Modeling the effect of centering of a spherical hydrodynamic suspension.pdf High-load float gyroscopes are widely used elements of the control system of moving objects. In turn, spherical hydrodynamic suspension is a sensitive element of a number of float gyroscopes [1]. The stability on the mobile equilibrium curve of a weakly loaded spherical suspension at low oscillatory Reynolds numbers was studied earlier [1]. Cylindrical hydrodynamic suspension is quickly centered with increasing oscillatory Reynolds number [2]. Experimental data [1] confirm this effect for spherical hydrodynamic suspension. The influence of Coriolis forces of inertia on the dynamics of viscous incompressible fluid increases significantly with increasing oscillatory Reynolds numbers [3-9]. Nonlinear effects can destabilize the Couette flow in the layers between rotating spheres [10-14], but this is true when their angular velocities are significantly different. The aim of this work is to simulate the centering effect of a spherical hydrodynamic suspension based on the asymptotic integration of the reduced Navier-Stokes equations for the isothermal flow of a viscous incompressible fluid in a rotating support layer. 1. Mathematical model We assume that the sensitivity axis coincides with the Oz axis of the Oxyz coordinate system, the origin of which is in the center of mass of the inner sphere, and all the decentering forces lie in the Oxy plane. In disregard of some terms of the order of the square of the small relative thickness of the spherical layer (which is justified in [2]) and the small compressibility of the liquid, the equations of equilibrium of the suspension in dimensionless variables follow from [15] and take the form 2ir тн(Р2 / P “Pig- a0) + I fo sin МОГ dLP^a ifl 1 (dvv / dx\\ e - p x-0 Ф 1 2 e ) = 0, x=0 r> 1 I Tt[3fi + C sin2 T>d$/2ъ (dv / dx)\\ / h = 0, h = (3 2[[(1 + P)2 + (32((u • e,r)2 - u2)]1/2 - 1] - u Іж=0 v = flv e + ih.e.. + v e , 1 Г Г Tt it tp tp? v = x(l + (3hxy\\ ■ V(s)/i - (1 + I3hxy2fj Vм ■ (h(l + (3hx)v)dx, 13 D.K. Andreichenko, K.P. Andreichenko, D. V. Melnichuk v\\ = 0, = 0, v\\ = -Qsind, v\\ = sind(u эіпф - и совф), ' ’ ^ Іш-0 ’ 4= Іж=0 Пж = 1 Х х Г u YП - 2w„ cos d - 2(3w sin d + F [v] = 0, _1 дРІ=о ! 1 1 + |3/іж д$ a 1 1 9pLo , 1 1 + [3/іж sin d Эф т Іш-0 dv,, --- + 2v cosd + FJv =0, Эф u 1)L J ’ «J = Pcos d(u simp -и совф), ѵ =Р(и эіпф + м совф), Wn =-2ѵ sin d + ѵ: + ѵ2, (1) 1 ' х г ѵ г п ФІ._1 ' ѵ ѵ т х тп 0 ф ^ ф7Ѵ/ V(s) • Ф(0)[ѵ] = 0, Ф(0)М = Чоі1 + + ѵ е )dx - -(! + р /іж)2Ц(0) х и, FM = f h dv P Эж 1 + р/іж К) v(s) hf WQdx о - xWV(s)/i ~ Pvv - (1 - р/іж) [i V(ѵ^.у(х: d) + (u0y - іи0іЕ)е_‘ч’г;Цір(а:, d). dv dx _J._d_ sind 5d (40) sin d), (sin d (vf / = 0, - i + 2vf cos d = 0, (8) dvj+) dx 1+)| =fl+) =0, г>1+)I =^+) =0. r \\x~0 ^ \\x - 0 'P \\x=0 Л \\x-1 -P \\x-i Subject to (7), the conditions (5) take the form Kj, +4J/07tsin2 V+)cK> = I (p2 / P - !)