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Multi Degree of Freedom Holonomic System
=========================================

In this example we demonstrate the use of the functionality provided in
:mod:`sympy.physics.mechanics` for deriving the equations of motion (EOM) of a
holonomic system that includes both particles and rigid bodies with contributing
forces and torques, some of which are specified forces and torques. The system
is shown below:

.. image:: multidof-holonomic.svg
   :align: center

The system will be modeled using :class:`~.System`. First we need to create the
:func:`~.dynamicsymbols` needed to describe the system as shown in the above
diagram. In this case, the generalized coordinates :math:`q_1` represent lateral
distance of block from wall, :math:`q_2` represents angle of the compound
pendulum from vertical, :math:`q_3`  represents angle of the simple pendulum
from the compound pendulum. The generalized speeds :math:`u_1` represents
lateral speed of block, :math:`u_2` represents lateral speed of compound
pendulum and :math:`u_3` represents angular speed of C relative to B.

We also create some :func:`~.symbols` to represent the length and mass of the
pendulum, as well as gravity and others. ::

    >>> from sympy import zeros, symbols
    >>> from sympy.physics.mechanics import *
    >>> q1, q2, q3, u1, u2, u3 = dynamicsymbols('q1, q2, q3, u1, u2, u3')
    >>> F, T = dynamicsymbols('F, T')
    >>> l, k, c, g, kT = symbols('l, k, c, g, kT')
    >>> ma, mb, mc, IBzz= symbols('ma, mb, mc, IBzz')

With all symbols defined, we can now define the bodies and initialize our
instance of :class:`~.System`. ::

    >>> wall = RigidBody('N')
    >>> block = Particle('A', mass=ma)
    >>> compound_pend = RigidBody('B', mass=mb)
    >>> compound_pend.central_inertia = inertia(compound_pend.frame, 0, 0, IBzz)
    >>> simple_pend = Particle('C', mass=mc)
    >>> system = System.from_newtonian(wall)
    >>> system.add_bodies(block, compound_pend, simple_pend)

Next, we connect the bodies using joints to establish the kinematics. Note that
we specify the intermediate frames for both particles, as particles do not have
an associated frame. ::

    >>> block_frame = ReferenceFrame('A')
    >>> block.masscenter.set_vel(block_frame, 0)
    >>> slider = PrismaticJoint('J1', wall, block, coordinates=q1, speeds=u1,
    ...                         child_interframe=block_frame)
    >>> rev1 = PinJoint('J2', block, compound_pend, coordinates=q2, speeds=u2,
    ...                 joint_axis=wall.z, child_point=l*2/3*compound_pend.y,
    ...                 parent_interframe=block_frame)
    >>> simple_pend_frame = ReferenceFrame('C')
    >>> simple_pend.masscenter.set_vel(simple_pend_frame, 0)
    >>> rev2 = PinJoint('J3', compound_pend, simple_pend, coordinates=q3,
    ...                 speeds=u3, joint_axis=compound_pend.z,
    ...                 parent_point=-l/3*compound_pend.y,
    ...                 child_point=l*simple_pend_frame.y,
    ...                 child_interframe=simple_pend_frame)

    >>> system.add_joints(slider, rev1, rev2)

Now we can apply loads (forces and torques) to the bodies, gravity acts on all
bodies, a linear spring and damper act on block and wall, a rotational linear
spring acts on C relative to B specified torque T acts on compound_pend and
block, specified force F acts on block. ::

    >>> system.apply_uniform_gravity(-g * wall.y)
    >>> system.add_loads(Force(block, F * wall.x))
    >>> spring_damper_path = LinearPathway(wall.masscenter, block.masscenter)
    >>> system.add_actuators(
    ...     LinearSpring(k, spring_damper_path),
    ...     LinearDamper(c, spring_damper_path),
    ...     TorqueActuator(T, wall.z, compound_pend, wall),
    ...     TorqueActuator(kT * q3, wall.z, compound_pend, simple_pend_frame),
    ... )

With the system setup, we can now form the equations of motion with
:class:`~.KanesMethod` in the backend. ::

