An inverse coefficient problem of identifying the flexural rigidity in damped Euler–Bernoulli beam from measured boundary rotation
We present a new mathematical model and method for identifying the unknown flexural rigidity r ( x ) in the damped Euler–Bernoulli beam equation ρ ( x ) w t t + μ ( x ) w t + ( r ( x ) w x x ) x x − ( T r ( x ) w x ) x = F ( x , t ) , ( x , t ) ∈ Ω T := ( 0 , ℓ ) × ( 0 , T ) , subject to the simply...
Saved in:
| Published in | Philosophical transactions of the Royal Society of London. Series A: Mathematical, physical, and engineering sciences Vol. 380; no. 2236; p. 20210358 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
14.11.2022
|
| Online Access | Get full text |
Cover
Loading…
| Summary: | We present a new mathematical model and method for identifying the unknown flexural rigidity
r
(
x
)
in the damped Euler–Bernoulli beam equation
ρ
(
x
)
w
t
t
+
μ
(
x
)
w
t
+
(
r
(
x
)
w
x
x
)
x
x
−
(
T
r
(
x
)
w
x
)
x
=
F
(
x
,
t
)
,
(
x
,
t
)
∈
Ω
T
:=
(
0
,
ℓ
)
×
(
0
,
T
)
, subject to the simply supported boundary conditions
w
(
0
,
t
)
=
w
x
x
(
0
,
t
)
=
0
,
w
(
ℓ
,
t
)
=
w
x
x
(
ℓ
,
t
)
=
0
, from the available measured boundary rotation
θ
(
t
)
:=
w
x
(
0
,
t
)
. We prove the existence of a quasi-solution and derive an explicit gradient formula for the Fréchet derivative of the Tikhonov functional
J
(
r
)
=
‖
w
x
(
0
,
⋅
;
r
)
−
θ
‖
L
2
(
0
,
T
)
2
. The results obtained here also form the basis of gradient-based computational methods for solving this class of inverse coefficient problems.
This article is part of the theme issue ‘Non-smooth variational problems and applications’. |
|---|---|
| Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 23 |
| ISSN: | 1364-503X 1471-2962 1471-2962 |
| DOI: | 10.1098/rsta.2021.0358 |