File size: 31,420 Bytes
9c6594c |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 |
"""
This module can be used to solve problems related
to 2D Cables.
"""
from sympy.core.sympify import sympify
from sympy.core.symbol import Symbol,symbols
from sympy import sin, cos, pi, atan, diff, Piecewise, solve, rad
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.solvers.solveset import linsolve
from sympy.matrices import Matrix
from sympy.plotting import plot
class Cable:
"""
Cables are structures in engineering that support
the applied transverse loads through the tensile
resistance developed in its members.
Cables are widely used in suspension bridges, tension
leg offshore platforms, transmission lines, and find
use in several other engineering applications.
Examples
========
A cable is supported at (0, 10) and (10, 10). Two point loads
acting vertically downwards act on the cable, one with magnitude 3 kN
and acting 2 meters from the left support and 3 meters below it, while
the other with magnitude 2 kN is 6 meters from the left support and
6 meters below it.
>>> from sympy.physics.continuum_mechanics.cable import Cable
>>> c = Cable(('A', 0, 10), ('B', 10, 10))
>>> c.apply_load(-1, ('P', 2, 7, 3, 270))
>>> c.apply_load(-1, ('Q', 6, 4, 2, 270))
>>> c.loads
{'distributed': {}, 'point_load': {'P': [3, 270], 'Q': [2, 270]}}
>>> c.loads_position
{'P': [2, 7], 'Q': [6, 4]}
"""
def __init__(self, support_1, support_2):
"""
Initializes the class.
Parameters
==========
support_1 and support_2 are tuples of the form
(label, x, y), where
label : String or symbol
The label of the support
x : Sympifyable
The x coordinate of the position of the support
y : Sympifyable
The y coordinate of the position of the support
"""
self._left_support = []
self._right_support = []
self._supports = {}
self._support_labels = []
self._loads = {"distributed": {}, "point_load": {}}
self._loads_position = {}
self._length = 0
self._reaction_loads = {}
self._tension = {}
self._lowest_x_global = sympify(0)
self._lowest_y_global = sympify(0)
self._cable_eqn = None
self._tension_func = None
if support_1[0] == support_2[0]:
raise ValueError("Supports can not have the same label")
elif support_1[1] == support_2[1]:
raise ValueError("Supports can not be at the same location")
x1 = sympify(support_1[1])
y1 = sympify(support_1[2])
self._supports[support_1[0]] = [x1, y1]
x2 = sympify(support_2[1])
y2 = sympify(support_2[2])
self._supports[support_2[0]] = [x2, y2]
if support_1[1] < support_2[1]:
self._left_support.append(x1)
self._left_support.append(y1)
self._right_support.append(x2)
self._right_support.append(y2)
self._support_labels.append(support_1[0])
self._support_labels.append(support_2[0])
else:
self._left_support.append(x2)
self._left_support.append(y2)
self._right_support.append(x1)
self._right_support.append(y1)
self._support_labels.append(support_2[0])
self._support_labels.append(support_1[0])
for i in self._support_labels:
self._reaction_loads[Symbol("R_"+ i +"_x")] = 0
self._reaction_loads[Symbol("R_"+ i +"_y")] = 0
@property
def supports(self):
"""
Returns the supports of the cable along with their
positions.
"""
return self._supports
@property
def left_support(self):
"""
Returns the position of the left support.
"""
return self._left_support
@property
def right_support(self):
"""
Returns the position of the right support.
"""
return self._right_support
@property
def loads(self):
"""
Returns the magnitude and direction of the loads
acting on the cable.
"""
return self._loads
@property
def loads_position(self):
"""
Returns the position of the point loads acting on the
cable.
"""
return self._loads_position
@property
def length(self):
"""
Returns the length of the cable.
"""
return self._length
@property
def reaction_loads(self):
"""
Returns the reaction forces at the supports, which are
initialized to 0.
"""
return self._reaction_loads
@property
def tension(self):
"""
Returns the tension developed in the cable due to the loads
applied.
