Improved Dutch Roll Approximation for Hypersonic Vehicle

Transcription

1 Sensos & Tansduces, ol. 173, Issue 6, June 14, Sensos & Tansduces 14 by IFSA Publishing, S. L. htt:// Imoved Dutch Roll Aoximation fo Hyesonic ehicle 1, * Liang-Liang Yin, 1 Yi-min Huang, 1 Chun-Zhen Sun, Jie Jia, 3, * Yong Yang 1 College of Automation Engineeing, Nanjing Univesity of Aeonautics and Astonautics, Nanjing, 16, China College of Infomation Engineeing, Nanchang Hangkong Univesity, Nanchang, 3369, China 3 School of Infomation Technology, Jiangxi Univesity of Finance and Economics, Nanchang 3313, China * yinliangshmily@16.com Received: 18 Ail 14 /Acceted: 31 May 14 /Published: 3 June 14 Abstact: An imoved dutch oll aoximation fo hyesonic vehicle is esented. Fom the new aoximations, the dutch oll fequency is shown to be a function of the stability axis yaw stability and the dutch oll daming is mainly effected by the oll daming atio. In additional, an imotant aamete called oll-to-yaw atio is obtained to descibe the dutch oll mode. Solution shows that lage-oll-to-yaw atio is the geneate chaacte of hyesonic vehicle, which esults the lage eo fo the actical aoximation. Pedictions fom the liteal aoximations deived in this ae ae comaed with actual numeical values fo s examle hyesonic vehicle, esults show the aoximations wok well and the eo is below %. Coyight 14 IFSA Publishing, S. L. Keywods: Hyesonic vehicle, Dutch oll, Liteal aoximation, Roll-to-yaw atio. 1. Intoduction Dutch oll is an oscillatoy intechange between side sli, oll and yaw that occus as a statically stable aicafts to eestablish lateal and diectional equilibium [1-3]. It is usually the most comlex mode of an aicaft. Dutch oll is chaacteized by the fequency and daming, which can be easily done by numeically detemining the eigenvalues and eigenvectos associated with the lineaized equations of motion [4-7]. Howeve, the eigenvalues and eigenvectos fo dutch oll deend on many aametes, and the natue of this deendence is not easily obsevable fom a numeical solution. Fo this eason, a liteal-fom aoximation is desied to evaluate the dutch oll. In addition, a liteal-fom solution has always been useful fo the otimization of flight contol system. Liteal aoximation to the dutch oll has taditionally been deived by consideing simlifying assumtions [8, 9]. The aoximation has been descibed in seveal textbooks and has been widely used ove the ast seveal decades [-13]. Howeve, esults obtained fom the taditional aoximation ae not in aticulaly good ageement with the numeical solution fo hyesonic vehicle, which flight with high angle of attack and lage mach numbes [14, 15]. In this ae, an imoved dutch oll liteal aoximation fo hyesonic vehicle is esented. In addition, a aamete called oll-to-yaw atio is 8 htt://

