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FIGURE 2.133 Vibration of dynamic amplification factor with frequency ratio.

Similarly, the ratio of the acceleration response relative to the ground acceleration may be expressed as

Da is called the dynamic acceleration magnification factor.

2.13.4 Response to Suddenly Applied Load

Consider the spring-mass damper system where a load Po is applied suddenly. The differential equation is given by

M€ + cx+kx = Po If the system is started at rest, the equation of motion is

Po k

If the system is undamped, then X = 0 and od = o, we have x = p0 [1 — cos oat ] k

The maximum displacement is 2(Po/k) corresponding to cos mdt = —1. Since Po/k is the maximum static displacement, the dynamic amplification factor is equal to 2. The presence of damping would naturally reduce the dynamic amplification factor and the force in the system.

2.13.5 Response to Time-Varying Loads

Some forces and ground motions that are encountered in practice are rather complex. In general, numerical analysis is required to predict the response of such effects, and the finite element method is one of the most common techniques to be employed in solving such problems.

The evaluation of responses due to time-varying loads can be carried out using the piecewise exact method. In using this method, the loading history is divided into small time intervals. Between these

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