## 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.