Page 342 - MATLAB an introduction with applications

Direct Numerical Integration Methods ——— 327

5. Compute{∆X t } and X t
{ } at time t:

∆
=
−
} [] { ∆K
{ X t } [ ] ( M –1 {∆F − X t } [] { ∆C X t } )
t
+
X t = t− ∆t } {∆X t }
{ } { X
6. Second iteration cycle:

+
{∆X t } = ∆t { ( X t –∆t } { })
X
t
2

=
+
{ } { X t− ∆t } {∆X t }
X
t
∆t } { })
{∆X } = { ( X + X
t
2 t –∆t t

, X
7. Compute {∆X t } in step 5 using {∆X t } { } and { ∆X t } from step 6.
t
{

X
8. Finally, compute { } from step 5, using { } and ∆X t } from step 7.
X
t
t
6.4.3 Fourth-Order Runge-Kutta Method
In the fourth-order Runge-Kutta method, the system of second-order differential Eq. (6.13) is converted
into state variable form. That is, both the displacements and velocities are treated as unknowns {y} defined
by

  {}
X
{} y =   ...(6.34)
X
  {}  
Using Eq. (6.34), Eq.(6.13) can be rewritten as

{} =− [ ] [ ]{} [ ] [ ]{} [] { ()M − 1 K X − M − 1 C X + m − 1 F t } ...(6.35)
X
Using the identity
{} {} y= y ...(6.36)
we obtain from Eqs. (6.35) and (6.36)

X 

  [] 0 [] I    {}  0 
X


{}

{} y =   =  − 1 − 1    +  − 1  ...(6.37)
}
X
C 
X
K −
M
M
M
F
 

  {}   − [ ] [ ] [ ] [ ] {}    [ ] { () t  



or {} []{} { *( )y = E y + F t } ...(6.38)
y =

That is {} { (, )ft y } ...(6.39)
y } is obtained from {} in such a way that the
y
In the Runge-Kutta method, an approximation to { t+ ∆t t
N
∆t
power series expansion of the approximation coincides, up to the terms of a certain order ( ) in the time
interval t∆ with the actual Taylor series expansion of (t + ∆ ) in powers of t∆ . This method has the
,
t
advantage that no initial values are required beyond the prescribed ones. The general fourth-order algorithms
are based on formulas of the form