Derivation using "Infenitesmal Strain Theory" from the Rod Theory

% Derive the representation of force in the body frame.

clc; clear;

syms alpha r F_e M_e l_t GJ EI GA EA M_y M_z F_y F_z Delta_theta_h;

assume([alpha r F_e M_e l_t GJ EI GA EA M_y M_z F_y F_z Delta_theta_h], 'real')

assume([alpha r F_e M_e l_t GJ EI GA EA M_y M_z F_y F_z Delta_theta_h], 'positive')

pi = sym(pi);

K = diag([ EI, EI, GJ, GA, GA, EA]) % this is the K matrix

K = 

R_0 = simplify(Rz(pi)*Rx(-pi/2 - alpha))

R_0 = 

p_0 = [r; 0; 0];

Adg0 = [R_0 zeros(3); skew(p_0)*R_0 R_0];

% the negative or possitive depending on the positive or negative face.

Notice that the and are downward in the global brame.

W = Adg0'*We + [0;0;Delta_theta_h*GJ;0;0;0 ] % the wrench in the body frame.

W = 

D_xi = Adg0*(K^-1*W);

D_l = simplify(l_t* D_xi(6)) % since Dl = l-l_star

D_l = 

D_l = collect(D_l, [F_e, Delta_theta_h])

D_l =