Nonlinear connected oscillations of rigid bodies

PROCEEDINGS
OF VIBRATION PROBLEMS
WARSAW, 3, 10 (1969)
NONLINEAR CONNECTED OSCILLATION S O F RIGID BODIES
NGUYEN VAN D A O (HANOI, VIET NAM)
1. Introduction
At present, the theory of oscillations of nonlinear systems has acquired special interest
and has achieved considerable development. In spite of the vast achievements of the
theory of oscillations, it has now come up against phenomena whose essence cannot
be fully explained by means of well-known models. Lately have been observed phenomena
which are the result of intense oscillations of rigid bodies in the directions of their coordi
nates not subject to external forces. These oscillations were first investigated by V. o.
K on o ne n k o [1-5].
In the present paper, we shall consider connected oscillations of a vibrator and of
rigid bodies accomplishing plane-parallel and spatial motions, and also of elastic beams
[11-19]. These systems with two, three, six and infinite degrees of freedom perform for
ced stationary oscillations characterized by constant amplitudes and frequencies, non-
stationary oscillations when passing through the zone of resonance and self-excited oscil
lations.
The basic problem is stated as follows: to find the conditions of,origin of the oscilla
tions of rigid bodies in the directions of their coordinates not subject to external forces,
to determine these oscillations and to investigate their stability.
We consider a nonlinear vibrator fixed to immobile foundations by three springs
of equal length /. The axes of these springs lie in a single vertical plane in which also is
contained the vibrator (Fig. 1).
2. Connected Oscillations of the Simplest Vibrator
y-í ẩ
I
F ig. 1
304 Nguyêỉỉ xăn Dao
The elasticity of springs is assumed to change by the linear law with stiffness coeffi

cients k?. The vibrator considered as a material point is subject to the harmonic external
force psin((ot+-@) directed unchangeably along the vertical axis.
The motion of this system, when k°i = £3 , is written by the equations:
Fi = hy+ -yk2xi + kiy'i- k 0xly,
x\ = 2ku }\ — k2, k0 = 2k- + k2, kt = ạ = Po/IM,
M is th e m a s s o f th e v ib r a to r a n d ỊZ is a s m a ll p o sit iv e p a r a m e te r c h a r a cte r iz in g th e s m a ll
ness o f the term s o f the function s Fi an d F2. It is ea sy to check that the system of Eqs.
(2.1) posseses a solution:
(2.3) X = 0, y # 0,
which corresponds to the oscillations of vibrator only in the direction y.
For the linear theory of oscillations, the solution (2.3) is always stable, and therefore
no o s c ill a tio n s of the vibrator in the direction X occur. However, the nonlinearity of the
equations of motion changes this situation. Under definite conditions, the solution (2.3)
may become unstable and the oscillation of the coordinate X arises.
The loss of stability of motion (2.3) is expected in the zone of resonance. First, we
investigate the simple subharmonic resonance of the second kind, when there is the correla
tion
y = A2 cosd2 + q* sin cot, ỳ = —Ằ2A2smd2Jt q*cocos(ưt9 q* = ^/(Ẩị—co2).
The transformed equations are:
x+Ằ]x = x,y),
ỹ + ĩịy — —fiF2ịỹ,x,y)-\-q%\nwí,
(2.2)
F j = hx + k 2x y + - j k 2x i — ,':zx}/2,
tf — -j-co2 = lue1.
We transform the system o f Eqs. (2.1) by means cf the formulae:
(2.4)
Ù) • (O . (i) 0} 0)
X = Ni COS — / + A il sin — t, X = — — Ni sin — 1+ - y Mị COS — t
These eq uatio ns belong to the standard form which is applied coveniently by the method
of perturbation theory [6].

Nonlinear connected oscillations o f rigid bodies 305
The solution of the system (2.5) is found in the form:
(2.6)
N, = N°t+n ơ,(/, N l M ữ„ A\, yị),
Mx = MĨ+fiU2Ọ, N°u M°, Aị, yị),
A2 = A\ f ịaUẠị, Aỉl Mĩ, Aị, yị),
y2 = yĩ + MUA(t, N°, Aị, yỉ),
where flU, are small periodic functions of time.
Quantities NĨ, M,°, Aị and YĨ are determined in the first approximation from the
equations:
= - I - ỳ M - k2 q * m - [*1 - ý K q'*2 -+ y k2( K + M ?2)]M \Ị + . ,
(2.7) ị (a íA + * 2g* )W ? _ Ị e i_ . i _ * 0 í * 2 f . | * 2« + A / f ) Ị ; v ỉ Ị f
which are obtained by averaging the right-hand sides of (2.5) according to time. Non
written terms in (2.7) will be equal to zero when A2 = 0.
The equation for A2 is independent of the rest of the equations, from which it follows
that the amplitude A 2 tends to zero. Therefore, below we are interested only in the equa
tions for NĨ and Af? in which the non-written terms are rejected.
The values NĨ and Ml in the stationary regimes of motion are determined as the roots
of the system of equations:
Hence we arrive at the amplitude of oscillation of the vibrator Ân the direction x:
Thus, we have obtained two forms of stationary oscilla tio ns which correspond to
the signs plus and minus before the radical. In order to explain which of these forms of
oscillations corresponds to the real stationary process, we investigate their stability. To
this end, we analyse the variational equations formed for the solution (2.9).
The result of investigation shows that of two forms of oscillations, the form with
large amplitude is stable and the form with small amplitude is unstable.
In Fig. 2 is represented the dependence of the quantity a2 on p for Q = 0.1, Q = 0.25
with diverse values H: 0.1, 0.15 and 0.3, where
(2.8)
(k2q*-wh)N°l ^ị2el- k 0q*2+ ~ k 2Aữ1]jMỈ = 0,

