Corrugated Shells geometry

Init symbols for sympy



In [1]:

    
from sympy import *
from geom_util import *
from sympy.vector import CoordSys3D
import matplotlib.pyplot as plt
import sys
sys.path.append("../")

%matplotlib inline

%reload_ext autoreload
%autoreload 2
%aimport geom_util



In [2]:

    
# Any tweaks that normally go in .matplotlibrc, etc., should explicitly go here
%config InlineBackend.figure_format='retina'
plt.rcParams['figure.figsize'] = (12, 12)

    
plt.rc('text', usetex=True)
    
plt.rc('font', family='serif')
# SMALL_SIZE = 42
# MEDIUM_SIZE = 42
# BIGGER_SIZE = 42
    
# plt.rc('font', size=SMALL_SIZE)          # controls default text sizes
# plt.rc('axes', titlesize=SMALL_SIZE)     # fontsize of the axes title
# plt.rc('axes', labelsize=MEDIUM_SIZE)    # fontsize of the x and y labels
# plt.rc('xtick', labelsize=SMALL_SIZE)    # fontsize of the tick labels
# plt.rc('ytick', labelsize=SMALL_SIZE)    # fontsize of the tick labels
# plt.rc('legend', fontsize=SMALL_SIZE)    # legend fontsize
# plt.rc('figure', titlesize=BIGGER_SIZE)  # fontsize of the figure title

init_printing()



In [3]:

    
N = CoordSys3D('N')
alpha1, alpha2, alpha3 = symbols("alpha_1 alpha_2 alpha_3", real = True, positive=True)

Cylindrical coordinates



In [4]:

    
R, L, ga, gv = symbols("R L g_a g_v", real = True, positive=True)



In [5]:

    
a1 = pi / 2 + (L / 2 - alpha1)/R

a2 = 2 * pi * alpha1 / L

x1 = (R + ga * cos(gv * a2)) * cos(a1)
x2 = alpha2
x3 = (R + ga * cos(gv * a2)) * sin(a1)

r = x1*N.i + x2*N.j + x3*N.k

r1=r.diff(alpha1)
r1









    Out[5]:





$$(\frac{1}{R} \left(R + g_{a} \cos{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right) \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + \frac{2 \pi}{L} g_{a} g_{v} \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \sin{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )})\mathbf{\hat{i}_{N}} + (\frac{1}{R} \left(R + g_{a} \cos{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right) \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} - \frac{2 \pi}{L} g_{a} g_{v} \sin{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )} \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )})\mathbf{\hat{k}_{N}}$$



In [6]:

    
z = 2*ga*gv*pi/L*sin(gv*a2)
w = 1 + ga/R*cos(gv*a2)

dr1x=w*sin(a1) - z*cos(a1)
dr1z=-w*cos(a1) - z*sin(a1)

r1 = dr1x*N.i + dr1z*N.k
r2 =N.j

mag=sqrt((w)**2+(z)**2)

nx = -dr1z/mag
nz = dr1x/mag

n = nx*N.i+nz*N.k

dnx=nx.diff(alpha1)
dnz=nz.diff(alpha1)

dn= dnx*N.i+dnz*N.k



In [7]:

    
Ralpha = r+alpha3*n

R1=r1+alpha3*dn
R2=Ralpha.diff(alpha2)
R3=n

Ralpha









    Out[7]:





$$(\frac{\alpha_{3} \left(\left(-1 - \frac{g_{a}}{R} \cos{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right) \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + \frac{2 \pi}{L} g_{a} g_{v} \sin{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )} \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right)}{\sqrt{\left(1 + \frac{g_{a}}{R} \cos{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right)^{2} + \frac{4 \pi^{2}}{L^{2}} g_{a}^{2} g_{v}^{2} \sin^{2}{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}}} - \left(R + g_{a} \cos{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right) \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )})\mathbf{\hat{i}_{N}} + (\alpha_{2})\mathbf{\hat{j}_{N}} + (\frac{\alpha_{3} \left(\left(1 + \frac{g_{a}}{R} \cos{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right) \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + \frac{2 \pi}{L} g_{a} g_{v} \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \sin{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right)}{\sqrt{\left(1 + \frac{g_{a}}{R} \cos{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right)^{2} + \frac{4 \pi^{2}}{L^{2}} g_{a}^{2} g_{v}^{2} \sin^{2}{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}}} + \left(R + g_{a} \cos{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right) \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )})\mathbf{\hat{k}_{N}}$$



