 1)¢r(1 - R)/(1 + R ) ,
(40)
k=l
T h o s e s o l u t i o n s o f e q u a t i o n (40) w h i c h a r e s t a b l e a r e g i v e n b y t h e i n e q u a l i t y (32) w h i c h m a y b e t r a n s f o r m e d into
I
(1 + R ) z I+I+R2+(1-R2)O'H
p(O)lq - r + 1
K-1
(41)
<1.
.o/., A
.o .o,o,,o..
//
//s,.b,. J
L
I
i
I
I
I
I
i
0.1
0,2
0.3
0./.
0.5
0.6
0.7
0.8
0.9
>
1.0
R
FIG. 5. Stable and unstable regions of the load factor Polpi as a function of the coefficient of restitution R ( 2wd = 0, - - . - - 21rd = 0.03). Pneumatic hammer k = 1.
480
U. BJORKENSTAM
PO//Pl0
Stsb,e
s°'"ti°ns I//~/
'
/
i
'
0,8
No solutions
0,6
0.4
0.2
•"" -
Unstable solutions
t >
0.1
0.2
0.3
0.4
0.5
0.6
0.7
08
09
R
1.0
FIG. 6. Stable and unstable regions of the load factor P0/Pt0 as a function of the coefficient of restitution R ( - s~ -> 10, - - . - - s¢r = 10.1, - - - - s~" = 10.01). Pneumatic hammer k = 10. I t is f o u n d f r o m t h i s i n e q u a l i t y t h a t all p o s s i b l e s t e a d y - s t a t e m o t i o n s , w i t h o n e i m p a c t p e r l o a d p e r i o d , a r e u n s t a b l e . P o s s i b l e v a l u e s o f t h e p h a s e a n g l e $ a r e d r a w n in F i g . 7, w h i l e p o s s i b l e v a l u e s o f t h e l o a d f a c t o r x a r e s h o w n in F i g . 8.
3~t/2
~=o.07/ ////// /./ //././// //
/,//
/p;//~
./
~/./
6.T /0.8 . 0.9
./
./
//
/
./
% % / i %. ~ ...-~
7 7 / . . " J / . . i.- ~ - J I/~
/
i" ./j
I.
I-- I " I . .!
:~/2 I
1.1
I
I
~
I
I
I
I
[
[
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1,9
2.0
)K
Fl6. 7. Phase angle ~b as a function of the load factor x for R =0.0,0.1 . . . . . 0.9 (. . . . unstable solutions). Hydraulic hammer. K
2.00
1.751.50
No solutions
~,,"
Unstable solutions 1.25
L
I
;
I
I
I
I
I
I
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.0
)R 1.0
FIG. 8. Unstable region of the load factor r as a function of the coefficient of resitution R ( 2¢rd = 0, - - . - - 2wd = 0.03).Hydraulic hammer.
Impact vibration of a bar
481
5. CONCLUSIONS The steady-state vibration of a bar has been analysed when the bar at each cycle of motion impacts an impenetrable and energy absorbing surface. The bar is loaded by a general periodic force and by a distributed linear viscous damping force. Two special cases are studied more closely, namely a pneumatically and a hydraulically driven hammer. It is found that for a pneumatic hammer it is always possible to find regions of stable motion, while the hydraulic hammer does not have any such stable regions. Because of the non-linear nature of the problem the solutions obtained here are not unique. In fact, if po/pk > 1, one mode of stable motion of the bar would be to remain in contact with the surface at all times, i.e. U ( L , t) =- O. Numerical experiments have been done in order to find out what form the steady-state motion will take for arbitrary initial conditions. A computer program was written to directly determine the integration constants of equation (8). The process is started by assuming an arbitrary set of initial values which determines the constants. The motion is then followed until the time when contact between the bar and the surface occurs. New integration