Spectral properties of non-homogeneous Timoshenko beam and its controllability
Controllability of slowly rotating non-homogeneous beam clamped to a disc is considered. It is assumed that at the beginning the beam remains at the position of rest and it is supposed to rotate by the given angle and achieve desired position. The rotor of propelling engine is in the middle of the d...
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Інститут прикладної математики і механіки НАН України
2007
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| Цитувати: | Spectral properties of non-homogeneous Timoshenko beam and its controllability / G.M. Sklyar, G. Szkibiel // Механика твердого тела: Межвед. сб. науч. тр. — 2007. — Вип 37. — С. 175-183. — Бібліогр.: 16 назв. — англ. |
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irk-123456789-279472011-10-25T12:19:27Z Spectral properties of non-homogeneous Timoshenko beam and its controllability Sklyar, G.M. Szkibiel, G. Controllability of slowly rotating non-homogeneous beam clamped to a disc is considered. It is assumed that at the beginning the beam remains at the position of rest and it is supposed to rotate by the given angle and achieve desired position. The rotor of propelling engine is in the middle of the disk. The movement is governed by the system of two di erential equations with non-constant coe cients: linear mass density, exural rigidity, rotational inertia and shear sti ness. To solve the problem of controllability, the spectrum of the operator generating the dynamics of the model is studied. Then the problem of controllability is reduced to the moment problem that is, in turn, solved with the use of the asymptotics of the spectrum and Ullrich Theorem. 2007 Article Spectral properties of non-homogeneous Timoshenko beam and its controllability / G.M. Sklyar, G. Szkibiel // Механика твердого тела: Межвед. сб. науч. тр. — 2007. — Вип 37. — С. 175-183. — Бібліогр.: 16 назв. — англ. 0321-1975 http://dspace.nbuv.gov.ua/handle/123456789/27947 531.38 en Механика твердого тела Інститут прикладної математики і механіки НАН України |
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Controllability of slowly rotating non-homogeneous beam clamped to a disc is considered. It is assumed that at the beginning the beam remains at the position of rest and it is supposed to rotate by the given angle and achieve desired position. The rotor of propelling engine is in the middle of the disk. The movement is governed by the system of two di erential equations with non-constant coe cients: linear mass density, exural rigidity, rotational inertia and shear sti ness. To solve the problem of controllability, the spectrum of the operator generating the dynamics of the model is studied. Then the problem of controllability is reduced to the moment problem that is, in turn, solved with the use of the asymptotics of the spectrum and Ullrich Theorem. |
| format |
Article |
| author |
Sklyar, G.M. Szkibiel, G. |
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Sklyar, G.M. Szkibiel, G. Spectral properties of non-homogeneous Timoshenko beam and its controllability Механика твердого тела |
| author_facet |
Sklyar, G.M. Szkibiel, G. |
| author_sort |
Sklyar, G.M. |
| title |
Spectral properties of non-homogeneous Timoshenko beam and its controllability |
| title_short |
Spectral properties of non-homogeneous Timoshenko beam and its controllability |
| title_full |
Spectral properties of non-homogeneous Timoshenko beam and its controllability |
| title_fullStr |
Spectral properties of non-homogeneous Timoshenko beam and its controllability |
| title_full_unstemmed |
Spectral properties of non-homogeneous Timoshenko beam and its controllability |
| title_sort |
spectral properties of non-homogeneous timoshenko beam and its controllability |
| publisher |
Інститут прикладної математики і механіки НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/27947 |
| citation_txt |
Spectral properties of non-homogeneous Timoshenko beam and its controllability / G.M. Sklyar, G. Szkibiel // Механика твердого тела: Межвед. сб. науч. тр. — 2007. — Вип 37. — С. 175-183. — Бібліогр.: 16 назв. — англ. |
| series |
Механика твердого тела |
| work_keys_str_mv |
AT sklyargm spectralpropertiesofnonhomogeneoustimoshenkobeamanditscontrollability AT szkibielg spectralpropertiesofnonhomogeneoustimoshenkobeamanditscontrollability |
| first_indexed |
2025-07-03T07:56:11Z |
| last_indexed |
2025-07-03T07:56:11Z |
| _version_ |
1836611663431729152 |
| fulltext |
ISSN 0321-1975. Ìåõàíèêà òâåðäîãî òåëà. 2007. Âûï. 37
ÓÄÊ 531.38
c©2007. G.M. Sklyar, G. Szkibiel
SPECTRAL PROPERTIES OF NON-HOMOGENEOUS
TIMOSHENKO BEAM AND ITS CONTROLLABILITY
Controllability of slowly rotating non-homogeneous beam clamped to a disc is considered. It
is assumed that at the beginning the beam remains at the position of rest and it is supposed
to rotate by the given angle and achieve desired position. The rotor of propelling engine is in
the middle of the disk. The movement is governed by the system of two di�erential equations
with non-constant coe�cients: linear mass density, �exural rigidity, rotational inertia and shear
sti�ness. To solve the problem of controllability, the spectrum of the operator generating the
dynamics of the model is studied. Then the problem of controllability is reduced to the moment
problem that is, in turn, solved with the use of the asymptotics of the spectrum and Ullrich
Theorem.