"""
return self._tension
def tension_at(self, x):
"""
Returns the tension at a given value of x developed due to
distributed load.
"""
if 'distributed' not in self._tension.keys():
raise ValueError("No distributed load added or solve method not called")
if x > self._right_support[0] or x < self._left_support[0]:
raise ValueError("The value of x should be between the two supports")
A = self._tension['distributed']
X = Symbol('X')
return A.subs({X:(x-self._lowest_x_global)})
def apply_length(self, length):
"""
This method specifies the length of the cable
Parameters
==========
length : Sympifyable
The length of the cable
Examples
========
>>> from sympy.physics.continuum_mechanics.cable import Cable
>>> c = Cable(('A', 0, 10), ('B', 10, 10))
>>> c.apply_length(20)
>>> c.length
20
"""
dist = ((self._left_support[0] - self._right_support[0])**2
- (self._left_support[1] - self._right_support[1])**2)**(1/2)
if length < dist:
raise ValueError("length should not be less than the distance between the supports")
self._length = length
def change_support(self, label, new_support):
"""
This method changes the mentioned support with a new support.
Parameters
==========
label: String or symbol
The label of the support to be changed
new_support: Tuple of the form (new_label, x, y)
new_label: String or symbol
The label of the new support
x: Sympifyable
The x-coordinate of the position of the new support.
y: Sympifyable
The y-coordinate of the position of the new support.
Examples
========
>>> from sympy.physics.continuum_mechanics.cable import Cable
>>> c = Cable(('A', 0, 10), ('B', 10, 10))
>>> c.supports
{'A': [0, 10], 'B': [10, 10]}
>>> c.change_support('B', ('C', 5, 6))
>>> c.supports
{'A': [0, 10], 'C': [5, 6]}
"""
if label not in self._supports:
raise ValueError("No support exists with the given label")
i = self._support_labels.index(label)
rem_label = self._support_labels[(i+1)%2]
x1 = self._supports[rem_label][0]
y1 = self._supports[rem_label][1]
x = sympify(new_support[1])
y = sympify(new_support[2])
for l in self._loads_position:
if l[0] >= max(x, x1) or l[0] <= min(x, x1):
raise ValueError("The change in support will throw an existing load out of range")
self._supports.pop(label)
self._left_support.clear()
self._right_support.clear()
self._reaction_loads.clear()
self._support_labels.remove(label)
self._supports[new_support[0]] = [x, y]
if x1 < x:
self._left_support.append(x1)
self._left_support.append(y1)
self._right_support.append(x)
self._right_support.append(y)
self._support_labels.append(new_support[0])
else:
self._left_support.append(x)
self._left_support.append(y)
self._right_support.append(x1)
self._right_support.append(y1)
self._support_labels.insert(0, new_support[0])
for i in self._support_labels:
self._reaction_loads[Symbol("R_"+ i +"_x")] = 0
self._reaction_loads[Symbol("R_"+ i +"_y")] = 0
def apply_load(self, order, load):
"""
This method adds load to the cable.
Parameters
==========
order : Integer
The order of the applied load.
- For point loads, order = -1
- For distributed load, order = 0
load : tuple
* For point loads, load is of the form (label, x, y, magnitude, direction), where:
label : String or symbol
The label of the load
x : Sympifyable
The x coordinate of the position of the load
y : Sympifyable
The y coordinate of the position of the load
magnitude : Sympifyable
The magnitude of the load. It must always be positive
direction : Sympifyable
The angle, in degrees, that the load vector makes with the horizontal
in the counter-clockwise direction. It takes the values 0 to 360,
inclusive.
