
Substituting Equation 8.46 into Equation 8.48 results in the asymptotic fractional error as a
linear function of v
n
T, that is,
e
v
n
0:5z(v
n
T), v
n
T 1(8:49)
From Equation 8.45, the damping ratio error e
z
is expressible as
e
z
¼ z* z ¼
ln
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 2
z
v
n
T þ (v
n
T)
2
q
v
n
*
T
z (8:50)
where v
n
*
is given in Equation 8.44. From Equation 8.47, the asymptotic approximation for e
z
is
e
z
0:5(z
2
1)v
n
T, v
n
T 1(8:51)
The asymptotic expressions for e
v
n
in Equation 8.49 and e
z
in Equation 8.51 are of order O(v
n
T)
when using Euler integration to simulate an underdamped second-order system.
A plot of the exact and asymptotic formulas for ...