Study of the Stability Properties for a General Shape of Damped Euler–Bernoulli Beams under Linear Boundary Conditions

Kouakou Kra Isaac, Teya; Gossrin Jean-Marc, Bomisso; Kidjegbo Augustin, Touré; Adama, Coulibaly

doi:https://doi.org/10.1155/2023/9939530

Abstract and Applied Analysis

On this page

Abstract Introduction Conclusion Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 9939530 | https://doi.org/10.1155/2023/9939530

Study of the Stability Properties for a General Shape of Damped Euler–Bernoulli Beams under Linear Boundary Conditions

Teya Kouakou Kra Isaac,¹Bomisso Gossrin Jean-Marc,²Touré Kidjegbo Augustin,¹and Coulibaly Adama³

Academic Editor: Jozef Banas

Received16 Jun 2023

Revised05 Sept 2023

Accepted04 Oct 2023

Published14 Nov 2023

Abstract

We study in this paper a general shape of damped Euler–Bernoulli beams with variable coefficients. Our main goal is to generalize several works already done on damped Euler–Bernoulli beams. We start by studying the spectral properties of a particular case of the system, and then we determine asymptotic expressions that generalize those obtained by other authors. At last, by adopting well-known techniques, we establish the Riesz basis property of the system in the general case, and the exponential stability of the system is obtained under certain conditions relating to the feedback coefficients and the sign of the internal damping on the interval studied of length .

1. Introduction

The stabilization of damped beams submitted to vibrations has been one of the main research topics in smart materials and structures. A beam has two spatially nonhomogeneous damping terms. The first is often called structural damping, while the second acts opposite to the velocity and is called viscous damping. When these damping coefficients are constant, it is known that viscous damping causes a constant attenuation rate for all frequencies of vibrations. In this paper, we study a general shape of Euler–Bernoulli beams with variable coefficients under the influence of a viscous damping. This beam is clamped at one end and controlled at its free end in force by a linear combination of the velocity, rotation, and velocity of rotation and in a moment by a linear combination of the velocity, velocity of rotation, and the position term. Let the transversal deviation at position and time . The equations are defined by the following:

The terms , and are positive given constants. The function is the mass density of the beam and the function is its flexural rigidity satisfying . Throughout this paper, we assume that

Moreover, is a continuous coefficient function of feedback damping that is assumed to satisfy the condition

Our interest in this work is to generalize the works carried out by several authors on damped Euler–Bernoulli beams.

Aouragh and Yebari [1] studied the system (1) without damping () and with the condition . In their article, the authors showed that there is a sequence of generalized eigenfunctions that form a Riesz basis of the appropriate Hilbert space and that there is exponential stability under certain conditions relating to the feedback coefficients.

For our part, we study the system (1) with the presence of a viscous damping and the condition . To do this, we analyze two situations as follows: Case 1: We consider the case where , which implies with the condition (2) that . Our investigation of this case allows us to deduce the results of asymptotic expressions by Guo [2], Koffi et al. [3], Touré et al. [4], and Wang et al. [5]. In these cases, the authors have concluded for the exponential stability. Case 2: We study the case where and . Here, first, we find a general asymptotic expression of the eigenvalues of the system where we could deduce those found by Jean-Marc et al. [6] and Aouragh and Yebari [1] in the constant case. Next, by proceeding by Wang [7], the exponential stability of the system (1) is obtained when γ is non-negative in [0,1], and we discuss the case that is continuous and indefinite on the interval [0, 1].

In the second case, we consider non-zero all the coefficients in the boundary conditions, which made the calculations very complex. The calculations have been carried out meticulously to obtain the expected result.

This present work reveals capital importance because it is a general case of the results obtained in several works [1–6].

During our analysis, we used the asymptotic method of Wang, which is essential for our study. This method comes from the work of Birkhoff [8, 9]. We also find in the literature several authors who have used this approach to study Euler–Bernoulli beam equations with variable coefficients (see Wang et al. [5], Wang [7], Guo [10], Guo and Wang [11]).

The contents of the paper are arranged as follows: in Section 2, we formulate the system (1) into an abstract Cauchy problem in Hilbert state space and discuss some basic properties of the system. In Section 3, we study the spectrum properties for each case. The Riesz basis property and the exponential stability in the general case are concluded in Section 4.

2. Semigroup Formulation

In this part, we study the well-posedness of the system (1) and deduce some properties of the operator of this system. We introduce this Hilbert spaces as follows:with the inner productwhere and . Here, denotes the corresponding norm. The spaces and are defined as follows:

Next, we define an unbounded linear operator as follows:with the domain

The system (1) can be formally written as a first-order evolution problemwhere and . Furthermore, we introduce the linear operator defined on by the following:

Notice that denotes the operator where . In the study of Aouragh and Yebari [1], it is shown that the linear operator generates a -semigroup of contractions on . Moreover, the linear operator is bounded on . In fact,where

Now, we deduce the well-posedness of the system through the following theorem.