constants are then determined in the same way as in equation (18) but here the time is measured continously. The process is repeated over and over so as to obtain the time behavior of the system. For the pneumatic hammer it was found that parameters corresponding to a point below the upper curve in Fig. 5 always give a steady-state motion having the here assumed form (i.e. if the U ( L , t ) = - 0 motion is included). For points where several forms of motion (several k-values) can exist it was found that all of these forms could be obtained if the initial conditions were properly chosen. However, if the initial values were chosen randomly the steady-state motion corresponding to the smallest allowed k-values was almost always (in more than 99% of the numerical experiments) obtained. In the case of a hydraulically driven hammer it was not possible to find any steady-state motion. The frequency of the impacts was not constant but fluctuated around the value of the load frequency. Depending on the chosen parameters but not on the initial conditions the impact frequency varied with a standard deviation of 5-10%. Since an operating rock drilling machine is always exposed to large perturbations the motion of the hydraulic hammer will be unstable. REFERENCES P. LIEBERand D. P. JENSEN, Trans. ASME 67, 523-530 (1945). C. GRUBIN,J. Appl. Mech. 23, 373-378 (1956). G. B. WARBURTON,J. Appl. Mech. 24, 322-324 (1957). S. F. MASRIand T. K. CAUGHLEY,J. Appl. Mech. 33, 586-592 (1966); J. Appl. Mech. 34, 253 (1967). S. F. MASRIand K. KAHYALInt. J. Non-Linear Mech. 9, 451-462 (1974). C. C. Fu and B. PAUL,Int. J. Solids Struct. 4, 897-905 0968). C. C. Fu and B. PAUL,3". Engng Ind. 91, 1175-1179 (1969). D. L. SIKARSKIEand B. PAUL,J. Engng Ind. 91,931-938 (1969). C. C. Fu, 3". Appl. Mech. 56, 743-749 (1969). I. G. RUSAKOVand A. A. KHARKEVICH,Zhur. Tekh. Fiz. 12(11-12), 715-721 (1942). P. S. LIVSHITS,Zhur. Tekh. Fiz. 22(6), 921-931 (1952). A. E. KOBRINSKn,Izvest. Akad. Naak. SSSR, Otdel, Tekh, Nouk, No, 5, 113-121 (1956). R. F. BRUNSHTEINand A. E. KOBRINSKn,Trudy, Inst. Maschinv, Semin. Teor. Mash. Mekh. 19(75) (1959). 14. R. F. BRUNSHTEINand A. E. KOBmNSKII,IZV. Akad. Nauk. SSSR, Otdel. tekh. Nauk, Mekh. Mashin. No. 5 (1960). 15. M. I. FEGIN,IZV. Akad. Nauk SSSR, Otdel. tekh, Nauk, Mekh. Maskinost. No. 5 (1960). 16. A. E. KOBRINSKII,Dynamics of Mechanisms with Elastic Connections and Impact Systems. Iliffe Books, London (1969).
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. ll. 12. 13.
Pergamon Press 1977. Printed in Great Britain
IMPACT
VIBRATION
OF
A BAR
U . B JORKENSTAM Department of Mechanical Engineering, Link6ping University, S-581 83 Link6ping, Sweden
(Received 14 January 1977; in revised form 23 April 1977) Summary--This paper treats the motion and dynamic stability of a bar having constant stiffness and mass distribution when subjected to a periodic driving force, a distributed and linear damping force and impact forces due to its motion and operation (such as would occur on, e.g. the hammer in a rock drilling machine). Comparisons have been made between the characteristics of a bar operated by either pneumatic or hydraulic means.