Introduction. The considered model of Timoshenko beam clamped to a disc
of driving motor was �rst described by X.J. Xiong [1]. The derivation of that
model was described also in monograph [2]. The controllability and stabilization
of homogeneous Timoshenko beam was elaborated by W. Krabs and G.M. Sklyar
�rst in [3] and then in a series of papers that was concluded with the monograph
[2]. The �rst author to consider controllability of non-homogeneous Timoshenko
beam was M. Shubov [4�6]. She considers the di�erent model of the beam, but
we used her ideas while deriving equations of current model.
In this paper we present our results published in [7] and [8]. Described are
theorems and some ideas of proofs. For detailed proofs we send reader to mentioned
articles.
1. Model equation and spectral operators connected to it.We consider
the motion of a beam in a horizontal plane. The left end of the beam is clamped
to the disk of a driving motor. We denote by r the radius of that disk and let
θ = θ(t) be the rotation angle considered as a function of time (t ≥ 0). Further
on, we assign to a (uniform) cross section at x, with 0 ≤ x ≤ 1 the following:
E(x) which is the �exural rigidity, K(x) � shear sti�ness, %(x) � mass of the cross
section and R(x) � rotary inertia. All of the above functions are assumed to be
real and bounded by two positive numbers. We also assume that their �rst and
second derivatives are bounded. The length of the beam is assumed to be 1. We
denote by w(x; t) the de�ection of the center line of the beam and by ξ(x, t) the
rotation angle of the cross section area at the location x and at time t. Then w and
ξ are governed by the following system of two hyperbolic di�erential equations:
%(x)ẅ(x, t)− (
K(x)(w′(x, t) + ξ(x, t))
)′ = −θ̈(t)%(x)(x + r),
R(x)ξ̈(x, t)− (E(x)ξ′(x, t))′ + K(x)(w′(x, t) + ξ(x, t)) = θ̈(t)R(x).
(1)
In addition to (1) we impose the following boundary conditions:
175
G.M. Sklyar, G. Szkibiel
w(0, t) = ξ(0, t) = 0,
w′(1, t) + ξ(1, t) = 0, ξ′(1, t) = 0
for t ≥ 0.
We consider L2([0, 1],C2) as an underlying set with the inner product
〈(
y1
z1
)
,
(
y2
z2
)〉
=
∫ 1
0
%(x)y1(x)y2(x)dx +
∫ 1
0
R(x)z1(x)z2(x)dx. (2)
Due to hypotheses imposed on % and R, the norm generated by (2) is equivalent
to the standard L2 norm.
Let H = H2
0 ([0, 1],C2) be the set of all functions from L2([0, 1],C2) (with the
inner product (2)) that are twice di�erentiable and whose value at 0 is (0, 0)T .
We de�ne the linear operator A : D(A) → H by the formula
A
(
y
z
)
=
−1
%
(K(y′ + z))′
− 1
R
(
(Ez′)′ −K(y′ + z)
)
, (3)
where K, E, %, R, y and z are functions of variable x ∈ [0, 1] and
D(A) =
{(
y
z
)
: y(0) = z(0) = 0, y′(1) + z(1) = z′(1) = 0
}
⊂ H.
