Page 230 - The Combined Finite-Discrete Element Method

THE COMBINED FINITE-DISCRETE ELEMENT SIMULATION 213

For a linear dynamic system, the necessary and sufficient condition for the scheme to be
stable is
k
h< (5.133)
ω n

where h is the actual time step employed, ω n is the highest frequency of the system and
k is a constant dependent on the damping supplied. For undamped systems k = 2, while
for an overdamped system, k decreases proportionally with increasing damping.
Several options for choosing C matrix of the system (126) are available.

• Mass proportional damping – underdamped system: if damping is introduced in the
form
C = 2 ω 1 M (5.134)

then equation (5.126) can be written as follows:

Kx + M¨x + 2 ω 1 M˙x = p (5.135)

By writing the solution in the form

n

x = u i (t)v i (5.136)
i=1

and exploiting the M-orthogonality and K-orthogonality of eigenvectors v i
T
T
v Mv j > 0for i = j;and v Mv j = 0for i = j;
i
i
(5.137)
T
T
v Kv j > 0for i = j;and v Kv j = 0for i = j;
i
i
The following equation for mode i is obtained:
T T T T
(v Kv i )u i + (v Mv i )¨u i + 2 ω 1 (v Mv i )˙u i = (v pv )u i (5.138)
i i i i i
After substituting
T T
i
i
2 v Kv i v pv i
ω = and q i = (5.139)
i T T
v Mv i v Mv i
i i
into (5.144), the equation for mode i can be written as follows:
2
ω i u i +¨u i + 2 ω 1 ω i u i = q i where (5.140)
ω i
ω 1
ξ i = is the damping ratio
ω i
The damping ratio for the lowest frequency is equal to 1. Thus the damping of lowest
frequency is equal to the critical damping. The damping ratio for all other frequencies is
smaller than 1, which means that all other frequencies are underdamped.