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) 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 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. [...]... /s Nonlinear connected oscillations o f rigid bodies F ig 315 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. . .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... 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... 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. .. 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 C ! of the system satisfies the correlation: O (Oi... 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 th e d irectio n o f the c o o rd in a te s Xy y , y , (p a n d 0, w hich are n o t su bject to e x te rn a l force and determine the amplitudes of these oscillations Replacing the variables by means of the formulae: * = ^,sin(wif—yi) +... = — ^ — (X4A22 —* 2 . 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,. 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. 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