\hzF`SSKJrJrJr SSKJr SSKJr SSK J r S/r "SS\\5r Sr g) )sympifyAddImmutableMatrix) EvalfMixin) Printable) prec_to_dpsDyadicc\rSrSrSrSrSr\S5rSr \ r Sr \ r Sr \ rS rS rS rS rS rSrSrSrSrSr\rSSjrSSjrSrSrSrSrSr Sr!Sr"Sr#g)r a]A 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/UlUS:Xa/n[U5S:wGaSn[UR5Hup4[USS5[URUS5:XdM5[USS5[URUS5:XdMeURUSUSS-USSUSS4URU'UR US5 Sn O US:wa2URR US5 UR US5 [U5S:waGMSnU[UR5:aURUSS:HURUSS:H-URUSS:H-(a-URR URU5 US-nUS- nU[UR5:aMgg)z 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 S/var/www/auris/envauris/lib/python3.13/site-packages/sympy/physics/vector/dyadic.py__init__Dyadic.__init__s Q;F&kQE!$)),1&#diil1o*>>VAYq\*c$))A,q/.BB$(IIaLOfQil$B$*1IaL&)A,$@DIIaLMM&),E-z   + fQi(&kQ #dii. 1aA%$))A,q/Q*>?YYq\!_)+   1.Q FA #dii. c[$)zReturns the class Dyadic. )r rs rfunc Dyadic.func@s  rc\[U5n[URUR-5$)zThe add operator for Dyadic. ) _check_dyadicr rrothers r__add__Dyadic.__add__Es$e$dii%**,--rc[UR5n[U5n[[ U55HnXUS-X#SX#S4X#'M [ U5$)a%Multiplies 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__Dyadic.__mul__Lsa&tyy/s7|$A!*Q-/A!*Q-)GJ%grcSSKJnJn [U[5(a{[ U5n[ S5nUR HSnUR H@nXESUS-USRUS5-USRUS5-- nMB MU U$U"U5nU"S5nUR H%nXESUS-USRU5-- nM' U$)azThe 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_vectorrr ) sympy.physics.vector.vectorr/r0 isinstancer r#rdotouter)rr%r/r0olrv2s rr3 Dyadic.doths, F eV $ $!%(EBYY**BA$A,!A$((2a5/:adjjA>OPPB% "%(EBYYdQqTkQqTXXe_55 rc*URSU- 5$)z0Divides the Dyadic by a sympifyable expression. r )r,r$s r __truediv__Dyadic.__truediv__s||AI&&rcUS:Xa [S5n[U5nUR/:XaUR/:XagUR/:XdUR/:Xag[UR5[UR5:H$)zKTests for equality. Is currently weak; needs stronger comparison testing rTF)r r#rsetr$s r__eq__ Dyadic.__eq__sk A:1IEe$ IIO%**"2ii2o5::#3499~UZZ00rc X:X+$Nr$s r__ne__ Dyadic.__ne__s   rc US-$NrArs r__neg__Dyadic.__neg__s byrcZURn[U5S:Xa [S5$/nUGH2nUSS:Xa@URSUR US5-S-UR US5-5 MMUSS:Xa@URSUR US5-S-UR US5-5 MUSS:wdMUR US5n[ US[ 5(aSU-nURS 5(aUSSnSnOSnURXe-UR US5-S-UR US5-5 GM5 S RU5nURS5(aUS SnU$URS 5(aUSSnU$) Nrr  + z\otimes rrF - (%s)- ) rrrr_printr2r startswithjoinrprinterarr5rarg_str str_startoutstrs r_latex Dyadic._latexs YY r7a<q6M Atqy %'..1"66D!..1./01 %!..1./%&"..1./0 1!..1.adC(($w.G%%c**%abkG %I %I )-qt0DD%&(/qt(<=>-0   U # #ABZF   s # #ABZF rc4^^Um"UU4SjS5nU"5$)Nc(>\rSrSrSrUU4SjrSrg)Dyadic._pretty..Fakerc >T RnT n[U5S:Xa [S5$T R(aSOSn/nUGHCnUSS:Xa;UR SUR US5UUR US5/5 MHUSS:Xa;UR SUR US5UUR US5/5 MUSS:wdM[ US[5(a&URUS5R5SnOUR US5nURS 5(aUSSnSn OSn UR XS UR US5UUR US5/5 GMF S RU5n U RS5(aU S Sn U $U RS 5(aU SSn U $) Nru⊗|r rJrrFrKrMrPrNrO) rrr _use_unicodeextenddoprintr2rrQparensrRrS) rrkwargsrVmppbarr5rrWrXrYerUs rrender#Dyadic._pretty..Fake.rendersVVr7a<q6M-4-A-A)sAtqy 5"%++ad"3"%"%++ad"3#56 1 5"%++ad"3"%"%++ad"3#561%adC00&)jj !!'&&,fhq'2G'*kk!A$&7G"--c22&-abkG(-I(-I 9s"%++ad"3"%"%++ad"3#569B$$U++#ABZF &&s++#ABZF rrAN)__name__ __module__ __qualname____firstlineno__baselinerj__static_attributes__)rirUsrFaker^sH- - rrrrA)rrUrrris ` @r_prettyDyadic._prettys 0 0 bv rcSU-U-$rErAr$s r__rsub__Dyadic.