Consider a ship of length \(130 ~m\) with a design speed of \(15~knots\). During the steady phase of the turning circle maneuver it is observed that the ship experiences a yaw rate of \(r = 0.01 ~rad/s\). If the ship expereinces a drift angle of \(5^{\circ}\), determine the longitudinal location of the pivot point with respect to the BCS.
A ship has non-dimensional pivot point location \(x'_c = 0.32\) during a steady turn. Using the relationships between \(R'\), \(\beta\), and \(x'_c\):
Consider a ship of length \(150 ~m\) with a design speed of \(20 ~knots\) executing a port turning circle maneuver. Assume that the ship has the following parameters:
Determine the sway and yaw acceleration of the ship in the first phase of the turn. Compare it with the values obtained when cross coupled effects are neglected.
Consider a box barge of length \(100 ~m\), breadth \(40 ~m\) and draft \(15 ~m\) operating with a design speed of \(20 ~knots\) executing a port turning circle maneuver. Assume that the ship has the following parameters:
Determine the heel of the vessel during the steady phase of the turn if center of gravity is located at the waterline. Assume that the rudder hydrodynamic foces act at \(5 ~m\) from the keel and the hull hydrodynamic forces act at the same height as the center of buoyancy. You may also assume that the density of seawater is \(\rho = 1025 ~kg/m^3\).
Consider a ship of length \(130 ~m\) with a design speed of \(15~knots\). During the steady phase of the turning circle maneuver it is observed that the ship experiences a yaw rate of \(r = 0.01 ~rad/s\). If the ship expereinces a drift angle of \(5^{\circ}\), determine the longitudinal location of the pivot point with respect to the BCS.
import numpy as np
def solve_problem_01():
L = 130
U = 15 * 0.514444
r = 0.01
R = U/r
bet = 5 * np.pi / 180
x_p = R * np.sin(bet)
print(f"The location of the pivot point is {x_p:.2f} m from BCS origin.")
solve_problem_01()The location of the pivot point is 67.26 m from BCS origin.A ship has non-dimensional pivot point location \(x'_c = 0.32\) during a steady turn. Using the relationships between \(R'\), \(\beta\), and \(x'_c\):
import numpy as np
def solve_problem_02():
xcp = 0.32
bet = 5 * np.pi / 180
Rp = xcp / np.sin(bet)
vp = -np.sin(bet)
vp_new = 2 * vp
bet_new = np.arcsin(-vp_new)
Rp_new = xcp / np.sin(bet_new)
print(f"Non-dimensional radius is {Rp:.2f} when drift angle is {bet*180/np.pi:.1f} degrees")
print(f"Non-dimensional radius is {Rp_new:.2f}, drift angle is {bet_new*180/np.pi:.1f} degrees")
solve_problem_02()Non-dimensional radius is 3.67 when drift angle is 5.0 degrees
Non-dimensional radius is 1.84, drift angle is 10.0 degreesConsider a ship of length \(150 ~m\) with a design speed of \(20 ~knots\) executing a port turning circle maneuver. Assume that the ship has the following parameters:
Determine the sway and yaw acceleration of the ship in the first phase of the turn. Compare it with the values obtained when cross coupled effects are neglected.
import numpy as np
def solve_problem_03():
L = 150
U = 20 * 0.51444
Yvdp = -0.03
Nrdp = -0.01
Yrdp = 0.002
Nvdp = -0.0035
Ndp = -0.004
Ydp = 0.005
xGp = 0.03
mp = 0.02
Izp = 0.005
delta = 35 * np.pi / 180
Amat = np.array([
[mp - Yvdp, -(Yrdp - mp*xGp)],
[-(Nvdp - mp*xGp), Izp - Nrdp]
])
bvec = np.array([
Ydp * delta,
Ndp * delta
])
xp, _, _, _ = np.linalg.lstsq(Amat, bvec, rcond=None)
x = xp.copy()
x[0] = x[0] * U**2 / L
x[1] = x[1] * U**2 / L**2
x_alt = np.zeros(2)
x_alt[0] = (Ydp * delta / (mp - Yvdp)) * U**2 / L
x_alt[1] = (Ndp * delta / (Izp - Nrdp)) * U**2 / L**2
print(f"The sway acceleration is {x[0]:.4f} m/s^2 and yaw acceleration is {x[1]:.4f} rad/s^2")
print(f"Neglecting cross coupling, the sway acceleration is {x_alt[0]:.4f} m/s^2 and yaw acceleration is {x_alt[1]:.4f} rad/s^2")
solve_problem_03()The sway acceleration is 0.0396 m/s^2 and yaw acceleration is -0.0008 rad/s^2
Neglecting cross coupling, the sway acceleration is 0.0431 m/s^2 and yaw acceleration is -0.0008 rad/s^2Consider a box barge of length \(100 ~m\), breadth \(40 ~m\) and draft \(15 ~m\) operating with a design speed of \(20 ~knots\) executing a port turning circle maneuver. Assume that the ship has the following parameters:
Determine the heel of the vessel during the steady phase of the turn if center of gravity is located at the waterline. Assume that the rudder hydrodynamic foces act at \(5 ~m\) from the keel and the hull hydrodynamic forces act at the same height as the center of buoyancy. You may also assume that the density of seawater is \(\rho = 1025 ~kg/m^3\).
import numpy as np
def solve_problem_04():
L = 100
B = 40
T = 15
U = 20*0.514444
g = 9.81
Yvp = -0.05
Nvp = 0.001
Yrp = 0.001
Nrp = -0.05
Ydp = 0.05
Ndp = -0.004
xGp = 0.02
rho = 1025
delta = 35 * np.pi / 180
mp = L * B * T / (0.5 * L**3)
Amat = np.array([
[-Yvp, -(Yrp - mp)],
[-Nvp, -(Nrp - mp*xGp)]
])
bvec = np.array([
Ydp * delta,
Ndp * delta
])
x, _, _, _ = np.linalg.lstsq(Amat, bvec, rcond=None)
vp = x[0]
rp = x[1]
KB = T/2
Ixx = L * B**3 / 12
Dsp = L * B * T
BM = Ixx / Dsp
KG = T
GM = KB + BM - KG
Y_hull = (Yvp * vp + Yrp * rp) * (0.5 * rho * L**2 * U**2)
Y_rudder = Ydp * delta * (0.5 * rho * L**2 * U**2)
K = Y_hull * T/2 + Y_rudder * (T - 5)
C44 = rho * g * Dsp * GM
heel = K / C44
print(f" The heel angle is {heel * 180 / np.pi:.2f} degrees")
solve_problem_04() The heel angle is 1.72 degrees