(k2q* f co/0 A/? f ^ - k o q ^ + ị- k tA ^N Ĩ = 0,
A? = +
(2.9)
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NguyéH văn Dao
Fat plots on the amplitude curves correspond to stable states.
When p increases from zero, the state of rest remains stable until the point s is reached.
Beginning from this point, the subharmonic oscillations of the vibrator in the direction X
appear. By further increase of frequency of external force, the amplitude of oscillations
grows at first along the curve STL At the point / a jump of amplitude occurs. The value
of the amplitude jumps down to the point M, and by further increase of frequency of
external force is chanced along the curve MN—that is, the amplitude tends to zero.
If we now begin to decrease the frequency of the external force, then the amplitude
of oscillations changes along the straight line MD. On reaching the point D, the value
of the amplitude passes to the point T and further is changed along the upper branch
of the resonant curve TS.
Note that in speaking about the change of frequency of external force we mean a very
slow change so that in practice at each moment of time the system can be treated as sta
tionary.
Analogously investigated are the simple principal resonance when the natural fre
quency ?.i of system is in the neighbourhood of quantity (0, together with the simple ultra
harmonic resonance. It is proved that in simple ultraharmonic resonance no oscillations
o f the vibrator in the direction X occur.
In different resonance cases—double and combinatorial — the averaging equations
have a complicated structure and, therefore, in these cases principal attention may be
concentrated on finding only the necessary conditions for the origin of the oscillation
o f the vibrator in the d irection o f the coo rd in ate X. It has been proved th at this oscilla tion
cannot arise in the cases of double subharmonic resonance when Ả] — -i- -\-uEi, ẰĨ = —5 +
rr nr
\-ịxe2 if n # m, n 7* 2m and n ^ 2, m 7* 3, in cases of principal ultraharmonic resonance

when
7* CO2 f pe l9 ẰỊ 7^ 4co2-f fxe2 or ẲỈ ^ 9oj2-\-fx€2 and when AỈ = n2cu2+ f ie j,
xị = cu2+/xe2(n > 1),
in cases of double ultraharmonic resonance when
= n2(o2+/xelt )\ — m2(X)2 ỊẦS2 if n ^ m, In 7* m, l—m+2n 7^ 0
and 1 + m —In # 0 ,
Nonlinear connected oscillations o f rigid bodies
307
and also in cases of ultrasubharmonic resonance if k\ = —Y + /^ 1
.2
Ằị = m2aj2+ỊẨ€2> n ^ 2 or a] = rtW +//£i, Ằị = -^2" + /^2 Ĩ /I, /w > 1,
3. Connected Nonstationary Oscillation when Passing Through a Resonance of Vibrator
We consider in this paragraph the nonstationary oscillation of a vibrator assuming that
the system investigated is subject to external harmonic force directed along the vertiẹal
axis y, and that the frequency of this force changes so that the system passes through
a resonance after a definite time.
Using the assumptions of the preceding paragraph, we can write the equations of
motion of the vibrator in the following form:
The momentary frequency v(t) = dOỊdt of the external force is assumed to be a slowly
varying linear function of time.
We shall consider a resonance of n-kind, assuming that the frequency of the external
force v(r) takes values which are in the neighbourhood of nX\, where n is a rational number
the ch oice o f w hich depends o n the kind o f resonance in vestigated — that is, when betw een
the frequencies v(t) and Ai we have the correlation
Bearing in mind the application of the asymptotic method of nonlinear mechanics
for the construction of approximate solutions of Eqs. (3.1), we transform them into
standard form by means of the formulae which reduce
X, X, y and ỳ to new variables
<*1, a2y V>1 and y2:
(3 .1 )

x+X]x = — X, y ),
ỳ + % y = — ự P Á ỳ , X, )0 + ?sinỡ(0.
(3 .3 )
X = —Xidi sin
The transformed equations are:
308 NguyéH văn Dao
+ - y k2a\ COS30 1 —kồai(ạ2 COS # 2+ 0 * sin ớ)2 COS 0 1 Ị COS 0 ị ,
= -J- |/ỉ(^*vcosỡ — Ằ2a2sin 0 2) + k2aĩ COS2 01+k^q* sinỚ4 ứ2cos 02)3
sin 0 2 »
-+ /Ci(<7* sin 0 - f a 2 COS (& 2Ỷ — k<ì c i\{q * sin ớ 4 a 2 COS 0 2) COS2 (p ì — q * ơ n 2 s in 0
q* dv \ - 6 d
+ ~

-T- cos 3 f cos </>2 > where <Ị>1 = — + y/j, (p2 = — -ị.y2.
ịx at ) n n
So far we have made only the substitution of variables and the equations obtained are
equivalent to the original system (3.1). However, Eqs. (3.4) are of such a form as may
conveniently be applied by the method of perturbation theory of nonlinear mechanics.
Now we take into account the study of single-frequency oscillations in the zone of
resonance, which is in the neighbourhood of ).ị. It is assumed that the natural frequencies
o f the system considered are linearly ind ep en d en t, so th a t between them there is n o co rre
latio n o f the form /lA j-f *2^2 = 0, w here /, are in tegers n o t sim ultaneously equal to zero.
The resonant oscillations of the vibrator in the neighbourhood of the frequency Ẳl9
with conditions as indicated above are for the most part characterized by the change
of the coordinates al and y>i. While the oscillations of other coordinates a2 and y>2 will
flow far from the resonance, they are sm all as com p ared w ith the oscillation s o f the coo r d i
nates ax and V>!, and therefore in the first a p p ro xim atio n w e m ay disregard th em .
Following the method of perturbation theory, approximate solutions of Eqs. (3.4)
are taken in the form:
(3.5) Qị = a+ịxV^t, a, yO, Vi = V>+/*V2(t, a, y),

where ỊÀVị{t9 a , y) is a small periodic fu n ctio n o f t.
The principal parts of the solution for a and ip are determined from the equations of
first approximation obtained by averaging the right-hand sides of Eqs. (3.4) in time.
We have for the principal resonance (n = 1):
(3.6)
4 = V W + ^ ( - Ỉ *.«■ -1 *.?••+ ý CM 2»).
for the subharmonic resonance of the second kind (n = 2):
(3.4)
CỊ* dv I
— k0a](q* sin d- a2 COS 0 2) COS2 &i — q*ơn2 sin 0 + — ~ COS ỚỊ
= Ằ2— ^ ~ + - A — \h(q*v COSỠ — ^2fl2sin02) + 4- ^2^1 COS2
ỔÍ 72 A2a2 I 2
Nonlinear connected oscillations o f rigid bodies
^09
and for n 7* 1 , 2 :
da h
From Eqs. (3.8), it can be seen that when n 7^ 1, 2, the amplitude a of the oscillations
teDds asym pto tically to zero w hen / 00, and therefore n o oscillations o f the vibrator
in th e d ir e c t io n X o c c u r .
Now, according to Eqs. (3.6) and (3.7) obtained, it is easy to construct the resonant
curves which characterize the change of amplitude and phase of oscillations in time.
The progress of the transient process can be calculated by numerical integration of
these equations. The result of such calculation for Eqs. (3.7) in the case X = 0.32; /j = 0.5;
/ 2 = 0.1 where X — ịxhịì^s 11 = ạk2q*l2X]y I2 = fẦ,kồq2J2X]y B = 3 fik2a/%X{ip = v(t)l 2A|,
and with velocity of change of frequency of external force •dpjdt = 0 .1 , dp/di = —0 .1
is represented in Figs. 3 and. 4.
a
b
Nguyên văn Dao
4. Connected Self-Excited Oscillation of a Vibrator

