APPENDIX A
APPENDIX A EQUATIONS OF MOTION USED IN LATERAL-DIRECTIONAL RFS STUDY The side-force and moment equations (ref. 11) a r e written in a set of orthogonal body-axis coordinates. The three equations written in conventional nonlinear form a r e as follows:
Side force -
Rolling moment - Yawing moment - These three equations a r e modified by assuming that, in still air, W a x - V p " " V To further reduce complexity with little loss in accuracy, the following approximations are made: 1. Inertia coupling terms in the moment equations a r e dropped V
2. p = -
V 3.
-CD sin p + C y cos p is approximated by Cy
APPENDIX A
APPENDIX A 4. V is used for u in the side-force equation The side-force equation then becomes m(b+ r V - p w ) = m g s i n c p c o s e + p V 1 2 SCY Dividing by mV,
$ + r - Q - v - g sin cp cos e + E c ~
Incorporating the approximations,
--c + g sin cpcos e + pa! - r
b - 2 m Y v
The rolling-moment equation is The yawing-moment equation is The three equations are then linearized about a trim flight condition (V = VT) where the initial p, r, (9, $, p , 6r, and 6a are zero. The resultant linearized perturbation equations are (assuming C y . = C
= C l . = 0)
P ys, P
Side force -
Rolling moment -
APPENDIX A
APPENDIX A Yawing moment - As a further modification to conserve analog equipment necessary for programing, the Euler angular rate relationship is solved for p as p = (s - J , s i n e Substituting the Euler expression for 4, p = (p - (q sin 9 + r cos ‘p) tan e Linearizing this expression and assuming tan eT = tan aT = aT, Differentiating , Substituting p and p into the three equations and rearranging, using the dimensional forms of the aerodynamic stability derivatives, yields the final form of the lateral- directional equations of motion. Note that the equations are now in terms of an earth reference, @ and v, rather than a body axis, p and b, as follows: Side force - Rolling moment - Yawing moment -
APPENDIX A
APPENDIX A By Laplace transforming the equations and using the variable s, the equations can be written as
Side force -
Rolling moment -
Yawing moment -
APPENDIX B
APPENDIX B APPLICATION OF ROOT-LOCUS METHODS TO ANALYSIS O F RESPONSE FEEDBACK LOOPS The root-locus method of analysis (refs. 7 and 8) directly relates feedback gain values and characteristic equation roots. It is these roots that determine the form of the response of the dynamic system. Generally, the characteristic equation for the lateral-directional mode is a fourth-order equation which, for the transport class of aircraft, usually factors into the following form (ref. 5): Each parenthetical term is referred to as a dynamic mode. The second-order term is the IXltch roll mode, and the two first-order terms are the roll and spiral modes, respectively. Adequate descriptions of the physical characteristics of these modes a r e found in most texts on aircraft dynamics. For the analyses of this paper it is sufficient to note that each parenthetical term in the equation in the Laplace variable (frequency domain) has an equivalent form in the time domain, as explained in refer- ences 7 and 8. The Dutch roll is an exponentially varying sinusoidal oscillation of the general form where h is an arbitrary phase angle. The roll and spiral modes are, respectively, t t -- exponential functions of the form e - F and e TS . The root-locus technique provides a coiwenier,t xethod of plotting the roots of the characteristic equation as a function of the particular feedback gain used. The resultant curve, presented on the s-plane, yields information about the dynamic stability of each of the modes. Figure 25 shows the three modes and indicates the parameters characteristic of the dynamics of each mode.
The basic aircraft characteristic equation roots are designated by X's and a r e referred to a s poles. The Dutch roll mode appears as two poles because the roots The of the second-order term in the characteristic equation a r e a complex pair.
magnitude of the distance between the origin and one Dutch roll pole is the undamped natural frequency in radians/second. The component of this distance along the imaginary axis is the damped frequency that would be measured on a time history.
The damping ratio of the Dutch roll is the cosine of the angle between the negative real axis and a line drawn from the origin to the pole. The roll and spiral mode time con- stants, in seconds, correspond to the inverse of the distance between the origin and the pole location. ju axis, the resultant time response If a root lies to the left of the is stable; whereas, roots falling to the right of the ju axis a r e unstable. Roots falling on the ju axis are termed neutrally stable.
APPENDIX B
APPENDIX B Graphical techniques used in plotting root-locus diagrams utilize the transfer function numerator roots (zeros) as well as the poles. These zero locations are 0 ' s and are shown herein on root-locus diagrams. The graphical designated by techniques are described in references 7 and 8.
Imaginary axis Stable jw Unstable
Dutch roll I (+)
I t
x
Dutch roll Figure 25.- Lateral-directional m o d e s as displayed on the complex o r s-plane.
APPENDIX C
APPENDIX C TWO-DEGREE -0 F-FREEDOM APPROXIMATION TO
THE DUTCH ROLL WITH THE % LOOP
By choosing only the dominant terms in the equation of motion, the side-force and yawing-moment equations become and -Npp+ (s - N r ) r = N6,6, allowing only p and r motions. NOW letting 6 , = 6rp + M r ,
-N p + (s - N r ) r = Ng 6 + e ) N ( j r r
P r rP o r the characteristic equation is
S2 - [Nr + Yp + ($)N~Js + Np + NrYp + E)(Ng,Yp - NpY6,) = 0
+ - p , the two-degree-of-freedom approximation
In similar fashion, writing 6 , = 6
rP (“p.1
to the Dutch roll can be used to obtain the characteristic equation
APPENDIX D
APPENDIX D 6,
ANALYTICAL EXPRESSION RELATING - TO
B
O F THE DUTCH ROLL The expanded characteristic equation general literal form is easily shown (ref. 5) to be + - 1 + -)s 1 3 Tr T s Comparing the coefficients of this equation with E
s 4 + A s E 3 + - s c 2 + - s + - D = o
A A A which is the general form of a fourth-order algebraic equation where the coefficient of s4 has been made equal to unity, and equating the s3 terms the highest order term of the two equations results in For most transport aircraft, - can be neglected with respect to the other two terms.
T S Also, to a very good approximation and so B
= 2 & p # - Lp
APPENDIX D
APPENDIX D After writing the feedback loop equation as .
and substituting into the lateral-directional equations of motion, the characteristic equation can be found. The coefficient of s4 term (A) is and the coefficient of s3 term (B) is The s4 and s3 coefficients can be sim lified by neglecting small triple-product pxz
, small numbers themselves, appear in terms; inertia terms, since - IXz and -
Ixx I z z
products usually much smaller than other terms in the sum; and terms involving aT, since the trim angle of attack is assumed to be small. Using the remaining significant terms, the first two terms of the characteristic equation are and But, as shown previously,
= 2E$W$ - Lp
A
APPENDIX D
APPENDIX D and, substituting the expressions derived for A and B, 6r and solving for 7 9 Equating the two expressions for P
APPENDIX E
APPENDIX E t APPROXMATION TO THE SPIRAL MODE TIME CONSTANT
WHEN 6a IS USED AS A FEEDBACK LOOP
4p * Reference 5 states that if the spiral mode constant T~ is very large, where D and E are the coefficients of the s1 and so terms, respectively, of the lateral-directional characteristic equation. If the auxiliary feedback equation is written as D, the coefficient of the s1 term, is The E t e r m i s
APPENDIX E
APPENDIX E ?
Neglecting small triple-product terms and assuming that inertia terms are small, ?
since Ixz << 1 for the JetStar,
Iz z
For the JetStar, and
D = LpNp - LpNp - & Lp +
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