(*(! + %)-%x), 2^n0£sin2 fdf(dv^ / dxyo =0.(10) In particular, solving the linear boundary value problem (8) and using (10), we find ( ту 04 (d) = 0(1 /-v/a), a ->• oo, d ^ it / 3,2"іт / 3; 04 (тг / 3) = 0(1), 04 (2тт / 3) = 0(1), a ->• oo. Requirements for the absence of singularities of the solution of equation (13) in the poles of the sphere and anti-symmetry with respect to the equator d = it / 2 lead to boundary conditions - (гА+) dd ' 0 = 0, »(+) = o. (14) When a ->■ oo, the asymptotic integration of the linear boundary value problem (13), (14) can be performed on the basis of the successive approximations method _d_ dd sm3d 0-Ч0-1 ^(4+)) d k+1 03 -04(t>((+)) , -U+A =o,U+A ' 13 /k dd'"9 'HI,, n ' l) >k+1 A = ,«=°. 0,1,2,..., (,/;>) =0 = 0, (15) On the first iteration we assume Ѳ,,1 + Ѳ tx -4, (-)1 - (-) tx 8cosd, and we find (see (10)) ”*) = - d cos fo ^ S3n2 о -> oo, 10 ’ 0x 9 On the second iteration, we find U0x = -? (p2 / P - iKr, (p2 / P - !)(! + %)• fit in ■ 2 Cl (+) 3 • , n / Г П 1971п/з+4061 . . 1971л/з-4061 L ddsm d»v 7 = -- * + G /Vo, 6 =-r- + *-1--, Jo F 10 p / 4 > p 17160n/2 17160л/2 CT -1 (50 . The coefficients ux, , vr, /у . / ■. . j>] of asymptotic series (4) are the solution of the problem /0 dtp £ sin dddper = 0, /Q dtp £ sin2 ddd (dv.^ / dx + (u0 • ег)дѵщ / дх)\\^ = О, дѵг /дх = - V(s) ■ ѵ + (u0 • er)V(s) • v0 - ®[(u0 • e,&)dv{>o / dx + (u0 • е^)дѵщ / dx], v = т>ае,ѳ + v e v0 = uf) е,ѳ + w e p = p(d,ip), V(s) • - (u0 • er)(v0) - ІЦ(0) X u4 = 0, (16) (17) (18) (19) "фсфі vo (4’0)) = sindIm(v(+)), + (24) f$(x, d) = -2a 1 sin d Im d2v^ / dx2 - 2 Re[(v£ ^ + - x sin d)dv^ / dx] - 2 Re(u,[) ^dv^ / dO) + + 2sin 1 d Im(u^ V^) + 2|dh)| ctgd, / (x,d) = -2ctgdRe(u,^u^)- - 2a 1 sin d Im d2v^ / dx2 - 2 Re[(v£ ^ i ж sin d)3?A+) / Зж] - 2 Re(v,j> ^ЗгА4^ / 3d). 1 2 / ф / V t) ^ As a result of the solution (24) we find (1|)+->]^, Е3(ж, d) = і(Е5(ж, cos d) + Е5(ж, - cos d)), V4(x1 d) = -ji(V5(x, cos d) - V5(x, - cos d)), V5(x, z) = sh(xsj2ioz) / sh yj2ioz. The functions V (1 - ж, 0) - К, . (ж, d) are odd in the variable x around the point ж = d . We find (dv/ dx) = - 2 а Re p(-+-> + sin d Im (Зг/4--* / dx) 2 (25) (26) (27) T = (p2 / p - 1ГІХ^-2. X = const = 0(1), а -i oo . (28) From (4), (11), (17), (27) and (28) we find the asymptotic behavior of the relative eccentricity u and slip Q as a function of the large oscillatory Reynolds numbers x = 0 - а sin 2 d Re 3[sin2 d f* dxv^ V4"-*] / 3d. From (11), (23) and (25) follows 2- + ... = I o(u2x + U2p )£ sin2 ddd Re p{+), а -► oo . From (18) and (26) we find О = п {и2 +V2.), C0 = - Re О = 7(1971^+4061) = о 404269 а -> сю . 