Python mechanics.LagrangesMethod类代码示例

def test_linearize_pendulum_lagrange_nonminimal():
    q1, q2 = dynamicsymbols('q1:3')
    q1d, q2d = dynamicsymbols('q1:3', level=1)
    L, m, t = symbols('L, m, t')
    g = 9.8
    # Compose World Frame
    N = ReferenceFrame('N')
    pN = Point('N*')
    pN.set_vel(N, 0)
    # A.x is along the pendulum
    theta1 = atan(q2/q1)
    A = N.orientnew('A', 'axis', [theta1, N.z])
    # Create point P, the pendulum mass
    P = pN.locatenew('P1', q1*N.x + q2*N.y)
    P.set_vel(N, P.pos_from(pN).dt(N))
    pP = Particle('pP', P, m)
    # Constraint Equations
    f_c = Matrix([q1**2 + q2**2 - L**2])
    # Calculate the lagrangian, and form the equations of motion
    Lag = Lagrangian(N, pP)
    LM = LagrangesMethod(Lag, [q1, q2], hol_coneqs=f_c, forcelist=[(P, m*g*N.x)], frame=N)
    LM.form_lagranges_equations()
    # Compose operating point
    op_point = {q1: L, q2: 0, q1d: 0, q2d: 0, q1d.diff(t): 0, q2d.diff(t): 0}
    # Solve for multiplier operating point
    lam_op = LM.solve_multipliers(op_point=op_point)
    op_point.update(lam_op)
    # Perform the Linearization
    A, B, inp_vec = LM.linearize([q2], [q2d], [q1], [q1d],
            op_point=op_point, A_and_B=True)
    assert A == Matrix([[0, 1], [-9.8/L, 0]])
    assert B == Matrix([])

def test_nonminimal_pendulum():
    q1, q2 = dynamicsymbols('q1:3')
    q1d, q2d = dynamicsymbols('q1:3', level=1)
    L, m, t = symbols('L, m, t')
    g = 9.8
    # Compose World Frame
    N = ReferenceFrame('N')
    pN = Point('N*')
    pN.set_vel(N, 0)
    # Create point P, the pendulum mass
    P = pN.locatenew('P1', q1*N.x + q2*N.y)
    P.set_vel(N, P.pos_from(pN).dt(N))
    pP = Particle('pP', P, m)
    # Constraint Equations
    f_c = Matrix([q1**2 + q2**2 - L**2])
    # Calculate the lagrangian, and form the equations of motion
    Lag = Lagrangian(N, pP)
    LM = LagrangesMethod(Lag, [q1, q2], hol_coneqs=f_c,
            forcelist=[(P, m*g*N.x)], frame=N)
    LM.form_lagranges_equations()
    # Check solution
    lam1 = LM.lam_vec[0, 0]
    eom_sol = Matrix([[m*Derivative(q1, t, t) - 9.8*m + 2*lam1*q1],
                      [m*Derivative(q2, t, t) + 2*lam1*q2]])
    assert LM.eom == eom_sol
    # Check multiplier solution
    lam_sol = Matrix([(19.6*q1 + 2*q1d**2 + 2*q2d**2)/(4*q1**2/m + 4*q2**2/m)])
    assert LM.solve_multipliers(sol_type='Matrix') == lam_sol

def test_linearize_pendulum_lagrange_minimal():
    q1 = dynamicsymbols('q1')                     # angle of pendulum
    q1d = dynamicsymbols('q1', 1)                 # Angular velocity
    L, m, t = symbols('L, m, t')
    g = 9.8

    # Compose world frame
    N = ReferenceFrame('N')
    pN = Point('N*')
    pN.set_vel(N, 0)

    # A.x is along the pendulum
    A = N.orientnew('A', 'axis', [q1, N.z])
    A.set_ang_vel(N, q1d*N.z)

    # Locate point P relative to the origin N*
    P = pN.locatenew('P', L*A.x)
    P.v2pt_theory(pN, N, A)
    pP = Particle('pP', P, m)

    # Solve for eom with Lagranges method
    Lag = Lagrangian(N, pP)
    LM = LagrangesMethod(Lag, [q1], forcelist=[(P, m*g*N.x)], frame=N)
    LM.form_lagranges_equations()

