Aircraft flight dynamics

242: 259: 12713: 276: 556: 8202: 10563: 85: 8358: 7082: 5013: 4846: 210: 3745: 224: 9021:(sideslip), r (yaw rate) and p (roll rate), with moments N (yaw) and L (roll), and force Y (sideways). There are nine stability derivatives relevant to this motion, the following explains how they originate. However a better intuitive understanding is to be gained by simply playing with a model airplane, and considering how the forces on each component are affected by changes in sideslip and angular velocity: 196: 43: 10263: 6785: 3467: 4302: 10558:{\displaystyle \left({\frac {N_{p}}{C}}{\frac {E}{A}}-{\frac {L_{p}}{A}}\right){\frac {d^{2}\beta }{dt^{2}}}+\left({\frac {L_{p}}{A}}{\frac {N_{r}}{C}}-{\frac {N_{p}}{C}}{\frac {L_{r}}{A}}\right){\frac {d\beta }{dt}}-\left({\frac {L_{p}}{A}}{\frac {N_{\beta }}{C}}-{\frac {L_{\beta }}{A}}{\frac {N_{p}}{C}}\right)\beta =0} 8420:

this analysis, they are discarded from the outset for the sake of clarity. In effect, we assume that the direction of the velocity of the particle due to the simultaneous roll and yaw rates does not change significantly throughout the motion. With this simplifying assumption, the acceleration of the particle becomes:

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A positive sideslip angle generates empennage incidence which can cause positive or negative roll moment depending on its configuration. For any non-zero sideslip angle dihedral wings causes a rolling moment which tends to return the aircraft to the horizontal, as does back swept wings. With highly

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Generally this oscillation is high frequency (hence short period) and is damped over a period of a few seconds. A real-world example would involve a pilot selecting a new climb attitude, for example 5° nose up from the original attitude. A short, sharp pull back on the control column may be used, and

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of the airplane about the z axis. Just as in the case of longitudinal stability it is desirable that the aircraft should tend to return to an equilibrium condition when subjected to some form of yawing disturbance. For this the slope of the yawing moment curve must be positive. An airplane possessing

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Made up of two terms, the force on this particle is first the proportional to rate of v change, the second is due to the change in direction of this component of velocity as the body moves. The latter terms gives rise to cross products of small quantities (pq, pr, qr), which are later discarded. In

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Yaw increases the speed of the outboard wing whilst reducing speed of the inboard one, causing a rolling moment to the inboard side. The contribution of the fin normally supports this inward rolling effect unless offset by anhedral stabilizer above the roll axis (or dihedral below the roll axis).

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Sideslip generates a sideforce from the fin and the fuselage. In addition, if the wing has dihedral, side slip at a positive roll angle increases incidence on the starboard wing and reduces it on the port side, resulting in a net force component directly opposite to the sideslip direction. Sweep

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If the stick is held fixed, the aircraft will not maintain straight and level flight (except in the unlikely case that it happens to be perfectly trimmed for level flight at its current altitude and thrust setting), but will start to dive, level out and climb again. It will repeat this cycle until

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Roll rate causes incidence at the fin, which generates a corresponding side force. Also, positive roll (starboard wing down) increases the lift on the starboard wing and reduces it on the port. If the wing has dihedral, this will result in a side force momentarily opposing the resultant sideslip

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Aircraft lack the symmetry between pitch and yaw, so that directional stability in yaw is derived from a different set of stability derivatives. The yaw plane equivalent to the short period pitch oscillation, which describes yaw plane directional stability is called Dutch roll. Unlike pitch plane

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The body frame is often of interest because the origin and the axes remain fixed relative to the aircraft. This means that the relative orientation of the Earth and body frames describes the aircraft attitude. Also, the direction of the force of thrust is generally fixed in the body frame, though

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The most common aeronautical convention defines roll as acting about the longitudinal axis, positive with the starboard (right) wing down. Yaw is about the vertical body axis, positive with the nose to starboard. Pitch is about an axis perpendicular to the longitudinal plane of symmetry, positive

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Longitudinal stability refers to the stability of an aircraft in pitch. For a stable aircraft, if the aircraft pitches up, the wings and tail create a pitch-down moment which tends to restore the aircraft to its original attitude. For an unstable aircraft, a disturbance in pitch will lead to an

10835: 7077:{\displaystyle {\frac {d^{2}\alpha }{dt^{2}}}-\left({\frac {Z_{\alpha }}{mU}}+{\frac {M_{q}}{B}}+(1+{\frac {Z_{q}}{mU}}){\frac {M_{\dot {\alpha }}}{B}}\right){\frac {d\alpha }{dt}}+\left({\frac {Z_{\alpha }}{mU}}{\frac {M_{q}}{B}}-{\frac {M_{\alpha }}{B}}(1+{\frac {Z_{q}}{mU}})\right)\alpha =0} 4890:

to describe the longitudinal motion, and then factorize it approximately into a high frequency mode and a low frequency mode. The approach adopted here is using qualitative knowledge of aircraft behavior to simplify the equations from the outset, reaching the result by a more accessible route.

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Roll creates counter rotational forces on both starboard and port wings whilst also generating such forces at the empennage. These opposing rolling moment effects have to be overcome by the aileron input in order to sustain the roll rate. If the roll is stopped at a non-zero roll angle the

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Yaw rate input at any roll angle generates rudder, fin and fuselage force vectors which dominate the resultant yawing moment. Yawing also increases the speed of the outboard wing whilst slowing down the inboard wing, with corresponding changes in drag causing a (small) opposing yaw moment.

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It is customary to derive the equations of motion by formal manipulation in what, to the engineer, amounts to a piece of mathematical sleight of hand. The current approach follows the pitch plane analysis in formulating the equations in terms of concepts which are reasonably familiar.

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in yaw is the same as the pitch stability; it resembles the short period pitch oscillation, with yaw plane equivalents to the pitch plane stability derivatives. For this reason, pitch and yaw directional stability are collectively known as the "weathercock" stability of the missile.

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Since the tail is operating in the flowfield of the wing, changes in the wing incidence cause changes in the downwash, but there is a delay for the change in wing flowfield to affect the tail lift, this is represented as a moment proportional to the rate of change of incidence:

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is typically speed stable. If speed increases, drag increases, which will reduce the speed back to equilibrium for its configuration and thrust setting. If speed decreases, drag decreases, and the aircraft will accelerate back to its equilibrium speed where thrust equals drag.

3740:{\displaystyle C_{D}=C_{Df}+C_{Dp}{\begin{cases}C_{Df}={\dfrac {D}{qS}}=-{\dfrac {1}{S}}\int _{\Sigma }C_{f}\mathbf {t} \bullet \mathbf {i_{w}} \,d\sigma \\C_{Dp}={\dfrac {D}{qS}}=-{\dfrac {1}{S}}\int _{\Sigma }(-C_{p})\mathbf {n} \bullet \mathbf {i_{w}} \,d\sigma \end{cases}}} 6746: 2572: 4807:

increasing pitching moment. Longitudinal static stability is the ability of an aircraft to recover from an initial disturbance. Longitudinal dynamic stability refers to the damping of these stabilizing moments, which prevents persistent or increasing oscillations in pitch.

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The Earth frame is a convenient frame to express aircraft translational and rotational kinematics. The Earth frame is also useful in that, under certain assumptions, it can be approximated as inertial. Additionally, one force acting on the aircraft, weight, is fixed in the

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The moment equations are a bit trickier. The trim condition is with the aircraft at an angle of attack with respect to the airflow. The body x-axis does not align with the velocity vector, which is the reference direction for wind axes. In other words, wind axes are not

4043: 596:, which are commonly referred to as Euler angles. This convention is described in detail below for the roll, pitch, and yaw Euler angles that describe the body frame orientation relative to the Earth frame. The other sets of Euler angles are described below by analogy. 8894: 12249: 12089: 9323:

Roll rate generates fin lift causing a yawing moment and also differentially alters the lift on the wings, thus affecting the induced drag contribution of each wing, causing a (small) yawing moment contribution. Positive roll generally causes positive

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has a long time constant, the pilot can intervene to effectively stabilize it, but an aircraft with an unstable Dutch roll would be difficult to fly. It is usual to design the aircraft with a stable Dutch roll mode, but slightly unstable spiral mode.

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of the aircraft, it must be negative for stability. The damping term is reduced by the downwash effect, and it is difficult to design an aircraft with both rapid natural response and heavy damping. Usually, the response is underdamped but stable.

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To keep the analysis (relatively) simple, the control surfaces are assumed fixed throughout the motion, this is stick-fixed stability. Stick-free analysis requires the further complication of taking the motion of the control surfaces into account.

126:, often principally relative to the atmospheric frame in normal flight, but also relative to terrain during takeoff or landing, or when operating at low elevation. The concept of attitude is not specific to fixed-wing aircraft, but also extends to 11238:

etc. In view of the accuracy with which stability derivatives can be calculated, this is an unnecessary pedantry, which serves to obscure the relationship between aircraft geometry and handling, which is the fundamental objective of this article.

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In case the configuration is asymmetrical respect to the XY plane, however, minimum drag differs from the parasitic drag. On these cases, a new approximate parabolic drag equation can be traced leaving the minimum drag value at zero lift value.

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The roll motion is characterized by an absence of natural stability, there are no stability derivatives which generate moments in response to the inertial roll angle. A roll disturbance induces a roll rate which is only canceled by pilot or

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adjust the orientation of a vehicle about its cg. A control system includes control surfaces which, when deflected, generate a moment (or couple from ailerons) about the cg which rotates the aircraft in pitch, roll, and yaw. For example, a

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In studying the trajectory, it is the direction of the velocity vector, rather than that of the body, which is of interest. The direction of the velocity vector when projected on to the horizontal will be called the track, denoted

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near zero, though an aircraft may be deliberately "sideslipped" to increase drag and descent rate during landing, to keep aircraft heading same as runway heading during cross-wind landings and during flight with asymmetric power.

12268: 9096:. However, anhedral may be used with high backsweep angles in high performance aircraft to offset the wing incidence effects of sideslip. Oddly enough this does not reverse the sign of the wing configuration's contribution to 9254:, thus creating a yawing moment counteracted only by the directional stiffness which would tend to point the aircraft's nose back into the wind in horizontal flight conditions. Under sideslip conditions at a given roll angle 10697: 5692: 5782: 5891: 11494: 12887:

Handling the Big Jets: An Explanation of the Significant Differences in Flying Qualities Between Jet Transport Aeroplanes and Piston Engined Transport Aeroplanes, Together with Some Other Aspects of Jet Transport

12581:(both products are positive). This is in direct conflict with the Dutch roll stability requirement, and it is difficult to design an aircraft for which both the Dutch roll and spiral mode are inherently stable. 8517: 4685: 374:

The other two reference frames are body-fixed, with origins moving along with the aircraft, typically at the center of gravity. For an aircraft that is symmetric from right-to-left, the frames can be defined as:

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Simply holding the stick still, when starting with the wings near level, an aircraft will usually have a tendency to gradually veer off to one side of the straight flightpath. This is the (slightly unstable)

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with the roll rate, and hence also with the sideslip. The motion consists of oscillations in roll and yaw, with the roll motion lagging 90 degrees behind the yaw. The wing tips trace out elliptical paths.

7425:. The acceleration along the trajectory is equal to the net x-wise force minus the component of weight. We should not expect significant aerodynamic derivatives to depend on the flight path angle, so only 7211: 4929:

will generally lead to oscillations about the new trim condition. If the oscillations are poorly damped the aircraft will take a long period of time to settle at the new condition, potentially leading to

6629: 2478: 4297:{\displaystyle E={\dfrac {C_{L}}{C_{D0}+kC_{L}^{2}}}\Rightarrow {\begin{cases}E_{max}={\dfrac {1}{2{\sqrt {kC_{D0}}}}}\\(C_{L})_{Emax}={\sqrt {\dfrac {C_{D0}}{k}}}\\(C_{Di})_{Emax}=C_{D0}\end{cases}}} 8354:(the mass is not distributed symmetrically about the yaw and roll axes). Consider the motion of an element of mass in position -z, x in the direction of the y-axis, i.e. into the plane of the paper. 6164: 1609: 156:(AOA). The roll angle is also known as bank angle on a fixed-wing aircraft, which usually "banks" to change the horizontal direction of flight. An aircraft is streamlined from nose to tail to reduce 11520: 8315: 4763:, there are six types of aircraft stability: speed stability, stick free static longitudinal stability, static lateral stability, directional stability, oscillatory stability, and spiral stability. 11208:

The damping term is dominated by the product of the roll damping and the yaw damping derivatives, these are both negative, so their product is positive. The Dutch roll should therefore be damped.

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is made up of the rate of change of sideslip angle and the rate of turn. Taking the latter as zero, assuming no effect on the trajectory, for the limited purpose of studying the Dutch roll:

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In addition to defining the reference frames, the relative orientation of the reference frames can be determined. The relative orientation can be expressed in a variety of forms, including:

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The various Euler angles relating the three reference frames are important to flight dynamics. Many Euler angle conventions exist, but all of the rotation sequences presented below use the

8714: 7414: 4918:) in pitch (generally via the elevator in a standard configuration fixed-wing aircraft) will generally lead to overshoots about the trimmed condition. The transition is characterized by a 5127: 5072: 2383: 1658: 8728: 7343: 3057: 2986: 12105: 11317: 2666: 9713: 6098: 2224: 10640: 11945: 2708: 2624: 7129: 1230: 1203: 6513: 6500: 6461: 5595: 11068: 9495:

swept wings the resultant rolling moment may be excessive for all stability requirements and anhedral could be used to offset the effect of wing sweep induced rolling moment.

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at significant sideslip), the longitudinal equations of motion (involving pitch and lift forces) may be treated independently of the lateral motion (involving roll and yaw).

9024: 8248:, which is the oscillation in roll and yaw, with the roll motion lagging yaw by a quarter cycle, so that the wing tips follow elliptical paths with respect to the aircraft. 3090: 2466: 1255: 9869: 9730: 2872: 2761: 7238: 6373: 6338: 6276: 11236: 11203: 11149: 11122: 9611: 9484: 9279: 9240: 9121: 9094: 9058: 8536: 7685: 10011: 10134: 8410: 3117: 2438: 2411: 1545: 1441: 1416: 1391: 7286: 7149: 6775: 6240: 4982: 4962: 11928: 11178:

is positive. Excessive dihedral can destabilize the Dutch roll, so configurations with highly swept wings require anhedral to offset the wing sweep contribution to

9019: 8340: 2903: 12518: 11711: 11684: 11417: 11348: 11176: 11095: 11025: 10672: 10589: 9644: 9575: 9532: 9450: 9414: 9380: 9349: 9313: 9198: 9155: 7971: 7944: 7917: 7531: 7504: 7477: 7450: 6415: 6311: 5186: 5159: 1068: 11391: 3780: 10985:{\displaystyle {\frac {{\frac {L_{\beta }}{A}}{\frac {N_{p}}{C}}-{\frac {L_{p}}{A}}{\frac {N_{\beta }}{C}}}{{\frac {N_{p}}{C}}{\frac {E}{A}}-{\frac {L_{p}}{A}}}}} 988:

The speed, height and trim angle of attack are different for each flight condition, in addition, the aircraft will be configured differently, e.g. at low speed

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angle between north and the horizontal component of the velocity vector, which describes which direction the aircraft is moving relative to cardinal directions.

1271: 12481:{\displaystyle {\frac {d\phi }{dt}}=mg{\frac {(L_{\beta }N_{r}-N_{\beta }L_{r})}{mU(L_{p}N_{\beta }-N_{p}L_{\beta })-Y_{\beta }(L_{r}N_{p}-L_{p}N_{r})}}\phi } 64: 1999: 1832: 1667: 3752:

There is a second typical decomposition taking into account the definition of the drag coefficient equation. This decomposition separates the effect of the

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opposes the inherent directional stiffness which tends to point the aircraft's nose back into the wind and always matches the sign of the yaw rate input.

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performs its proper function and maintains the angle of attack near its nominal value. The two states which are mainly affected are the flight path angle

1004: 10830:{\displaystyle {\frac {{\frac {L_{p}}{A}}{\frac {N_{r}}{C}}-{\frac {N_{p}}{C}}{\frac {L_{r}}{A}}}{{\frac {N_{p}}{C}}{\frac {E}{A}}-{\frac {L_{p}}{A}}}}} 4940:

pitch oscillation; it arises from the tendency of a stable aircraft to point in the general direction of flight. It is very similar in nature to the

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The motion of a body through a flow is considered, in flight dynamics, as continuum current. In the outer layer of the space that surrounds the body

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is anhedral or fin is below the roll axis. Lateral force components resulting from dihedral or anhedral wing lift differences has little effect on

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If the geometry of the body is fixed and in case of symmetric flight (β=0 and Q=0), pressure and friction coefficients are functions depending on:

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back of the wings has the same effect on incidence, but since the wings are not inclined in the vertical plane, backsweep alone does not affect

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is a handling mode, analogous to the short period pitch oscillation, any effect it might have on the trajectory may be ignored. The body rate

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rolling moment resulting from sideslip induced yaw rate. Longitudinal stability could be ensured or improved by minimizing the latter effect.

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If the configuration of the plane is symmetrical respect to the XY plane, minimum drag coefficient equals to the parasitic drag of the plane.

11428: 906:-plane and the aircraft longitudinal axis and, among other things, is an important variable in determining the magnitude of the force of lift 4933:. If the short period mode is unstable it will generally be impossible for the pilot to safely control the aircraft for any period of time. 8430: 4606: 3339:{\displaystyle {\begin{cases}C_{D}=C_{D}(\alpha ,M,Re)\\C_{L}=C_{L}(\alpha ,M,Re)\\E=E(\alpha ,M,Re)={\dfrac {C_{L}}{C_{D}}}\\\end{cases}}} 5537:{\displaystyle Z_{f}=m{\frac {dw_{f}}{dt}}=m{\frac {dU}{dt}}\sin(\theta -\alpha )+mU{\frac {d(\theta -\alpha )}{dt}}\cos(\theta -\alpha )} 5367:{\displaystyle X_{f}=m{\frac {du_{f}}{dt}}=m{\frac {dU}{dt}}\cos(\theta -\alpha )-mU{\frac {d(\theta -\alpha )}{dt}}\sin(\theta -\alpha )} 3936: 11821: 11728: 8057: 11350:

is negative, so the roll rate will decay with time. The roll rate reduces to zero, but there is no direct control over the roll angle.

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tendency. Anhedral wing and or stabilizer configurations can cause the sign of the side force to invert if the fin effect is swamped.

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When performing the rotations described earlier to obtain the body frame from the Earth frame, there is this analogy between angles:

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Asymmetric aircraft have analogous body-fixed frames, but different conventions must be used to choose the precise directions of the

6741:{\displaystyle {\frac {dq}{dt}}={\frac {M_{q}}{B}}q+{\frac {M_{\alpha }}{B}}\alpha +{\frac {M_{\dot {\alpha }}}{B}}{\dot {\alpha }}} 2567:{\displaystyle \lambda ={\dfrac {\mu }{\rho }}{\sqrt {\dfrac {\pi }{2R\theta }}}={\dfrac {M}{Re}}{\sqrt {\dfrac {k\pi }{2}}}\equiv } 10003:

The inertial moment due to the roll acceleration is considered small compared with the aerodynamic terms, so the equations become:

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When performing the rotations described above to obtain the body frame from the Earth frame, there is this analogy between angles:

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The wind frame is a convenient frame to express the aerodynamic forces and moments acting on an aircraft. In particular, the net

11647:{\displaystyle {\frac {d\mu }{dt}}={\frac {Y_{\beta }}{mU}}\beta +{\frac {Y_{r}}{mU}}r+{\frac {Y_{p}}{mU}}p+{\frac {g}{U}}\phi } 11211:

The motion is accompanied by slight lateral motion of the center of gravity and a more "exact" analysis will introduce terms in

12599: 3145:. Aerodynamic efficiency, defined as the relation between lift and drag coefficients, will depend on those parameters as well. 6108: 1554: 8259: 5797:

frame so we must resolve the fixed axes forces into wind axes. Also, we are only concerned with the force along the z-axis:

7981: 7825: 1460: 812:γ: is the angle between horizontal and the velocity vector, which describes whether the aircraft is climbing or descending. 12804: 9616:

rolling moment induced by the ensuing sideslip should return the aircraft to the horizontal unless exceeded in turn by the

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Lift due to incidence, this is negative because the z-axis is downwards whilst positive incidence causes an upwards force.

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about the new trim. There is very little change in the trajectory over the time it takes for the oscillation to damp out.

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see frequent use in flight dynamics. The first coordinate system has an origin fixed in the reference frame of the Earth:

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intervention. This takes place with insignificant changes in sideslip or yaw rate, so the equation of motion reduces to:

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is the lift increment due to speed increment, it is also negative because lift acts in the opposite sense to the z-axis.

12524: 11027:, the roll damping derivative, which is always negative, so the denominators of these two expressions will be positive. 9281:

will tend to point the nose into the sideslip direction even without rudder input, causing a downward spiraling flight.

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will be negligible. However viscosity effects will have to be considered when analysing the flow in the nearness of the

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increases or decreases the lift generated by the wings when it pitches nose up or down by increasing or decreasing the

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The yaw plane translational equation, as in the pitch plane, equates the centripetal acceleration to the side force.