In the literature, well know n is th e m echanical unidirectional m odel o f an self-e xcie d
system w ith endless band: a h eavy load fastened to the im m ob ile p oint by a sp rin g ies
on an endless horizontal band m ov in g under a load w ith constan t v elo city .
In the present paper, we con sider the m ultidirectional m od el o f a self-excited system,
assum ing that the vibrator is fastened to three im m obile p o ints by springs o f equal length,
Fig. 5.
The self-excited oscillations arise in the system considered in consequence of the action
of the force of sliding f r ic ti o n Tị at the point of contact of the vibrator with the band.
This force is a function of relative velocity V — ỳ I = l(v 0—ỳ) and is conveniently repre
sented in the form Tị = mlT(v0—ỳ), where V = Iv0 is the linear velocity of the bard,
ỳị = lỳ is the v elocity o f the vibrator.
It is assumed that the force of sliding friction and also the nonlinear terms are smill
quantities of the first order. Later, we shall show the smallness of the terms enumerated
by the small parameter ỊJL.
Using the notations of the preceding paragraphs, we can write the equations of tie
vibrator in the fo llow ing form :
The second equation of this system, which describes the self-excited oscillation of tie
vibrator in the direction y, has been studied exhaustively in the unidirectional modd.
Our task will be concluded by investigating the origin of the oscillation of the vibrator
in the direction of the coordinate X .
In the system investigated, intense oscillations of the vibrator in the direction X a'e
expected in conditions of internal resonances. Therefore, we assume that between tie
natural frequencies Xi and Ă2 of the self-excited system there is a correlation:
F ig . 5.
( 4 . 1 )
X \ X ] x = — f i F ị ( x , X , y ) ,
ỹ+Ằịy= -/i[F2-T(v0-ỳ)].
Nonlinear connected oscillations o f rigid bodies
311
(4.2) )ị = Xị-{ne,

w h e r e e is a q u a n t i t y o f d e t u n in g o f f r e q u e n c ie s.
It is easy to prove that in the other resonance relations no oscillation of the coordinate
X occurs.
By substituting the variables according to the formulae
x = A lcos 9lf X = - Ằ lÀ 1 s in ớ i, dị = Ằyt+Yu
(4.3)
y = À2cosỡ2t ỷ = —Ằ2À2siũ 02, 02 = hit-ryii
Eqs. (4.1) are reduced to the standard form:
dAi __ r • a dyi ự
(44) ■ = ~ [F2-T(v0 \ A2,42sinỡ2)]sinỡ2,
= ỵ Ị - [Fi-T(v0 ị ẰIA2siũd2)]cos02.
Of course, the self-excited oscillation in the system under consideration depends
essentially on the characteristic of friction Tịụ) and may be only on the decreased branch
of the curve
(4.5) r(u)= T (u)ihu = 0,
on which Eqs. (4.1) describe system with “negative” friction.
For the determination of the approximate solution of the system (4.4), we shall use
the method of perturbation theory, according to which the solution of this system is found
in the form:
(4 Ai = tfi ±t*Uị(t, au a2, r u A ) ,
Yj = rj-rf*Uj+i(t> a2, r l9 r 2)y i ,j = 1 ,2 ,
where /ẤƯk are small periodic functions of /, and the principal parts aiy ưị of the solution
in the first a p pr ox im a tion are determ ined by the eq u a tion s o btained from (4 .4 ) by ave
raging their right-hand sides in time.
The form of the averaging equations depends on the concrete form of the functions
T(u). We examine certain typical characteristics of friction. First, we assume that the
characteristic of friction has the form:
(4.7) T(u) = A0signw—hu.
Putting this expression into the system (4.4), and averaging their right-hand sides in
time, we arrive at:

for a2 < Vo A
dt 2Ằ2
(4.8) da2 _ -ịẮ
dt Ih
— ~2\ [~~^2+ k2ơ2sin{2ri
d r ' - It f ~ k2\aị-k o ấ ìi Jt2ữ2cos(2rx- A ) Ị ,
(h- h ^ 2a2+ ~ * 2 aỉ sin(2 r , —r 2)l,
4 Problem y drgaủ
312
Sguyêrỉ văn Dao
i n
dt~ = ~ĨX~cT i3^ i —2k°aĩfl2 +^20?cos(2A - r 2)];
and for c2 > Vo P'2
(4.9)
dal
- k [- h~df
drx
dt
~~ 2A2
da2
-f*
át
- 2X1
dr2
V
dt 8 Ằ2a2
£+■ ~ k^áị — k0a\\-k2a2cos{2ri ~ r 2)j,
(A—Ai)/2 a2+ 4 r ^ sin (2 /\ —A ) f COS arc sin -5^ -
4 71 ^2^2
[3K!fl|-2Ả:oơ?a2 f * 2 ofcos(2 r , - r 2)].

at o/.2 ^2
Thus, the oscillating process will consist of two stages. In the first stage of motic
when the amplitude of self-excited oscillation a2 of the vibrator in the direction y de>
loping from zero remains smaller than fc’oM?, the oscillations of the system investigat
are governed by Eqs. (4.8). The self-excited oscillations continue to develop, amplitu
a2 achieves the value v0/á2, and from this moment of time the phenomenon begins to
described by Eqs. (4.9). The further course of the change of Qi, 71! leads to some stationa
values, for which ảũiịảt = dr{\dt = 0 .
Hence we obtain for the determination of ax and a2 the equations:
ỵ
7>k1a\+ % Ea]— \ 2 k ì a2-\-4kQa ,ị a ị 0.
Taking into account the binding between the quantities k0y kx and k2 expressed
the formulae (2.2) and (4.2), we have in the case of exact resonance (e = 0):
aỉ = —
(A
(4.10)
n1)/.2a2+ - cosarcsin
7 1 Ả 2 @2
a2>
(4.11)
0.51tf2> a2 —
6 1 .5 //0 c o s a re s i n ( v j x 2 a 2)
" Ằ2n {l5 .3 8 V :_ 1 6.38/r)
Following these formulae, the curves of the dependence of the q u a n tity A2 = Ả
o n Vq = v/l, for the case fci = 125 s~2, k 2 = 103 S"2, /1 = 0.7 S '1, hi = 5 w ith dive
values /z0, are represented in Fig. 6 . All the amplitude curves lie in the semi-plane x2a2 >
and touch the straight line X2a2 = ^0 only in the origin of the coordinates.
4 5 6
v-a-trV
Fio. 6.