    # Linearize
    A, B, inp_vec = LM.linearize([q1], [q1d], A_and_B=True)

    assert A == Matrix([[0, 1], [-9.8*cos(q1)/L, 0]])
    assert B == Matrix([])

def test_rolling_disc():
    # Rolling Disc Example
    # Here the rolling disc is formed from the contact point up, removing the
    # need to introduce generalized speeds. Only 3 configuration and 3
    # speed variables are need to describe this system, along with the
    # disc's mass and radius, and the local gravity.
    q1, q2, q3 = dynamicsymbols('q1 q2 q3')
    q1d, q2d, q3d = dynamicsymbols('q1 q2 q3', 1)
    r, m, g = symbols('r m g')

    # The kinematics are formed by a series of simple rotations. Each simple
    # rotation creates a new frame, and the next rotation is defined by the new
    # frame's basis vectors. This example uses a 3-1-2 series of rotations, or
    # Z, X, Y series of rotations. Angular velocity for this is defined using
    # the second frame's basis (the lean frame).
    N = ReferenceFrame('N')
    Y = N.orientnew('Y', 'Axis', [q1, N.z])
    L = Y.orientnew('L', 'Axis', [q2, Y.x])
    R = L.orientnew('R', 'Axis', [q3, L.y])

    # This is the translational kinematics. We create a point with no velocity
    # in N; this is the contact point between the disc and ground. Next we form
    # the position vector from the contact point to the disc's center of mass.
    # Finally we form the velocity and acceleration of the disc.
    C = Point('C')
    C.set_vel(N, 0)
    Dmc = C.locatenew('Dmc', r * L.z)
    Dmc.v2pt_theory(C, N, R)

    # Forming the inertia dyadic.
    I = inertia(L, m / 4 * r**2, m / 2 * r**2, m / 4 * r**2)
    BodyD = RigidBody('BodyD', Dmc, R, m, (I, Dmc))

    # Finally we form the equations of motion, using the same steps we did
    # before. Supply the Lagrangian, the generalized speeds.
    BodyD.set_potential_energy(- m * g * r * cos(q2))
    Lag = Lagrangian(N, BodyD)
    q = [q1, q2, q3]
    q1 = Function('q1')
    q2 = Function('q2')
    q3 = Function('q3')
    l = LagrangesMethod(Lag, q)
    l.form_lagranges_equations()
    RHS = l.rhs()
    RHS.simplify()
    t = symbols('t')

    assert (l.mass_matrix[3:6] == [0, 5*m*r**2/4, 0])
    assert RHS[4].simplify() == (-8*g*sin(q2(t)) + 5*r*sin(2*q2(t)
        )*Derivative(q1(t), t)**2 + 12*r*cos(q2(t))*Derivative(q1(t), t
        )*Derivative(q3(t), t))/(10*r)
    assert RHS[5] == (-5*cos(q2(t))*Derivative(q1(t), t) + 6*tan(q2(t)
        )*Derivative(q3(t), t) + 4*Derivative(q1(t), t)/cos(q2(t))
        )*Derivative(q2(t), t)

def test_dub_pen():

    # The system considered is the double pendulum. Like in the
    # test of the simple pendulum above, we begin by creating the generalized
    # coordinates and the simple generalized speeds and accelerations which
    # will be used later. Following this we create frames and points necessary
    # for the kinematics. The procedure isn't explicitly explained as this is
    # similar to the simple  pendulum. Also this is documented on the pydy.org
    # website.
    q1, q2 = dynamicsymbols('q1 q2')
    q1d, q2d = dynamicsymbols('q1 q2', 1)
    q1dd, q2dd = dynamicsymbols('q1 q2', 2)
    u1, u2 = dynamicsymbols('u1 u2')
    u1d, u2d = dynamicsymbols('u1 u2', 1)
    l, m, g = symbols('l m g')

    N = ReferenceFrame('N')
    A = N.orientnew('A', 'Axis', [q1, N.z])
    B = N.orientnew('B', 'Axis', [q2, N.z])

    A.set_ang_vel(N, q1d * A.z)
    B.set_ang_vel(N, q2d * A.z)

    O = Point('O')
    P = O.locatenew('P', l * A.x)
    R = P.locatenew('R', l * B.x)

    O.set_vel(N, 0)
    P.v2pt_theory(O, N, A)
    R.v2pt_theory(P, N, B)

    ParP = Particle('ParP', P, m)
    ParR = Particle('ParR', R, m)