6170: 12727: 8889:{\displaystyle N=-{\frac {dp}{dt}}\int xzdm+{\frac {dr}{dt}}\int x^{2}+y^{2}dm=-E{\frac {dp}{dt}}+C{\frac {dr}{dt}}} 6313:

Lift due to pitch rate, arises from the increase in tail incidence, hence is also negative, but small compared with

4313: 3367: 12244:{\displaystyle p={\frac {(L_{\beta }N_{r}-L_{r}N_{\beta })}{(L_{p}N_{\beta }-N_{p}L_{\beta })}}{\frac {d\mu }{dt}}} 8666: 7361: 5078: 5023: 2349: 1618: 823:μ: represents a rotation of the lift force around the velocity vector, which may indicate whether the airplane is 12978: 7296: 4734: 2992: 2921: 12896: 12614: 12258:

Substituting for sideslip and roll rate in the force equation results in a first order equation in roll angle:

11263: 2631: 12084:{\displaystyle \beta ={\frac {(L_{r}N_{p}-L_{p}N_{r})}{(L_{p}N_{\beta }-N_{p}L_{\beta })}}{\frac {d\mu }{dt}}} 9672: 6060: 5793:

distribution on the body, and are referred to the velocity vector. But the velocity (wind) axes set is not an

2191: 12660: 7241: 6379: 6100:, the pitch rate. The equations of motion, with all forces and moments referred to wind axes are, therefore: 145:

comes from a force applied at a distance forward or aft of the cg, causing the aircraft to pitch up or down.

10602: 4787:". The aircraft will be speed unstable, because a decrease in speed will cause a further decrease in speed. 6612:{\displaystyle {\frac {d\alpha }{dt}}=\left(1+{\frac {Z_{q}}{mU}}\right)q+{\frac {Z_{\alpha }}{mU}}\alpha } 2673: 2589: 304: 98:

vehicle orientation and control in three dimensions. The three critical flight dynamics parameters are the

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this mode of stability will always point towards the relative wind, hence the name weathercock stability.

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According to λ there are three possible rarefaction grades and their corresponding motions are called:

134:, where the flight dynamics involved in establishing and controlling attitude are entirely different. 11033: 9990:{\displaystyle A{\frac {dp}{dt}}-E{\frac {dr}{dt}}=L_{\beta }\beta -L_{r}{\frac {d\beta }{dt}}+L_{p}p} 9851:{\displaystyle C{\frac {dr}{dt}}-E{\frac {dp}{dt}}=N_{\beta }\beta -N_{r}{\frac {d\beta }{dt}}+N_{p}p} 7973:

is the lift coefficient (assumed constant because the incidence is constant), we have, approximately:

3459:, can be decomposed in two ways. First typical decomposition separates pressure and friction effects: 12799:

Flightwise Volume 2 - Aircraft Stability And Control, Chris Carpenter 1997, Airlife Publishing Ltd.,

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increase the coupling between short period and phugoid modes, so that these will modify the phugoid.

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JSBsim, An open source, platform-independent, flight dynamics & control software library in C++

12721: 8649:{\displaystyle \delta mx{\frac {dv}{dt}}=-{\frac {dp}{dt}}xz\delta m+{\frac {dr}{dt}}x^{2}\delta m} 7805:{\displaystyle {\frac {d^{2}u}{dt^{2}}}-{\frac {X_{u}}{m}}{\frac {du}{dt}}-{\frac {Z_{u}g}{mU}}u=0} 6222:

We are only concerned with perturbations in forces and moments, due to perturbations in the states

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angle between north and the projection of the aircraft longitudinal axis onto the horizontal plane;

10123:{\displaystyle -C{\frac {d^{2}\beta }{dt^{2}}}=N_{\beta }\beta -N_{r}{\frac {d\beta }{dt}}+N_{p}p} 8903:

where N is the yawing moment, E is a product of inertia, and C is the moment of inertia about the

12644: 10243:{\displaystyle E{\frac {d^{2}\beta }{dt^{2}}}=L_{\beta }\beta -L_{r}{\frac {d\beta }{dt}}+L_{p}p} 8351: 7216: 6351: 6316: 6254: 4591: 961:

In analyzing the stability of an aircraft, it is usual to consider perturbations about a nominal

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angle between the velocity vector and the projection of the aircraft longitudinal axis onto the x

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A good attempt for the induced drag coefficient is to assume a parabolic dependency of the lift

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Furthermore, the flight is assumed to take place in still air, and the aircraft is treated as a

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Jerking the stick sideways and returning it to center causes a net change in roll orientation.

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These may be expressed as a second order equation in flight path angle or speed perturbation:

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is known as the parasitic drag coefficient and it is the base drag coefficient at zero lift. C

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where the tangent line from the coordinate origin touches the drag coefficient equation plot.

3099: 2420: 2393: 1527: 1423: 1398: 1373: 1135:{\displaystyle \mathbf {F} _{A}=\int _{\Sigma }(-\Delta p\mathbf {n} +\mathbf {f} )\,d\sigma } 12968: 12680: 8218: 7271: 7134: 6760: 6754: 6243: 6225: 4967: 4947: 4816: 3913:{\displaystyle C_{D}=C_{D0}+C_{Di}{\begin{cases}C_{D0}=(C_{D})_{C_{L}=0}\\C_{Di}\end{cases}}} 2724:

Depending on the compressibility of the flow, different kinds of currents can be considered:

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DTIC ADA124610: Fixed Wing Stability and Control Theory and Flight Test Techniques. Revision

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Since the lift is very much greater than the drag, the phugoid is at best lightly damped. A

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In many flight dynamics applications, the Earth frame is assumed to be inertial with a flat

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axis - positive in the direction of the velocity vector of the aircraft relative to the air

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equilibrium state. The equilibrium roll angle is known as wings level or zero bank angle.

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There is an additional yawing moment due to the offset of the particle in the y direction:

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The delayed downwash effect gives the tail more lift and produces a nose down moment, so

4783:, as speed decreases, drag increases (and vice versa). This is known as the "back of the 2728: 2413: 1353:{\displaystyle \mathbf {F} _{A}=-(\mathbf {i} _{w}D+\mathbf {j} _{w}Q+\mathbf {k} _{w}L)} 179:

Roll, pitch and yaw refer to rotations about the respective axes starting from a defined

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mode of missile or rocket configurations. The motion involves mainly the pitch attitude

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Stability is the ability of the aircraft to counteract disturbances to its flight path.

258: 12917:

Elementary Flight Dynamics with an Introduction to Bifurcation and Continuation Methods

12690: 12496: 8190: 2152:{\displaystyle \equiv C_{L}={\dfrac {L}{qS}}=-{\dfrac {1}{S}}\int _{\Sigma }\,d\sigma } 1985:{\displaystyle \equiv C_{Q}={\dfrac {Q}{qS}}=-{\dfrac {1}{S}}\int _{\Sigma }\,d\sigma } 1820:{\displaystyle \equiv C_{d}={\dfrac {D}{qS}}=-{\dfrac {1}{S}}\int _{\Sigma }\,d\sigma } 861: 275: 99: 1010:

The following considers perturbations about a nominal straight and level flight path.

12973: 12928: 12892: 12864: 12800: 12634: 12585: 7260:

the pilot intervenes. This long period oscillation in speed and height is called the

6054: 4915: 4780: 4746: 2765: 1050: 1038: 878:-plane, which describes whether there is a lateral component to the aircraft velocity 527: 490:-plane, positive determined by the right hand rule (generally, positive to the right) 391:

axis - positive out the nose of the aircraft in the plane of symmetry of the aircraft

107: 30:

This article is about flight dynamics for aircraft. For general flight dynamics, see

6417:

Pitching moment due to pitch rate - the pitch damping term, this is always negative.

12624: 10676: 7422: 5980: 3753: 3356: 3352: 3130: 3126: 1994: 1662: 1454: 794: 572: 531: 520: 294: 12885: 12670: 6375: 4911: 3142: 3134: 3092: 2235: 2177:

In absence of thermal effects, there are three remarkable dimensionless numbers:

989: 883: 567: 431: 300: 264: 215: 153: 142: 127: 31: 12629: 12948: 9208:

Yawing generates side forces due to incidence at the rudder, fin and fuselage.

8343: 8206: 6246:

determined from the flight condition. The possible stability derivatives are:

4711: 2718: 2574: 2295: 783:

rotation around the aircraft longitudinal axis after rotating by yaw and pitch.

742:. The reference frame that results after the three rotations is the body frame. 247: 201: 157: 137: 12521:

is positive or negative. The denominator is usually negative, which requires

9382:

because the wing axis is normally closely aligned with the center of gravity.

7350:

which means the centripetal force is equal to the perturbation in lift force.

5687:{\displaystyle X_{f}=-mU{\frac {d(\theta -\alpha )}{dt}}\sin(\theta -\alpha )} 3772:

is known as the induced drag coefficient and it is produced by the body lift.

12962: 12784: 12685: 12609: 10255:

This becomes a second order equation governing either roll rate or sideslip:

9250:

Sideslip in the absence of rudder input causes incidence on the fuselage and

5777:{\displaystyle Z_{f}=mU{\frac {d(\theta -\alpha )}{dt}}\cos(\theta -\alpha )} 4919: 4822: 962: 775: 180: 8189:

with fixed speed would help. Heavy damping of the pitch rotation or a large

6505:

The equations of motion, with small perturbation forces and moments become:

4984:(alpha). The direction of the velocity vector, relative to inertial axes is 12953: 12675: 11659:

Roll rates and yaw rates are expected to be small, so the contributions of

7265: 5193: 4937: 4895: 2440: 993: 601: 577: 542: 467:

axis, in the plane of symmetry of the aircraft, positive below the aircraft

407:

axis, in the plane of symmetry of the aircraft, positive below the aircraft

555: 12604: 11395: 5973: 4941: 4595: 3138: 2469: 2186: 806: 705:. This results in another intermediate reference frame with axes denoted 5886:{\displaystyle Z=-Z_{f}\cos(\theta -\alpha )+X_{f}\sin(\theta -\alpha )} 11717: 11489:{\displaystyle {\frac {d\mu }{dt}}={\frac {Y}{mU}}+{\frac {g}{U}}\phi } 9655: 8245: 8235: 4784: 2908: 2876: 2839: 1018: 816: 582: 131: 103: 12863:(5. ed., internat. ed.). Boston : McGraw-Hill. pp. 274–275. 8718:

The yawing moment is found by summing over all particles of the body:

4729:

This relation is reasonably accurate for 0.3 < M < 0.7 and when

11420:(psi). The force equation of motion includes a component of weight: 11252: 10684: 9352: 9251: 8512:{\displaystyle {\frac {dv}{dt}}=-{\frac {dp}{dt}}z+{\frac {dr}{dt}}x} 8357: 8201: 8049:

The period of the phugoid, T, is obtained from the coefficient of u:

7506:

is the drag increment with increased speed, it is negative, likewise

4680:{\displaystyle C_{p}={\frac {C_{p0}}{\sqrt {|1-{M_{\infty }}^{2}|}}}} 2802: 2714: 751: 84: 12782: 4027:{\displaystyle C_{Di}=kC_{L}^{2}\Rightarrow C_{D}=C_{D0}+kC_{L}^{2}} 942:

Between the three reference frames there are hence these analogies:

646:. This results in an intermediate reference frame with axes denoted 9722:

The yaw and roll equations, with the stability derivatives become:

8904: 7288:(gamma) and speed. The small perturbation equations of motion are: 5794: 5790: 2172: 1521: 530:

can be divided into components along the wind frame axes, with the

95: 11896:{\displaystyle L_{\beta }\beta +L_{r}{\frac {d\mu }{dt}}+L_{p}p=0} 11803:{\displaystyle N_{\beta }\beta +N_{r}{\frac {d\mu }{dt}}+N_{p}p=0} 8113:{\displaystyle {\frac {2\pi }{T}}={\sqrt {\frac {2g^{2}}{U^{2}}}}} 4733:

it becomes ∞ which is impossible physical situation and is called

27:

Science of air vehicle orientation and control in three dimensions

8996: 7261: 7254: 4899: 980: 8993:

where L is the rolling moment and A the roll moment of inertia.

965:

state. So the analysis would be applied, for example, assuming:

209: 42: 12654: 6242:

and q, and their time derivatives. These are characterized by

5012: 4585: 1034: 1030: 223: 4740: 314:

Origin - arbitrary, fixed relative to the surface of the Earth

195: 10570:

The equation for roll rate is identical. But the roll angle,

8226:

motions, the lateral modes involve both roll and yaw motion.

7206:{\displaystyle M_{\alpha }<{\frac {Z_{\alpha }}{mU}}M_{q}} 5197: 2386: 324: 1178:

Difference between static pressure and free current pressure

771:

angle between the aircraft longitudinal axis and horizontal;

9497: 5554: 5188:

are the inertial axes components of velocity. According to

4821:

Directional or weathercock stability is concerned with the

4290: 3906: 3733: 3359:. This relation is known as the drag coefficient equation: 3332: 1517: 336: 9023: 11716:

The sideslip and roll rate vary gradually, so their time

6753:

These may be manipulated to yield as second order linear

3925: 1232:

tangential stress vector practised by the air on the body

11398:). The body orientation is called the heading, denoted 8364:

If the roll rate is p, the velocity of the particle is:

8244:

Applying an impulse via the rudder pedals should induce

6159:{\displaystyle {\frac {d\alpha }{dt}}=q+{\frac {Z}{mU}}} 1604:{\displaystyle \equiv C_{p}={\dfrac {p-p_{\infty }}{q}}} 367:-plane, though the Earth frame can also be considered a 12828:. Motion Imagery Standards Board (MISB). Archived from 8310:{\displaystyle {\frac {d\beta }{dt}}={\frac {Y}{mU}}-r} 7131:

to be small compared with unity, so the coefficient of

8213:

wings, which make it less stable but more manoeuvrable

8037:{\displaystyle Z_{u}={\frac {2W}{U}}={\frac {2mg}{U}}} 7887:{\displaystyle Z={\frac {1}{2}}\rho U^{2}c_{L}S_{w}=W} 5597:

is negligible over the period of the oscillation, so:

1503:{\displaystyle \equiv q={\tfrac {1}{2}}\,\rho \,V^{2}} 1474: 1060:

The expression to calculate the aerodynamic force is:

1055: 519:

some aircraft can vary this direction, for example by

12527: 12506: 12271: 12108: 11948: 11916: 11824: 11731: 11692: 11665: 11523: 11431: 11405: 11379: 11329: 11266: 11217: 11184: 11157: 11130: 11103: 11076: 11036: 11006: 10855: 10700: 10660: 10605: 10577: 10266: 10137: 10014: 9872: 9733: 9675: 9625: 9592: 9556: 9513: 9465: 9431: 9395: 9361: 9330: 9294: 9260: 9221: 9179: 9136: 9102: 9075: 9039: 9007: 8979:{\displaystyle L=A{\frac {dp}{dt}}-E{\frac {dr}{dt}}} 8920: 8731: 8669: 8539: 8433: 8377: 8328: 8262: 8136: 8060: 7984: 7952: 7925: 7905: 7828: 7688: 7610: 7547: 7512: 7485: 7458: 7431: 7364: 7299: 7274: 7219: 7157: 7137: 7098: 6788: 6763: 6632: 6516: 6474: 6435: 6396: 6354: 6319: 6292: 6257: 6228: 6173: 6111: 6063: 5994: 5960:{\displaystyle Z=-mU{\frac {d(\theta -\alpha )}{dt}}} 5909: 5808: 5701: 5608: 5563: 5381: 5211: 5167: 5140: 5081: 5026: 4990: 4970: 4950: 4609: 4500: 4410: 4316: 4210: 4136: 4054: 4046: 3939: 3783: 3663: 3640: 3561: 3538: 3470: 3370: 3304: 3156: 3102: 3075: 2995: 2924: 2885: 2848: 2811: 2774: 2737: 2678: 2676: 2636: 2634: 2594: 2592: 2544: 2526: 2502: 2489: 2481: 2451: 2423: 2396: 2352: 2317: 2303: 2257: 2243: 2205: 2194: 2043: 2020: 2002: 1876: 1853: 1835: 1711: 1688: 1670: 1639: 1621: 1575: 1557: 1530: 1463: 1426: 1401: 1376: 1274: 1240: 1213: 1186: 1158: 1071: 592:

convention. This convention corresponds to a type of

4894:

The two longitudinal motions (modes) are called the

2909:

Drag coefficient equation and aerodynamic efficiency

12783:Defense Technical Information Center (1981-11-01). 11720:are ignored. The yaw and roll equations reduce to: 5557:. By the nature of the motion, the speed variation 949:

Pitch / Flight path / Attack angle (Y axis, wing)

