In [17]:

    

from sympy import *
alpha1, K, L, gv, ga, a = symbols("alpha_1 K L g_a g_v a")
init_printing()


    

In [18]:

    

a1 = pi / 2 + K * (L / 2 - alpha1)

r1 = (1 / K + ga * cos(gv * a1)) * cos(a1)
r2 = (1 / K + ga * cos(gv * a1)) * sin(a1)

dr1 = diff(r1, alpha1)
dr2 = diff(r2, alpha1)

n1 = -dr2 / sqrt(dr1**2 + dr2**2)
n2 = dr1 / sqrt(dr1**2 + dr2**2)

dn1 = sympify('K**2*g_a*g_v*(-K*g_a*g_v*sin(g_a*(K*(L/2 - alpha_1) + pi/2))*cos(K*(L/2 - alpha_1)) - K*(g_v*cos(g_a*(K*(L/2 - alpha_1) + pi/2)) + 1/K)*sin(K*(L/2 - alpha_1)))*(K*g_a**2*g_v*sin(g_a*(K*L - 2*K*alpha_1 + pi))/2 - K*g_v*sin(g_a*(K*L - 2*K*alpha_1 + pi))/2 - sin(g_a*(K*L/2 - K*alpha_1 + pi/2)))/((-K*g_a*g_v*sin(K*(L/2 - alpha_1))*sin(g_a*(K*(L/2 - alpha_1) + pi/2)) + K*(g_v*cos(g_a*(K*(L/2 - alpha_1) + pi/2)) + 1/K)*cos(K*(L/2 - alpha_1)))**2 + (K*g_a*g_v*sin(g_a*(K*(L/2 - alpha_1) + pi/2))*cos(K*(L/2 - alpha_1)) + K*(g_v*cos(g_a*(K*(L/2 - alpha_1) + pi/2)) + 1/K)*sin(K*(L/2 - alpha_1)))**2)**(3/2) + (K**2*g_a**2*g_v*cos(K*(L/2 - alpha_1))*cos(g_a*(K*(L/2 - alpha_1) + pi/2)) - 2*K**2*g_a*g_v*sin(K*(L/2 - alpha_1))*sin(g_a*(K*(L/2 - alpha_1) + pi/2)) + K**2*(g_v*cos(g_a*(K*(L/2 - alpha_1) + pi/2)) + 1/K)*cos(K*(L/2 - alpha_1)))/sqrt((-K*g_a*g_v*sin(K*(L/2 - alpha_1))*sin(g_a*(K*(L/2 - alpha_1) + pi/2)) + K*(g_v*cos(g_a*(K*(L/2 - alpha_1) + pi/2)) + 1/K)*cos(K*(L/2 - alpha_1)))**2 + (K*g_a*g_v*sin(g_a*(K*(L/2 - alpha_1) + pi/2))*cos(K*(L/2 - alpha_1)) + K*(g_v*cos(g_a*(K*(L/2 - alpha_1) + pi/2)) + 1/K)*sin(K*(L/2 - alpha_1)))**2)')
dn2 = sympify('K**2*g_a*g_v*(-K*g_a*g_v*sin(K*(L/2 - alpha_1))*sin(g_a*(K*(L/2 - alpha_1) + pi/2)) + K*(g_v*cos(g_a*(K*(L/2 - alpha_1) + pi/2)) + 1/K)*cos(K*(L/2 - alpha_1)))*(K*g_a**2*g_v*sin(g_a*(K*L - 2*K*alpha_1 + pi))/2 - K*g_v*sin(g_a*(K*L - 2*K*alpha_1 + pi))/2 - sin(g_a*(K*L/2 - K*alpha_1 + pi/2)))/((-K*g_a*g_v*sin(K*(L/2 - alpha_1))*sin(g_a*(K*(L/2 - alpha_1) + pi/2)) + K*(g_v*cos(g_a*(K*(L/2 - alpha_1) + pi/2)) + 1/K)*cos(K*(L/2 - alpha_1)))**2 + (K*g_a*g_v*sin(g_a*(K*(L/2 - alpha_1) + pi/2))*cos(K*(L/2 - alpha_1)) + K*(g_v*cos(g_a*(K*(L/2 - alpha_1) + pi/2)) + 1/K)*sin(K*(L/2 - alpha_1)))**2)**(3/2) + (K**2*g_a**2*g_v*sin(K*(L/2 - alpha_1))*cos(g_a*(K*(L/2 - alpha_1) + pi/2)) + 2*K**2*g_a*g_v*sin(g_a*(K*(L/2 - alpha_1) + pi/2))*cos(K*(L/2 - alpha_1)) + K**2*(g_v*cos(g_a*(K*(L/2 - alpha_1) + pi/2)) + 1/K)*sin(K*(L/2 - alpha_1)))/sqrt((-K*g_a*g_v*sin(K*(L/2 - alpha_1))*sin(g_a*(K*(L/2 - alpha_1) + pi/2)) + K*(g_v*cos(g_a*(K*(L/2 - alpha_1) + pi/2)) + 1/K)*cos(K*(L/2 - alpha_1)))**2 + (K*g_a*g_v*sin(g_a*(K*(L/2 - alpha_1) + pi/2))*cos(K*(L/2 - alpha_1)) + K*(g_v*cos(g_a*(K*(L/2 - alpha_1) + pi/2)) + 1/K)*sin(K*(L/2 - alpha_1)))**2)')