    ParP.potential_energy = - m * g * l * cos(q1)
    ParR.potential_energy = - m * g * l * cos(q1) - m * g * l * cos(q2)
    L = Lagrangian(N, ParP, ParR)
    lm = LagrangesMethod(L, [q1, q2], bodies=[ParP, ParR])
    lm.form_lagranges_equations()

    assert simplify(l*m*(2*g*sin(q1) + l*sin(q1)*sin(q2)*q2dd
        + l*sin(q1)*cos(q2)*q2d**2 - l*sin(q2)*cos(q1)*q2d**2
        + l*cos(q1)*cos(q2)*q2dd + 2*l*q1dd) - lm.eom[0]) == 0
    assert simplify(l*m*(g*sin(q2) + l*sin(q1)*sin(q2)*q1dd
        - l*sin(q1)*cos(q2)*q1d**2 + l*sin(q2)*cos(q1)*q1d**2
        + l*cos(q1)*cos(q2)*q1dd + l*q2dd) - lm.eom[1]) == 0
    assert lm.bodies == [ParP, ParR]

def test_disc_on_an_incline_plane():
    # Disc rolling on an inclined plane
    # First the generalized coordinates are created. The mass center of the
    # disc is located from top vertex of the inclined plane by the generalized
    # coordinate 'y'. The orientation of the disc is defined by the angle
    # 'theta'. The mass of the disc is 'm' and its radius is 'R'. The length of
    # the inclined path is 'l', the angle of inclination is 'alpha'. 'g' is the
    # gravitational constant.
    y, theta = dynamicsymbols('y theta')
    yd, thetad = dynamicsymbols('y theta', 1)
    m, g, R, l, alpha = symbols('m g R l alpha')

    # Next, we create the inertial reference frame 'N'. A reference frame 'A'
    # is attached to the inclined plane. Finally a frame is created which is attached to the disk.
    N = ReferenceFrame('N')
    A = N.orientnew('A', 'Axis', [pi/2 - alpha, N.z])
    B = A.orientnew('B', 'Axis', [-theta, A.z])

    # Creating the disc 'D'; we create the point that represents the mass
    # center of the disc and set its velocity. The inertia dyadic of the disc
    # is created. Finally, we create the disc.
    Do = Point('Do')
    Do.set_vel(N, yd * A.x)
    I = m * R**2 / 2 * B.z | B.z
    D = RigidBody('D', Do, B, m, (I, Do))

    # To construct the Lagrangian, 'L', of the disc, we determine its kinetic
    # and potential energies, T and U, respectively. L is defined as the
    # difference between T and U.
    D.set_potential_energy(m * g * (l - y) * sin(alpha))
    L = Lagrangian(N, D)

    # We then create the list of generalized coordinates and constraint
    # equations. The constraint arises due to the disc rolling without slip on
    # on the inclined path. Also, the constraint is holonomic but we supply the
    # differentiated holonomic equation as the 'LagrangesMethod' class requires
    # that. We then invoke the 'LagrangesMethod' class and supply it the
    # necessary arguments and generate the equations of motion. The'rhs' method
    # solves for the q_double_dots (i.e. the second derivative with respect to
    # time  of the generalized coordinates and the lagrange multiplers.
    q = [y, theta]
    coneq = [yd - R * thetad]
    m = LagrangesMethod(L, q, coneq)
    m.form_lagranges_equations()
    rhs = m.rhs()
    rhs.simplify()
    assert rhs[2] == 2*g*sin(alpha)/3

def test_simp_pen():
    # This tests that the equations generated by LagrangesMethod are identical
    # to those obtained by hand calculations. The system under consideration is
    # the simple pendulum.
    # We begin by creating the generalized coordinates as per the requirements
    # of LagrangesMethod. Also we created the associate symbols
    # that characterize the system: 'm' is the mass of the bob, l is the length
    # of the massless rigid rod connecting the bob to a point O fixed in the
    # inertial frame.
    q, u = dynamicsymbols('q u')
    qd, ud = dynamicsymbols('q u ', 1)
    l, m, g = symbols('l m g')

    # We then create the inertial frame and a frame attached to the massless
    # string following which we define the inertial angular velocity of the
    # string.
    N = ReferenceFrame('N')
    A = N.orientnew('A', 'Axis', [q, N.z])
    A.set_ang_vel(N, qd * N.z)

