Advanced Methods of Structural Analysis

14.3 Free Vibrations of Systems with Finite Number Degrees of Freedom 535

Equations (e) divide byA 1 .Let 2 DA 2 =A 1 ; 3 D A 3 =A 1. Equations for

modes become

.9:8572i/9:42852iC3:85723iD0;

9:4285C.13:7142i/2i9:42853iD0;

3:85729:42852iC.9:8572i/3iD0:

(f)

1.Eigenvalue 1 D0:3804

.9:85720:3804/9:42852iC3:85723iD0;

9:4285C.13:71420:3804/ 2i9:42853iD0:

Solution of this equation 2 D1:4142;  3 D1:0.

The same procedure should be repeated for 2 D6:0and 3 D27:0482.

The modal matrixˆis defined as

ˆD

2

4

11 1

1:4142 0:0 1:4142

1  11

3

(^5) (g)
Same mode shape coefficients have been obtained by force method.
Example 14.4.Design diagram of multistore frame is presented in Fig.14.13a. The
cross bars are absolutely rigid bodies; theirs masses are shown in design diagram.
Flexural sriffness of the vertical members areEIand masses of the struts are ignored.
Calculate the frequencies of vibrations and find the corresponding mode shapes.
Solution.The primary system is shown in Fig.14.13b. For computation of unit
reactions, we need to construct the bending moment diagrams due to unit diplace-
ments of the introduced constraints and then to consider the equilibrium condition
for each cross-bar.
Bending moment diagram caused by unit displacement of the constraint 1 is
shown in Fig.14.13c. Elastic curve is shown by dotted line. Since cross bars are
absolutely rigid members, then jointscannot be rotated and each vertical member
should be considered as fixed-fixed member. In this case, specified ordinates are
6i=h. Bending moment diagram is shown on the extended fibers. Now we need to
show free-body diagram for each vertical member. The sections are passes infinitely
close to the bottom and lower joints. Bending moments are6i=h. Both moments
may be equilibrate by two forces12i=h^2. These forces should be transmitted on both
cross-bars. Positive unit reactionsr 11 ;r 21 ,andr 31 are shown by dotted arrows.
Equilibrium condition for each cross-bar leads to the follofing unit reactions
r 11 D 2
12i
h^2
D 24
i
h^2
;r 21 D 24
i
h^2
;r 31 D0; iD
EI
h
: