def test_adding_elements(self):
        conn = RigidConnection('conn')
        body = RigidBody('body', mass=1.235)
        s = System()
        s.add_leaf(conn)
        conn.add_leaf(body)
        s.setup()

        # Should have dict of elements
        self.assertEqual(s.elements, {'conn': conn, 'body': body})

        # Number of states:
        #   6 ground
        # + 6 constraints on conn
        # + 6 <node-0>   between conn and body
        # ---
        #  18
        self.assertEqual(s.lhs.shape, (18, 18))
        for vec in (s.rhs, s.qd, s.qdd):
            self.assertEqual(len(vec), 18)

        # Check there are no dofs
        self.assertEqual(len(s.q.dofs), 0)
        self.assertEqual(len(s.qd.dofs), 0)
        self.assertEqual(len(s.qdd.dofs), 0)

    def test_three_rigid_elements_as_disc_have_ends_in_right_place(self):
        length = 20.0
        offset = 5.0

        # Make 3 elements spaced by 120 deg about z axis
        system = System()
        elements = []
        for i in range(3):
            rotmat = rotations(('z', i * 2*pi/3))
            offset_vector = dot(rotmat, [offset, 0, 0])
            conn = RigidConnection('offset%d' % i, offset_vector, rotmat)
            element = RigidConnection('element%d' % i, [length, 0, 0])
            elements.append(element)
            system.add_leaf(conn)
            conn.add_leaf(element)
        system.setup()

        r = offset
        R = offset + length
        assert_aae(elements[0].rp, [r,     0,           0])
        assert_aae(elements[1].rp, [-r/2,  r*sqrt(3)/2, 0])
        assert_aae(elements[2].rp, [-r/2, -r*sqrt(3)/2, 0])
        assert_aae(elements[0].rd, [R,     0,           0])
        assert_aae(elements[1].rd, [-R/2,  R*sqrt(3)/2, 0])
        assert_aae(elements[2].rd, [-R/2, -R*sqrt(3)/2, 0])

    def test_inertia_when_offset_axially(self):
        density = 230.4
        length = 20.0
        offset = 5.0
        element = _mock_rigid_uniform_beam(density, length)
        conn = RigidConnection('offset', offset=[offset, 0, 0])
        joint = FreeJoint('joint')

        system = System()
        system.add_leaf(joint)
        joint.add_leaf(conn)
        conn.add_leaf(element)
        system.setup()

        # Calculate reduced system to get rigid body matrices
        rsys = ReducedSystem(system)

        # Expected values: rod along x axis
        m = density * length
        Iy = m * (length**2 / 12 + (length/2 + offset)**2)
        expected_mass = m * eye(3)
        expected_inertia = diag([0, Iy, Iy])
        expected_offdiag = zeros((3, 3))

        # Y accel -> positive moment about Z
        # Z accel -> negative moment about Y
        expected_offdiag[2, 1] = +m * (length/2 + offset)
        expected_offdiag[1, 2] = -m * (length/2 + offset)

        assert_aae(rsys.M[:3, :3], expected_mass)
        assert_aae(rsys.M[3:, 3:], expected_inertia)
        assert_aae(rsys.M[3:, :3], expected_offdiag)
        assert_aae(rsys.M[:3, 3:], expected_offdiag.T)

    def test_1dof_nonlinear_system(self):
        s = System()
        j = PrismaticJoint('joint', [0, 0, 1])
        k = 0.45  # quadratic stiffness coefficient
        j.internal_force = lambda el, t: -k * el.xstrain[0]**2
        b = RigidBody('body', 10)
        s.add_leaf(j)
        j.add_leaf(b)
        s.setup()

        # Linearise around z0 = 0: stiffness should be zero
        linsys = LinearisedSystem.from_system(s, z0=0)
        assert_aae(linsys.M, [[10.0]])
        assert_aae(linsys.C, [[0.0]])
        assert_aae(linsys.K, [[0.0]])

        # Linearise about z0 = 2: stiffness should be 2kx
        linsys = LinearisedSystem.from_system(s, z0=[2])
        assert_aae(linsys.M, [[10.0]])
        assert_aae(linsys.C, [[0.0]])
        assert_aae(linsys.K, [[2 * k * 2]])