    # Next, we create the point O and fix it in the inertial frame. We then
    # locate the point P to which the bob is attached. Its corresponding
    # velocity is then determined by the 'two point formula'.
    O = Point('O')
    O.set_vel(N, 0)
    P = O.locatenew('P', l * A.x)
    P.v2pt_theory(O, N, A)

    # The 'Particle' which represents the bob is then created and its
    # Lagrangian generated.
    Pa = Particle('Pa', P, m)
    Pa.set_potential_energy(- m * g * l * cos(q))
    L = Lagrangian(N, Pa)

    # The 'LagrangesMethod' class is invoked to obtain equations of motion.
    lm = LagrangesMethod(L, [q])
    lm.form_lagranges_equations()
    RHS = lm.rhs()
    assert RHS[1] == -g*sin(q)/l

def test_lagrange_2forces():
    ### Equations for two damped springs in serie with two forces

    ### generalized coordinates
    qs = q1, q2 = dynamicsymbols('q1, q2')
    ### generalized speeds
    qds = q1d, q2d = dynamicsymbols('q1, q2', 1)

    ### Mass, spring strength, friction coefficient
    m, k, nu = symbols('m, k, nu')

    N = ReferenceFrame('N')
    O = Point('O')

    ### Two points
    P1 = O.locatenew('P1', q1 * N.x)
    P1.set_vel(N, q1d * N.x)
    P2 = O.locatenew('P1', q2 * N.x)
    P2.set_vel(N, q2d * N.x)

    pP1 = Particle('pP1', P1, m)
    pP1.potential_energy = k * q1**2 / 2

    pP2 = Particle('pP2', P2, m)
    pP2.potential_energy = k * (q1 - q2)**2 / 2

    #### Friction forces
    forcelist = [(P1, - nu * q1d * N.x),
                 (P2, - nu * q2d * N.x)]
    lag = Lagrangian(N, pP1, pP2)

    l_method = LagrangesMethod(lag, (q1, q2), forcelist=forcelist, frame=N)
    l_method.form_lagranges_equations()

    eq1 = l_method.eom[0]
    assert eq1.diff(q1d) == nu
    eq2 = l_method.eom[1]
    assert eq2.diff(q2d) == nu

def test_linearize_rolling_disc_lagrange():
    q1, q2, q3 = q = dynamicsymbols("q1 q2 q3")
    q1d, q2d, q3d = qd = dynamicsymbols("q1 q2 q3", 1)
    r, m, g = symbols("r m g")

    N = ReferenceFrame("N")
    Y = N.orientnew("Y", "Axis", [q1, N.z])
    L = Y.orientnew("L", "Axis", [q2, Y.x])
    R = L.orientnew("R", "Axis", [q3, L.y])

    C = Point("C")
    C.set_vel(N, 0)
    Dmc = C.locatenew("Dmc", r * L.z)
    Dmc.v2pt_theory(C, N, R)

    I = inertia(L, m / 4 * r ** 2, m / 2 * r ** 2, m / 4 * r ** 2)
    BodyD = RigidBody("BodyD", Dmc, R, m, (I, Dmc))
    BodyD.potential_energy = -m * g * r * cos(q2)

    Lag = Lagrangian(N, BodyD)
    l = LagrangesMethod(Lag, q)
    l.form_lagranges_equations()

    # Linearize about steady-state upright rolling
    op_point = {q1: 0, q2: 0, q3: 0, q1d: 0, q2d: 0, q1d.diff(): 0, q2d.diff(): 0, q3d.diff(): 0}
    A = l.linearize(q_ind=q, qd_ind=qd, op_point=op_point, A_and_B=True)[0]
    sol = Matrix(
        [
            [0, 0, 0, 1, 0, 0],
            [0, 0, 0, 0, 1, 0],
            [0, 0, 0, 0, 0, 1],
            [0, 0, 0, 0, -6 * q3d, 0],
            [0, -4 * g / (5 * r), 0, 6 * q3d / 5, 0, 0],
            [0, 0, 0, 0, 0, 0],
        ]
    )

    assert A == sol

# Other system variables
m, l, g = symbols('m l g')

# Set up the reference frames
# Reference frame A set up in the plane perpendicular to the page containing
# segment OP
N = ReferenceFrame('N')
A = N.orientnew('A', 'Axis', [theta, N.z])

# Set up the points and particles
O = Point('O')
P = O.locatenew('P', l * A.x)

Pa = Particle('Pa', P, m)