NOTATION Young's modulus cross-sectional area mass per unit of length length spatial coordinate displacement time force per unit of length load damping factor wave propagation velocity dimensionless spatial coordinate dimensionless displacement dimensionless time dimensionless load dimensionless damping factor dimensionless wave propagation velocity transformed dimensionless displacement A~, B~ constants of integration ¢k phase angle R coefficient of restitution K load factor E A M L X U T F P D C x u t p d s
1. I N T R O D U C T I O N
IMPACT vibration of systems having one or two degrees of freedom has been investigated in a large number of articles. The results have been applied to mainly acceleration dampers ~-4 and vibrating hammers. 6-9 Only in one case has a continuous system been studied) All these systems are loaded by a periodic driving force. However, only the first two terms of its Fourier expansion are considered, i.e. the constant term and the first harmonic term. The motion of the systems has mostly been determined by the method suggested by Warburton) The steady-state solution is sought by replacing the initial value problem approach by a boundary value problem approach, i.e. a boundary value problem is formulated such that the general solution of the equations of motion satisfies conditions at the beginning and at the end of each period between impacts. A relation between the period of the driving force and the impact period has to be assumed. Due to the non-linearity of the problem, it is theoretically possible to have more than one steady-state solution. But not all of the solutions can exist for longer times. Marsi and Caughey 4 have derived a method of finding the asymptotically stable regions of the solution. Several Soviet authors have dealt with symmetrical and asymmetrical two body systems moving subjected to periodic collisions, t°-~5 Kobrinskii has written a textb o o k 16 o n the subject. There the method of analysing the stability of periodic states is based on the application of finite differences. MS Vol. 19, No. 8---C
471
472
u. BJORKENSTAM
This paper treats the motion and dynamic stability of a periodically impacting bar (see Fig. 1). The bar is assumed to have constant stiffness and mass distribution and is loaded by a periodic force and a distributed linear viscous damping force. The methods derived by Warburton, Marsi and Caughey and Kobrinskii are generalized to hold for a continuous system and a general periodic force. There are several technical applications of the model system studied. Here it is used in order to determine the motion and stability of the hammer in a pneumatic and in a hydraulic driven rock drilling machine.
r~P(T)
///////// FIG. 1. The studied model system. 2. MOTION OF THE SYSTEM Assuming the same impact period as force period possible steady-states of motion for the axially loaded bar in Fig. 1 will be derived in this section. The partial differential equation describing the longitudinal motion U ( X , T ) of a thin rod is
EA
02U(X, T) oEu(x, T) OX 2 + F(X, T) = M OT 2
(la)
Here the distribution of mass M and stiffness E A are assumed to be constants of the spatial coordinate X, while the distributed damping force is assumed to be proportional to the velocity of the longitudinal motion, hence F(X, T)= -DaU(X,
T)/aT.
(lb)
For convenience dimensionless quantities are introduced as x = XIL u = UIL t=
TO
(2)
p = p[(ML2fl) d = DI(MII) s = C/(Lfl),
where L is the length of the bar and l~ is the frequency of the first harmonic of the periodic driving force while C = ~ / ( E A I M ) is the longitudinal wave propagation velocity. The equation of motion (1) then transforms into
Impact vibration of a bar
02u(x, t) + dOU(X' t) 0-~ Ot
473
s2 02u(x, t) O. --------T-Ox =
(3)
A steady-state solution will be attempted assuming that the period of impacting equals that of the first harmonic. The time the bar is in contact with the surface is assumed to be much less than the periodic time of the first harmonic of the excitation. With the dimensionless time t counted from one impact at steady-state motion the boundary conditions are, for 0 < t < 2~r,
Ou(x, t)] OX
= I ~=o
(4)
Ou(x, t) = O. OX x=l Because of the nature of the boundary conditions, a solution will be attempted by means of a finite cosine transform defined by l
ti(n, t) = fo u(x, t) cos nlrx dx.
(5)
A transformation of (3) together with the boundary conditions (4) lead to
02a(x, t) Ot 2
+
d
Oa(n, t) + (smr)Za(n, t) Ot
= [Ou(x, t) ]~ I _ - - - ~ x cos nlrx + ncru(x, t) sin ncrx o= p(t).
(6)
The periodic load function may be represented by the real component of the Fourier series
p(t) = f~ Pk e ik"+~),
(7)
k=O
where the complex coefficients Pk contain the information regarding the phase angles of the various harmonics while ~ is the phase lag between the period of the force and the impacting period. The general solution of the ordinary differential equation (6) then is the real component of oo
ti(n, t) = e-dt/2(An e-at+ B. e +ia~)+ ~'~ H,~pk e ik(t+¢')
(8)
k=0
where
Hnk = 1/[(sn~r) 2 + ikd - k 2] A = ~/[(smr) 2 - d2/4].
