Page 337 - MATLAB an introduction with applications

322 ——— MATLAB: An Introduction with Applications

The Runge-Kutta method is self-starting and has the advantage that no initial values are needed beyond
the prescribed values. A brief discussion of its basis is represented here. In the Runge-Kutta method, the
second-order differential equation is first reduced to two first-order equations. Consider the differential
equation for the single degree of freedom system given in Eq. (6.1). Equation (6.1) can be rewritten as

X = 1   F ( ) – CX – KX  = f ( , X , ) ...(6.10)
t
t
X

M
By letting X = X and X = X , Eq. (6.10) can be reduced to the following two first-order equations;
1
2

X = X 2
1

X = ( f X X 2 , ) ...(6.11)
,
t
2
1
By defining
() 
 Xt
1
() 
Xt = 
()
 Xt 
2
 x 2 
() = 
and Ft 
(, x
t
 fx 1 2 , ) 
the following recurrence formula is obtained to find the values of Xt at mesh or grids points t according
()
i
to the fourth order Runge-Kutta method. We omit the details of the derivation of the method.
1
X = X +   K + 2K + 2K + K 
4
i+
1
i
6 1 2 3
where K = hF (X i , )
t
1
i
1 1
,t +
K = hF (X + K 1 i ) h
i
2
2 2
1 1
K = hF (X + K 2 ,t + ) h ...(6.12)
i
3
i
2 2
1
,t
and K = hF (X + K 3 i+ 1 )
i
4
2
Although the Runge-Kutta method does not require the computation of derivatives beyond the first, its
higher accuracy is obtained by four evaluations of the first derivatives to obtain agreement with the Taylor
series solution through terms of order h . Since the fourth-order Runge-Kutta method is an explicit method,
4
the maximum time step is usually governed by stability considerations. The change in time step can be
easily implemented between iterations and hence the method can be considered as an inherently stable
method. The main drawback of the method is that each forward step requires several computations of the
functions thus increasing the computational cost. The Runge-Kutta method is applicable and extendable to
a system of differential equations.
6.3 MULTI-DEGREE OF FREEDOM SYSTEM
The general form of the equations of motion for a multi-degree of freedom system are written as

[ ]{} []{} []{} { ()M X + C X + K X = F t } ...(6.13)