746:Based on the rotations and axes conventions above: 12574:{\displaystyle L_{\beta }N_{r}>N_{\beta }L_{r}} 12573: 12512: 12499:or decay, depending on whether the coefficient of 12480: 12243: 12083: 11922: 11895: 11802: 11705: 11678: 11646: 11488: 11411: 11385: 11342: 11311: 11230: 11197: 11170: 11143: 11116: 11089: 11062: 11019: 10984: 10829: 10666: 10634: 10583: 10557: 10242: 10122: 9989: 9850: 9707: 9638: 9605: 9569: 9526: 9478: 9444: 9408: 9374: 9343: 9307: 9273: 9234: 9192: 9149: 9115: 9088: 9052: 9013: 8978: 8888: 8708: 8648: 8511: 8404: 8334: 8309: 8172: 8112: 8036: 7965: 7938: 7911: 7886: 7804: 7665:{\displaystyle m{\frac {du}{dt}}=X_{u}u-mg\gamma } 7664: 7595: 7525: 7498: 7471: 7444: 7423:acceleration due to gravity at the Earth's surface 7408: 7337: 7280: 7232: 7205: 7151:(the 'stiffness' term) will be positive, provided 7143: 7123: 7076: 6769: 6740: 6611: 6494: 6455: 6409: 6367: 6332: 6305: 6270: 6234: 6209: 6158: 6092: 6040: 5979:The moment equation is the time derivative of the 5959: 5885: 5776: 5686: 5589: 5536: 5366: 5180: 5153: 5121: 5066: 5002: 4976: 4956: 4679: 4574: 4484:{\displaystyle C_{Dmin}=C_{DM}\neq (C_{D})_{CL=0}} 4483: 4389: 4296: 4026: 3912: 3739: 3430: 3338: 3111: 3084: 3051: 2980: 2897: 2866: 2829: 2792: 2755: 2702: 2660: 2618: 2566: 2460: 2432: 2405: 2377: 2330: 2279:{\displaystyle \equiv Re={\dfrac {\rho Vl}{\mu }}} 2278: 2218: 2151: 1984: 1819: 1652: 1603: 1539: 1502: 1435: 1410: 1385: 1352: 1249: 1224: 1197: 1170: 1134: 10687:" and "damping" terms to be positive. These are: 7089:This represents a damped simple harmonic motion. 6041:{\displaystyle M=B{\frac {d^{2}\theta }{dt^{2}}}} 4905: 3444:The aerodynamic efficiency has a maximum value, E 3351:It is also possible to get the dependency of the 787: 608: 599: 12960: 9060:Side force due to side slip (in absence of yaw). 8907:. A similar reasoning yields the roll equation: 8173:{\displaystyle T={\frac {2\pi U}{{\sqrt {2}}g}}} 7596:{\displaystyle mU{\frac {d\gamma }{dt}}=-Z_{u}u} 4829: 4795: 4575:{\displaystyle C_{D}=C_{DM}+k(C_{L}-C_{LM})^{2}} 2331:{\displaystyle \equiv Kn={\dfrac {\lambda }{l}}} 2173:Dimensionless parameters and aerodynamic regimes 855: 12657:(An open source flight dynamics software model) 7353:For the speed, resolving along the trajectory: 6210:{\displaystyle {\frac {dq}{dt}}={\frac {M}{B}}} 4886:It is common practice to derive a fourth order 4790: 347:axis - positive towards the center of the Earth 8997:Lateral and longitudinal stability derivatives 6378:due to incidence - the static stability term. 5972:In words, the wind axes force is equal to the 4390:{\displaystyle C_{Dmin}=(C_{D})_{CL=0}=C_{D0}} 3431:{\displaystyle C_{D}=C_{D}(C_{L},M,Re)\equiv } 8709:{\displaystyle {\frac {dr}{dt}}y^{2}\delta m} 7409:{\displaystyle m{\frac {du}{dt}}=X-mg\gamma } 6053:where M is the pitching moment, and B is the 4037:Aerodynamic efficiency is now calculated as: 7264:mode. This is analyzed by assuming that the 5122:{\displaystyle w_{f}=U\sin(\theta -\alpha )} 5067:{\displaystyle u_{f}=U\cos(\theta -\alpha )} 4586:Variation of parameters with the Mach number 2586:Continuum current (negligible rarefaction): 2378:{\displaystyle a={\sqrt {kR\theta }}\equiv } 1653:{\displaystyle \equiv C_{f}={\dfrac {f}{q}}} 946:Yaw / Heading / Sideslip (Z axis, vertical) 7338:{\displaystyle mU{\frac {d\gamma }{dt}}=-Z} 4853:It has been suggested that this section be 4741:Aerodynamic force in a specified atmosphere 3133:are functions depending exclusively on the 3052:{\displaystyle C_{f}=C_{f}(\alpha ,M,Re,P)} 2981:{\displaystyle C_{p}=C_{p}(\alpha ,M,Re,P)} 2670:Free molecular current (high rarefaction): 2628:Transition current (moderate rarefaction): 1449: 1029:Three forces act on an aircraft in flight: 10653:is a damped simple harmonic motion, so is 8217:With a symmetrical rocket or missile, the 4936:This damped harmonic motion is called the 2169:in every point on the considered surface. 1205:outer normal vector of the element of area 12757:Learn how and when to remove this message 11366: 11312:{\displaystyle A{\frac {dp}{dt}}=L_{p}p.} 7817:Now lift is very nearly equal to weight: 3723: 3612: 2661:{\displaystyle {\dfrac {M}{Re}}\approx 1} 2142: 1975: 1810: 1489: 1485: 1125: 12949:MIXR - mixed reality simulation platform 12927:(1st ed.). Oxford: Pergamon Press. 12922: 12891:(3rd ed.). Air Registration Board. 12858: 12720:This article includes a list of general 9708:{\displaystyle {\frac {d\beta }{dt}}=-r} 8356: 8200: 6093:{\displaystyle {\frac {d\theta }{dt}}=q} 4810: 2219:{\displaystyle \equiv M={\dfrac {V}{a}}} 554: 434:(generally, positive out the right wing) 371:with origin at the center of the Earth. 67:of all important aspects of the article. 12925:Aircraft dynamic stability and response 11505:is the gravitational acceleration, and 8346:, Y the side force and r the yaw rate. 14: 12961: 12915:NK Sinha and N Ananthkrishnan (2013), 12883: 12817: 12815: 12813: 12600:Acronyms and abbreviations in avionics 10635:{\displaystyle {\frac {d\phi }{dt}}=p} 9649: 5200:, so the forces in inertial axes are: 3926:Parabolic and generic drag coefficient 3756:in the equation, obtaining two terms C 63:Please consider expanding the lead to 12877: 11512:Including the stability derivatives: 9242:Yawing moment due to sideslip forces. 4881: 2703:{\displaystyle {\dfrac {M}{Re}}\gg 1} 2619:{\displaystyle {\dfrac {M}{Re}}\ll 1} 12706: 5789:But the forces are generated by the 4839: 1044: 952:Roll / Bank / nothing (X axis, nose) 335:axis - positive in the direction of 323:axis - positive in the direction of 36: 12810: 7946:is the wing area, W the weight and 7124:{\displaystyle {\frac {Z_{q}}{mU}}} 1225:{\displaystyle \mathbf {f} \equiv } 1198:{\displaystyle \mathbf {n} \equiv } 1056:Components of the aerodynamic force 1024: 614:First, rotate the Earth frame axes 442:Origin - airplane center of gravity 430:-plane, positive determined by the 382:Origin - airplane center of gravity 288: 160:making it advantageous to keep the 24: 12919:, CRC Press, Taylor & Francis. 12726:it lacks sufficient corresponding 11242: 11030:Considering the "stiffness" term: 7213:. This expression is dominated by 6495:{\displaystyle M_{\dot {\alpha }}} 6456:{\displaystyle M_{\dot {\alpha }}} 4898:pitch oscillation (SPPO), and the 4766: 4657: 3679: 3577: 2059: 1892: 1727: 1589: 1263:projected on wind axes we obtain: 1241: 1159: 1103: 1092: 122:. These are collectively known as 83: 25: 12990: 12942: 9123:(compared to the dihedral case). 5590:{\displaystyle m{\frac {dU}{dt}}} 559:Mnemonics to remember angle names 12711: 11063:{\displaystyle -L_{p}N_{\beta }} 11000:The denominator is dominated by 9577:Rolling moment due to roll rate. 9496: 9022: 8196: 7536:The equations of motion become: 5011: 4844: 4779:However, in slow flight, due to 3717: 3713: 3704: 3606: 3602: 3593: 2133: 2129: 2120: 2100: 2096: 2087: 1966: 1962: 1953: 1933: 1929: 1920: 1801: 1797: 1788: 1768: 1764: 1755: 1334: 1316: 1298: 1277: 1215: 1188: 1118: 1110: 1074: 274: 257: 240: 222: 208: 194: 41: 12909: 9534:Rolling moment due to yaw rate. 9486:Rolling moment due to sideslip. 9315:Yawing moment due to roll rate. 8526:The yawing moment is given by: 5003:{\displaystyle \theta -\alpha } 3119:considered point of the surface 2830:{\displaystyle 0.8<M<1.2} 2793:{\displaystyle 0.3<M<0.8} 2729:Incompressible subsonic current 1171:{\displaystyle \Delta p\equiv } 956: 55:may be too short to adequately 12852: 12793: 12776: 12615:Aircraft flight control system 12469: 12423: 12407: 12361: 12350: 12304: 12215: 12169: 12164: 12118: 12055: 12009: 12004: 11958: 11353: 9416:Yawing moment due to yaw rate. 7057: 7026: 6907: 6876: 5943: 5931: 5880: 5868: 5846: 5834: 5771: 5759: 5739: 5727: 5681: 5669: 5649: 5637: 5531: 5519: 5499: 5487: 5469: 5457: 5361: 5349: 5329: 5317: 5299: 5287: 5116: 5104: 5061: 5049: 4906:Short-period pitch oscillation 4725:is the freestream Mach number. 4670: 4640: 4563: 4533: 4463: 4449: 4353: 4339: 4253: 4236: 4187: 4173: 4105: 3974: 3864: 3850: 3700: 3684: 3422: 3394: 3297: 3276: 3260: 3239: 3209: 3188: 3085:{\displaystyle \alpha \equiv } 3046: 3019: 2975: 2948: 2461:{\displaystyle \theta \equiv } 2139: 2083: 2067: 2064: 1972: 1916: 1900: 1897: 1807: 1751: 1735: 1732: 1347: 1293: 1250:{\displaystyle \Sigma \equiv } 1122: 1097: 788:From Earth frame to wind frame 609:From Earth frame to body frame 65:provide an accessible overview 13: 1: 12702: 12661:Longitudinal static stability 8229: 7242:longitudinal static stability 6382:requires this to be negative. 4830:Dynamic stability and control 4796:Longitudinal static stability 4598:by the relation given below: 2867:{\displaystyle 1.2<M<5} 2766:Compressible subsonic current 2756:{\displaystyle 0<M<0.3} 856:From wind frame to body frame 168: 11151:is usually negative, whilst 9157:Side force due to roll rate. 6502:is expected to be negative. 4791:Static stability and control 4751: 2181:Compressibility of the flow: 937: 476:axis - perpendicular to the 460:axis - perpendicular to the 416:axis - perpendicular to the 400:axis - perpendicular to the 305:Cartesian coordinate systems 7: 12847:File:MISB Standard 0601.pdf 12592: 9200:Side force due to yaw rate. 7233:{\displaystyle M_{\alpha }} 6368:{\displaystyle M_{\alpha }} 6333:{\displaystyle Z_{\alpha }} 6271:{\displaystyle Z_{\alpha }} 6057:about the pitch axis. Let: 4735:Prandtl–Glauert singularity 369:spherical coordinate system 250:or heading angle definition 10: 12995: 12859:Anderson, John D. (2005). 11231:{\displaystyle Y_{\beta }} 11198:{\displaystyle L_{\beta }} 11144:{\displaystyle L_{\beta }} 11117:{\displaystyle N_{\beta }} 10675:, but the roll must be in 9606:{\displaystyle L_{\beta }} 9479:{\displaystyle L_{\beta }} 9274:{\displaystyle N_{\beta }} 9235:{\displaystyle N_{\beta }} 9116:{\displaystyle Y_{\beta }} 9089:{\displaystyle Y_{\beta }} 9053:{\displaystyle Y_{\beta }} 8233: 7252: 7248: 5010:. The velocity vector is: 4833: 4814: 4799: 1829:Lateral force coefficient 1257:adequate reference surface 1048: 292: 172: 29: 18:Flight dynamics (aircraft) 12884:Davies, David P. (1971). 12620:Aircraft flight mechanics 11070:will be positive because 4931:Pilot-induced oscillation 3438:drag coefficient equation 2161:It is necessary to know C 971:Straight and level flight 130:such as helicopters, and 12923:Babister, A. W. (1980). 12769: 11124:is positive by design. 10683:Stability requires the " 8405:{\displaystyle v=-pz+xr} 5196:are proportional to the 3125:Under these conditions, 3112:{\displaystyle P\equiv } 2433:{\displaystyle R\equiv } 2406:{\displaystyle k\equiv } 2290:Rarefaction of the flow: 1540:{\displaystyle \equiv S} 1450:Aerodynamic coefficients 1436:{\displaystyle L\equiv } 1411:{\displaystyle Q\equiv } 1386:{\displaystyle D\equiv } 992:may be deployed and the 12741:more precise citations. 12645:Flight control surfaces 11097:is always negative and 7281:{\displaystyle \gamma } 7144:{\displaystyle \alpha } 6770:{\displaystyle \alpha } 6235:{\displaystyle \alpha } 4977:{\displaystyle \alpha } 4957:{\displaystyle \theta } 4888:characteristic equation 4592:Coefficient of pressure 3455:The drag coefficient, C 175:Aircraft principal axes 12979:Flight control systems 12861:Introduction to flight 12575: 12514: 12482: 12245: 12085: 11924: 11923:{\displaystyle \beta } 11897: 11804: 11707: 11680: 11648: 11490: 11413: 11387: 11367:Spiral mode trajectory 11344: 11313: 11232: 11199: 11172: 11145: 11118: 11091: 11064: 11021: 10986: 10831: 10668: 10636: 10585: 10559: 10244: 10124: 9991: 9852: 9709: 9640: 9607: 9571: 9528: 9480: 9446: 9410: 9376: 9345: 9309: 9275: 9236: 9194: 9151: 9117: 9090: 9054: 9015: 9014:{\displaystyle \beta } 8980: 8890: 8710: 8650: 8513: 8406: 8361: 8336: 8335:{\displaystyle \beta } 8311: 8214: 8174: 8114: 8038: 7967: 7940: 7913: 7888: 7806: 7666: 7597: 7527: 7500: 7473: 7446: 7410: 7339: 7282: 7234: 7207: 7145: 7125: 7078: 6771: 6742: 6613: 6496: 6457: 6411: 6369: 6334: 6307: 6272: 6236: 6211: 6160: 6094: 6042: 5961: 5887: 5778: 5688: 5591: 5538: 5368: 5182: 5155: 5123: 5068: 5004: 4978: 4964:(theta) and incidence 4958: 4923:simple harmonic motion 4859:Aircraft dynamic modes 4836:Aircraft dynamic modes 4802:Longitudinal stability 4681: 4576: 4485: 4391: 4298: 4028: 3914: 3741: 3432: 3340: 3113: 3086: 3053: 2982: 2899: 2898:{\displaystyle 5<M} 2868: 2831: 2794: 2757: 2704: 2662: 2620: 2568: 2462: 2434: 2407: 2379: 2332: 2280: 2230:Viscosity of the flow: 2220: 2153: 1986: 1821: 1654: 1605: 1541: 1504: 1437: 1412: 1387: 1354: 1251: 1226: 1199: 1172: 1136: 974:Turn at constant speed 845:(Flight path vs pitch) 560: 88: 12681:Stability derivatives 12576: 12515: 12513:{\displaystyle \phi } 12483: 12246: 12086: 11925: 11898: 11805: 11708: 11706:{\displaystyle Y_{p}} 11681: 11679:{\displaystyle Y_{r}} 11649: 11491: 11414: 11412:{\displaystyle \psi } 11388: 11345: 11343:{\displaystyle L_{p}} 11314: 11233: 11200: 11173: 11171:{\displaystyle N_{p}} 11146: 11119: 11092: 11090:{\displaystyle L_{p}} 11065: 11022: 11020:{\displaystyle L_{p}} 10987: 10832: 10669: 10667:{\displaystyle \phi } 10637: 10586: 10584:{\displaystyle \phi } 10560: 10245: 10125: 9992: 9853: 9710: 9641: 9639:{\displaystyle L_{r}} 9608: 9572: 9570:{\displaystyle L_{p}} 9529: 9527:{\displaystyle L_{r}} 9481: 9447: 9445:{\displaystyle N_{r}} 9411: 9409:{\displaystyle N_{r}} 9377: 9375:{\displaystyle N_{p}} 9346: 9344:{\displaystyle N_{p}} 9310: 9308:{\displaystyle N_{p}} 9276: 9237: 9195: 9193:{\displaystyle Y_{r}} 9152: 9150:{\displaystyle Y_{p}} 9118: 9091: 9055: 9016: 8981: 8891: 8711: 8651: 8514: 8407: 8360: 8337: 8312: 8219:directional stability 8204: 8175: 8115: 8039: 7968: 7966:{\displaystyle c_{L}} 7941: 7939:{\displaystyle S_{w}} 7914: 7912:{\displaystyle \rho } 7889: 7807: 7667: 7598: 7528: 7526:{\displaystyle Z_{u}} 7501: 7499:{\displaystyle X_{u}} 7474: 7472:{\displaystyle Z_{u}} 7447: 7445:{\displaystyle X_{u}} 7411: 7340: 7283: 7235: 7208: 7146: 7126: 7079: 6772: 6755:differential equation 6743: 6614: 6497: 6458: 6412: 6410:{\displaystyle M_{q}} 6370: 6335: 6308: 6306:{\displaystyle Z_{q}} 6273: 6244:stability derivatives 6237: 6212: 6161: 6095: 6043: 5962: 5888: 5779: 5689: 5592: 5539: 5369: 5183: 5181:{\displaystyle w_{f}} 5156: 5154:{\displaystyle u_{f}} 5124: 5069: 5005: 4979: 4959: 4817:Directional stability 4811:Directional stability 4682: 4577: 4486: 4392: 4299: 4029: 3915: 3742: 3433: 3341: 3114: 3087: 3054: 2983: 2900: 2869: 2832: 2795: 2758: 2705: 2663: 2621: 2569: 2463: 2435: 2408: 2380: 2333: 2281: 2221: 2154: 1987: 1822: 1655: 1606: 1542: 1505: 1438: 1413: 1388: 1355: 1252: 1227: 1200: 1173: 1137: 558: 293:Further information: 281:roll angle definition 87: 12823:"MISB Standard 0601" 12525: 12504: 12269: 12106: 11946: 11914: 11822: 11729: 11690: 11663: 11521: 11429: 11403: 11386:{\displaystyle \mu } 11377: 11327: 11264: 11215: 11182: 11155: 11128: 11101: 11074: 11034: 11004: 10853: 10698: 10658: 10603: 10575: 10264: 10135: 10012: 9870: 9731: 9673: 9623: 9590: 9554: 9511: 9463: 9429: 9393: 9359: 9328: 9292: 9258: 9219: 9177: 9134: 9100: 9073: 9037: 9005: 8918: 8729: 8667: 8537: 8431: 8375: 8326: 8260: 8134: 8058: 7982: 7950: 7923: 7919:is the air density, 7903: 7826: 7686: 7608: 7545: 7510: 7483: 7479:need be considered. 