Repeat Problem 10.24 using a different set of DOFs—those defined in Problem 9.15. Show that the natural vibration frequencies and modes determined using the two sets of DOFs are the same.
For the system defined in Problem 9.14, m = 90 kips/g, k = 1.5 kips/in., and b = 25 ft.
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(a) Determine the natural vibration frequencies and modes.
(b) Normalize each mode so that the modal mass Mn has unit value. Sketch these modes.
Figure P9.14 shows a uniform slab supported on four columns rigidly attached to the slab and clamped at the base. The slab has a total mass m and is rigid in plane and out of plane. Each column is of circular cross section, and its second moment of cross-sectional area about any diametrical axis is as noted. With the DOFs selected as ux , uy , and uθ at the center of the slab, and using influence coefficients:
(a) Formulate the mass and stiffness matrices in terms of m and the lateral stiffness k = 12 /h3 of the smaller column; h is the height.
(b) Formulate the equations of motion for ground motion in (i) the x-direction, (ii) the y-direction, and (iii) the direction d–b.
Repeat Problem 9.14 using the second set of DOFs shown in Fig. P9.15.