\hzSrSSKJr SSKJrJr SSKJrJrJ r J r J r J r J r Jr SSKJr SSKJr SSKJr SSKJr "S S 5rg ) zA This module can be used to solve problems related to 2D Cables. )sympify)Symbolsymbols)sincospiatandiff Piecewisesolverad)sqrt)linsolve)Matrix)plotc\rSrSrSrSr\S5r\S5r\S5r \S5r \S5r \S 5r \S 5r \S 5rS rS rSrSrSrSrSrSrSrSrSrSrg)Cablea 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]} cT/Ul/Ul0Ul/Ul00S.Ul0UlSUl0Ul0Ul[S5Ul [S5Ul SUl SUl USUS:Xa [S5eUSUS:Xa [S5e[US5n[US5nX4/URUS'[US5n[US5nXV/URUS'USUS:aURRU5 URRU5 URRU5 URRU5 URRUS5 URRUS5 OURRU5 URRU5 URRU5 URRU5 URRUS5 URRUS5 URH?nSUR[!SU-S -5'SUR[!SU-S -5'MA g) a5 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 ) distributed point_loadrNz$Supports can not have the same labelz(Supports can not be at the same locationR__x_y) _left_support_right_support _supports_support_labels_loads_loads_position_length_reaction_loads_tensionr_lowest_x_global_lowest_y_global _cable_eqn _tension_func ValueErrorappendr)self support_1 support_2x1y1x2y2is _/var/www/auris/envauris/lib/python3.13/site-packages/sympy/physics/continuum_mechanics/cable.py__init__Cable.__init__)sh&  !&(; ! ! '  ' ! Q<9Q< 'CD D q\Yq\ )GH H Yq\ " Yq\ "(*xy|$ Yq\ " Yq\ "(*xy|$ Q<)A, &    % %b )    % %b )    & &r *    & &r *  ' ' ! 5  ' ' ! 5    % %b )    % %b )    & &r *    & &r *  ' ' ! 5  ' ' ! 5%%A:;D a !6 7:;D a !6 7&cUR$)z? Returns the supports of the cable along with their positions. )rr,s r4supportsCable.supportsks ~~r7cUR$)z+ Returns the position of the left support. )rr9s r4 left_supportCable.left_supportss !!!r7cUR$)z, Returns the position of the right support. )rr9s r4 right_supportCable.right_supportzs """r7cUR$)zG Returns the magnitude and direction of the loads acting on the cable. )r!r9s r4loads Cable.loadss {{r7cUR$)z> Returns the position of the point loads acting on the cable. )r"r9s r4loads_positionCable.loads_position ###r7cUR$)z" Returns the length of the cable. )r#r9s r4length Cable.lengths ||r7cUR$)zJ Returns the reaction forces at the supports, which are initialized to 0. )r$r9s r4reaction_loadsCable.reaction_loadsrHr7cUR$)zF Returns the tension developed in the cable due to the loads applied. )r%r9s r4tension Cable.tensions }}r7c$SURR5;a [S5eXRS:dXRS:a [S5eURSn[ S5nUR X1UR- 05$)zN Returns the tension at a given value of x developed due to distributed load. rz4No distributed load added or solve method not calledrz1The value of x should be between the two supportsX)r%keysr*rrrsubsr&)r,xArSs r4 tension_atCable.tension_ats  2 2 4 4ST T ""1% %-?-?