How Do Length of Pylon and Rotor Disk Radius Impact Angular Frequency and Torsional Stiffness in Whirl Flutter Analysis for a 2 DOF Propeller System?

Question

Actually, I'm having trouble implementing the plot of the flutter region for the whirl prop system with 2 DOF (Influences of structural damping and propeller—pivot point distance on whirl flutter stability) and My plot isn't getting a hump.

The explanation I would like to give about my solution is that first, I calculated the eigenvalues of the system's equation of motion. I did this by writing the characteristic polynomial equation and then examining its roots.

However, What I am most doubtful about is the method of calculating the moment of inertia of the rotor disk, angular frequency and torsional Stiffnesses. But not sure if the length of the pylon and the radius of a rotor disk impact the angular frequency or stiffness.

Attached you can check my MATLAB script.

I would highly appreciate any suggestions you might have

clc clear

% Constants with specific and random values

Dp = 3; %[m] 
Rp = Dp/2; %[m] rotor radius
Ap = pi*(Rp^2); %[m^2] 

m = 100; % total mass of propeller

J0_yaw = 1/4*m*Rp^2; % Mass yaw moment of inertia
J0_pitch = 1/4*m*Rp^2; % Mass pitch moment of inertia
J0x = 1/2*m*Rp^2; %Mass roll moment of inertia

rho = 1.225; %[kg/m^3]

J_av= 0.6;
Omega = 100; %rotational speed


f_pitch= 0.1:0.05:70;
f_yaw= f_pitch;

a = 0.01:0.05:3; %pivot point distance

     unstable = [];

for i = 1:length(f_pitch)
            for l = 1:length(a)

J_pitch = J0_pitch + (m * a(l)^2); J_yaw = J0_yaw + (m * a(l)^2); Jx = J0x;

    % Calculate stiffness for the current frequency

    w_omega_pitch = 2*pi*f_pitch(i);
    w_omega_yaw = w_omega_pitch;

    K_psi = (w_omega_pitch^2) * J_pitch * a(l);
    K_theta = (w_omega_yaw^2) * J_yaw * a(l);

% Propeller rotational speeds
   V=Omega*J_av*Rp; %[rad/s]
   q_inf = 1/2*rho*V^2; %[rad/s]

% Propeller aerodynamics, air density, and aero derivatives
c_y_theta = 0.08;
c_z_theta = -0.26;
c_m_theta = 0.01;
c_n_theta = -0.075;

c_y_q = -0.13;
c_z_q = -0.03;
c_m_q = -0.05;
c_n_q = -0.01;

c_z_psi = c_y_theta;
c_m_psi = -c_n_theta;
c_n_r = c_m_q;
c_z_r = c_y_q;
c_y_psi = -c_z_theta;
c_n_psi = c_m_theta;
c_m_r = -c_n_q;
c_y_r = -c_z_q;

gamma_theta = 0.03;
gamma_psi = 0.03;

D_aero = [-1/2*c_m_q-a(l)^2/Dp^2*c_z_theta, 1/2*(a(l)/Dp)*c_y_q - a(l)/Dp*c_n_theta-a(l)^2/Dp^2*c_y_theta;
              -1/2*(a(l)/Dp)*c_y_q+(a(l)/Dp)*(c_n_theta)+a(l)^2/Dp^2*c_y_theta, -1/2*c_m_q-a(l)^2/Dp^2*c_z_theta];

K_aero = [a(l)/Dp*c_z_theta, c_n_theta + a(l)/Dp*c_y_theta;
              -c_n_theta-a(l)/Dp*c_y_theta, a(l)/Dp*c_z_theta];


M = [J_pitch, 0; 0, J_yaw];
D = [K_psi*gamma_theta/w_omega_yaw, 0; 0, K_theta*gamma_psi/w_omega_pitch];
K = [K_psi, 0; 0, K_theta];
G = [0, Jx*Omega; -Jx*Omega, 0];

Kbar = (K + (q_inf * Ap * Dp * K_aero));
Cbar = D + G + q_inf * Ap * (Dp^2 / V) * D_aero;

M11 = M(1, 1); M22 = M(2, 2);
C11 = Cbar(1, 1); C12 = Cbar(1, 2); C21 = Cbar(2, 1); C22 = Cbar(2, 2);
K11 = Kbar(1, 1); K12 = Kbar(1, 2); K21 = Kbar(2, 1); K22 = Kbar(2, 2);

P0 = M11*M22;
P1 = (- C11*M22*1j - C22*M11*1j);
P2 = (C11*C22*(1j)^2 - K22*M11 - K11*M22 - C12*C21*(1j)^2);
P3 = (C11*K22*1j - C12*K21*1j - C21*K12*1j + C22*K11*1j);
P4 = K11*K22 - K12*K21;

P = roots([P0, P1, P2, P3, P4]);

        r = 1 * P;

%Flutter for k = 1:length(r) if (real(r(k)) > 0) if (imag(r(k)) <= 0) unstable = [unstable; a(l)/Rp, w_omega_pitch/Omega]; end end end

    end
    end

% Plot the Flutter and divergence maps figure; hold on; scatter(unstable(:,1), unstable(:,2), 'filled'); xlabel('a/Rp'); ylabel('w/omega'); title('pivot point distance on whirlflutter stability'); legend('Flutter area'); hold off;

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Answer 1

It is too complex to be answered here. But very roughly, just an intuitive qualitative estimate here.

The longer the pilon (same cross section), the greater the period of vibration of the assembly, the reduced stiffness, and the larger the impact forces on the wing and the bigger the effect on the aerodynamics of the airplane.

If we consider the restoring force of the pilon as gravity on the propeller acting as a gyro, the longer the pilon, the longer the procession period, and the larger the radius of the procession.

Radius of a rotor disk is related by a power of two to its moment of inertia, $I= mr^2 \ $. Its angular momentum $L=I\omega \ $is larger and its frequency smaller; it affects more torque when it flutters or has unsymmetrical torque loading due to the AOA of the wing.