534 14 Dynamics of Elastic Systems
The introduced constraints 1, 2, 3which prevent to displacementyiare shown in
Fig.14.12a.
For calculation of unit reactions, we need to construct bending moment diagram
due to unit displacements of each introduced constraint. These diagrams are pre-
sented in Fig.14.12b; they are constructed using Table A.18. Since a structure is
symmetrical, then bendingmoment diagram caused byunit displacement constraint
3 is not shown.
Calculation of unit reactions has no difficulties. All shear and unit reactions have
multiplierEI=a^3.
SinceaDl=4, then the stiffness matrix
SD64 EI
l^32
49:8572 9:4285 3:8572
9:4285 13:7142 9:4285
3:8572 9:4285 9:85723(^5) : (a)
Since all masses are equal, then (14.10) may be rewritten as
2
4
9:8572 9:4285 3:8572
9:4285 13:7142 9:4285
3:8572 9:4285 9:8572
3
(^5) D0; (b)
where parameterDm!^2. Note that expressions for eigenvaluefor displacement
and force methods
D1=mı 0!^2
are different.
The eigenvalues present the roots of equation (b);in increasing orderthey are
1 D0:3804
EI
a^3
; 2 D6:0
EI
a^3
; 3 D27:0482
EI
a^3
: (c)
Now we can calculate the frequencies of vibration which corresponds to eigen-
values.
1 D0:3804aEI 3 Dm! 12!! 12 D0:3804maEI 3 D0:3804 (^64) mlEI 3 D24:345mlEI 3 ;
2 D6:0aEI 3 Dm! 22!! 22 D6:0maEI 3 D6:0 (^64) mlEI 3 D (^384) mlEI 3 ;
3 D27:0482aEI 3 Dm! 32!! 32 D27:0482maEI 3 D27:0482 (^64) mlEI 3 D1731:08mlEI 3 :
(d)
Same frequencies have been obtained by force method.
For each i-th eigenvalue the set of equation for calculation of amplitudes is
.9:8572i/A 1 9:4285A 2 C3:8572A 3 D0;
9:4285A 1 C.13:7142i/A 2 9:4285A 3 D0;
3:8572A 1 9:4285A 2 C.9:8572i/A 3 D0:
(e)