Nonlinear connected oscillations o f rigid bodies
313
Comparing these curves, it can be seen that with a definite value of velocity of motion
of the band K, the increase of the constant component of friction h—that is, the increase
of “coulomb” friction—reduces only slightly the stationary amplitude of the self-excited
oscillations.
The diagrams of dependence A2 on Vo for values h = 0.7, 1.4 and 2.8 in cases /*0 = 20,
hỵ = 5 are represented in Fig. 7. From these diagrams it can be seen that the increase of
linear friction, also insignificant, reduces the stationary amplitude of the self-excited
oscillations.
1 2 3 4 5 6
V0 -1Ơ”V
F ig . 7.
Taking into account the linear friction with h > hi is an effective method of limiting
the amplitude of self-excited oscillations, since in this case da2/dt < 0 .
Analogously, it is not difficult to consider self-excited oscillations with the character
istic of friction of the form:
(4.12) T(u) = /j0 signw—/*! W+A3W3,
and of other forms.
We conclude this Section by investigating the case of the motion of a band of high
velocity as compared to the velocity of the vibrator itself- -that is, when V— \ỷịị > 0,
assuming that the characteristic of friction has the form:
(4.13) T(u) = hoihiU+h'ttf+hsU*.
First, we transform the equations of motion (4.1) into the symmetric complex form by
means of the substitution of variables:
J x = u x - \ v u X = i k i f a — V i ) ,
y = u2+ v 2, ỳ = iẰ2(ụ2—v2), i2 = - 1 .
The transformed equations have the form:
( 4 .1 5 ) w, = ÌẰSUS f ~ J f s , v t = - i Ằ av , —
where

/ 1 = ex+hx+k2xy+ Y k2x3—k0xy2,
(4.16) /2 = hy± ~ k2x2i-kỵy^-ko^y-TiVo-ỳ),
Ả2 — , s = 1,2.
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Nguyên văn Dao
Having in mind the application of the same method of perturbation theory used in
the preceding paragraphs, we transform Eqs. (4.15) into the standard form; then we ave
rage them in time. To this end, we make use of the formulae reflecting the character of
the expected motion:
( 4 .1 7 ) u, = C , e ' V ,
where c, and D, are new functions of time.
The equations for Cj and Ds have the form:
dt 21, Jt'
(4.18)1 dD, - i n iX
dt 2ẰS
The recommended approximate solution of this system is found n the form:
(4 19) ^ 5 ~~ /^1> fii)Ĩ
Ds = psi / x ư s+2(t, (Xl, p2).
The quantities a5 and Px are determined from the following averaging equations:
ih).ị)z{ +/c2a2/9i + -y *2Ịaỉ/31-2Ẩr0a1a2/32Ị,
(4.20)
day
if*
dt =
I
d h
-if*
dt
h
da.2

ifi
dt =
2*2
dp 2
- i / x
dt
2ẰI
(e -ihĩ.ùPi t k2p2Xi r ~ k2pị Xị-ỉkoPiX2P21»
1 >
The stationary regimes of motion satisfy the conditions doCi/dt = dPildt = 0, from
which the amplitudes of oscillations are obtained as:
of = - ~r- (k 0- ỳ k ị -t-9kị k 2)a ị,
(4.21) ->k2
aị = [kah — 6k2T'{vQ)-h ị/kị ] 9klk2).
If kí — 0 -that is, if the springs 1 and 3 are absent, these formulae reduce to well-
known expressions received in the investigation of unidirectional model [9].
Following the formulae (4.21) the curves characterizing the dependence of quantities
and a2 on the velocity V of motion of the band are represented in Fig. 8 for the case:
f e i- 1 2 5 s - 2, k2= 1 0 V 2, A = 0.7, h0 = 50, Ax = 10, A2 = — 1.5, h = 0 .0 5 .
From this figure one can see that the self-excited oscillations may occur only on the
decrease branch of the curve r(u) = 0 and the amplitudes of these oscillations reach its
maximum value with a certain mean velocity of motion of the band, namely for
V = 100 cm /s.
Nonlinear connected oscillations o f rigid bodies
315
F ig . 8.
5. Connected Plane-Parallel Oscillations of a Rigid Body
We consider the plane-parallel oscillations of a rigid body, that is such oscillations
for which all points of body move in planes parallel to immobile plane (Fig. 9). It is known
that the investigation of plane parallel motion of a body is reduced to the investigation

of motion of plane figure moved in its plane.
At every time the position of the rigid body is determined by three parameters: coordi
nates X , y of its center of mass and its rotation (p round this center.
The considered body was fastened to the immobile foundations by means of two
systems of springs of the same length / with the stiffness coefficients ku k2t kA. The
316
Nguyên văn Dao
external force is assumed direct to the vertical axis y and immediately excites the oscilla
tion of this coordinate. We shall find the conditions for origin of oscillations of the rigid
b o d y in the directions X a n d (p.
We limit by considering such oscillations for which the following approximate for-
1
mulae are received: sinẹ? = (p— T ọ?3, cosẹ? = 1— -JT-9 and we shall retain in the equa-
6 2
tio ns o f m otion on ly term s o f degree low er than fourth for X, y and (p.
The equations of motion of rigid body for the case of the symmetric springs kị == kị
and k2 = ki are written in the form:
x+Xịx+m^y — /X&U
(5.1) ip + licp+JiX = n$2,
ỷ + Ằịy—ạsintu/ = /U&3,
where
= -Ih^x-tntxy+miycp- -jm 4xi+m6(pi+m7x2(pjrmixy2—mlt\x(pi+mty2<p,
= -2ỗxỹ-{-J<,xy-\-J2í>y<p— + y Ji<p3+Jix2cp+Jtxy2+J2lx<p1—Jlty2<p,
^5'2^ = —2hiỳ— y m4x1+ms<p* + msx<p— y m\y' +mtxiy+miixy<p-m{ty<pi^
rrij = — , Jj —— , = - ị. h i= — , Ằ21 = nti, Ai = / a, xị = m4;
tn T J ÌĨỈ
m is the mass of the rigid body, J is the moment of inertia of the body in relation to its
principal axis GỊ. The constants Cj are the linear combinations of coefficients of rigidity—
for example, Cị = fcj+&4, c2 = kỵ—kị,
It is easy to verify that for linear theory the equations of motion of a rigid body have

a unique stable stationary solution X = (f = 0, y 7* 0, which corresponds to the absence
of oscillations of a rigid body in the dừection X and (p. Therefore, the linear theory of
oscillations does not enable us to discover and elucidate a new phenomenon of a real
system—the connected oscillation of a rigid body.
First, we consider a simple subharmonic resonance of the second type, in which the
natural frequency CO! of the system satisfies the correlation:
(5.3) (Oi = Y w+fd,
where e is a quantity of detuning of frequencies, and C0L is determined as a root of the
equation:
(5.4) —J\) = 0.
We now transform the equations of motion (5.1) so that the generated equations
of the transformed equations have co/2, 0 )2 and A3 as their natural frequencies. With that
in view we add in the two parts of the first equations of the system (5.1) the corresponding
terms:
ỊMứe[(X\—(ứDx+miỲỈKừýị—ù)2i) and ỉMứt[Ọị—coỉ)<p+Jix]Ị(ù)ị—a>ỉ),
Nonlinear connected oscillations o f rigid bodies
317
and write them in the form:
(5.5)
X + OLiX + U 2<p = /U&Ĩ,
'<p+a}(p i-a4x = ụ&ĩ,
ỷ - ị - Ả l y — q sin U)t = n ® 3;
here,
(5.6)
a l — — A §)4

