o 8VaK@sTddlmZmZmZddlmZmZddlm Z dgZ Gddde eZ ddZ dS) )sympifyAddImmutableMatrix) EvalfMixin prec_to_dps) PrintableDyadicc@seZdZdZdZddZeddZddZd d Z d d Z d dZ ddZ ddZ ddZddZddZddZddZddZdd Zd!d"Zd#d$ZeZe Zd8d&d'Zd8d(d)Zd*d+Zd,d-Zd.d/Zd0d1Zd2d3Ze Z eZ!d4d5Z"d6d7Z#d%S)9rayA Dyadic object. See: https://en.wikipedia.org/wiki/Dyadic_tensor Kane, T., Levinson, D. Dynamics Theory and Applications. 1985 McGraw-Hill A more powerful way to represent a rigid body's inertia. While it is more complex, by choosing Dyadic components to be in body fixed basis vectors, the resulting matrix is equivalent to the inertia tensor. FcCsxg|_|dkr g}t|dkr|d}t|jD]L\}}t|ddt|j|dkrbt|ddt|j|dkrb|j|d|dd|dd|ddf|j|<||dd}nq|dkrv|j|d||dt|dksd}|t|jkr|j|ddk|j|ddkB|j|ddkBr|j|j||d8}|d7}|t|jksdSdS)a1 Just like Vector's init, you shouldn't call this unless creating a zero dyadic. zd = Dyadic(0) Stores a Dyadic as a list of lists; the inner list has the measure number and the two unit vectors; the outerlist holds each unique unit vector pair. rN)argslen enumeratestrremoveappend)selfinlistZaddedivr=/usr/lib/python3/dist-packages/sympy/physics/vector/dyadic.py__init__s:  "    "zDyadic.__init__cCstS)zReturns the class Dyadic. )rrrrrfunc=sz Dyadic.funccCst|}t|j|jS)zThe add operator for Dyadic. ) _check_dyadicrr rotherrrr__add__BszDyadic.__add__c Csddlm}m}t|trEt|}td}t|jD](\}}t|jD]\}}||d|d|d|d@|d|dB7}q#q|S||}|d}t|jD]\}}||d|d|d|@7}qR|S)aThe inner product operator for a Dyadic and a Dyadic or Vector. Parameters ========== other : Dyadic or Vector The other Dyadic or Vector to take the inner product with Examples ======== >>> from sympy.physics.vector import ReferenceFrame, outer >>> N = ReferenceFrame('N') >>> D1 = outer(N.x, N.y) >>> D2 = outer(N.y, N.y) >>> D1.dot(D2) (N.x|N.y) >>> D1.dot(N.y) N.x rVector _check_vectorr r )sympy.physics.vector.vectorrr isinstancerrr r ) rrrr olrrZi2Zv2rrr__and__Gs 6"zDyadic.__and__cCs|d|S)z0Divides the Dyadic by a sympifyable expression. r )__mul__rrrr __truediv__kzDyadic.__truediv__cCs\|dkrtd}t|}|jgkr|jgkrdS|jgks"|jgkr$dSt|jt|jkS)z[Tests for equality. Is currently weak; needs stronger comparison testing rTF)rrr setrrrr__eq__osz Dyadic.__eq__cCsXdd|jD}t|D]\}}t|||d||d||df||<q t|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) cSsg|]}|qSrr.0rrrr sz"Dyadic.__mul__..rr r )r r rr)rrZnewlistrrrrrr%s   zDyadic.__mul__cCs ||k SNrrrrr__ne__s z Dyadic.__ne__cCs|dSNrrrrr__neg__szDyadic.__neg__c Cs|j}t|dkr tdSg}t|D]\}}||ddkr9|d|||dd|||dq||ddkr[|d|||dd|||dq||ddkr|||d}t||dtryd|}|d r|dd}d}nd}||||||dd|||dqd |}|dr|d d}|S|d r|dd}|S) Nrr  + z\otimes r r0 - (%s)- ) r r rr r_printr"r startswithjoin rprinterarr#rrarg_str str_startoutstrrrr_latexsL       z Dyadic._latexcs|Gfddd}|S)NcseZdZdZfddZdS)zDyadic._pretty..Fakerc sj}}t|dkrtdSjrdnd}g}t|D]\}}||ddkr@|d|||d||||dgq||ddkr`|d|||d||||dgq||ddkrt||dtr| ||d d} n |||d} | d r| dd} d} nd} || | d |||d||||dgqd |} | dr| d d} | S| d r| dd} | S) Nru⊗|r r2r r0r3r5r8r6r7) r r rZ _use_unicoder extendZdoprintr"rr9Zparensr:r;) rr kwargsr>ZmppZbarr#rrr?r@rAer=rrrendersX         z#Dyadic._pretty..Fake.renderN)__name__ __module__ __qualname__ZbaselinerHrrFrrFakesrLr)rr=rLrrFr_prettys1zDyadic._prettycCsXddlm}m}||}|d}t|jD]\}}||d|d|d|@7}q|S)aThe inner product operator for a Vector or Dyadic, and a Dyadic This is for: Vector dot Dyadic Parameters ========== other : Vector The vector we are dotting with Examples ======== >>> from sympy.physics.vector import ReferenceFrame, dot, outer >>> N = ReferenceFrame('N') >>> d = outer(N.x, N.x) >>> dot(N.x, d) N.x rrr r )r!rr r r )rrrr r#rrrrr__rand__s "zDyadic.