Lanchester damper effect for quenching the self-excited vibrations of mechanical systems

'roceedings of the NationỉU Centre for SdentiSc Research of Vietnam, Vol. 4, No 1, (1992) (3-24)
L A N C H E S T E R D A M P E R E F F E C T F O R Q U E N C H IN G
T H E S E L F -E X C IT E D V IB R A T IO N S
O F M E C H A N IC A L S Y ST E M S
N g u y e n V a n D ao
Institute of Mechanics NCSR of Vietnam
Su m m ary . The Lanchester damper effect for quenching both free and forced self-excited
vibrations of the mechanical systems with one, two and many degrees of freedom is investi
gated by means of the asymptotic method of averaging. Many quantitative estimations for
the stationary amplitudes of vibrations and theừ stability are given.
In recent years a lot of papers concerned with the dynamic absorber effect for quench
ing self-exciied vibrations of mechanical systems with finite degrees .of freedom and also
system s w ith distributed param eters have been published [1-20]. In paper [19] certain
characteristics of the behaviour of the Lanchester damper for the Q ue nch ing of self-excited
vibrations were established. In the present paper the analysis will be extended system at
ically for mechanical system s with one, two and many degrees of freedom.
Fig. 1 show s a self-excited system which consists of the main mass M and spring
with rigidity c and acted on by a “negative” damping force R. The self-excited vibration
of the m ass M is to be suppressed by means of a Lanchester damper with mass m and
d a m p in g m e c h a n ism (A).
1. IN T R O D U C T IO N
✓
\
R
M _ J X
S e/ f ~ e xc ite d
system
Fig. 1. Schematic diagram of a self-excitcd vibrating man M and a Lancheiter damper
4
NGUYEN VAN DAO
Usually, the “negative” friction force is of the form

= hi>0, 1 = 1,3, (1)
he re a n d .s u b se q u e n tly e is a sm a ll p o s itiv e p a ra m e te rs c h a ra c te riz in g th e s m a lln e ss o f th e
te rm s . T h e c oe fficie nt hi is a c o m b in a tio n o f lin e a r fr ic t io n a c tin g on m a ss M a nd th e
lin e a r p a rt o f th e e x citin g “ n e g a tiv e ” f ric tio n fo rce .
The g o ve rn in g e q u a tio n s o f the s yste m u n de r c o n sid e ra tio n are
M i + cx = —A( X — t>) + e[h\% — A3Ì 3),
ủ = V, (2)
mi) -f À (v — z ) = 0.
O b v io u sly , i f A is fin ite th en th e re w ill be no s elf-e x cite d v ib ra tio n o f th e m a ss M b ec au se
th e zero s o lu tio n X = u = 0 o f e q u at io n s (2 ) w ill be s ta b le . T h is c ase is n o t o f in t ere st fo r
this theoretical study. So, it will be supposed that the damping coefficient of Lanchester
d am p e r A is a s m a ll q u a n tit y o f o rd e r £.
It is n ec ess ary to e x am in e s e pa ra te ly tw o c a s e s : a st ro n g ab s o rb e r w h en th e r a tio o f
m asse s -77 is fin it e a nd w e ak o ne w hen th is r a tio is sm a ll.
M
2. S T R O N G L A N C H E S T E R D A M P E R
The motion equations in this case take the form
MX + ci = f [ - A(i - t>) -f /iii - /13X3] ,
u = vt
mi) = e\[i — v)
When £ = 0 equations (3) have a partial periodic solution of type
X = a c o s f u f - f \p), J1 = —
M
X = - a t ư sin[vt T
v = t) = 0,
u = fc,
(3)
(4)
where a, 6 are constants.
For £ 7^ 0 but sm all, the formulae (4 ) are considered as the transformation to new

variables a(i), 1/>(*), 6(i) w ith'the additional condition
â(í) cos(u>i -f _a(t)0(i) sin(u;i + = ,0. (5 )
Substituting the expressions (4) in equations (3) and solving with respect to the derivatives
ả, V' gives
LAN CHESTER DAMPER EFFECT FOR QUENCHING
c _ .
u a = “ — / s i ll <p:
M
w h ere
tocnp = — — F COS
M
tp u t + \p,
F = u (A - A i)a s in V? 4- /i3U 3 a3 sin 3 (p.
(6)
(7)
T h e d e riv a t iv e s g ive n b y e q u a tio n s (6 ) a re s m a ll a nd , c on se q ue n tly , a, rp a re e ss e n tia lly
c o n sta n t o v er an in t e rv a l w h ere <p ch an ge s by 2tt. T h e a p p ro x im a tio n w h ic h is no w to be
in t ro d u c e d consists of a ve ra g in g the rig h t s id es of e q u atio n s (6 ) o ve r 27T in <p. S o, in th e
fir st a p p ro x im a tio n w e h av e f ollo w in g av era ge d e qu a tio n s
ea
2M
a = - /li + - / i 3a;2a 2'), = 0.
(8)
T o d e te rm in e th e s te a dy s ta te re sp on se o f e q u atio n (8 ) w e p u t à = 0. T h e re is a
t riv ia l s o lu tio n , b u t a n o n t riv ia l so lu tio n a lso e x ist s, w it h a 0 g ive n by
A 0 = -/ỉ 31»;2 a~ = hi — A,
4
(9)
w h ere th e ze ro s u b s c rip t d en o te s the s ta tio n a ry s o lu tio n . S t a b ilit y w ill be in v es tig a te d by
s tu d y in g th e b e h a v io u r fo r s m a ll p e r tu r b a tio n o f the ste ad y sta te re sp on se . It is easy to