__rsub__sT U""rcdURn[U5S:XaURS5$/nUGH1nUSS:XaCURSURUS5-S-URUS5-S-5 MPUSS:XaCURSURUS5-S-URUS5-S-5 MUSS:wdMURUS5n[ US[ 5(aS U-nUSS :XaUSS nS nOS nURXe-S-URUS5-S-URUS5-S-5 GM4 SR U5nURS 5(aUSS nU$URS5(aUSS nU$)zPrinting method. rr z + (rar)rFz - (rLrMNrKrJz*(rNrOrP)rrrQrr2rrSrRrTs r _sympystrDyadic._sympystrs YY r7a<>>!$ $ Atqy &7>>!A$#77#=!..1./14561 &7>>!A$#77#=!..1./14561!..1.adC(($w.G1:$%abkG %I %I )-4!..1./ 'qt 457:;<).   U # #ABZF   s # #ABZF rc*URUS-5$)zThe subtraction operator. rF)r&r$s r__sub__Dyadic.__sub__*s||EBJ''rcSSKJn U"U5n[S5nURH1nX4SUSR USR U55-- nM3 U$)a:Returns the dyadic resulting from the dyadic vector cross product: Dyadic x Vector. Parameters ========== other : Vector Vector to cross with. Examples ======== >>> from sympy.physics.vector import ReferenceFrame, outer, cross >>> N = ReferenceFrame('N') >>> d = outer(N.x, N.x) >>> cross(d, N.y) (N.x|N.z) r)r0r r)r1r0r rr4cross)rr%r0r5rs rr Dyadic.cross.sX$ >e$ AYA A$!A$**adjj&79: :B rNc SSKJn U"XU5$)agExpresses this Dyadic in alternate frame(s) The first frame is the list side expression, the second frame is the right side; if Dyadic is in form A.x|B.y, you can express it in two different frames. If no second frame is given, the Dyadic is expressed in only one frame. Calls the global express function Parameters ========== frame1 : ReferenceFrame The frame to express the left side of the Dyadic in frame2 : ReferenceFrame If provided, the frame to express the right side of the Dyadic in Examples ======== >>> from sympy.physics.vector import ReferenceFrame, outer, dynamicsymbols >>> from sympy.physics.vector import init_vprinting >>> init_vprinting(pretty_print=False) >>> N = ReferenceFrame('N') >>> q = dynamicsymbols('q') >>> B = N.orientnew('B', 'Axis', [q, N.z]) >>> d = outer(N.x, N.x) >>> d.express(B, N) cos(q)*(B.x|N.x) - sin(q)*(B.y|N.x) r)express)sympy.physics.vector.functionsr)rframe1frame2rs rrDyadic.expressJs@ ;tV,,rc UcUn[UVVs/sH,nUH"oCRU5RU5PM$ M. snn5RSS5$s snnf)aiReturns the matrix form of the dyadic with respect to one or two reference frames. Parameters ---------- reference_frame : ReferenceFrame The reference frame that the rows and columns of the matrix correspond to. If a second reference frame is provided, this only corresponds to the rows of the matrix. second_reference_frame : ReferenceFrame, optional, default=None The reference frame that the columns of the matrix correspond to. Returns ------- matrix : ImmutableMatrix, shape(3,3) The matrix that gives the 2D tensor form. Examples ======== >>> from sympy import symbols, trigsimp >>> from sympy.physics.vector import ReferenceFrame >>> from sympy.physics.mechanics import inertia >>> Ixx, Iyy, Izz, Ixy, Iyz, Ixz = symbols('Ixx, Iyy, Izz, Ixy, Iyz, Ixz') >>> N = ReferenceFrame('N') >>> inertia_dyadic = inertia(N, Ixx, Iyy, Izz, Ixy, Iyz, Ixz) >>> inertia_dyadic.to_matrix(N) Matrix([ [Ixx, Ixy, Ixz], [Ixy, Iyy, Iyz], [Ixz, Iyz, Izz]]) >>> beta = symbols('beta') >>> A = N.orientnew('A', 'Axis', (beta, N.x)) >>> trigsimp(inertia_dyadic.to_matrix(A)) Matrix([ [ Ixx, Ixy*cos(beta) + Ixz*sin(beta), -Ixy*sin(beta) + Ixz*cos(beta)], [ Ixy*cos(beta) + Ixz*sin(beta), Iyy*cos(2*beta)/2 + Iyy/2 + Iyz*sin(2*beta) - Izz*cos(2*beta)/2 + Izz/2, -Iyy*sin(2*beta)/2 + Iyz*cos(2*beta) + Izz*sin(2*beta)/2], [-Ixy*sin(beta) + Ixz*cos(beta), -Iyy*sin(2*beta)/2 + Iyz*cos(2*beta) + Izz*sin(2*beta)/2, -Iyy*cos(2*beta)/2 + Iyy/2 - Iyz*sin(2*beta) + Izz*cos(2*beta)/2 + Izz/2]]) rO)Matrixr3reshape)rreference_framesecond_reference_framerjs r to_matrixDyadic.to_matrixmsdV " )%4 "?.?a,HIuuT{q),*?.