7456: 7429: 7362: 7297: 7272: 7240:, which defines the 7217: 7155: 7135: 7096: 6786: 6761: 6630: 6514: 6472: 6433: 6394: 6352: 6317: 6290: 6255: 6226: 6171: 6109: 6061: 5992: 5907: 5806: 5699: 5606: 5561: 5379: 5209: 5165: 5138: 5079: 5024: 4988: 4968: 4948: 4715:pressure coefficient 4702:pressure coefficient 4700:is the compressible 4607: 4498: 4408: 4314: 4044: 3937: 3781: 3468: 3368: 3154: 3100: 3073: 2993: 2922: 2883: 2846: 2809: 2772: 2735: 2674: 2632: 2590: 2479: 2449: 2421: 2394: 2350: 2301: 2241: 2192: 2000: 1833: 1668: 1619: 1614:Friction coefficient 1555: 1550:Pressure coefficient 1528: 1520:surface, in case of 1461: 1457:of the free current 1424: 1399: 1374: 1272: 1238: 1211: 1184: 1156: 1069: 977:Approach and landing 892:: angle between the 719:Finally, rotate the 106:about the vehicle's 12666:Rigid body dynamics 12650:Helicopter dynamics 12640:Dynamic positioning 10592:(phi) is given by: 9650:Equations of motion 5190:Newton's Second Law 4100: 4023: 3973: 2414:specific heat ratio 673:Second, rotate the 150:fixed-wing aircraft 12696:1902 Wright Glider 12691:Weathervane effect 12571: 12510: 12497:exponential growth 12478: 12241: 12081: 11920: 11893: 11800: 11703: 11676: 11644: 11486: 11409: 11383: 11340: 11309: 11228: 11195: 11168: 11141: 11114: 11087: 11060: 11017: 10982: 10827: 10664: 10632: 10581: 10555: 10240: 10120: 9987: 9848: 9705: 9636: 9603: 9567: 9524: 9476: 9442: 9406: 9372: 9351:values unless the 9341: 9305: 9271: 9232: 9190: 9147: 9113: 9086: 9050: 9011: 8976: 8886: 8706: 8646: 8509: 8402: 8362: 8332: 8307: 8215: 8191:rotational inertia 8170: 8110: 8034: 7963: 7936: 7909: 7884: 7802: 7662: 7593: 7523: 7496: 7469: 7442: 7406: 7335: 7278: 7230: 7203: 7141: 7121: 7074: 6767: 6738: 6609: 6492: 6453: 6407: 6365: 6330: 6303: 6268: 6232: 6207: 6156: 6090: 6038: 5957: 5883: 5774: 5684: 5587: 5534: 5364: 5178: 5151: 5119: 5064: 5000: 4974: 4954: 4910:A short input (in 4882:Longitudinal modes 4677: 4572: 4481: 4387: 4294: 4289: 4229: 4167: 4103: 4086: 4024: 4009: 3959: 3910: 3905: 3737: 3732: 3672: 3654: 3570: 3552: 3428: 3336: 3331: 3327: 3109: 3082: 3049: 2978: 2895: 2877:Hypersonic current 2864: 2840:Supersonic current 2827: 2790: 2753: 2700: 2692: 2658: 2650: 2616: 2608: 2564: 2558: 2540: 2519: 2498: 2458: 2430: 2403: 2375: 2328: 2326: 2276: 2274: 2216: 2214: 2149: 2052: 2034: 1982: 1885: 1867: 1817: 1720: 1702: 1650: 1648: 1601: 1599: 1537: 1500: 1483: 1433: 1408: 1383: 1350: 1247: 1222: 1195: 1168: 1132: 1001:asymmetric designs 561: 541:direction and the 100:angles of rotation 94:is the science of 89: 12767: 12766: 12759: 12635:Crosswind landing 12473: 12290: 12239: 12219: 12079: 12059: 11869: 11776: 11713:will be ignored. 11639: 11623: 11595: 11567: 11542: 11481: 11468: 11450: 11288: 10980: 10977: 10957: 10947: 10928: 10911: 10891: 10874: 10825: 10822: 10802: 10792: 10773: 10756: 10736: 10719: 10624: 10539: 10522: 10502: 10485: 10460: 10435: 10418: 10398: 10381: 10356: 10317: 10297: 10287: 10222: 10173: 10102: 10053: 9969: 9920: 9894: 9830: 9781: 9755: 9694: 8974: 8948: 8884: 8858: 8797: 8759: 8688: 8628: 8593: 8567: 8504: 8478: 8452: 8299: 8281: 8168: 8162: 8108: 8107: 8074: 8032: 8011: 7843: 7791: 7761: 7741: 7721: 7632: 7572: 7386: 7324: 7191: 7119: 7092:We should expect 7055: 7024: 7004: 6987: 6957: 6932: 6925: 6905: 6871: 6851: 6821: 6735: 6724: 6717: 6694: 6671: 6651: 6604: 6571: 6535: 6488: 6449: 6205: 6192: 6154: 6130: 6082: 6055:moment of inertia 6036: 5955: 5751: 5661: 5585: 5511: 5449: 5423: 5341: 5279: 5253: 4879: 4878: 4781:lift-induced drag 4747:Aerodynamic force 4675: 4674: 4230: 4228: 4166: 4163: 4102: 3671: 3653: 3569: 3551: 3326: 2803:Transonic current 2691: 2649: 2607: 2559: 2557: 2539: 2520: 2518: 2497: 2370: 2325: 2273: 2213: 2051: 2033: 1884: 1866: 1719: 1701: 1647: 1598: 1512:Proper reference 1482: 1051:Aerodynamic force 1045:Aerodynamic force 1039:aerodynamic force 1005:symmetric designs 933:(nothing vs roll) 927:(attack vs pitch) 917:(sideslip vs yaw) 600:Transformations ( 594:Tait-Bryan angles 573:Direction cosines 568:Rotation matrices 528:aerodynamic force 124:aircraft attitude 108:center of gravity 82: 81: 16:(Redirected from 12986: 12938: 12903: 12902: 12881: 12875: 12874: 12856: 12850: 12844: 12842: 12840: 12835:on 24 March 2017 12834: 12827: 12819: 12808: 12797: 12791: 12790: 12780: 12762: 12755: 12751: 12748: 12742: 12737:this article by 12728:inline citations 12715: 12714: 12707: 12625:Aircraft heading 12580: 12578: 12577: 12572: 12570: 12569: 12560: 12559: 12547: 12546: 12537: 12536: 12519: 12517: 12516: 12511: 12487: 12485: 12484: 12479: 12474: 12472: 12468: 12467: 12458: 12457: 12445: 12444: 12435: 12434: 12422: 12421: 12406: 12405: 12396: 12395: 12383: 12382: 12373: 12372: 12353: 12349: 12348: 12339: 12338: 12326: 12325: 12316: 12315: 12302: 12291: 12289: 12281: 12273: 12250: 12248: 12247: 12242: 12240: 12238: 12230: 12222: 12220: 12218: 12214: 12213: 12204: 12203: 12191: 12190: 12181: 12180: 12167: 12163: 12162: 12153: 12152: 12140: 12139: 12130: 12129: 12116: 12090: 12088: 12087: 12082: 12080: 12078: 12070: 12062: 12060: 12058: 12054: 12053: 12044: 12043: 12031: 12030: 12021: 12020: 12007: 12003: 12002: 11993: 11992: 11980: 11979: 11970: 11969: 11956: 11929: 11927: 11926: 11921: 11902: 11900: 11899: 11894: 11883: 11882: 11870: 11868: 11860: 11852: 11850: 11849: 11834: 11833: 11809: 11807: 11806: 11801: 11790: 11789: 11777: 11775: 11767: 11759: 11757: 11756: 11741: 11740: 11712: 11710: 11709: 11704: 11702: 11701: 11685: 11683: 11682: 11677: 11675: 11674: 11653: 11651: 11650: 11645: 11640: 11632: 11624: 11622: 11614: 11613: 11604: 11596: 11594: 11586: 11585: 11576: 11568: 11566: 11558: 11557: 11548: 11543: 11541: 11533: 11525: 11495: 11493: 11492: 11487: 11482: 11474: 11469: 11467: 11456: 11451: 11449: 11441: 11433: 11418: 11416: 11415: 11410: 11392: 11390: 11389: 11384: 11349: 11347: 11346: 11341: 11339: 11338: 11318: 11316: 11315: 11310: 11302: 11301: 11289: 11287: 11279: 11271: 11237: 11235: 11234: 11229: 11227: 11226: 11204: 11202: 11201: 11196: 11194: 11193: 11177: 11175: 11174: 11169: 11167: 11166: 11150: 11148: 11147: 11142: 11140: 11139: 11123: 11121: 11120: 11115: 11113: 11112: 11096: 11094: 11093: 11088: 11086: 11085: 11069: 11067: 11066: 11061: 11059: 11058: 11049: 11048: 11026: 11024: 11023: 11018: 11016: 11015: 10991: 10989: 10988: 10983: 10981: 10979: 10978: 10973: 10972: 10963: 10958: 10950: 10948: 10943: 10942: 10933: 10930: 10929: 10924: 10923: 10914: 10912: 10907: 10906: 10897: 10892: 10887: 10886: 10877: 10875: 10870: 10869: 10860: 10857: 10836: 10834: 10833: 10828: 10826: 10824: 10823: 10818: 10817: 10808: 10803: 10795: 10793: 10788: 10787: 10778: 10775: 10774: 10769: 10768: 10759: 10757: 10752: 10751: 10742: 10737: 10732: 10731: 10722: 10720: 10715: 10714: 10705: 10702: 10673: 10671: 10670: 10665: 10641: 10639: 10638: 10633: 10625: 10623: 10615: 10607: 10590: 10588: 10587: 10582: 10564: 10562: 10561: 10556: 10545: 10541: 10540: 10535: 10534: 10525: 10523: 10518: 10517: 10508: 10503: 10498: 10497: 10488: 10486: 10481: 10480: 10471: 10461: 10459: 10451: 10443: 10441: 10437: 10436: 10431: 10430: 10421: 10419: 10414: 10413: 10404: 10399: 10394: 10393: 10384: 10382: 10377: 10376: 10367: 10357: 10355: 10354: 10353: 10340: 10336: 10335: 10325: 10323: 10319: 10318: 10313: 10312: 10303: 10298: 10290: 10288: 10283: 10282: 10273: 10249: 10247: 10246: 10241: 10236: 10235: 10223: 10221: 10213: 10205: 10203: 10202: 10187: 10186: 10174: 10172: 10171: 10170: 10157: 10153: 10152: 10142: 10129: 10127: 10126: 10121: 10116: 10115: 10103: 10101: 10093: 10085: 10083: 10082: 10067: 10066: 10054: 10052: 10051: 10050: 10037: 10033: 10032: 10022: 9996: 9994: 9993: 9988: 9983: 9982: 9970: 9968: 9960: 9952: 9950: 9949: 9934: 9933: 9921: 9919: 9911: 9903: 9895: 9893: 9885: 9877: 9857: 9855: 9854: 9849: 9844: 9843: 9831: 9829: 9821: 9813: 9811: 9810: 9795: 9794: 9782: 9780: 9772: 9764: 9756: 9754: 9746: 9738: 9714: 9712: 9711: 9706: 9695: 9693: 9685: 9677: 9645: 9643: 9642: 9637: 9635: 9634: 9612: 9610: 9609: 9604: 9602: 9601: 9576: 9574: 9573: 9568: 9566: 9565: 9533: 9531: 9530: 9525: 9523: 9522: 9500: 9485: 9483: 9482: 9477: 9475: 9474: 9451: 9449: 9448: 9443: 9441: 9440: 9415: 9413: 9412: 9407: 9405: 9404: 9381: 9379: 9378: 9373: 9371: 9370: 9350: 9348: 9347: 9342: 9340: 9339: 9314: 9312: 9311: 9306: 9304: 9303: 9280: 9278: 9277: 9272: 9270: 9269: 9241: 9239: 9238: 9233: 9231: 9230: 9199: 9197: 9196: 9191: 9189: 9188: 9156: 9154: 9153: 9148: 9146: 9145: 9122: 9120: 9119: 9114: 9112: 9111: 9095: 9093: 9092: 9087: 9085: 9084: 9059: 9057: 9056: 9051: 9049: 9048: 9026: 9020: 9018: 9017: 9012: 8985: 8983: 8982: 8977: 8975: 8973: 8965: 8957: 8949: 8947: 8939: 8931: 8895: 8893: 8892: 8887: 8885: 8883: 8875: 8867: 8859: 8857: 8849: 8841: 8824: 8823: 8811: 8810: 8798: 8796: 8788: 8780: 8760: 8758: 8750: 8742: 8715: 8713: 8712: 8707: 8699: 8698: 8689: 8687: 8679: 8671: 8655: 8653: 8652: 8647: 8639: 8638: 8629: 8627: 8619: 8611: 8594: 8592: 8584: 8576: 8568: 8566: 8558: 8550: 8518: 8516: 8515: 8510: 8505: 8503: 8495: 8487: 8479: 8477: 8469: 8461: 8453: 8451: 8443: 8435: 8411: 8409: 8408: 8403: 8341: 8339: 8338: 8333: 8316: 8314: 8313: 8308: 8300: 8298: 8287: 8282: 8280: 8272: 8264: 8179: 8177: 8176: 8171: 8169: 8167: 8163: 8158: 8155: 8144: 8119: 8117: 8116: 8111: 8109: 8106: 8105: 8096: 8095: 8094: 8081: 8080: 8075: 8070: 8062: 8043: 8041: 8040: 8035: 8033: 8028: 8017: 8012: 8007: 7999: 7994: 7993: 7972: 7970: 7969: 7964: 7962: 7961: 7945: 7943: 7942: 7937: 7935: 7934: 7918: 7916: 7915: 7910: 7893: 7891: 7890: 7885: 7877: 7876: 7867: 7866: 7857: 7856: 7844: 7836: 7811: 7809: 7808: 7803: 7792: 7790: 7782: 7778: 7777: 7767: 7762: 7760: 7752: 7744: 7742: 7737: 7736: 7727: 7722: 7720: 7719: 7718: 7705: 7701: 7700: 7690: 7671: 7669: 7668: 7663: 7646: 7645: 7633: 7631: 7623: 7615: 7602: 7600: 7599: 7594: 7589: 7588: 7573: 7571: 7563: 7555: 7532: 7530: 7529: 7524: 7522: 7521: 7505: 7503: 7502: 7497: 7495: 7494: 7478: 7476: 7475: 7470: 7468: 7467: 7451: 7449: 7448: 7443: 7441: 7440: 7415: 7413: 7412: 7407: 7387: 7385: 7377: 7369: 7344: 7342: 7341: 7336: 7325: 7323: 7315: 7307: 7287: 7285: 7284: 7279: 7239: 7237: 7236: 7231: 7229: 7228: 7212: 7210: 7209: 7204: 7202: 7201: 7192: 7190: 7182: 7181: 7172: 7167: 7166: 7150: 7148: 7147: 7142: 7130: 7128: 7127: 7122: 7120: 7118: 7110: 7109: 7100: 7083: 7081: 7080: 7075: 7064: 7060: 7056: 7054: 7046: 7045: 7036: 7025: 7020: 7019: 7010: 7005: 7000: 6999: 6990: 6988: 6986: 6978: 6977: 6968: 6958: 6956: 6948: 6940: 6938: 6934: 6933: 6928: 6927: 6926: 6918: 6911: 6906: 6904: 6896: 6895: 6886: 6872: 6867: 6866: 6857: 6852: 6850: 6842: 6841: 6832: 6822: 6820: 6819: 6818: 6805: 6801: 6800: 6790: 6776: 6774: 6773: 6768: 6747: 6745: 6744: 6739: 6737: 6736: 6728: 6725: 6720: 6719: 6718: 6710: 6703: 6695: 6690: 6689: 6680: 6672: 6667: 6666: 6657: 6652: 6650: 6642: 6634: 6618: 6616: 6615: 6610: 6605: 6603: 6595: 6594: 6585: 6577: 6573: 6572: 6570: 6562: 6561: 6552: 6536: 6534: 6526: 6518: 6501: 6499: 6498: 6493: 6491: 6490: 6489: 6481: 6462: 6460: 6459: 6454: 6452: 6451: 6450: 6442: 6416: 6414: 6413: 6408: 6406: 6405: 6380:Static stability 6374: 6372: 6371: 6366: 6364: 6363: 6339: 6337: 6336: 6331: 6329: 6328: 6312: 6310: 6309: 6304: 6302: 6301: 6277: 6275: 6274: 6269: 6267: 6266: 6241: 6239: 6238: 6233: 6216: 6214: 6213: 6208: 6206: 6198: 6193: 6191: 6183: 6175: 6165: 6163: 6162: 6157: 6155: 6153: 6142: 6131: 6129: 6121: 6113: 6099: 6097: 6096: 6091: 6083: 6081: 6073: 6065: 6047: 6045: 6044: 6039: 6037: 6035: 6034: 6033: 6020: 6016: 6015: 6005: 5981:angular momentum 5966: 5964: 5963: 5958: 5956: 5954: 5946: 5926: 5892: 5890: 5889: 5884: 5861: 5860: 5827: 5826: 5783: 5781: 5780: 5775: 5752: 5750: 5742: 5722: 5711: 5710: 5693: 5691: 5690: 5685: 5662: 5660: 5652: 5632: 5618: 5617: 5596: 5594: 5593: 5588: 5586: 5584: 5576: 5568: 5543: 5541: 5540: 5535: 5512: 5510: 5502: 5482: 5450: 5448: 5440: 5432: 5424: 5422: 5414: 5413: 5412: 5399: 5391: 5390: 5373: 5371: 5370: 5365: 5342: 5340: 5332: 5312: 5280: 5278: 5270: 5262: 5254: 5252: 5244: 5243: 5242: 5229: 5221: 5220: 5187: 5185: 5184: 5179: 5177: 5176: 5160: 5158: 5157: 5152: 5150: 5149: 5128: 5126: 5125: 5120: 5091: 5090: 5073: 5071: 5070: 5065: 5036: 5035: 5015: 5009: 5007: 5006: 5001: 4983: 4981: 4980: 4975: 4963: 4961: 4960: 4955: 4874: 4871: 4848: 4847: 4840: 4823:static stability 4686: 4684: 4683: 4678: 4676: 4673: 4668: 4667: 4662: 4661: 4660: 4643: 4638: 4637: 4636: 4624: 4619: 4618: 4581: 4579: 4578: 4573: 4571: 4570: 4561: 4560: 4545: 4544: 4526: 4525: 4510: 4509: 4490: 4488: 4487: 4482: 4480: 4479: 4461: 4460: 4445: 4444: 4429: 4428: 4396: 4394: 4393: 4388: 4386: 4385: 4370: 4369: 4351: 4350: 4335: 4334: 4303: 4301: 4300: 4295: 4293: 4292: 4286: 4285: 4270: 4269: 4251: 4250: 4231: 4224: 4223: 4211: 4209: 4204: 4203: 4185: 4184: 4168: 4165: 4164: 4162: 4161: 4146: 4137: 4131: 4130: 4104: 4101: 4099: 4094: 4079: 4078: 4065: 4064: 4055: 4033: 4031: 4030: 4025: 4022: 4017: 4002: 4001: 3986: 3985: 3972: 3967: 3952: 3951: 3919: 3917: 3916: 3911: 3909: 3908: 3902: 3901: 3885: 3884: 3877: 3876: 3862: 3861: 3846: 3845: 3825: 3824: 3809: 3808: 3793: 3792: 3754:lift coefficient 3746: 3744: 3743: 3738: 3736: 3735: 3722: 3721: 3720: 3707: 3699: 3698: 3683: 3682: 3673: 3664: 3655: 3652: 3641: 3635: 3634: 3611: 3610: 3609: 3596: 3591: 3590: 3581: 3580: 3571: 3562: 3553: 3550: 3539: 3533: 3532: 3512: 3511: 3496: 3495: 3480: 3479: 3437: 3435: 3434: 3429: 3406: 3405: 3393: 3392: 3380: 3379: 3357:lift coefficient 3353:drag coefficient 3345: 3343: 3342: 3337: 3335: 3334: 3328: 3325: 3324: 3315: 3314: 3305: 3238: 3237: 3225: 3224: 3187: 3186: 3174: 3173: 3143:Reynolds numbers 3137:of the body and 3131:lift coefficient 3118: 3116: 3115: 3110: 3091: 3089: 3088: 3083: 3058: 3056: 3055: 3050: 3018: 3017: 3005: 3004: 2987: 2985: 2984: 2979: 2947: 2946: 2934: 2933: 2904: 2902: 2901: 2896: 2873: 2871: 2870: 2865: 2836: 2834: 2833: 2828: 2799: 2797: 2796: 2791: 2762: 2760: 2759: 2754: 2709: 2707: 2706: 2701: 2693: 2690: 2679: 2667: 2665: 2664: 2659: 2651: 2648: 2637: 2625: 2623: 2622: 2617: 2609: 2606: 2595: 2573: 2571: 2570: 2565: 2560: 2553: 2545: 2543: 2541: 2538: 2527: 2521: 2517: 2503: 2501: 2499: 2490: 2467: 2465: 2464: 2459: 2439: 2437: 2436: 2431: 2412: 2410: 2409: 2404: 2384: 2382: 2381: 2376: 2371: 2360: 2337: 2335: 2334: 2329: 2327: 2318: 2285: 2283: 2282: 2277: 2275: 2269: 2258: 2225: 2223: 2222: 2217: 2215: 2206: 2158: 2156: 2155: 2150: 2138: 2137: 2136: 2123: 2118: 2117: 2105: 2104: 2103: 2090: 2082: 2081: 2063: 2062: 2053: 2044: 2035: 2032: 2021: 2015: 2014: 1995:Lift coefficient 1991: 1989: 1988: 1983: 1971: 1970: 1969: 1956: 1951: 1950: 1938: 1937: 1936: 1923: 1915: 1914: 1896: 1895: 1886: 1877: 1868: 1865: 1854: 1848: 1847: 1826: 1824: 1823: 1818: 1806: 1805: 1804: 1791: 1786: 1785: 1773: 1772: 1771: 1758: 1750: 1749: 1731: 1730: 1721: 1712: 1703: 1700: 1689: 1683: 1682: 1663:Drag coefficient 1659: 1657: 1656: 1651: 1649: 1640: 1634: 1633: 1610: 1608: 1607: 1602: 1600: 1594: 1593: 1592: 1576: 1570: 1569: 1546: 1544: 1543: 1538: 1509: 1507: 1506: 1501: 1499: 1498: 1484: 1475: 1455:Dynamic pressure 1442: 1440: 1439: 1434: 1417: 1415: 1414: 1409: 1392: 1390: 1389: 1384: 1359: 1357: 1356: 1351: 1343: 1342: 1337: 1325: 1324: 1319: 1307: 1306: 1301: 1286: 1285: 1280: 1256: 1254: 1253: 1248: 1231: 1229: 1228: 1223: 1218: 1204: 1202: 1201: 1196: 1191: 1177: 1175: 1174: 1169: 1141: 1139: 1138: 1133: 1121: 1113: 1096: 1095: 1083: 1082: 1077: 1025:Forces of flight 839:(heading vs yaw) 727:axes around the 714: 693: 687:axes around the 685: 678: 661: 656: 651: 521:thrust vectoring 295:axes conventions 289:Reference frames 278: 267:angle definition 261: 244: 226: 212: 198: 77: 74: 68: 45: 37: 21: 12994: 12993: 12989: 12988: 12987: 12985: 12984: 12983: 12959: 12958: 12945: 12935: 12912: 12907: 12906: 12899: 12882: 12878: 12871: 12857: 12853: 12838: 12836: 12832: 12825: 12821: 12820: 12811: 12798: 12794: 12789:. pp. V-5. 