In [8]:

    
R1









    Out[8]:





$$(\alpha_{3} \left(\frac{1}{\left(\left(1 + \frac{g_{a}}{R} \cos{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right)^{2} + \frac{4 \pi^{2}}{L^{2}} g_{a}^{2} g_{v}^{2} \sin^{2}{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right)^{\frac{3}{2}}} \left(\left(-1 - \frac{g_{a}}{R} \cos{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right) \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + \frac{2 \pi}{L} g_{a} g_{v} \sin{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )} \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right) \left(\frac{2 \pi g_{a} g_{v}}{L R} \left(1 + \frac{g_{a}}{R} \cos{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right) \sin{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )} - \frac{8 \pi^{3}}{L^{3}} g_{a}^{2} g_{v}^{3} \sin{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )} \cos{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right) + \frac{1}{\sqrt{\left(1 + \frac{g_{a}}{R} \cos{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right)^{2} + \frac{4 \pi^{2}}{L^{2}} g_{a}^{2} g_{v}^{2} \sin^{2}{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}}} \left(- \frac{1}{R} \left(-1 - \frac{g_{a}}{R} \cos{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right) \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + \frac{4 \pi g_{a} g_{v}}{L R} \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \sin{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )} + \frac{4 g_{a}}{L^{2}} \pi^{2} g_{v}^{2} \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right)\right) + \left(1 + \frac{g_{a}}{R} \cos{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right) \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + \frac{2 \pi}{L} g_{a} g_{v} \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \sin{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )})\mathbf{\hat{i}_{N}} + (\alpha_{3} \left(\frac{1}{\left(\left(1 + \frac{g_{a}}{R} \cos{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right)^{2} + \frac{4 \pi^{2}}{L^{2}} g_{a}^{2} g_{v}^{2} \sin^{2}{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right)^{\frac{3}{2}}} \left(\left(1 + \frac{g_{a}}{R} \cos{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right) \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + \frac{2 \pi}{L} g_{a} g_{v} \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \sin{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right) \left(\frac{2 \pi g_{a} g_{v}}{L R} \left(1 + \frac{g_{a}}{R} \cos{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right) \sin{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )} - \frac{8 \pi^{3}}{L^{3}} g_{a}^{2} g_{v}^{3} \sin{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )} \cos{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right) + \frac{1}{\sqrt{\left(1 + \frac{g_{a}}{R} \cos{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right)^{2} + \frac{4 \pi^{2}}{L^{2}} g_{a}^{2} g_{v}^{2} \sin^{2}{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}}} \left(\frac{1}{R} \left(1 + \frac{g_{a}}{R} \cos{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right) \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} - \frac{4 \pi g_{a} g_{v}}{L R} \sin{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )} \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + \frac{4 g_{a}}{L^{2}} \pi^{2} g_{v}^{2} \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right)\right) - \left(-1 - \frac{g_{a}}{R} \cos{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right) \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} - \frac{2 \pi}{L} g_{a} g_{v} \sin{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )} \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )})\mathbf{\hat{k}_{N}}$$



In [9]:

    
R2









    Out[9]:





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



In [10]:

    
R3









    Out[10]:





$$(\frac{\left(-1 - \frac{g_{a}}{R} \cos{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right) \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + \frac{2 \pi}{L} g_{a} g_{v} \sin{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )} \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}}{\sqrt{\left(1 + \frac{g_{a}}{R} \cos{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right)^{2} + \frac{4 \pi^{2}}{L^{2}} g_{a}^{2} g_{v}^{2} \sin^{2}{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}}})\mathbf{\hat{i}_{N}} + (\frac{\left(1 + \frac{g_{a}}{R} \cos{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right) \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + \frac{2 \pi}{L} g_{a} g_{v} \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \sin{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}}{\sqrt{\left(1 + \frac{g_{a}}{R} \cos{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right)^{2} + \frac{4 \pi^{2}}{L^{2}} g_{a}^{2} g_{v}^{2} \sin^{2}{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}}})\mathbf{\hat{k}_{N}}$$