dn1 = dn1.subs(K*(L/2 - alpha1) + pi/2, a1)
dn2 = dn2.subs(K*(L/2 - alpha1) + pi/2, a1)

dn1


    

    
Out[18]:





$$\frac{K^{2} g_{a} g_{v} \left(- K g_{a} g_{v} \sin{\left (a_{1} g_{a} \right )} \cos{\left (K \left(\frac{L}{2} - \alpha_{1}\right) \right )} - K \left(g_{v} \cos{\left (a_{1} g_{a} \right )} + \frac{1}{K}\right) \sin{\left (K \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right) \left(\frac{K g_{v}}{2} g_{a}^{2} \sin{\left (g_{a} \left(K L - 2 K \alpha_{1} + \pi\right) \right )} - \frac{K g_{v}}{2} \sin{\left (g_{a} \left(K L - 2 K \alpha_{1} + \pi\right) \right )} - \sin{\left (g_{a} \left(\frac{K L}{2} - K \alpha_{1} + \frac{\pi}{2}\right) \right )}\right)}{\left(\left(- K g_{a} g_{v} \sin{\left (K \left(\frac{L}{2} - \alpha_{1}\right) \right )} \sin{\left (a_{1} g_{a} \right )} + K \left(g_{v} \cos{\left (a_{1} g_{a} \right )} + \frac{1}{K}\right) \cos{\left (K \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (a_{1} g_{a} \right )} \cos{\left (K \left(\frac{L}{2} - \alpha_{1}\right) \right )} + K \left(g_{v} \cos{\left (a_{1} g_{a} \right )} + \frac{1}{K}\right) \sin{\left (K \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right)^{2}\right)^{\frac{3}{2}}} + \frac{K^{2} g_{a}^{2} g_{v} \cos{\left (K \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos{\left (a_{1} g_{a} \right )} - 2 K^{2} g_{a} g_{v} \sin{\left (K \left(\frac{L}{2} - \alpha_{1}\right) \right )} \sin{\left (a_{1} g_{a} \right )} + K^{2} \left(g_{v} \cos{\left (a_{1} g_{a} \right )} + \frac{1}{K}\right) \cos{\left (K \left(\frac{L}{2} - \alpha_{1}\right) \right )}}{\sqrt{\left(- K g_{a} g_{v} \sin{\left (K \left(\frac{L}{2} - \alpha_{1}\right) \right )} \sin{\left (a_{1} g_{a} \right )} + K \left(g_{v} \cos{\left (a_{1} g_{a} \right )} + \frac{1}{K}\right) \cos{\left (K \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (a_{1} g_{a} \right )} \cos{\left (K \left(\frac{L}{2} - \alpha_{1}\right) \right )} + K \left(g_{v} \cos{\left (a_{1} g_{a} \right )} + \frac{1}{K}\right) \sin{\left (K \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right)^{2}}}$$

In [5]:

    

dn1


    

    
Out[5]:





$$K \left(\frac{L}{2} - \alpha_{1}\right) + \frac{\pi}{2}$$

In [6]:

    

r1


    

    
Out[6]:





$$- \left(g_{v} \cos{\left (g_{a} \left(K \left(\frac{L}{2} - \alpha_{1}\right) + \frac{\pi}{2}\right) \right )} + \frac{1}{K}\right) \sin{\left (K \left(\frac{L}{2} - \alpha_{1}\right) \right )}$$

In [7]:

    

r2


    