# Set up velocities
A.set_ang_vel(N, thetad * N.z)
O.set_vel(N, 0)
P.v2pt_theory(O, N, A)

# Set up the lagrangian
L = Lagrangian(N, Pa)

# Create the list of forces acting on the system
fl = [(P, g * m * N.x)]

# Create the equations of motion using lagranges method
l = LagrangesMethod(L, [theta], forcelist=fl, frame=N)

pprint(l.form_lagranges_equations())

    v0: 20}

N = ReferenceFrame('N')
B = N.orientnew('B', 'axis', [q3, N.z])


O = Point('O')
S = O.locatenew('S', q1*N.x + q2*N.y)
S.set_vel(N, S.pos_from(O).dt(N))

#Is = m/12*(l**2 + w**2)
Is = symbols('Is')
I = inertia(B, 0, 0, Is, 0, 0, 0)
rb = RigidBody('rb', S, B, m, (I, S))
rb.set_potential_energy(0)


L = Lagrangian(N, rb)
lm = LagrangesMethod(
    L, q, nonhol_coneqs = [q1d*sin(q3) - q2d*cos(q3) + l/2*q3d])
lm.form_lagranges_equations()
rhs = lm.rhs()
print('{} = {}'.format(msprint(q1d.diff(t)),
    msprint(rhs[3].simplify())))
print('{} = {}'.format(msprint(q2d.diff(t)),
    msprint(rhs[4].simplify())))
print('{} = {}'.format(msprint(q3d.diff(t)),
    msprint(rhs[5].simplify())))
print('{} = {}'.format('λ', msprint(rhs[6].simplify())))

def LEq(L,a):
	LM = LagrangesMethod(L,[a])

	return LM.form_lagranges_equations()

from sympy.physics.vector import ReferenceFrame
from sympy.physics.mechanics import inertia, msprint
from sympy.physics.mechanics import Point, RigidBody
from sympy.physics.mechanics import Lagrangian, LagrangesMethod
from sympy import symbols

q = q1, q2, q3, q4, q5, q6 = dynamicsymbols("q1:7")
m, g, k, px, Ip = symbols("m g k px Ip")
N = ReferenceFrame("N")
B = N.orientnew("B", "body", [q4, q5, q6], "zyx")

A = Point("A")
S = A.locatenew("S", q1 * N.x + q2 * N.y + q3 * N.z)
P = S.locatenew("P", px * B.x)
A.set_vel(N, 0)
S.set_vel(N, S.pos_from(A).dt(N))
P.v2pt_theory(S, N, B)

Ixx = Ip / 2
Iyy = Ip / 2
Izz = Ip
I = inertia(B, Ixx, Iyy, Izz, 0, 0, 0)
rb = RigidBody("rb", S, B, m, (I, S))
rb.set_potential_energy(-m * g * (rb.masscenter.pos_from(A) & N.z) + k / 2 * (P.pos_from(A)).magnitude() ** 2)

L = Lagrangian(N, rb)
print("{} = {}\n".format("L", msprint(L)))

lm = LagrangesMethod(L, q)
print(msprint(lm.form_lagranges_equations()))

p2 = O.locatenew('p2', r2)

O.set_vel(N, 0)
p1.set_vel(N, p1.pos_from(O).dt(N))
p2.set_vel(N, p2.pos_from(O).dt(N))

P1 = Particle('P1', p1, 2*m)
P2 = Particle('P2', p2, m)

P1.set_potential_energy(0)
P2.set_potential_energy(P2.mass * g * (p2.pos_from(O) & N.y))

L1 = Lagrangian(N, P1, P2)
print('{} = {}'.format('L1', msprint(L1)))

lm1 = LagrangesMethod(L1, [s, theta])
lm1.form_lagranges_equations()
rhs = lm1.rhs()
t = symbols('t')
print('{} = {}'.format(msprint(sd.diff(t)), msprint(rhs[2].simplify())))
print('{} = {}\n'.format(msprint(thetad.diff(t)), msprint(rhs[3].simplify())))

# part b
r1 = s*N.x + h*N.y
r2 = (s + l*cos(theta))*N.x + (h + l*sin(theta))*N.y

p1 = O.locatenew('p1', r1)
p2 = O.locatenew('p2', r2)
p1.set_vel(N, p1.pos_from(O).dt(N))
p2.set_vel(N, p2.pos_from(O).dt(N))