— ( u ) \ — Ằ ] ) ] / ( u > ị (ứ ị ) , 3 -2 —
7‘ “ “4" ) / ("Ỉ-ÍOỈ),.
= 0 !+ft)£[(A^ — co|)x+Wi9j]/(a»2—CL>f),
0 * = 0 2 + &>£[(^2— toĩty Jl XV(to2 &>?)•

The particular periodic solution with period 4ĩt/co of the system (5.5) will be found
in the form of the series:
(5.7) X = x0+fzxi+/*2x2+ (p = (po+Wi+ụ2<P2 + y = yo+Wi+/*2yz+
The functions * 0 , (po and y0 satisfying the generated equations of the system (5.5)
take the form:
The constants Mi and Ml, determining approximations to the amplitudes of the
oscillations of a rigid body in directions X and <p, will be found from the periodic
conditions of the functions Xi, 9?! and yL. It can easily be seen that these functions
satisfy the following system:
ỳi+Aịyi = ^30»
where 0*0 = ®k(x = Xq, y = yo, (p = <Po)-
The periodic solution with the period 47r/co of this system is represented in the form:
Putting these expressions into the system (5.9), it is easily found that the necessary
conditions for existence of a periodic solution of the system (5.9) are:
(5.8)
(5.9)
*i+ ai*i+a2ẹ>i = <£fo,
+<*39?! 4 <*4*1 = #20,
(5.10)
(5.11)
where
hiujMi + q0(m4—m5di)M2
j +3 Ị - ^■mA+m(>dl-\-m1dì-mXođ^ MỊM2
318
Nguyêẵ văn Dao
— w 8( 1 + í/i)^Ồ H Y ~ T
L CO 2 — Cư ĩ
A/i,
M.A/I
/ 2 = — i Ị-/íicoA/2+ y ?o(wu—^ ^ A / i Ị + 3 Ị — y m4+/?j6í/J+m 7 </1—m,o<íf

(5.12) + í- y rt?8(l +ííl)?ổ +- 2' 2 (^ ỉ+ ^ ? ^ l—“>!) -^ 2»
I z u>2—
/ 3 = — i ỏiíodi M \ -Ị—— ^0(^5 ~r ^1*^20) ~!~ 3>ịdiJỊ— Y Jị~\~dĩ Jĩ\~\~~Ỵ d\J%^ M ị M 2
+ [-5- ụ* — diJiì)ql-ì 2
_
2 (^2^1—ơ)\dì +/i)| Afi,
I ^ co 2 CO] I
/4 = ĩ Ịcoổ!^ M 2+ - y #0(^5 +"^1 + 3 ịdxJ, -ỵ /ậ+ế/ị2 / 21+ ”y dịJtj Mị MỈ
+ k 8 — ^1^18)^ổn 2 ~ 2” (^2^1““^2^1+^l)l M2 .
L
2
a>|—cot J
From Eqs. (5.11), we obtain the following expressions for determination of the ampli
tude and phase of oscillations of a rigid body in the directions X and (p:
A1 — (2nòơ)£-ị n2qị±x'riìql-H2),
O/ỈI
(5.13) J H
y = -r- arc COS — >
' 2 «4?0
w h e r e
*1 = 4 ^3— - y -/4+ ^ 1 / 21+ - y <*?/*+ - y mAd1- m (td\d1- m 1dxd2+m^địd2j
(5.14) /72 = — +^l)> ^2 =
/23 = (cưỉ—OJ 2)/ Af, /74 = Ji-\-d\J2Q—Ttiidxdi-^m+diy
H — 2a)(hld2—ôìdl), A2 = MxMly Ml = M2 = Ẩe~£.
Further investigation of the stability of stationary oscillations indicates that only
t h e s ig n p lu s b e fo r e t h e r a d ic a l c o r r e s p o n d s t o t h e s t a b le r e g im e o f m o t i o n .
It is not difficult to prove that in the case of simple subharmonic resonance of n type
(n > 2), when C0j = íứ/rĩy no oscillations of a rigid body in the directions X and <p occur.
L e t u s n o w c o n s id e r t h e s im p l e p r in c ip a l r fe s o n a n ce a s s u m i n g t h a t o n e o f n a t u r a l
frequencies of the system under investigation, for example (02, is equal to the frequency

o f the external force CO. The equation s o f m o tio n (5.1) are, b y m eans o f substituting the
v a r i a b l e s :
X = i41Ịsin01 + ơlcoscư2f+Ơ 2smco2f,
X = Ả iC ử iC O sd ỵ — ^ U ị S Ì n ^ t + ^ U i C o s ^ t ,
(5.15) <p = dlAìsìnOl+d2UlcoscD2t+d2U2SÌncú2t,
ỉp = dì(ú1A l c o s 0 i — cỈ2(O2UiS\n<ú2Ĩ+a>2d2U2COSU)2t9
y = j43sin03+Ợ2SÌnco2f, ỷ = Ằ}A3 Cosd3+q2 <ứ2cosci)2t,
Nonlinear connected oscillations o f rigid bodies
319
where
= ?/(*!— ỡi = «i/+yi» 03 = ■M+yj,
transformed into the system
(<p2—rfi0 i)coscơ2/,
dt
_
r~
toiiậi d-i)
(0 ;
du2
(tf>2
dt
tolidy — dl)
dAi
(0 ;
dt
3
1
■»3
3
1

dyl
-A*
(5.16) — (0 2- d 20 l)cosOif
at Ioi{di—a2)Ai
Q - ị * , ™ * , . 1 r = m ; 0 '*1" " -
for which the method of perturbation theory is used. Following this method, the variables
-4«, y,, Í/] and t/ 2 in the first approximation are determined by averaging equations of
the form:
dUv ịi
(5.17)
dt 2 a>2W — d2)
dU2 _ fjL
dt 2oj2{di — d2)
xUl- — xlUiB2-3x,qlU2
kUiH

-ị ^lU[B2-ị-X2(Ị2^l
+ •••,
■f
dA\ Ẵ dA3 ,
= —flOiAu -jf- = —M l^ 3 ,
where
iff ^ 1 " dt
X = 2co2(42< W iM , x2 = i-[rf2y18- y 8 + /W8rf1(i+rf2)], i*2 = ơ? + ơ 22,
*1 = ~2 —3/zi^2-t"3*/i Ị

