WOLFRAM NOTEBOOK

WOLFRAM|DEMONSTRATIONS PROJECT

Boundary Control of a 1D Wave Equation by the Filtered Finite-Difference Method

​
filtering:
filtered
not filtered
N: number of nodes
20
k
1
: viscous damping gain
0
k
3
: boundary damping gain
1
k
4
: normal mode-displacement
1
k
5
: normal mode-velocity
1
T
F
: end time
0
string vibrations at
T
F
= 0 (×
-3
10
​).
tip velocity (×
-3
10
​)
total energy (×
-4
10
​)
In this Demonstration, the boundary control/stabilization of a wave equation is modeled as a string. The string is clamped on the left and free on the right. The box on the right, used to better visualize the tip displacement and the bending angle of the string, is hypothetical. This box rotates while keeping its normal vector, on the side connected to the string, parallel to the last segment of the string.
Two types of damping to suppress vibrations are considered in this Demonstration. The first is a distributed viscous damping, and the second is boundary damping injected through the right end. The solutions are filtered by adding a viscosity term (structural damping type) to the wave equation.
Following is the continuous partial differential equation (PDE) considered:
w
tt
+
k
1
w
t
-
w
xx
=0
0≤x≤L=1,t≥0
w(0,t)=
w
x
(1,t)+
k
3
w
t
(1,t)=0
​​
w(x,0)=
-3
10
sin
(2
k
4
-1)πx
2
​​
w
t
(x,0)=
-3
10
sin
(2
k
5
-1)πx
2
.
​​
The discretized version is:
w
i,tt
+
k
1
w
i,t
-
w
i+1
-2
w
i
+
w
i-1
2
h
-
k
2
2
N
w
i+1,t
-2
w
i,t
+
w
i-1,t
1
=0
i=1,2,…,N,t≥0
w(0,t)=
w
N
-
w
N-1
+
k
3
1
N
w
N,t
=0
​​
w
i
(0)=
-3
10
sin
(2
k
4
-1)πi
2N
​​
w
i,t
(0)=
-3
10
sin
(2
k
5
-1)πi
2N
​​
where
N
is the number of discrete nodes,
h=
L
N
=
1
N
,
x
i
=i
1
N
and
w
i
(t)≈w(
x
i
,t)
.
The constants
k
1
and
k
3
are non-negative feedback gains.
The following values can be set with the controls:
• filtering: whether or not the system has a filtering term
(
k
2
∈{0,1})
.
•
N
: the number of nodes in the discretization of the interval
[0,L]
(N∈{20,30,40})
.
•
k
1
: viscous damping gain
(
k
1
∈[0,10])
.
•
k
3
: boundary damping gain
(
k
3
∈[1,10])
.
•
k
4
: normal mode-displacement
(
k
4
∈{m|1≤m≤nandmisodd})
.
•
k
5
: normal mode-velocity
(
k
5
∈{m|1≤m≤nandmisodd})
.
•
T
F
: end time
(
T
F
∈[0,10])
.
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