__rand__cCs d||Sr/rrrrr__rsub__s zDyadic.__rsub__cCsTddlm}||}td}t|jD]\}}||d||dA|dB7}q|S)aFor a cross product in the form: Vector x Dyadic Parameters ========== other : Vector The Vector that we are crossing this Dyadic with Examples ======== >>> from sympy.physics.vector import ReferenceFrame, outer, cross >>> N = ReferenceFrame('N') >>> d = outer(N.x, N.x) >>> cross(N.y, d) - (N.z|N.x) rr r r r!r rr r rrr r#rrrrr__rxor__ "zDyadic.__rxor__c Cs|j}t|dkr|dSg}t|D]\}}||ddkr<|d|||dd|||ddq||ddkr`|d|||dd|||ddq||ddkr|||d}t||dtr~d |}|dd kr|dd }d }nd }|||d|||dd|||ddqd|}|d r|dd }|S|dr|dd }|S)zPrinting method. rr z + (rCr )r0z - (r4r5Nr3r2z*(r6r7r8) r r r9r rr"rr;r:r<rrr _sympystr4s@  88       zDyadic._sympystrcCs||dS)zThe subtraction operator. r0)rrrrr__sub__Ur'zDyadic.__sub__cCsTddlm}||}td}t|jD]\}}||d|d|d|AB7}q|S)aFor a cross product in the form: Dyadic x Vector. Parameters ========== other : Vector The Vector that we are crossing this Dyadic 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) rrPr r rQrRrrr__xor__YrTzDyadic.__xor__NcCsddlm}||||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.functionsrY)rZframe1Zframe2rYrrrrYws zDyadic.expresscs,dur|tfdd|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 >>> from sympy.physics.vector import ReferenceFrame, Vector >>> Vector.simp = True >>> 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)) >>> 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]]) Ncs&g|]}D] }||qqSr)dot)r+rjsecond_reference_framerrrr,sz$Dyadic.to_matrix..r7)MatrixZreshape)rZreference_framer^rr]r to_matrixs ,zDyadic.to_matrixc stfdd|jDtdS)z(Calls .doit() on each term in the Dyadiccs4g|]}t|djdi|d|dfgqS)rr r r)rdoitr*hintsrrr,,zDyadic.doit..rsumr r)rrcrrbrras z Dyadic.doitcCsddlm}|||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)rZrg)rframergrrrdts  z Dyadic.dtcCs<td}|jD]}|t|d|d|dfg7}q|S)zReturns a simplified Dyadic.rr r )rr simplify)routrrrrrjs &zDyadic.simplifycs tfdd|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) cs4g|]}t|dji|d|dfgqS)rr r )rsubsr*r rErrr,rdzDyadic.subs..rre)rr rErrmrrls z Dyadic.subscCsBt|stdtd}|jD]\}}}|||||B7}q|S)z/Apply a function to each component of a Dyadic.z`f` must be callable.r)callable TypeErrorrr )rfrkabcrrr applyfuncs zDyadic.applyfunccCsP|js|Sg}|jD]}t|}|djt|d|d<|t|q t|S)Nr)n)r listZevalfrrtupler)rZprecnew_argsr new_inlistrrr _eval_evalfs zDyadic._eval_evalfcCs@g}|jD]}t|}|d||d<|t|qt|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)r rvxreplacerrwr)rZrulerxrryrrrr{s ' zDyadic.xreplacer-)$rIrJrK__doc__Z is_numberrpropertyrrr$r&r)r%r.r1rBrMrNrOrSrVrWrX__radd____rmul__rYr`rarirjrlrtr[Zcrossrzr{rrrrrsD & $$6!  #2  cCst|ts td|S)NzA Dyadic must be supplied)r"rro)rrrrrLs rN) Zsympy.core.backendrrrr_Zsympy.core.evalfrrZsympy.printing.defaultsr__all__rrrrrrs  H