VIBRATION ANALYSIS OF VIBRATORY GYROS. C. S. Chou*, C. O. Chang* and F. H. Shieh +

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1 IV14 irn Autrli 9-1 July 007 VIBRATION ANALYI OF VIBRATORY GYRO.. hou*. O. hng* nd F. H. hieh + *Intitute of Applied Mechnic Ntionl Tiwn Univerity No.1 ec. 4 Rooevelt RodTipei Tiwn R.O. hin + hung-hn Intitute of cience nd Technology Toyun Tiwn R.O. hin chouc@pring.im.ntu.edu.tw Abtrct The purpoe of thi pper i to invetigte the vibrtion of vibrtory gyrocope with imperfection. A liner model due to imperfection of mteril or mnufcturing tolernce of vibrtory gyrocope i etblihed. The opertion of vibrtory gyrocope without or with imperfection re decribed under free vibrtion. Effect of the imperfection in term of dmping gyrocopic tiffne nd circultion in the governing eqution re nlyed with multiple time cle method. Effect of the reulting ngulr frequency vrition only nd nioelticity re invetigted vi the vrition of the ellipticl orbit of reference point on the element reltive to coordinte ytem fixed on the gyro. 1. INTRODUTION Vibrtion gyrocope bed on the modl vibrtion pttern of the ring or the hemiphericl hell move when the hell i rotted bout it xi nd thi movement provide meure of the pplied rte of turn. Hemiphericl reontor gyrocope excited with electrottic field i not only expenive nd hrd to fbricte. We re developing new type of hemiphericl reontor gyrocope which i excited with dicrete piezocermic ctution nd ening element bonded on the outer urfce nd cloe to the rim of the hell. A liner error model due to imperfection of mteril or mnufcturing tolernce of the hell in term of dmping gyrocopic tiffne nd circultory w etblihed [1]. The behviour of the reulting ngulr frequency vrying only nd nioelticity re invetigted vi the vrition of the ellipticl orbit of reference point on the element reltive to coordinte ytem fixed on the gyro.. IDEAL VIBRATORY GYROOPE Governing eqution for vibrtory gyrocope cn be written in generl form && x + ω x GΩ y & && y + ω y GΩx. & (1) where x nd y repreent et of orthogonl generl coordinte of point on the longitudinl xi of the vibrting member meured in plne fixed in the gyro nd norml to the xi ω i the rdil frequency of the vibrtory gyrocope Ω(<<ω) i the rotting peed of the gyro nd G i the ening coefficient. Eqution (3) i normlized with

2 IV July 007 irn Autrli d d Ω * t ωt Ω εω x( t) ( t) y( t) Y( t) ω dt dt ω () to obtin * + εgω Y * Y + Y εgω (3) where ε i mll prmeter in perturbtion method nd ω1..1 Ellipticl orbit For non-rotting ce; ω0 the generl olution of Eq. (3) i given by ( t) Acoϕ co( t + ϑ) Binϕin( t + ϑ) (4) Y( t) Ainϕ co( t + ϑ) + Bcoϕin( t + ϑ). The motion of the point (Y) i n ellipe in the -Y plne hown in Figure. Prmeter A B ϕϑ re contnt tht depend on the initil condition. Prmeter A B ϕ define the hpe nd orienttion of the ellipe nd ϑ the orbitl ngle. Figure Ellipticl orbit The normlized generl energy E NG nd ngulr momentum re A NG given by 1 ( ENG + Y + & + Y & ) ANG Y& Y &. (5) nd 1 ( ENG A + B ) ANG AB. The motion of the point (Y) i tright line(b0) when the ngulr momentum i zero.. Angulr velocity ening For non-zero ngulr velocity perturbtion technique bed on the method of two time cle i ued to olve the governing eqution [3]. The vibrtion olution of n idel vibrtory gyro i ( r1co( εgω t) + rin( εgω t))co t + ( r3co( εgω t) r4in( εgω t)) + O( ε)in t Y ( rco( εgω t) r1in( εgω t))co t + ( r4co( εgω t) + r3in( εgω t)) + O( ε)in t. In form of the ellipticl orbit generl olution we hve (6) cot0 + in T0 Y Y cot0 + Y in T0. (7) Acoϕ coϑ Binϕin ϑ Acoϕinϑ Binϕco ϑ Y Ainϕ coϑ+ Bcoϕin ϑ Y Ainϕinϑ+ Bcoϕco ϑ. (8) The behviour of the orbit prmeter i invetigted by differentiting Eq. (8) to obtin

