APPENDIX B: DYNAMICS AND CONTROL



APPENDIX B: DYNAMICS AND CONTROL

B.1 Empennage Sizing

The vertical and horizontal tail volume coefficients, CVT and CHT are given by the following equations [Ref. 2.1], where the subscripts HT, VT, and W refer to horizontal tail, vertical tail and wing, respectively.

L = moment arm

S = planform area

b = span

c = mean aerodynamic chord

Table B.1: Tail Volume Coefficient Determination

[pic]

After determining suitable values for the tail volume coefficients based on past AAE 451 designs (see Table B.1), the tail surface areas were determined by plugging in the appropriate values for the wing dimensions given by the results of the sizing code (see Appendix G.1). The moment arms were determined according to a fuselage length of 5.9 ft. The fuselage was constrained to this length to ensure that it would fit into a compact car, as required by the mission specification.

B.2 Static Margin, Center of Gravity, and Aerodynamic Center

The static margin desired for the SID-5 was 10%. Static margin is defined by the following equation [Ref. 2.5].

xac = position of the aircraft aerodynamic center, or neutral point nondimensionalized by the wing chord

xcg = position of the center of gravity nondimensionalized by the wing chord

The position of the aerodynamic center was supplied by the aerodynamic analysis, so the position of the center of gravity was solved for given the desired static margin of 10%. Once this location was known, the internal equipment was placed in the fuselage to achieve this center of gravity location.

B.3 Static Stability Analysis And Aircraft Dynamic Model Construction

The following code was used to compute pertinent static stability derivatives for the static stability analysis as well as for building up the dynamic model for the purposes of implementing feedback. The dynamic model construction part of the code basically calculates the parameters needed to build the short period approximation of the aircraft transfer function as described by Roskam [Ref. 2.5].

%AAE 451 - Dynamic Modeling Code

%Applicable References (note the first number in parentheses is the number by which the given reference will be called in the code comments)

% (1)[Ref. 1.2] Roskam’s Airplane Design Series

% (2)[Ref. 2.1] Aircraft design: A Conceptual Approach 3rd Ed. by Daniel P. Raymer

% (3)[Ref. 2.6] Mark Peters Master's Thesis

% (4)[Ref. 2.5] Roskam’s Airplane Flight Dynamics and Automatic Flight Controls

clear all

close all

%Input necessary parameters

rho=.002308; %Air density at West Lafayette, IN [slugs/ft^3]

U1=25; %Cruise flight speed [ft/s]

q1=0.5.*rho.*(U1.^2); %Dynamic pressure [lbs/ft^2]

Xac_h=3.684; %Nondimensional station of the HT aero. center (rel. to nose, divided by wing chord)

E=0.4; %Elevator Flap-to-chord ratio

tau_e=0.6; %elevator effectiveness (Calculated from Ref (1) p. 61)

VHT=0.45; %HT volume coefficient

CL_alpha_h=3.3218; %Lift curve slope of the HT (calculated according to NACA 0006 chars. using Ref (3))

eta_h=0.9; %Dynamic pressure ratio at the HT (Assumed 0.9 to account for large amount of interference from wings at tail)

S_h=3.336; %Planform Area of the horizontal tail [ft^2]

S=20.09; %Planform Area of the wings [ft^2]

c=1.511; %Mean geometric chord [ft]

%Xcg=.742; %Nondimensional station of the aircraft CG (relative to nose, divided by wing chord)

deps_dalph=.56; %Downwash derivative (change in downwash at HT due to change in angle of attack)

SM=0.10; %Static Margin Desired

CL_alpha=3.94; %Airplane lift curve slope

CL_alpha_w=5.61; %Wing lift curve slope [rad^-1]

CD1=0.1204; %Airplane Drag coeff. at flight speed U1

Xac_wingbody=1.05; %Station of wing-body aero. center (assumed to be at wing aero. center (=.23c_wing))

m=0.2950; %Airplane mass [slugs]

I_yy=2.2; %Pitch moment of inertia [slugs-ft^2]

%Calculate the non-dimensional static stability derivatives

%Needed for Trim code

CL_ih = CL_alpha_h*eta_h*(S_h/S);

Cm_ih = -1.*CL_alpha_h.*eta_h.*VHT;

CL_ih_deg=CL_ih.*pi/180;

Cm_ih_deg=Cm_ih.*pi/180;

Cm_delta_e = -1.*CL_alpha_h.*eta_h.*VHT.*tau_e; %Change in mom. coeff. due to elev. defl.

Cm_delta_e_deg=Cm_delta_e*pi/180;

CL_delta_e = CL_alpha_h.*eta_h.*(S_h./S).*tau_e; %Change in lift coeff. due to elev. defl.

