126
5 Continuous Systems
by using the arguments as for Eq. 5.5. The latter equalities of Eq. 5.25 give the
following ordinary differential equations:
¨
q(t) + ω
2 q(t) = 0 (Initial value problem) and
D[ϕ(x)] = ϕ
(x) + ρ
2 ϕ(x) = 0 (Boundary value problem)
(5.26)
for q(t) and ϕ(x) with solutions
q(t) = A cos(ω t) + B sin(ω t)
ϕ(x) = C 1 cos(ρ x) + C 2 sin(ρ x),
(5.27)
where
ρ
2
=
m ω 2
G A
.
(5.28)
The boundary conditions are used to find the constants C 1 and C 2 in the expression
of ϕ(x), construct the modal shapes, and find the modal frequencies. We illustrate
this construction by the following example.
Example 5.2 Consider the shear beam in Fig. 5.3 with length l. The boundary
conditions are v(0, t) = 0 and Q(l, t) = 0. The latter condition and Eq. 5.21 imply
τ (x, t) = 0 so that γ (x, t) = ∂v(x, t)/∂x = 0 at x = l, which shows that the
condition v (l, t) = 0 can be substituted for Q(l, t) = 0.
The boundary condition v(0, t) = ϕ(0) q(t) = 0, t ≥ 0, at x = 0 implies
ϕ(0) = 0 so that C 1 = 0 and, as a result, ϕ(x) = C 2 sin(ρ x). The boundary
condition v (l, t) = ϕ (l) q(t) = 0, t ≥ 0, at x = l implies ϕ(l) = C 2 sin(ρ l) = 0.
Since C 2 = 0 is not possible for non-zero initial conditions (v(x, t) = 0 at all times
if C 2 = 0), we require sin(ρ l) = 0. This equation has the (countable) infinite set of
solutions
ρ n l =
(2 n − 1) π
2
so that
m l 2 ω 2
n
G A
=
(2 n − 1) π
2
2
, which gives
ω n =
(2 n − 1) π
2 l
G A
m
, n = 1, 2, . . . .
(5.29)
The above values of ρ can also be found by considering the boundary conditions
simultaneously. This approach yields homogeneous system of linear equations,
1
0
ρ sin(ρ l) ρ cos(ρ l)
C 1
C 2
= 0,