p ro ve t h a t th e z ero s o lu tio n a = 0 is stab le if
A - hi > 0, (10)
an d the n o n tr iv ia ] s o lu tio n d ete rm in e d by fo rm u la (9 ) is s ta b le if A - hỵ < 0 . T h e d epe n 
de nce o f th e s t a tio n a r y a m p lit u d e a<) on th e d am p in g c oe fficie nt A is re p re se n te d in fig. 2.
So fo r a st ro n g L a n c h e st e r d a m p e r in cr ea sin g th e d a m p in g fo rce (A ) le ad s to de cre a sin g
th e a m p lit u d e o f th e s e lf- ex cite d v ib ra t io n o f m ass M .
Fig. 2. The dcpcndcncc of amplitude of vibrating mass hi from the damping coefficient X in
the case of a strong Lanchcstcr damper
3. W E AK LAN CH ESTE R D A M P E R
Let us consider now the -case when the damper mass m is small of order c. The
motion equations then become
6 NGUYEN VAN DAO
(11)
(12)
M x + cx = e[ — A(x — t>) + hịX — /13Ì 3],
Ú = V,
mi) = \(x — v).
The solution of the system of equations (11) will be found in the form
I 2 ^
x = aco s V?, (p = ut + vs w = - 7 7,
M
X = —aa> sin V?,
u = au;(i? cos <p + G simp),
V = cujj2(—E sin V? + G cos £>),
where
_ A2 mA . .
= "T v™ ” TZ 0 0 ’ ( )
w(A- *+■ m*u/**J A- -f m-uj~
an d ơ, yị> a re fu n c tio n s o f tim e s a tis fy in g th e c o n d itio n (5).
S u b s tit u t in g th e e xp re s sio n s (12) in to e q u at io n s (11) a n d s o lv in g th e m fo r th e d e r iv a 

tiv es g ive s
itJCL — [a(x — v) — hịX 4- /13Ì 3] sin <Py
• e 3 (14)
wcl\Ịj = ^ [A(i — u) — M 4- /13 X ] cos V,.
These equations belong to the standard form to which the averaging technique is applied
[21]. So in the first approximation the right sides of equations (14) can be replaced by
their averaged values over one cycle of vibration:
à = — z [a(uE — 1) + hi — -h^uj2 a2] ,
x (15)
' ị . - ệ L v C - a .
The equations (15) have the following stationary solutions:
1) T h e ze ro s o lu tio n a = 0, w h ich is s ta b le if
X(uE - 1) + hx = h1 - 7T~— < 0- (16)
À -r
In the o p p o site case th e z ero s o lu tio n is u n st a b le an d th e m a ss M v ib ra t e s w it h a s t a tio n a r y V
a m p litu d e d e te rm in e d b y :
2 )
3 , *1 1 , m2(jj2X . .
An = jA sw OỈ - fci - m aw r + A3 > °- ( 1 7)
LANCHESTER da m per e f fe ct for quenching
7
Fig- s. Some typical curvcs given the dependence of amplitude of vibration on the
rn Ị c
damping coefficient A in the cast of a weak Lanchestcr damper. Curve 1: hi > y W curve 2:
m r r
k,< 2 v i r
This relation is plotted in fig. 3 giving the dependence of the am plitude of stationary
self-excited vibration on the dam ping coefficient A. The m inim um of the am plitudes cor
responds to the value
A = A. = mu/, (18)

and
- hi - ịrnM. (19)
If the parameters of the system under consideration are chosen so that
X = m' J ỉ ' 1201
then the self-excited vibration of the main mass M is com pletely suppressed.
So, for a weak Lanchester damper the suppressing effect is achieved only with some
intermediate values (A.) of the dam ping force (see Fig. 3).
4. S T R O N G L A N C H E S T E R D A M P E R E F F E C T F O R A S E L F - E X C I T E D
v i b r a t i o n a l s y s t e m w i t h t w o d e g r e e s o f f r e e d o m
We consider now the self-excited vibrational system w ith two degrees of freedom
(fig. 4) which consists of two vibrating masses mi and m 2 w ith elastic elem ents Cl, c 12
and viscous dam ping (/i12) between them. The exciting “negative” dam ping force R is
8
NGUYEN VAN DAO
su p p o se d to be o f fo rm (1). To s up p re ss th e s elf-e xc ite d v ib r a tio n o f th ese m a ss es tl
L a n c h e ste r d a m p e r (m , A) is used .
Fig. 4 ■ Two degrees of freedom self-cxcited system carrying a LanchtsUr damper
D e n o tin g th e d is p la ce m e n ts o f m as ses m L, m 2 a nd d a m p e r m fro m th e ir p o sitio n s o f
e q u ilib r iu m by xi} 12 a nd u re sp e ct iv e ly , the m o tio n e q ua tio n s o f th e sy ste m c on sid ere d
fo r th e ca se o f str on g d a m p e r ( — is fin ite ) ca n be w r itte n in the fo rm
m,
m l*l + C1X1 + C i2(J i - x 2 ) = - th i2{xi - i 2) +
m 2X2 + Ci2(xo — Xl) = —£^12(^2 “ *l) ~ sA(io — v),
ủ = V,
m ũ = —c A (u — Jo),
w here all kinds of friction forces are su p p o sed to be sm a ll qu an titie s of order 5.
B e fo re s tu d y in g the v ib ra tio n s g ove rn ed by e q u a tio n s (21) le t us e xa m in e th e se lf
e xc ite d v ib r a tio n s o f a tw o d egree s o f fre ed om sy ste m w ith o u t a L a nc h e ste r d a m p e r, whose
m o tio n is d e sc rib e d b y e q u at io n s :
m Lx L + (cL + cI2)ji - ci2x2 = -fffci3(ii - is ) + j