(9)
According to the method suggested by Warburton 3 the initial value problem approach is now replaced by a boundary value problem approach. This is done by studying the steady-state motion of the system at the beginning and at the end of a period between impacts. If the dynamic stiffness of the surface in Fig. 1 is larger than the dynamic stiffness of the bar then the boundary conditions are
u(x, 0 +) = u(x, 27r-)
Ou(X,ot t) t=o÷ = - R Ou(x,ot t) t=z,,-
(10)
474
U. BJORKENSTAM
where 0 ~< R ~< 1. The time of impact will be equal to 2 L / C , which is the time it takes for a w a v e to travel twice the length of the bar. Since the duration of impact is assumed to be much less than the periodic time of the excitation, the dimensionless stiffness s 2= [C[(L~)] 2 must be much larger than 1/w 2. If, e.g. the surface in Fig. 1 is rigid then R is equal to 1. Assuming on the other hand that the surface is another infinitely long bar with the same modulus of elasticity and the same density as the impacting bar but with a different area As then R = (As - A)/(As + A). Hence, As + A gives R = 0 while As -> A gives R = 1. As has to be larger than A since the dynamic stiffness of the surface is to be larger than that of the bar. The t r a n s f o r m e d boundary conditions are ti(n, 0 +) = ti(n, 2w-) O/~(n, t) = - R Oa(n, t) Ot t:0 + Ot z:2,~-
(11)
where n = 1, 2, 3 . . . . . F r o m equations (8) and (11) it follows that A0 is an arbitrary constant and that Bo = (2 ~rpo/ d)l(1 - e -2,~a) A. = (1 - e-"d+~2"a)Q. B. = - ( 1 - e-"a-i2"A)Q.
(12)
where
Q, = (1 + R ) ~, H,,kpkik eiU'/{(d/2 + ih)(1 + R e-"a-i2~a)(1 - e -~a+i2~A) k=0 -
(d/2 - iA)(1 + R e-~d+iz'~a)(1 -- e-~a-i2~a)}.
(13)
The boundary conditions for n = 0 also give the relation f r o m which the unknown phase angle ~ m a y be determined (1 + R ) ~ Hokpkik eik* = 2wp0(1 + R e-2"d)/(l -- e-2"a).
(14)
k=0
If a value of ~ can be found which satisfies equation (14), then a corresponding one-impact-per-force-period steady-state solution also exists. Knowing the constants of integration the dimensionless displacement is obtained by taking the inverse transformation of (8)
u(x,
t) = a(0, t) + 2 ~ a(n, t) cos n,rx.
(15)
n=l
H e r e a(0, t) is a rigid body motion while the other terms describe the d e f o r m a t i o n of the bar. It follows f r o m (9) and (13) that for an undamped system resonance will occur when s=k/(nzr)
n--1,2,3 .....
(16)
3. STABILITY OF THE MOTION A possible state of motion was determined in the last section. H e r e the stability of that motion will be investigated. The method of analysing the stability of the periodic impact vibrations used here is based on the regularities of the variations in small
475
Impact vibration of a bar
perturbations building up over a number of successive intervals. If the magnitude of the small perturbations remain finite, the corresponding unperturbed motion will be considered stable; if it is discovered that the small perturbations have a tendency to increase infinitely the corresponding unperturbed motion will be regarded as unstable. Following one of the successive impacts, let a small arbitrary disturbance be introduced into the periodic motion, such that the displacement at the next impact differ from the steady-state value u(x, t) = F(x, t, An, Bn, ok). The displacement for the perturbed motion for the ith interval after the initial perturbation can be written u(x, th = F(x, t, An.t, Bn.~, qbt), where An.t= An + AAn.I and Bn.i = Bn + ABn,t are new constants of integration and ~b~= 4~ + A~bt is a new phase lag. If AAn.i, ABn,t and A~t tend to zero when l tends to infinity the disturbed motion will return to its steady-state form and is stable. As before, the dimensionless time is measured from the moment of the last impact. The accumulated effect of the perturbations is given by the perturbations AAn,t and ABn,t while the cumulative perturbation of the phase of the ith impact period is Aq~t = 6~bl + 8~b2+ • • • + &kt. All the quantities AAn.t, ABn.~, Ac~t and &kt are assumed to be small. Hence, terms of the second order of magnitude and smaller, e.g. AAn.t6fft+~, will be neglected in the stability analyses. The condition that the surface is impenetrable gives u(1,0+)~ = u(1, 0 + ),
(17)
while the condition of continuity of two adjacent intervals gives u(x, 0+)l+l = U(X, ( 2 w + 84't+l)-)l
dU(X, t)t+l = - R Ou(x, t)t Ot t=o÷ Ot
(18) t =(2'tr +84,1+1)-"
From equations (17) and (15) it follows that 00
AA0,, + ABo,, + Atb, ~ Hokpkik e 'k* + 2 ~ ( - 1)n k=l
n=l
(AAn,t + ABn,t + A~b~~ Hnkpkik e 'k*) = 0.