12781: 12777: 12772: 12763: 12752: 12746: 12743: 12733:Please help to 12732: 12716: 12712: 12705: 12700: 12671:Rotation matrix 12595: 12565: 12561: 12555: 12551: 12542: 12538: 12532: 12528: 12526: 12523: 12522: 12505: 12502: 12501: 12463: 12459: 12453: 12449: 12440: 12436: 12430: 12426: 12417: 12413: 12401: 12397: 12391: 12387: 12378: 12374: 12368: 12364: 12354: 12344: 12340: 12334: 12330: 12321: 12317: 12311: 12307: 12303: 12301: 12282: 12274: 12272: 12270: 12267: 12266: 12231: 12223: 12221: 12209: 12205: 12199: 12195: 12186: 12182: 12176: 12172: 12168: 12158: 12154: 12148: 12144: 12135: 12131: 12125: 12121: 12117: 12115: 12107: 12104: 12103: 12071: 12063: 12061: 12049: 12045: 12039: 12035: 12026: 12022: 12016: 12012: 12008: 11998: 11994: 11988: 11984: 11975: 11971: 11965: 11961: 11957: 11955: 11947: 11944: 11943: 11915: 11912: 11911: 11878: 11874: 11861: 11853: 11851: 11845: 11841: 11829: 11825: 11823: 11820: 11819: 11785: 11781: 11768: 11760: 11758: 11752: 11748: 11736: 11732: 11730: 11727: 11726: 11697: 11693: 11691: 11688: 11687: 11670: 11666: 11664: 11661: 11660: 11631: 11615: 11609: 11605: 11603: 11587: 11581: 11577: 11575: 11559: 11553: 11549: 11547: 11534: 11526: 11524: 11522: 11519: 11518: 11473: 11460: 11455: 11442: 11434: 11432: 11430: 11427: 11426: 11404: 11401: 11400: 11378: 11375: 11374: 11369: 11356: 11334: 11330: 11328: 11325: 11324: 11297: 11293: 11280: 11272: 11270: 11265: 11262: 11261: 11245: 11243:Roll subsidence 11222: 11218: 11216: 11213: 11212: 11189: 11185: 11183: 11180: 11179: 11162: 11158: 11156: 11153: 11152: 11135: 11131: 11129: 11126: 11125: 11108: 11104: 11102: 11099: 11098: 11081: 11077: 11075: 11072: 11071: 11054: 11050: 11044: 11040: 11035: 11032: 11031: 11011: 11007: 11005: 11002: 11001: 10968: 10964: 10962: 10949: 10938: 10934: 10932: 10931: 10919: 10915: 10913: 10902: 10898: 10896: 10882: 10878: 10876: 10865: 10861: 10859: 10858: 10856: 10854: 10851: 10850: 10813: 10809: 10807: 10794: 10783: 10779: 10777: 10776: 10764: 10760: 10758: 10747: 10743: 10741: 10727: 10723: 10721: 10710: 10706: 10704: 10703: 10701: 10699: 10696: 10695: 10659: 10656: 10655: 10616: 10608: 10606: 10604: 10601: 10600: 10576: 10573: 10572: 10530: 10526: 10524: 10513: 10509: 10507: 10493: 10489: 10487: 10476: 10472: 10470: 10469: 10465: 10452: 10444: 10442: 10426: 10422: 10420: 10409: 10405: 10403: 10389: 10385: 10383: 10372: 10368: 10366: 10365: 10361: 10349: 10345: 10341: 10331: 10327: 10326: 10324: 10308: 10304: 10302: 10289: 10278: 10274: 10272: 10271: 10267: 10265: 10262: 10261: 10231: 10227: 10214: 10206: 10204: 10198: 10194: 10182: 10178: 10166: 10162: 10158: 10148: 10144: 10143: 10141: 10136: 10133: 10132: 10111: 10107: 10094: 10086: 10084: 10078: 10074: 10062: 10058: 10046: 10042: 10038: 10028: 10024: 10023: 10021: 10013: 10010: 10009: 9978: 9974: 9961: 9953: 9951: 9945: 9941: 9929: 9925: 9912: 9904: 9902: 9886: 9878: 9876: 9871: 9868: 9867: 9839: 9835: 9822: 9814: 9812: 9806: 9802: 9790: 9786: 9773: 9765: 9763: 9747: 9739: 9737: 9732: 9729: 9728: 9686: 9678: 9676: 9674: 9671: 9670: 9652: 9630: 9626: 9624: 9621: 9620: 9597: 9593: 9591: 9588: 9587: 9561: 9557: 9555: 9552: 9551: 9518: 9514: 9512: 9509: 9508: 9470: 9466: 9464: 9461: 9460: 9436: 9432: 9430: 9427: 9426: 9400: 9396: 9394: 9391: 9390: 9366: 9362: 9360: 9357: 9356: 9335: 9331: 9329: 9326: 9325: 9299: 9295: 9293: 9290: 9289: 9265: 9261: 9259: 9256: 9255: 9226: 9222: 9220: 9217: 9216: 9184: 9180: 9178: 9175: 9174: 9141: 9137: 9135: 9132: 9131: 9107: 9103: 9101: 9098: 9097: 9080: 9076: 9074: 9071: 9070: 9044: 9040: 9038: 9035: 9034: 9006: 9003: 9002: 9001:The states are 8999: 8966: 8958: 8956: 8940: 8932: 8930: 8919: 8916: 8915: 8876: 8868: 8866: 8850: 8842: 8840: 8819: 8815: 8806: 8802: 8789: 8781: 8779: 8751: 8743: 8741: 8730: 8727: 8726: 8694: 8690: 8680: 8672: 8670: 8668: 8665: 8664: 8634: 8630: 8620: 8612: 8610: 8585: 8577: 8575: 8559: 8551: 8549: 8538: 8535: 8534: 8496: 8488: 8486: 8470: 8462: 8460: 8444: 8436: 8434: 8432: 8429: 8428: 8376: 8373: 8372: 8327: 8324: 8323: 8291: 8286: 8273: 8265: 8263: 8261: 8258: 8257: 8238: 8232: 8199: 8157: 8156: 8145: 8143: 8135: 8132: 8131: 8101: 8097: 8090: 8086: 8082: 8079: 8063: 8061: 8059: 8056: 8055: 8018: 8016: 8000: 7998: 7989: 7985: 7983: 7980: 7979: 7957: 7953: 7951: 7948: 7947: 7930: 7926: 7924: 7921: 7920: 7904: 7901: 7900: 7872: 7868: 7862: 7858: 7852: 7848: 7835: 7827: 7824: 7823: 7783: 7773: 7769: 7768: 7766: 7753: 7745: 7743: 7732: 7728: 7726: 7714: 7710: 7706: 7696: 7692: 7691: 7689: 7687: 7684: 7683: 7641: 7637: 7624: 7616: 7614: 7609: 7606: 7605: 7584: 7580: 7564: 7556: 7554: 7546: 7543: 7542: 7517: 7513: 7511: 7508: 7507: 7490: 7486: 7484: 7481: 7480: 7463: 7459: 7457: 7454: 7453: 7436: 7432: 7430: 7427: 7426: 7421:where g is the 7378: 7370: 7368: 7363: 7360: 7359: 7316: 7308: 7306: 7298: 7295: 7294: 7273: 7270: 7269: 7257: 7251: 7224: 7220: 7218: 7215: 7214: 7197: 7193: 7183: 7177: 7173: 7171: 7162: 7158: 7156: 7153: 7152: 7136: 7133: 7132: 7111: 7105: 7101: 7099: 7097: 7094: 7093: 7047: 7041: 7037: 7035: 7015: 7011: 7009: 6995: 6991: 6989: 6979: 6973: 6969: 6967: 6966: 6962: 6949: 6941: 6939: 6917: 6916: 6912: 6910: 6897: 6891: 6887: 6885: 6862: 6858: 6856: 6843: 6837: 6833: 6831: 6830: 6826: 6814: 6810: 6806: 6796: 6792: 6791: 6789: 6787: 6784: 6783: 6762: 6759: 6758: 6727: 6726: 6709: 6708: 6704: 6702: 6685: 6681: 6679: 6662: 6658: 6656: 6643: 6635: 6633: 6631: 6628: 6627: 6596: 6590: 6586: 6584: 6563: 6557: 6553: 6551: 6544: 6540: 6527: 6519: 6517: 6515: 6512: 6511: 6480: 6479: 6475: 6473: 6470: 6469: 6441: 6440: 6436: 6434: 6431: 6430: 6401: 6397: 6395: 6392: 6391: 6376:Pitching moment 6359: 6355: 6353: 6350: 6349: 6324: 6320: 6318: 6315: 6314: 6297: 6293: 6291: 6288: 6287: 6262: 6258: 6256: 6253: 6252: 6227: 6224: 6223: 6197: 6184: 6176: 6174: 6172: 6169: 6168: 6146: 6141: 6122: 6114: 6112: 6110: 6107: 6106: 6074: 6066: 6064: 6062: 6059: 6058: 6029: 6025: 6021: 6011: 6007: 6006: 6004: 5993: 5990: 5989: 5947: 5927: 5925: 5908: 5905: 5904: 5856: 5852: 5822: 5818: 5807: 5804: 5803: 5743: 5723: 5721: 5706: 5702: 5700: 5697: 5696: 5653: 5633: 5631: 5613: 5609: 5607: 5604: 5603: 5577: 5569: 5567: 5562: 5559: 5558: 5503: 5483: 5481: 5441: 5433: 5431: 5415: 5408: 5404: 5400: 5398: 5386: 5382: 5380: 5377: 5376: 5333: 5313: 5311: 5271: 5263: 5261: 5245: 5238: 5234: 5230: 5228: 5216: 5212: 5210: 5207: 5206: 5172: 5168: 5166: 5163: 5162: 5145: 5141: 5139: 5136: 5135: 5086: 5082: 5080: 5077: 5076: 5031: 5027: 5025: 5022: 5021: 4989: 4986: 4985: 4969: 4966: 4965: 4949: 4946: 4945: 4914:terminology an 4912:control systems 4908: 4884: 4875: 4869: 4866: 4849: 4845: 4838: 4832: 4819: 4813: 4804: 4798: 4793: 4771:An aircraft in 4769: 4767:Speed stability 4761:David P. Davies 4754: 4743: 4723: 4709: 4699: 4669: 4663: 4656: 4652: 4651: 4650: 4639: 4629: 4625: 4623: 4614: 4610: 4608: 4605: 4604: 4588: 4566: 4562: 4553: 4549: 4540: 4536: 4518: 4514: 4505: 4501: 4499: 4496: 4495: 4466: 4462: 4456: 4452: 4437: 4433: 4415: 4411: 4409: 4406: 4405: 4378: 4374: 4356: 4352: 4346: 4342: 4321: 4317: 4315: 4312: 4311: 4288: 4287: 4278: 4274: 4256: 4252: 4243: 4239: 4233: 4232: 4216: 4212: 4208: 4190: 4186: 4180: 4176: 4170: 4169: 4154: 4150: 4145: 4141: 4135: 4120: 4116: 4109: 4108: 4095: 4090: 4071: 4067: 4066: 4060: 4056: 4053: 4045: 4042: 4041: 4018: 4013: 3994: 3990: 3981: 3977: 3968: 3963: 3944: 3940: 3938: 3935: 3934: 3928: 3904: 3903: 3894: 3890: 3887: 3886: 3872: 3868: 3867: 3863: 3857: 3853: 3838: 3834: 3827: 3826: 3817: 3813: 3801: 3797: 3788: 3784: 3782: 3779: 3778: 3771: 3767: 3763: 3759: 3731: 3730: 3716: 3712: 3711: 3703: 3694: 3690: 3678: 3674: 3662: 3645: 3639: 3627: 3623: 3620: 3619: 3605: 3601: 3600: 3592: 3586: 3582: 3576: 3572: 3560: 3543: 3537: 3525: 3521: 3514: 3513: 3504: 3500: 3488: 3484: 3475: 3471: 3469: 3466: 3465: 3458: 3451: 3447: 3401: 3397: 3388: 3384: 3375: 3371: 3369: 3366: 3365: 3355:respect to the 3330: 3329: 3320: 3316: 3310: 3306: 3303: 3264: 3263: 3233: 3229: 3220: 3216: 3213: 3212: 3182: 3178: 3169: 3165: 3158: 3157: 3155: 3152: 3151: 3135:angle of attack 3101: 3098: 3097: 3093:angle of attack 3074: 3071: 3070: 3013: 3009: 3000: 2996: 2994: 2991: 2990: 2942: 2938: 2929: 2925: 2923: 2920: 2919: 2911: 2884: 2881: 2880: 2847: 2844: 2843: 2810: 2807: 2806: 2773: 2770: 2769: 2736: 2733: 2732: 2683: 2677: 2675: 2672: 2671: 2641: 2635: 2633: 2630: 2629: 2599: 2593: 2591: 2588: 2587: 2546: 2542: 2531: 2525: 2507: 2500: 2488: 2480: 2477: 2476: 2450: 2447: 2446: 2422: 2419: 2418: 2395: 2392: 2391: 2359: 2351: 2348: 2347: 2316: 2302: 2299: 2298: 2259: 2256: 2242: 2239: 2238: 2236:Reynolds number 2204: 2193: 2190: 2189: 2175: 2168: 2164: 2132: 2128: 2127: 2119: 2113: 2109: 2099: 2095: 2094: 2086: 2077: 2073: 2058: 2054: 2042: 2025: 2019: 2010: 2006: 2001: 1998: 1997: 1965: 1961: 1960: 1952: 1946: 1942: 1932: 1928: 1927: 1919: 1910: 1906: 1891: 1887: 1875: 1858: 1852: 1843: 1839: 1834: 1831: 1830: 1800: 1796: 1795: 1787: 1781: 1777: 1767: 1763: 1762: 1754: 1745: 1741: 1726: 1722: 1710: 1693: 1687: 1678: 1674: 1669: 1666: 1665: 1638: 1629: 1625: 1620: 1617: 1616: 1588: 1584: 1577: 1574: 1565: 1561: 1556: 1553: 1552: 1529: 1526: 1525: 1494: 1490: 1473: 1462: 1459: 1458: 1452: 1425: 1422: 1421: 1400: 1397: 1396: 1375: 1372: 1371: 1338: 1333: 1332: 1320: 1315: 1314: 1302: 1297: 1296: 1281: 1276: 1275: 1273: 1270: 1269: 1239: 1236: 1235: 1214: 1212: 1209: 1208: 1187: 1185: 1182: 1181: 1157: 1154: 1153: 1117: 1109: 1091: 1087: 1078: 1073: 1072: 1070: 1067: 1066: 1058: 1053: 1047: 1027: 959: 940: 904: 897: 877: 873: 858: 790: 712: 691: 683: 676: 668: 659: 654: 649: 634: 626: 619: 611: 606: 550: 539: 513: 488: 481: 474: 465: 458: 449: 432:right-hand rule 428: 421: 414: 405: 398: 389: 365: 358: 345: 333: 321: 297: 291: 286: 285: 284: 283: 282: 279: 270: 269: 268: 262: 253: 252: 251: 245: 234: 233: 232: 231: 230: 227: 219: 218: 213: 205: 204: 199: 177: 171: 154:angle of attack 143:pitching moment 138:Control systems 128:rotary aircraft 110:(cg), known as 92:Flight dynamics 78: 72: 69: 62: 50:This article's 46: 35: 32:Flight dynamics 28: 23: 22: 15: 12: 11: 5: 12992: 12982: 12981: 12976: 12971: 12957: 12956: 12951: 12944: 12943:External links 12941: 12940: 12939: 12934:978-0080247687 12933: 12920: 12911: 12908: 12905: 12904: 12897: 12876: 12869: 12851: 12809: 12792: 12774: 12773: 12771: 12768: 12765: 12764: 12719: 12717: 12710: 12704: 12701: 12699: 12698: 12693: 12688: 12683: 12678: 12673: 12668: 12663: 12658: 12652: 12647: 12642: 12637: 12632: 12627: 12622: 12617: 12612: 12607: 12602: 12596: 12594: 12591: 12568: 12564: 12558: 12554: 12550: 12545: 12541: 12535: 12531: 12509: 12493: 12492: 12491: 12490: 12489: 12488: 12477: 12471: 12466: 12462: 12456: 12452: 12448: 12443: 12439: 12433: 12429: 12425: 12420: 12416: 12412: 12409: 12404: 12400: 12394: 12390: 12386: 12381: 12377: 12371: 12367: 12363: 12360: 12357: 12352: 12347: 12343: 12337: 12333: 12329: 12324: 12320: 12314: 12310: 12306: 12300: 12297: 12294: 12288: 12285: 12280: 12277: 12256: 12255: 12254: 12253: 12252: 12251: 12237: 12234: 12229: 12226: 12217: 12212: 12208: 12202: 12198: 12194: 12189: 12185: 12179: 12175: 12171: 12166: 12161: 12157: 12151: 12147: 12143: 12138: 12134: 12128: 12124: 12120: 12114: 12111: 12096: 12095: 12094: 12093: 12092: 12091: 12077: 12074: 12069: 12066: 12057: 12052: 12048: 12042: 12038: 12034: 12029: 12025: 12019: 12015: 12011: 12006: 12001: 11997: 11991: 11987: 11983: 11978: 11974: 11968: 11964: 11960: 11954: 11951: 11919: 11907: 11906: 11905: 11904: 11892: 11889: 11886: 11881: 11877: 11873: 11867: 11864: 11859: 11856: 11848: 11844: 11840: 11837: 11832: 11828: 11814: 11813: 11812: 11811: 11799: 11796: 11793: 11788: 11784: 11780: 11774: 11771: 11766: 11763: 11755: 11751: 11747: 11744: 11739: 11735: 11700: 11696: 11673: 11669: 11657: 11656: 11655: 11654: 11643: 11638: 11635: 11630: 11627: 11621: 11618: 11612: 11608: 11602: 11599: 11593: 11590: 11584: 11580: 11574: 11571: 11565: 11562: 11556: 11552: 11546: 11540: 11537: 11532: 11529: 11509:is the speed. 