Dynamic absorber for drilling instrument

I C N O X I A W W \ ni!J)Ai>r.r>i / V 1 9 0 7 / \ \ PROCEEDINGS IF THE ELEVENTH INTERNATIONAL CONFERENCE ON NONLINEAR OSCILLATIONS JANOS BOLYAI MATHEMATICAL SOCIETY B u d a p e st, 1 98 7 DYNAMIC ABSORỈiĩ R FOR DRILLING IKSTHULIEKT Nguyen Van Dao For damping the self-excited oscillation of the instrument (1) when drilling a spccicil mechanism [1] which consists o f an absorber tube (2) v;ith vibroda’-qping layer (3) is used (fie* !)• In this paper i t turned out that for strong absorber increasing the frictio n of layer (3) leads to essential decrease of the oscillation of the instrument. For the case of weak absorber, for optimal extinguishing of that oscillatio n i t is necessary to choose the suitable valuo of the friction. $1- We ale absorber Let us consider the weak absorber expressed, in the smallness of the stifnoss c2 and the moment of inertia Ig and Governed by equations friction force between instrument and cleiT.il (4). In the case Df strong absorber when Co , I o are fin ite the motion t _ t— 2 quations take form I 1 + C1 ^ 1 = e [~h ( - Q p ) + 0 ^ ) ] * ( 1 .1 ) here y f L ( Q p = CÍ 0^ - P0^ , a ,(5 y 0 is the moment of - 694 - 1 2 0 2 + c2 02 = -eh( 02 - 0 1) . (1.2) and the damping condition of oscillation of the instrum ent (1) is [1] h ^ a (1.3) The solution of (1.1) w ill be found in the form 0^ = acoscj) , Ỡ£ = a(Mcos<£> + Nsỉncị} ) , <p = 6-^t+Ỷ", (l*zO h| c ù ỵ h g ^ C c O i - U ) | ) hf ỏof + (6 o | - L ỏ ịỷ h |o j2 + (0 ) | - 0 )f)2 , , 2 ® i h h w i = T • **2 = 7 * ỉ 2 Substituting those expressions into (1.1) and averaginc on tim e th e rec eiv e d eq ua tions we o bta in da = ( a g _ - l p a ) 2a2 ) , (1.5) dt 2IX 1+Ơ h 4 1 aầ 2 t §— h íOỵHa , ơ2 = CủỊ / 1^(0)ị - CủỊ)2 . d t 21-^ Prom here i t fo llo w s th e s ta tio n a r y am p litudes a=const: 1) a = 0 , stable if a z_ h / (1 + o^li2) , (1.6) 2 ) a ỊẺ 0 , 2 p w ? a 2 = a ĩ ì _ ^ 0 ( f i s . 2 ) ( 1 . 7 ) 4 A 1 + a h Obviously, for optimal damping the oscillation of the instru m ent th e most advantageous v alue of h w il l be h = 1 /a Now consider the more complicated case •when the absorber tube (2) elastic ally (Cjp) connect0(1 v.ith inrjtrup.c-nt (1). In this case, the equations of iroticn will be I 1 + C 1 Ỡ 1 n G£~C12< Ỡ 1 “ " hC + 3l(- ' ( ( 1 - a) «• • * I2 0 2 + (c2 + c]2) 0 2 + h 02 = c12 9 X + h Ũ1 . The solution of these conations cell be found in the Torn Dj - acoscị} , ^ 2 = &(Fcosộ + Qsỉn ộ ) , (1.9) h2 iủị + o?2ĩ?(yị ~coỊ) hùì^iLừị -(ùị) x 2 1 v2 m 2n2 1 ,2 n 2 ,2 , y M2x2 h C0£ -h ^2*^2 ~ 1 1 ~2 ^2 *" 1 2 ° 2 + c 12 2 Averaged eauationn for amplitude a and phase- ijr will be . c t i) , ĩị(C ủ ị~ C ô h 2h 7 ~ p y đ a = . - i a [ a l ẵ l _ 2 5 H o ^ a 2 ], ( 1 .1 0 ) l d t 2 I 1 ù ị h 2 * 4 <iỵ2 - a ệ ) 2 * /s ù) a ílỶ = _J2_ [c (1 _ p) _ hq ] , 1 dt 2 lj The stationary amplitudes arc : ĩ ị c c ù ị - c £ ) \ 1) a = 0 , stable if <x ^ - — ,(1.11) p o 0 , p p <— h ỂOi + ij A Y j - a i l ) 2) a p 0 determined by O P - t o ? ) 2h 0 C ú ỉa = a - — ■ — • /1 r ^ o o d h2 ư ị + if(arf-a)f) The amplitude curve a = a(h) has minimum at value - u}G - 1 $ 2- Strong absorb01- ( 2 . 1 ) T h e m o t io n e q u a t i o n s f o r s t r o n r : absorber e l o s t i c -illy connected with instrument (1) will be : h § 1 + C1 Ỡ 1 + c 1 2 ( 0 1 - 0 2 } = £ [ - b( I 2 + c 2 0 2 + c 1 2 ^ = ~ e h ( ^ 2 “ ^ 1 ^ * The solu tion of those equations is found in the form 0 X = a-^co sT Ị^ + a 2 c o s t \ 2 t = ỉ-ỵa i C0G7l i + /.2 a 2 C 0 S 1^2 * - n lt * f t , a . ^ . I* = — ; > v . <3.2> 2 v 2 It is not difficult to receive the follov/inc averr^oil equations for amplitudes Slid phase Ỷ± : CLSl-1 _ _ / o 7 o o _ o o ^ 1 “ 7 7 = o / ỵ ) l i - ^ j3 (-n .1 a 1 + 2 Í22-'2)] , dt 2 Q = 1 k 2 n 2 a 2 [ a - ( l - /-2 )2 li - 5 p ( ứ | a | + 2 ) ] , (2,3) 2 (It 2 4 ị - ịr 2 = 0 . T h e s y s t e m ( 2 ,5 ) U 0SG 0S 3 C 3 t h o s t a t i c i : 3 : : 7 s o l u t i o n : 1 ) a -1 = ap = 0 , s t a b l e i f tt Z- ( 1 ~ iL ) c h raid OS / - ( 1 - L>)'“ 2 ) r j = 0 > p j ;■ u » A , - '■ r -Q V'-f - - ( 1 - t - '/ ' h ^ 0 s t a b l e i f ll - -L x f p f i r i ^ Ằ - ( 1 - Ụ 2jl ] • 5 ) R j ~ 0 , r , / 0 , J p - o: - ( 1 - ^ 0 s t a b l e i f J- I— r . il d 3 p o p ^ r , / -Ị I s P.Ị -1 ' \} 2 <> L « - C-1* L1 ) ii J - 4 <- 2 4) Cl-, /■ 0 , 0 o /- 0 but ii; is always unstable. TliUoj iiic.vec-r; the frictio n h leads to the d e c r e a s e o f t h e a r .p l it u d o a ( f i e . 3 ) . o fc. a f i a . 2 . 98 7 DYNAMIC ABSORỈiĩ R FOR DRILLING IKSTHULIEKT Nguyen Van Dao For damping the self-excited oscillation of the instrument (1) when drilling a spccicil mechanism [1] which consists o f an absorber. turned out that for strong absorber increasing the frictio n of layer (3) leads to essential decrease of the oscillation of the instrument. For the case of weak absorber, for optimal extinguishing. friction. $1- We ale absorber Let us consider the weak absorber expressed, in the smallness of the stifnoss c2 and the moment of inertia Ig and Governed by equations friction force between instrument