-B)BPQ Q MM- ( 3KvvqD111344r7cURSURS- S-URSURS- S-- S-nX:a [S5eXlg)a This method specifies the length of the cable Parameters ========== length : Sympifyable The length of the cable Examples ======== >>> from sympy.physics.continuum_mechanics.cable import Cable >>> c = Cable(('A', 0, 10), ('B', 10, 10)) >>> c.apply_length(20) >>> c.length 20 rrr?z@length should not be less than the distance between the supportsN)rrr*r#)r,rJdists r4 apply_lengthCable.apply_lengthss&##A&)<)+>q+AAAEFILN =_` ` r7cXR;a [S5eURRU5nURUS-S-nURUSnURUSn[ US5n[ US5nUR H2n U S[ Xu5:dU S[Xu5::dM)[S5e URRU5 URR5 URR5 URR5 URRU5 Xx/URUS'XW:aURRU5 URRU5 URRU5 URRU5 URRUS5 OURRU5 URRU5 URRU5 URRU5 URRSUS5 URH?nSUR[!SU-S-5'SUR[!SU-S-5'MA g ) a This method changes the mentioned support with a new support. Parameters ========== label: String or symbol The label of the support to be changed new_support: Tuple of the form (new_label, x, y) new_label: String or symbol The label of the new support x: Sympifyable The x-coordinate of the position of the new support. y: Sympifyable The y-coordinate of the position of the new support. Examples ======== >>> from sympy.physics.continuum_mechanics.cable import Cable >>> c = Cable(('A', 0, 10), ('B', 10, 10)) >>> c.supports {'A': [0, 10], 'B': [10, 10]} >>> c.change_support('B', ('C', 5, 6)) >>> c.supports {'A': [0, 10], 'C': [5, 6]} z&No support exists with the given labelrrrz>The change in support will throw an existing load out of rangerrrN)rr*r indexrr"maxminpoprclearrr$remover+insertr) r,label new_supportr3 rem_labelr/r0rVyls r4change_supportCable.change_supportsZ<  &EF F  & &u -((!A#q1 ^^I &q ) ^^I &q ) KN # KN #%%Ats1z!QqTSZ%7 !abb& 5!   " !!# ""$ ##E**+{1~& 6    % %b )    % %b )    & &q )    & &q )  ' ' A 7    % %a (    % %a (    & &r *    & &r *  ' ';q> :%%A:;D a !6 7:;D a !6 7&r7cUS:Xa[URS5S:wa [S5eUSnX0RS;a [S5e[US5n[US5nX@RS:dX@R S:a [S 5e[US 5n[US 5nXg/URSU'XE/UR U'gUS:Xaf[UR 5S:wa [S 5eUSnX0RS;a [S5e[US5nX`RSU'g[S 5e)aV 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': {}} rrzDistributed load already existsrzLabel already existsrrz2The load should be positioned between the supportszPoint load(s) already existzOrder should be either -1 or 0N)lenr!r*rrrr")r,orderloadrgrVrj magnitude directions r4 apply_loadCable.apply_loadsVB B;4;;}-.!3 !BCCGE L11 !788Q AQ A&&q))Q1C1CA1F-F !UVVQ(IQ(I09/EDKK %e ,+,&D  ' aZ4''(A- !>??GE M22 !788Q(I09KK &u -=> >r7cUHn[UR5S:XaCX RS;a[SU-S-5eURSR U5 M_X RS;a[SU-S-5eURSR U5 URR U5 M g)a 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': {}} rrzError removing load z: no such load exists disrtibutedrN)rrr"r!r*rc)r,argsr3s r4 remove_loadsCable.remove_loads~s,A4''(A-KK 66$%;a%?BY%YZZKK .2215KK 55$%;a%?BY%YZZKK -11!4((,,Q/r7c d[UR5S:wGa1[URR5SS9nUR UR S5 UR SUR S5 URR5 SnSnSnSnSUl /n[S5n[S[U5S- 5GHnn U S:XarU=R[URSURX)SS- S-URSURX)SS- S--5- sl OU=R[URX)S- SSURX)SS- S-URX)S- SSURX)SS- S--5- sl U [U5S- :XaqU=R[URSURX)SS- S-URSURX)SS- S--5- sl X0RSX)SS[!