Nonlinear connected oscillations of rigid bodies

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.. The motion of t

O F VIBRA TIO N PRO BLEM S

W A RSA W , 3, 10 (1969)

N O NLIN EA R CONNECTED O SC IL L A T IO N S O F R IG ID BODIES

N G U Y EN VAN D A O (H A N O I, 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 o f 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 o n o n e 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 ẩI

304 N guyêỉỉ xăn Dao

The elasticity of springs is assumed to change by the linear law with stiffness coeffi­cients 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:

which corresponds to the oscillations o f 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 i l l a t i o 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 o f 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 correla­tion

The transformed equations are:

X = Ni COS — / + A il sin — t , X = — — Ni sin — 1 + - y Mị COS — t

These e q u a tio n s belong to the standard form which is applied coveniently by the method

o f p e rtu rb a tio n theory [6].

Nonlinear connected oscillations o f rig id 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),

M x = M Ĩ + f i U 2Ọ, N°u M°, Aị, yị),

A2 = A\ f ịaUẠị, Aỉ l M ĩ, Aị, yị),

where flU, are small periodic functions o f 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 í A + * 2 g * ) 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 o f (2.5) according to time Non­

written terms in (2.7) will be equal to zero when A 2 = 0.

The equation for A 2 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 N Ĩ and M l in the stationary regimes of motion are determined as the roots

of the system of equations:

Hence we arrive at the amplitude of oscillation o f the vibrator Ân the direction x :

Thus, we have obtained two forms of stationary o s c illa tio n s 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 o f 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 o f the quantity a2 on p for Q = 0.1, Q = 0.25 with diverse values H: 0.1, 0.15 and 0.3, where

306 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 o f 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 M N — 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 t h e v i b r a t o r i n t h e d i r e c t i o n X o c c u r

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 th e v ib r a to r in th e d ir e c tio n o f the c o o r d in a t e X It h a s b een p r o v e d th a t th is o sc illa tio n cannot arise in the cases of double subharmonic resonance when Ả] — -i- -\-uEi, ẰĨ = — 5 +

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" + /^2Ĩ / I , /w > 1,

3 Connected Nonstationary Oscillation when Passing Through a Resonance o f 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 c h o ic e o f w h ic h d ep en d s o n th e k ind o f r e so n a n c e in v e stig a te d — th a t is, w hen b e tw e e n

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

308 NguyéH văn Dao

+ - y k 2 a \ COS30 1 —k ồa i ( ạ 2 COS # 2+ 0 * sin ớ)2 COS 0 1 Ị COS 0 ị ,

= -J- |/ỉ(^*vcosỡ — Ằ2a2sin 0 2) + k2aĩ C O S 2 0 1 + k ^ q * sinỚ4 ứ2c o s 0 2) 3

resonance, which is in the neighbourhood of ).ị It is assumed that the natural frequencies

o f th e sy stem con sid ered are lin ea rly in d e p e n d e n t, s o th a t b etw e en th e m there is n o c o r r e ­

la tio n o f th e fo rm /lA j-f * 2^2 = 0 , w h ere /, a re in te g er s n o t sim u lta n eo u sly eq u al t o ze r o

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

flo w far fro m the reson an ce, th e y are sm a ll a s c o m p a r e d w ith th e o s c illa tio n s o f th e c o o r d i­

n a te s a x an d V>!, an d therefore in th e first a p p r o x im a t io n w e m ay d isregard th e m

Following the method of perturbation theory, approximate solutions of Eqs (3.4) are taken in the form:

Nonlinear connected oscillations o f rigid bodies ^09

and for n 7* 1,2:

From Eqs (3.8), it can be seen that when n 7^ 1, 2, the amplitude a of the oscillations

teDds a s y m p to tic a lly to zero w h e n / 00, a n d th e re fo r e n o o sc illa tio n s o f th e v ib ra to r

/ 2 = 0.1 where X — ịxhịì^s 1 1 = ạ k 2q*l2X]y I2 = fẦ,kồq2J2X]y B = 3 fik2a/%X{i p = 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

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4 Connected Self-Excited Oscillation of a Vibrator

In th e literature, w ell k n o w n is th e m ech a n ica l u n id ir ectio n a l m o d e l o f an s e lf- e x c ie d system w ith e n d less b an d : a h e a v y lo a d fasten ed to th e im m o b ile p o in t by a s p r in g ies

o n an en d less h o r iz o n ta l b an d m o v in g under a lo a d w ith c o n s ta n t v e lo c ity

In th e p resen t p a p e r, w e c o n sid e r th e m u ltid ire ctio n a l m o d e l o f a se lf-e x c ite d system ,

a ssu m in g that th e vib rator is fa ste n e d to three im m o b ile p o in ts b y sp rin g s o f equal length,

F ig 5.