    >>> system.form_eoms(explicit_kinematics=True)
    Matrix([
    [                                -c*u1(t) - k*q1(t) + 2*l*mb*u2(t)**2*sin(q2(t))/3 - l*mc*(-sin(q2(t))*sin(q3(t)) + cos(q2(t))*cos(q3(t)))*Derivative(u3(t), t) - l*mc*(-sin(q2(t))*cos(q3(t)) - sin(q3(t))*cos(q2(t)))*(u2(t) + u3(t))**2 + l*mc*u2(t)**2*sin(q2(t)) - (2*l*mb*cos(q2(t))/3 + mc*(l*(-sin(q2(t))*sin(q3(t)) + cos(q2(t))*cos(q3(t))) + l*cos(q2(t))))*Derivative(u2(t), t) - (ma + mb + mc)*Derivative(u1(t), t) + F(t)],
    [-2*g*l*mb*sin(q2(t))/3 - g*l*mc*(sin(q2(t))*cos(q3(t)) + sin(q3(t))*cos(q2(t))) - g*l*mc*sin(q2(t)) + l**2*mc*(u2(t) + u3(t))**2*sin(q3(t)) - l**2*mc*u2(t)**2*sin(q3(t)) - mc*(l**2*cos(q3(t)) + l**2)*Derivative(u3(t), t) - (2*l*mb*cos(q2(t))/3 + mc*(l*(-sin(q2(t))*sin(q3(t)) + cos(q2(t))*cos(q3(t))) + l*cos(q2(t))))*Derivative(u1(t), t) - (IBzz + 4*l**2*mb/9 + mc*(2*l**2*cos(q3(t)) + 2*l**2))*Derivative(u2(t), t) + T(t)],
    [                                                                                                                                                                        -g*l*mc*(sin(q2(t))*cos(q3(t)) + sin(q3(t))*cos(q2(t))) - kT*q3(t) - l**2*mc*u2(t)**2*sin(q3(t)) - l**2*mc*Derivative(u3(t), t) - l*mc*(-sin(q2(t))*sin(q3(t)) + cos(q2(t))*cos(q3(t)))*Derivative(u1(t), t) - mc*(l**2*cos(q3(t)) + l**2)*Derivative(u2(t), t)]])

    >>> system.mass_matrix_full
    Matrix([
    [1, 0, 0,                                                                                            0,                                                                                            0,                                                     0],
    [0, 1, 0,                                                                                            0,                                                                                            0,                                                     0],
    [0, 0, 1,                                                                                            0,                                                                                            0,                                                     0],
    [0, 0, 0,                                                                                 ma + mb + mc, 2*l*mb*cos(q2(t))/3 + mc*(l*(-sin(q2(t))*sin(q3(t)) + cos(q2(t))*cos(q3(t))) + l*cos(q2(t))), l*mc*(-sin(q2(t))*sin(q3(t)) + cos(q2(t))*cos(q3(t)))],
    [0, 0, 0, 2*l*mb*cos(q2(t))/3 + mc*(l*(-sin(q2(t))*sin(q3(t)) + cos(q2(t))*cos(q3(t))) + l*cos(q2(t))),                                         IBzz + 4*l**2*mb/9 + mc*(2*l**2*cos(q3(t)) + 2*l**2),                           mc*(l**2*cos(q3(t)) + l**2)],
    [0, 0, 0,                                        l*mc*(-sin(q2(t))*sin(q3(t)) + cos(q2(t))*cos(q3(t))),                                                                  mc*(l**2*cos(q3(t)) + l**2),                                               l**2*mc]])

    >>> system.forcing_full
    Matrix([
    [                                                                                                                                                                           u1(t)],
    [                                                                                                                                                                           u2(t)],
    [                                                                                                                                                                           u3(t)],
    [                  -c*u1(t) - k*q1(t) + 2*l*mb*u2(t)**2*sin(q2(t))/3 - l*mc*(-sin(q2(t))*cos(q3(t)) - sin(q3(t))*cos(q2(t)))*(u2(t) + u3(t))**2 + l*mc*u2(t)**2*sin(q2(t)) + F(t)],
    [-2*g*l*mb*sin(q2(t))/3 - g*l*mc*(sin(q2(t))*cos(q3(t)) + sin(q3(t))*cos(q2(t))) - g*l*mc*sin(q2(t)) + l**2*mc*(u2(t) + u3(t))**2*sin(q3(t)) - l**2*mc*u2(t)**2*sin(q3(t)) + T(t)],
    [                                                                                -g*l*mc*(sin(q2(t))*cos(q3(t)) + sin(q3(t))*cos(q2(t))) - kT*q3(t) - l**2*mc*u2(t)**2*sin(q3(t))]])
