Page 233 - The Combined Finite-Discrete Element Method

216 TEMPORAL DISCRETISATION

then equation (5.126) can be written as follows:
2
Kx + M¨x + K˙x = p (5.151)
ω n
By writing the solution in the form

n

x = u i (t)v i (5.152)
i=1
and exploiting the M-orthogonality and K-orthogonality of eigenvectors, the following
equation for mode i is obtained:

2
T T T T
(v Kv i )u i + (v Mv i )¨u i + (5.153)
i i (v Kv i )˙u i = (v pv )u i
i
i
i
ω n
After substituting
T T
i
i
2 v Kv i v pv i
ω = T and q i = T (5.154)
i
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 i ω i ˙u i = q i where (5.155)
ω n
ω i
ξ i = is damping ratio
ω n
The damping ratio for the highest frequency is equal to 1, i.e. the damping of highest
frequency is critical damping. The damping ratio for all other frequencies is smaller than
1, which means that all other frequencies are underdamped. Convergence of the mode i
to the zero energy state (state of rest) is given by
ω i
− ω i t
e ω n (5.156)
The time needed to reach a static solution is therefore proportional to

ω i const ω n
ω i t s = const ⇒ t s = (5.157)
ω n ω i ω i
This means that higher frequencies (modes) reach a static solution quicker than lower
frequency modes.
Since the critical time step is proportional to 1/ω n , the total number of time steps
required to obtain a static solution is given by

2

t s const ω n ω n
n = = ω n = const (5.158)
h ω i ω i ω i