Draw



In [17]:

    
import plot

%aimport plot

# x1 = (R + alpha3 + ga * cos(gv * a2)) * cos(a1)
# x2 = alpha2
# x3 = (R + alpha3 + ga * cos(gv * a2)) * sin(a1)

x1 = Ralpha.dot(N.i)
x3 = Ralpha.dot(N.k)

alpha1_x = lambdify([R, L, ga, gv, alpha1, alpha3], x1, "numpy")
alpha3_z = lambdify([R, L, ga, gv, alpha1, alpha3], x3, "numpy")

R_num = 1/0.8
L_num = 2
h_num = 0.01
ga_num = 0.01
gv_num = 20

x1_start = 0
x1_end = L_num
x3_start = -h_num/2
x3_end = h_num/2

def alpha_to_x(a1, a2, a3):
    x=alpha1_x(R_num, L_num, ga_num, gv_num, a1, a3)
    z=alpha3_z(R_num, L_num, ga_num, gv_num, a1, a3)
    return x, 0, z
    

plot.plot_init_geometry_2(x1_start, x1_end, x3_start, x3_end, alpha_to_x)









    












    





<matplotlib.figure.Figure at 0x1cc24a5c438>



In [12]:

    
%aimport plot

R3_1=R3.dot(N.i)
R3_3=R3.dot(N.k)

R3_1_x = lambdify([R, L, ga, gv, alpha1, alpha3], R3_1, "numpy")
R3_3_z = lambdify([R, L, ga, gv, alpha1, alpha3], R3_3, "numpy")

def R3_to_x(a1, a2, a3):
    x=R3_1_x(R_num, L_num, ga_num, gv_num, a1, a3)
    z=R3_3_z(R_num, L_num, ga_num, gv_num, a1, a3)
    return x, 0, z

plot.plot_vectors(x1_start, x1_end, 0, alpha_to_x, R3_to_x)



In [13]:

    
%aimport plot

R1_1=r1.dot(N.i)
R1_3=r1.dot(N.k)

R1_1_x = lambdify([R, L, ga, gv, alpha1, alpha3], R1_1, "numpy")
R1_3_z = lambdify([R, L, ga, gv, alpha1, alpha3], R1_3, "numpy")

def R1_to_x(a1, a2, a3):
    x=R1_1_x(R_num, L_num, ga_num, gv_num, a1, a3)
    z=R1_3_z(R_num, L_num, ga_num, gv_num, a1, a3)
    return x, 0, z

plot.plot_vectors(x1_start, x1_end, 0, alpha_to_x, R1_to_x)

Lame params



In [15]:

    
A=mag
q=w/R+ga*(2*pi*gv/L)**2*cos(gv*a2)
K=(q*w+2*z*z/R)/(mag**3)

H1 = A*(1+alpha3*K)
H2=S(1)
H3=S(1)

H=[H1, H2, H3]
DIM=3
dH = zeros(DIM,DIM)
for i in range(DIM):
    dH[i,0]=H[i].diff(alpha1)
    dH[i,1]=H[i].diff(alpha2)
    dH[i,2]=H[i].diff(alpha3)
    
    
trigsimp(H1)









    Out[15]:





$$\left(\frac{\alpha_{3}}{\left(\left(1 + \frac{g_{a}}{R} \cos{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right)^{2} + \frac{4 \pi^{2}}{L^{2}} g_{a}^{2} g_{v}^{2} \sin^{2}{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right)^{\frac{3}{2}}} \left(\left(1 + \frac{g_{a}}{R} \cos{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right) \left(\frac{1}{R} \left(1 + \frac{g_{a}}{R} \cos{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right) + \frac{4 g_{a}}{L^{2}} \pi^{2} g_{v}^{2} \cos{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right) + \frac{8 \pi^{2} g_{a}^{2} g_{v}^{2}}{L^{2} R} \sin^{2}{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right) + 1\right) \sqrt{\left(1 + \frac{g_{a}}{R} \cos{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}\right)^{2} + \frac{4 \pi^{2}}{L^{2}} g_{a}^{2} g_{v}^{2} \sin^{2}{\left (\frac{2 \pi}{L} \alpha_{1} g_{v} \right )}}$$