3 IV July 007 irn Autrli A coϕcoϑ coϕinϑ+ Y inϕcoϑ Y inϕin ϑ B inϕinϑ inϕcoϑ+ Y coϕinϑ+ Y coϕco ϑ 1 ϕ [ ( in co co in A ϕ ϑ+ B ϕ ϑ0 ) + ( Ainϕinϑ+ Bcoϕco ϑ) A B (9) + Y ( Acoϕcoϑ+ Binϕin ϑ) + Y ( Acoϕinϑ+ Binϕco ϑ)] 1 ϑ [ ( co in in co ) ( co co in in ) A ϕ ϑ B ϕ ϑ + A ϕ ϑ+ B ϕ ϑ A B + Y ( Ainϕinϑ+ Bcoϕco ϑ) + Y ( Ainϕcoϑ Bcoϕin ϑ)]. The olution of the rotting gyro repreent in ellipticl orbit form we hve the mplitude term 1 ( r1co( GΩ T1) + rin( GΩ T1)) b1 ( r3co( GΩ T1) r4in( GΩ T1)) Y ( rco( GΩ T1) r1in( GΩ T1)) Y b ( r4co( GΩ T1) + r3in( GΩ T1)). Eqution (10) i differentited with repect to T 1 we obtin (10) GΩ Y GΩ Y Y GΩ Y GΩ. (11) The chnge rte of the orbit i obtined * A 0 B 0 ϕ GΩ ϑ 0. (1) The chnge rte of the ϕ i proportionl to the rotting peed of the gyro nd G i the ening coefficient. The orbit of the point (Y) i no longer n ellipe but h roete ppernce hown in Figure. Figure The preceion of the orbit of gyro with non-zero rottion 3. NONIDEAL VIBRATORY GYROOPE A liner error model due to imperfection of mteril or mnufcturing tolernce of the gyro in term of dmping c gyrocopic g tiffne k nd circultion h in the governing eqution re written * && + εgω Y & - ε( c& -c& +gy& gy& + k k + hy hy ) (13) * Y&& + Y + εgω & - ε( cy+c & Y+g & & + g& + ky + ky + h + h). Error re umed reltively mll nd re treted through ε mll perturbtion. ubcript c nd re repreenting ymmetricl nd nti-ymmetricl term repectively. Prmeter c c g g k k h nd h re umed contnt in time. Ech term in the liner eqution Eq. (13) cn be conidered eprtely nd then uperpoed. Eqution of error due to the ymmetric prt of dmping c on non-rotting (Ω0) gyro re && + - εc & Y && + Y - εcy &. (14)

4 IV July 007 irn Autrli A Ac B Bc ϕ 0 ϑ 0. (15) Eqution for error due to the nti-ymmetric prt of dmping c we hve + εc Y + Y- εcy. (16) ( A + B )inϕ ABinϕ A caco ϕ B cbco ϕ ϕ c. ϑ c A B A B (17) imilrly we obtin orbit chnge rte due to error of tiffne k nd k repectively A 0 B 0 ϕ 0 ϑ k (18) AB co ϕ ( A + B )co ϕ nd A kbin ϕ B kain ϕ ϕ k. ϑ k (19) A B A B Orbit chnge rte due to error of gyrocopic g nd g repectively ( A + B )coϕ ABcoϕ A g Ain ϕ B gbin ϕ ϕ g. ϑ g A B A B (0) nd A 0 B 0 ϕ g ϑ 0. (1) Orbit chnge rte due to error of circultory h nd h repectively ABin ϕ ( A + B )in ϕ A h Bco ϕ B haco ϕ ϕ h. ϑ h A B A B () nd A Bh B Ah ϕ 0 ϑ 0. (3) We obtin A 0 c h 0 0 A B 0 h c 0 0 B + ϕ g ϕ ϑ k ϑ ca coϕ ga inϕ + kb inϕ + hb coϕ (4) cb coϕ + gb inϕ ka inϕ ha coϕ + ( A + B )in ϕ ( A + B )co ϕ AB co ϕ AB in ϕ c g + k h A B A B A B A B ABin ϕ ABco ϕ ( A + B ) co ϕ ( A + B )in ϕ c + g k + h A B A B A B A B The contnt term in the RH of Eq. (4); the gyrocopic nti-ymmetric error g chnge only ϕ but not A B nd the phe ngle ϑ the other contnt term k chnge the frequency only hown in Figure 3 nd 3b. For zero rte input (Ω0) nd without k nd g we hve: && + - ε ( c-c & +gy & & k+ hy hy ) Y&& + Y - ε ( cy+c & Y+g & & + ky + h + h). (5) A perturbtion technique bed on the method of two time cle i ued to olved bove eqution to obtin exp[ ( c ζ ) T1]( N co( T0 T1) + N in( T0 T1)) + exp[ ( c + ζ) T1]( Pco( T0 + T1) + Pin( T0 + T1)) Y exp[ ( c ζ ) T1]( YN co( T0 T1) + YN in( T0 T1)) + exp[ ( c + ζ) T1]( YPco( T0 + T1) + YPin( T0 + T1)). (6) The ubcript in Eq. (6) re: for coine for ine N for P for + nd