mnXn + 012(12 ~ Xi ) = -*£^12(^2 ~ i i ) •
U s in g th e t ra n s fo r m a tio n in to th e n o rm al m o des £ i, £2 •
*1 — %1 + s2»
x2 = <?lĩi -r &2s2ỉ
(23)
w here
n
o J
%
i = 1; 2,
c X c X 2 ^12
m i ĨÌĨ2
\j ( Ci’2 ~ 2 " — ) +
y V mi mọ /
m l m 2
4c?,
(24)
w e h ave
LàNCHESTER damper effect for quenching
9
•ii + n ỉíi = r r t ^ i + cri ^ ) l
Ml
s2 *+■ = 7 7 ” (^1 + ^ 2^ 2)1
Mọ
(25)
^here
Fa = -/i12(ii - x2) + /iiii - hzil,
F2 = - h i 2[Ì2 - ii), AÍ,- = mi + ơt2mọ, X = 1 , 2 .
Introducing the new variables a, 6, 0 , <f) connected with the co-ordinates {2 by relations
(26)

£1 = a cos 0, (i = —aO 1 sin 0, Ớ = H it -t- t/>,
£0 = 6 cos ry, £0 = —6 0 2 sin »7, rj = fiot -r
with the additional conditions
ả cos 0 — dip sin 0 = 0, 6 cos rj — bộ sin 77 = 0,
equations (25) are replaced by
O i ủ = “ T 7 - ( i \ + ơi/ọỊsin tf,
Ml
Cliarp = — ~ r ( F i 1- ơ i Ạ ) COS 0,
M l
0 26 = +ơ2F2)sinr),
Mo
Qnbặ) = - *77“ (^*1 •+■ f**) cos ĨỊ.
A/~
:7)
Since a, t/j, ò, ự) are slowly varying variables, then in the first approximation the
right sides of equations (27) can be averaged over the time, considering those variables as
constants. T he averaged equations will be
a — -
o 0
“■ hi *+- [ơI — 1)"/112 + — /13 Oj a ~ ■+• - h z fo b -] ,
o Af [ “ ^ + (Ơ2 1)2^12 + -hztiib2 + - / ^ n ^ a 2] ,
z Ai 1 4 L
^ = Ậ = 0.
The zero solution a = 6 = 0 of the equations Í28) corresponds to the equilibrium of masses
rnA and m2. This solution is stable if
Ml = hỵ — (ơi — l)2 /ii2 < 0 ) . .
ỵ2 = hi — (ơn — 1)2/ijl2 < 0.
If these conditions are not satisfied simultaneously then the equilibrium of the masses will
be unstable and vibrations m ay occur.
The vibration of the first mode with frequency ill and amplitude Oo is determined

10 NGUYEN VAN DAO
b y
b = 0,
A = — hi — (ơi — 1)2A12,
4
(30 )
which is stable if
Ả> - [hi - (ơ2 - l ) 2/iia ] — -B- (3 1)
T h e v ib r a t io n o f th e se co nd m o de w it h fre q ue n cy n 2 a nd a m p lit u d e 6() is d e te rm in e d
b y
a = 0,
B = \hzĩì\bị = hx - (ơ2 - l) 2/i!2, ^ ^
4
w h ic h is s ta b le i f
f l> Ì [/n -( ơl- l ) 3h13]= ÌA . (33)
It is e as y to p ro v e th a t sim u lta n e o u s v ib r a tio n in tw o m o de s is u n sta b le .
W e r e tu r n n ow to th e e q u atio n s (21) for a s tr on g d a m p e r. W h e n 5 = 0 fir st tw o
e q u a tio n s o f th e s ys te m (21) are c ou p led a nd th e la s t tw o e q u at io n s b eco m e
ủ = Vị V = 0
a n d th e y h a ve a s o lu tio n o f th e fo rm
X j = a c o s (n xt + íp) -Ị- bcos(ilot + <£),
X\ = — aCli sin (n ii + ip) — tfiosinfOoi -f ^),
X2 = ơia cos(Qii + 0) + ÒƠ2 cos(ÍÌ2t + <£)» (34)
x 2 = —a H iơ i s in (Q it + 0 ) — bíÌ2ơ2 sin (n 2Í ■+■ <£),
v = v = 0, Ui = u«> = c on st,
h ere th e n o t a tio n s (2 4 ) are u sed .
C o n s id e rin g fo rm u la e (3 4 ) as th e tr an s fo rm a tio n in t o th e new v a ria b le s a, t/>, b, w e
h av e a sy s te m o f e q u at io n s w h ic h is s im ila r to th e e q u a tio n s (2 7 ) a nd th e c o rre s p o n d in g
a ve ra g ed e q u a tio n s w ill be
à = — — [ — h i + Ớ ị \ + (ơi - 1)2Aì2 -f ~ h z ĩ í \ a 2 + - ^ n ^ b 2 ],