It is easy to see that D(A) is dense in H. Using the de�ned operator A and putting
f1(x, t) = −θ̈(t)(r + x), f2(x, t) = θ̈(t), (4)
we can rewrite the equations (1) in the vector form
(
ẅ(x, t)
ξ̈(x, t)
)
+ A
(
w(x, t)
ξ(x, t)
)
=
(
f1(x, t)
f2(x, t)
)
. (5)
It follows readily that the operator A : D(A) → H is positive, symmetric,
invertible and self-adjoint [7]. Therefore there exists the unique weak solution to
(1) given by
(
w(x, t)
ξ(x, t)
)
=
∞∑
j=1
1√
λj
∫ t
0
〈(
f1(·, s)
f2(·, s)
)
,
(
yj
zj
)〉
sin
√
λj(t− s)ds
(
yj(x)
zj(x)
)
. (6)
The inner product we use here is de�ned in (2), the functions f1 and f2 are de�ned
in (4) and
(
yj
zj
)
for j ∈ N are the eigenvectors of the operator A that correspond
176
Spectral properties of non-homogeneous Timoshenko beam
to eigenvalues λj . Also we notice, that the �rst (time) derivative of the above
solution is
(
ẇ(x, t)
ξ̇(x, t)
)
=
∞∑
j=1
∫ t
0
〈(
f1(·, s)
f2(·, s)
)
,
(
yj
zj
)〉
cos
√
λj(t− s)ds
(
yj(x)
zj(x)
)
. (7)
To use the solution (6) for the controllability purposes, we need to know at least
approximate location of eigenvalues of the operator A.
2. Asymptotic behaviour of eigenvalues. In order to study the location
of eigenvalues of the operator A de�ned by (3), we need to consider the following
system of spectral equations
−(
K(x)(y′(x) + z(x))
)′ = λ%(x)y(x),
−(
(E(x)z′(x)
)′ + K(x)(y′(x) + z(x)) = λR(x)z(x)
with the boundary conditions
The asymptotic location of eigenvalues are given by the following theorem [7].
Theorem 1. The set of eigenvalues of the operator A is given by formulas
λ(0)
n =
(∫ 1
0
√
%(t)
K(t)
dt
)−2 (π
2
+ nπ + ε(0)
n
)2
, (8)
λ(1)
n =
(∫ 1
0
√
R(t)
E(t)
dt
)−2 (π
2
+ nπ + ε(1)
n
)2
(9)
with ε
(0)
n , ε
(1)
n → 0 as n →∞. Thus the spectrum of the operator A asymptotically
splits into two sets � Λ(0), whose elements are described by (8) and Λ(1) containing
all elements of the form (9).
The proof of the above theorem is solving the spectral equation. It is brought
to the system of two integral equations that is, in turn, solved with the use
of Neumann series. It is shown in [7] that the eigenvalues are at most double.
However, we suspect that the eigenvalues are asymptotically single, but we do
not have a rigorous proof of that fact.
3. Rest to rest controllability.Given the beam, whose movement is described
by (1), we want to rotate it from the state of rest at time t = 0 to the state of
rest at the time t = T > 0. Thus we have the following boundary conditions:
w(x, 0) = ẇ(x, 0) = ξ(x, 0) = ξ̇(x, 0) = 0,
w(x, T ) = ẇ(x, T ) = ξ(x, T ) = ξ̇(x, T ) = 0
(10)
for x ∈ [0, 1]. The beam is controlled by motor that rotates it from angle 0 to
θT . The control is given by angular acceleration θ̈(t) and our goal is to �nd that
177
G.M. Sklyar, G. Szkibiel
function. The beginning position of rest means that the motor does not work, i.e.
the function θ is a member of H2
0 (0, T ), where
H2
0 (0, T ) =
{
f ∈ H2(0, T ) : f(0) = ḟ(0) = 0
}
.
At the end of the movement the beam is at the position θ(T ) = θT and the motor
does not move, i.e. θ̇(T ) = 0.
Thus, to solve the problem of controllability from rest to rest, we need for
given time T > 0 and angle θT ∈ R, θT 6= 0 to �nd a function θ ∈ H2
0 (0, T ) with
θ(T ) = θT , θ̇(T ) = 0. (11)
Using weak solution (6) and its derivative (7) we arrive at the conclusion that
conditions (10) are equivalent to
1√
λn
∫ T
0
〈(
f1(·, t)
f2(·, t)
)
,
(
yn
zn
)〉
sin(T − t)
√
λndt = 0
and ∫ T
0
〈(
f1(·, t)
f2(·, t)
)
,
(
yn
zn
)〉
cos(T − t)
√
λndt = 0
for all n ∈ N. Here the eigenvalues λn are not yet distinguish on those that belong
to Λ(0) and Λ(1). Next, upon putting
an =
∫ 1
0
R(x)zn(x)dx−
∫ 1
0
%(x)(r + x)yn(x)dx (12)
we obtain 〈(
f1(x, t)
f2(x, t)
)
,
(
yn(x)
zn(x)
)〉
= anθ̈(t)
for all positive integer n. We recall that the above inner product is de�ned by the
formula (2).