(19)
k=l
The transformed initial values (18) give for n = 0 e -2~rd
AAo,t+l - AA0,t +
AB0./+t
ABo.t e -2~rd
-
+
21rpo 1 -- ~'~,~d ~ - d
(A4.+l - A,bt) = 0
(20)
and
R
AB0,t+l + R a/J0.t - ~ e -2*rd- 2¢rp0 1 - (1 + R)Ackt+l(1/d ) ~
e -2~rd e -2¢d (A~I+I
- A~/)
(21)
HokPk(ik) 2 e ik* = O.
k=l
For n > 0 it is found that AAn,I+1
- - -Z-L- a- l n , I e
-¢td-i2~rA. - r ABn .t+t- ABn.t e -'a-i2"~
=
0
(22)
and AAn,l+l + AAn,t R
e -*rd-12*rA
--
(ABn.~+l + ABn.I R e-"d+i2"~) _~ 0.
(23)
476
U. BJORKENSTAM
In deriving equations (22) and (23) approximations have been made based on the assumption that s and hence also A are much larger than unity. Equations (19)--(23) constitute a system of linear homogeneous finite difference equations in the unknowns AAn,~, ABn,~and A4~. The solution is sought in the following form AAn,t = an~ :t ABn,I = / 3 n ~ l
A,bt = v~ :t.
(24)
The system of algebraic equations obtained when substituting (24) into (19)-(23) has non-trivial solutions an, /3n and 3' only if the determinant of the system is equal to zero. The system determinant can be decomposed into a product of infinitely many smaller determinants ~
~
~:~(S0t + 2 ~ (-1)nSn ~) n=l e_2,rd (~1+1 _ ~l)2,ffp0 L_1 ~ _ 2~d J
~l+l__~l ~l+l_~le-2,r d 0
R e -2,~a ~/+1 + ~tR e-2#d _(~1+1_ ~l)21rp0__ ~ 1-~
X fi ~+l--~l e-~'d-i27rA ~l+l--~le-~rd+i2~A
~t(1 + R)S02
I
,.=l ~+l + fl R e-,.d-12~.~, _(~H + ~lR e-.,d÷i2.~,) = o where
(25)
o~ Sn I = ~ , Hnkpkik e ik¢'
k=l
0o Sn2 = (l/d) ~ Hnkpk(ik) 2 e ik'~. k=l
(26)
The solution of the problem of the stability of the periodic motion is reduced to a calculation of the quantity ~. If for all roots ~i of equation (25) 161 < 1 then the motion is stable. The roots are found by studying the characteristic equations obtained when each of the determinants in (25) are put equal to zero. The characteristic equations will, after elemination of roots equal to zero, be of the form (27)
an2~ 2 + ant~ + ano = O.
According to Schur's theorem, the necessary and sufficient conditions that both the roots of the second order equations lie inside the unit circle are given by the inequalities lanola,2[ < 1
[anl/(ano+ a.2)l < 1.
(28)
The first determinant gives a02 = 1 +(1 +R)oao~ = - 1 - R 2 e -2,~d
e1- -2~
2;rd
d
®
(1 + R) 2 ~ Hok(pdpo)(ik) z e i~ k=l
-(1 - R e-2'~a)(1 + R)tr aoo = R e-2~d[R -- (1 + R)o']
(29)
Impact vibration of a bar
477
where o- = (l[Tr) ~ ~ (--1)nHnk(Pk/po)ik e 'k~ n~l k=l
(30)
while the remaining determinants give an2 = 1
a~l = -(1 - R) e-"a cos 2IrA ano =
-R
(31)
e -2~rd.
