Civil Engineering Reference
In-Depth Information
Thus,
-1
t
⎡
t
⎤
⎛
Δ
⎞
(
)
⎛
Δ
⎞
2
2
t
2
t
r
=+
M C
Δ + −Δ
R M Kr
−−
M Cr
(9.174)
⎜
⎟
⎢
⎜
⎟
⎥
1
0
0
−
1
2
2
⎝
⎠
⎝
⎠
⎣
⎦
The stability of the second central difference method may be evaluated by considering
an undamped and unloaded single-degree-of freedom system (se Eqs. 9.167 and 9.168)
with mass
2
i
M
, stiffness
K
and eigen-frequency
K
M
ω
=
i
i
1
⎫
(
)
Mr
2
r
r
r
0
−+
+
=
⎪
i
k
1
k
k
1
i k
+
−
2
t
Δ
⎪
⎪
⎬
⎪
(9.175)
(
)
22
r
t
2
r
r
0
⇒
+Δ
ω
−
+
=
⎪
⎪
⎭
k
1
i
k
k
1
+
−
ra
λ
t
(where
a
is the amplitude), will render
which, for a harmonic motion
=
⎫
⎪
⎪
(
)
(
)
(
)
t
t
t
t
λ
+Δ
22
λ
t
λ
−Δ
ae
k
t
2
ae
ae
k
0
+Δ
ω
−
k
+
=
⎪
⎪
⎪
i
(
)
2
(
)
t
22
t
λ
Δ
λ
Δ
⇒
e
+Δ
t
ω
−
2
e
+ =
1
0
(9.176)
⎬
⎪
i
(
)
⎫
λ
Δ
t
⎪
e
1
⎪
⎡
⎤⎪
1
22
22
⇒
=
2
− Δ± ΔΔ−
t
ωωω
t
t
4
⎬
i
i
i
(
)
⎢
⎥⎪
2
⎣
⎦
λ
Δ
t
e
⎪
⎪
⎭
⎭
2
and thus the response at
t
is given by
1
+
(
)
(
)
(
)
t
t
⎡
⎤
λ
+Δ
λ
t
λ
Δ
t
λ
t
λ
Δ
t
λ
Δ
t
k
r
=
e
=
e
k
e
=
e
k
c
e
+
c
e
⎦
(9.177)
1
⎢
1
1
⎥
+
⎣
1
2
where
c
and
c
are constants dependant on initial conditions. In a second order
equation
2
x
x
0
xx
αβγ
++=
the product of the roots
⋅
=
γ α
. Thus, it is seen from
12
(
) (
)
t
t
λ
Δ
λ
Δ
Eq. 9.176 that
e
⋅
e
=
1
. It may be taken for granted that both roots are
1
2
distinct. A positive radicand in the solution in Eq. 9.176 will render two real roots, and
since the product of the two roots is unity one of them must be larger that one, and thus,
the solution is consistently growing, i.e. it is unstable. A negative radicand on the other
hand will render complex roots, and the product of the two roots can only be unity if
they are complex conjugates and both has an absolute value equal to one. Thus, the
solution is numerically stable if