5 IV July 007 irn Autrli Er 1 Er 1 ζ + Ei + Er 0 + Ei + Er 0. E c + g + h h k E g h + c k. (7) Y Y r i P N P N [ g + ( h + h ) ]( cγ + cγ ) + cγ cγ [ g + ( h + h ) ]( cγ cγ ) cγ cγ P ( ζ + )[ g + ( h+ h) ] ( ζ + )[ g + ( h+ h) ] [ g + ( h + h ) ]( cγ cγ ) cγ + cγ [ g + ( h + h )]( cγ cγ ) + cγ + cγ N ( ζ + )[ g + ( h+ h) ] ( ζ + )[ g + ( h+ h) ] c1γ6 cγ7+ c3γ5 c4γ1 c1γ7+ cγ6+ c3γ1+ c4γ5 Y P ( ζ + ) ( ζ + ) c1γ6+ cγ7+ c3γ+ c4γ1 c1γ7+ cγ6+ c3γ1 c4γ Y. N (8) ( ζ + ) ( ζ + ) γ ζ γ ζ ζ 1 c k + c k γ + ζ + + ζ + ζ ζ ( h h)[ ( c k ) ck] g[ ( c k ) ck] γ + ζ ζ + + ζ + + ζ ( h h)[ ( c k ) ck] g[ ( c k ) ck] γ5 ζ + + ζc + k γ6 ζ g + ( h + h) γ7 g ζ( h + h). c 1 c c 3 nd c 4 re integrl contntdetermining by the initil condition. (9) () (b) Figure 3 Orbit chnge due to gyrocopic error g () nd tiffne error k >0 (b) nd Y re combined with four hrmonic vibrtion Eq. (6). Angulr frequencie of the vibrtion re vried from 1(normlized) to 1+ε nd 1-ε. For free vibrtion c ζ Effect of ngulr frequency vrie only ( 0ζ 0) For ngulr frequency vrie only ce Eq. (7) become Er c + g + h h k < 0. Ei gh + ck 0. (30) Then we hve ζ 0 ( c + g + h h k ) 0. (31) Thi i the ce of ngulr frequency vrie only. ubtitute Eq. (30) into Eq.(7) nd Eq. (6) to obtin exp( ct)( co( T T) + in( T T) + co( T + T) + in( T + T)) 1 N 0 1 N 0 1 P 0 1 P 0 1 Y exp( ct 1)( YN co( T0 T1) + YN in( T0 T1) + YPco( T0 + T1) + YPin( T0 + T1)) (3) where c 1( + k ) + c 3( h h ) c c + c g ( ) 4( ) N cc + c g c + k c h h N

6 IV July 007 irn Autrli c1( k) c3( h h) + cc c4g cc 1 c3g c( k) + c4( h h) P P c1( h + h) + c3( k) + cg + c4c c1g + c3c + c( h + h) c4( k) YN YN (33) c1( h + h) + c3( + k) cg c4c c1g c3c c( h + h) c4( + k) YP YP. nd Y re combined motion of four hrmonic ocilltion with mplitude of liner combintion of c 1 c c 3 nd c 4 nd lo of function of the coefficient of error. For ech given initil vlue there re two elliptic orbit hving me orienttion ngle correpond to 1+ε nd 1-ε hown in Fig. 4. ()1-ε (b) 1+ε Figure 4 Orbit for ngulr frequencie () 1-ε nd (b) 1+ε with initil vlue of 1. For et of given c 1 c c 3 nd c 4 there re two orbit of frequencie 1-ε nd 1+ε with me orbit ngle ϑ tht of individul c 1 c c 3 nd c 4 nd thee two orbit combine to form the trjectory in Y plne hown in Fig. 5. Figure 5 Orbit with frequencie 1-ε nd 1+ε nd trjectorie in Y plne In Figure 5 when the orbit with ngulr frequency 1+ε reched the pogee A the orbit of 1-ε i lg behind nd with origin t A rther thn O. The trjectory i formed by the orbit of ngulr frequency 1-ε with origin t ech point of the trjectorie of ngulr frequency 1+ε. Mjor xe of the orbit of ngulr frequencie 1-ε nd1+ε re referred the tiff xi nd the oft xi repectively. The orienttion ngle of orbit of 1-ε nd1+ε re given by

