STRUCTURAL MASS, DAMPING, AND RESTRAINT
51
Example Problem 2.10. Consider the explicit forms for the cable restraint
functions q(v) of the spread moored ship depicted in Figure 1.6 and previously
introduced in Example Problem 2.1. As in the original experimental study and
analysis by O’Brien and Muga (1964), traditional English units are used to
express the numerical results. The subject ship was an LST (Landing Ship,
Tank) moored in 45 ft of water in the Gulf of Mexico. The calculations for the
separate restraint functions q(v) for surge and sway were based on the catenary
theory just discussed and the actual mooring geometry of ail seven chains for this
experimental study. The results are shown in Figure 2.22. These results typify a
hard spring nonlinear restraint System and are similar to the two-cable example
shown in Figure 2.21. When each curve of Figure 2.22 is fit to the odd order
cubic polynomial of équation (2.69), the restraining force for surge (longitudinal)
displacement and for sway (latéral) displacement become, respectively
q(v) = 20, 300i> + 400r3 1b
(2.70)
q(y) = 12,700u + 950v31b
(2.71)
The coefficient of v in each case is Aq, or the slope of the curve at v — 0.
Example Problem 2.11. The purposes of this example are to compute the
virtual masses for the moored ship described in Example Problem 2.10, and to
set up the uncoupled équations of motion in surge and sway. Traditional English
units are employed for the purpose of clarifying the unit of mass. As previously,
the ship is assumed to be a rigid body with an actual mass mu. Based on its
given displacement (weight) of 4400 long tons, the ship’s actual mass is
mo =------ r~,—ô x 4400 long tons x 2240:;----------- = 3.06 x 105slug
32.1 ft/sec2
long ton
in which the mass unit of lb-sec2/ft is designated as slug. Experimental evidence
shows that for surge motion only, the virtual mass is approximately 15 percent
greater than m0, or m = 1.15m0 = 3.52 x 105 slug; and for sway motion only,
the virtual mass for this ship (unstreamlined for sway) is about twice mq, or
m = 2m0 = 6.12 x 105 slug. For either motion, the governing équations are
of the form of équation (2.2). Assume negligible damping, /(û) = 0, and calm
seas, p(t) = 0. Under these conditions, with q(v) given by équations (2.70) and
(2.71), and with the virtual masses just calculated, the respective équations of
motion for surge and sway are as follows:
3.52 x 105v + 20.300r + 400v3 = 0
(2.72)
6.12 x 105ü + 12,700v + 950v3 = 0
(2.73)
In these équations of motion, v has units of ft/sec2 and v has units of ft.
51
Example Problem 2.10. Consider the explicit forms for the cable restraint
functions q(v) of the spread moored ship depicted in Figure 1.6 and previously
introduced in Example Problem 2.1. As in the original experimental study and
analysis by O’Brien and Muga (1964), traditional English units are used to
express the numerical results. The subject ship was an LST (Landing Ship,
Tank) moored in 45 ft of water in the Gulf of Mexico. The calculations for the
separate restraint functions q(v) for surge and sway were based on the catenary
theory just discussed and the actual mooring geometry of ail seven chains for this
experimental study. The results are shown in Figure 2.22. These results typify a
hard spring nonlinear restraint System and are similar to the two-cable example
shown in Figure 2.21. When each curve of Figure 2.22 is fit to the odd order
cubic polynomial of équation (2.69), the restraining force for surge (longitudinal)
displacement and for sway (latéral) displacement become, respectively
q(v) = 20, 300i> + 400r3 1b
(2.70)
q(y) = 12,700u + 950v31b
(2.71)
The coefficient of v in each case is Aq, or the slope of the curve at v — 0.
Example Problem 2.11. The purposes of this example are to compute the
virtual masses for the moored ship described in Example Problem 2.10, and to
set up the uncoupled équations of motion in surge and sway. Traditional English
units are employed for the purpose of clarifying the unit of mass. As previously,
the ship is assumed to be a rigid body with an actual mass mu. Based on its
given displacement (weight) of 4400 long tons, the ship’s actual mass is
mo =------ r~,—ô x 4400 long tons x 2240:;----------- = 3.06 x 105slug
32.1 ft/sec2
long ton
in which the mass unit of lb-sec2/ft is designated as slug. Experimental evidence
shows that for surge motion only, the virtual mass is approximately 15 percent
greater than m0, or m = 1.15m0 = 3.52 x 105 slug; and for sway motion only,
the virtual mass for this ship (unstreamlined for sway) is about twice mq, or
m = 2m0 = 6.12 x 105 slug. For either motion, the governing équations are
of the form of équation (2.2). Assume negligible damping, /(û) = 0, and calm
seas, p(t) = 0. Under these conditions, with q(v) given by équations (2.70) and
(2.71), and with the virtual masses just calculated, the respective équations of
motion for surge and sway are as follows:
3.52 x 105v + 20.300r + 400v3 = 0
(2.72)
6.12 x 105ü + 12,700v + 950v3 = 0
(2.73)
In these équations of motion, v has units of ft/sec2 and v has units of ft.