Question

Hi! I only need the solution for Fig. 9.7.

This is Problem 10.11 from textbook: Dynamics of Structures by Anil K. Chopra, fifth edition

(I don't care if you copy the solution manual because I cannot find it online for free! But if you do, please add some notes or insight into tricky steps please!)

Thanks in advance! :)

For the three-story shear buildings shown in Figs. P9.7 and P9.8: (a) Determine the natural vibration frequencies and modes, express the frequencies in terms of m, El, and h. Sketch the modes and identify the associated natural frequencies. (b) Verify that the modes satisfy the orthogonality properties. (c) Normalize each mode so that the modal mass Mn has unit value. Sketch these normalized modes. Compare these modes with those obtained in part (a) and comment on the differences. m/2 Rigid beams Pl EI EI 1m EI 1m El 2h Figure P9.7

Answer 1

stiffhess matrix Apply ul-1 , แ2 0 and determine ka "3 31 K21 =-k Apply "2-1, りー0 and determine ka 32-k Apply = 1, ul-u2 = 0 and determine ki . "3-1 13-0

The stiffness matrix is 2 1 0 k=kl-1 2-1 0- 1 Mass matrix m u, 13 23 13 The mass matrix is 05

24EI 0 1 Natural Frequencies and modes 2-λ -1 24 EI -1 - 052 where 24 EI and det [k-dam-0 so the frequency equation is The solution gives λ1-2-V3 = 0.2679, λ2-2 and λ3-2 +-3-3.7321. The corresponding natural trequencies are 6.9282 a, 9.464 1 mh The mode shapes can be determined from 05 05 0.866 -0.866 0.5 0.5 First mode Second mode Third mode

b) Model orthogonality 05 1 05 Wn0з = m(05 0.866 1기 ll-0866) = 0 í 05 05 -0.866 0 05 24EI 011 1 24 EI 05 0.866 1〉1-1 2 -ik-0.866 = 0 05 24 EI (-1 0 1)-1 2-1 -0866 =0 Thus the computed modes satisfy the orthogonality properties. Normalize each modes so that M1 05 =m(05 0866 )| |10866 1 05 - 15m Divide by $1.5m, 4.2 by T5m and фз by 1.5m toobtain the normalized modes: 0866 These modes are scalar multiples of the modes in (a); the shapes of the two sets of modes are the same