1 4 2 (3 5)
6 = — [ — /li + Ơ2 A + (cT2 — 1)2/*12 ■+■ -hrfinb2 + r^ nỉa 2].
2AÍ2 4 z
F ro m th e e q u a tio n s (3 5 ) one c an see th e fo llo w in g s te a d y sta te s o f m o tio n :
LANCHESTER* DAMPER EFFECT POR QUENCHING 11
1 ) T h e e q u ilib r iu m Xi = i ị = i 2 = X2 = 0 (a = b = o) is s ta b le if
(36 )
hi
-
Xi
< 0,
hi - < 0,
where
^1 = (ơi - 1)2^12 +
~ [ơ2 ~~ l ) 3^12 ơ2^‘
(37)
2 ) S e lf- ex c ite d v ib r a tio n o f m a ss es mi an d m? w it h fre q u en cy n ! a n d a m p lit u d e ao
determined by
6 = 0,
3 o n (38)
^0 — —/13OfaQ = hi — (ơ i — 1)**^12 ~ Ơ^À,
4
which is stable if
Ao > ị[hx - h ) = jB o . (3 3)
3 ) S e lf-e x cite d v ib ra tio n o f m a sse s m i an d m 2 w it h fre q ue n cy n 2 an d a m p litu d e 60
determined by
a = 0 ,
3 (40)
So = - h 9nĩbỊ = hỵ - (ơ2 - 1) /lÍ2 — Ơ-ỒÀ,
4

which is stable if
Bo > 2 ^°' (^)
4) T h e s im u lta n e o u s v ib r a tio n o f tw o m o de s w it h fre q ue n cie s n : , is u n sta b le .
F ro m th e fo r m u la e (3 8 ) and (4 0 ) it is e vid e n t th a t fo r a s tr o n g L a n c n e st e r d a m p e r,
in creasing its da m p in g coefficien t (A) lead s to th e d ecrease o f sta tion ary am plitu de o f th e
se lf-e xc ite d v ib r a tio n o f masses m i an d mo.
5 . W E A K L A N C H E S T E R D A M P E R E F F E C T
F O R A S Y S T E M W I T H T W O D E G R E E S O F F R E E D O M
In this section the case when the mass m of Lanchester damper is small quantity of
order e is investigated. In this case the differential equations of motion (2 1 ) become
miXi + (cI + c12)z 1 - c12x2 = sFit
+ c12( i 2 - Ix ) = s\F2 - X[x2 - v)Ị,
" I 42)
u = V,
m i) -f Xv = A Ì2,
w h ere fu n c tio n s Fi a n d F3 a re o f fo rm ( 2 5 ). T h e d iffe re nc e be tw e en th e e q u a tio n (2 1 ) a n d
(4 2 ) is t h a t in th e la s t e q u a tio n o f th e s ys te m (4 2 ) th ere is no s m a ll p a ra m e te r c .
U s in g t he n o r m a l c o -o rd in a te s (2 3 ) w e ca n tr an s fo rm firs t tw o e q u a tio n s o f (4 2 ) in to
th e fo rm
12 NQUYEK VAN DAO
(43)
£i + nỈÉi = ^ “ 1^1 + ^i[F2 - A(io - v)] j,
Ỉ2 + ^2^2 = + Ơ2 [f2 - - v )] 1»
an d th e la s t tw o e q u a tio n s o f (4 2 ) b eco m e
m ũ + Ati = A(<7a£ i 4- 02s2)> (44)
h ere th e n o t a tio n s (2 4 ) are used .
In t ro d u c in g th e ne w v a ria b le s ax, ao, £>1, ^2 co n ne cte d w ith th e co -o rd in a te s £1, $2
by th e fo rm u la e
£1 = ai cos V?1 , Í 1 = -<*1 ^ 1 sin <Pi, <iPi == H it + \pit
£2 = a2 COS <P2, Ỉ2 — — ^2 ^2 sin <p2j v^2 = ^2^ ■+* ^2» (45)