2 Sensos & Tansduces, ol. 173, Issue 6, June 14, obtained to descibe the mode, which scaled the elative amlitudes between oll ate and yaw ate. Solution shows that most of the hyesonic vehicles have lage oll-to-yaw atio chaacte, because of the high angle of attack and slende shae. The ae also oint out that the lage oll-to-yaw atio is the mainly eason fo the oo ageement between taditional aoiate and numeical solution. Finally, edictions fom the imoved liteal aoximation ae comaed with actual numeical values fo an examle hyesonic vehicle. The esult indicates the imoved aoximation woks well and the eos fo dutch oll fequency, daming and ollto-yaw atio ae all within %.. Lateal-Diectional Dynamics fo Hyesonic ehicle The lateal-diectional equations of motion fo hyesonic vehicle ae well defined fo the fom Δ x AΔx, (1) whee Δ x is the etubed state vecto, Δ x [ Δ, Δφ, Δ, Δ ] T, () The lateal-diectional system matix is given by Eq. (3). Y A N g Y + sinα 1 N Y tanα, (3) N The chaacteistic equation fo this system can be witten as in tems of eigenvalues as ( λr )( λs)( ξω ω) si A s s s + +, (4) whee λ R is the ole of oll mode, λ S is the ole of sial mode, ω is the dutch oll fequency and ξ is dutch oll daming. Dutch oll is the most imotant and comlex mode in these thee modes and the taditional dutch oll aoximation can be found in many textbooks on flight dynamics to be N Y ω d N + ( Y ) N, ξd (5) ωd This dutch oll aoximation has been used in actical alication fo the ast seveal decades. (about UA suesonic vehicle). Howeve It is quite aaent fom Table 3, that the aoximation is athe inaccuate fo both ω and ξ fo hyesonic vehicle. 3. Imoved Dutch Roll Aoximation fo Hyesonic ehicle The taditional aoximation fo dutch oll is obtained by neglecting the olling ate. The assumtion is easonable fo low seed aicafts, which have big wingsan and dutch oll amlitude fo the olling ate is smalle than that fo the yawing ate. Howeve, fo hyesonic vehicles, which usually have slende body and flight with high angle of attack, dutch oll amlitude fo the olling ate is much lage than the yawing ate. In this section, we develo a aamete called oll-to-yaw atio, which is defined as the atio of the amlitudes of the olling ate and yawing ate in dutch oll. Results show that hyesonic vehicle usually have big oll-to-yaw atio and the olling ate could not be neglected. To incease the ecision fo dutch oll, we develo an imoved dutch oll aoximation that includes the effects of olling ate Dutch oll Fequency ω Fo dutch oll fequency aoximation, we neglect the gavity and daming tem, because these values ae little and have little effect on dutch oll fequency. Hence, the lateal-diectional system matix can be witten as A Y sinα 1 tanα, (6) N The chaacteistic equation becomes Y + si A s s s N L sinα, (7) Hee we have lost two oots, one esulting fom the assumtion of no gavity and the othe esulting fom the assumtion of no damingtems (the sial and olling modes, esectively). Fom Eq. 4 and Eq. 7, the dutch oll fequency is ω N sinα, (8) Fom Eq. 7, the equiements fo latealdiectional static stability is N sinα >, (9) If we neglected the oduct of inetia I xz, the Eq. 9 can be educes to the equivalent condition. C ndyn, >, () 9

3 Sensos & Tansduces, ol. 173, Issue 6, June 14, whee I C C C sinα, (11) zz ndyn, n l I xx Equation 11 indicates that C ndyn, is diectly elated to the genealized static stability equiement, and the taditional lateal-diectional stability citeion C n is only valid when α. 4.. Roll-to-Yaw Ratio Roll-to-yaw atio is defined as the atio of the amlitudes of the olling ate and yawing ate in dutch oll mode. Fo aicafts with low oll-to-yaw atio, yawing motion is much lage than olling motion and vice vesa. Roll-to-yaw atio is equal to the atio of olling ate and yawing ate in eigenvecto associate with dutch oll mode. The lateal-diectional eigenvectos is given by Table 1. Table 1. The lateal eigenvectos. Roll mode Sial mode Dutch oll mode v 11 v 1 v 13 v 14 φ v 1 v v 3 v 4 v 31 v 3 v 33 v 34 v 41 v 4 v 43 v 44 g s Y si A s N Δ () s and Δ () s can be obtained by qi and Thus, qj Y Y sinα + 1, (15) s N s N s Δ 43 s tanα s, (16) s sinα Δ 44 s 1 s( s( s ) sinα), (17) s s( s ) sinα s ξω + jω 1 ξ d d ξω ω + ξω sinα jω 1 ξ ( ξω + ) d d d d d (18) The oll-to-yaw atio is defined by the amlitudes of as follows Thus, v34, (1) v44 The atio of the x i and x j elements in the eigenvecto associated with mode k have been denoted by x () i ( 1 i j Δqi s ), x j Δqj() s (13) modek s λ whee Δqi () s and Δ qj () s ae the minos of the chaacteistic deteminant of the system. Thus, Δ Δ Y Neglect the tems s ξω + jωd Y, and deteminant of the system becomes k a ξ, (14), chaacteistic ( ξω ) ( ) sin 1 ( ) d ωd + ξωd α + ωd ξ ξωd + ( L sin ) ω + α + ω d d Fom Eq. 8 and Eq. 19, the oll-to-yaw atio is 1, d N sinα 1+ N N cos α (19) () The oll-to-yaw atio is mainly effected by the atio of lateal static stability and diectional static stability in stability axes. Diectional static stability of hyesonic vehicles is usually weak, fo the vetical stabilize being shaded by the body at high angle of attack. In addition, the hyesonic vehicle most has a slende shae. Hence, the oll-to-yaw atio is vey lage. Results fom Table indicate that the 3

