g.GhddlmZmZmZddlmZddlmZddl m Z dgZ GddeeZ dZ dS) )sympifyAddImmutableMatrix) EvalfMixin) Printable) prec_to_dpsDyadicceZdZdZdZdZedZdZeZ dZ e Z dZ e Z dZd Zd Zd Zd Zd ZdZdZdZdZeZddZddZdZdZdZdZdZdZ dZ!dS)r ayA Dyadic object. See: https://en.wikipedia.org/wiki/Dyadic_tensor Kane, T., Levinson, D. Dynamics Theory and Applications. 1985 McGraw-Hill A more powerful way to represent a rigid body's inertia. While it is more complex, by choosing Dyadic components to be in body fixed basis vectors, the resulting matrix is equivalent to the inertia tensor. Fc>g|_|dkrg}t|dkrOd}t|jD]\}}t|ddt|j|dkrt|ddt|j|dkrd|j|d|ddz|dd|ddf|j|<||dd}n|dkr;|j|d||dt|dkOd}|t|jkr|j|ddk|j|ddkz|j|ddkzr*|j|j||dz}|dz }|t|jkdSdS)a2 Just like Vector's init, you should not call this unless creating a zero dyadic. zd = Dyadic(0) Stores a Dyadic as a list of lists; the inner list has the measure number and the two unit vectors; the outerlist holds each unique unit vector pair. rN)argslen enumeratestrremoveappend)selfinlistaddedivs W/var/www/html/ai-engine/env/lib/python3.11/site-packages/sympy/physics/vector/dyadic.py__init__zDyadic.__init__s Q;;F&kkQE!$),,  11&&#dil1o*>*>>>VAYq\**c$)A,q/.B.BBB$(IaLOfQil$B$*1IaL&)A,$@DIaLMM&),,,EEzz   +++ fQi(((&kkQ #di..  1aA%$)A,q/Q*>?Yq\!_)+    1...Q FA #di..      ctS)zReturns the class Dyadic. )r rs rfuncz Dyadic.func@s  rcXt|}t|j|jzS)zThe add operator for Dyadic. ) _check_dyadicr rrothers r__add__zDyadic.__add__Es&e$$di%*,---rct|j}t|}t|D]4\}}|||dz||d||df||<5t |S)aMultiplies the Dyadic by a sympifyable expression. Parameters ========== other : Sympafiable The scalar to multiply this Dyadic with Examples ======== >>> from sympy.physics.vector import ReferenceFrame, outer >>> N = ReferenceFrame('N') >>> d = outer(N.x, N.x) >>> 5 * d 5*(N.x|N.x) rr r )listrrrr )rr"newlistrrs r__mul__zDyadic.__mul__Lsu&ty//g&& ) )DAq'!*Q-/A!!*Q-)GAJJgrcddlm}m}t|trt |}t d}|jD]d}|jD]Z}||d|dz|d|dz|d|dzz }[enP||}|d}|jD]2}||d|dz|d|zz }3|S)aThe inner product operator for a Dyadic and a Dyadic or Vector. Parameters ========== other : Dyadic or Vector The other Dyadic or Vector to take the inner product with Examples ======== >>> from sympy.physics.vector import ReferenceFrame, outer >>> N = ReferenceFrame('N') >>> D1 = outer(N.x, N.y) >>> D2 = outer(N.y, N.y) >>> D1.dot(D2) (N.x|N.y) >>> D1.dot(N.y) N.x r)Vector _check_vectorr r ) sympy.physics.vector.vectorr)r* isinstancer r rdotouter)rr"r)r*olrv2s rr-z Dyadic.doths , FEEEEEEE eV $ $ 6!%((EBY Q Q*QQB!A$A,!A$((2a5//:adjjA>O>OPPBBQ Q"M%((EBY 6 6adQqTkQqTXXe__55 rc2|d|z S)z0Divides the Dyadic by a sympifyable expression. r )r'r!s r __truediv__zDyadic.