MZdR!XddlmZmZmZmZmZmZddlmZddl m Z dgZ GddeZ y))Body Lagrangian KanesMethodLagrangesMethod RigidBodyParticle)_Methods)Matrix JointsMethodceZdZdZdZedZedZedZedZ edZ edZ ed Z ed Z ed Zed Zd ZdZdZdZdZdZefdZddZy)r a%Method for formulating the equations of motion using a set of interconnected bodies with joints. Parameters ========== newtonion : Body or ReferenceFrame The newtonion(inertial) frame. *joints : Joint The joints in the system Attributes ========== q, u : iterable Iterable of the generalized coordinates and speeds bodies : iterable Iterable of Body objects in the system. loads : iterable Iterable of (Point, vector) or (ReferenceFrame, vector) tuples describing the forces on the system. mass_matrix : Matrix, shape(n, n) The system's mass matrix forcing : Matrix, shape(n, 1) The system's forcing vector mass_matrix_full : Matrix, shape(2*n, 2*n) The "mass matrix" for the u's and q's forcing_full : Matrix, shape(2*n, 1) The "forcing vector" for the u's and q's method : KanesMethod or Lagrange's method Method's object. kdes : iterable Iterable of kde in they system. Examples ======== This is a simple example for a one degree of freedom translational spring-mass-damper. >>> from sympy import symbols >>> from sympy.physics.mechanics import Body, JointsMethod, PrismaticJoint >>> from sympy.physics.vector import dynamicsymbols >>> c, k = symbols('c k') >>> x, v = dynamicsymbols('x v') >>> wall = Body('W') >>> body = Body('B') >>> J = PrismaticJoint('J', wall, body, coordinates=x, speeds=v) >>> wall.apply_force(c*v*wall.x, reaction_body=body) >>> wall.apply_force(k*x*wall.x, reaction_body=body) >>> method = JointsMethod(wall, J) >>> method.form_eoms() Matrix([[-B_mass*Derivative(v(t), t) - c*v(t) - k*x(t)]]) >>> M = method.mass_matrix_full >>> F = method.forcing_full >>> rhs = M.LUsolve(F) >>> rhs Matrix([ [ v(t)], [(-c*v(t) - k*x(t))/B_mass]]) Notes ===== ``JointsMethod`` currently only works with systems that do not have any configuration or motion constraints. cDt|tr|j|_n||_||_|j |_|j |_|j|_ |j|_ |j|_ d|_yN) isinstancerframe_joints_generate_bodylist_bodies_generate_loadlist_loads _generate_q_q _generate_u_u_generate_kdes_kdes_method)self newtonionjointss F/usr/lib/python3/dist-packages/sympy/physics/mechanics/jointsmethod.py__init__zJointsMethod.__init__Ns| i &"DJ"DJ ..0 --/ ""$""$((*  c|jS)zList of bodies in they system.)rrs r bodieszJointsMethod.bodies]||r"c|jS)zList of loads on the system.)rr$s r loadszJointsMethod.loadsbs{{r"c|jSz$List of the generalized coordinates.)rr$s r qzJointsMethod.qg wwr"c|jS)zList of the generalized speeds.)rr$s r uzJointsMethod.ulr,r"c|jSr*)rr$s r kdeszJointsMethod.kdesqszzr"c.|jjS)z)The "forcing vector" for the u's and q's.)method forcing_fullr$s r r3zJointsMethod.forcing_fullvs{{'''r"c.|jjS)z&The "mass matrix" for the u's and q's.)r2mass_matrix_fullr$s r r5zJointsMethod.mass_matrix_full{s{{+++r"c.|jjS)zThe system's mass matrix.)r2 mass_matrixr$s r r7zJointsMethod.mass_matrixs{{&&&r"c.