à i COS <PI — aiự>i sin ^>1 = 0, 02 COS <P2 — ao02 sin <P2 = Oj
we o b t a in
m ũ -r Ail = “ AỊơiaxHi sin v^i “H ơoơoOo sin ^>2)» (46)
a nd , s o lv in g fo r th e d e riv a tiv e s, w e ha ve
A iá i = + ơ i[F ọ - À (i2 - u)] I sin P i ,
nxa + c7i[F2 - À(i2 - w)]Ị cosv?i,
n 2ả 2 = - - ^ ị - Ị F ỵ + ơ 2 [/2 - À (i 2 - 1> ) ]| sin
ũ->a2'Ộ2 — ~ T T \ Fí + ơ-JF ọ - A(iọ - v )] Ị COS <p2 ,
ÀZ2 V *
w h e re th e fu n c tio n u is th e s olu tio n of e q u atio n (46):
\ ơ ì ai
u = 00" xõ(mfli sinv? 1 + AcospJ-i-
m*\ìị -+ A-
2fl2 (m flo sin v?2 ■+■ A cos £>2). (48)
(47)
F o llo w in g th e a s y m p t o tic m e th o d o f a v era g in g [2 1] in th e fir st a p p ro x im a tio n th e
r ig h t h a n d s id es o f th e e q u a tio n s ( 47) m a y be re p la ce d b y t h e ir ave ra ge d v alu e s o ve r o ne
c y cle o f v i b r a t i o n :
LANCHESTER DAMPER EFFECT FOR QUENCHING
13
ea 1
ỮA ' ~ n T x
Cd2
a 2 = -
2 Mr
Ị -/ii + (ơi - l)7hí2 + -h s tíịa * + - h & ị o ĩ + Ơ1 m2Q2 ^ ^2 I »
Ị —Al + (ơ2 — 1)2^12 + Ị^3^2a2 + “ /l3^ia? + ^ m2n2 + À* } *
(49)
c m H i cr][A2 d i
f liV i = -

2 M 1 (m 2n ỉ + A2)’
emĩÌ2Ớị\7a2
a2^ 2 = 2 A /2(m 2n ^ + Ã2) ‘
In c o m p a ris o n w ith th e fo rm u la e (2 8 ) w h en th e L a nc h est er d a m p e r is ab sen t, h ere th e re
e xis ts th e “ a d d it io n a l” f ric tio n fo rce o f ty pe
ơ\rr?n ? • m ọn2 + A3 . i= l, 2 . (50 )
F ro m the e q u a tio n s (4 9 ) o ne can fin d th e fo llo w in g s te a dy s ta te s o f m o tio n
1) T h e e q u ilib riu m o f m a sse s mi a nd mo w h ich c o rre sp o n d s to a i = ao = 0 is s ta b le
if
L /_ »o . ơ ?m 2f l? • A
h i - (ơ i - 1) *2 - “ ĩ c ị s T \ ỉ < ° '
m ní + A (51)
cr?m2n | Ằ 1 '
fci - {02 - I)7hi - i < 0-
O b v io u s ly , th e L a n c h e st er d a m p e r (A ) sta b iliz e s the e q u ilib r iu m o f th e m a sse s m i an d m 2.
2 ) M a ss es m a a n d m 2 v ib r a te w it h fre q ue n cy ĩìỵ an d a m p litu d e g ive n b y th e fo rm u la e
a2 = 0,
J 3 ft2 2 , *2, ơJm2nỊA (52)
/ lx _ j / . 3n ? a ? - /li - (<7, - 1 ) ^ 3 - J n 3 + A2 .
if th e rig h t s id e o f th e e x p re ss io n (5 2 ) is p o s itiv e . T h is v ib r a tio n is s ta b le if
ẢI > ~A2 (53)
(see (5 4 ) fo r A2).
3 ) T h e m as se s mi a n d m 2 v ib r a te w it h fre q ue n cy fto a n d a m p lit u d e d e te rm in e d by
fo rm u la e
14
NGUYEN van Dao
1
M > 2 1*
4) T h e s im u lt an e o u s v ib ra t io n o f m a sse s a n d m 2 w it h tw o fre q u e n c y nA, n2 is
u n sta b le .

F ro m th e fo rm u la e (5 2 ) a n d (5 4 ) on e ca n see th e rô le o f th e d a m p in g m e c h an is m
(A ) o f th e L a n c h e st e r d a m p e r in q u en ch in g th e v ib r a t io n o f m a ss es m i, m 2- A s in th e
case o f th e sin g le m a ss sy ste m c on sid ere d in se ctio n 3 , h e re h ig h e ffe ct ive n es s o f th e w ea k
L a n ch e st er d a m p e r c o rre s p o n d s o nly w ith som e in t erm e d ia te v a lu e o f A :
À = A. = m a x {m fl;} . (55)
*=1.2
if the right hand side of (54) is positive. This vibration is stable if
6. S T R O N G L A N C H E S T E R D A M P E R E F F E C T F O R
A S E L F - E X C I T E D S Y S T E M W I T H N D E G R E E S O F F R E E D O M
T h e r e su lt s o b ta in e d in th e p re v io u s se ct io n s m a y be e x te n d ed to a m o re c o m p lic a te d
m e c h a n ic al sy ste m w it h N d eg rees o f fre ed om (fig. 5 ). T h e g o ve rn in g e q u a tio n s fo r th is
sy ste m a re o f th e fo rm
c
,2
/77
Li
a
m
A
j "M -i, N
I
I
m2
-JJC;
t p 1 f i f i
/77/
- J - 2/
Ldnchesftr Jd/77pcr
s elf - excited s y s t e m
with N de cr ee s o f f re edo m