Hence, the necessary and sufficient conditions which give stable motion are
IR
e -2,dR
-
(1 + R)o"
1 - e -2~d 1 + - -
2~'d
IR e-2~dl <
l+R2e-2'~a+(1-Re-2"a)(l+R)cr
r
<1
1
(1 - R) e -'~d cos
I
0o (1 + R) 2 ~ Hok(pk/po)(ik) 2 e ik~ k=l
1 - R e-2"a
I
<1.
(32)
The first, third and fourth inequalities always hold for d > 0 or R < 1, while the second inequality gives the condition from which the stability of the motion, found in Section 2, may be determined. 4. E X A M P L E S
The general equations derived in Sections 2 and 3 will here be used in two simple examples taken from percussive rock drilling. The length of the hammer in an ordinary rock drilling machine is 6-8 times the diameter and can be represented by the model system in Fig. 1. Rock drilling machines are usually either pneumatically or hydraulically driven. The motion and stability of these two kinds of hammers will be compared. ptt)
Po )t
FIG. 2. L o a d function for the p n e u m a t i c h a m m e r (
experimental curve, - - - -
approximate
curve).
A typical load history for one period of a pneumatically driven hammer is shown in Fig. 2. The experimental curve may be approximated by the cosine function
478
U. BJORKENSTAM
p(t) = Po+ pk cos k(t + d~)
(33)
where in practice k should be equal to 1. However, it is of theoretical interest to also study k larger than 1, i.e. a one impact per the kth force period. The load history for a hydraulic hammer can be represented with good accuracy by the square-wave function in Fig. 3. It is described by the real component of the Fourier series
p ( t ) = q {K-- l +(217r) ~ [sinkKcr + i (coskKcr--1)]e'k(~'÷~)[k}.
(34)
k=l
For an undamped pneumatic hammer it follows, from equations (14) and (33) when d ~ 0, that
(35)
sin k~k = ¢r(kpo/pk)(1 - R)/(1 + R). p(t)
+q
% K~
2~
-q
FIG. 3. Load function for the hydraulic hammer.
This equation gives the possible values of the phase angle ~b, which corresponds to a steady-state motion with one impact per the kth load period. According to (32) and (33) the obtained motion is stable provided the inequality
I1 + (1 + R )2/(1 + R 2+ (1 -
R 2)crp)(pk/pok) cos
k4,l <
1,
(36)
where c~
o'p = (l/Tr) ~ (--1)nHnk(Pk/pO) sin k~b n=l
1 -R
~-~ ~
k2(-1)
~
1 + R ~/7'=1(sncr) 2 - k 2'
(37)
is satisfied. The phase angle ~k, derived from equation (35), is drawn in Fig. 4 as a function of the load factor po/pk for different values of the coefficient of restitution R. Values of ~b corresponding to unstable forms of motion, according to (36), are marked with a dashed line. From equation (35) the upper bound of the load factor po/pk follows as
kpo/Pk <--(l/Tr)(1 + R)/(1 - R)
(38)
while the inequality (36) gives the upper and lower bounds of stable forms of motion
Impact vibration of a bar
479
k¢
~AAX~
3~/4
~r12
t Illl Jlllj/ /I I] / I IIII i, / // IIIIi//
/ /
[ ~}t///. I'//// / _ . / /
~i'-"
!1!!////
/
~
/1
j
l
I"1" / /
Y
/
>
I
2
3
4
5
6
k .o/p 1
FIG. 4. Phase angle 4, as a function of the load factor PolP~for R =0.0,0.1 . . . . . 0.9 ( stable solutions, - - - - unstable solutions). Pneumatic hammer. as
