5.2 Shear Beams
127
Fig. 5.4 First three modal
shapes of a simply supported
beam with l = 1
0
0.2
0.4
0.6
0.8
1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
x
n (x)
which admits non-trivial solutions if the determinant of its matrix vanishes, i.e.,
cos(ρ l) = 0, which is the previous equation for ρ. The solutions ϕ(x) corresponding to the above values of ρ are
ϕ n (x) = sin
(2 n − 1) π
2 l
x
, n = 1, 2, . . . ,
(5.30)
and are referred to as modal shapes corresponding to the modal frequencies {ω n }
in Eq. 5.29. The first three modal shapes are shown in Fig. 5.4. Note that the modes
satisfy the boundary conditions, i.e., ϕ n (0) = 0 and ϕ (l) = 0, an expected property.
As mentioned previously, (1) the modal shapes are orthogonal in the sense
l
0
ϕ n (x) ϕ r (x) dx =
l
2
δ nr , r = 1, 2, . . . ,
(5.31)
(2) the constant C 2 remains undetermined so that the function ϕ(x) can be found
up to a multiplicative constant, which is set unity in the expression of ϕ n (x), (3) the
functions {q n (t)} are projections of the displacement function v(x, t) on the basis
functions {ϕ n (x)}, (4) the functions ϕ n (x) q n (t) satisfy Eq. 5.23 by construction, and
(5) the modal shapes {ϕ n (x)} span the solution space so that
v(x, t) =
∞
n=1
ϕ n (x) q n (t) =
∞
n=1
sin
(2 n − 1) π
2 l
x
q n (t).
(5.32)
We reemphasize that the representation of the displacement function v(x, t) in
Eq. 5.32 is conceptually similar to that of the displacement vector of MDOF