In [24]:

    

from sympy import *
from sympy.vector import CoordSys3D
N = CoordSys3D('N')
alpha, beta, K, L, ga, gv = symbols("alpha beta K L g_a g_v")
af=Function('a')

init_printing()


    

In [25]:

    

a = pi / 2 + K * (L / 2 - alpha)


x = (1 / K + ga * cos(gv * a)) * cos(a)
y = beta
z = (1 / K + ga * cos(gv * a)) * sin(a)


    

In [26]:

    

r = x*N.i + y*N.j + z*N.k

r.subs(pi / 2 + K * (L / 2 - alpha), af(alpha)).subs(K * (L / 2 - alpha), af(alpha)-pi / 2)


    

    
Out[26]:





$$(- \left(- g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} - \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )})\mathbf{\hat{i}_{N}} + (\beta)\mathbf{\hat{j}_{N}} + (\left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )})\mathbf{\hat{k}_{N}}$$

In [28]:

    

dr_dalpha = r.diff(alpha)
dr_dalpha.subs(pi / 2 + K * (L / 2 - alpha), af(alpha)).subs(K * (L / 2 - alpha), af(alpha)-pi / 2)


    

    
Out[28]:





$$(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )})\mathbf{\hat{i}_{N}} + (K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )})\mathbf{\hat{k}_{N}}$$

In [30]:

    

dr_dbeta = r.diff(beta)
dr_dbeta


    

    
Out[30]:





$$\mathbf{\hat{j}_{N}}$$

In [47]:

    

A=dr_dalpha.magnitude()
A.subs(pi / 2 + K * (L / 2 - alpha), af(alpha)).subs(K * (L / 2 - alpha), af(alpha)-pi / 2)


    

    
Out[47]:





$$\sqrt{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}}$$

In [56]:

    

e_alpha=dr_dalpha / A
e_alpha = e_alpha.subs(pi / 2 + K * (L / 2 - alpha), af(alpha)).subs(K * (L / 2 - alpha), af(alpha)-pi / 2)
e_alpha


    

    
Out[56]:





$$(\frac{K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}}{\sqrt{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}}})\mathbf{\hat{i}_{N}} + (\frac{K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}}{\sqrt{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}}})\mathbf{\hat{k}_{N}}$$

In [54]:

    

B=dr_dbeta.magnitude()
B


    

    
Out[54]:





$$1$$

In [55]:

    

e_beta=dr_dbeta / B
e_beta


    

    
Out[55]:





$$\mathbf{\hat{j}_{N}}$$

In [64]:

    

n_temp = e_alpha.cross(e_beta)
n=n_temp.normalize()
n = n.subs(pi / 2 + K * (L / 2 - alpha), af(alpha)).subs(K * (L / 2 - alpha), af(alpha)-pi / 2)
n = sympify(n)
n


    

    
Out[64]:





$$(- \frac{K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}}{\sqrt{\frac{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2}}{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}} + \frac{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}}{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}}} \sqrt{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}}})\mathbf{\hat{i}_{N}} + (\frac{K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}}{\sqrt{\frac{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2}}{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}} + \frac{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}}{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}}} \sqrt{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}}})\mathbf{\hat{k}_{N}}$$

In [58]:

    

e_alpha.diff(alpha)


    

    
Out[58]:





$$(\frac{1}{\left(\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}\right)^{\frac{3}{2}}} \left(- \frac{1}{2} \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right) \left(2 K g_{a} g_{v}^{2} \sin{\left (a{\left (\alpha \right )} \right )} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + 4 K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + 2 K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )}\right) - \frac{1}{2} \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right) \left(2 K g_{a} g_{v}^{2} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} - 4 K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + 2 K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )}\right)\right) \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right) + \frac{K g_{a} g_{v}^{2} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} - 2 K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )}}{\sqrt{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}}})\mathbf{\hat{i}_{N}} + (\frac{1}{\left(\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}\right)^{\frac{3}{2}}} \left(- \frac{1}{2} \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right) \left(2 K g_{a} g_{v}^{2} \sin{\left (a{\left (\alpha \right )} \right )} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + 4 K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + 2 K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )}\right) - \frac{1}{2} \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right) \left(2 K g_{a} g_{v}^{2} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} - 4 K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + 2 K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )}\right)\right) \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right) + \frac{K g_{a} g_{v}^{2} \sin{\left (a{\left (\alpha \right )} \right )} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + 2 K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )}}{\sqrt{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}}})\mathbf{\hat{k}_{N}}$$

In [59]:

    

e_alpha.diff(beta)


    

    
Out[59]:





$$\mathbf{\hat{0}}$$

In [60]:

    

e_beta.diff(alpha)


    

    
Out[60]:





$$\mathbf{\hat{0}}$$

In [61]:

    

e_beta.diff(beta)


    