__truediv__s||AI&&&rc|dkrtd}t|}|jgkr |jgkrdS|jgks |jgkrdSt|jt|jkS)z[Tests for equality. Is currently weak; needs stronger comparison testing rTF)r r rsetr!s r__eq__z Dyadic.__eq__sx A::1IIEe$$ IOO%*"2"24i2oo5:#3#3549~~UZ00rc||k SNr!s r__ne__z Dyadic.__ne__s5=  rc |dzSNr8rs r__neg__zDyadic.__neg__s byrc|j}t|dkrtdSg}t|D]\}}||ddkr]|d|||dzdz|||dzu||ddkr]|d|||dzdz|||dz||ddkr|||d}t ||dtrd|z}|d r |dd}d}nd}|||z|||dzdz|||dzd |}|dr |d d}n|d r |dd}|S) Nrr  + z\otimes r r< - (%s)- ) rrrrr_printr,r startswithjoin rprinterarr/rrarg_str str_startoutstrs r_latexz Dyadic._latexsW Y r77a<<q66M bMM B BDAq!uQx1}} %'..Aq":"::[H!..Aq2234444AqR %!..Aq223%&"..Aq2234444 AqQ!..Aq22beAh,,/$w.G%%c**&%abbkG %II %I )g-r!uQx0H0HH%&(/r!uQx(@(@ABBB   U # # ABBZFF   s # # ABBZF rc>|Gfdd}|S)Nc eZdZdZfdZdS)Dyadic._pretty..Fakerc   j} }t|dkrtdS jrdnd}g}t |D]\}}||ddkrX|d|||d||||dgp||ddkrX|d|||d||||dg||ddkrt||dtr:| ||d d} n!|||d} | d r | dd} d} nd} || | d |||d||||dgd |} | dr | d d} n| d r | dd} | S) Nru⊗|r r?r r<r@rBrErCrD) rrr _use_unicoderextenddoprintr,rrFparensrGrH)rrkwargsrKmppbarr/rrrLrMrNerJs rrenderz#Dyadic._pretty..Fake.renders|Vr77a<<q66M-4-AJ))s%bMM::DAq!uQx1}} 5"%++beAh"7"7"%"%++beAh"7"7#9:::: AqR 5"%++beAh"7"7"%"%++beAh"7"7#9::::AqQ%beAh44<&)jj "1a'*'**0&((1'6GG'*kk"Q%(&;&;G"--c22.&-abbkG(-II(-I 9gs"%++beAh"7"7"%"%++beAh"7"7#9::: $$U++(#ABBZFF&&s++(#ABBZF rN)__name__ __module__ __qualname__baseliner])r\rJsrFakerRs8H- - - - - - - - rrbr8)rrJrbr\s ` @r_prettyzDyadic._prettysN 0 0 0 0 0 0 0 0 0 0 0 btvv rcd|z|zSr;r8r!s r__rsub__zDyadic.__rsub__sT U""rc|j}t|dkr|dSg}t|D]\}}||ddkr`|d|||dzdz|||dzdzx||ddkr`|d|||dzdz|||dzdz||ddkr|||d}t ||dt rd |z}|dd kr |dd }d }nd }|||zdz|||dzdz|||dzdzd|}|d r |dd }n|dr |dd }|S)zPrinting method. rr z + (rTr )r<z - (rArBNr@r?z*(rCrDrE) rrrFrrr,rrHrGrIs r _sympystrzDyadic._sympystrsp Y r77a<<>>!$$ $ bMM @ @DAq!uQx1}} &7>>"Q%(#;#;;cA!..Aq223589::::AqR &7>>"Q%(#;#;;cA!..Aq223589::::AqQ!..Aq22beAh,,/$w.G1:$$%abbkG %II %I )g-4!..Aq223 'r!uQx 8 89;>?@@@   U # # ABBZFF   s # # ABBZF rc2||dzS)zThe subtraction operator. r<)r#r!s r__sub__zDyadic.__sub__*s||EBJ'''rcddlm}||}td}|jD]B}||d|d|d|zz }C|S)aReturns the dyadic resulting from the dyadic vector cross product: Dyadic x Vector. Parameters ========== other : Vector Vector to cross with. Examples ======== >>> from sympy.physics.vector import ReferenceFrame, outer, cross >>> N = ReferenceFrame('N') >>> d = outer(N.x, N.x) >>> cross(d, N.y) (N.x|N.z) r)r*r r )r+r*r rr.cross)rr"r*r/rs rrlz Dyadic.cross.s{$ >===== e$$ AYY ; ;A !A$!A$**adjj&7&799: :BB rNc(ddlm}||||S)aExpresses this Dyadic in alternate frame(s) The first frame is the list side expression, the second frame is the right side; if Dyadic is in form A.x|B.y, you can express it in two different