— rĩĩị-^r ĩTĩ^dị-^ ìyi-ỉdi ^ìo^ỉỊí
and the non-written terms will be equal to zero when Aị = Ai — 0 .
The equations for and yi3 are independent of the rest of the equations. From which
it can be seen that the amplitude of oscillations A\ and Ai tend to zero. Therefore, in

what follows we shall be interested in the first two equations of the system (5.17) in which
the non-written terms arc rejected.
The stationary values of U\ and Ư2 are determined from the equations dUJdt =
= dUijdt = 0, which for determination of the amplitudes of the oscillations of a rigid
body in the directions X and (p give the following expression:
(5.18) B2 = - - ( - 2 x ìql± \/xịqị-xi).
X[
Since only such amplitude as corresponds to a stable regime will be of practical in
terest, it is necessary to verify stability of the solution obtained. Analysis of the variation
al equations for the solutions (5.18) gives us the following condition of stability:
320
Nguyên văn Dao
(5.19)
Xi0*1 £2+ 8x292) > 0 ,
from which we see that the form of oscillations with big amplitude is stable and the form
with small amplitude is unstable.
It is easy to prove that for the system under consideration no oscillations of rigid body
in the directions X an d (p occur in th e case o f sim ple ultraharm o nic resonance.
Since in the double and combinatorial resonances, the averaging equations are compli
cated in form, we can obtain only the necessary conditions for the origin of oscillations
o f a r ig id b o d y in t h e d ir e c tio n s o f t h e c o o r d in a t e s X a n d (p.
For double subharmonic resonances when among the natural frequencies (x){ and Xi
of the system, and frequency CO of external force we have the correlations:
(5.20) C0 i = ~ c o +ftel9 Ằ) = — co-ị/ẨS},
Kị ac3
oscillations o f a rigid body in the directions X and 90 may arise in the cases:
1) — k3 = 2 or kỵ — 4, &3 = 2, when the necessary condition (n\-{-n\)x
x(^4 +nị) 9* 0 is satisfied,
2 ) kị = k 3 = 2 , when the n ecessary c ond ition n2 # 0 is satisfied,
3) kị = 2, ki = 3 or kl = 6 , k3 = 3, when n4 ^ 0,

(5.21) 4) 2/r3 = kị = 4,6, when 7* 0,
5) = ki= 3, when n2rt5 7* 0.
In other cases, no oscillations o f a b ody in the directio ns X and (p occur.
For double ultraharmonjc resonances, when
(5.22) ơ)\ = P\C0 + fi£[, A3 = p 20) + fi£ 3 ,
it has been proved that the oscillations of a rigid body in the directions X and (p o ccur
in the following cases:
1) p { = p 3 = 3 or P i = 2, p 3 = 3, when the necessary condition n2 ^ 0 is sa-
with small amplitude of external force, no oscillations of a body in the directions X and 97
occur.
We also consider the combinatorial resonances. It has been found that the oscillations
of coordinates X and (p may arise in the cases:
tisfied,
(5.23) 2) 2Pi = P i, when n4n6 # 0,
3) Pi = Pi # 3 or 1—/>3+ 2/?! = 0 or l+/?3—2px = 0, when /I2rt5 7* 0.
For double principal resonances, when
(5.24)
(5.25) 2) c/>! ~ CO, A3 £ or » CO, A3 £ 3co only when n2 ^ 0,
3) ơ)l X 0), A3 « 2cư only when (rtỉ+nỉXrtl+rtỉ) # 0,
4) U>1 ~ -y (O, A3 * 2 a> only when (ffỉ+rtỉ)(rtỉ+n?) ^ 0 ,
Nonlinear connected oscillations o f rigid bodies
321
5) C ủị zz -ị CO, A3 ^ 3co only when nA ^ 0,
2 1
6) C0i ~ -Ỵ CO, Ẩ3 Í: — w only when n2 # 0.
It should be observed here that in the cases under consideration, the parameters n2,
tt4, and fl6, where
(5.26)
n5 = /Wg + </iW15—rf?/w18,
play an important role in the excitation of oscillations of the coordinates X and 97. They

are called fundamental parameters. If oscillations of rigid bodies in the direction X and (p
are not desired, we can change the parameters of the oscillating system so that the relevant
fundamental parameters are equal to zero.
6. Oscillations of a Rigid Body Connected Spatially
Let us next consider the system shown in Fig. 10. The rigid body is attached to the
immobile foundation by means of twelve springs of equal length / with coefficients of
rigidity /ru, kll9 kA3.
The position of the body at any time is adequately determined by six parameters:
the coordinates X G , y c , Z G of its center of mass and the three Euler angles 93, y), 6,
chosen by A. N. K r y lo v [10].
The body under consideration is acted upon by an external harmonic force which
is invariably directed along the vertical axis z. Limiting our investigations to small oscilla
tions of the coordinates, we retain in the equations of motion of a rigid body only the
terms of degree smaller than three relatively to
X , y, 2, <p, 0 and their derivatives.
322
Nguyên vân Dao
The eq uation s o f m otion o f a rigid b o dy can be represented in the form :
mx+cLix-ị c12y>-f cli(p+hlx+ ~ Cuy2+ ý ci5z2 + cl(ly2+ +cl2i<pd = 0 ,
mỳ.+ c2iy+c226+c2ì<p+h2ỷ+ y C 2^ + | c i j Z 2-( c26y2+ +C222<p0 = 0,
(6.1) m i + c 31z+c32v>-f cììd+hì'z+ ^-0 ^ + ^■cìiy1-\-cì6Ỷ+ -f Cì22<f>0 = psiũítít,
ÁÓ-\-dnd + di2y+dnz+dl4y)+dlỉ(p + ôlỏ+(A — B+C)ỳị + +di26(pd = 0,
By)-\-d2iv+d22x+d2iZ-\-d24<P+d2i6 + 02V-lr(A — B— C)ịè+ +d22b<p6 = 0,
Cip + d^cp+duX-ị duyi dì4y+ d3iO + ô3ị + (B — A — C)ipé+ +di26<pO = 0 ,
in w h ic h
[x=xc/l, y = yd>, z = za/l, A = Aoll2, B = B0/l2, c = c 0//2,
and Ao, B 0 , c 0 are the principal m o m ents o f inertia o f the body. The con stants Cịj anc
dij a r e t h e l in e a r c o m b in a t i o n s o f th e s t i f fn e s s c o e f f ic ie n ts k i j— f o r e x a m p l e , cn = Ấr,
2
-|-

