4.2 Direct Integration Techniques
141
(3) Based on the values of ˙
x i + 1 and x i + 1 in step (2), a new ¨
x i + 1 is computed
¨
x i + 1 =
F i + 1 (t)
m
−
c
m
˙
x i + 1 −
k
m
x i + 1
(4.27)
(4) Steps (2) and (3) are repeated, beginning with a newly computed value of ¨
x i + 1 ,
or with an extrapolated value, until satisfactory convergence is attained.
With δ =
1
2
and β =
1
6
, Eqs. (4.25) and (4.26) reduce to Eqs. (4.16) and
(4.17) respectively, corresponding to the linear acceleration method.
Newmark has suggested that a choice of β between
1
4
and
1
6
will give sufficiently
satisfactory and accurate value ( with δ =
1
2 ). The value of β =
1
6
corresponds
to a parabolic variation of the acceleration.
In Newmark’s β-method, equilibrium conditions at time station (i + 1) is considered, unlike the finite difference method, where the equilibrium is considered at time
i. A great elegance of the method lies, in that no starting procedures are needed,
since displacements, velocities and accelerations at time t i + 1 are expressed in terms
of same quantities at time t i only. Greater accuracy and faster convergence in this
method are attained by shortening the period t/T.
Example 4.5 Compute the response of the SDF system, which is initially at rest by
Newmark’s β-method. The data in consistent units are as follows:
k = 72, m = 8, ζ = 0.1 and F(t) = 480 t.
The natural angular frequency of the system is
p =
72
8
= 3
n = 0.1 × 3 = 0.3
T =
2π
p
=
2π
3
= 2.0944s
Assume T = 0.1T ∼ = 0.2 s, β =
1
6
and δ =
1
2
.
At t = 0, x 1 = 0, ˙
x 1 = 0 and ¨
x 1 = 0, further, F(t) = 0 at t = 0
˙
x 2 = ˙
x 1 +
1
2
( ¨
x 1 + ¨
x 2 ) t = 0.1 ¨
x 2
(4.28)
x 2 = x 1 + ˙
x 1 t +
1
3
¨
x 1 t
2
+
1
6
¨
x 2 t
2
x 2 = 0.0067 ¨
x 2
(4.29)
141
(3) Based on the values of ˙
x i + 1 and x i + 1 in step (2), a new ¨
x i + 1 is computed
¨
x i + 1 =
F i + 1 (t)
m
−
c
m
˙
x i + 1 −
k
m
x i + 1
(4.27)
(4) Steps (2) and (3) are repeated, beginning with a newly computed value of ¨
x i + 1 ,
or with an extrapolated value, until satisfactory convergence is attained.
With δ =
1
2
and β =
1
6
, Eqs. (4.25) and (4.26) reduce to Eqs. (4.16) and
(4.17) respectively, corresponding to the linear acceleration method.
Newmark has suggested that a choice of β between
1
4
and
1
6
will give sufficiently
satisfactory and accurate value ( with δ =
1
2 ). The value of β =
1
6
corresponds
to a parabolic variation of the acceleration.
In Newmark’s β-method, equilibrium conditions at time station (i + 1) is considered, unlike the finite difference method, where the equilibrium is considered at time
i. A great elegance of the method lies, in that no starting procedures are needed,
since displacements, velocities and accelerations at time t i + 1 are expressed in terms
of same quantities at time t i only. Greater accuracy and faster convergence in this
method are attained by shortening the period t/T.
Example 4.5 Compute the response of the SDF system, which is initially at rest by
Newmark’s β-method. The data in consistent units are as follows:
k = 72, m = 8, ζ = 0.1 and F(t) = 480 t.
The natural angular frequency of the system is
p =
72
8
= 3
n = 0.1 × 3 = 0.3
T =
2π
p
=
2π
3
= 2.0944s
Assume T = 0.1T ∼ = 0.2 s, β =
1
6
and δ =
1
2
.
At t = 0, x 1 = 0, ˙
x 1 = 0 and ¨
x 1 = 0, further, F(t) = 0 at t = 0
˙
x 2 = ˙
x 1 +
1
2
( ¨
x 1 + ¨
x 2 ) t = 0.1 ¨
x 2
(4.28)
x 2 = x 1 + ˙
x 1 t +
1
3
¨
x 1 t
2
+
1
6
¨
x 2 t
2
x 2 = 0.0067 ¨
x 2
(4.29)