Advanced Methods of Structural Analysis

528 14 Dynamics of Elastic Systems

ı 12 Dı 21 D

Z

M 1 M 2

EI

dxD

11

768

l^3

EI

;

ı 13 Dı 31 D

Z

M 1 M 3

EI

dxD

7

768

l^3

EI

;ı 23 Dı 32 Dı 12 Dı 21 D

11

768

l^3

EI

:

Letı 0 Dl^3 =768EI. Matrix of unit displacementsF(Flexibility matrix) is

FDŒıikD

2

4

ı 11 ı 12 ı 13

ı 21 ı 22 ı 23

ı 31 ı 32 ı 33

3

(^5) Dı 0
2
4
9117
11 16 11
7119
3
(^5) Dı 0 F 0 :
Equations (14.4) with unknown amplitudesAiof massmiare

m 1 ı 11!^2  1

A 1 Cm 2 ı 12!^2 A 2 Cm 3 ı 13!^2 A 3 D0;
m 1 ı 21!^2 A 1 C

m 2 ı 22!^2  1

A 2 Cm 3 ı 23!^2 A 3 D0;
m 1 ı 31!^2 A 1 Cm 2 ı 32!^2 A 2 C

m 3 ı 33!^2  1

A 3 D0:
(a)
In our case all massesmi Dm.Dividebymı 0!^2 and denoteD1=mı 0!^2.
Equations for amplitudesAi
.9/ A 1 C11A 2 C7A 3 D0;
11A 1 C.16/ A 2 C11A 3 D0;
7A 1 C11A 2 C.9/ A 3 D0:
(b)
Frequency equation becomes
detŒF 0 IDdet
2
4
9 11 7
11 1611
7119 
3
(^5) D0;
whereIis unit matrix. Eigenvalues are in descending order
 1 D31:5563;
 2 D2:0;
 3 D0:44365:
(c)
Verification:
1.The sum of the eigenvalues is 1 C 2 C 3 D31:5563C2:0C0:44365D 34 ;
On the other hand, the trace of the matrixTr.F 0 /D 9 C 16 C 9 D 34.
2.The multiplication of the eigenvalues is 1  2  3 D31:55632:00:44365D 28 ;
On the other hand, detF 0 D 28. (Note, that determinant of unit displacement
matrix is strictly positive, i.e. detF>0).