•+^21+^32"f > dll — —^13 + ^23 + ^33 —^43| •••
The first three equations of the system (6.1) describe the motion of the center of mas:
G of a rigid body, and the last three equations of this system describe the rotation 0
a body around its center of mass.
From the system (6 .1) it can be seen that, in general, the co ordinates X, y , z, 0, y , <,
are intimately associated one with another and therefore the oscillations of the coordinat
z will in the general case excite the oscillations of the remaining coordinates. However
by suitable choice of the system of springs, it is possible to ensure that under the actio:
o f an external force directed alon g the vertical axis 2 , the oscillations o f oth er co ordinate
are not excited.
In fact, after coosing for example kn — k22 = kn — kl2 = kn = kn = kil
kli = ^23 = ^33 = £«3. the system of Eqs. (6.1) becomes:
x + c ° n x + c ° n y) = - f i F i ( x , X , y , 2, e , Y>, ỳ ) ,
ỹ - \- c ị xy + c ị 2 Q = - ạ ĩ t i ỳ , X , y , z , 0, y , <p),
z + c h z = - ạ F } ( z , X, y , e, y), < p )± qsm a > t,
(6.2) e+dỉìd+dỉiy = -ụF A(e, ỳi, if), ip, X, y, z, e, V, 9>),
V + V>+C$2 X = - n F i ( ỏ , ỳ), <p, 0, ặ , X, y , z , e, y>, <p),
ỷ+ d}Ị<p = —ụF„(ỏ, ỳ), ị>, 0' xp, X, y, z, 9, y>,<p),
where /í is a sm all p ositiv e param eter, and <5? = 0 J A , (5° = 0 J B ,
<5j = Ỗ3/C, hi - hk/m, c°kj = ckj/m, dỊi = dJA, = » d%i = -jr I
[iFt = h°x + c°n0xz+cniy(p+c,inz\p+ci22<p6,
fiF2 = hịỳ+cịnXẹ+clnyi+clvĩQ+cịnĩixp,
nFi = hịz+ ~ c ị 4** + — cW +^W +cW +CHp+CiioXrp+cluyO ,
Nonlinear connected oscillations o f rigid bodies 323
fiFẬ = ỗ"ớ+ Ị l - ^ Ị 1 - -jj<PV>+ ^VKP+ỗì^ỊV>V>+dĩisX<P
(6.3) Jr<f\ìlyz-\-cĩỊ2izQ-\-(fỈ2ịV0,
ụFs = ỗịỳ- Ịl— ^ -^-Ịp× +
+ 4ẵi*>’9H-í&iZy' + í&69>0i
/xF6 = <5“ỷ_ Ịl + A _ -^jỳ>0- Ịl - -§Ị ớ ỷ - ^ vỡ-ỗ? -£ V®
+ ^ ]2*>' +<*?,6x6 + d°ìtiyyi cTliỉ29H ^ 2Jy0 .

Obviously, the system (6.2) has a particular solution of the form:
(6.4) x = y= y = (p = 0 = 0, 2 ^ 0 ,
which corresponds to oscillation of a rigid body in the direction z.
Later, we shall find the conditions of origin of the intense oscillations of a rigid body
in the direction of the c oordinates Xy y, y, (p and 0, w hich are n ot subject to ex tern al force
and determine the amplitudes of these oscillations.
Replacing the variables by means of the formulae:
* = ^,sin(wif—yi) + ^3sin(a>5 /—y5),
X = cưịAiCOSÌcUií—y ,) + cosA 5cos(cư5í- y 5),
y = / í 2 s i n ( í o 2 / — y 2) + A 4sin(co4 t — y 4),
ỷ = (ư2y42cos(cư2ỉ —y2)+<°4'44cos(co4t - y4),
z = /43 sin(í')3r — y3)-f 12 sinco/,
(6.5) z = a>3 /43cos(tojf —y3) + coQcosaư,
Ớ = *2>Min(a;2/—y2)+X4'44sin(w4/—y4),
0 = x2o)2.42cos(^2Í—y2) 4-*4fc>4/44COs(co4r — y4),
V = y41 sin(cư1 r—yi) 4 x5 ^ 5 SÌn(a>5/—y5),
ỹ = /4] COS ( t o i/ — }>]) f XịCOsA sCOsCíOị í — y 5) ,
9? = /ẩ6sin(co6/—y6)>
ỷ = co6A6cos((!j(,t—y0)>
where col9 a>6 are determined as the roots of the equations:
✓g wi,5“"(^ll+^21)^1,54“ — C]2^22 = 0» cof = C31,
a>2,4 — (dll *t“^2l)w2,4 + ^?|C21—^ 2^22 = 0 » = ^31 >
we obtain for the new variables the following equations:
dAx (xsF^Fs) , V _ (*3^1- F5)
- = ỊẤ —7-— cos(a>! t—yi), ~ ỊẨ -“ 7-7 —-^-xSin(a>i/—yi),
dt r ( o f a —x5) v ' 7 dt #s)
dA2 (xj4F2—F4) x (x4F2—F4)
4 i r = - — F )C 0 S ((0 3/ — y 3) , = F j s m i a v - y j ) .
(6 .7 ) đĩ a>3 dt a>3 /l3
324 Nguyêrì văn Dao

dAt (X2F2-F * )
__
% dyA i?(iF1- F Ậ)
<£4 5 (xịFi — Fị) / X ày Ị (k\F\—Fs) . ,
l * i - Ỳ - - 1Ò
in which
coĩ-tíx ^ _ co22- c°2ì _ C0Ỉ - C Ỉ 1 ^ _ c o f-c ? !
*1 — 0 » *2 — -0 » *4 ~ r° ~ 9 Xĩ ~ r°
c 12 C22 ^22 ^12
Now, we consider subharmonic resonance of the second type, assuming that the nat
ural frequency 0)2 of the system satisfies the correlation:
(6.8) 0J2 = -ị <oi file.
In the first approximation, we have:
(6.9) Aj = ajf yj = r j9 j = 1, 2, 6 ,
where Oj and /y are determined by the averaging equations:
dax (*5h\-xxồị) dr,
dt 2 (* ,-x s) *
dũ2 * 4 ^ 2 — * 2 ^ 1 [x 4 (c 2 1 3 ~ l"x 2 g 219) — ^ 1 17 — x 2 ^ 23] COS 2.7^2
A 2(x2—*4) ơ2 4ío2(*2“ *4) *
dr2 [^4(C213 + ^2^219)"~^117 — *2^23]0Sin2A
(6 .10) ^ =