We remark here that for the controllability from rest to rest, the condition
an 6= 0 for all positive integer n (the formulas for an an are given by (12)) is
not necessary. It becomes crucial while considering controllability from rest to
arbitrary condition. The values of those parameters depend on the radius r of the
disc (in general, on the ratio radius to the length of the beam). However, it was
proved in [8] that there are only countable many values of r for which some of
an's are zeroes.
Employing (12) and well-known trigonometric formulas we obtain
an
∫ T
0
θ̈(t) sin(t
√
λn)dt = 0,
an
∫ T
0
θ̈(t) cos t
√
λndt = 0.
(13)
178
Spectral properties of non-homogeneous Timoshenko beam
Now we notice that if θ ∈ H2
0 (0, T ), then the conditions (11) are equivalent to
∫ T
0
θ̈(t)dt = 0,
∫ T
0
tθ̈(t)dt = −θT .
(14)
Gathering (13) and (14) we obtain that the problem of controllability from
rest to rest is equivalent to the following moment problem.
Moment Problem. Find u ∈ L2(0, T ) such that for all n ∈ N the conditions
∫ T
0
u(t) cos t
√
λndt = 0,
∫ T
0
u(t) sin t
√
λndt = 0;
∫ T
0
u(t)dt = 0,
∫ T
0
tu(t)dt = −θT
are satis�ed.
To �nd the solution to the stated Moment Problem we consider the system
{
t, 1, cos t
√
λ
(i)
n , sin t
√
λ
(i)
n : n ∈ N, i ∈ {0, 1}
}
. (15)
Here the λ
(i)
n 's are the eigenvalues described in Theorem 1. Just for convenience,
we rewrite the system (15) in the form
V ∪ {t} , where V =
{
1, cos t
√
λ
(i)
n , sin t
√
λ
(i)
n : n ∈ N, i ∈ {0, 1}
}
.
Let W be the closure of the linear span over V . Using the Theorem of Russel [9],
one can prove [7] that the above system is minimal for T > 2(J (0) + J (1)), where
J (0) =
∫ 1
0
√
%(x)
K(x)
dx and J (1) =
∫ 1
0
√
R(x)
E(x)
dx.
Let f(t) = t. Considering T given above, we have the existence of exactly one
h0 ∈ W such that
‖h0 − f‖2 ≤ ‖h− f‖2 for all h ∈ W ,
179
G.M. Sklyar, G. Szkibiel
where ‖ · ‖2 denotes the L2-norm in L2(0, T ). For that h0 we have
∫ T
0
(f(t)− h0(t))h(t)dt = 0 for all h ∈ W .
In particular, the above implies
∫ T
0
(f(t)− h0(t))dt = 0,
∫ T
0
(f(t)− h0(t)) cos t
√
λ
(i)
n dt = 0, n ∈ N, i ∈ {0, 1} ,
∫ T
0
(f(t)− h0(t)) sin t
√
λ
(i)
n dt = 0, n ∈ N, i ∈ {0, 1}
and ∫ T
0
(f(t)− h0(t))f(t)dt = ‖h0 − f‖2
2 > 0.
Upon de�ning
u(t) = − θT
‖h0 − f‖2
2
(f(t)− h0(t))
for t ∈ [0, T ] we receive u ∈ L2(0, T ) that solves the Moment Problem.
Thus we have the following theorem:
Theorem 2. The problem of controllability from rest to rest is solvable if
T > 2
(∫ 1
0
√
%(x)
K(x)
dx +
∫ 1
0
√
R(x)
E(x)
dx
)
.