11499: 11498: 11497: 11496: 11485: 11480: 11477: 11472: 11466: 11463: 11459: 11454: 11448: 11445: 11440: 11437: 11408: 11382: 11368: 11365: 11355: 11352: 11337: 11333: 11322: 11321: 11320: 11319: 11308: 11305: 11300: 11296: 11292: 11286: 11283: 11278: 11275: 11269: 11244: 11241: 11225: 11221: 11192: 11188: 11165: 11161: 11138: 11134: 11111: 11107: 11084: 11080: 11057: 11053: 11047: 11043: 11039: 11014: 11010: 10998: 10997: 10996: 10995: 10994: 10993: 10976: 10971: 10967: 10961: 10956: 10953: 10946: 10941: 10937: 10927: 10922: 10918: 10910: 10905: 10901: 10895: 10890: 10885: 10881: 10873: 10868: 10864: 10843: 10842: 10841: 10840: 10839: 10838: 10821: 10816: 10812: 10806: 10801: 10798: 10791: 10786: 10782: 10772: 10767: 10763: 10755: 10750: 10746: 10740: 10735: 10730: 10726: 10718: 10713: 10709: 10663: 10647: 10646: 10645: 10644: 10643: 10642: 10631: 10628: 10622: 10619: 10614: 10611: 10580: 10568: 10567: 10566: 10565: 10554: 10551: 10548: 10544: 10538: 10533: 10529: 10521: 10516: 10512: 10506: 10501: 10496: 10492: 10484: 10479: 10475: 10468: 10464: 10458: 10455: 10450: 10447: 10440: 10434: 10429: 10425: 10417: 10412: 10408: 10402: 10397: 10392: 10388: 10380: 10375: 10371: 10364: 10360: 10352: 10348: 10344: 10339: 10334: 10330: 10322: 10316: 10311: 10307: 10301: 10296: 10293: 10286: 10281: 10277: 10270: 10253: 10252: 10251: 10250: 10239: 10234: 10230: 10226: 10220: 10217: 10212: 10209: 10201: 10197: 10193: 10190: 10185: 10181: 10177: 10169: 10165: 10161: 10156: 10151: 10147: 10140: 10130: 10119: 10114: 10110: 10106: 10100: 10097: 10092: 10089: 10081: 10077: 10073: 10070: 10065: 10061: 10057: 10049: 10045: 10041: 10036: 10031: 10027: 10020: 10017: 10001: 10000: 9999: 9998: 9986: 9981: 9977: 9973: 9967: 9964: 9959: 9956: 9948: 9944: 9940: 9937: 9932: 9928: 9924: 9918: 9915: 9910: 9907: 9901: 9898: 9892: 9889: 9884: 9881: 9875: 9862: 9861: 9860: 9859: 9847: 9842: 9838: 9834: 9828: 9825: 9820: 9817: 9809: 9805: 9801: 9798: 9793: 9789: 9785: 9779: 9776: 9771: 9768: 9762: 9759: 9753: 9750: 9745: 9742: 9736: 9720: 9719: 9718: 9717: 9716: 9715: 9704: 9701: 9698: 9692: 9689: 9684: 9681: 9651: 9648: 9633: 9629: 9600: 9596: 9583: 9582: 9581: 9580: 9579: 9578: 9564: 9560: 9540: 9539: 9538: 9537: 9536: 9535: 9521: 9517: 9492: 9491: 9490: 9489: 9488: 9487: 9473: 9469: 9439: 9435: 9422: 9421: 9420: 9419: 9418: 9417: 9403: 9399: 9369: 9365: 9338: 9334: 9321: 9320: 9319: 9318: 9317: 9316: 9302: 9298: 9268: 9264: 9248: 9247: 9246: 9245: 9244: 9243: 9229: 9225: 9206: 9205: 9204: 9203: 9202: 9201: 9187: 9183: 9163: 9162: 9161: 9160: 9159: 9158: 9144: 9140: 9110: 9106: 9083: 9079: 9066: 9065: 9064: 9063: 9062: 9061: 9047: 9043: 9010: 8998: 8995: 8991: 8990: 8989: 8988: 8987: 8986: 8972: 8969: 8964: 8961: 8955: 8952: 8946: 8943: 8938: 8935: 8929: 8926: 8923: 8901: 8900: 8899: 8898: 8897: 8896: 8882: 8879: 8874: 8871: 8865: 8862: 8856: 8853: 8848: 8845: 8839: 8836: 8833: 8830: 8827: 8822: 8818: 8814: 8809: 8805: 8801: 8795: 8792: 8787: 8784: 8778: 8775: 8772: 8769: 8766: 8763: 8757: 8754: 8749: 8746: 8740: 8737: 8734: 8705: 8702: 8697: 8693: 8686: 8683: 8678: 8675: 8661: 8660: 8659: 8658: 8657: 8656: 8645: 8642: 8637: 8633: 8626: 8623: 8618: 8615: 8609: 8606: 8603: 8600: 8597: 8591: 8588: 8583: 8580: 8574: 8571: 8565: 8562: 8557: 8554: 8548: 8545: 8542: 8524: 8523: 8522: 8521: 8520: 8519: 8508: 8502: 8499: 8494: 8491: 8485: 8482: 8476: 8473: 8468: 8465: 8459: 8456: 8450: 8447: 8442: 8439: 8417: 8416: 8415: 8414: 8413: 8412: 8401: 8398: 8395: 8392: 8389: 8386: 8383: 8380: 8352:principal axes 8344:sideslip angle 8342:(beta) is the 8331: 8320: 8319: 8318: 8317: 8306: 8303: 8297: 8294: 8290: 8285: 8279: 8276: 8271: 8268: 8234:Main article: 8231: 8228: 8207:Antonov An-225 8198: 8195: 8183: 8182: 8181: 8180: 8166: 8161: 8154: 8151: 8148: 8142: 8139: 8123: 8122: 8121: 8120: 8104: 8100: 8093: 8089: 8085: 8078: 8073: 8069: 8066: 8047: 8046: 8045: 8044: 8031: 8027: 8024: 8021: 8015: 8010: 8006: 8003: 7997: 7992: 7988: 7960: 7956: 7933: 7929: 7908: 7897: 7896: 7895: 7894: 7883: 7880: 7875: 7871: 7865: 7861: 7855: 7851: 7847: 7842: 7839: 7834: 7831: 7815: 7814: 7813: 7812: 7801: 7798: 7795: 7789: 7786: 7781: 7776: 7772: 7765: 7759: 7756: 7751: 7748: 7740: 7735: 7731: 7725: 7717: 7713: 7709: 7704: 7699: 7695: 7675: 7674: 7673: 7672: 7661: 7658: 7655: 7652: 7649: 7644: 7640: 7636: 7630: 7627: 7622: 7619: 7613: 7603: 7592: 7587: 7583: 7579: 7576: 7570: 7567: 7562: 7559: 7553: 7550: 7520: 7516: 7493: 7489: 7466: 7462: 7439: 7435: 7419: 7418: 7417: 7416: 7405: 7402: 7399: 7396: 7393: 7390: 7384: 7381: 7376: 7373: 7367: 7348: 7347: 7346: 7345: 7334: 7331: 7328: 7322: 7319: 7314: 7311: 7305: 7302: 7277: 7253:Main article: 7250: 7247: 7227: 7223: 7200: 7196: 7189: 7186: 7180: 7176: 7170: 7165: 7161: 7140: 7117: 7114: 7108: 7104: 7087: 7086: 7085: 7084: 7073: 7070: 7067: 7063: 7059: 7053: 7050: 7044: 7040: 7034: 7031: 7028: 7023: 7018: 7014: 7008: 7003: 6998: 6994: 6985: 6982: 6976: 6972: 6965: 6961: 6955: 6952: 6947: 6944: 6937: 6931: 6924: 6921: 6915: 6909: 6903: 6900: 6894: 6890: 6884: 6881: 6878: 6875: 6870: 6865: 6861: 6855: 6849: 6846: 6840: 6836: 6829: 6825: 6817: 6813: 6809: 6804: 6799: 6795: 6766: 6751: 6750: 6749: 6748: 6734: 6731: 6723: 6716: 6713: 6707: 6701: 6698: 6693: 6688: 6684: 6678: 6675: 6670: 6665: 6661: 6655: 6649: 6646: 6641: 6638: 6622: 6621: 6620: 6619: 6608: 6602: 6599: 6593: 6589: 6583: 6580: 6576: 6569: 6566: 6560: 6556: 6550: 6547: 6543: 6539: 6533: 6530: 6525: 6522: 6487: 6484: 6478: 6466: 6465: 6464: 6463: 6448: 6445: 6439: 6421: 6420: 6419: 6418: 6404: 6400: 6386: 6385: 6384: 6383: 6362: 6358: 6344: 6343: 6342: 6341: 6327: 6323: 6300: 6296: 6282: 6281: 6280: 6279: 6265: 6261: 6231: 6220: 6219: 6218: 6217: 6204: 6201: 6196: 6190: 6187: 6182: 6179: 6166: 6152: 6149: 6145: 6140: 6137: 6134: 6128: 6125: 6120: 6117: 6089: 6086: 6080: 6077: 6072: 6069: 6051: 6050: 6049: 6048: 6032: 6028: 6024: 6019: 6014: 6010: 6003: 6000: 5997: 5976:acceleration. 5970: 5969: 5968: 5967: 5953: 5950: 5945: 5942: 5939: 5936: 5933: 5930: 5924: 5921: 5918: 5915: 5912: 5896: 5895: 5894: 5893: 5882: 5879: 5876: 5873: 5870: 5867: 5864: 5859: 5855: 5851: 5848: 5845: 5842: 5839: 5836: 5833: 5830: 5825: 5821: 5817: 5814: 5811: 5787: 5786: 5785: 5784: 5773: 5770: 5767: 5764: 5761: 5758: 5755: 5749: 5746: 5741: 5738: 5735: 5732: 5729: 5726: 5720: 5717: 5714: 5709: 5705: 5694: 5683: 5680: 5677: 5674: 5671: 5668: 5665: 5659: 5656: 5651: 5648: 5645: 5642: 5639: 5636: 5630: 5627: 5624: 5621: 5616: 5612: 5583: 5580: 5575: 5572: 5566: 5547: 5546: 5545: 5544: 5533: 5530: 5527: 5524: 5521: 5518: 5515: 5509: 5506: 5501: 5498: 5495: 5492: 5489: 5486: 5480: 5477: 5474: 5471: 5468: 5465: 5462: 5459: 5456: 5453: 5447: 5444: 5439: 5436: 5430: 5427: 5421: 5418: 5411: 5407: 5403: 5397: 5394: 5389: 5385: 5374: 5363: 5360: 5357: 5354: 5351: 5348: 5345: 5339: 5336: 5331: 5328: 5325: 5322: 5319: 5316: 5310: 5307: 5304: 5301: 5298: 5295: 5292: 5289: 5286: 5283: 5277: 5274: 5269: 5266: 5260: 5257: 5251: 5248: 5241: 5237: 5233: 5227: 5224: 5219: 5215: 5175: 5171: 5148: 5144: 5132: 5131: 5130: 5129: 5118: 5115: 5112: 5109: 5106: 5103: 5100: 5097: 5094: 5089: 5085: 5074: 5063: 5060: 5057: 5054: 5051: 5048: 5045: 5042: 5039: 5034: 5030: 4999: 4996: 4993: 4973: 4953: 4907: 4904: 4883: 4880: 4877: 4876: 4852: 4850: 4843: 4834:Main article: 4831: 4828: 4815:Main article: 4812: 4809: 4800:Main article: 4797: 4794: 4792: 4789: 4768: 4765: 4753: 4750: 4742: 4739: 4727: 4726: 4721: 4717: 4712:incompressible 4707: 4704: 4697: 4690: 4689: 4688: 4687: 4672: 4666: 4659: 4655: 4649: 4646: 4642: 4635: 4632: 4628: 4622: 4617: 4613: 4587: 4584: 4583: 4582: 4569: 4565: 4559: 4556: 4552: 4548: 4543: 4539: 4535: 4532: 4529: 4524: 4521: 4517: 4513: 4508: 4504: 4492: 4491: 4478: 4475: 4472: 4469: 4465: 4459: 4455: 4451: 4448: 4443: 4440: 4436: 4432: 4427: 4424: 4421: 4418: 4414: 4398: 4397: 4384: 4381: 4377: 4373: 4368: 4365: 4362: 4359: 4355: 4349: 4345: 4341: 4338: 4333: 4330: 4327: 4324: 4320: 4305: 4304: 4291: 4284: 4281: 4277: 4273: 4268: 4265: 4262: 4259: 4255: 4249: 4246: 4242: 4238: 4235: 4234: 4227: 4222: 4219: 4215: 4207: 4202: 4199: 4196: 4193: 4189: 4183: 4179: 4175: 4172: 4171: 4160: 4157: 4153: 4149: 4144: 4140: 4134: 4129: 4126: 4123: 4119: 4115: 4114: 4112: 4107: 4098: 4093: 4089: 4085: 4082: 4077: 4074: 4070: 4063: 4059: 4052: 4049: 4035: 4034: 4021: 4016: 4012: 4008: 4005: 4000: 3997: 3993: 3989: 3984: 3980: 3976: 3971: 3966: 3962: 3958: 3955: 3950: 3947: 3943: 3927: 3924: 3923: 3922: 3921: 3920: 3907: 3900: 3897: 3893: 3889: 3888: 3883: 3880: 3875: 3871: 3866: 3860: 3856: 3852: 3849: 3844: 3841: 3837: 3833: 3832: 3830: 3823: 3820: 3816: 3812: 3807: 3804: 3800: 3796: 3791: 3787: 3769: 3765: 3761: 3757: 3750: 3749: 3748: 3747: 3734: 3729: 3726: 3719: 3715: 3710: 3706: 3702: 3697: 3693: 3689: 3686: 3681: 3677: 3670: 3667: 3661: 3658: 3651: 3648: 3644: 3638: 3633: 3630: 3626: 3622: 3621: 3618: 3615: 3608: 3604: 3599: 3595: 3589: 3585: 3579: 3575: 3568: 3565: 3559: 3556: 3549: 3546: 3542: 3536: 3531: 3528: 3524: 3520: 3519: 3517: 3510: 3507: 3503: 3499: 3494: 3491: 3487: 3483: 3478: 3474: 3456: 3449: 3448:, respect to C 3445: 3442: 3441: 3440: 3439: 3427: 3424: 3421: 3418: 3415: 3412: 3409: 3404: 3400: 3396: 3391: 3387: 3383: 3378: 3374: 3349: 3348: 3347: 3346: 3333: 3323: 3319: 3313: 3309: 3302: 3299: 3296: 3293: 3290: 3287: 3284: 3281: 3278: 3275: 3272: 3269: 3266: 3265: 3262: 3259: 3256: 3253: 3250: 3247: 3244: 3241: 3236: 3232: 3228: 3223: 3219: 3215: 3214: 3211: 3208: 3205: 3202: 3199: 3196: 3193: 3190: 3185: 3181: 3177: 3172: 3168: 3164: 3163: 3161: 3123: 3122: 3121: 3120: 3108: 3105: 3095: 3081: 3078: 3062: 3061: 3060: 3059: 3048: 3045: 3042: 3039: 3036: 3033: 3030: 3027: 3024: 3021: 3016: 3012: 3008: 3003: 2999: 2988: 2977: 2974: 2971: 2968: 2965: 2962: 2959: 2956: 2953: 2950: 2945: 2941: 2937: 2932: 2928: 2910: 2907: 2906: 2905: 2894: 2891: 2888: 2874: 2863: 2860: 2857: 2854: 2851: 2837: 2826: 2823: 2820: 2817: 2814: 2800: 2789: 2786: 2783: 2780: 2777: 2763: 2752: 2749: 2746: 2743: 2740: 2719:boundary layer 2711: 2710: 2699: 2696: 2689: 2686: 2682: 2668: 2657: 2654: 2647: 2644: 2640: 2626: 2615: 2612: 2605: 2602: 2598: 2580: 2579: 2578: 2577: 2575:mean free path 2563: 2556: 2552: 2549: 2537: 2534: 2530: 2524: 2516: 2513: 2510: 2506: 2496: 2493: 2487: 2484: 2474: 2473: 2472: 2457: 2454: 2444: 2429: 2426: 2416: 2402: 2399: 2374: 2369: 2366: 2363: 2358: 2355: 2339: 2338: 2324: 2321: 2315: 2312: 2309: 2306: 2296:Knudsen number 2292: 2291: 2287: 2286: 2272: 2268: 2265: 2262: 2255: 2252: 2249: 2246: 2232: 2231: 2227: 2226: 2212: 2209: 2203: 2200: 2197: 2183: 2182: 2174: 2171: 2166: 2162: 2148: 2145: 2141: 2135: 2131: 2126: 2122: 2116: 2112: 2108: 2102: 2098: 2093: 2089: 2085: 2080: 2076: 2072: 2069: 2066: 2061: 2057: 2050: 2047: 2041: 2038: 2031: 2028: 2024: 2018: 2013: 2009: 2005: 1981: 1978: 1974: 1968: 1964: 1959: 1955: 1949: 1945: 1941: 1935: 1931: 1926: 1922: 1918: 1913: 1909: 1905: 1902: 1899: 1894: 1890: 1883: 1880: 1874: 1871: 1864: 1861: 1857: 1851: 1846: 1842: 1838: 1816: 1813: 1809: 1803: 1799: 1794: 1790: 1784: 1780: 1776: 1770: 1766: 1761: 1757: 1753: 1748: 1744: 1740: 1737: 1734: 1729: 1725: 1718: 1715: 1709: 1706: 1699: 1696: 1692: 1686: 1681: 1677: 1673: 1646: 1643: 1637: 1632: 1628: 1624: 1597: 1591: 1587: 1583: 1580: 1573: 1568: 1564: 1560: 1536: 1533: 1497: 1493: 1488: 1481: 1478: 1472: 1469: 1466: 1451: 1448: 1447: 1446: 1445: 1444: 1432: 1429: 1419: 1407: 1404: 1394: 1382: 1379: 1363: 1362: 1361: 1360: 1349: 1346: 1341: 1336: 1331: 1328: 1323: 1318: 1313: 1310: 1305: 1300: 1295: 1292: 1289: 1284: 1279: 1261: 1260: 1259: 1258: 1246: 1243: 1233: 1221: 1217: 1206: 1194: 1190: 1179: 1167: 1164: 1161: 1145: 1144: 1143: 1142: 1131: 1128: 1124: 1120: 1116: 1112: 1108: 1105: 1102: 1099: 1094: 1090: 1086: 1081: 1076: 1057: 1054: 1049:Main article: 1046: 1043: 1026: 1023: 986: 985: 984: 983: 978: 975: 972: 958: 955: 954: 953: 950: 947: 939: 936: 935: 934: 928: 918: 908: 907: 902: 895: 880: 879: 875: 871: 857: 854: 853: 852: 851:(Bank vs Roll) 846: 840: 830: 829: 814: 803: 789: 786: 785: 784: 772: 760: 744: 743: 717: 671: 666: 632: 624: 617: 610: 607: 605: 598: 586: 585: 580: 575: 570: 548: 537: 511: 494: 493: 492: 491: 486: 479: 472: 468: 463: 456: 452: 447: 443: 437: 436: 435: 426: 419: 412: 408: 403: 396: 392: 387: 383: 363: 356: 351: 350: 349: 348: 343: 339: 331: 327: 319: 315: 290: 287: 280: 273: 272: 271: 263: 256: 255: 254: 246: 239: 238: 237: 236: 235: 228: 221: 220: 214: 207: 206: 200: 193: 192: 191: 190: 189: 173:Main article: 170: 167: 162:sideslip angle 80: 79: 59:the key points 49: 47: 40: 26: 9: 6: 4: 3: 2: 12991: 12980: 12977: 12975: 12972: 12970: 12967: 12966: 12964: 12955: 12952: 12950: 12947: 12946: 12936: 12930: 12926: 12921: 12918: 12914: 12913: 12900: 12894: 12890: 12889: 12880: 12872: 12870:9780071238182 12866: 12862: 12855: 12848: 12831: 12824: 12818: 12816: 12814: 12806: 12805:1 85310 870 7 12802: 12796: 12788: 12787: 12779: 12775: 12761: 12758: 12750: 12747:February 2009 12740: 12736: 12730: 12729: 12723: 12718: 12709: 12708: 12697: 12694: 12692: 12689: 12687: 12686:Static margin 12684: 12682: 12679: 12677: 12674: 12672: 12669: 12667: 12664: 12662: 12659: 12656: 12653: 12651: 12648: 12646: 12643: 12641: 12638: 12636: 12633: 12631: 12630:Aircraft bank 12628: 12626: 12623: 12621: 12618: 12616: 12613: 12611: 12610:Steady flight 12608: 12606: 12603: 12601: 12598: 12597: 12590: 12587: 12582: 12566: 12562: 12556: 12552: 12548: 12543: 12539: 12533: 12529: 12520: 12507: 12498: 12475: 12464: 12460: 12454: 12450: 12446: 12441: 12437: 12431: 12427: 12418: 12414: 12410: 12402: 12398: 12392: 12388: 12384: 12379: 12375: 12369: 12365: 12358: 12355: 12345: 12341: 12335: 12331: 12327: 12322: 12318: 12312: 12308: 12298: 12295: 12292: 12286: 12283: 12278: 12275: 12265: 12264: 12263: 12262: 12261: 12260: 12259: 12235: 12232: 12227: 12224: 12210: 12206: 12200: 12196: 12192: 12187: 12183: 12177: 12173: 12159: 12155: 12149: 12145: 12141: 12136: 12132: 12126: 12122: 12112: 12109: 12102: 12101: 12100: 12099: 12098: 12097: 12075: 12072: 12067: 12064: 12050: 12046: 12040: 12036: 12032: 12027: 12023: 12017: 12013: 11999: 11995: 11989: 11985: 11981: 11976: 11972: 11966: 11962: 11952: 11949: 11942: 11941: 11940: 11939: 11938: 11937: 11936: 11934: 11930: 11917: 11890: 11887: 11884: 11879: 11875: 11871: 11865: 11862: 11857: 11854: 11846: 11842: 11838: 11835: 11830: 11826: 11818: 11817: 11816: 11815: 11797: 11794: 11791: 11786: 11782: 11778: 11772: 11769: 11764: 11761: 11753: 11749: 11745: 11742: 11737: 11733: 11725: 11724: 11723: 11722: 11721: 11719: 11714: 11698: 11694: 11671: 11667: 11641: 11636: 11633: 11628: 11625: 11619: 11616: 11610: 11606: 11600: 11597: 11591: 11588: 11582: 11578: 11572: 11569: 11563: 11560: 11554: 11550: 11544: 11538: 11535: 11530: 11527: 11517: 11516: 11515: 11514: 11513: 11510: 11508: 11504: 11483: 11478: 11475: 11470: 11464: 11461: 11457: 11452: 11446: 11443: 11438: 11435: 11425: 11424: 11423: 11422: 11421: 11419: 11406: 11397: 