metacentric height, (a) Obtain the equation of motion describing the ship\'s rol

Question

metacentric height, (a) Obtain the equation of motion describing the ship's rolling motion in terms of the angle theta. (b) The given parameters arc the ship's weight W, its metacentric height A, and its moment of inertia I about the center of gravity. Obtain an expression for the natural frequency of the rolling motion. In the system shown in Figure P4.16, the input is the angular displacement phi of the end of the shaft, and the output is the angular displacement theta of the inertia I. The shafts have torsional stiffnesses k1 and k2. The equilibrium position corresponds to phi = theta = 0. Derive the equation of motion and find the transfer function Theta(s)/Phi(s). In Figure P4.17, assume that the cylinder rolls without slipping. The spring is at its free length when x and y arc zero, (a) Derive the equation of motion in terms of x, with y(t) as the input, (b) Suppose that m = 10 kg, R = 0.3 m, k = 1000 N/m, and that y(t) is a unit-step function. Solve for x(t) if x(0) = (0) = 0. In Figure P4.18 when x1 = x2 = 0 the springs are at their free lengths. Derive the equations of motion. Figure P4.16 Figure P4.17 Figure P4.18 In Figure P4.19 model the three shafts as massless torsional springs. When theta1 = theta2 = 0 the springs arc at their free lengths. Derive the equations of motion with the torque T2 as the input. In Figure P4.20 when theta1 = theta2 = 0 the spring is at its free length. Derive the equations of motion, assuming small angles. Figure P4.19 Figure P4.20

Explanation / Answer

by taking moment about a fixed point

body itself moment is= force * distance = Ig*(theta)

there are three different portion with different stifness ( k1, k2 and thier central part)

moment 1 = force * perpendicular distance from point = k1*(fi)

simlarly moment 2 = k2*(fi)

and moment 3 = I * (theta)

sum of moment of these three portions is equal & opposite to body itself moment so we can write it

Ig(theta)

Ig(theta) = -k1(fi) - k2(fi) - I(theta)

simplify

Ig(theta) + k1(fi) + k2(fi) + I(theta)= 0

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