4. Controllability from rest to arbitrary position. We assume that at
time t = 0 the beam remains at the position of rest, i.e.
w(x, 0) = ẇ(x, 0) = ξ(x, 0) = ξ̇(x, 0) = θ(0) = θ̇(0) = 0
for x ∈ [0, 1]. At a given time T , we need to achieve the following position
w(x, T ) = wT (x), ẇ(x, T ) = ẇT (x),
ξ(x, T ) = ξT (x), ξ̇(x, T ) = ξ̇T (x),
(16)
where functions wT , ẇT , ξT , ξ̇T de�ned on [0, 1] are given. The problem of
controllability from rest to arbitrary position is:
Problem of controllability. Given time T > 0, numbers θT , θ̇T ∈ R and
position (16), �nd a function θ ∈ H2
0 (0, T ) satisfying
θ(T ) = θT , θ̇(T ) = θ̇T (17)
180
Spectral properties of non-homogeneous Timoshenko beam
and such that the weak solution (6) of (1) satis�es (16).
Consideration similar to the above one leads to following moment problem.
Moment Problem. Find u ∈ L2(0, T ) such that for all n ∈ N and k ∈ {0, 1}
the conditions
∫ T
0
u(t) cos t
√
λ
(k)
n dt = ċ(k)
n ,
∫ T
0
u(t) sin t
√
λ
(k)
n dt = c(k)
n ,
∫ T
0
u(t)dt = θ̇T ,
∫ T
0
tu(t)dt = θT
are satis�ed.
Here (c(k)
n )n and (ċ(k)
n )n are sequences whose values depend on end conditions (16)
and on eigenvalues.
The stated problem is divided onto three cases.
Case 1: J(0) = J(1) = J .
In the solution the Ullrich Theorem [10] is used and the �nal result is given
by theorem [8]:
Theorem 3. Provided
∫ 1
0
√
%(x)
K(x)
dx =
∫ 1
0
√
R(x)
E(x)
dx and T ≥ 4
∫ 1
0
√
%(x)
K(x)
> dx,
the problem of controllability from the state of rest to arbitrary position is solvable
if and only if the following condition is satis�ed
∞∑
n=1
|cn0|2 +
∣∣∣∣∣∣
cn0 − cn1√
λ
(0)
n −
√
λ
(1)
n
∣∣∣∣∣∣
2
< ∞
with cnk = (π/J)(ċ(k)
n + ic
(k)
n ).
Case 2: J(1)/J(0) =
p
q
is rational number and p and q are relatively prime
positive odd integers.
Without loss of generality, we may assume that J (1)/J (0) = γ > 1. Here the
Ullrich-type Theorem [11] in place of Ullrich Theorem is used . The �nal result is
the following theorem [7]
Theorem 4. If
∫ 1
0
√
%(x)
K(x)
dx
/∫ 1
0
√
R(x)
E(x)
dx =
p
q
with p, q relatively prime
odd positive integers and
T ≥ 2
(∫ 1
0
√
%(x)
K(x)
dx +
∫ 1
0
√
R(x)
E(x)
dx
)
,
181
G.M. Sklyar, G. Szkibiel
the problem of controllability from the state of rest to arbitrary position is solvable
if and only if the condition
∞∑
n=1
|cn0|2 + |cn1|2 +
∣∣∣∣∣∣
c((1−q)/2)+qn,0 − c((1−p)/2)+pn,1√
λ
(0)
((1−q)/2)+qn −
√
λ
(1)
((1−p)/2)+pn
∣∣∣∣∣∣
2
< ∞
is satis�ed.
Case 3: J(1)/J(0) =
p
q
is rational number, p and q are relatively prime positive
integers and exactly one of them is even.
We proceed in the same way like in Case 2 and �nally get the following theorem
Theorem. If
∫ 1
0
√
%(x)
K(x)
dx
/ ∫ 1
0
√
R(x)
E(x)
dx =
p
q
with p, q relatively prime
positive integers, from which exactly one is even and
T ≥ 2
(∫ 1
0
√
%(x)
K(x)
dx +
∫ 1
0
√
R(x)
E(x)
dx
)
,
the problem of controllability from the state of rest to arbitrary position is solvable
if and only if the condition
∞∑
n=−∞
(|cn0|2 + |cn1|2
)
< ∞.
is satis�ed.
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Inform. � 2000. � 17. � P. 375�395.
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Institute of Mathematics, Szczecin University, Poland
sklar@sus.univ.szczecin.pl, szkibiel@poczta.onet.pl
Received 18.06.07
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