    
Out[61]:





$$\mathbf{\hat{0}}$$

In [62]:

    

n.diff(alpha)


    

    
Out[62]:





$$(- \frac{\left(- \frac{1}{2} \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right) \left(2 K g_{a} g_{v}^{2} \sin{\left (a{\left (\alpha \right )} \right )} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + 4 K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + 2 K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )}\right) - \frac{1}{2} \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right) \left(2 K g_{a} g_{v}^{2} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} - 4 K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + 2 K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )}\right)\right) \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)}{\sqrt{\frac{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2}}{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}} + \frac{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}}{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}}} \left(\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}\right)^{\frac{3}{2}}} - \frac{K g_{a} g_{v}^{2} \sin{\left (a{\left (\alpha \right )} \right )} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + 2 K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )}}{\sqrt{\frac{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2}}{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}} + \frac{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}}{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}}} \sqrt{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}}} - \frac{1}{\left(\frac{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2}}{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}} + \frac{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}}{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}}\right)^{\frac{3}{2}} \sqrt{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}}} \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right) \left(- \frac{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2}}{2 \left(\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}\right)^{2}} \left(- \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right) \left(2 K g_{a} g_{v}^{2} \sin{\left (a{\left (\alpha \right )} \right )} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + 4 K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + 2 K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )}\right) - \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right) \left(2 K g_{a} g_{v}^{2} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} - 4 K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + 2 K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )}\right)\right) - \frac{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}}{2 \left(\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}\right)^{2}} \left(- \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right) \left(2 K g_{a} g_{v}^{2} \sin{\left (a{\left (\alpha \right )} \right )} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + 4 K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + 2 K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )}\right) - \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right) \left(2 K g_{a} g_{v}^{2} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} - 4 K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + 2 K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )}\right)\right) - \frac{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right) \left(2 K g_{a} g_{v}^{2} \sin{\left (a{\left (\alpha \right )} \right )} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + 4 K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + 2 K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )}\right)}{2 \left(\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}\right)} - \frac{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right) \left(2 K g_{a} g_{v}^{2} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} - 4 K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + 2 K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )}\right)}{2 \left(\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}\right)}\right))\mathbf{\hat{i}_{N}} + (\frac{\left(- \frac{1}{2} \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right) \left(2 K g_{a} g_{v}^{2} \sin{\left (a{\left (\alpha \right )} \right )} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + 4 K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + 2 K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )}\right) - \frac{1}{2} \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right) \left(2 K g_{a} g_{v}^{2} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} - 4 K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + 2 K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )}\right)\right) \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)}{\sqrt{\frac{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2}}{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}} + \frac{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}}{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}}} \left(\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}\right)^{\frac{3}{2}}} + \frac{K g_{a} g_{v}^{2} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} - 2 K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )}}{\sqrt{\frac{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2}}{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}} + \frac{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}}{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}}} \sqrt{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}}} + \frac{1}{\left(\frac{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2}}{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}} + \frac{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}}{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}}\right)^{\frac{3}{2}} \sqrt{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}}} \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right) \left(- \frac{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2}}{2 \left(\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}\right)^{2}} \left(- \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right) \left(2 K g_{a} g_{v}^{2} \sin{\left (a{\left (\alpha \right )} \right )} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + 4 K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + 2 K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )}\right) - \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right) \left(2 K g_{a} g_{v}^{2} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} - 4 K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + 2 K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )}\right)\right) - \frac{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}}{2 \left(\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}\right)^{2}} \left(- \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right) \left(2 K g_{a} g_{v}^{2} \sin{\left (a{\left (\alpha \right )} \right )} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + 4 K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + 2 K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )}\right) - \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right) \left(2 K g_{a} g_{v}^{2} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} - 4 K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + 2 K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )}\right)\right) - \frac{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right) \left(2 K g_{a} g_{v}^{2} \sin{\left (a{\left (\alpha \right )} \right )} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + 4 K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + 2 K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )}\right)}{2 \left(\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}\right)} - \frac{\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right) \left(2 K g_{a} g_{v}^{2} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} - 4 K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )} + 2 K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )} \frac{d}{d \alpha} a{\left (\alpha \right )}\right)}{2 \left(\left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \sin{\left (a{\left (\alpha \right )} \right )} - K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \cos{\left (a{\left (\alpha \right )} \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{v} a{\left (\alpha \right )} \right )} \cos{\left (a{\left (\alpha \right )} \right )} + K \left(g_{a} \cos{\left (g_{v} a{\left (\alpha \right )} \right )} + \frac{1}{K}\right) \sin{\left (a{\left (\alpha \right )} \right )}\right)^{2}\right)}\right))\mathbf{\hat{k}_{N}}$$

In [63]:

    

n.diff(beta)


    

    
Out[63]:





$$\mathbf{\hat{0}}$$

In [ ]: