Transform the Wrench to the Body Frame

% Derive the representation of force in the body frame.

clc; clear;

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

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

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

pi = sym(pi);

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

K = 

R_0 = Rz(pi)*Rx(-pi/2)*Rx(-alpha)% This is R_0

R_0 = 

p_0 = [r; 0; 0];

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

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

Fe = [0;0;F_e];

Me = [0;0;M_e];

We = [Me; Fe]; %

W = Ad'*We %+ [0;0;tau_a;0;0;0 ] % the wrench in the body frame.

W = 

Derivation of TCA model using the Kirchhoff-Love's Equation

clc; clear;

syms F_e M_e alpha M_t M_b Delta_theta GJ EI r_h r alpha_h alpha_star r_star DLH l_t F_e

assume(alpha,'real')

M_t = M_e*sin(alpha) + F_e*r*cos(alpha) % twising moment along tangent direction.

M_t = 

M_b = M_e*cos(alpha) - F_e*r*sin(alpha); % bending moment along binormal direction.

f1 = M_b/(EI) == cos(alpha)^2/r - cos(alpha_h)^2/r_h

f1 = 

f2 = M_t/(GJ) == sin(alpha)*cos(alpha)/r - sin(alpha_h)*cos(alpha_h)/r_h

f2 = 

[Me, Fe] = solve(f1, f2,[M_e, F_e]);

Me = simplify(Me,'Steps',30);

Me = collect(Me, [GJ, EI])

Me = 

Fe = simplify(Fe,'Steps',25);

Fe = collect(Fe, [GJ, EI])

Fe =