Page 200 - The Combined Finite-Discrete Element Method

THE CENTRAL DIFFERENCE TIME INTEGRATION SCHEME 183

which can be written in matrix form:
2
vh 1 −h k/m vh
= (5.19)
2
x 11 − h k/m x
next current
Thus the central difference scheme for damping a free linear system can be reduced to a
recursive formula. For the scheme to be stable, it is necessary that the spectral radius of
the recursive linear operator

2
1 −h k/m
A = 2 (5.20)
1 1 − h k/m
is not greater than one. The associated eigenvalue problem yields the following equation:

2
1 − λ −h k/m

2 = 0 (5.21)
1 1 − h k/m − λ
which results in the following characteristic equation:

2
2
(1 − λ)(1 − h k/m − λ) + h k/m = 0 (5.22)
or
2
2
λ + λ(h k/m − 2) + 1 = 0 (5.23)
Solution of this equation is given by
2 # 2 2
−(h k/m − 2) ± (h k/m − 2) − 4
λ 1,2 = (5.24)
2
which for
2
h k/m = 4 (5.25)

results in
λ 1,2 =−1 (5.26)

and the spectral radius is equal to 1. For

2
h k/m < 4 (5.27)
equation (5.24) yields

#
2
−(h k/m − 2) ± i 4 − (h k/m − 2) 2
2
λ 1,2 = (5.28)
2
and the spectral radius is given by
1 2 2 2 2
#
max λ 1,2 = 2 (h k/m − 2) + [4 − (h k/m − 2) ] = 1 (5.29)