11393: 11380: 11364: 11362: 11351: 11335: 11331: 11306: 11303: 11298: 11294: 11290: 11284: 11281: 11276: 11273: 11267: 11260: 11259: 11258: 11257: 11256: 11254: 11248: 11240: 11223: 11219: 11209: 11206: 11190: 11186: 11163: 11159: 11136: 11132: 11109: 11105: 11082: 11078: 11055: 11051: 11045: 11041: 11037: 11028: 11012: 11008: 10974: 10969: 10965: 10959: 10954: 10951: 10944: 10939: 10935: 10925: 10920: 10916: 10908: 10903: 10899: 10893: 10888: 10883: 10879: 10871: 10866: 10862: 10849: 10848: 10847: 10846: 10845: 10844: 10819: 10814: 10810: 10804: 10799: 10796: 10789: 10784: 10780: 10770: 10765: 10761: 10753: 10748: 10744: 10738: 10733: 10728: 10724: 10716: 10711: 10707: 10694: 10693: 10692: 10691: 10690: 10689: 10688: 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9935: 9930: 9926: 9922: 9916: 9913: 9908: 9905: 9899: 9896: 9890: 9887: 9882: 9879: 9873: 9866: 9865: 9864: 9863: 9845: 9840: 9836: 9832: 9826: 9823: 9818: 9815: 9807: 9803: 9799: 9796: 9791: 9787: 9783: 9777: 9774: 9769: 9766: 9760: 9757: 9751: 9748: 9743: 9740: 9734: 9727: 9726: 9725: 9724: 9723: 9702: 9699: 9696: 9690: 9687: 9682: 9679: 9669: 9668: 9667: 9666: 9665: 9664: 9663: 9661: 9657: 9647: 9631: 9627: 9619: 9615: 9598: 9594: 9562: 9558: 9550: 9549: 9548: 9547: 9546: 9545: 9544: 9519: 9515: 9507: 9506: 9505: 9504: 9503: 9502: 9501: 9499: 9471: 9467: 9459: 9458: 9457: 9456: 9455: 9454: 9453: 9437: 9433: 9401: 9397: 9389: 9388: 9387: 9386: 9385: 9384: 9383: 9367: 9363: 9354: 9336: 9332: 9300: 9296: 9288: 9287: 9286: 9285: 9284: 9283: 9282: 9266: 9262: 9253: 9227: 9223: 9215: 9214: 9213: 9212: 9211: 9210: 9209: 9185: 9181: 9173: 9172: 9171: 9170: 9169: 9168: 9167: 9142: 9138: 9130: 9129: 9128: 9127: 9126: 9125: 9124: 9108: 9104: 9081: 9077: 9045: 9041: 9033: 9032: 9031: 9030: 9029: 9028: 9027: 9025: 9008: 8994: 8970: 8967: 8962: 8959: 8953: 8950: 8944: 8941: 8936: 8933: 8927: 8924: 8921: 8914: 8913: 8912: 8911: 8910: 8909: 8908: 8906: 8880: 8877: 8872: 8869: 8863: 8860: 8854: 8851: 8846: 8843: 8837: 8834: 8831: 8828: 8825: 8820: 8816: 8812: 8807: 8803: 8799: 8793: 8790: 8785: 8782: 8776: 8773: 8770: 8767: 8764: 8761: 8755: 8752: 8747: 8744: 8738: 8735: 8732: 8725: 8724: 8723: 8722: 8721: 8720: 8719: 8716: 8703: 8700: 8695: 8691: 8684: 8681: 8676: 8673: 8643: 8640: 8635: 8631: 8624: 8621: 8616: 8613: 8607: 8604: 8601: 8598: 8595: 8589: 8586: 8581: 8578: 8572: 8569: 8563: 8560: 8555: 8552: 8546: 8543: 8540: 8533: 8532: 8531: 8530: 8529: 8528: 8527: 8506: 8500: 8497: 8492: 8489: 8483: 8480: 8474: 8471: 8466: 8463: 8457: 8454: 8448: 8445: 8440: 8437: 8427: 8426: 8425: 8424: 8423: 8422: 8421: 8399: 8396: 8393: 8390: 8387: 8384: 8381: 8378: 8371: 8370: 8369: 8368: 8367: 8366: 8365: 8359: 8355: 8353: 8347: 8345: 8329: 8304: 8301: 8295: 8292: 8288: 8283: 8277: 8274: 8269: 8266: 8256: 8255: 8254: 8253: 8252: 8249: 8247: 8242: 8237: 8227: 8223: 8220: 8212: 8208: 8203: 8197:Lateral modes 8194: 8192: 8188: 8164: 8159: 8152: 8149: 8146: 8140: 8137: 8130: 8129: 8128: 8127: 8126: 8102: 8098: 8091: 8087: 8083: 8076: 8071: 8067: 8064: 8054: 8053: 8052: 8051: 8050: 8029: 8025: 8022: 8019: 8013: 8008: 8004: 8001: 7995: 7990: 7986: 7978: 7977: 7976: 7975: 7974: 7958: 7954: 7931: 7927: 7906: 7881: 7878: 7873: 7869: 7863: 7859: 7853: 7849: 7845: 7840: 7837: 7832: 7829: 7822: 7821: 7820: 7819: 7818: 7799: 7796: 7793: 7787: 7784: 7779: 7774: 7770: 7763: 7757: 7754: 7749: 7746: 7738: 7733: 7729: 7723: 7715: 7711: 7707: 7702: 7697: 7693: 7682: 7681: 7680: 7679: 7678: 7659: 7656: 7653: 7650: 7647: 7642: 7638: 7634: 7628: 7625: 7620: 7617: 7611: 7604: 7590: 7585: 7581: 7577: 7574: 7568: 7565: 7560: 7557: 7551: 7548: 7541: 7540: 7539: 7538: 7537: 7534: 7518: 7514: 7491: 7487: 7464: 7460: 7437: 7433: 7424: 7403: 7400: 7397: 7394: 7391: 7388: 7382: 7379: 7374: 7371: 7365: 7358: 7357: 7356: 7355: 7354: 7351: 7332: 7329: 7326: 7320: 7317: 7312: 7309: 7303: 7300: 7293: 7292: 7291: 7290: 7289: 7275: 7267: 7263: 7256: 7246: 7243: 7225: 7221: 7198: 7194: 7187: 7184: 7178: 7174: 7168: 7163: 7159: 7138: 7115: 7112: 7106: 7102: 7090: 7071: 7068: 7065: 7061: 7051: 7048: 7042: 7038: 7032: 7029: 7021: 7016: 7012: 7006: 7001: 6996: 6992: 6983: 6980: 6974: 6970: 6963: 6959: 6953: 6950: 6945: 6942: 6935: 6929: 6922: 6919: 6913: 6901: 6898: 6892: 6888: 6882: 6879: 6873: 6868: 6863: 6859: 6853: 6847: 6844: 6838: 6834: 6827: 6823: 6815: 6811: 6807: 6802: 6797: 6793: 6782: 6781: 6780: 6779: 6778: 6764: 6756: 6732: 6729: 6721: 6714: 6711: 6705: 6699: 6696: 6691: 6686: 6682: 6676: 6673: 6668: 6663: 6659: 6653: 6647: 6644: 6639: 6636: 6626: 6625: 6624: 6623: 6606: 6600: 6597: 6591: 6587: 6581: 6578: 6574: 6567: 6564: 6558: 6554: 6548: 6545: 6541: 6537: 6531: 6528: 6523: 6520: 6510: 6509: 6508: 6507: 6506: 6503: 6485: 6482: 6476: 6446: 6443: 6437: 6429: 6428: 6427: 6426: 6425: 6402: 6398: 6390: 6389: 6388: 6387: 6381: 6377: 6360: 6356: 6348: 6347: 6346: 6345: 6325: 6321: 6298: 6294: 6286: 6285: 6284: 6283: 6263: 6259: 6251: 6250: 6249: 6248: 6247: 6245: 6229: 6202: 6199: 6194: 6188: 6185: 6180: 6177: 6167: 6150: 6147: 6143: 6138: 6135: 6132: 6126: 6123: 6118: 6115: 6105: 6104: 6103: 6102: 6101: 6087: 6084: 6078: 6075: 6070: 6067: 6056: 6030: 6026: 6022: 6017: 6012: 6008: 6001: 5998: 5995: 5988: 5987: 5986: 5985: 5984: 5982: 5977: 5975: 5951: 5948: 5940: 5937: 5934: 5928: 5922: 5919: 5916: 5913: 5910: 5903: 5902: 5901: 5900: 5899: 5877: 5874: 5871: 5865: 5862: 5857: 5853: 5849: 5843: 5840: 5837: 5831: 5828: 5823: 5819: 5815: 5812: 5809: 5802: 5801: 5800: 5799: 5798: 5796: 5792: 5768: 5765: 5762: 5756: 5753: 5747: 5744: 5736: 5733: 5730: 5724: 5718: 5715: 5712: 5707: 5703: 5695: 5678: 5675: 5672: 5666: 5663: 5657: 5654: 5646: 5643: 5640: 5634: 5628: 5625: 5622: 5619: 5614: 5610: 5602: 5601: 5600: 5599: 5598: 5581: 5578: 5573: 5570: 5564: 5556: 5552: 5528: 5525: 5522: 5516: 5513: 5507: 5504: 5496: 5493: 5490: 5484: 5478: 5475: 5472: 5466: 5463: 5460: 5454: 5451: 5445: 5442: 5437: 5434: 5428: 5425: 5419: 5416: 5409: 5405: 5401: 5395: 5392: 5387: 5383: 5375: 5358: 5355: 5352: 5346: 5343: 5337: 5334: 5326: 5323: 5320: 5314: 5308: 5305: 5302: 5296: 5293: 5290: 5284: 5281: 5275: 5272: 5267: 5264: 5258: 5255: 5249: 5246: 5239: 5235: 5231: 5225: 5222: 5217: 5213: 5205: 5204: 5203: 5202: 5201: 5199: 5195: 5194:accelerations 5191: 5173: 5169: 5146: 5142: 5113: 5110: 5107: 5101: 5098: 5095: 5092: 5087: 5083: 5075: 5058: 5055: 5052: 5046: 5043: 5040: 5037: 5032: 5028: 5020: 5019: 5018: 5017: 5016: 5014: 4997: 4994: 4991: 4971: 4951: 4943: 4939: 4934: 4932: 4926: 4924: 4921: 4917: 4913: 4903: 4901: 4897: 4892: 4889: 4873: 4864: 4860: 4856: 4851: 4842: 4841: 4837: 4827: 4824: 4818: 4808: 4803: 4788: 4786: 4782: 4777: 4774: 4773:cruise flight 4764: 4762: 4759:According to 4757: 4749: 4748: 4738: 4736: 4732: 4724: 4718: 4716: 4713: 4705: 4703: 4695: 4694: 4693: 4664: 4653: 4647: 4644: 4633: 4630: 4626: 4620: 4615: 4611: 4603: 4602: 4601: 4600: 4599: 4597: 4593: 4567: 4557: 4554: 4550: 4546: 4541: 4537: 4530: 4527: 4522: 4519: 4515: 4511: 4506: 4502: 4494: 4493: 4476: 4473: 4470: 4467: 4457: 4453: 4446: 4441: 4438: 4434: 4430: 4425: 4422: 4419: 4416: 4412: 4404: 4403: 4402: 4382: 4379: 4375: 4371: 4366: 4363: 4360: 4357: 4347: 4343: 4336: 4331: 4328: 4325: 4322: 4318: 4310: 4309: 4308: 4282: 4279: 4275: 4271: 4266: 4263: 4260: 4257: 4247: 4244: 4240: 4225: 4220: 4217: 4213: 4205: 4200: 4197: 4194: 4191: 4181: 4177: 4158: 4155: 4151: 4147: 4142: 4138: 4132: 4127: 4124: 4121: 4117: 4110: 4096: 4091: 4087: 4083: 4080: 4075: 4072: 4068: 4061: 4057: 4050: 4047: 4040: 4039: 4038: 4019: 4014: 4010: 4006: 4003: 3998: 3995: 3991: 3987: 3982: 3978: 3969: 3964: 3960: 3956: 3953: 3948: 3945: 3941: 3933: 3932: 3931: 3898: 3895: 3891: 3881: 3878: 3873: 3869: 3858: 3854: 3847: 3842: 3839: 3835: 3828: 3821: 3818: 3814: 3810: 3805: 3802: 3798: 3794: 3789: 3785: 3777: 3776: 3775: 3774: 3773: 3755: 3727: 3724: 3708: 3695: 3691: 3687: 3675: 3668: 3665: 3659: 3656: 3649: 3646: 3642: 3636: 3631: 3628: 3624: 3616: 3613: 3597: 3587: 3583: 3573: 3566: 3563: 3557: 3554: 3547: 3544: 3540: 3534: 3529: 3526: 3522: 3515: 3508: 3505: 3501: 3497: 3492: 3489: 3485: 3481: 3476: 3472: 3464: 3463: 3462: 3461: 3460: 3453: 3425: 3419: 3416: 3413: 3410: 3407: 3402: 3398: 3389: 3385: 3381: 3376: 3372: 3364: 3363: 3362: 3361: 3360: 3358: 3354: 3321: 3317: 3311: 3307: 3300: 3294: 3291: 3288: 3285: 3282: 3279: 3273: 3270: 3267: 3257: 3254: 3251: 3248: 3245: 3242: 3234: 3230: 3226: 3221: 3217: 3206: 3203: 3200: 3197: 3194: 3191: 3183: 3179: 3175: 3170: 3166: 3159: 3150: 3149: 3148: 3147: 3146: 3144: 3140: 3136: 3132: 3128: 3106: 3103: 3096: 3094: 3079: 3076: 3069: 3068: 3067: 3066: 3065: 3043: 3040: 3037: 3034: 3031: 3028: 3025: 3022: 3014: 3010: 3006: 3001: 2997: 2989: 2972: 2969: 2966: 2963: 2960: 2957: 2954: 2951: 2943: 2939: 2935: 2930: 2926: 2918: 2917: 2916: 2915: 2914: 2892: 2889: 2886: 2878: 2875: 2861: 2858: 2855: 2852: 2849: 2841: 2838: 2824: 2821: 2818: 2815: 2812: 2804: 2801: 2787: 2784: 2781: 2778: 2775: 2767: 2764: 2750: 2747: 2744: 2741: 2738: 2730: 2727: 2726: 2725: 2722: 2720: 2716: 2697: 2694: 2687: 2684: 2680: 2669: 2655: 2652: 2645: 2642: 2638: 2627: 2613: 2610: 2603: 2600: 2596: 2585: 2584: 2583: 2576: 2561: 2554: 2550: 2547: 2535: 2532: 2528: 2522: 2514: 2511: 2508: 2504: 2494: 2491: 2485: 2482: 2475: 2471: 2455: 2452: 2445: 2443:by mass unity 2442: 2427: 2424: 2417: 2415: 2400: 2397: 2390: 2389: 2388: 2372: 2367: 2364: 2361: 2356: 2353: 2346: 2345: 2344: 2343: 2342: 2322: 2319: 2313: 2310: 2307: 2304: 2297: 2294: 2293: 2289: 2288: 2270: 2266: 2263: 2260: 2253: 2250: 2247: 2244: 2237: 2234: 2233: 2229: 2228: 2210: 2207: 2201: 2198: 2195: 2188: 2185: 2184: 2180: 2179: 2178: 2170: 2159: 2146: 2143: 2124: 2114: 2110: 2106: 2091: 2078: 2074: 2070: 2055: 2048: 2045: 2039: 2036: 2029: 2026: 2022: 2016: 2011: 2007: 2003: 1996: 1992: 1979: 1976: 1957: 1947: 1943: 1939: 1924: 1911: 1907: 1903: 1888: 1881: 1878: 1872: 1869: 1862: 1859: 1855: 1849: 1844: 1840: 1836: 1827: 1814: 1811: 1792: 1782: 1778: 1774: 1759: 1746: 1742: 1738: 1723: 1716: 1713: 1707: 1704: 1697: 1694: 1690: 1684: 1679: 1675: 1671: 1664: 1660: 1644: 1641: 1635: 1630: 1626: 1622: 1615: 1611: 1595: 1585: 1581: 1578: 1571: 1566: 1562: 1558: 1551: 1547: 1534: 1531: 1523: 1519: 1515: 1510: 1495: 1491: 1486: 1479: 1476: 1470: 1467: 1464: 1456: 1430: 1427: 1420: 1418:Lateral force 1405: 1402: 1395: 1380: 1377: 1370: 1369: 1368: 1367: 1366: 1344: 1339: 1329: 1326: 1321: 1311: 1308: 1303: 1290: 1287: 1282: 1268: 1267: 1266: 1265: 1264: 1244: 1234: 1219: 1207: 1192: 1180: 1165: 1162: 1152: 1151: 1150: 1149: 1148: 1129: 1126: 1114: 1106: 1100: 1088: 1084: 1079: 1065: 1064: 1063: 1062: 1061: 1052: 1042: 1040: 1036: 1032: 1022: 1020: 1015: 1011: 1008: 1006: 1002: 997: 996:may be down. 995: 994:undercarriage 991: 982: 979: 976: 973: 970: 969: 968: 967: 966: 964: 963:steady flight 951: 948: 945: 944: 943: 932: 929: 926: 922: 919: 916: 913: 912: 911: 905: 898: 891: 888: 887: 882: 881: 869: 866: 864: 860: 859: 850: 847: 844: 841: 838: 835: 834: 833: 828: 826: 821: 819: 815: 813: 809: 808: 804: 801: 797: 796: 792: 791: 782: 778: 777: 773: 770: 766: 765: 761: 758: 754: 753: 749: 748: 747: 741: 737: 733: 731: 726: 722: 718: 715: 708: 704: 700: 696: 694: 686: 679: 672: 669: 662: 652: 645: 641: 637: 635: 627: 620: 613: 612: 603: 597: 595: 591: 584: 581: 579: 576: 574: 571: 569: 566: 565: 564: 557: 553: 551: 544: 540: 533: 529: 524: 522: 516: 514: 505: 503: 499: 489: 482: 475: 469: 466: 459: 453: 450: 444: 441: 440: 438: 433: 429: 422: 415: 409: 406: 399: 393: 390: 384: 381: 380: 378: 377: 376: 372: 370: 366: 359: 346: 340: 338: 334: 328: 326: 322: 316: 313: 312: 310: 309: 308: 306: 302: 296: 277: 266: 260: 249: 243: 225: 217: 211: 203: 197: 188: 184: 182: 181:steady flight 176: 166: 163: 159: 155: 151: 146: 144: 139: 135: 133: 129: 125: 121: 117: 113: 109: 105: 101: 97: 93: 86: 76: 66: 60: 58: 53: 48: 44: 39: 38: 33: 19: 12969:Aerodynamics 12924: 12916: 12910:Bibliography 12886: 12879: 12860: 12854: 12837:. Retrieved 12830:the original 12807:, Figure 2.6 12795: 12785: 12778: 12753: 12744: 12725: 12676:Ship motions 12583: 12500: 12494: 12257: 11932: 11910: 11909:Solving for 11908: 11715: 11658: 11511: 11506: 11502: 11500: 11399: 11373: 11370: 11360: 11357: 11323: 11249: 11246: 11210: 11207: 11029: 10999: 10682: 10654: 10650: 10648: 10571: 10569: 10254: 10002: 9721: 9659: 9653: 9617: 9613: 9584: 9541: 9493: 9423: 9322: 9249: 9207: 9164: 9067: 9000: 8992: 8902: 8717: 8662: 8525: 8418: 8363: 8348: 8321: 8250: 8243: 8239: 8224: 8216: 8184: 8124: 8048: 7898: 7816: 7676: 7535: 7420: 7352: 7349: 7258: 7091: 7088: 6752: 6504: 6467: 6422: 6221: 6052: 5978: 5971: 5897: 5788: 5550: 5548: 5133: 4938:short period 4935: 4927: 4909: 4896:short period 4893: 4885: 4867: 4820: 4805: 4778: 4770: 4758: 4755: 4744: 4730: 4728: 4719: 4691: 4594:varies with 4589: 4399: 4306: 4036: 3929: 3751: 3454: 3443: 3350: 3124: 3063: 2912: 2723: 2712: 2581: 2441:gas constant 2340: 2176: 2160: 1993: 1828: 1661: 1612: 1548: 1511: 1453: 1364: 1262: 1146: 1059: 1028: 1016: 1012: 1009: 998: 987: 960: 957:Design cases 941: 930: 924: 920: 914: 909: 900: 893: 889: 885: 867: 862: 848: 842: 836: 831: 822: 817: 811: 805: 799: 793: 780: 774: 768: 762: 756: 750: 745: 739: 735: 729: 728: 724: 720: 710: 706: 702: 698: 689: 688: 681: 674: 664: 657: 647: 643: 639: 630: 629: 622: 615: 602:Euler angles 589: 587: 578:Euler angles 562: 546: 535: 525: 517: 509: 506: 501: 497: 495: 484: 477: 470: 461: 454: 445: 424: 417: 410: 401: 394: 385: 373: 361: 354: 352: 341: 329: 317: 311:Earth frame 301:right-handed 298: 185: 178: 147: 136: 123: 119: 115: 111: 91: 90: 70: 54: 52:lead section 12739:introducing 12605:Aeronautics 12586:spiral mode 12495:This is an 11718:derivatives 11361:spiral mode 11354:Spiral mode 10992:(stiffness) 5974:centripetal 4942:weathercock 4596:Mach number 2470:temperature 2187:Mach number 999:Except for 807:Flight path 628:around the 583:Quaternions 552:direction. 515:direction. 