(1 + R ) 2 X/[4(1 + R 2 + (1 -
R2)o~)21k 2 + Ir2(1 -
R2) 2] <
kpopk< ¢tl 11+ RR
(39)
I n Fig. 5 t h e s e b o u n d a r i e s a r e d r a w n f o r a n u n d a m p e d p n e u m a t i c h a m m e r w i t h k = 1. T h e b o u n d a r i e s f o r a s l i g h t l y d a m p e d s y s t e m w i t h t h e r e l a t i v e d a m p i n g 2~rd = 0.03 a r e a l s o s h o w n . S i n c e scr >> 1 a n d t h e l o w e s t r e s o n a n c e f r e q u e n c y is o b t a i n e d , a c c o r d i n g to e q u a t i o n (16), f o r s~r~ 1 n o i n f l u e n c e o f t h e s t i f f n e s s s 2 o f t h e b a r o n t h e s t a b i l i t y o f t h e m o t i o n c a n b e f o u n d w h e n k = 1. F o r k l a r g e r t h a n 1 t h e e f f e c t o f t h e s t i f f n e s s is e s s e n t i a l , as s e e n f r o m Fig. 6 w h e r e k - - ' 1 0 is s t u d i e d f o r d i f f e r e n t v a l u e s o f sir. P o s s i b l e v a l u e s o f t h e p h a s e a n g e l 4, f o r a n u n d a m p e d d y d r a u l i c h a m m e r f o l l o w f r o m e q u a t i o n s (14) a n d (34)
(2/¢r) ~ , [sin krcr sin k~b + ( c o s k r c r - 1) c o s k~b] = (K - 1)¢r(1 - R)/(1 + R ) ,
(40)
k=l
T h o s e s o l u t i o n s o f e q u a t i o n (40) w h i c h a r e s t a b l e a r e g i v e n b y t h e i n e q u a l i t y (32) w h i c h m a y b e t r a n s f o r m e d into
I
(1 + R ) z I+I+R2+(1-R2)O'H
p(O)lq - r + 1
K-1
(41)
<1.
.o/., A
.o .o,o,,o..
//
//s,.b,. J
L
I
i
I
I
I
I
i
0.1
0,2
0.3
0./.
0.5
0.6
0.7
0.8
0.9
>
1.0
R
FIG. 5. Stable and unstable regions of the load factor Polpi as a function of the coefficient of restitution R ( 2wd = 0, - - . - - 21rd = 0.03). Pneumatic hammer k = 1.
480
U. BJORKENSTAM
PO//Pl0
Stsb,e
s°'"ti°ns I//~/
'
/
i
'
0,8
No solutions
0,6
0.4
0.2
•"" -
Unstable solutions
t >
0.1
0.2
0.3
0.4
0.5
0.6
0.7
08
09
R
1.0
FIG. 6. Stable and unstable regions of the load factor P0/Pt0 as a function of the coefficient of restitution R ( - s~ -> 10, - - . - - s¢r = 10.1, - - - - s~" = 10.01). Pneumatic hammer k = 10. I t is f o u n d f r o m t h i s i n e q u a l i t y t h a t all p o s s i b l e s t e a d y - s t a t e m o t i o n s , w i t h o n e i m p a c t p e r l o a d p e r i o d , a r e u n s t a b l e . P o s s i b l e v a l u e s o f t h e p h a s e a n g l e $ a r e d r a w n in F i g . 7, w h i l e p o s s i b l e v a l u e s o f t h e l o a d f a c t o r x a r e s h o w n in F i g . 8.
3~t/2
~=o.07/ ////// /./ //././// //
/,//
/p;//~
./
~/./
6.T /0.8 . 0.9
./
./
//
/
./
% % / i %. ~ ...-~
7 7 / . . " J / . . i.- ~ - J I/~
/
i" ./j
I.
I-- I " I . .!
:~/2 I
1.1
I
I
~
I
I
I
I
[
[
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1,9
2.0
)K
Fl6. 7. Phase angle ~b as a function of the load factor x for R =0.0,0.1 . . . . . 0.9 (. . . . unstable solutions). Hydraulic hammer. K
2.00
1.751.50
No solutions
~,,"
Unstable solutions 1.25
L
I
;
I
I
I
I
I
I
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.0
)R 1.0
FIG. 8. Unstable region of the load factor r as a function of the coefficient of resitution R ( 2¢rd = 0, - - . - - 2wd = 0.03).Hydraulic hammer.
