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Equations 3.42a, and 3.42b can be rewritten as xi = XstiPiSvi

in which Xti is equivalent to static displacement and Svi is the velocity response spectrum corresponding to the ith mode. For undamped vibration, we simply let e-p,p,t and e-Pipi(t- D) be a unity in Equations 3.40 to 3.42. The actual displacement response is obtained from

As mentioned before, [F] and [X] have significant physical meaning because they show how much contribution is made from each mode to the total displacement.

For example, the structure shown in Figure 3.13 has p1 = 2.218 rad/s, p2 = 10.781 rad/s, and the applied forces F1(t) = F1f(t), F2(t) = F2f(t), where F1 = 20 k, F2 = 30 k, and f(t) = 1 - (t/Z) are given in the accompanying figure. Find the displacement contributed from each individual mode and the total response. Assume initial conditions are zero. [F11 F21] = [0.340 0.293], [F12 F22] = [-0.293 0.340]. Based on Equations 3.41 and 3.43a, the displacement response is i\

x1(due to the first mode)+ x1(due to the second mode) x2(due to the first mode)+ x2(due to the second mode)

The results obtained from Equation 3.44 are illustrated in Figure 3.14 and Figure 3.15. This example reveals how the individual modes contribute to the total response. In this case, the first mode is indeed contributing the most. For tall buildings, in general, the first several fundamental modes are essential in affecting response behavior and are practically needed in design. This example also reveals that FriFr is measuring how much of the applied force contributing to the rth mode. Thus Fr,Fr, XnFr, F^Mr, and XriMr in Equations 3.41a, 3.41b and 3.42a, 3.41b are all measuring how much the rth mode participates

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