Theorem 2.1. The operator defined by Equations (9) and (10) generates a -semigroup of contraction on and has compact resolvent and Therefore, the spectrum consists entirely of isolated eigenvalues.

Proof. Let ). Integrating twice by parts and taking real part, we obtain the following:

Thus, is a dissipative operator.

Moreover, the linear operator is bounded on and generates a C₀-semigroup of contractions. Then, according to perturbation theory by a bounded linear operator (see Pazy [12] P.76), the operator is an infinitesimal generator of a C₀-semigroup of contractions , satisfying to for and .

Now, we show that the operator exists. For any , we must find a unique such that . This leads us to the following system:

A direct calculation shows thatwhere and are defined as follows:

Obviously, we have . Therefore, we get the following:

Thus, exists and consequently we have and Sobolev’s embedding theorem implies that is a compact operator on .

3. General Spectral Analysis

3.1. Study of Particular Case:

In this part, we study the eigenvalue problem of the operator in order to determine the behavior of these eigenvalues. The system (1) becomesand

Our work shall follow the results from Wang [5]. Let be an eigenvalue of the operator and let , the corresponding eigenfunction.where the prime refers to the derivative with respect to space variables. This equation leads to the following system:

First, we use a space transformation to transform the coefficient function appearing with in the first expression of Equation (23) into a constant (see Guo [10]).

A direct computation gives this system as follows:where

Next, we make an invertible state transformation by Naimark [13].

The function satisfies the following:with

By setting , the first equation of the system (29) becomes the following:

Now, to solve the eigenvalue problem (29), we divide the complex plane into eight distinct sectors by Naimark [13].

Let , and be the roots of characteristic equation with

In sector S₁, we name the roots as follows:that satisfy the inequalities (34). Similarly, the choices can also be made for other sectors. In the following, we study the asymptotic behavior of the eigenvalues specific to sectors S₁ and S₂ because this will be verified in the other sectors with similar arguments. For this, we will use the following lemma:

Lemma 3.1. For , with | large enough, the equation

has four linearly independent asymptotic fundamental solutionsand hence their derivatives for and are given by the following:where for with

Proof. The reader can refer to the study of Touré et al. [4].

For convenience, we introduce the notation

Lemma 3.2. For , If , then we have the following inequalities:
, , when .

Furthermore, substituting Equations (37) and (38) into the boundary conditions (29), we obtain asymptotic expressions for the boundary conditions for large enough as follows:where

Notice that , is the eigenvalue of Equation (29) if and only if satisfies the characteristic equation as follows:

By substitution, the following expression is obtained:

In sector S₁, the choices are such that

Developing the determinant, and after a straightforward computation, we obtain the following:

Noting thatthe characteristic determinant in sector S₁ becomes the following:

Then, we have , and , (see Jean-Marc et al. [6]).

We notice that . Then, the following theorem follows directly.

Theorem 3.3. The boundary conditions of the eigenvalue problem (29) are strongly regular. Therefore, the eigenvalues are asymptotically simple and separated.

Now, we study the asymptotic behavior for the eigenvalues of problem (29). The equation implieswhich can also be rewritten as follows:

The equation below:has the solutions given by the following:

Let be the solution of Equation (53) According to Rouché’s theorem (see Naimark [13]), we obtain the following expression:where is a sufficiently large positive integer.

By substituting it in Equation (52) and using the equality we get the following:

Moreover, expanding the exponential function according to its Taylor series, we get the following:

Note that and in sector and . So, we have the following:where , with large enough.

Now, in sector S₂, the eigenvalues of the problem (29) can be obtained by a similar computation with the choices.such that:

Hence, in the sector S₂, the characteristic determinant is as follows:

By a similar calculation to the one done in sector S₁, we have the following:with a large enough integer. In the sector S₂, using and , we obtain for

Remark 3.4. By referring to Naimark [13], we can say that the eigenvalues generated by the sectors coincide with those determined in the sectors and .

Combining with Equations (59) and (63), we obtain the asymptotic expression

Theorem 3.5. An asymptotic expression of the eigenvalues of the problem (29) is given by the following:where k , with a large enough integer, andMoreover, with sufficiently large modulus are simple and distinct except for finitely many of them, and satisfy

From Equation (67), we find these results already known. The following examples illustrate the special cases carried out in certain works:

Example 1. For and and from Equation (20), we get the system as follows:The asymptotic expression (67) becomes the following:This case corresponds to the one studied by Wang et al. [5], where the result (69) is obtained.

Example 2. Suppose that . Then, the system (20) becomes the following:This system has been studied by Touré et al. [4]. The same asymptotic expression (67) is obtained.

