The motions of moored vessels under the influence of an external force, such as the one induced by a passing ship, can be determined from the solution of the equation of motion. Fig.17 shows the conventional mooring rig layout of a ship. To simplify the calculation, this arrangement is idealised to heading/stern lines and the breast lines, which are treated parallel and perpendicular to the centreline of the moored ship always (Fig.18). All the ropes in the direction of x (or y) are considered of the same length, tension, material, size and construction. These assumptions enable one to treat the mooring system reaction in both directions independently.

　

　

　

 Considering the system as a spring-mass one with damping, it being oscillatory in nature, the equation of motion in x-direction can be written as

　

　

 where κx=σbd²nr/δx, X(t) is the interaction force in surge mode as a function of time t, m is the mass of the moored ship, mx' is the added mass of the moored ship in surge, x is the displacement of the moored ship along its axis, Dx is the damping coefficient in longitudinal motion, κx is the rope spring constant, σb is the breaking load of the rope, d is the rope diameter, nr is the number of ropes and δx is elongation of the stem/head lines.

　

 By knowing all the coefficients and force, the above differential equation can be solved for the moored ship's longitudinal displacement x. The mooring rope force can be obtained by taking the product of the rope (spring) constant kx and x. Damping term in the equation is usually very small. That is, the oscillation in x-direction keeps on continuing for a longer time. But, the importance here is amplitude of the force rather than the period of damping out.

　

 While considering the generation of the equation of motion in y-direction (ie., with the breast lines in mind), the contributions coming from both sway and yaw motions must be included. The formulations of the equation of motion are dealt with independently and while estimating the breast line forces, the resultant effect of sway and yaw displacements are considered.

　

　

where

　

 Solution of Eq.(3.2) and (3.3) give, respectively, the sway (y) and yaw displacements (ψ). The resultant linear displacements of the ship at the position of stern and bow breast lines are given by Eqs.(3.4) and (3.5) below

　

　

 where Y(t) and N(t) are the interaction sway force and yaw moment as a function of time t, Iψ is the mass moment of inertia of the moored ship about z-axis, my' is the added mass of the moored ship in sway, Iψ' is the added mass moment of inertia of the ship about z-axis, y and ψ are the displacement of the moored ship sway and yaw, Dy and Dψ are the damping coefficient in sway and yaw modes, ky and kψ are the rope spring or restoring constants in sway and yaw modes, δy is elongation of the breast lines and Ib is the spacing between stern and bow breast lines.

　

 In order to investigate the response of a flexibly moored ship to the loads induced by a passing ship, the equations of motion shown above need to be solved. A knowledge of the coefficients in these equations, which includes the ship mass, added mass and damping coefficients in surge and sway modes, ship mass moment of inertia, added mass moment of inertia and damping coefficients in yaw mode and the interaction forces and moment, are required for its solution. Similar investigation on mooring loads of a ship due to the passage of ships with different sizes, speeds and lateral separation distances can be found in [1]. Fig.19 shows a case of it, where the lateral mooring line (rope constant = 1609 kN/m) forces of a ship (Lm=257m) due to the passing of a ship of LP=302m at a speed of 7.0 knots with a separation distance of 30.0m through a water depth of 1.15 times the moored ship draft are presented. The calculated and measured values of Remery compare well with the present ones, where the lateral lines are taken as breast lines on either side of the mid-ship. In athwart-ship direction the oscillation dies out fast due to high damping in the sway mode. The excitation forces and moment used here are the experimental values given in Figs.2 to 4. The values of motion displacements and velocities are taken as zero initially (ie. at t=0).