        # Test setting z0 in another way
        linsys = LinearisedSystem.from_system(s, z0={'joint': [4.2]})
        assert_aae(linsys.M, [[10.0]])
        assert_aae(linsys.C, [[0.0]])
        assert_aae(linsys.K, [[2 * k * 4.2]])

    def test_rigid_body_with_no_dofs(self):
        s = System()
        b = RigidBody('body', 23.7)
        s.add_leaf(b)
        s.setup()

        # Calculate reduced system to get rigid body matrices
        rsys = ReducedSystem(s)
        self.assertEqual(rsys.M.shape, (0, 0))
        self.assertEqual(rsys.Q.shape, (0,))

    def test_accounts_for_offset_centre_of_mass_in_applied_force(self):
        # check applied force due to gravity is correct
        b = RigidBody("body", mass=5.6, Xc=[1.2, 3.4, 5.4])
        s = System(gravity=9.81)  # need a System to define gravity
        s.add_leaf(b)
        s.setup()

        b.calc_mass()
        b.calc_external_loading()
        assert_array_equal(b.applied_forces, b.mass * 9.81 * array([0, 0, -1, -3.4, 1.2, 0]))

    def test_nonzero_prescribed_acceleration(self):
        # Test reduction where a prescribed acceleration is non-zero:
        # two sliders in series, with a mass on the end. If the second
        # slider's acceleration is prescribed, the first slider's DOF
        # sees an inertial force corresponding to the acceleration of
        # the mass.
        mass = 36.2
        s = System()
        s1 = PrismaticJoint('slider1', [1, 0, 0])
        s2 = PrismaticJoint('slider2', [1, 0, 0])
        b = RigidBody('body', mass)
        s.add_leaf(s1)
        s1.add_leaf(s2)
        s2.add_leaf(b)
        s.setup()

        s.prescribe(s2, acc=0)

        # With hinge angle = 0, no generalised inertial force
        rsys = ReducedSystem(s)
        assert_aae(rsys.M, mass)
        assert_aae(rsys.Q, 0)

        # With hinge angle = 90deg, do see generalised inertial force
        s.prescribe(s2, acc=2.3)
        rsys = ReducedSystem(s)
        assert_aae(rsys.M, mass)
        assert_aae(rsys.Q, -mass * 2.3)

    def test_call(self):
        s = System()
        c = RigidConnection('conn', [1, 0, 0])
        h = Hinge('hinge', [0, 1, 0])
        b = RigidBody('body', 1)
        s.add_leaf(h)
        h.add_leaf(c)
        c.add_leaf(b)
        s.setup()

        # Set hinge angle
        h.xstrain[0] = 0.82
        h.vstrain[0] = 1.2
        h.astrain[0] = -0.3
        s.update_kinematics()
        s.solve_reactions()

        # Test load outputs
        out = LoadOutput('node-1')
        assert_array_equal(out(s), s.joint_reactions['node-1'])

        out = LoadOutput('node-1', local=True)
        F = s.joint_reactions['node-1']
        assert_array_equal(out(s), np.r_[np.dot(b.Rp.T, F[:3]),
                                         np.dot(b.Rp.T, F[3:])])

    def test_dofs_subset(self):
        s = System()
        j = FreeJoint('joint')
        s.add_leaf(j)
        s.setup()

        # 2 nodes, 6 constraints, 6 dofs
        self.assertEqual(len(s.q), 2 * 12 + 6 + 6)
        self.assertEqual(len(s.qd), 2 * 6 + 6 + 6)
        self.assertEqual(len(s.qdd), 2 * 6 + 6 + 6)
        self.assertEqual(len(s.q.dofs), 6)
        self.assertEqual(len(s.qd.dofs), 6)
        self.assertEqual(len(s.qdd.dofs), 6)

 def test_print_functions(self):
     # Not very good tests, but at least check they run without errors
     joint = FreeJoint('joint')
     body = RigidBody('body', mass=1.235)
     s = System()
     s.add_leaf(joint)
     joint.add_leaf(body)
     s.setup()
     s.print_states()
     s.print_info()

    def test_solve_reactions(self):
        # Check it calls the Element method in the right order: down
        # the tree from leaves to base. It must also reset reactions.
        s = System()
        c0 = RigidConnection('c0')
        c1 = RigidConnection('c1')
        c2 = RigidConnection('c2')
        b1 = RigidBody('b1', 1)
        b2 = RigidBody('b2', 1)
        s.add_leaf(c0)
        c0.add_leaf(c1)
        c0.add_leaf(c2)
        c1.add_leaf(b1)
        c2.add_leaf(b2)
        s.setup()