CL_delta_e_deg=CL_delta_e*pi/180;

CL_h_delta_e = (CL_alpha_h/pi)*((acos(1-2*E))+2*sqrt(E*(1-E)));

CL_h_delta_e_deg=CL_h_delta_e*pi/180;

%This neutral point (airplane aero. center) calculation is taken from Ref. (3)

Xnp = (Xac_wingbody + ((CL_alpha_h./CL_alpha_w).*eta_h.*(S_h./S).*Xac_h.*(1-deps_dalph)))./(1 + (CL_alpha_h./CL_alpha_w).*eta_h.*(S_h./S).*(1-deps_dalph));

Xcg = Xnp - SM;

Cm_alpha = -CL_alpha.*(Xnp - Xcg); %Change in mom. coeff. due to change in angle of attack

Cm_alphadot = -2.*CL_alpha_h.*eta_h.*VHT.*(Xac_h - Xcg).*deps_dalph;

%Change in mom. coeff. due to rate of change of angle of attack

Cm_q = -2.2*CL_alpha_h.*eta_h.*VHT.*(Xac_h - Xcg); %Change in mom. coeff. due to pitch rate

%Calculate Dimensional Derivatives

M_delta_e = (q1.*S.*c.*Cm_delta_e)./(I_yy); %Pitch angular accel. per unit elev. defl. [(rad/s^2)/rad]

Z_delta_e = -(q1.*S.*CL_delta_e)./m; %Vertical accel. per unit elev. defl. [(ft/s^2)/rad]

M_alphadot = (q1.*S.*(c.^2).*Cm_alphadot)./(2.*I_yy.*U1); %Pitch ang. accel. per unit rate of change of angle of attack [(rad/s^2)/rad/s]

M_alpha = (q1.*S.*c.*Cm_alpha)./I_yy; %Pitch ang. accel. per unit angle of attack [(rad/s^2)/rad]

Z_alpha = (-q1.*S.*(CL_alpha+CD1))./m; %Vert. accel. per unit angle of attack [(ft/s^2)/rad]

M_q = (q1.*S.*(c.^2).*Cm_q)./(2.*I_yy.*U1); %Pitch ang. accel. per unit pitch rate [(rad/s^2)/rad/s]

%Calculate elements of the Short Period Mode Transfer function between from elevator delf. to pitch rate. These equation come from Ref (4)

n1=((U1.*M_delta_e)+(Z_delta_e.*M_alphadot))./U1; %numerator term multiplying 's^1'

n0=((M_alpha.*Z_delta_e)-(Z_alpha.*M_delta_e))./U1; %remaining numerator term multiplying 's^0'

w_n_sp=sqrt(((Z_alpha.*M_q)./U1)-M_alpha); %approx. short period nat'l freq. [rad/s]

damp_sp=-(M_q+(Z_alpha./U1)+M_alphadot)./(2.*w_n_sp); %approx. short period damping ratio

d2=1;

d1=2.*damp_sp.*w_n_sp;

d0=(w_n_sp.^2);

%Set up for root locus

num=[-n1 -n0];

den=[d2 d1 d0];

figure

rlocus(num,den)

%Yaw Derivatives

eta_v=0.9;

VVT=0.04;

CL_alpha_v=CL_alpha_h;

tau_r=0.6;

S_v=1.3;

b=6.6;

L_v=5.6-(.6637.*c);

CnBeta = eta_v*VVT*CL_alpha_v - 0.05895*(Sfs*ful/Sref/bw);

Cndelr = -CL_alpha_v*tau_r*eta_v*S_v*L_v/S/b;

B.4 Verification of Horizontal and Vertical Tail Sizes

The following code was used to perform a Class II analysis on the horizontal and vertical tail sizes in order to verify that they were large enough. The methods described by Roskam [Ref. 1.2] are used for the vertical tail analysis while the horizontal tail analysis is original.

clear all

close all

%Longitudinal X - Plot

VHT=0.45; %HT volume coeff.

SHT=2.5:0.01:5; %Horizontal Tail Area Range we want to test [ft^2]

l_fuse=5.85; %Fuselage Length [ft]

S=20.09; %Planform Area of the wings [ft^2]

c=1.511; %Mean geometric chord [ft]

b=6.6483;

Xac_wingbody=1.05; %Station of aero. center for wing body (at 0.23Cwing)[nondim]

CL_alpha=3.94; %Airplane lift curve slope

CL_alpha_w=5.61; %Wing lift curve slope [rad^-1]

CL_alpha_h=3.3218;

eta_h=0.9;

deps_dalph=0.56; %Downwash derivative (change in downwash at HT due to change in angle of attack)

SM_desired=.10;