•
<&J3 _ 1 0 daA (*2^1— *4 <5?) da5 _ Xihĩ—XỊỗị
dt ~ 2 3*6’ ~ s r ị2(*4- x 2) *4’ "ữ T “ 2(k5- * i )“ ° 5'
1 £0 dr I dr5 - _ I 2 2\
dt ~ 2 * = * = 0 ’ e-«K “5 - “ )•
From the equations it can be seen that the quantities fli, 0 3 , «4 , Ỡ3, and a6 tend asympto
tically to zero. The equations for and r2 become, by means of the formulae u2 = 02 COS r2,
t>2 = Qi sin A *

-ậ - = ,í[(tf1 + //2)K2+ £t>2],
= /Uf-tt/j + Ctfj —
(6.11)
in which
- *2(f2-* ,)? ’ 7/2 - 4 ^ ( 1 -* ,) M<5iJ+^A.)-d?n-xi<ff2al.
It can easily be seen that
(6.12) e2+ H Ỉ-H ĩ> 0
Nonlinear connected oscillations o f rigid bodies
325
is the necessary and sufficient condition for the stability of the solution u2 = v2 = 0 of
the system (6.11) and therefore is a condition for the stability of the solution (6.4) of
the system (6.2).
'From the inequality (6.12), it is obvious that the viscous damping forces characterized
by th e coefficients <?! an d
<?2
a n d th e q u a n tity o f d e tu n in g o f th e fre q u en c ies e w ill in cre a se ;
by contrast, the amplitude of the external force ( 0 will diminish the stability of the so
lution (6.4).
If the condition of stability (6.12) is not satisfied, then as shown in paper [18], oscilla
tions of a rigid body in the directions y and 0 will occur. These oscillations in the first ‘
approximation are determined by the equation:
y = /40sin Ị y i-yo|.
CO \
2 '-» )■
z = Qsincot,
where
1 E
A i~ -j-(e 2Q2- e 3), tangyo = „ TT.
e1 lii — n 2
e i = C 2 2 [ “ " < í l 7 " ~ * 2 ^ 1 2 3 + t f 4 ( C 2 l 3 + * 2 C Ĩ l 9 ) ] ị ~ 2 C ? 5 + * 2 c 3 1 6 + * 2 c 3 s j 2 + C O 2 — C O 2 I *

e2 =

— (^4^213(^213 + ^2^219)— (^2 — ^4)^219! + — ^ 23)) »
X4 — X2
e3 = — ^ — (X4A22—* 2 <5?2) + 4 co2£2.
Further investigation of the stability of stationary oscillations (6.13) gives us the follow
ing conditions of stability:
(6.15) ee, < 0 , e(e2 Qĩ- e ì) < 0.
7. Nonlinear Connected Oscillations of Systems with Distributed Parameters
Now, we have to consider Ịthe connected transverse oscillations of beams having uni
form cross-sections s and subjected to the external force Q(x, /) directed along the vertical
axis z. The elasticity of the beam is assumed to change by Hook’s law. However, we shall
take into account the elastic deformation of its longitudinal axis, whereby the oscillations
of the beams become nonlinear [2 0 ].
For the deduction of Ịthe {differential equations of motion of the beam, we shall use
the Hamilton-Ostrogradsky principle of stationary action; to that end it is necessary to
express the deformation work Ay the kinetic energy T and the work E of the external
force Q(x, t) through the displacements of the beam. We have:
(6.13)
0 — X2ẨQ sin
326
Nguyêh văn Dao
'-f/[£+i(£H(£)T*+4 /H £ M £íb
r-Ị/W),+(£H£ĩh
E = Ị Q(x, t)zdx,
0
u, 2 being the components o f displacement o f the beam at section X, Q being the mass
of unit length of the beam and Jy = ffy 2dydz, Jx = / fz 2dydz being the axial moments
of inertia of the diametrical section of the beam in relation to the axes y and z.
According to the Hamilton-Ostrogradsky principle of stationary action, we have:

Ố I (A- E -T )d t = 0.
to
Whence we obtain:
<rz d if £u 1 I cy y Ì I 8 z\2] Sz\ CAZ
Q~s? t e \[ J x + ~2\Tx) + ~2\ex) \~8x\ + ‘ y Tx*~~q ^x' /)#
Taking into account a small damping force with components—pH 4^-, —pHx —pH2-ỊT- $
ot ổt ỏt
and assuming that the nonlinear terms are small in comparison with the linear terms,
we can write the differential equations of motion of the beam in the form
82u
* 8t2
<7'1) e-ĩỉr^Ej, dÁy
dr
d2 z
dx4
d4z
al equations of motion of the beam in the form:
= (iị-Q ỈỈl^ - + E S
8x 1 + 2
Sz
+
n m w
+ {?(*, 0 ,
where /X is a small parameter.
Let us consider the transverse oscillations of a beam in the directions y and z. Substi
tuting u = 0 in Eqs. (7.1), we have:
(7.2)
Nonlinear connected oscillations o f rigid bodies
3
Here, the following designations were used:

2 _ [EJ. ịp. _ Ely
a —
Q Q
J We shall find the conditions for the origin of the connected oscillations of the beai
in the direction of the coordinate y which is not subject to external force.
The solution of the system of Eqs. (7.2) is found in the form:
(7.3)
00
y = ỵ Xk(x)Tk(t),
ẮC-1
CO 00
z - ỵ ỵ k(x)skự)± ỵ Xk{x)h(t),
1 A- 1
where Xk(x) are the eigenfunctions satisfying the end conditions for any particular beam
and
t t
A — ~ ^ ' ĩ [sinmịò/J"Nkcosmịbtdí~cosmịbíỊNksinmlbtc/t),
0 0
I 11
Nk= ị QXk dx Ị J Xịdx-
Ỏ I 0
mk are eigenvalues. Substituting the values of y and z into Eqs. (7.2), and comparing the
coefficients of xki we obtain the following equations for Tk and sk:
Tk + ojị Tk = —fxHi Tk+fẲ&k(Ti, 7 *2, , S ị, s 2, ••• y 0 >
( 7 - 4 ) T A . . * V .
Sk+GiSk — —nH2(Sk -\-Iì)+ỊÀxFk(Ti, Ti, Sly s 2, Of
0* = / F*XkdxIịX ịd x , wk = ị F*XkdxlịX ỉdx,
0 I 0 / 0
F. _ r I „ ^ p. p I rf r .2 a2™* 02
ri; — rj i-/7| ——, r2§ — /^2+ **2 ■> ù)k — —ỹ4—> — ỹi—

Equations (7.4) are by substitution of the variables
Ợ 5) Tk = f*sin(a>*/+9>*), 7* = ct>*f*cos(a>*/-fỹ>k),
S’* = rjk siũ(ữk t+04), 5* = ữ k YỊk COS (Qk t+6k),
transformed into the following form:
ỉh = -/*Wi£*cos2ỹ* + -^-<Pằcosỹẳ,
cư*
5 Problem y drgaA