Chapter 9 Dynamic stability analysis III Lateral motion (Lectures 33 and 34)

Transcription

1 Pof. E.G. Tulapukaa Stability and contol Chapte 9 Dynamic stability analysis Lateal motion (Lectues 33 and 34) Keywods : Lateal dynamic stability - state vaiable fom of equations, chaacteistic equation and its oots ; motions indicated by oots - ole subsidence, spial mode, Dutch oll ; stability diagam. Topics 9. ntoduction 9. State vaiable fom of lateal dynamic stability equations 9.3 Roots of stability quatic fo geneal aviation aiplane Navion 9.3. teative solution fo lateal stability quatic 9.4 Lateal dynamic stability analysis which includes angula displacement in yaw ( Δψ ) as a state vaiable 9.5 Response indicated by oots of chaacteistic equation of lateal motion- change in flight diection, oll subsidence, spial mode and Dutch oll 9.6 Lateal-diectional esponse of geneal aviation aiplane (Navion) 9.7 Appoximations to modes of lateal motion 9.8 Two paamete stability diagam fo lateal motion Refeences Execises Dept. of Aeospace Engg., T Madas

2 Pof. E.G. Tulapukaa Stability and contol Chapte 9 Dynamic stability analysis Lateal motion - Lectue 33 Topics 9. ntoduction 9. State vaiable fom of lateal dynamic stability equations 9.3 Roots of stability quatic fo geneal aviation aiplane Navion Example teative solution fo lateal stability quatic 9.4 Lateal dynamic stability analysis which includes angula displacement in yaw ( Δψ ) as a state vaiable 9.5 Response indicated by oots of chaacteistic equation of lateal motion- change in flight diection, oll subsidence, spial mode and Dutch oll 9. ntoduction n chapte 7 the small petubation equations fo the longitudinal and lateal motions with contols fixed wee deived. Expessions fo the stability deivatives wee also obtained. The analysis of the longitudinal dynamic stability was dealt with in Chapte 8. With this backgound, the lateal dynamic stability is biefly discussed in this chapte. The distinguishing featues of lateal dynamic stability ae highlighted. 9. State vaiable fom of lateal dynamic stability equation The equations of motion fo the lateal case (Eqs.7.88, 7.89, 7.90) ae epoduced hee fo eady efeence. d ( -Y ) Δv - (u -Y )Δ - g cosθ Δφ = Y Δδ dt d d -L' Δv + ( - L' )Δp - [ + L' ]Δ v 0 0 δ xz v p dt xx dt (7.88) = L' δa Δδ a + L' δ Δδ (7.89) Dept. of Aeospace Engg., T Madas

3 Stability and contol Pof. E.G. Tulapukaa d d - N Δv - ( + N ) Δp - [ - N ] Δ xz v p dt dt = Nδa Δδ a +Nδ Δδ (7.90) These equations can be put in the state vaiable fom using the following steps: Δv = Y V Δv + Yp Δp-(u0 - Y )Δ + gcosθ0δφ + Y δ Δδ (9.) d Δp = L Δv + L Δp + Δ + L Δ + L Δδ + L Δδ (9.) XZ V p δa a δ XX dt Δ = N Δv + N Δp + d Δp + N Δ + N Δδ + N Δδ XZ V p δa a δ ZZ dt (9.3) φ Δp (9.4) The tem [d(δ) / dt ] o Δ occus on the ight hand side of Eq.(9.). ts expession, given by Eq.(9.3), is used to eplace it. The esulting equation would have the following tem on the ight hand side: xz xx zz d Δp dt Taking it to the left hand side yields: d (- ) Δp = (L' + N )Δv + (L' + N )Δp dt xz xz xz v v p p xx zz xx xx + (L' + N )Δ + (L' + N )Δδ + (L' + N ) Δδ xz xz xz δa δa a δ δ xx xx xx The following notations ae intoduced hee (Ref.., chapte 4) : * L' v * Nv L v =, N v =, etc. xz xz (- ) (- ) xx zz xx zz Use of the notations given in Eq.(9.6), endes Eq.(9.5) as: Δp = (L + N )Δv + (L + N ) Δp * xz * * xz * v v p p xx xx + (L + N )Δ + (L + N ) Δδ + (L + N )Δδ * xz * * xz * * xz * δa δa a δ δ xx xx xx (9.5) (9.6) (9.7) Dept. of Aeospace Engg., T Madas 3