* For uniformly distributed load, load is of the form (label, magnitude)
label : String or symbol
The label of the load
magnitude : Sympifyable
The magnitude of the load. It must always be positive
Examples
========
For a point load of magnitude 12 units inclined at 30 degrees with the horizontal:
>>> from sympy.physics.continuum_mechanics.cable import Cable
>>> c = Cable(('A', 0, 10), ('B', 10, 10))
>>> c.apply_load(-1, ('Z', 5, 5, 12, 30))
>>> c.loads
{'distributed': {}, 'point_load': {'Z': [12, 30]}}
>>> c.loads_position
{'Z': [5, 5]}
For a uniformly distributed load of magnitude 9 units:
>>> from sympy.physics.continuum_mechanics.cable import Cable
>>> c = Cable(('A', 0, 10), ('B', 10, 10))
>>> c.apply_load(0, ('X', 9))
>>> c.loads
{'distributed': {'X': 9}, 'point_load': {}}
"""
if order == -1:
if len(self._loads["distributed"]) != 0:
raise ValueError("Distributed load already exists")
label = load[0]
if label in self._loads["point_load"]:
raise ValueError("Label already exists")
x = sympify(load[1])
y = sympify(load[2])
if x > self._right_support[0] or x < self._left_support[0]:
raise ValueError("The load should be positioned between the supports")
magnitude = sympify(load[3])
direction = sympify(load[4])
self._loads["point_load"][label] = [magnitude, direction]
self._loads_position[label] = [x, y]
elif order == 0:
if len(self._loads_position) != 0:
raise ValueError("Point load(s) already exist")
label = load[0]
if label in self._loads["distributed"]:
raise ValueError("Label already exists")
magnitude = sympify(load[1])
self._loads["distributed"][label] = magnitude
else:
raise ValueError("Order should be either -1 or 0")
def remove_loads(self, *args):
"""
This methods removes the specified loads.
Parameters
==========
This input takes multiple label(s) as input
label(s): String or symbol
The label(s) of the loads to be removed.
Examples
========
>>> from sympy.physics.continuum_mechanics.cable import Cable
>>> c = Cable(('A', 0, 10), ('B', 10, 10))
>>> c.apply_load(-1, ('Z', 5, 5, 12, 30))
>>> c.loads
{'distributed': {}, 'point_load': {'Z': [12, 30]}}
>>> c.remove_loads('Z')
>>> c.loads
{'distributed': {}, 'point_load': {}}
"""
for i in args:
if len(self._loads_position) == 0:
if i not in self._loads['distributed']:
raise ValueError("Error removing load " + i + ": no such load exists")
else:
self._loads['disrtibuted'].pop(i)
else:
if i not in self._loads['point_load']:
raise ValueError("Error removing load " + i + ": no such load exists")
else:
self._loads['point_load'].pop(i)
self._loads_position.pop(i)
def solve(self, *args):
"""
This method solves for the reaction forces at the supports, the tension developed in
the cable, and updates the length of the cable.
Parameters
==========
This method requires no input when solving for point loads
For distributed load, the x and y coordinates of the lowest point of the cable are
required as
x: Sympifyable
The x coordinate of the lowest point
y: Sympifyable
The y coordinate of the lowest point
Examples
========
For point loads,
>>> from sympy.physics.continuum_mechanics.cable import Cable
>>> c = Cable(("A", 0, 10), ("B", 10, 10))
>>> c.apply_load(-1, ('Z', 2, 7.26, 3, 270))
>>> c.apply_load(-1, ('X', 4, 6, 8, 270))
>>> c.solve()
>>> c.tension
{A_Z: 8.91403453669861, X_B: 19*sqrt(13)/10, Z_X: 4.79150773600774}
>>> c.reaction_loads
{R_A_x: -5.25547445255474, R_A_y: 7.2, R_B_x: 5.25547445255474, R_B_y: 3.8}
>>> c.length
5.7560958484519 + 2*sqrt(13)
For distributed load,
>>> from sympy.physics.continuum_mechanics.cable import Cable