    
Out[7]:





$$\left(g_{v} \cos{\left (g_{a} \left(K \left(\frac{L}{2} - \alpha_{1}\right) + \frac{\pi}{2}\right) \right )} + \frac{1}{K}\right) \cos{\left (K \left(\frac{L}{2} - \alpha_{1}\right) \right )}$$

In [16]:

    

dn1


    

    
Out[16]:





$$\frac{K^{2} g_{a} g_{v} \left(- K g_{a} g_{v} \sin{\left (g_{a} \left(K \left(\frac{L}{2} - \alpha_{1}\right) + \frac{\pi}{2}\right) \right )} \cos{\left (K \left(\frac{L}{2} - \alpha_{1}\right) \right )} - K \left(g_{v} \cos{\left (g_{a} \left(K \left(\frac{L}{2} - \alpha_{1}\right) + \frac{\pi}{2}\right) \right )} + \frac{1}{K}\right) \sin{\left (K \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right) \left(\frac{K g_{v}}{2} g_{a}^{2} \sin{\left (g_{a} \left(K L - 2 K \alpha_{1} + \pi\right) \right )} - \frac{K g_{v}}{2} \sin{\left (g_{a} \left(K L - 2 K \alpha_{1} + \pi\right) \right )} - \sin{\left (g_{a} \left(\frac{K L}{2} - K \alpha_{1} + \frac{\pi}{2}\right) \right )}\right)}{\left(\left(- K g_{a} g_{v} \sin{\left (K \left(\frac{L}{2} - \alpha_{1}\right) \right )} \sin{\left (g_{a} \left(K \left(\frac{L}{2} - \alpha_{1}\right) + \frac{\pi}{2}\right) \right )} + K \left(g_{v} \cos{\left (g_{a} \left(K \left(\frac{L}{2} - \alpha_{1}\right) + \frac{\pi}{2}\right) \right )} + \frac{1}{K}\right) \cos{\left (K \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{a} \left(K \left(\frac{L}{2} - \alpha_{1}\right) + \frac{\pi}{2}\right) \right )} \cos{\left (K \left(\frac{L}{2} - \alpha_{1}\right) \right )} + K \left(g_{v} \cos{\left (g_{a} \left(K \left(\frac{L}{2} - \alpha_{1}\right) + \frac{\pi}{2}\right) \right )} + \frac{1}{K}\right) \sin{\left (K \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right)^{2}\right)^{\frac{3}{2}}} + \frac{K^{2} g_{a}^{2} g_{v} \cos{\left (K \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos{\left (g_{a} \left(K \left(\frac{L}{2} - \alpha_{1}\right) + \frac{\pi}{2}\right) \right )} - 2 K^{2} g_{a} g_{v} \sin{\left (K \left(\frac{L}{2} - \alpha_{1}\right) \right )} \sin{\left (g_{a} \left(K \left(\frac{L}{2} - \alpha_{1}\right) + \frac{\pi}{2}\right) \right )} + K^{2} \left(g_{v} \cos{\left (g_{a} \left(K \left(\frac{L}{2} - \alpha_{1}\right) + \frac{\pi}{2}\right) \right )} + \frac{1}{K}\right) \cos{\left (K \left(\frac{L}{2} - \alpha_{1}\right) \right )}}{\sqrt{\left(- K g_{a} g_{v} \sin{\left (K \left(\frac{L}{2} - \alpha_{1}\right) \right )} \sin{\left (g_{a} \left(K \left(\frac{L}{2} - \alpha_{1}\right) + \frac{\pi}{2}\right) \right )} + K \left(g_{v} \cos{\left (g_{a} \left(K \left(\frac{L}{2} - \alpha_{1}\right) + \frac{\pi}{2}\right) \right )} + \frac{1}{K}\right) \cos{\left (K \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{a} \left(K \left(\frac{L}{2} - \alpha_{1}\right) + \frac{\pi}{2}\right) \right )} \cos{\left (K \left(\frac{L}{2} - \alpha_{1}\right) \right )} + K \left(g_{v} \cos{\left (g_{a} \left(K \left(\frac{L}{2} - \alpha_{1}\right) + \frac{\pi}{2}\right) \right )} + \frac{1}{K}\right) \sin{\left (K \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right)^{2}}}$$

In [7]:

    

dn2


    