4 Stability and contol Pof. E.G. Tulapukaa Similaly, eliminating expession fo Δ : Δ = (N + L )Δv + (N + L ) Δp * xz * * xz * v v p p p fom.h.s of Eq. (9.3) and simplifying gives the following + (N + L )Δ + (N + L )Δδ + (N + L ) Δδ * xz * * xz * * xz * δa δa a δ δ The final fom of the equations fo lateal motion in the state vaiable fom is: (9.8) Yv Yp -u 0+Y gcosθ0 Δv * xz * * xz * * xz * Δv L v+ Nv L p+ Np L + N 0 Δp Δp xx xx xx = Δ * xz * * xz * * xz * Δ N v+ Lv N p+ Lp N + L 0 Δ zz zz Δ Y δ * xz * * xz * Δδa + L δa+ Nδa L δ + Nδ Δδ xx xx * xz * * xz * N δa+ Lδa N δ + L δ This can be put in the fom: (9.9) X = A.X +B.η (9.0) When the contol deflections ( δ a and δ in the pesent case) ae fixed, η = 0 and Eq.(9.0) educes to: X = A.X (9.) As pointed out in section 8.0, Eq(9.) has the following solution. λ A 0 (9.) Whee λ s ae the eigen values of matix A. Expanding Eq.(9.), the following chaacteistic equation fo the lateal motion is obtained. Dept. of Aeospace Engg., T Madas 4

5 Stability and contol Pof. E.G. Tulapukaa 4 3 A λ + B λ + C λ + D λ + E = 0 (9.3) 9.3 Roots of stability quatic fo geneal aviation aiplane (Navion) This section illustates the pocess of setting up matix A and then obtaining the eigen values. Example 9. The geneal aviation aiplane (Navion) discussed in example 8. has the following lateal stability deivatives (Ref..4, section X ). Y v = s -,Y β = m s -, Y p = 0, Y = 0, L v = m - s -, L β = s -, L p = s -, L =.93 s -, N v = m - s -, N β = s -, N p = s -, N = s -, xx = 40.9 kg m, zz = kg m, xz = 0. The matix A in Eqs.(9.9) and (9.0) in this case is A = (9.4) The eigen values of this matix ae the oots of the lateal stability quatic. Using MATLAB softwae the oots ae: , , ± i teative solution of lateal stability quatic The chaacteistic equation (Eq. 9.3) is called lateal stability quatic. Refeence.5, pat chapte, contains an iteative method to solve it. The case of Navion is consideed to illustate the pocedue. Using the stability deivatives in example 9., the following chaacteistic equation is obtained. 4 3 λ λ λ λ = 0 (9.5) As mentioned in section 8.5, the iteative technique, fo obtaining oots of a quatic, depends on the elative magnitudes of the coefficients of the tems in Dept. of Aeospace Engg., T Madas 5

6 Stability and contol Pof. E.G. Tulapukaa the polynomial. Fo lateal stability quatic a technique diffeent fom that used in section 8.5 is employed. t is seen that the coefficient E is much smalle than D. Hence, a small oot is expected. When the oot is small, the tems with powes of λ would be vey small in Eq.(9.5) and the fist appoximation λ () can be witten as: () λ = - E / D = / = To get the second appoximation to λ, Eq.(9.5) is witten as: 3 (λ ) (λ λ λ ) = 0 f λ () wee the exact oot, then the emainde should be zeo. n this case it is The second appoximation is obtained as: λ () = / = Substituting in Eq.(9.5), gives: 3-6 (λ )(λ λ λ )+.7 0 (9.6) Since, the emainde is vey small λ = is taken as the fist oot. To get the othe thee oots, the cubic pat of Eq.(9.6) is now consideed i.e. 3 λ λ λ = 0 (9.7) To solve the Eq.(9.7), it is assumed that the second oot ( λ ) is lage. Then, fo the fist appoximation ( λ () ), the second and thid tems in Eq.(9.7) can be ignoed i.e. 3 () λ λ = 0 o λ = To get the second appoximation to λ, Eq.(9.7) is divided by λ and λ () is substituted in the neglected tems. This gives : () λ = = ( ) ( ) n a simila fashion, (3) (4) λ = and λ = ae obtained. The last two values ae faily close to each othe and λ = is taken as the second oot. Substituting fo λ and λ in Eq.(9.5) leads to: Dept. of Aeospace Engg., T Madas 6