>>> c=Cable(("A", 0, 40),("B", 100, 20))
>>> c.apply_load(0, ("X", 850))
>>> c.solve(58.58)
>>> c.tension
{'distributed': 36465.0*sqrt(0.00054335718671383*X**2 + 1)}
>>> c.tension_at(0)
61717.4130533677
>>> c.reaction_loads
{R_A_x: 36465.0, R_A_y: -49793.0, R_B_x: 44399.9537590861, R_B_y: 42868.2071025955}
"""
if len(self._loads_position) != 0:
sorted_position = sorted(self._loads_position.items(), key = lambda item : item[1][0])
sorted_position.append(self._support_labels[1])
sorted_position.insert(0, self._support_labels[0])
self._tension.clear()
moment_sum_from_left_support = 0
moment_sum_from_right_support = 0
F_x = 0
F_y = 0
self._length = 0
tension_func = []
x = symbols('x')
for i in range(1, len(sorted_position)-1):
if i == 1:
self._length+=sqrt((self._left_support[0] - self._loads_position[sorted_position[i][0]][0])**2 + (self._left_support[1] - self._loads_position[sorted_position[i][0]][1])**2)
else:
self._length+=sqrt((self._loads_position[sorted_position[i-1][0]][0] - self._loads_position[sorted_position[i][0]][0])**2 + (self._loads_position[sorted_position[i-1][0]][1] - self._loads_position[sorted_position[i][0]][1])**2)
if i == len(sorted_position)-2:
self._length+=sqrt((self._right_support[0] - self._loads_position[sorted_position[i][0]][0])**2 + (self._right_support[1] - self._loads_position[sorted_position[i][0]][1])**2)
moment_sum_from_left_support += self._loads['point_load'][sorted_position[i][0]][0] * cos(pi * self._loads['point_load'][sorted_position[i][0]][1] / 180) * abs(self._left_support[1] - self._loads_position[sorted_position[i][0]][1])
moment_sum_from_left_support += self._loads['point_load'][sorted_position[i][0]][0] * sin(pi * self._loads['point_load'][sorted_position[i][0]][1] / 180) * abs(self._left_support[0] - self._loads_position[sorted_position[i][0]][0])
F_x += self._loads['point_load'][sorted_position[i][0]][0] * cos(pi * self._loads['point_load'][sorted_position[i][0]][1] / 180)
F_y += self._loads['point_load'][sorted_position[i][0]][0] * sin(pi * self._loads['point_load'][sorted_position[i][0]][1] / 180)
label = Symbol(sorted_position[i][0]+"_"+sorted_position[i+1][0])
y2 = self._loads_position[sorted_position[i][0]][1]
x2 = self._loads_position[sorted_position[i][0]][0]
y1 = 0
x1 = 0
if i == len(sorted_position)-2:
x1 = self._right_support[0]
y1 = self._right_support[1]
else:
x1 = self._loads_position[sorted_position[i+1][0]][0]
y1 = self._loads_position[sorted_position[i+1][0]][1]
angle_with_horizontal = atan((y1 - y2)/(x1 - x2))
tension = -(moment_sum_from_left_support)/(abs(self._left_support[1] - self._loads_position[sorted_position[i][0]][1])*cos(angle_with_horizontal) + abs(self._left_support[0] - self._loads_position[sorted_position[i][0]][0])*sin(angle_with_horizontal))
self._tension[label] = tension
tension_func.append((tension, x<=x1))
moment_sum_from_right_support += self._loads['point_load'][sorted_position[i][0]][0] * cos(pi * self._loads['point_load'][sorted_position[i][0]][1] / 180) * abs(self._right_support[1] - self._loads_position[sorted_position[i][0]][1])
moment_sum_from_right_support += self._loads['point_load'][sorted_position[i][0]][0] * sin(pi * self._loads['point_load'][sorted_position[i][0]][1] / 180) * abs(self._right_support[0] - self._loads_position[sorted_position[i][0]][0])
label = Symbol(sorted_position[0][0]+"_"+sorted_position[1][0])
y2 = self._loads_position[sorted_position[1][0]][1]
x2 = self._loads_position[sorted_position[1][0]][0]
x1 = self._left_support[0]
y1 = self._left_support[1]