The self-excited oscillations arise in the system considered in consequence of the action

of the force of sliding f r i c t i o n Tị at the point of contact of the v ib ra to r with the band

This force is a function of relative velocity V — ỳ I = l(v 0—ỳ) and is conveniently repre­sented in the form Tị = mlT(v 0—ỳ), where V = Iv0 is the linear velocity of the bard,

ỳ ị = l ỳ is the v e lo c ity o f the vib ra to r.

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

v ib ra to r in th e fo llo w in g form :

The second equation of this system, which describes the self-excited oscillation o f 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 i g 5.

( 4 1 ) X \ X ] x = — f i F ị ( x , X , y ) ,

ỹ + Ằ ị y = - / i [ F 2- T ( v 0- ỳ ) ]

Nonlinear connected oscillations o f rigid bodies 311

y = À2cosỡ2t ỷ = —Ằ 2À2siũ 02, 02 = hit-ryii

Eqs (4.1) are reduced to the standard form:

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 th e first a p p r o x im a tio n a re d eterm ined by th e e q u a tio n s o b ta in e d fr o m ( 4 4 ) by ave­

r a g in g their r ig h t-h a n d sid e s in tim e.

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:

Putting this expression into the system (4.4), and averaging their right-hand sides in

time, we arrive at:

312 Sguyêrỉ văn Dao

described by Eqs (4.9) The further course of the change of Qi, 71! leads to some stationa values, for which ảũiịảt = d r {\dt = 0

Hence we obtain for the determination of ax and a2 the equations:

ỵ

7>k1a \ + % E a ] — \ 2 k ì a 2 - \ - 4 k Qa ,ị a ị 0

Taking into account the binding between the quantities k 0y k x and k 2 expressed

the formulae (2.2) and (4.2), we have in the case of exact resonance (e = 0):

o n Vq = v/l, fo r the ca se 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 x 2a2 >

and touch the straight line X 2a2 = ^0 only in the origin of the coordinates

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

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:

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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:

where c, and D, are new functions of time.

The equations for Cj and Ds have the form:

The quantities a5 and Px are determined from the following averaging equations:

ih).ị)z{ +/c2a2/9i + -y *2Ịaỉ/31-2Ẩr0a1a2/32Ị,

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-9k ị k 2) a ị,

If k í — 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 a 2 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, k 2 = 1 0 V 2, A = 0.7, h0 = 50, Ax = 10, A 2 = — 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 i g 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: coordi­nates 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 k u 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

tio n s o f m o tio n o n ly term s o f d eg re e lo w e r th a n fo u r th for X, y an d (p.

The equations of motion of rigid body for the case of the symmetric springs kị == kị

and k 2 = ki are written in the form:

x+ X ịx+ m ^y — /X&U

ỷ + Ằị y—ạsintu/ = /U&3,

where

= - I h ^ x - tn tx y + m iy c p - - j m 4xi +m6(pi +m7x2(pjrmi xy2—mlt\x(pi +m ty 2<p,

= - 2 ỗ xỹ-{-J<,xy-\-J 2í>y<p— + y Ji<p 3+Ji x2cp+Jt xy2+J2lx<p1—Jlty 2<p,

^5'2^ = —2hiỳ— y m4x1+ms<p* + msx<p— y m\y' +mt x iy + m iixy<p-m{ty<pi^

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:

where e is a quantity o f detuning o f frequencies, and C0L is determined as a ro o t o f the

equation:

We now transform the equations of motion (5.1) so that the generated equations

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:

= 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 = x 0+fzxi +/*2x2+ (p = ( p o + W i + ụ 2<P 2 + y = yo+ W i+ /*2y z+

The functions * 0 , (po and y 0 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 y L. It can easily be seen that these functions satisfy the following system:

ỳi+Aịyi = ^30»

where 0*0 = ®k(x = X q , 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+ a i* i+ a2ẹ>i = <£fo,+<*39?! 4 <*4*1 = #20,

318 Nguyêẵ văn Dao

— w 8( 1 + í/i) ^ Ồ H Y ~ T

M.A/I/ 2 = — i Ị-/íicoA /2+ y ?o(wu—^ ^ A / i Ị + 3 Ị — y m 4 +/?j 6 í/J+ m 7 </ 1 —m,o<íf

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:

(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 i d e r t h e s i m p l e p r i n c i p a l r f e s o n a n c e 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 th e ex te rn a l fo rc e CO T h e e q u a tio n s o f m o t io n (5 1 ) are, b y m ean s o f su b stitu tin g th e

v a r i a b l e s :

X = i41Ịsin01 + ơl coscư2f+Ơ2smco2f,

X = Ả i C ử i C O s d ỵ — ^ U ị S Ì n ^ t + ^ U i C o s ^ t ,

(5.15) <p = dl Aì sìnOl +d2Ul coscD2t+d2U2SÌncú2t,

ỉp = d ì (ú 1A l c o s 0 i — c Ỉ2 (O 2UiS\n < ú2Ĩ+ a> 2d2U2C O SU )2t9

y = j43sin03+Ợ2SÌnco2f, ỷ = Ằ}A 3 Cosd3+q 2 <ứ2cosci)2t,