7 Stability and contol Pof. E.G. Tulapukaa (λ )(λ )(λ λ+5.676). Hence, the fou oots ae: , , i.335 which ae almost the same as those obtained by using matix A and the MATLAB Softwae. 9.4 Lateal dynamic stability analysis which includes angula displacement in yaw (ΔΨ) as a state vaiable n the analysis pesented in section 9., the quantity Δφ is included as a vaiable since, it is diectly involved in Eq.(7.88). The angula displacement in yaw ( ) can also be included. n this case the following additional govening equation is obtained. Δ = (9.8) Refeence. (chapte 6) and Ref..5 (chapte 7 pat ) follow this appoach. n this case Eq.(9.9) would get modified as: Yv Yp -u 0+Y gcosθ0 0 Δv * xz * * xz * * Δv xz * L v+ Nv L p+ Np L + N 0 0 Δp xx xx Δp xx = Δ * xz * * xz * * xz * Δ N v+ Lv N p+ Lp N + L 0 0 Δ zz zz Δ Δψ Δψ Yδ * xz * * xz * Δδa + L δa+ Nδa L δ + N δ Δδ xx xx * xz * * xz * N δa+ Lδa N δ + L δ The chaacteistic equation coesponding to this set of equations would be : (9.9) Aλ + Bλ + Cλ + Dλ + Eλ = 0 (9.0) This equation has λ = 0 as the fifth oot in addition to the fou oots discussed ealie. Dept. of Aeospace Engg., T Madas 7

8 Pof. E.G. Tulapukaa Stability and contol Note: Simila esult (i.e. a fifth degee polynomial as the chaacteistic equation), would have been obtained, if Eqs.(7.88),(7.89) and (7.90) wee solved assuming : Δv = ρ e λt, Δφ = ρ e λt, Δ = ρ 3 e λt (9.) 9.5. Response indicated by oots of chaacteistic equation of lateal motion change of flight diection, oll subsidence, spial mode and Dutch oll The chaacteistic equation fo the lateal motion (Eq.9.0) has five oots. The motions indicated by these can be biefly descibed as follows. ) λ equal to zeo oot: Fom Eq.(9.) it is obseved that when, λ is zeo the solution would be: Δv = ρ, Δφ = ρ, Δ = ρ 3. (9.) Substituting these in Eqs.(7.88),(7.89) and (7.90) shows that Δv equals 0 but Δφ and ΔΨ need not be zeo. Thus the zeo oot, which indicates neutal stability, could lead to the aiplane attaining a constant angle of bank (Δφ) and / o yaw (ΔΨ). Figue 9. shows a change in flight diection as a esult of the zeo oot. This featue is discussed again, in section 9.6, when the esponse of the geneal aviation aiplane to a lateal distubance is consideed. t may be pointed out that appopiate contol action is needed to coect the constant values of Δφ and Δ. Dept. of Aeospace Engg., T Madas 8

9 Stability and contol Pof. E.G. Tulapukaa Fig.9. Diectional divegence and spial divegence ) Lage negative oot: t is called oll subsidence. t is heavily damped and does not pose any poblem. ) Small negative oot: The lightly damped eal oot is called spial mode. When the oot is positive i.e. unstable, the aiplane loses altitude, gains speed, banks moe and moe with inceasing tun ate. The flight path is a slowly tightening spial motion (Fig.9.). The diffeence between spial motion and spin is that in the fome the angle of attack is below stall and the contol sufaces ae effective. Section 0. biefly deals with spin. V) The complex oot indicating oscillatoy motion: The motion coesponding to this oot is called Dutch oll. t can be biefly descibed as follows. a) Let the distubance cause oll to left. b) Due to advese yaw, this oll esults in yaw to ight, i.e. sideslip to the ight. Dept. of Aeospace Engg., T Madas 9

10 Pof. E.G. Tulapukaa Stability and contol c) Because of the dihedal effect, the sideslip causes a estoing moment causing oll to ight. d) Subsequently aiplane yaws to left and olls to left. e) This sequence of events continues and when the eal pat of the oot is negative the amplitude of the oscillatoy motion deceases with time. f) Duing this oscillatoy motion the aiplane is all along moving fowad (Fig.9.). The motion is simila to the weaving motion of an ice skate. This mode is called Dutch oll. Accoding to Wikipedia, skating epetatively to the ight and to the left is efeed to as Dutch oll. J.Hunsake appeas to be the fist to efe the afoesaid oscillatoy motion of the aiplane as Dutch oll. Fig. 9. Dutch oll Dept. of Aeospace Engg., T Madas 0

11 Stability and contol Pof. E.G. Tulapukaa Remak: Dutch oll is an undesiable mode as it causes discomfot to the passenges in civil aiplanes and may esult in missing the enemy tagets in militay aiplanes. t should have adequate damping. Dept. of Aeospace Engg., T Madas