Page 232 - The Combined Finite-Discrete Element Method

THE COMBINED FINITE-DISCRETE ELEMENT SIMULATION 215

an external force which influences considerably the path through which the state of rest
is reached.
• Mass proportional damping – overdamped system: if damping is introduced in the form

C = 2 ω n M (5.144)

then equation (5.126) can be written as follows:

Kx + M¨x + 2 ω n M˙x = p (5.145)

After applying the modal decomposition as explained above, the governing equation for
mode i can be written in the form

2
ω i u i +¨u i + 2 ω n ω i ˙u i = q i where (5.146)
ω i
ω n
ξ i = is the damping ratio
ω i
The damping ratio for the highest frequency is equal to 1, which means that the damping
of the highest frequency mode is equal to the critical damping. The damping ratio for all
other frequencies is greater than 1, which means that all other frequencies are overdamped.
The lowest frequency is highly overdamped, and its convergence to the zero energy state
is given by
ω 1 ω 1
− t − ω 1 t
e 2ξ 1 = e 2 ω n (5.147)
The time needed to reach a static solution is therefore proportional to

ω 1 ω n
− ω 1 t s = const ⇒ t s = const 2 (5.148)
ω n ω 1
As the critical time step for such a highly overdamped system is still proportional to
1/ω n (because the highest frequency is not overdamped), the total number of time steps
required to obtain static solution is given by

2
t s ω n ω n
n = = const 2 ω n = const (5.149)
h ω ω
1
The beneficial feature of the scheme is that no overshooting is present, however, it comes
at the price of a greatly reduced performance of the scheme in terms of the total CPU
time required. On the other hand, only the highest frequency of the system has to be
evaluated to calculate the damping matrix, which is usually the easiest to estimate. It is
important to note that this very high mass proportional damping comes as external force,
and can lead to results which are not physical, i.e. can lead to convergence to a wrong
static solution.
• Stiffness proportional damping: if damping is introduced in the form
2
C = K (5.150)
ω n