frames. If no second frame is given, the Dyadic is expressed in only one frame. Calls the global express function Parameters ========== frame1 : ReferenceFrame The frame to express the left side of the Dyadic in frame2 : ReferenceFrame If provided, the frame to express the right side of the Dyadic in Examples ======== >>> from sympy.physics.vector import ReferenceFrame, outer, dynamicsymbols >>> from sympy.physics.vector import init_vprinting >>> init_vprinting(pretty_print=False) >>> N = ReferenceFrame('N') >>> q = dynamicsymbols('q') >>> B = N.orientnew('B', 'Axis', [q, N.z]) >>> d = outer(N.x, N.x) >>> d.express(B, N) cos(q)*(B.x|N.x) - sin(q)*(B.y|N.x) r)express)sympy.physics.vector.functionsrn)rframe1frame2rns rrnzDyadic.expressJs+@ ;:::::wtVV,,,rcn|tfd|DddS)aReturns the matrix form of the dyadic with respect to one or two reference frames. Parameters ---------- reference_frame : ReferenceFrame The reference frame that the rows and columns of the matrix correspond to. If a second reference frame is provided, this only corresponds to the rows of the matrix. second_reference_frame : ReferenceFrame, optional, default=None The reference frame that the columns of the matrix correspond to. Returns ------- matrix : ImmutableMatrix, shape(3,3) The matrix that gives the 2D tensor form. Examples ======== >>> from sympy import symbols, trigsimp >>> from sympy.physics.vector import ReferenceFrame >>> from sympy.physics.mechanics import inertia >>> Ixx, Iyy, Izz, Ixy, Iyz, Ixz = symbols('Ixx, Iyy, Izz, Ixy, Iyz, Ixz') >>> N = ReferenceFrame('N') >>> inertia_dyadic = inertia(N, Ixx, Iyy, Izz, Ixy, Iyz, Ixz) >>> inertia_dyadic.to_matrix(N) Matrix([ [Ixx, Ixy, Ixz], [Ixy, Iyy, Iyz], [Ixz, Iyz, Izz]]) >>> beta = symbols('beta') >>> A = N.orientnew('A', 'Axis', (beta, N.x)) >>> trigsimp(inertia_dyadic.to_matrix(A)) Matrix([ [ Ixx, Ixy*cos(beta) + Ixz*sin(beta), -Ixy*sin(beta) + Ixz*cos(beta)], [ Ixy*cos(beta) + Ixz*sin(beta), Iyy*cos(2*beta)/2 + Iyy/2 + Iyz*sin(2*beta) - Izz*cos(2*beta)/2 + Izz/2, -Iyy*sin(2*beta)/2 + Iyz*cos(2*beta) + Izz*sin(2*beta)/2], [-Ixy*sin(beta) + Ixz*cos(beta), -Iyy*sin(2*beta)/2 + Iyz*cos(2*beta) + Izz*sin(2*beta)/2, -Iyy*cos(2*beta)/2 + Iyy/2 - Iyz*sin(2*beta) + Izz*cos(2*beta)/2 + Izz/2]]) Ncjg|]/}D]*}||+0Sr8)r-).0rjsecond_reference_framers r z$Dyadic.to_matrix..sO...a,..qquuT{{q))....rrD)Matrixreshape)rreference_framervs` `r to_matrixzDyadic.to_matrixms\V " )%4 ".....?...///6wq!