angle_with_horizontal = -atan((y2 - y1)/(x2 - x1))
tension = -(moment_sum_from_right_support)/(abs(self._right_support[1] - self._loads_position[sorted_position[1][0]][1])*cos(angle_with_horizontal) + abs(self._right_support[0] - self._loads_position[sorted_position[1][0]][0])*sin(angle_with_horizontal))
self._tension[label] = tension
tension_func.insert(0,(tension, x<=x2))
self._tension_func = Piecewise(*tension_func)
angle_with_horizontal = pi/2 - angle_with_horizontal
label = self._support_labels[0]
self._reaction_loads[Symbol("R_"+label+"_x")] = -sin(angle_with_horizontal) * tension
F_x += -sin(angle_with_horizontal) * tension
self._reaction_loads[Symbol("R_"+label+"_y")] = cos(angle_with_horizontal) * tension
F_y += cos(angle_with_horizontal) * tension
label = self._support_labels[1]
self._reaction_loads[Symbol("R_"+label+"_x")] = -F_x
self._reaction_loads[Symbol("R_"+label+"_y")] = -F_y
elif len(self._loads['distributed']) != 0 :
if len(args) == 0:
raise ValueError("Provide the lowest point of the cable")
lowest_x = sympify(args[0])
self._lowest_x_global = lowest_x
a = Symbol('a', positive=True)
c = Symbol('c')
# augmented matrix form of linsolve
M = Matrix(
[[(self._left_support[0]-lowest_x)**2, 1, self._left_support[1]],
[(self._right_support[0]-lowest_x)**2, 1, self._right_support[1]],
])
coefficient_solution = list(linsolve(M, (a, c)))
if len(coefficient_solution) ==0 or coefficient_solution[0][0]== 0:
raise ValueError("The lowest point is inconsistent with the supports")
A = coefficient_solution[0][0]
C = coefficient_solution[0][1] + coefficient_solution[0][0]*lowest_x**2
B = -2*coefficient_solution[0][0]*lowest_x
self._lowest_y_global = coefficient_solution[0][1]
lowest_y = self._lowest_y_global
# y = A*x**2 + B*x + C
# shifting origin to lowest point
X = Symbol('X')
Y = Symbol('Y')
Y = A*(X + lowest_x)**2 + B*(X + lowest_x) + C - lowest_y
temp_list = list(self._loads['distributed'].values())
applied_force = temp_list[0]
horizontal_force_constant = (applied_force * (self._right_support[0] - lowest_x)**2) / (2 * (self._right_support[1] - lowest_y))
self._tension.clear()
tangent_slope_to_curve = diff(Y, X)
self._tension['distributed'] = horizontal_force_constant / (cos(atan(tangent_slope_to_curve)))
label = self._support_labels[0]
self._reaction_loads[Symbol("R_"+label+"_x")] = self.tension_at(self._left_support[0]) * cos(atan(tangent_slope_to_curve.subs(X, self._left_support[0] - lowest_x)))
self._reaction_loads[Symbol("R_"+label+"_y")] = self.tension_at(self._left_support[0]) * sin(atan(tangent_slope_to_curve.subs(X, self._left_support[0] - lowest_x)))
label = self._support_labels[1]
self._reaction_loads[Symbol("R_"+label+"_x")] = self.tension_at(self._left_support[0]) * cos(atan(tangent_slope_to_curve.subs(X, self._right_support[0] - lowest_x)))
self._reaction_loads[Symbol("R_"+label+"_y")] = self.tension_at(self._left_support[0]) * sin(atan(tangent_slope_to_curve.subs(X, self._right_support[0] - lowest_x)))
def draw(self):
"""
This method is used to obtain a plot for the specified cable with its supports,
shape and loads.
Examples
========
For point loads,
>>> from sympy.physics.continuum_mechanics.cable import Cable
>>> c = Cable(("A", 0, 10), ("B", 10, 10))
>>> c.apply_load(-1, ('Z', 2, 7.26, 3, 270))
>>> c.apply_load(-1, ('X', 4, 6, 8, 270))
>>> c.solve()
>>> p = c.draw()
>>> p # doctest: +ELLIPSIS
Plot object containing:
[0]: cartesian line: Piecewise((10 - 1.37*x, x <= 2), (8.52 - 0.63*x, x <= 4), (2*x/3 + 10/3, x <= 10)) for x over (0.0, 10.0)
...