|jjS)zThe system's forcing vector.)r2forcingr$s r r9zJointsMethod.forcings{{"""r"c|jS)z3Object of method used to form equations of systems.)rr$s r r2zJointsMethod.methodr&r"cg}|jD]U}|j|vr|j|j|j|vs;|j|jW|Sr)rchildappendparent)rr%joints r rzJointsMethod._generate_bodylistsY\\ ,E{{&( ekk*||6) ell+  ,  r"cbg}|jD]}|j|j|Sr)r%extendr()r load_listbodys r rzJointsMethod._generate_loadlists2 KK )D   TZZ ( )r"cg}|jD]3}|jD]"}||vr td|j|$5t |S)Nz'Coordinates of joints should be unique.)r coordinates ValueErrorr=r )rq_indr? coordinates r rzJointsMethod._generate_qs[\\ )E#// ) &$%NOO Z( ) ) e}r"cg}|jD]3}|jD]"}||vr td|j|$5t |S)Nz"Speeds of joints should be unique.)rspeedsrFr=r )ru_indr?speeds r rzJointsMethod._generate_usX\\ $E $E>$%IJJ U# $ $ e}r"ctddgj}|jD]}|j|j}|S)Nr)r Trcol_joinr0)rkd_indr?s r rzJointsMethod._generate_kdess@1b!##\\ 1E__UZZ0F 1 r"c g}|jD]}|jrpt|j|j|j |j |j|jf}|j|_|j|t|j|j|j }|j|_|j||Sr) r% is_rigidbodyrname masscenterrmasscentral_inertiapotential_energyr=r)rbodylistrCrbparts r _convert_bodieszJointsMethod._convert_bodiessKK &D  tyy$//4::tyy))4??;=&*&;&;## 4??DIIF(,(=(=%% &r"c|j}t|trFt|jg|}|||j |j ||j|_nE||j|j |j|j|j ||_|jj}|S)a7Method to form system's equation of motions. Parameters ========== method : Class Class name of method. Returns ======== Matrix Vector of equations of motions. Examples ======== This is a simple example for a one degree of freedom translational spring-mass-damper. >>> from sympy import S, symbols >>> from sympy.physics.mechanics import LagrangesMethod, dynamicsymbols, Body >>> from sympy.physics.mechanics import PrismaticJoint, JointsMethod >>> q = dynamicsymbols('q') >>> qd = dynamicsymbols('q', 1) >>> m, k, b = symbols('m k b') >>> wall = Body('W') >>> part = Body('P', mass=m) >>> part.potential_energy = k * q**2 / S(2) >>> J = PrismaticJoint('J', wall, part, coordinates=q, speeds=qd) >>> wall.apply_force(b * qd * wall.x, reaction_body=part) >>> method = JointsMethod(wall, J) >>> method.form_eoms(LagrangesMethod) Matrix([[b*Derivative(q(t), t) + k*q(t) + m*Derivative(q(t), (t, 2))]]) We can also solve for the states using the 'rhs' method. >>> method.rhs() Matrix([ [ Derivative(q(t), t)], [(-b*Derivative(q(t), t) - k*q(t))/m]]) )rGrKkd_eqs forcelistr%) r\ issubclassrrrr+r(rr.r0r2 _form_eoms)rr2rYLsolns r form_eomszJointsMethod.form_eomssZ'') fo .4::11A!!TVVTZZ4::NDL!$**DFF$&&QUQZQZ.2jjKDL{{%%' r"Nc:|jj|S)axReturns equations that can be solved numerically. Parameters ========== inv_method : str The specific sympy inverse matrix calculation method to use. For a list of valid methods, see :meth:`~sympy.matrices.matrices.MatrixBase.inv` Returns ======== Matrix Numerically solvable equations. See Also ======== sympy.physics.mechanics.kane.KanesMethod.rhs: KanesMethod's rhs function. sympy.physics.mechanics.lagrange.LagrangesMethod.rhs: LagrangesMethod's rhs function. ) inv_method)r2rhs)rrfs r rgzJointsMethod.rhss6{{*55r"r)__name__ __module__ __qualname____doc__r!propertyr%r(r+r.r0r3r5r7r9r2rrrrrr\rrdrgr"r r r sBH ((,,''##   +5n6r"N) sympy.physics.mechanicsrrrrrrsympy.physics.mechanics.methodr sympy.core.backendr __all__r rmr"r rrs+993%  N68N6r"