Information Technology Reference
In-Depth Information
TABLE 10.1
Generalized Dynamic Stiffness Matrix k i dyn
for a Beam Element
V a
M a
V b
M b
w a
θ a
w b
θ b
=
k i dyn
p i
k i dyn
v i
p i
p
=
The expressions for k i dyn
and p i
are given as k i
and p i
in Chapter 4, Table 4.4. The definitions of
λ
,
η
,
and
ζ
are expanded to include dynamic-related properties.
2
λ = (
k
ρω
)/
EI
2
η = (
k
ρω
)/(
k s GA
)
k + ρ
r y ω
2
ζ = (
N
)/
EI
where
ω
is the natural frequency (rad/sec)
ρ
is the mass per unit length (mass/length)
r y
is the radius of gyration (length)
Other definitions are given in Chapter 4, Tables 4.3 and 4.4.
P
=
Axial force
r y =
Radius of gyration
Magnitude of the distributed applied moment (force-length/length)
These relations, which include the effects of shear deformation and rotary inertia, as
well as bending, are the Timoshenko beam equations. The governing equations are reduced
to those for a Rayleigh 2
c
=
beam (bending, rotary inertia) by setting 1
/(
k s GA
)
equal to zero.
For a shear beam (bending, shear deformation), set
2
r y
θ
ρ
t 2 =
0
For free vibrations, set the applied loadings M T , p z , and c equal to zero in Eq. (10.41),
and let the motion be harmonic, e.g., let
The resulting equations
can be solved exactly and placed in stiffness matrix format. In Chapter 4, this was ac-
complished by solving the equations in transfer matrix form and then converting the
results to a stiffness matrix. This procedure led to the stiffness matrix of Table 10.1. Since
the stiffness matrix of Table 10.1 includes the effects of inertia, it is an element dynamic
stiffness matrix, k i dyn .
The exact stiffness matrix of Table 10.1 can also be obtained by using the exact solution
of the free vibration form of Eq. (10.41) to compose u
w(
x, t
) = w(
x
)
sin
ω
t
.
Nv . Substitution of exact shape
functions N in Eq. (10.40) leads to this exact stiffness matrix.
The element dynamic stiffness matrix can be assembled in the same fashion as the
usual stiffness matrix, giving the global dynamic stiffness matrix K dyn . The characteristic
equation, from which the natural frequencies can be computed, would take the form
det K dyn (ω) =
=
0 (10.42)
where the boundary conditions have been employed and resulted in a reduced K dyn .A
numerical determinant search can be utilized to find the frequencies.
2 Lord Rayleigh or John William Strutt (1842-1919) was an English physicist and chemist and a Nobel Laureate
(1904) for his discovery of argon. He served as a professor at Cambridge University and the Royal Institution in
London. His contributions were in acoustics, hydrodynamics, and optics.
Search WWH ::




Custom Search