Aircraft pitch angle response to elevator inputs

Is it possible to form a simulink block diagram based upon the image below. The code below are the equations for the given variables, where delta eta is the elevator input.

% short period 

% constant variables
rho = 0.905;
S = 64.8;
c = 2.51;
a = 5.3;
h_n = 0.63;
h_fwd = 0.18;
h_ac = 0.27;
I_yy = 136182.4308;
V_bar_t = 0.72;
a_1 = 4.33;
a_2 = 2.16;
a_3 = 0.47;
DWG = 0.4;
d_T = 11.2259;
V_cruise = 138.283;

% damping ratio
zeta = sqrt((rho*S*c)/(8*(h_n-h_fwd)*a*I_yy))*(V_bar_t*a_1*d_T);
% natural frequency
omega_n_cruise = V_cruise*sqrt(((rho*S*c*a*(h_n-h_fwd))/(2*I_yy)));
% damped frequency
omega_d_cruise = omega_n_cruise*sqrt(1-(zeta^2));
% elevator sensitivity
k_eta = -(V_bar_t*a_2)/((h_n - h_fwd)*a);
% change in pitching moment 
delta_C_M = ((h_fwd - h_ac)*delta_C_L) - (V_bar_t*delta_C_L_T);
% change in lift coefficient
delta_C_L = a*delta_theta;
% change in tail lift coefficient
delta_C_L_T = a_1*delta_alpha_T + a_2*delta_eta;
% based on the following relationships 
delta_alpha = delta_theta;
delta_beta = 0;
% change in angle of attack
delta_alpha_T = (delta_theta(1-DWG)) + ((q*d_T)/V_cruise);
% angle of attack
alpha_T = (alpha(1-DWG)) + psi_T;

Answer 1

[Sam Chak]

Hi @Daniel Jackson

If all the equations are correct, you can use the MATLAB Function block to enter each equation.

https://www.mathworks.com/help/simulink/slref/matlabfunction.html

For an example, refer to:

https://www.mathworks.com/help/simulink/ug/what-is-a-matlab-function-block.html

Nevertheless, I recommend simulating the aircraft pitch dynamics in MATLAB before constructing the blocks in Simulink. By the way, I don't see the aircraft pitch dynamics in your code. Don't miss it.

[Daniel Jackson]

Dear @Sam Chak

I've interpreted what needed to be done incorrectly, therefore it's a different set of equations which need to be used. Is there a way to plot the the aircraft pitch angle response to elevator using simulink with these equations

[Daniel Jackson]

@Sam Chak

The image attached below is what I have so far, based of the following code. However the pitch rate and pitch graphs do not look right, do you know where I have gone wrong? Step 1 is: Step time = 1, Initial value = 0 and final value = 5. Step 2 is: Step time = 4, Initial value = 5, and final value = 0. The bottom two are just the opposite of these two.

% short period 

% constant variables
rho = 0.905;
S = 64.8;
c = 2.51;
a = 5.3;
h_n = 0.63;
h_fwd = 0.18;
h_ac = 0.27;
I_yy = 136182.4308;
g = 9.81;
m = 18000;
C_D_0 = 0.011;
k = 0.041263048;
V_bar_t = 0.72;
a_1 = 4.33;
a_2 = 2.16;
a_3 = 0.47;
DWG = 0.4;
d_T = 11.2259;
V_cruise = 138.283;

% damping ratio
zeta = sqrt((rho*S*c)/(8*(h_n-h_fwd)*a*I_yy))*(V_bar_t*a_1*d_T);

% omegas for cruise velocity
omega_n_cruise = V_cruise*sqrt(((rho*S*c*a*(h_n-h_fwd))/(2*I_yy)));

omega_d_cruise = omega_n_cruise*sqrt(1-(zeta^2));

k_eta = (-1*V_bar_t*a_2)/((h_n - h_fwd)*a);

eta_coefficient = k_eta * (omega_n_cruise)^2;
theta = (omega_n_cruise)^2;
theta_dot = 2*zeta*omega_n_cruise;

[Sam Chak]

Hi @Daniel Jackson,

The 2nd-order differential equation of the pitch dynamics that you posted is linear. Thus, you can use a single state-space block to represent that system. The values for the parameters , ζ, and can be calculated by hand or computed in MATLAB.

https://www.mathworks.com/help/simulink/slref/statespace.html

Please write out the state-space model. This should be covered in the Aircraft course.