Affiliation:
1. Department of Mathematics, Kocaeli University, Şehit Ekrem District, Altunşehir Str., Ayazma Villalari, No. 22. Bahčecik - Başiskele, Kocaeli, 41030, Turkey
Abstract
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’.
Subject
General Physics and Astronomy,General Engineering,General Mathematics
Cited by
2 articles.
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1. Inverting mechanical and variable-order parameters of the Euler–Bernoulli beam on viscoelastic foundation;Journal of Inverse and Ill-posed Problems;2024-01-30
2. Non-smooth variational problems and applications;Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences;2022-09-26