        # Check elements' iter_reactions() are called
        def mock_iter_reactions(element):
            calls.append(element)
        calls = []
        import types
        for el in s.elements.values():
            el.iter_reactions = types.MethodType(mock_iter_reactions, el)

        # Test
        s.joint_reactions[:] = 3
        s.solve_reactions()
        self.assertEqual(calls, [b2, c2, b1, c1, c0])
        assert_aae(s.joint_reactions, 0)

    def test_1dof_linear_system(self):
        s = System()
        j = PrismaticJoint('joint', [0, 0, 1])
        j.stiffness = 5.0
        j.damping = 2.3
        b = RigidBody('body', 10)
        s.add_leaf(j)
        j.add_leaf(b)
        s.setup()

        linsys = LinearisedSystem.from_system(s)

        assert_array_equal(linsys.M, [[10.0]])
        assert_array_equal(linsys.C, [[2.3]])
        assert_array_equal(linsys.K, [[5.0]])

def rigid_body_mass_matrix(element):
    joint = FreeJoint('joint')
    system = System()
    system.add_leaf(joint)
    joint.add_leaf(element)
    system.setup()

    for el in joint.iter_leaves():
        system.prescribe(el, 0, 0)
    system.update_kinematics()
    rsys = ReducedSystem(system)
    return rsys.M

    def setUp(self):
        # FE model for beam - no modes, i.e. rigid
        self.x = x = linspace(0, self.length, 20)
        fe = BeamFE(x, density=2, EA=0, EIy=0, EIz=0)

        # Build the elements
        self.shaft = Hinge('shaft', [1, 0, 0])

        self.roots = []
        self.blades = []
        self.pitch_bearings = []
        for ib in range(1):
            R = rotations(('x', ib*2*pi/3), ('y', -pi/2))
            root_offset = dot(R, [self.root_length, 0, 0])
            root = RigidConnection('root%d' % (ib+1), root_offset, R)
            bearing = Hinge('pitch%d' % (ib+1), [1, 0, 0])
            blade = ModalElementFromFE('blade%d' % (ib+1), fe, 0)

            self.shaft.add_leaf(root)
            root.add_leaf(bearing)
            bearing.add_leaf(blade)

            self.roots.append(root)
            self.blades.append(blade)
            self.pitch_bearings.append(bearing)

        # Build system
        self.system = System()
        self.system.add_leaf(self.shaft)
        self.system.setup()
        self.system.update_kinematics()    # Set up nodal values initially
        self.system.update_matrices()

    def setUp(self):
        # FE model for beam
        x = linspace(0, self.length, 20)
        fe = BeamFE(x, density=self.density, EA=0, EIy=1, EIz=0)
        fe.set_boundary_conditions('C', 'F')
        self.beam = ModalElementFromFE('beam', fe, 0)

        # Set loading - in negative Z direction
        load = np.zeros((len(x), 3))
        load[:, 2] = -self.force
        self.beam.loading = load

        # Hinge with axis along Y axis
        self.hinge = Hinge('hinge', [0, 1, 0])
        self.hinge.internal_torque = self.hinge_torque

        # Build system
        self.system = System()
        self.system.add_leaf(self.hinge)
        self.hinge.add_leaf(self.beam)
        self.system.setup()

        if not self.free_beam:
            # Prescribe hinge to be fixed
            self.system.prescribe(self.hinge)

        # Initial calculations
        self.recalc()

    def test_single_rigid_body(self):
        mass = 36.2
        inertia = np.diag([75.4, 653, 234])
        s = System()
        j = FreeJoint('joint')
        b = RigidBody('body', mass, inertia)
        s.add_leaf(j)
        j.add_leaf(b)
        s.setup()

        # Calculate reduced system to get rigid body matrices
        rsys = ReducedSystem(s)
        self.assertEqual(rsys.M.shape, (6, 6))
        self.assertEqual(rsys.Q.shape, (6,))
        assert_aae(rsys.M[:3, :3], mass * np.eye(3))
        assert_aae(rsys.M[3:, 3:], inertia)
        assert_aae(rsys.M[3:, :3], 0)
        assert_aae(rsys.M[:3, 3:], 0)
        assert_aae(rsys.Q, 0)