>>> p.show()
For uniformly distributed loads,
>>> from sympy.physics.continuum_mechanics.cable import Cable
>>> c=Cable(("A", 0, 40),("B", 100, 20))
>>> c.apply_load(0, ("X", 850))
>>> c.solve(58.58)
>>> p = c.draw()
>>> p # doctest: +ELLIPSIS
Plot object containing:
[0]: cartesian line: 0.0116550116550117*(x - 58.58)**2 + 0.00447086247086247 for x over (0.0, 100.0)
[1]: cartesian line: -7.49552913752915 for x over (0.0, 100.0)
...
>>> p.show()
"""
x = Symbol("x")
annotations = []
support_rectangles = self._draw_supports()
xy_min = min(self._left_support[0],self._lowest_y_global)
xy_max = max(self._right_support[0], max(self._right_support[1],self._left_support[1]))
max_diff = xy_max - xy_min
if len(self._loads_position) != 0:
self._cable_eqn = self._draw_cable(-1)
annotations += self._draw_loads(-1)
elif len(self._loads['distributed']) != 0 :
self._cable_eqn = self._draw_cable(0)
annotations += self._draw_loads(0)
if not self._cable_eqn:
raise ValueError("solve method not called and/or values provided for loads and supports not adequate")
cab_plot = plot(*self._cable_eqn,(x,self._left_support[0],self._right_support[0]),
xlim=(xy_min-0.5*max_diff,xy_max+0.5*max_diff),
ylim=(xy_min-0.5*max_diff,xy_max+0.5*max_diff),
rectangles=support_rectangles,show= False,annotations=annotations, axis=False)
return cab_plot
def _draw_supports(self):
member_rectangles = []
xy_min = min(self._left_support[0],self._lowest_y_global)
xy_max = max(self._right_support[0], max(self._right_support[1],self._left_support[1]))
max_diff = xy_max - xy_min
supp_width = 0.075*max_diff
member_rectangles.append(
{
'xy': (self._left_support[0]-supp_width,self._left_support[1]),
'width': supp_width,
'height':supp_width,
'color':'brown',
'fill': False
}
)
member_rectangles.append(
{
'xy': (self._right_support[0],self._right_support[1]),
'width': supp_width,
'height':supp_width,
'color':'brown',
'fill': False
}
)
return member_rectangles
def _draw_cable(self,order):
xy_min = min(self._left_support[0],self._lowest_y_global)
xy_max = max(self._right_support[0], max(self._right_support[1],self._left_support[1]))
max_diff = xy_max - xy_min
if order == -1 :
x,y = symbols('x y')
line_func = []
sorted_position = sorted(self._loads_position.items(), key = lambda item : item[1][0])
for i in range(len(sorted_position)):
if(i==0):
y = ((sorted_position[i][1][1] - self._left_support[1])*(x-self._left_support[0]))/(sorted_position[i][1][0]- self._left_support[0]) + self._left_support[1]
else:
y = ((sorted_position[i][1][1] - sorted_position[i-1][1][1] )*(x-sorted_position[i-1][1][0]))/(sorted_position[i][1][0]- sorted_position[i-1][1][0]) + sorted_position[i-1][1][1]
line_func.append((y,x<=sorted_position[i][1][0]))
y = ((sorted_position[len(sorted_position)-1][1][1] - self._right_support[1])*(x-self._right_support[0]))/(sorted_position[i][1][0]- self._right_support[0]) + self._right_support[1]
line_func.append((y,x<=self._right_support[0]))
return [Piecewise(*line_func)]
elif order == 0:
x0 = self._lowest_x_global
diff_force_height = max_diff*0.075
a,c,x,y = symbols('a c x y')
parabola_eqn = a*(x-x0)**2 + c - y