Fig. 5. Lanckattr damper attacked to a self-czciud system with N degrees of freedom
+ (cx + c12) x i - ci2X
2
= efi,
m2x2 + (ci2 + 003)^2 C12X1 — C23^3 — £/2»
mNỈN + Ctf-i.Ar(x/sr - ZAT-i) = e f tf y
ủ = V,
(56)
m v = e \( x# — v),
LANCHESTER d am pe r e f f e ct for quenching
15
where
/1 = ^1*1 - fcai? + ^12(^2 - ii)i
/2 = ^1 2 ^1 (^ 12 ■+■ ^23)^2 + ^23*3>
(57)
/ at = fc N - i.js r( ijV - l - Ztf ) + A (v - 2at).
T h e o th er param eters are clear from the fig. 5. The exciting “negative” viscous dam ping
force is supposed to be of the form (1 ).
The characteristic equation of hom ogeneous part of the first N equations of the
sy ste m (5 6 ) is
D ( n 2 ) = 0, (58)
w he re
c 1 -f c 12 — rn\n —c 12
— C \ 2 C 1 2 c ° 3 — T T lo O " . .
cK-l.N - m A'
• n 2
It is s u pp o s ed that all roots of the equation (58) are simple and they are denoted by
Qj, , ĩiị,. Let ơtjỉ = A(Hy) be the algebraic supplem ent of the elem ent at the i-th
c olu m n and la s t line of the determinant z>(Oj). Using the transformation into the normal
modes £1 , Ỉ2 , , ÍA' •

= ^ I = 4 jT ’ j = 1 . 2
y=i Ơ1
we ha ve
(59)
where
y = 1,2, ,7V,
The N equations (59) and the last two equations of (56) governe the m otion of the system
under consideration in the case of the strong Lanchester damper. W hen E = 0 these
equations have a solution of the type
Zi = ay cos <p_,, (j = - a y f lj sin v=>y,
V = V *= 0, u = uo, j = 1,2

= fijt +
(60)
16
NGUYEN VAN DAO
C o n s id e r in g (6 0 ) as th e fo rm u la e o f t ra n s fo rm a tio n in t o n ew v a ria b le s ay a n d rpj w it
a d d it io n a l c o n d it io n s
an d s u b s tit u tin g th e e x p re ssio n s (6 0 ) in to e q ua tio n s (5 9 ) a n d so lv in g fo r th e d eriv ativ e
Qy, rpj we o b ta in
A v e ra g in g the rig h t h a n d sid e s o f e q ua tio n s (6 1 ) in tim e we h av e in the fir st a p p ro x im a t e
F ro m th e e q u a tio n s (6 2 ) o ne can fin d the fo llo w in g s ta t io n a r y re gim e s of v ib r a tio n
1 ) T h e e q u ilib r iu m ay = 0 (; = 1,2, , N) w h ic h is st ab le if
2 ) T h e h a r m o n ic v ib r a tio n o f j - no rm al m od e w ith fre q ue n cy Clj a n d w it h th
a m p lit u d e d et erm in e d b y
i. e. th e n o rm a l v ib r a ti o n w it h th e a m p litu d e , w h ic h is h a lf s m a lle r th a n th e o th er n o rm s
v ib ra tio n , is U n sta b le.
3 ) S im u lta n e o u s v ib ra tio n is in som e n o rm al m o de s, b u t t h is is u n sta b le .
T h e fo rm u la e (6 4 ) a nd (6 5 ) sh o w th a t th e d a m p in g m e ch a n is m (A) o f th e L an ch e ste
d a m p e r s t a b iliz e s th e e q u ilib r iu m a nd th a t fo r st ro n g d a m p e r in cre a sin g A le ad s to th

d ec rea se o f a m p lit u d e s o f se lf-e x cite d v ib ra t io n s o f th e m as se s m i, m2j , .
àj COS <Pj - djipj sin <Pj = 0, j = 1, 2, , N t
(61
(62
V'j =0, ; = 1, 2, , N,
w h ere
(63
h ,- H 3- d ^ A < 0, ; = 1,2
(6-4
A, = 0,
Aj = hi. - Hj - < # '5 À,
(65
w h ic h is s ta b le if
A, > ị [ h x - Hi - d{f \) = l-A„ (t,y =
1.2, ■ , N, I # ;■),
(66
LANCHESTER DAMPER EFFECT POE QUENCHING 17
7. W EAK LANCH ESTE R D AM PER EFFECT FOR
A SY ST EM W ITH N DEG REES OP FR EEDO M
It is supposed that the damper mass m is small of order c. Then the last two
equations of the system (56) become
Therefore
N
m ũ + Xủ = Xitf = -A d^ajClj&hi <pj.
y =i
\ **ya j / • * \
“ - * E -m? n +Ỉ5 lm“” + n j e“ p '-) -
j= l 1 3
(67)
(68)

Substituting this expression into the equations (61) and averaging their right sides in time
gives
an ; < # ,s A
m ‘
(69)
; _ cmĩlj(ijd^ X2
G;^J - “ 2My(m2^ + A2) *
From the equation (69) it follows that
1) The equilibrium of masses (i. e. ay = o) is stable if
A
fei-gy- - W T T T <0, y = 1,2, , A'- (70)
m-W y + A*
2) M asses m A, m2, , m x vibrate with frequency n ; and am plitude determined by
Ai = 0, » = 1 ,2 , , Ar, t # y,
m2ĩ ìự ^ X (71)
Aj — hi Hj 0 2 p »
m-Wy -f A-
which is stable if
3) Vibration of masses


with various frequencies, but this is unstable.
From the formula (71) one can see that a weak Lanchester damper is highly effective
only with som e interm ediate values of the viscous damping coefficient A :
A = À* = max {mil.}.
»=: 1-r JV ;
(73)
18
NGUYEN van Dao
8. LANCHESTER DAMPER EFFECT FOR