        # Now if some freedoms are prescribed, don't appear in reduced system
        s.prescribe(j, part=[1, 2, 3, 4, 5])  # only x-translation is free now
        rsys = ReducedSystem(s)
        self.assertEqual(rsys.M.shape, (1, 1))
        self.assertEqual(rsys.Q.shape, (1,))
        assert_aae(rsys.M[0, 0], mass)
        assert_aae(rsys.Q, 0)

    def test_simple_prescribed_integration(self):
        s = System()
        h = Hinge('hinge', [0, 1, 0])
        s.add_leaf(h)
        s.setup()

        s.prescribe(h)
        w = h.vstrain[0] = 0.97  # rad/s

        integ = Integrator(s)
        t, y = integ.integrate(9.0, 0.1)

        # Check time vector and shape of result
        assert_array_equal(t, np.arange(0, 9.0, 0.1))
        self.assertEqual(len(y), 1)
        self.assertEqual(y[0].shape, (len(t), 1))

        # Result should be y = wt, but wrapped to [0, 2pi)
        assert_aae(y[0][:, 0], (w * t) % (2 * np.pi))

        # Check asking for velocity and acceleration works
        h.xstrain[0] = s.time = 0.0  # reset
        integ = Integrator(s, ('pos', 'vel', 'acc'))
        t, y = integ.integrate(1.0, 0.1)
        assert_array_equal(t, np.arange(0, 1.0, 0.1))
        self.assertEqual(len(y), 3)
        for yy in y:
            self.assertEqual(yy.shape, (len(t), 1))
        assert_aae(y[0][:, 0], w * t)
        assert_aae(y[1][:, 0], w)
        assert_aae(y[2][:, 0], 0)

    def test_callback(self):
        s = System()
        s.setup()

        # Exponential decay: qd = -A q
        def callback(system, ti, q_struct, q_other):
            self.assertIs(system, s)
            self.assertEqual(len(q_other), 1)
            return -q_other

        integ = Integrator(s)
        t, y = integ.integrate(9.0, 0.1, extra_states=np.ones(1),
                               callback=callback)

        # Check time vector and shape of result
        assert_array_equal(t, np.arange(0, 9.0, 0.1))
        self.assertEqual(len(y), 1)
        self.assertEqual(y[0].shape, (len(t), 1))
        assert_aae(y[0][:, 0], np.exp(-t))

    def test_assemble(self):
        # Test system:
        #
        #                   /-- m2
        #  m1 -----conn----|
        #                   \-- m3
        s = System()
        m1 = RigidBody('m1', 1.3)
        m2 = RigidBody('m2', 3.4)
        m3 = RigidBody('m3', 7.5)
        conn = RigidConnection('conn')
        s.add_leaf(conn)
        s.add_leaf(m1)
        conn.add_leaf(m2)
        conn.add_leaf(m3)
        s.setup()

        # Check starting mass matrices of elements are as expected
        assert_aae(np.diag(m1.mass_vv[:3, :3]), 1.3)
        assert_aae(np.diag(m2.mass_vv[:3, :3]), 3.4)
        assert_aae(np.diag(m3.mass_vv[:3, :3]), 7.5)

        # Initially make system matrix empty for testing
        s.lhs[:, :] = 0
        assert_aae(s.lhs, 0)

        # After assembly, the mass matrices are put in the correct places:
        #   0:6  -> m1 node
        #   6:12 -> conn constraints
        #  12:18 -> m2 node
        #  12:18 -> m3 node
        s.assemble()
        M = s.lhs.copy()
        # Subtract expected mass
        M[0:6, 0:6] -= m1.mass_vv
        M[12:18, 12:18] -= m2.mass_vv + m3.mass_vv
        # Subtract expected constraints
        M[0:6, 6:12] -= conn.F_vp
        M[6:12, 0:6] -= conn.F_vp
        M[12:18, 6:12] -= conn.F_vd
        M[6:12, 12:18] -= conn.F_vd
        assert_aae(M, 0)

def build_system():
    # Calculate inertia, stiffness and damping
    I2 = mass * length**2
    k = (2*np.pi*natfreq)**2 * I2
    c = 2 * damping_factor * I2 * (2*np.pi*natfreq)

    # Build model
    hinge = Hinge('hinge', [0,0,1])
    hinge.stiffness = k
    hinge.damping = c
    link = RigidConnection('link', [length, 0, 0])
    body = RigidBody('body', mass)

    system = System()
    system.add_leaf(hinge)
    hinge.add_leaf(link)
    link.add_leaf(body)
    system.setup()

    return system