XhzFhddlmZmZmZddlmZddlmZddl 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 m/var/www/tools.fuzzalab.pt/emblema-extractor/venv/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$$di%*,---rct|j}t|}tt |D]1}|||dz||d||df||<2t |S)aMultiplies the Dyadic by a sympifyable expression. Parameters ========== other : Sympafiable The scalar to multiply this Dyadic with Examples ======== >>> from sympy.physics.vector import ReferenceFrame, outer >>> N = ReferenceFrame('N') >>> d = outer(N.x, N.x) >>> 5 * d 5*(N.x|N.x) rr r )listrrrangerr )rr"newlistrs r__mul__zDyadic.__mul__Lsw&ty//s7||$$ ) )A'!*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$$ IOO%*"2"24i2oo5:#3#3549~~UZ00rc||k SNr!s r__ne__z Dyadic.__ne__s5=  rc |dzSNr9rs r__neg__zDyadic.__neg__s byrc|j}t|dkrtdSg}|D]{}|ddkrQ|d||dzdz||dz`|ddkrQ|d||dzdz||dz|ddkr||d}t |dt rd|z}|d r |dd}d}nd}|||z||dzdz||dz}d |}|dr |d d}n|d r |dd}|S) Nrr  + z\otimes r r= - (%s)- ) rrrr_printr-r startswithjoinrprinterarr0rarg_str str_startoutstrs r_latexz Dyadic._latexs Y r77a<<q66M  > >Atqyy %'..1"6"66D!..1../00001 %!..1../%&"..1../0000 1!..1..adC((/$w.G%%c**&%abbkG %II %I )g-qt0D0DD%&(/qt(<(<=>>>   U # # ABBZFF   s # # ABBZF rc>|Gfdd}|S)Nc eZdZdZfdZdS)Dyadic._pretty..Fakerc \ j} }t|dkrtdS jrdnd}g}|D]}|ddkrL|d||d|||dg[|ddkrL|d||d|||dg|ddkrt |dtr4||d d}n||d}| d r |dd}d} nd} || |d ||d|||dgd |} | dr | d d} n| d r | dd} | S) Nru⊗|r r@r r=rArCrFrDrE) rrr _use_unicodeextenddoprintr-rrGparensrHrI) rrkwargsrLmppbarr0rrMrNrOerKs rrenderz#Dyadic._pretty..Fake.rendersBVr77a<<q66M-4-AJ))s66Atqyy 5"%++ad"3"3"%"%++ad"3"3#56666 1 5"%++ad"3"3"%"%++ad"3"3#566661%adC008&)jj !!'&'&&,fhhq'2GG'*kk!A$&7&7G"--c22.&-abbkG(-II(-I 9gs"%++ad"3"3"%"%++ad"3"3#5666 $$U++(#ABBZFF&&s++(#ABBZF rN)__name__ __module__ __qualname__baseliner^)r]rKsrFakerSs8H- - - - - - - - rrcr9)rrKrcr]s ` @r_prettyzDyadic._prettysN 0 0 0 0 0 0 0 0 0 0 0 btvv rcd|z|zSr<r9r!s r__rsub__zDyadic.__rsub__sT U""rc|j}t|dkr|dSg}|D]~}|ddkrT|d||dzdz||dzdzc|ddkrT|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 + (rUr )r=z - (rBrCNrAr@z*(rDrErF)rrrGrr-rrIrHrJs r _sympystrzDyadic._sympystrs5 Y r77a<<>>!$$ $  < >!A$#7#77#=!..1../14566661 &7>>!A$#7#77#=!..1../14566661!..1..adC((/$w.G1:$$%abbkG %II %I )g-4!..1../ 'qt 4 457:;<<<   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+r0rs rrmz 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.functionsro)rframe1frame2ros rrozDyadic.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]*}||+0Sr9)r.).0rjsecond_reference_framers r z$Dyadic.to_matrix..sO...a,..qquuT{{q))....rrE)Matrixreshape)rreference_framerws` `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 r9)r doit)rurhintss rrxzDyadic.doit..sY(((YQqTY////1qt<=>>(((rrsumrr )rrs `rrz 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)rpr)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)rurrrZs rrxzDyadic.subs..sX(((YQqTY7771qtDEFF(((rrr)rrrZs ``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*-- . . . .hrr8)"r_r`ra__doc__ is_numberrpropertyrr#__radd__r(__rmul__r.__and__r3r6r:r>rPrdrfrirkrm__xor__ror|rrrrrrrr9rrr 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)sympyrrrrysympy.core.evalfrsympy.printing.defaultsrmpmath.libmp.libmpfr__all__r r r9rrrs9999999999''''''------++++++ *P P P P P Y P P P fr