}} =rc `tfd|jDtdS)z(Calls .doit() on each term in the Dyadicc pg|]2}t|djdi|d|dfg3S)rr r r8)r doit)rtrhintss rrwzDyadic.doit..sY(((YQqTY////1qt<=>>(((rrsumrr )rrs `rr~z Dyadic.doitsF((((!Y((()/44 4rc&ddlm}|||S)aTake the time derivative of this Dyadic in a frame. This function calls the global time_derivative method Parameters ========== frame : ReferenceFrame The frame to take the time derivative in Examples ======== >>> from sympy.physics.vector import ReferenceFrame, outer, dynamicsymbols >>> from sympy.physics.vector import init_vprinting >>> init_vprinting(pretty_print=False) >>> N = ReferenceFrame('N') >>> q = dynamicsymbols('q') >>> B = N.orientnew('B', 'Axis', [q, N.z]) >>> d = outer(N.x, N.x) >>> d.dt(B) - q'*(N.y|N.x) - q'*(N.x|N.y) r)time_derivative)ror)rframers rdtz Dyadic.dts)2 CBBBBBtU+++rctd}|jD]<}|t|d|d|dfgz }=|S)zReturns a simplified Dyadic.rr r )r rsimplify)routrs rrzDyadic.simplifysUQii ; ;A 6AaDMMOOQqT1Q489:: :CC rcdtfd|jDtdS)a5Substitution on the Dyadic. Examples ======== >>> from sympy.physics.vector import ReferenceFrame >>> from sympy import Symbol >>> N = ReferenceFrame('N') >>> s = Symbol('s') >>> a = s*(N.x|N.x) >>> a.subs({s: 2}) 2*(N.x|N.x) c pg|]2}t|dji|d|dfg3S)rr r )r subs)rtrrrYs rrwzDyadic.subs..sX(((YQqTY7771qtDEFF(((rrr)rrrYs ``rrz Dyadic.subssN (((((!Y((()/44 4rct|stdtd}|jD]*\}}}|||||zz }+|S)z/Apply a function to each component of a Dyadic.z`f` must be callable.r)callable TypeErrorr rr.)rfrabcs r applyfunczDyadic.applyfuncsi{{ 5344 4Qiiy ' 'GAq! 11Q441771::& &CC rc|js|Sg}t|}|jD]R}t|}|d||d<|t |St |S)Nr)n)rrr%evalfrtupler )rprecnew_argsdpsr new_inlists r _eval_evalfzDyadic._eval_evalfsy K$i / /FfJ"1IOOcO22JqM OOE*-- . . . .hrcg}|jD]Q}t|}|d||d<|t |Rt |S)a Replace occurrences of objects within the measure numbers of the Dyadic. Parameters ========== rule : dict-like Expresses a replacement rule. Returns ======= Dyadic Result of the replacement. Examples ======== >>> from sympy import symbols, pi >>> from sympy.physics.vector import ReferenceFrame, outer >>> N = ReferenceFrame('N') >>> D = outer(N.x, N.x) >>> x, y, z = symbols('x y z') >>> ((1 + x*y) * D).xreplace({x: pi}) (pi*y + 1)*(N.x|N.x) >>> ((1 + x*y) * D).xreplace({x: pi, y: 2}) (1 + 2*pi)*(N.x|N.x) Replacements occur only if an entire node in the expression tree is matched: >>> ((x*y + z) * D).xreplace({x*y: pi}) (z + pi)*(N.x|N.x) >>> ((x*y*z) * D).xreplace({x*y: pi}) x*y*z*(N.x|N.x) r)rr%xreplacerrr )rrulerrrs rrzDyadic.xreplaceslPi / /FfJ&qM22488JqM OOE*-- . . . .hrr7)"r^r_r`__doc__ is_numberrpropertyrr#__radd__r'__rmul__r-__and__r2r5r9r=rOrcrerhrjrl__xor__rnr{r~rrrrrrr8rrr r s  I$$$LX... H4H"""JG'''111 !!!"""H444l###"""H(((4G!-!-!-!-F/=/=/=/=b444 ,,,8444&    - - - - - rcNt|tstd|S)NzA Dyadic must be supplied)r,r r)r"s rr r s( eV $ $53444 LrN)sympyrrrrxsympy.core.evalfrsympy.printing.defaultsrmpmath.libmp.libmpfr__all__r r r8rrrs9999999999''''''------++++++ *P P P P P Y P P P fr