one unit of length, which, in fact, is the force caused by a unit side sway of the frame. The mass is in equilibrium under the action of two equal and opposite forces: the weight, W, acts downward and the spring force, Kxst, upward. The term xst denotes static deflection, which is the amount of movement from the undeformed position to the equilibrium position where the displacement of the mass is measured during the vibration. The motion equation of the structure is

in which p = \JK/M, where p is the angular frequency with the unit of rad/s; M = W/g; and g is the gravity of acceleration. Note that the units of K and M must be consistent. The relationships between frequency and period may be expressed as

The solution of Equation 3.1 is x = A sin pt + B cos pt (3.2)

where the integration constants A and B should be determined by using the information of motion as the known displacement, x, and velocity, x, at any time, t. The displacement and velocity may be given at the same time, say xt0 and xt0, or at a different time, xt0 at t0 and xt1 at t1. Let us assume that x and x are given as xt0 and xt0 at t0, then Equation 3.2 becomes x = xt0 cosp(t — t0)+ —sinp(t — t0) (3.3)

When the original time is measured from the instant that the mass is in one of the extreme positions, the initial displacement xo is X (X denotes amplitude) and the initial velocity is zero (as the physical condition should be). When time is measured from the instant that the mass is in the neutral position, the initial conditions are x = 0 and x = x0. If the origin is located at t0 units of time after the mass passes the neutral position with the initial conditions of x = xt0. The general expression becomes x = xt0 cos(pt — g) + — sin(pt — g) (3.4a)

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