4 Sensos & Tansduces, ol. 173, Issue 6, June 14, oll-to-yaw atio of a hyesonic vehicle eaches as lage as 15 when the AOA eaches 3. This also oints out why the taditional aoximation is not satisfactoy fo hyesonic vehicle Dutch oll Daming ξ Fo the lage oll-to-yaw atio, dutch oll of a hyesonic vehicle consists mainly of sidesliling and olling, and the yawing contibutes little to the dutch oll. With neglecting the yawing ate, we lead to a thee-degees-of-feedom dutch aoximation. The simlified thid-ode system matix is as follows. Y g sinα A 1, (1) The chaacteistic olynomial is now obtained vey easily by the system matix Y Y g + + +, () 3 si A s s ( L ) s( L Lsin α) L This olynomial has a oot coesond to the oll subsidence ole and two quadatic oots coesond to the dutch oll oles. Hence, the olynomial could be witten as si A s s s ( + ξω + ω)( λr), (3) Fom Eq. and Eq. 3, we obtain Y ξ ω + d, (4) g ω d Accoding to Eq. 8 and Eq. 4, the dutch oll fequency and daming is given by ω N sinα Y g, (5) ξ + ω ω 3 ω The exession indicates that the dutch oll daming is dominated by the deivative ( is vey lage fo hyesonic vehicle), which is the behavio of aicafts with lage oll-to-yaw atio. The daming fomula tends to be moe accuate than taditional aoximation. This will be illustated late, by examle. The daming fomula is helful fo stability agument of dutch oll. Fo a conventional aicaft, it is effective to incease the dutch oll daming by feeding back yawing ate to udde, which inceases N as given by Eq. 6. Howeve, fo a hyesonic vehicle, it will be moe effective to feed back olling ate to aileon, which to incease as given by Eq Numeical Illustation As an illustation, edictions fom the dutch oll aoximation in this ae ae now comaed with exact numeical values fo an examle hyesonic vehicle. Conside the aicaft at α 3 and H 7km having 7, L 9.63, N.1 L 5.6, L 1.39 N 5.6, Y.83 Y.76, 4.45 N, Y.64 Fo this aicaft and flight condition, the exact numeical solution obtained numeically fom the lineaized equation of motion gives 14.3, ω 1.16, 9.7 ξ The aoximate solution obtain by using Eq. and Eq. 5 esults in 14.5, ω 1.16, 1. ξ Fom the above esults, the oll-to-yaw atio, fequency and daming edicted by the imoved dutch oll aoximation agee vey closely with the exact solution. The oll-to-yaw eo is by %, the fequency eo is by % and the daming eo is by 5 %. Table and Table 3 give the comaed esults between the imoved aoximation and the taditional aoximation, with the angle of attack vaying between 17 and 3 degees, the altitude vaying between 85 and 55 km and the mach numbes vaying between and 17. Table. Comaing of oll-to-yaw atio to numeical values. H,α,M Numeical Aoximate a solution solution Eo (85,3,) % (77,3,1) % (7,3,) % (65,5,19) % (6,5,18) % (55,,17) % 31