points = [(self._left_support[0],self._left_support[1]),(self._right_support[0],self._right_support[1])]
equations = []
for px, py in points:
equations.append(parabola_eqn.subs({x: px, y: py}))
solution = solve(equations, (a, c))
parabola_eqn = solution[a]*(x-x0)**2 + solution[c]
return [parabola_eqn, self._lowest_y_global - diff_force_height]
def _draw_loads(self,order):
xy_min = min(self._left_support[0],self._lowest_y_global)
xy_max = max(self._right_support[0], max(self._right_support[1],self._left_support[1]))
max_diff = xy_max - xy_min
if(order==-1):
arrow_length = max_diff*0.1
force_arrows = []
for key in self._loads['point_load']:
force_arrows.append(
{
'text': '',
'xy':(self._loads_position[key][0]+arrow_length*cos(rad(self._loads['point_load'][key][1])),\
self._loads_position[key][1] + arrow_length*sin(rad(self._loads['point_load'][key][1]))),
'xytext': (self._loads_position[key][0],self._loads_position[key][1]),
'arrowprops': {'width': 1, 'headlength':3, 'headwidth':3 , 'facecolor': 'black', }
}
)
mag = self._loads['point_load'][key][0]
force_arrows.append(
{
'text':f'{mag}N',
'xy': (self._loads_position[key][0]+arrow_length*1.6*cos(rad(self._loads['point_load'][key][1])),\
self._loads_position[key][1] + arrow_length*1.6*sin(rad(self._loads['point_load'][key][1]))),
}
)
return force_arrows
elif (order == 0):
x = symbols('x')
force_arrows = []
x_val = [self._left_support[0] + ((self._right_support[0]-self._left_support[0])/10)*i for i in range(1,10)]
for i in x_val:
force_arrows.append(
{
'text':'',
'xytext':(
i,
self._cable_eqn[0].subs(x,i)
),
'xy':(
i,
self._cable_eqn[1].subs(x,i)
),
'arrowprops':{'width':1, 'headlength':3.5, 'headwidth':3.5, 'facecolor':'black'}
}
)
mag = 0
for key in self._loads['distributed']:
mag += self._loads['distributed'][key]
force_arrows.append(
{
'text':f'{mag} N/m',
'xy':((self._left_support[0]+self._right_support[0])/2,self._lowest_y_global - max_diff*0.15)
}
)
return force_arrows
def plot_tension(self):
"""
Returns the diagram/plot of the tension generated in the cable at various points.
Examples
========
For point loads,
>>> from sympy.physics.continuum_mechanics.cable import Cable
>>> c = Cable(("A", 0, 10), ("B", 10, 10))
>>> c.apply_load(-1, ('Z', 2, 7.26, 3, 270))
>>> c.apply_load(-1, ('X', 4, 6, 8, 270))
>>> c.solve()
>>> p = c.plot_tension()
>>> p
Plot object containing:
[0]: cartesian line: Piecewise((8.91403453669861, x <= 2), (4.79150773600774, x <= 4), (19*sqrt(13)/10, x <= 10)) for x over (0.0, 10.0)
>>> p.show()
For uniformly distributed loads,
>>> from sympy.physics.continuum_mechanics.cable import Cable
>>> c=Cable(("A", 0, 40),("B", 100, 20))
>>> c.apply_load(0, ("X", 850))
>>> c.solve(58.58)
>>> p = c.plot_tension()
>>> p
Plot object containing:
[0]: cartesian line: 36465.0*sqrt(0.00054335718671383*X**2 + 1) for X over (0.0, 100.0)
>>> p.show()
"""
if len(self._loads_position) != 0:
x = symbols('x')
tension_plot = plot(self._tension_func, (x,self._left_support[0],self._right_support[0]), show=False)
else:
X = symbols('X')
tension_plot = plot(self._tension['distributed'], (X,self._left_support[0],self._right_support[0]), show=False)
return tension_plot
|