A FORCED SELF-EXCITED VIBRATING SYSTEM
In this section it is assumed that the main mass M (fig. 1), beside the “negative”
d a m p in g fo rce R , is ac te d on b y a h a rm o n ic fo rce eP sin w ith sm a ll a m p litu d e eP an d w ith
frequency 7 which is close to the natural frequency u>, a namely (the primary resonance)
' “’ “ S ' |74>
w h ere A is the d et u n in g b etw e en tw o fre q ue n cie s 7 an d u/; p , 7 are c o n sta n ts . T h e m a in
goal here is to estim ate the effectiveness of the Lanchester dam per in suppressing the
vibration of the mass M under the simultaneous excitation of the mentioned above forces.
The governing equations are
Mx + cx = eR(x) + £A(v — i) H- ePsin^t,
ủ = V, (75)
m v = e \ ( i - t>),
for the strong damper, and
MX + cx = c/?(z) + £À(ư — x) + cPsin'■yi,
ủ = w, (76)
mi) = A(x — v),
for the weak damper.
L e t us co n sid e r firs t th e case o f th e s tr o n g d a m p e r. W he n 5 = 0 th e e q u a tio n s (75 )
has a periodic solution of the form
I = acosịyt -f lỷ), X = - a 7 SÌn(7 1 4- Ộ),
(77)
l> = X) = 0, u = Uy = const.
U s in g a a nd Ip as ne w v a ria b le s a n d s u b s tit u tin g the e xp re ssio n s (77) in to e q u a tio n s (75)
and s olv ing for th e d eriv a tive s <2, 0 w e ob tain
là = - ~ - Ọ s in p,
. \ <-S)
where
<p = nri 4- li ’ ,
(79)
Q = - A i -t i?(X) -r Ax -t- psin -)t.

In the first approximation the right hand sides of equations (78) can be replaced by their
averaged values over one cycle 2n of the variable <p:
LANCHESTER DAMPER EFFECT FOR QUENCHING
It is noted t h a t h ere the in te re s tin g ca se w ill be
19
kl - X > 0.
(81)
To find the stationary solution we put in equations (80) à = \ịỉ = 0 so that
(82)
here the zero subscript denotes the stationary values. Excluding the phase 00 we obtain
the following approximate formula:
I *>
_
**) ~
where V = — .
u~
The relation (84) gives the dependence of the amplitude a<, of stationary vibration
of mass M on the frequency 7 and is plotted ill fig. 6, where A = ịhyui2áỉr The form of
the resonance curve depends on the amplitude p of external force. When p = 0 there
exists pure self-excited vibration of mass M. The resonance curve degenerates into a point
I (u- - 1, A% - hị - A). So
characterizes the amplitude of the pure self-excited vibration of mass M. For sufficiently
small values of p (see branches 1., fig. 6) the resonance curve consists of two separate
brunches: one is closed around the point 1 and the other lies above the axis Ư2. When
p~ = p; = - A)3/ 81/<3 th e re so na nc e curve redu ce s to a c ro ss in g b ra n c h (se c curve
2, Fig. 6). For p > p. the resonance curve has only one branch lying outside the curve 2
(see curve 3, Fig. 6).
We study now the stability of stationary solution a,,, v:t» equations (80). By
introducing a perturbation of a„ and v/„
(83)

SoK'ing equation (83) for frequency *7 of forced excitation we have
A, — h[ — A
a = a() + ỏa, V = Vu -r f v
lin e a r iz in g e q u at io n s (80) in Say Sip and s u b st itu t in g the e xp re ss io n s for 511100, C05 Ạ, (82),
th e fo llo w in g set o f firs t o rd e r d iffe re n tia l e q ua tio n s w ith constant co effic ien ts w il l be fo un d
20
NGUYEN VAN DAO
T h e s t a b ilit y o f th e s ta t io n a r y so lu tio n do, ^0 d ep en d s o n th e e ig en v alu e s o f th e c oe ffic ien t
m a t rix o f th ese v a ria tio n a l e q ua tio n s w h ose c h a ra c te ris tic e q ua tio n is o f th e fo rm
dW
A2 + £7[3Ji3u;2a{; — 2(/ii - Aj]A. + £2T“2 =
o a 2
w h e re w is d e te rm in e d by fo rm u la (8 3 ) an d w = 0 is th e e q u atio n o f re so n an ce c u rv e . Fo r
a s y m p t o tic s ta b ility one w o u ld re q uire a ll th e e ige n va lue s to h a ve n eg a tiv e re a l p a r ts , so
th a t
Fig. 6. Frequency-amplitude diagram for equation (84) and the stability region of stationary
vibration. Tht hatching area is unstable. Point I: p=0; curve 1: 0 , p < p.; ciirvc 2: p = Pm =
4 [(A; — A)5 / 8 1 /13] curve S: p > P'
T h e lo cu s o f p o in t s w h ic h s ep a ra te s s ta b le a nd u n sta b le s o lu tio n s is d e te rm in ed by c o n d i
tio n
d W , ,
dA = (88>
*2 1
w h ic h is th e e q u a tio n o f an e llip se w it h ce n te r at I/2 = 1, A = A. a nd se m i-a xe s -A .,
3 3
cA*
. So , th e h a tc h ed a re a in fig. 6 is u n sta b le .
LANCHESTER damper eff ec t for quenching 21
It is worth mentioning that in the corresponding system (fig. 1) without the Lanch-
e ste r d a m p e r, th e d ep en d en ce o f th e a m p litu d e b of m a ss M o n th e r a tio fre q ue n cy V — —