    
Out[7]:





$$\frac{K^{2} g_{a} g_{v} \left(- K g_{a} g_{v} \sin{\left (K \left(\frac{L}{2} - a_{1}\right) \right )} \sin{\left (g_{a} \left(K \left(\frac{L}{2} - a_{1}\right) + \frac{\pi}{2}\right) \right )} + K \left(g_{v} \cos{\left (g_{a} \left(K \left(\frac{L}{2} - a_{1}\right) + \frac{\pi}{2}\right) \right )} + \frac{1}{K}\right) \cos{\left (K \left(\frac{L}{2} - a_{1}\right) \right )}\right) \left(\frac{K g_{v}}{2} g_{a}^{2} \sin{\left (g_{a} \left(K L - 2 K a_{1} + \pi\right) \right )} - \frac{K g_{v}}{2} \sin{\left (g_{a} \left(K L - 2 K a_{1} + \pi\right) \right )} - \sin{\left (g_{a} \left(\frac{K L}{2} - K a_{1} + \frac{\pi}{2}\right) \right )}\right)}{\left(\left(- K g_{a} g_{v} \sin{\left (K \left(\frac{L}{2} - a_{1}\right) \right )} \sin{\left (g_{a} \left(K \left(\frac{L}{2} - a_{1}\right) + \frac{\pi}{2}\right) \right )} + K \left(g_{v} \cos{\left (g_{a} \left(K \left(\frac{L}{2} - a_{1}\right) + \frac{\pi}{2}\right) \right )} + \frac{1}{K}\right) \cos{\left (K \left(\frac{L}{2} - a_{1}\right) \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{a} \left(K \left(\frac{L}{2} - a_{1}\right) + \frac{\pi}{2}\right) \right )} \cos{\left (K \left(\frac{L}{2} - a_{1}\right) \right )} + K \left(g_{v} \cos{\left (g_{a} \left(K \left(\frac{L}{2} - a_{1}\right) + \frac{\pi}{2}\right) \right )} + \frac{1}{K}\right) \sin{\left (K \left(\frac{L}{2} - a_{1}\right) \right )}\right)^{2}\right)^{\frac{3}{2}}} + \frac{K^{2} g_{a}^{2} g_{v} \sin{\left (K \left(\frac{L}{2} - a_{1}\right) \right )} \cos{\left (g_{a} \left(K \left(\frac{L}{2} - a_{1}\right) + \frac{\pi}{2}\right) \right )} + 2 K^{2} g_{a} g_{v} \sin{\left (g_{a} \left(K \left(\frac{L}{2} - a_{1}\right) + \frac{\pi}{2}\right) \right )} \cos{\left (K \left(\frac{L}{2} - a_{1}\right) \right )} + K^{2} \left(g_{v} \cos{\left (g_{a} \left(K \left(\frac{L}{2} - a_{1}\right) + \frac{\pi}{2}\right) \right )} + \frac{1}{K}\right) \sin{\left (K \left(\frac{L}{2} - a_{1}\right) \right )}}{\sqrt{\left(- K g_{a} g_{v} \sin{\left (K \left(\frac{L}{2} - a_{1}\right) \right )} \sin{\left (g_{a} \left(K \left(\frac{L}{2} - a_{1}\right) + \frac{\pi}{2}\right) \right )} + K \left(g_{v} \cos{\left (g_{a} \left(K \left(\frac{L}{2} - a_{1}\right) + \frac{\pi}{2}\right) \right )} + \frac{1}{K}\right) \cos{\left (K \left(\frac{L}{2} - a_{1}\right) \right )}\right)^{2} + \left(K g_{a} g_{v} \sin{\left (g_{a} \left(K \left(\frac{L}{2} - a_{1}\right) + \frac{\pi}{2}\right) \right )} \cos{\left (K \left(\frac{L}{2} - a_{1}\right) \right )} + K \left(g_{v} \cos{\left (g_{a} \left(K \left(\frac{L}{2} - a_{1}\right) + \frac{\pi}{2}\right) \right )} + \frac{1}{K}\right) \sin{\left (K \left(\frac{L}{2} - a_{1}\right) \right )}\right)^{2}}}$$

In [1]:

    

%matplotlib inline
%run main3.py


    

    













    













    













    













    













    













    













    













    




86.92912659267772
112.45861240311208
135.1676624088808
154.9158415108017
172.42072923554107
188.16511278632797
202.49222851595405
215.6223104918045
227.69930454607115
238.8169918618502
249.0336246685265
258.3796886656492
266.85963240529753
274.4426843431842
281.03292416824513
286.44244027410826
290.41638382395877
292.8710901813417
295.08133731309476
298.70535302141366