Impact vibration of a bar
481
5. CONCLUSIONS The steady-state vibration of a bar has been analysed when the bar at each cycle of motion impacts an impenetrable and energy absorbing surface. The bar is loaded by a general periodic force and by a distributed linear viscous damping force. Two special cases are studied more closely, namely a pneumatically and a hydraulically driven hammer. It is found that for a pneumatic hammer it is always possible to find regions of stable motion, while the hydraulic hammer does not have any such stable regions. Because of the non-linear nature of the problem the solutions obtained here are not unique. In fact, if po/pk > 1, one mode of stable motion of the bar would be to remain in contact with the surface at all times, i.e. U ( L , t) =- O. Numerical experiments have been done in order to find out what form the steady-state motion will take for arbitrary initial conditions. A computer program was written to directly determine the integration constants of equation (8). The process is started by assuming an arbitrary set of initial values which determines the constants. The motion is then followed until the time when contact between the bar and the surface occurs. New integration constants are then determined in the same way as in equation (18) but here the time is measured continously. The process is repeated over and over so as to obtain the time behavior of the system. For the pneumatic hammer it was found that parameters corresponding to a point below the upper curve in Fig. 5 always give a steady-state motion having the here assumed form (i.e. if the U ( L , t ) = - 0 motion is included). For points where several forms of motion (several k-values) can exist it was found that all of these forms could be obtained if the initial conditions were properly chosen. However, if the initial values were chosen randomly the steady-state motion corresponding to the smallest allowed k-values was almost always (in more than 99% of the numerical experiments) obtained. In the case of a hydraulically driven hammer it was not possible to find any steady-state motion. The frequency of the impacts was not constant but fluctuated around the value of the load frequency. Depending on the chosen parameters but not on the initial conditions the impact frequency varied with a standard deviation of 5-10%. Since an operating rock drilling machine is always exposed to large perturbations the motion of the hydraulic hammer will be unstable. REFERENCES P. LIEBERand D. P. JENSEN, Trans. ASME 67, 523-530 (1945). C. GRUBIN,J. Appl. Mech. 23, 373-378 (1956). G. B. WARBURTON,J. Appl. Mech. 24, 322-324 (1957). S. F. MASRIand T. K. CAUGHLEY,J. Appl. Mech. 33, 586-592 (1966); J. Appl. Mech. 34, 253 (1967). S. F. MASRIand K. KAHYALInt. J. Non-Linear Mech. 9, 451-462 (1974). C. C. Fu and B. PAUL,Int. J. Solids Struct. 4, 897-905 0968). C. C. Fu and B. PAUL,3". Engng Ind. 91, 1175-1179 (1969). D. L. SIKARSKIEand B. PAUL,J. Engng Ind. 91,931-938 (1969). C. C. Fu, 3". Appl. Mech. 56, 743-749 (1969). I. G. RUSAKOVand A. A. KHARKEVICH,Zhur. Tekh. Fiz. 12(11-12), 715-721 (1942). P. S. LIVSHITS,Zhur. Tekh. Fiz. 22(6), 921-931 (1952). A. E. KOBRINSKn,Izvest. Akad. Naak. SSSR, Otdel, Tekh, Nouk, No, 5, 113-121 (1956). R. F. BRUNSHTEINand A. E. KOBRINSKn,Trudy, Inst. Maschinv, Semin. Teor. Mash. Mekh. 19(75) (1959). 14. R. F. BRUNSHTEINand A. E. KOBmNSKII,IZV. Akad. Nauk. SSSR, Otdel. tekh. Nauk, Mekh. Mashin. No. 5 (1960). 15. M. I. FEGIN,IZV. Akad. Nauk SSSR, Otdel. tekh, Nauk, Mekh. Maskinost. No. 5 (1960). 16. A. E. KOBRINSKII,Dynamics of Mechanisms with Elastic Connections and Impact Systems. Iliffe Books, London (1969).
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