Page 351 - MATLAB an introduction with applications

336 ——— MATLAB: An Introduction with Applications

6.5.4 Park Stiffly Stable Method
The Park Stiffly Stable method is an accurate method for low frequency ranges and stable for all higher-
frequency components. Using a linear combination of the following two difference formulas derives the
velocity in the Park Stiffly Stable method:
1
X t+ ∆ = 6 t ∆ [11X t+ ∆t − 18X + 9X t− ∆ t t − 2X t− 2 t ∆ ] ...(6.76)
t
1
X t+ ∆ = 2 t ∆ ( 2X t+ ∆t − 4X + X t− ∆ t t ) ...(6.77)
t
The linear combination of (6.76) and (6.77) gives
1
X t+ ∆ = 4 t ∆ (3X t+ ∆t − 4X + X t− ∆ t t )
t
1
+ (11X − 18X + 9X − 2X ) ...(6.78)
12 t ∆ t+ ∆ t t− ∆t t t− 2 t ∆
1
or X t+ ∆ = 6 t ∆ (10X t+ ∆t − 15X + 6X t− ∆ t t − X t− 2 t ∆ ) ...(6.79)
t
Similarly, for the acceleration, we obtain
1

X t+ ∆ = 6 t ∆ (10X t+ ∆t t − 15X + 6X t−∆ t –X t –2 t ∆ ) ...(6.80)
t
The difference formulas in the Park Stiffly method are given by
}
1
−
−
{ X t+ ∆ } = 6 t ∆   10 { X t+ ∆t t } 15 X t + 6 t− ∆ t } { X t− 2 t ∆   ...(6.81)
{ } { X
1
−
}
−
{ X t+ ∆ } = 6 t ∆   10 {X t+ ∆t t } 15 X t + { t− ∆ t } {X t− 2 t ∆  ...(6.82)
{ } 6 X
We consider Eq. (6.13) to obtain solution for the displacements, velocities and accelerations at time t + t∆ .
{

By substituting the expressions for { X t+ ∆ } and X t+ ∆t t } from (6.81) and (6.82), respectively, into (6.13), we get
=
| M { | X t+∆ t } { t+∆ t } ...(6.83)
F

F


where the effective mass matrix M  and the effective force vector { t+∆ t } are given by
10
100

M  = 36 t ∆ 2 [ ] − 6 t ∆ [] [ ] ...(6.84)
C +
K
M


15 1 1 }
M
M
X
F
M
{ t+∆ t } = 6 t ∆ [ ]{ } − ∆t [ ]{ X t− ∆t } + 6 t ∆ [ ]{ X t− 2 t
∆
t
 150 15   10 1 
+   36 t ∆ 2 [ ] + 6 t ∆ [] { } −   6 t ∆ 2 [ ] + ∆t [] {X t − ∆t } ...(6.85)
M
X
M
C
C



t


1
1
+    36 t ∆ 2 [ ] + 6 t ∆ [] {X t−∆ }
M
C

2 t

The solution of Eq.(6.83) gives {X t+ ∆t } , which is then substituted in Eq.(6.82) to obtain velocities. The
values of { X t+ ∆t } are then obtained by the use of Eq.(6.81). Note that in the Park Stiffly stable method, the