%Horizontal Tail Parameters

lamdah=0.6; %found from Raymer p85

ARh=3; %found from Raymer p85

b_htail=sqrt(ARh.*SHT);

Croot_htail=2.*SHT./(b_htail.*(1+lamdah));

Ctip_htail=lamdah.*Croot_htail;

Cbar_htail=(2./3).*Croot_htail.*(1+lamdah+(lamdah.^2))./(1+lamdah);

sweep_leh=atan((Croot_htail-Ctip_htail)./(b_htail./2)); %leading edge sweep (rad)

sweep_qch=atan((tan(sweep_leh))- ((1-lamdah)./(ARh.*(1+lamdah)))); %sweep for the quarter chord (rad)

[x,num]=size(SHT);

L_htail_req=((VHT.*c.*S)./SHT)./c; %Moment Arm required by HT to achieve desired tail volume coeff. [ft]

for i = 1:num

for j=1:5

Xcg(j,i)=.61; %guess cg location [ft]

end

end

for i=1:5

L_htail_max_poss(i,:)=(l_fuse-(Croot_htail-(0.4.*Croot_htail)-(Croot_htail-0.75.*Cbar_htail))-(Xcg(i,:).*c))./c; %nondimensional

Xac_h(i,:)=L_htail_max_poss(i,:)+Xcg(i,:); %nondimensional

Xnp(i,:)=(Xac_wingbody + ((CL_alpha_h./CL_alpha_w).*eta_h.*(SHT./S).*Xac_h(i,:).*(1-

deps_dalph)))./(1 + (CL_alpha_h./CL_alpha_w).*eta_h.*(SHT./S).*Xac_h(i,:).*(1-deps_dalph));

SM(i,:)=Xnp(i,:)-Xcg(i,:);

if abs(SM(i,:)-SM_desired)= 0.057 rad-1 corresponds to a vertical tail area of around 0.75 square feet. This is well below the design point of 1.3 square feet, indicating that the vertical tail was adequate to ensure lateral stability according to the weathercock stability criterion. Weathercock stability refers to the tendency of a plane to point into the direction of relative wind. From this analysis it could be argued that the vertical tail size could have been reduced. However, it was decided to leave the vertical tail size at 1.3 square feet to be conservative, trading extra weathercock stability for a minor increase in drag and weight.

B.4 Trim Analysis

The following code was used to construct the aircraft trim diagram, according to the method described by Roskam [Ref. 2.5].

%Trim Diagram Plotting Script--K. Hout

clear all

%all values use degrees

alpha=-3:0.5:11;

Xcg=1.0487;

Xacw=1.05;

Xach=3.684;

c=1.511;

S=20.09;

etah=0.9;

Sh=3.34;

depsdalph=0.56; %per deg

Cmoaf=0.0115; %2D Cmo of the airfoil

Cm_alpha=-0.0069;

Cm_ih=-0.0235;

Cm_delta_e=-0.0141;

CLo=0.5;

CL_alpha=0.0688;

CL_ih=0.0087;

CL_delta_e=0.0052;

CL_alpha_h=3.3218*pi/180; %per deg

CLo_h=0;

CL_h_delta_e=0.0434; %per deg

inc=0; %HT incidence

incw=3; %wing incidence

AR=2.2;

Cmw=Cmoaf.*(AR/(AR+2)); %Raymer 16.16

Kf=.005;

Wf=.4167;

Lf=5.85;

Cm_alpha_fus=(Kf*(Wf^2)*Lf)./(c*S); %per deg (Raymer 16.22)

%%NEED TO MOVE CMo to CG

for delta_e=-20

CL=CLo + CL_alpha.*(alpha+incw) + CL_ih.*inc + CL_delta_e.*delta_e;

CLh=CLo_h + CL_alpha_h*(alpha+inc-depsdalph*alpha) + CL_h_delta_e.*delta_e;

CM=(CL.*(Xcg-Xacw)) + Cmw + Cm_alpha_fus*alpha - (etah.*(Sh./S).*CLh.*(Xach-Xcg)); %(Raymer 16.7)

figure(1)

plot(alpha,CL,'-')

hold

figure(2)

plot(CM,CL,'-')

hold

end

for delta_e=-15

CL=CLo + CL_alpha.*(alpha+incw) + CL_ih.*inc + CL_delta_e.*delta_e;

CLh=CLo_h + CL_alpha_h*(alpha+inc-depsdalph*alpha) + CL_h_delta_e.*delta_e;

CM=(CL.*(Xcg-Xacw)) + Cmw + Cm_alpha_fus*alpha - (etah.*(Sh./S).*CLh.*(Xach-Xcg)); %(Raymer 16.7)

%figure(1)

%plot(alpha,CL,' ................
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