["URSX)SS-S- 5-[%URSURX)SS- 5-- nX0RSX)SS['["URSX)SS-S- 5-[%URSURX)SS- 5-- nXPRSX)SS[!["URSX)SS-S- 5-- nX`RSX)SS['["URSX)SS-S- 5-- n[)X)SS -X)S-S-5n URX)SSn URX)SSn Sn SnU [U5S- :XaURSnURSn O4URX)S-SSnURX)S-SSn [+X- X- - 5nU*[%URSURX)SS- 5[!U5-[%URSURX)SS- 5['U5--- nUURU 'UR UX:*45 X@RSX)SS[!["URSX)SS-S- 5-[%URSURX)SS- 5-- nX@RSX)SS['["URSX)SS-S- 5-[%URSURX)SS- 5-- nGMq [)USSS -USS-5n URUSSSn URUSSSn URSnURSn [+X- X- - 5*nU*[%URSURUSSS- 5[!U5-[%URSURUSSS- 5['U5--- nUURU 'UR SUX:*45 [-U6Ul["S- U- nUR Sn ['U5*U-UR0[)S U -S -5'U['U5*U-- n[!U5U-UR0[)S U -S -5'U[!U5U-- nUR Sn U*UR0[)S U -S -5'U*UR0[)S U -S -5'g[URS 5S:wGa[U5S:Xa [3S5e[5US5nUUl[)SSS9n[)S5n[9URSU- S-SURS/URSU- S-SURS//5n[;[=UUU455n[U5S:Xd USSS:Xa [3S5eUSSnUSSUSSUS---nSUSS-U-nUSSUlUR>n[)S5n[)S5nUUU-S--UUU---U-U- nRS RA55nUSnUURSU- S--SURSU- -- nURR5 [CUU5nU[![+U55- URS 'UR Sn UREURS5[![+URGUURSU- 555-UR0[)S U -S -5'UREURS5['[+URGUURSU- 555-UR0[)S U -S -5'UR Sn UREURS5[![+URGUURSU- 555-UR0[)S U -S -5'UREURS5['[+URGUURSU- 555-UR0[)S U -S -5'gg)a 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} rcUSS$Nrritems r4Cable.solve..W[\]W^_`War7keyrrVrr_rrrrz%Provide the lowest point of the cableaT)positivecz2The lowest point is inconsistent with the supportsrSYN)$rrr"sorteditemsr+r rfr%rdr#rrangerrrr!rrabsrrr r r)r$r*rr&rlistrr'valuesr rXrU) r,r{sorted_positionmoment_sum_from_left_supportmoment_sum_from_right_supportF_xF_y tension_funcrVr3rgr2r1r0r/angle_with_horizontalrPlowest_xrrMcoefficient_solutionrWCBlowest_yrSr temp_list applied_forcehorizontal_force_constanttangent_slope_to_curves r4r Cable.solves8^ t## $ )$T%9%9%?%?%AIabO  " "4#7#7#: ;  " "1d&:&:1&= > MM   !+, (,- )CCDLL A1c/21456LL$(:(:1(=@T@TUdUghiUj@klm@n(nqr'rvzwIwIJKwLOSOcOcdsdvwxdyOz{|O}w}@AvA(A#BBLLL$(<(<_qS=QRS=T(UVW(X[_[o[opqCDEqF\GHI\J)JMN(NRVRfRfgvyzwzg{|}g~R@ARBEIEYEYZiZlmnZoEpqrEsRsvwQw(w#xxLO,Q..LL$(;(;A(>AUAUVeVhijVkAlmnAo(ors'sw{xKxKLMxNQUQeQefufxyzf{Q|}~QxBCwC(C#DDL, L0I/J\]^J_0`ab0cfijlosozoz|HpIJYJ\]^J_p`abpckcfikigj1jmpquqCqCDEqFIMI]I]^m^pqr^sItuvIwqwmx1xx,, L0I/J\]^J_0`ab0cfijlosozoz|HpIJYJ\]^J_p`abpckcfikigj1jmpquqCqCDEqFIMI]I]^m^pqr^sItuvIwqwmx1xx,{{<01CA1FGJSQSVZVaVabnVopqCDEqFWGHIWJRJMPRPNQQQ{{<01CA1FGJSQSVZVaVabnVopqCDEqFWGHIWJRJMPRPNQQQ1!4S819Ma9PPQ))/*$> !56DD !,,Hs As A1x>@AI%aLM)6$:M:Ma:PS[:[^_9_)_deimi|i|}~iCKjKeL)M % MM   !