439:Wind frame 379:Body frame 12963:Categories 12898:0903083019 12722:references 12703:References 12584:Since the 10677:quadrature 9656:Dutch roll 8246:Dutch roll 8236:Dutch roll 8230:Dutch roll 4870:April 2024 4785:drag curve 1037:, and the 1019:rigid body 543:lift force 532:drag force 169:Background 132:dirigibles 104:dimensions 12557:β 12534:β 12508:ϕ 12476:ϕ 12447:− 12419:β 12411:− 12403:β 12385:− 12380:β 12336:β 12328:− 12313:β 12279:ϕ 12228:μ 12211:β 12193:− 12188:β 12160:β 12142:− 12127:β 12068:μ 12051:β 12033:− 12028:β 11982:− 11950:β 11918:β 11858:μ 11836:β 11831:β 11765:μ 11743:β 11738:β 11642:ϕ 11570:β 11555:β 11531:μ 11484:ϕ 11439:μ 11407:ψ 11381:μ 11253:autopilot 11224:β 11191:β 11137:β 11110:β 11056:β 11038:− 10960:− 10921:β 10894:− 10867:β 10837:(damping) 10805:− 10739:− 10685:stiffness 10662:ϕ 10613:ϕ 10579:ϕ 10547:β 10515:β 10505:− 10495:β 10463:− 10449:β 10401:− 10338:β 10300:− 10211:β 10192:− 10189:β 10184:β 10155:β 10091:β 10072:− 10069:β 10064:β 10035:β 10016:− 9958:β 9939:− 9936:β 9931:β 9897:− 9819:β 9800:− 9797:β 9792:β 9758:− 9700:− 9683:β 9599:β 9472:β 9353:empennage 9267:β 9252:empennage 9228:β 9109:β 9082:β 9046:β 9009:β 8951:− 8835:− 8800:∫ 8762:∫ 8739:− 8701:δ 8641:δ 8602:δ 8573:− 8541:δ 8458:− 8385:− 8330:β 8302:− 8270:β 8187:propeller 8150:π 8068:π 7907:ρ 7846:ρ 7764:− 7724:− 7660:γ 7651:− 7578:− 7561:γ 7404:γ 7395:− 7330:− 7313:γ 7276:γ 7226:α 7179:α 7164:α 7139:α 7066:α 7017:α 7007:− 6975:α 6946:α 6923:˙ 6920:α 6839:α 6824:− 6803:α 6765:α 6733:˙ 6730:α 6715:˙ 6712:α 6697:α 6687:α 6607:α 6592:α 6524:α 6486:˙ 6483:α 6447:˙ 6444:α 6361:α 6326:α 6264:α 6230:α 6119:α 6071:θ 6018:θ 5941:α 5938:− 5935:θ 5917:− 5878:α 5875:− 5872:θ 5866:⁡ 5844:α 5841:− 5838:θ 5832:⁡ 5816:− 5769:α 5766:− 5763:θ 5757:⁡ 5737:α 5734:− 5731:θ 5679:α 5676:− 5673:θ 5667:⁡ 5647:α 5644:− 5641:θ 5623:− 5529:α 5526:− 5523:θ 5517:⁡ 5497:α 5494:− 5491:θ 5467:α 5464:− 5461:θ 5455:⁡ 5359:α 5356:− 5353:θ 5347:⁡ 5327:α 5324:− 5321:θ 5303:− 5297:α 5294:− 5291:θ 5285:⁡ 5114:α 5111:− 5108:θ 5102:⁡ 5059:α 5056:− 5053:θ 5047:⁡ 4998:α 4995:− 4992:θ 4972:α 4952:θ 4752:Stability 4658:∞ 4648:− 4547:− 4447:≠ 4106:⇒ 3975:⇒ 3728:σ 3709:∙ 3688:− 3680:Σ 3676:∫ 3660:− 3617:σ 3598:∙ 3578:Σ 3574:∫ 3558:− 3426:≡ 3280:α 3243:α 3192:α 3107:≡ 3080:≡ 3077:α 3023:α 2952:α 2715:viscosity 2695:≫ 2653:≈ 2611:≪ 2562:≡ 2551:π 2515:θ 2505:π 2495:ρ 2492:μ 2483:λ 2468:absolute 2456:≡ 2453:θ 2428:≡ 2401:≡ 2385:speed of 2373:≡ 2368:θ 2320:λ 2305:≡ 2271:μ 2261:ρ 2245:≡ 2196:≡ 2147:σ 2125:∙ 2092:∙ 2071:− 2060:Σ 2056:∫ 2040:− 2004:≡ 1980:σ 1958:∙ 1925:∙ 1904:− 1893:Σ 1889:∫ 1873:− 1837:≡ 1815:σ 1793:∙ 1760:∙ 1739:− 1728:Σ 1724:∫ 1708:− 1672:≡ 1623:≡ 1590:∞ 1582:− 1559:≡ 1532:≡ 1487:ρ 1465:≡ 1431:≡ 1406:≡ 1381:≡ 1291:− 1245:≡ 1242:Σ 1220:≡ 1193:≡ 1166:≡ 1160:Δ 1130:σ 1104:Δ 1101:− 1093:Σ 1089:∫ 938:Analogies 884:angle of 187:nose up. 102:in three 73:July 2018 57:summarize 12974:Avionics 12888:Handling 12845:Also at 12593:See also 9618:downward 8905:yaw axis 8211:anhedral 5795:inertial 5791:pressure 863:sideslip 709:, where 707:x",y",z" 663:, where 545:in the − 534:in the − 12735:improve 7262:phugoid 7255:Phugoid 7249:Phugoid 5553:is the 4916:impulse 4900:phugoid 4863:Discuss 4710:is the 3064:where: 2341:where: 1514:surface 1365:where: 1147:where: 981:Takeoff 931:(φ = 0) 825:turning 795:Heading 734:by the 697:by the 638:by the 590:z-y'-x" 12931: 12895: 12867: 12803: 12724:, but 12655:JSBSim 11903:(roll) 11501:where 9997:(roll) 9654:Since 9614:upward 8322:where 7899:where 5549:where 5198:forces 5192:, the 5134:where 4920:damped 4692:where 1522:planes 1035:thrust 1031:weight 886:attack 810:angle 798:angle 779:angle 767:angle 755:angle 738:angle 701:angle 642:angle 504:axes. 299:Three 12839:1 May 12833:(PDF) 12826:(PDF) 12770:Notes 11810:(yaw) 9858:(yaw) 4857:into 4855:moved 4731:M = 1 3760:and C 2387:sound 2165:and C 990:flaps 865:angle 820:angle 764:Pitch 713:' 699:pitch 692:' 684:' 677:' 660:' 655:' 650:' 325:north 265:pitch 216:pitch 112:pitch 12929:ISBN 12893:ISBN 12865:ISBN 12841:2015 12801:ISBN 12549:> 11931:and 11686:and 8209:had 8205:The 8125:Or: 7452:and 7266:SSPO 7169:< 5898:Or: 5555:mass 4745:see 4590:The 3141:and 3139:Mach 3129:and 3127:drag 2890:< 2859:< 2853:< 2822:< 2816:< 2785:< 2779:< 2748:< 2742:< 1518:wing 1443:Lift 1393:Drag 1003:(or 915:β, ψ 849:μ, φ 843:γ, θ 837:σ, ψ 818:Bank 776:Roll 736:roll 732:axis 723:and 711:y"=y 695:axis 680:and 665:z'=z 636:axis 621:and 500:and 337:east 229:roll 158:drag 118:and 116:roll 10649:If 6757:in 5863:sin 5829:cos 5754:cos 5664:sin 5514:cos 5452:sin 5344:sin 5282:cos 5099:sin 5044:cos 4865:) 4861:. ( 3764:. C 3446:max 2850:1.2 2825:1.2 2813:0.8 2788:0.8 2776:0.3 2751:0.3 752:Yaw 640:yaw 248:yaw 202:yaw 120:yaw 96:air 12965:: 12812:^ 11935:: 11396:mu 11363:. 11205:. 6777:: 5983:: 4902:. 4737:. 4708:p0 3770:Di 3766:D0 3762:Di 3758:D0 2879:: 2842:: 2805:: 2768:: 2731:: 2721:. 1524:) 1041:. 1033:, 1021:. 923:, 874:,y 868:β: 800:σ: 781:φ: 769:θ: 757:ψ: 730:x" 725:z" 721:y" 658:,z 653:,y 523:. 303:, 148:A 114:, 12937:. 12901:. 12873:. 12849:. 12843:. 12760:) 12754:( 12749:) 12745:( 12731:. 12567:r 12563:L 12553:N 12544:r 12540:N 12530:L 12470:) 12465:r 12461:N 12455:p 12451:L 12442:p 12438:N 12432:r 12428:L 12424:( 12415:Y 12408:) 12399:L 12393:p 12389:N 12376:N 12370:p 12366:L 12362:( 12359:U 12356:m 12351:) 12346:r 12342:L 12332:N 12323:r 12319:N 12309:L 12305:( 12299:g 12296:m 12293:= 12287:t 12284:d 12276:d 12236:t 12233:d 12225:d 12216:) 12207:L 12201:p 12197:N 12184:N 12178:p 12174:L 12170:( 12165:) 12156:N 12150:r 12146:L 12137:r 12133:N 12123:L 12119:( 12113:= 12110:p 12076:t 12073:d 12065:d 12056:) 12047:L 12041:p 12037:N 12024:N 12018:p 12014:L 12010:( 12005:) 12000:r 11996:N 11990:p 11986:L 11977:p 11973:N 11967:r 11963:L 11959:( 11953:= 11933:p 11891:0 11888:= 11885:p 11880:p 11876:L 11872:+ 11866:t 11863:d 11855:d 11847:r 11843:L 11839:+ 11827:L 11798:0 11795:= 11792:p 11787:p 11783:N 11779:+ 11773:t 11770:d 11762:d 11754:r 11750:N 11746:+ 11734:N 11699:p 11695:Y 11672:r 11668:Y 11637:U 11634:g 11629:+ 11626:p 11620:U 11617:m 11611:p 11607:Y 11601:+ 11598:r 11592:U 11589:m 11583:r 11579:Y 11573:+ 11564:U 11561:m 11551:Y 11545:= 11539:t 11536:d 11528:d 11507:U 11503:g 11479:U 11476:g 11471:+ 11465:U 11462:m 11458:Y 11453:= 11447:t 11444:d 11436:d 11394:( 11336:p 11332:L 11307:. 11304:p 11299:p 11295:L 11291:= 11285:t 11282:d 11277:p 11274:d 11268:A 11220:Y 11187:L 11164:p 11160:N 11133:L 11106:N 11083:p 11079:L 11052:N 11046:p 11042:L 11013:p 11009:L 10975:A 10970:p 10966:L 10955:A 10952:E 10945:C 10940:p 10936:N 10926:C 10917:N 10909:A 10904:p 10900:L 10889:C 10884:p 10880:N 10872:A 10863:L 10820:A 10815:p 10811:L 10800:A 10797:E 10790:C 10785:p 10781:N 10771:A 10766:r 10762:L 10754:C 10749:p 10745:N 10734:C 10729:r 10725:N 10717:A 10712:p 10708:L 10651:p 10630:p 10627:= 10621:t 10618:d 10610:d 10553:0 10550:= 10543:) 10537:C 10532:p 10528:N 10520:A 10511:L 10500:C 10491:N 10483:A 10478:p 10474:L 10467:( 10457:t 10454:d 10446:d 10439:) 10433:A 10428:r 10424:L 10416:C 10411:p 10407:N 10396:C 10391:r 10387:N 10379:A 10374:p 10370:L 10363:( 10359:+ 10351:2 10347:t 10343:d 10333:2 10329:d 10321:) 10315:A 10310:p 10306:L 10295:A 10292:E 10285:C 10280:p 10276:N 10269:( 10238:p 10233:p 10229:L 10225:+ 10219:t 10216:d 10208:d 10200:r 10196:L 10180:L 10176:= 10168:2 10164:t 10160:d 10150:2 10146:d 10139:E 10118:p 10113:p 10109:N 10105:+ 10099:t 10096:d 10088:d 10080:r 10076:N 10060:N 10056:= 10048:2 10044:t 10040:d 10030:2 10026:d 10019:C 9985:p 9980:p 9976:L 9972:+ 9966:t 9963:d 9955:d 9947:r 9943:L 9927:L 9923:= 9917:t 9914:d 9909:r 9906:d 9900:E 9891:t 9888:d 9883:p 9880:d 9874:A 9846:p 9841:p 9837:N 9833:+ 9827:t 9824:d 9816:d 9808:r 9804:N 9788:N 9784:= 9778:t 9775:d 9770:p 9767:d 9761:E 9752:t 9749:d 9744:r 9741:d 9735:C 9703:r 9697:= 9691:t 9688:d 9680:d 9660:r 9632:r 9628:L 9595:L 9563:p 9559:L 9520:r 9516:L 9468:L 9438:r 9434:N 9402:r 9398:N 9368:p 9364:N 9337:p 9333:N 9301:p 9297:N 9263:N 9224:N 9186:r 9182:Y 9143:p 9139:Y 9105:Y 9078:Y 9042:Y 8971:t 8968:d 8963:r 8960:d 8954:E 8945:t 8942:d 8937:p 8934:d 8928:A 8925:= 8922:L 8881:t 8878:d 8873:r 8870:d 8864:C 8861:+ 8855:t 8852:d 8847:p 8844:d 8838:E 8832:= 8829:m 8826:d 8821:2 8817:y 8813:+ 8808:2 8804:x 8794:t 8791:d 8786:r 8783:d 8777:+ 8774:m 8771:d 8768:z 8765:x 8756:t 8753:d 8748:p 8745:d 8736:= 8733:N 8704:m 8696:2 8692:y 8685:t 8682:d 8677:r 8674:d 8644:m 8636:2 8632:x 8625:t 8622:d 8617:r 8614:d 8608:+ 8605:m 8599:z 8596:x 8590:t 8587:d 8582:p 8579:d 8570:= 8564:t 8561:d 8556:v 8553:d 8547:x 8544:m 8507:x 8501:t 8498:d 8493:r 8490:d 8484:+ 8481:z 8475:t 8472:d 8467:p 8464:d 8455:= 8449:t 8446:d 8441:v 8438:d 8400:r 8397:x 8394:+ 8391:z 8388:p 8382:= 8379:v 8305:r 8296:U 8293:m 8289:Y 8284:= 8278:t 8275:d 8267:d 8165:g 8160:2 8153:U 8147:2 8141:= 8138:T 8103:2 8099:U 8092:2 8088:g 8084:2 8077:= 8072:T 8065:2 8030:U 8026:g 8023:m 8020:2 8014:= 8009:U 8005:W 8002:2 7996:= 7991:u 7987:Z 7959:L 7955:c 7932:w 7928:S 7882:W 7879:= 7874:w 7870:S 7864:L 7860:c 7854:2 7850:U 7841:2 7838:1 7833:= 7830:Z 7800:0 7797:= 7794:u 7788:U 7785:m 7780:g 7775:u 7771:Z 7758:t 7755:d 7750:u 7747:d 7739:m 7734:u 7730:X 7716:2 7712:t 7708:d 7703:u 7698:2 7694:d 7657:g 7654:m 7648:u 7643:u 7639:X 7635:= 7629:t 7626:d 7621:u 7618:d 7612:m 7591:u 7586:u 7582:Z 7575:= 7569:t 7566:d 7558:d 7552:U 7549:m 7519:u 7515:Z 7492:u 7488:X 7465:u 7461:Z 7438:u 7434:X 7401:g 7398:m 7392:X 7389:= 7383:t 7380:d 7375:u 7372:d 7366:m 7333:Z 7327:= 7321:t 7318:d 7310:d 7304:U 7301:m 7222:M 7199:q 7195:M 7188:U 7185:m 7175:Z 7160:M 7116:U 7113:m 7107:q 7103:Z 7072:0 7069:= 7062:) 7058:) 7052:U 7049:m 7043:q 7039:Z 7033:+ 7030:1 7027:( 7022:B 7013:M 7002:B 6997:q 6993:M 6984:U 6981:m 6971:Z 6964:( 6960:+ 6954:t 6951:d 6943:d 6936:) 6930:B 6914:M 6908:) 6902:U 6899:m 6893:q 6889:Z 6883:+ 6880:1 6877:( 6874:+ 6869:B 6864:q 6860:M 6854:+ 6848:U 6845:m 6835:Z 6828:( 6816:2 6812:t 6808:d 6798:2 6794:d 6722:B 6706:M 6700:+ 6692:B 6683:M 6677:+ 6674:q 6669:B 6664:q 6660:M 6654:= 6648:t 6645:d 6640:q 6637:d 6601:U 6598:m 6588:Z 6582:+ 6579:q 6575:) 6568:U 6565:m 6559:q 6555:Z 6549:+ 6546:1 6542:( 6538:= 6532:t 6529:d 6521:d 6477:M 6438:M 6403:q 6399:M 6357:M 6340:. 6322:Z 6299:q 6295:Z 6260:Z 6203:B 6200:M 6195:= 6189:t 6186:d 6181:q 6178:d 6151:U 6148:m 6144:Z 6139:+ 6136:q 6133:= 6127:t 6124:d 6116:d 6088:q 6085:= 6079:t 6076:d 6068:d 6031:2 6027:t 6023:d 6013:2 6009:d 6002:B 5999:= 5996:M 5952:t 5949:d 5944:) 5932:( 5929:d 5923:U 5920:m 5914:= 5911:Z 5881:) 5869:( 5858:f 5854:X 5850:+ 5847:) 5835:( 5824:f 5820:Z 5813:= 5810:Z 5772:) 5760:( 5748:t 5745:d 5740:) 5728:( 5725:d 5719:U 5716:m 5713:= 5708:f 5704:Z 5682:) 5670:( 5658:t 5655:d 5650:) 5638:( 5635:d 5629:U 5626:m 5620:= 5615:f 5611:X 5582:t 5579:d 5574:U 5571:d 5565:m 5551:m 5532:) 5520:( 5508:t 5505:d 5500:) 5488:( 5485:d 5479:U 5476:m 5473:+ 5470:) 5458:( 5446:t 5443:d 5438:U 5435:d 5429:m 5426:= 5420:t 5417:d 5410:f 5406:w 5402:d 5396:m 5393:= 5388:f 5384:Z 5362:) 5350:( 5338:t 5335:d 5330:) 5318:( 5315:d 5309:U 5306:m 5300:) 5288:( 5276:t 5273:d 5268:U 5265:d 5259:m 5256:= 5250:t 5247:d 5240:f 5236:u 5232:d 5226:m 5223:= 5218:f 5214:X 5174:f 5170:w 5161:, 5147:f 5143:u 5117:) 5105:( 5096:U 5093:= 5088:f 5084:w 5062:) 5050:( 5041:U 5038:= 5033:f 5029:u 4872:) 4868:( 4722:∞ 4720:M 4706:C 4698:p 4696:C 4671:| 4665:2 4654:M 4645:1 4641:| 4634:0 4631:p 4627:C 4621:= 4616:p 4612:C 4568:2 4564:) 4558:M 4555:L 4551:C 4542:L 4538:C 4534:( 4531:k 4528:+ 4523:M 4520:D 4516:C 4512:= 4507:D 4503:C 4477:0 4474:= 4471:L 4468:C 4464:) 4458:D 4454:C 4450:( 4442:M 4439:D 4435:C 4431:= 4426:n 4423:i 4420:m 4417:D 4413:C 4383:0 4380:D 4376:C 4372:= 4367:0 4364:= 4361:L 4358:C 4354:) 4348:D 4344:C 4340:( 4337:= 4332:n 4329:i 4326:m 4323:D 4319:C 4283:0 4280:D 4276:C 4272:= 4267:x 4264:a 4261:m 4258:E 4254:) 4248:i 4245:D 4241:C 4237:( 4226:k 4221:0 4218:D 4214:C 4206:= 4201:x 4198:a 4195:m 4192:E 4188:) 4182:L 4178:C 4174:( 4159:0 4156:D 4152:C 4148:k 4143:2 4139:1 4133:= 4128:x 4125:a 4122:m 4118:E 4111:{ 4097:2 4092:L 4088:C 4084:k 4081:+ 4076:0 4073:D 4069:C 4062:L 4058:C 4051:= 4048:E 4020:2 4015:L 4011:C 4007:k 4004:+ 3999:0 3996:D 3992:C 3988:= 3983:D 3979:C 3970:2 3965:L 3961:C 3957:k 3954:= 3949:i 3946:D 3942:C 3899:i 3896:D 3892:C 3882:0 3879:= 3874:L 3870:C 3865:) 3859:D 3855:C 3851:( 3848:= 3843:0 3840:D 3836:C 3829:{ 3822:i 3819:D 3815:C 3811:+ 3806:0 3803:D 3799:C 3795:= 3790:D 3786:C 3725:d 3718:w 3714:i 3705:n 3701:) 3696:p 3692:C 3685:( 3669:S 3666:1 3657:= 3650:S 3647:q 3643:D 3637:= 3632:p 3629:D 3625:C 3614:d 3607:w 3603:i 3594:t 3588:f 3584:C 3567:S 3564:1 3555:= 3548:S 3545:q 3541:D 3535:= 3530:f 3527:D 3523:C 3516:{ 3509:p 3506:D 3502:C 3498:+ 3493:f 3490:D 3486:C 3482:= 3477:D 3473:C 3457:D 3450:L 3423:) 3420:e 3417:R 3414:, 3411:M 3408:, 3403:L 3399:C 3395:( 3390:D 3386:C 3382:= 3377:D 3373:C 3322:D 3318:C 3312:L 3308:C 3301:= 3298:) 3295:e 3292:R 3289:, 3286:M 3283:, 3277:( 3274:E 3271:= 3268:E 3261:) 3258:e 3255:R 3252:, 3249:M 3246:, 3240:( 3235:L 3231:C 3227:= 3222:L 3218:C 3210:) 3207:e 3204:R 3201:, 3198:M 3195:, 3189:( 3184:D 3180:C 3176:= 3171:D 3167:C 3160:{ 3104:P 3047:) 3044:P 3041:, 3038:e 3035:R 3032:, 3029:M 3026:, 3020:( 3015:f 3011:C 3007:= 3002:f 2998:C 2976:) 2973:P 2970:, 2967:e 2964:R 2961:, 2958:M 2955:, 2949:( 2944:p 2940:C 2936:= 2931:p 2927:C 2893:M 2887:5 2862:5 2856:M 2819:M 2782:M 2745:M 2739:0 2698:1 2688:e 2685:R 2681:M 2656:1 2646:e 2643:R 2639:M 2614:1 2604:e 2601:R 2597:M 2555:2 2548:k 2536:e 2533:R 2529:M 2523:= 2512:R 2509:2 2486:= 2425:R 2398:k 2365:R 2362:k 2357:= 2354:a 2323:l 2314:= 2311:n 2308:K 2267:l 2264:V 2254:= 2251:e 2248:R 2211:a 2208:V 2202:= 2199:M 2167:f 2163:p 2144:d 2140:] 2134:w 2130:k 2121:t 2115:f 2111:C 2107:+ 2101:w 2097:k 2088:n 2084:) 2079:p 2075:C 2068:( 2065:[ 2049:S 2046:1 2037:= 2030:S 2027:q 2023:L 2017:= 2012:L 2008:C 1977:d 1973:] 1967:w 1963:j 1954:t 1948:f 1944:C 1940:+ 1934:w 1930:j 1921:n 1917:) 1912:p 1908:C 1901:( 1898:[ 1882:S 1879:1 1870:= 1863:S 1860:q 1856:Q 1850:= 1845:Q 1841:C 1812:d 1808:] 1802:w 1798:i 1789:t 1783:f 1779:C 1775:+ 1769:w 1765:i 1756:n 1752:) 1747:p 1743:C 1736:( 1733:[ 1717:S 1714:1 1705:= 1698:S 1695:q 1691:D 1685:= 1680:d 1676:C 1645:q 1642:f 1636:= 1631:f 1627:C 1596:q 1586:p 1579:p 1572:= 1567:p 1563:C 1535:S 1516:( 1496:2 1492:V 1480:2 1477:1 1471:= 1468:q 1428:L 1403:Q 1378:D 1348:) 1345:L 1340:w 1335:k 1330:+ 1327:Q 1322:w 1317:j 1312:+ 1309:D 1304:w 1299:i 1294:( 1288:= 1283:A 1278:F 1216:f 1189:n 1163:p 1127:d 1123:) 1119:f 1115:+ 1111:n 1107:p 1098:( 1085:= 1080:A 1075:F 925:θ 921:α 903:w 901:y 899:, 896:w 894:x 890:α 876:w 872:w 827:. 740:φ 716:. 703:θ 690:y 682:z 675:x 670:. 667:E 648:x 644:ψ 633:E 631:z 625:E 623:y 618:E 616:x 604:) 549:w 547:z 538:w 536:x 512:E 510:z 508:+ 502:z 498:x 487:w 485:z 483:, 480:w 478:x 473:w 471:y 464:w 462:x 457:w 455:z 448:w 446:x 427:b 425:z 423:, 420:b 418:x 413:b 411:y 404:b 402:x 397:b 395:z 388:b 386:x 364:E 362:y 360:, 357:E 355:x 344:E 342:z 332:E 330:y 320:E 318:x 75:) 71:( 61:. 34:. 20:)

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Knowledge

Aircraft flight dynamics

Index