1 ()\\ 0ж 1 0і//’ Q 1« 7) p\\x=0 = (U0y + 4je*V+)(^) + Ку - ІѢхК^Р^К®): -‘О у l,u'0x г/+) sin2 dp^ + - ^ dp G dx vr,i)Jx: d, ip) = К + iUox)e^vty (x, d) + («„ - iu0x)(Г^гН (x, d), ( )H = ( where do the boundary value problem with respect to p^^d), b, 3) come from Ф dv(+) (U0y+m0x)fM о (~aox +*(! + %))> (32) = -(1 + (Зж) 2 sin 1 d[S(sin dff (1 + (Зж)v^dx) / - 2v)}1 cos d - 2(3wy ■' sin d = 0, P \\x~0 “r b=0 i'> h=l 2 " > |ж=і Wr+)L=1 = -ysinO; sm-1 d[5(sind^+)))/5d + *(^+))] = y(l + (3)2 sind. We assume that (28) is true. When a »1, we look for a solution of (33) in the form V o,ip = vl+f fx, d) + o(l) in the area away from borders ж = 0 and ж = 1; 45 = 45%)(n^)+o(i), 4+) = oi1), ь = X'J®near ж = о; 45 = 45%)(^ + °(1)’ 4+) = Sin d + o(l), с = (1 - x)sfo near ж = 1. Boundary conditions are set for functions f and v f'f1. boundary conditions for v^e) are obtained from the conditions of matching asymptotic representations (34). Similarly (12), matching is possible everywhere except for the narrow neighborhoods of the support layer corresponding to d = тт / 3 and d = 2tt / 3 . These linear boundary value problems follow from (34) and (33) 18 Modeling the effect of centering of a spherical hydrodynamic suspension = -(1 + (Зж) 2 sin 1 TL>[f?(sin d (1 + $x)vj^dx) / [+,е) sin d + 2(3(1 + (3 x^if* vj’e)dx = 0, - (1 + (3x)~1dp('+') / df - + 2vJ,e') cos d + 2(3(1 + (3x)_19(sin d v^dx) / L_r3e 2 УѴ$ la;=o ^ Ф L_ U> (37) Ф 2 ^ "Ф \\x=0 X \\x=0 From (37) and (32) follows h=07 \\x=0 (U + iu )J7 df sin2 dp(+) = I (p / P - 1) (-a + i(l + a)) +0(1/Jo), о > 1. (38) Oy CU'JO r 3 Vr2 / г Ог V Oy'' 1 _ Since 3 «с 1, a further solution (35) is sufficient to hold up to Of S2), Up to small order 0(3), x'fJ> and v '1 are independent of x , and v- 1} is a linear function of x . That is ((3r/+’e)) = - i p sin d+о(з2), ^з(і+зжг1 /; vj^dx) = i 3 («■ From (39) and (35) we find dp- - i (ui+’e) \\ + 2 / v.(.+’e) \\ cos d + 8 - dd - ' \\ ^ , dd ,(+,d (39) i (vj’e) ^ + 2 ^+’e) ^ cos d + 3 sin d (vj’e) ^ + = 7T / 2 The solution of (41), (42) has the form (^+’e)) = -(?+Sp)cos^- From (43), (40), (38), (4) we find Jo sin2 ®dfp(+) = -г(А + Pi 3), (42) (43) 19 D.K. Andreichenko, K.P. Andreichenko, D. V. Melnichuk U0y; 0) ; (3)x,(l + %)u = u0a 2+о(ст 2), ct^oo, u0=(«0an _ 20 /-. 11 _ 20 /-. 11 u0x - 1^(1 yyP)X%r; u0y ~ ip (1 у Taking into account (4) and (29) we can assume Q = О03ст-7/2+ст-4), o^oo. Conclusion Based on the methods of asymptotic integration in the case when the decentering force is orthogonal to the sensitivity axis, the fast centering of the spherical hydrodynamic suspension at large values of the oscillatory Reynolds number is shown. It is shown that, up to the constant factor, the asymptotics of the dependence of the relative eccentricity on the oscillatory Reynolds number is similar to the results obtained earlier for cylindrical hydrodynamic suspension. However, the asymptotics of a similar dependence for the dimensionless sliding of the inner sphere is characterized by an order of magnitude smaller.

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