Lecture Notes: Introduction to Finite Element Method Chapter 7. Structural Vibration and Dynamics
© 1999 Yijun Liu, University of Cincinnati
172
VI. Transient Response Analysis
(Dynamic Response/Time-History Analysis)
•
Structure response to arbitrary, time-dependent loading.
f(t)
t
u(t)
t
Compute responses by integrating through time:
t
0
t
1
t
2
t
n
t
n+1
u
1
u
2
u
n
u
n+1
t
Lecture Notes: Introduction to Finite Element Method Chapter 7. Structural Vibration and Dynamics
© 1999 Yijun Liu, University of Cincinnati
173
Equation of motion at instance
n
t , n = 0, 1, 2, 3,
⋅⋅⋅
:
.
n
n
n
n
f
Ku
u
C
u
M
=
+
+
&
&&
Time increment:
∆
t=t
n+1
-t
n
, n=0, 1, 2, 3,
⋅⋅⋅
.
There are two categories of methods for transient analysis.
A. Direct Methods (Direct Integration Methods)
•
Central Difference Method
Approximate using finite difference:
)
2
(
)
(
1
),
(
2
1
1
1
2
1
1
−
+
−
+
+
−
∆
=
−
∆
=
n
n
n
n
n
n
n
t
t
u
u
u
u
u
u
u
&
&
&
Dynamic equation becomes,
,
)
(
2
1
)
2
(
)
(
1
1
1
1
1
2
n
n
n
n
n
n
n
t
t
f
Ku
u
u
C
u
u
u
M
=
+
−
∆
+
+
−
∆
−
+
−
+
which yields,
)
(
1
t
n
F
Au
=
+
where
( )
( )
( )
∆
−
∆
−
∆
−
−
=
∆
+
∆
=
−
.
2
1
1
2
)
(
,
2
1
1
1
2
2
2
n
n
n
t
t
t
t
t
t
u
C
M
u
M
K
f
F
C
M
A
Lecture Notes: Introduction to Finite Element Method Chapter 7. Structural Vibration and Dynamics
© 1999 Yijun Liu, University of Cincinnati
174
u
n+1
is calculated from u
n
& u
n-1
, and solution is
marching from
,
,
1
,
,
1
,
0
L
L
+
n
n
t
t
t
t
until convergent.
This method is unstable if
∆
t is too large.
•
Newmark Method:
Use approximations:
[
]
[
]
,
)
1
(
)
(
,
2
)
2
1
(
2
)
(
1
1
1
1
2
1
+
+
+
+
+
+
−
∆
+
≈
=
→
+
−
∆
+
∆
+
≈
n
n
n
n
n
n
n
n
n
n
t
t
t
u
u
u
u
u
u
u
u
u
u
&&
&&
&
&
L
&&
&&
&&
&
γ
γ
β
β
where
β
&
γ
are chosen constants. These lead to
)
(
1
t
n
F
Au
=
+
where
).
,
,
,
,
,
,
,
,
(
)
(
,
)
(
1
1
2
n
n
n
n
t
f
t
t
t
u
u
u
M
C
f
F
M
C
K
A
&&
&
∆
=
∆
+
∆
+
=
+
β
γ
β
β
γ
This method is unconditionally stable if
4
1
,
2
1
.,
.
e
.
2
1
2
=
=
≥
≥
β
γ
γ
β
g
which gives the constant average acceleration method.
Direct methods can be expensive! (the need to
compute A
-1
, often repeatedly for each time step).
Lecture Notes: Introduction to Finite Element Method Chapter 7. Structural Vibration and Dynamics
© 1999 Yijun Liu, University of Cincinnati
175
B. Modal Method
First, do the transformation of the dynamic equations using
the modal matrix before the time marching:
),
(
2
,
)
(
1
t
p
z
z
z
t
z
i
i
i
i
i
i
i
m
i
i
i
=
+
+
Φ
=
=
∑
=
ω
ω
ξ
&
&&
z
u
u
i = 1,2,
⋅⋅⋅
, m.
Then, solve the uncoupled equations using an integration
method. Can use, e.g., 10%, of the total modes (m= n/10).
•
Uncoupled system,
•
Fewer equations,
•
No inverse of matrices,
•
More efficient for large problems.
Comparisons of the Methods
Direct Methods
Modal Method
•
Small model
•
More accurate (with small
∆
t)
•
Single loading
•
Shock loading
•
…
•
Large model
•
Higher modes ignored
•
Multiple loading
•
Periodic loading
•
…
Lecture Notes: Introduction to Finite Element Method Chapter 7. Structural Vibration and Dynamics
© 1999 Yijun Liu, University of Cincinnati
176
Cautions in Dynamic Analysis
•
Symmetry: It should not be used in the dynamic analysis
(normal modes, etc.) because symmetric structures can
have antisymmetric modes.
•
Mechanism, rigid body motion means
ω
= 0. Can use
this to check FEA models to see if they are properly
connected and/or supported.
•
Input for FEA: loading F(t) or F(
ω
) can be very
complicated in real applications and often needs to be
filtered first before used as input for FEA.
Examples
Impact, drop test, etc.