ABSTRACT
Equations (4.1) to (4.3) combine to give
∂ ∂ ′ ′
∂ ′ ∂ ′
+
∂ ∂ ′ ′
∂ ′ ∂ ′
+ ρ ′ ∂ ′ ∂ ′ =x EI x
y
x x F
y
x x
y
t ( ) ( ) 0
2 (4.4)
Let
′ = ′ ρ ′ = ρ ′EI x EI l x x r x( ) ( ), ( ) ( )0 0 (4.5)
where EI0 is the maximum of EI, and ρ0 is the maximum of ρ. Consider a harmonic vibration with frequency ω′, i.e.,
′ = ′ ′ ′ω ′y w x e( ) i t (4.6)
By normalizing all length variables by the beam length L (e.g., x = x′/L), and the time by ρL EI/2 0 0 , and dropping the primes, Equation (4.4) becomes
+ − ω =
d
dx l x
d w
dx a
d w
dx r x w( ) ( ) 0
Here,
=
′ ω = ′ω ρa F L
EI L EI, /
are nondimensional compressive force and nondimensional frequency, respectively. At the ends of the beam, the classical boundary conditions are as follows:
For a clamped end (C),
= =w
dw
dx 0, 0 (4.9)
For a pinned end (P),
= =w
d w
dx 0, 0
2 (4.10)
dx´
dy´
m´+ dm´
V´+ dV´
V´
m´
F´
F´
+ = =
d
dx l
d w
dx a
dw
dx
d w
dx 0, 0
2 (4.11)
For a sliding end (S),
+ = =
d
dx l
d w
dx a
dw
dx
dw
dx 0, 0
2 (4.12)
Other nonclassical boundary conditions include the elastically supported end, where one of the boundary conditions is
∓
+ =
d
dx l
d w
dx a
dw
dx
k L
EI w
0 (4.13)
Here, k1 is the elastic spring constant, and the top and bottom signs refer to left or right ends, respectively. For a rotational spring,
= ±
l
d w
dx
k L
EI
dw
dx
0 (4.14)
where k2 is the rotational spring constant. For a mass M attached at the end
+ = ± ρ
ω
d
dx l
d w
dx a
dw
dx
M
L w
2 (4.15)
Equation (4.7) needs two boundary conditions at each end. For nontrivial solutions, the eigenvalues or the nondimensional frequencies ω are determined.