5 Sensos & Tansduces, ol. 173, Issue 6, June 14, Table 3. Comaing of dutch oll to numeical values. Dutch Roll Fequency Dutch Roll Daming Flight Condition Numeical solution Imove Aoximation Taditional Aoximation Numeical solution Imove Aoximation Taditional Aoximation H,α,M a value value eo value eo value value eo value eo (85,3,) * * * (77,3,1) * -4-1.* * (7,3,) * * * (65,5,19) * * * (6,5,18) * -4-3.* * (55,,17) * * * Table shows how the oll-to-yaw atio given by Eq. comaes with the exact numeical solution fo a boad ange of flight condition. It indicates that the aoximate oll-to-yaw atio given by Eq. is in good ageement with the exact solution. In addition, as the angle of alha gows, the oll-to-yaw atio becomes moe and moe lage. This is because the vetical stabilize is shaded by the body and the diectional static stability is low as the alha inceases. A comaison of the imoved dutch oll aoximation with the exact numeical values is shown in Table 3, which also contains a comaison of the taditional aoximation. Clealy, the imoved aoximation seems to edict the actual values well, and is definitely sueio to the taditional aoximation. The fequency eo is by % and the daming eo is by 6 %. Note that, with the decease in angle of attack, the daming eo fo the imoved dutch oll aoximation inceases. This is because the oll-toyaw atio deceases as the angle of attack deceases. As the oll-to-yaw atio is lowe, the imoved aoximation becomes wose, which base on a lage oll-to-yaw atio. Fo examle, if we decule the N in the ealie examle, the aicaft becomes a small oll-to-yaw atio tye. In this case, the exact numeical values is ξ 1. The aoximate values by Eq. 5 is ξ 1. We found that, the eo will incease to 5 % when the oll-to-yaw atio is less than.. Nevetheless, nealy all hyesonic vehicles have oll-to-yaw atio moe than.. 6. Conclusions An imoved dutch oll aoximation fo hyesonic vehicles has been develoed. The taditional dutch oll aoximation neglects the olling ate, which is valid only fo aicafts with small oll-to-yaw atio. Howeve, hyesonic vehicles ae a classical tye of aicafts with lage oll-to-yaw, fo which the taditional aoximation is invalid. The imoved dutch oll aoximation consides the effect of olling ate, and has highe accuacy as the oll-to-yaw atio inceases. Results show that the imoved dutch oll aoximation woks well unde a wide ange of flight conditions. Acknowledgements This wok was suoted by the Fundamental Reseach Fund (56XZA8), by the National Natual Science Foundation of China (61631, 61634), by the Postdoctoal Foundation of China (1M5593), by the Aeosace Science Foundation (1561), by the Natual Science Foundation of Jiangxi Povince (114BAB1, 13BAB5), by the Young Scientist Foundation of Jiangxi Povince (1BCB317), and by the Science and Technology Reseach Poject of the Education Deatment of Jiangxi Povince (No. GJJ133). Refeences [1]. L. Huang, Z. S. Duan and J. Y. Yang, Challenges of contol science in nea sace hyesonic aicafts, Contol Theoy & Alications, ol. 8, Octobe 11, []. N. M. Qi, Q. H. Zhou and C. M Qin, The Six DOF Model of Hyesonic ehicle and Couling Chaacteization Analysis, Jounal of Pojectiles, Rockets, Missiles and Guidance, ol. 3, 1, [3]. Phili Calhoun, An enty flight contols analysis fo a eusable launch vehicle, in Poceedings of the 38 th Aeosace Sciences Meeting and Exhibit,. [4]. M. T. Moul, J. W. Paulson, Dynamic Lateal Behavio of High-Pefomance Aicaft, NASA, RM- L58E16, [5]. D. Siqueia, P. Paglione and F. Moeia, Robust fixed stuctue outut feedback flight contol law synthesis and analysis using singula stuctued value, Aeosace Science and Technology, ol. 3, 13,

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