7 IV July 007 irn Autrli 1 YN coϑnl + YN inϑ Y nl 1 P coϑpl + YP inϑ pl ϕnl tn ( ). ϕ pl tn ( ). (34) N coϑnl + N inϑnl P coϑpl + P inϑpl The orienttion ngle of the ellipe i ocillting between the tiff nd the oft xe. 3. Effect of nioelticity on orbit Anioelticity effect i prticulr ce of ngulr frequency chnging only ( 0ζ 0). The tiffne mtrix i written in term of ioelticity nd nioelticity k 0 k h h 0 k + h h k (35) + The ioelticity effect i conidered in the contnt term in Eq. (6). For the nioelticity c 0 c 0 g 0. (36) Eqution (7) become E h h k E 0. (37) r i When E r <0 ζ 0 ( h h k ) 0. (38) N co( T0 T1) + N in( T0 T1) + P co( T0 + T1) + P in( T0 + T1) (40) Y YN co( T0 T1) + YN in( T0 T1) + YP co( T0 + T1) + YP in( T0 + T1). c1( + k) + c3( h h) c( + k) + c4( h h) where N N c1( k) c3( h h) c( k) c4( h h) P P (41) c1( h + h) c3( k) c( h + h) c4( k ) YN YN c1( h + h) + c3( + k) c( h + h) + c4( + k) YP YP. ( cc 3 cc 1 4)( + h h k ) We hve NYN NYN 0 (4) ( cc 3 cc 1 4)( + h h k ) PYP PYP 0. The ocilltion of ngulr frequencie 1-ε nd 1+ε re become line orbit with direction of b 1 ( N Y N ) nd b ( P Y P ) repectively hown in Figure 6 The incline ngle of the line ocilltion of ngulr frequencie 1-ε nd 1+ε re tn ϕnl ( h + h ) /( + k ) tn ϕnl ( h + h ) /( k ). (43) In generl the initil vlue re not limit the motion on the principl xe b 1 nd b the vibrtion i the combintion of the vibrtion on the two mjor xe determined by the initil vlue of c 1 c c 3 nd c 4 hown in Figure 8. (0) k + Y(0)( h h) [ (0)co( T1)]co( T0) + [ in( T1)]in( T0) (44) Y(0) k (0)( h + h) Y [ Y(0)co( T1)]co( T0) + [ in( T1)]in( T0). The mplitude i lowly vrying periodic function when T1 nπ n 01 K the orbit become line with incline ngle Y/Y(0)/(0). Thi i one of the mjor xe of motion A. when T 1 increing the mplitude of cot 0 decree nd int 0 incree nd the orbit become line gin whent1 nπ + π / n 01 K with incline ngle

8 IV July 007 irn Autrli Y Y(0) k (0)( h + h). Thi i lo one of the mjor xe of motion E. The incline (0) k + Y(0)( h h) ngle of the orbit i ocillting between thoe two mjor xe. () 1-ε (b) 1+ε Figure 6 Orbit of ngulr frequencie of () 1-ε nd (b) 1+ε Figure 8 Orbit with nioelticity UMMARY The contitutive eqution of homogenou nd niotropic thin hell re derived in n invrint form. Multiple time cle method i ued to derive the preceion of the free vibrting hell with non-zero rottion. A reference point on the vibrting hell reltive to coordinte ytem fixed on the hell upporting frme move in n ellipticl orbit with period inverely proportionl to the rotting peed of the hell. Liner error model due to imperfection of mteril or mnufcturing tolernce of the hell in term of dmping gyrocopic tiffne nd circultory re etblihed to derive the governing eqution. The effect of the reulting ngulr frequency vrying only nd the nioelticity re invetigted. ANKNOWLEDGEMENT Thi work i upported by the RO Ntionl cience ouncil under Grnt No E REFERENE [1] hou.. F. H. hieh nd. O. hng Vibrtion Anlyi of n Imperfect Hemiphericl Gyro Inter-noie 006 IN Dec. 006 Hwii UA.