//6wq!} =.s3A c [URVs/sH+n[USR"S0UD6USUS4/5PM- sn[S55$s snf)z(Calls .doit() on each term in the Dyadicrr rrA)sumrr doit)rhintsrs rr Dyadic.doitsa!YY(&QqTYY//1qt<=>&()/4 4(2AcSSKJn U"X5$)a'Take the time derivative of this Dyadic in a frame. This function calls the global time_derivative method Parameters ========== frame : ReferenceFrame The frame to take the time derivative in Examples ======== >>> from sympy.physics.vector import ReferenceFrame, outer, dynamicsymbols >>> from sympy.physics.vector import init_vprinting >>> init_vprinting(pretty_print=False) >>> N = ReferenceFrame('N') >>> q = dynamicsymbols('q') >>> B = N.orientnew('B', 'Axis', [q, N.z]) >>> d = outer(N.x, N.x) >>> d.dt(B) - q'*(N.y|N.x) - q'*(N.x|N.y) r)time_derivative)rr)rframers rdt Dyadic.dts2 Ct++rc[S5nURH,nU[USR5USUS4/5- nM. U$)zReturns a simplified Dyadic.rr r)r rsimplify)routrs rrDyadic.simplifysIQiA 6AaDMMOQqT1Q489: :C rc [URVs/sH+n[USR"U0UD6USUS4/5PM- sn[S55$s snf)zSubstitution on the Dyadic. Examples ======== >>> from sympy.physics.vector import ReferenceFrame >>> from sympy import Symbol >>> N = ReferenceFrame('N') >>> s = Symbol('s') >>> a = s*(N.x|N.x) >>> a.subs({s: 2}) 2*(N.x|N.x) rr r)rrr subs)rrrfrs rr Dyadic.subsse !YY(&QqTYY771qtDEF&()/4 4(rc[U5(d [S5e[S5nURH"up4nX!"U5UR U5-- nM$ U$)z/Apply a function to each component of a Dyadic.z`f` must be callable.r)callable TypeErrorr rr4)rfrabcs r applyfuncDyadic.applyfuncsR{{34 4QiyyGA! 1Q41771:& &C! rcUR(dU$/n[U5nURH=n[U5nUSRUS9US'UR [ U55 M? [ U5$)Nr)n)rrr)evalfrtupler )rprecnew_argsdpsr new_inlists r _eval_evalfDyadic._eval_evalfsjyyK$iiFfJ"1IOOcO2JqM OOE*- . hrc/nURH?n[U5nUSRU5US'UR[ U55 MA [ U5$)a Replace occurrences of objects within the measure numbers of the Dyadic. Parameters ========== rule : dict-like Expresses a replacement rule. Returns ======= Dyadic Result of the replacement. Examples ======== >>> from sympy import symbols, pi >>> from sympy.physics.vector import ReferenceFrame, outer >>> N = ReferenceFrame('N') >>> D = outer(N.x, N.x) >>> x, y, z = symbols('x y z') >>> ((1 + x*y) * D).xreplace({x: pi}) (pi*y + 1)*(N.x|N.x) >>> ((1 + x*y) * D).xreplace({x: pi, y: 2}) (1 + 2*pi)*(N.x|N.x) Replacements occur only if an entire node in the expression tree is matched: >>> ((x*y + z) * D).xreplace({x*y: pi}) (z + pi)*(N.x|N.x) >>> ((x*y*z) * D).xreplace({x*y: pi}) x*y*z*(N.x|N.x) r)rr)xreplacerrr )rrulerrrs rrDyadic.xreplacesWPiiFfJ&qM2248JqM OOE*- . hr)rr@)$rlrmrnro__doc__ is_numberrpropertyr r&__radd__r,__rmul__r3__and__r9r=rBrGrZrsrvrzr}r__xor__rrrrrrrrrrqrArrr r s I$L. H4H"JG'1 !"H4l#"H(4G!-F/=b4 ,84&  - rcF[U[5(d [S5eU$)NzA Dyadic must be supplied)r2r r)r%s rr#r#s eV $ $344 LrN)sympyrrrrsympy.core.evalfrsympy.printing.defaultsrmpmath.libmp.libmpfr__all__r r#rArrrs399'-+ *P Y P fr