U)
is given by the formula
1/2 = ~ ^ 2' (89)
T h e d iffe re n ce b etw e en th e fo rm u la e (8 4) a nd (8 9 ) is t h a t in st ea d o f hi in (8 9 ) one h as
Jii - A in (84).
To compare the maxima of amplitudes of vibrations of mass M in the case (a) with
ajid (b) without the Lanchester damper it is necessary to find these maxima (ama* and
bmax) fr o m th e fo rm u la e (8 4 ) -an d (8 9 ). T h e y are th e g re at est v a lues a a nd b w h ic h m ak e
vanish the expressions under the square root in the formulae (84) and (89), i. e. they
satisfy the following equations
-hzP2 = A{A -A .), (90)
4
j /i3P 2 = B[B - B.), (91)
4
where
3 3
Á a J A.% K\ A,
4 4
B = % w 2fc2, B. = hi, Bmax = 3-h 3wH7max.
4 4
E q u a tio n s (9 0 ) a n d (91) c an be so lve d g ra p h ic a lly b y d ra w in g in the ( z ,Y ) -p la n e the
s tr a ig h t lin e (fig . 7 ) :
n = z-h3p2,
4
and two cubic parabolas
Y2 = z(z - A.)2,
n = *{* - B.)2
The intersection of Y\ and Ỉ2 and of Yi and y3 gives the root of equations (90) and
(91). The greatest value of these roots determines the maximum of the amplitudes of
corresponding vibration (see fig . 7). The abscissas of the Tightest points of intersection of

the straight line Yi with the curves y2 and y3 are the values Amo* and Bmax- We can take
approximately
22
NGUYEN VAN DAO
So, the dem inution D i of the maximum of the amplitudes of vibration of mass M by
means of a strong Lanchester damper is characterized by À :
D i = A. (92)
Fig. 7. Graphical solutions of equations (90) and (91)
We consider now the case of a weak Lanchester d am per with the governing equations
(76). The solution of these equations will be found in the same form as (77) but in this
case the function u satisfies the differential equation
mủ + Aủ = Ai = — Aa*) sin 'P,
and therefore,
x2a m~ia A .
u = T 7 T 3 ? cos p \ 2 V'Z? 2 sin V93)
A2 4- Tri~i A2 -f m -7 2
Equations for a and \f> now are of the form
t
M
where
n ni-\ A _ n • ma~Ị2\ 2 m2a~Ị*\
Ọ. = J?(i + Ax + PSÌH7Í + 3 -r f-^ C 0 S 9 + -T T T rrrin p .
m- 7* -f A - m -7* -h A '
Averaging the right sides of equations (94) we have in the first approximation the following
equations
Qm sin <p, 7at/> = 7 7 Ọ* cos V?,
M
(9 4 )
LANCHESTER da m per e f fe c t fo r quenching
23

7<i = -
e r , 3 3 3 m2a7 3 A ]
; c /. ma72A2 . V
The stationary solution ao, v>0 is given by
(95)
/ 3 , 2-2 , m 37 2A ^ n / A n r r2OoA2
( 4^37 ag - hx + - ^ 2 ^ 3 j ^ o = - f cos 00, Aao + ^ f ^ = PsinV-o.
Eliminating the phase V'o and solving for the frequency 7 of external force we have the
following approximate expression
2 emuj2\ 2 e [p* " ' m2u2 A 77
^ ^ 7 1 o / o 0 ToT I / 0 A / ^ ^ \ ^ 0 *lỵ T o 2 \ 2 / *
A/u;2 (m2 u/2 -f A2) M u'2 y Oq -m2a/2 + A2 Í9 0 Ị
7
V — — .
UJ
By analogous discussion for the formula (92), here one can see that the diminution D 2 of
the maxim um of the amplitudes of vibration of mass M with the help of a weak Lanchester
damper is characterized by
2 ° A
_ m u r X . _v
Da = - ụ -~ - y . (97)
m-uJ~ -f A*
The biggest value of Do corresponds to À = mtư :
D 2m ,« = \ rn“ ' (9 8)
C O N C LU S IO N
The analysis shows that the Lanchester damper is suitable for quenching the self
excited vibrations of m echanical system s with one, two and several degrees of freedom but
it is necessary to distinguish between two different kinds of da m per: the strong damper
and the weak one. T he first acts on the principle of increasing the damping coefficient
(A), while the weak damper works effectively only with some intermediate values of this

coefficient.
.ACKNOWLEDGEMENT
The author wishes to express his deep appreciation to Prof. Dr. Peter Hagedorn for
helpful discussions during the preparation of this paper and for enabling him to make use
of all facilities of the Institute of Mechanics of TH Darm stadt, FRG.
The support of D A A D is gratefully acknowledged.