%)!QZ "+DDQgLhHi+jDMM- (((+E<@OODL^L^_`La>> 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() rVrrrorzRsolve method not called and/or values provided for loads and supports not adequater[F)xlimylim rectanglesshow annotationsaxis)r_draw_supportsrbrr'rarrrr" _draw_cabler( _draw_loadsr!r*r)r,rVrsupport_rectanglesxy_minxy_maxmax_diffcab_plots r4draw Cable.drawPsH 3K !002T''*4+@+@AT((+S1D1DQ1GHZHZ[\H]-^_? t## $ )"..r2DO ++B/ /K ]+ , 1"..q1DO ++A. .Kqr rgg!,>,>q,A$BUBUVWBX)Yg$S\1&X2EF$S\1&X2EF#5Ek`eg r7c/n[URSUR5n[URS[URSURS55nX2- nSU-nUR URSU- URS4UUSSS.5 UR URSURS4UUSSS.5 U$)Nrr333333?brownF)xywidthheightcolorfill)rbrr'rarr+)r,member_rectanglesrrr supp_widths r4rCable._draw_supportssT''*4+@+@AT((+S1D1DQ1GHZHZ[\H]-^_?8^   ))!,Z78J8J18MN##      **1-d.A.A!.DE##   ! r7c [URSUR5n[URS[URSURS55nX2- nUS:XGa[ S5upV/n[ URR5SS9n[[U55Hn U S:XaXXSSURS- XPRS- -XSSURS- - URS-nOPXSSXS- SS- XXU S- SS- -XSSXS- SS- - XS- SS-nURXeXSS:*45 M U[U5S- SSURS- XPRS- -UW SSURS- - URS-nURXeURS:*45 [U6/$US:XaURn US-n [ S5uppVXU - S --U -U- nURSURS4URSURS4/n/nUH*unnURURUUUU055 M, [UX45nUU XZ- S --UU -nXRU - /$g) Nrrrozx ycUSS$rrrs r4r#Cable._draw_cable..rr7rrza c x yr)rbrr'rarrrr"rrrrr+r r&rUr )r,rsrrrrVrj line_funcrr3x0diff_force_heightrr parabola_eqnpoints equationspxpysolutions r4rCable._draw_cablesdT''*4+@+@AT((+S1D1DQ1GHZHZ[\H]-^_? B;%.CAI$T%9%9%?%?%AIabO3/0a4),Q/2T5G5G5JJQOaOabcOdMdehwhz{|h}~iACGCUCUVWCXiXY\`\n\nop\qqA),Q/2_qS5I!5LQ5OOSTefghehUijkUlmnUoSoptCtFGHtIJKtLN]`a^aNbcdNefgNhthil{~|l@ABlCDElFFA  !'9!'!BC!Y/H#A;qy08A;>L "7"7:K"KL Lr7c V[URSUR5n[URS[URSURS55nX2- nUS:XGaUS-n/nUR SGHcnUR SURUSU[[UR SUS55--URUSU[[UR SUS55--4URUSURUS4SSSSS .S .5 UR SUSnUR US 3URUSUS -[[UR SUS55--URUSUS -[[UR SUS55--4S .5 GMf U$US:XGa2[S5n /n[SS5V s/sH7oRSURSURS- S- U --PM9 n n U HZn UR SU URSRX54U URSRX54SSSSS .S.5 M\ SnUR SHnXR SU- nM UR US3URSURS-S- URUS-- 4S .5 U$gs sn f)Nrrrog?rrpblack)r headlength headwidth facecolor)textrxytext arrowpropsNg?)rrrV g @)rrrrrz N/mrg333333?)rbrr'rarr!r+r"rr rrrr(rU) r,rsrrr arrow_length force_arrowsrmagrVr3x_vals r4rCable._draw_loadssT''*4+@+@AT((+S1D1DQ1GHZHZ[\H]-^_? "9#CNsSVW[WbWbcoWpqtWuvwWxSyOz>zz|1$ qj ALlqrstvlwxlwgh''*t/B/B1/EdFXFXYZF[/[]_._ab-bblwEx##! OOA.33A8"  OOA.33A8/0cs`g%h  C{{=1{{=1#662   !U$<--a01D1DQ1GGJ4K`K`cklpcpKpq   ;ys>L&c2[UR5S:wa>[S5n[URXR SUR S4SS9nU$[S5n[URSX0R SUR S4SS9nU$)an 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() rrVF)rrSr)rrr"rrr)rrr%)r,rV tension_plotrSs r4 plot_tensionCable.plot_tensionsB t## $ ) A !3!3a8J8J18MdNaNabcNd5elqrL A }!=BTBTUVBWX\XkXklmXn?ov{|Lr7) r(rr#r!r"r&r'r$rr rr%r)N)__name__ __module__ __qualname____firstlineno____doc__r5propertyr:r=r@rCrFrJrMrPrXr]rlrwr|r rrrrr__static_attributes__rr7r4rrs4@rs3 ',AAA9+!a a r7