Example 3. Suppose that . Equation (20) is equivalent to the following:Guo [2] studied this system. His work yielded a similar asymptotic expression as Equation (67) in the uniform case .

Example 4. For and , we get the following system:This case was investigated by Koffi et al. [3], where the asymptotic expression (67) was found.
Note that in the cited examples, the authors proved that the systems are exponentially stable.

3.2. Study of General Case: and

In this subsection, we study the system (1) with the conditions and . Following the previous approach, we determine a general asymptotic expression of the eigenvalues of the system (1) and deduce those established by Aouragh and Yebari [1] and Touré et al. [4].

Let be an eigenvalue of the operator of the system (1).

The system (73) is rewritten as follows:

First, by using the space transformation as follows:we havewhere and are defined in Equations (26) and (27)

Then, by making the invertible state transformationthe system (76) can be written as follows, for any where

By setting , the first equation of the system (79) becomes the following:which has four linearly independent asymptotic fundamental solutions given by Equations (37) and (38). To solve the eigenvalue problem (79), we make a study in the sector which is defined like this

Let , and be the roots of characteristic equation that are arranged so that

In sector , the choices are the same as in Equation (35). Substituting Equations (37) and (38) into the boundary conditions (79) we obtain asymptotic expressions for the boundary conditions for large enough with

By substituting the expression (84) into (44), the characteristic determinant of the eigenvalue problem (79) is given by the following:

Developing the determinant, and after a straightforward computation, we obtain the following:with

Then, we have , , and .

We notice that . Then, the following theorem follows directly.

Theorem 3.6. The boundary conditions of the eigenvalue problem (79) are strongly regular. Therefore, the eigenvalues are asymptotically simple and separated.

Now, we study the asymptotic behavior for the eigenvalues of problem (79). The equation implieswhich can also be rewritten as follows:

The equation has the solutions given by the following:

Let be the solution of Equation (93). According to Rouché’s theorem (see Naimark [13]), we obtain the following expression:where is a sufficiently large positive integer.

By substituting it in Equation (92) and using the equality we get the following:

Moreover, expanding the exponential function according to its Taylor series, we get the following:

Note that and in sector and . So, we have the following:where , with large enough.

By applying the same proof to the sector , the eigenvalues of the problem (79) can be obtained by a similar computation with the same choices as in Equation (60). Hence, in the sector , the characteristic determinant is as follows:

By a calculation similar to the one done in sector S₁, we have the following:

In the sector S₂, using and , we obtain the following:where k , with a large enough integer.

Theorem 3.7. Consider that and , then an asymptotic expression of the eigenvalues of the problem (79) is given bywhere k , with a large enough integer, andMoreover, with sufficiently large modulus are simple and distinct except for finitely many of them, and satisfywhere

According to the condition (2), the constant is positive. From Equation (105), we find the following results:

Example 1. Suppose that and . Then, the system (1) becomes the following:This system has been studied by Jean-Marc et al. [6]. If , then and Equation (105) becomes the following:The asymptotic expression (107) is equivalent to the one in Theorem 8 by Jean-Marc et al. [6].

Example 2. Suppose that and , the system (1) is equivalent to the following:which has been studied by Aouragh and Yebari [1], where they get the same result as Equation (105) in the uniform case

Remark 3.8. In the cited examples, the authors proved that the systems are exponentially stable.

4. Riesz Basis Property and Exponential Stability of the System in the General Case

In this section, we consider the system (1) in the general case with the conditions and

4.1. Riesz Basis Property of the System

Here, we discuss the Riesz basis property of the eigenfunctions of the operator of the system by following an idea due to Wang et al. [5]. We begin by considering a bounded invertible operator defined on by the following:with

Then, we define the ordinary differential operator as follows:

Let be the Hilbert space defined by Wang [7] and define the operator in by the following:

Let be an eigenvalue of and the corresponding eigenfunction. Then we obtain and satisfies as follows:

Now by takingwe see that and satisfies

Hence, we have that:

Theorem 4.1. Let operator be defined by Equations (9) and (10)Then the eigenvalues of operatorare all simple except for finitely many of them, and the generalized eigenfunctions of operatorform a Riesz basis for the Hilbert state space.

Proof. We have shown that the boundary problem (79) is strongly regular (see Theorem 3.6). Therefore, the eigenvalues are separated and simple except for finitely many of them. Thus, the first statement follows. Moreover, Theorem 4.1.1 in the study of [7] ensures that the strong regularity of the boundary problem leads to the sequence of generalized eigenfunctions of operator forms a Riesz basis for . Since is bounded and invertible on , it follows that also forms a Riesz basis on .

4.2. Exponential Stability of the System (1)

Referring to the study of Curtain and Zwart [14], we note that the Riesz basis property implies the spectrum-determined growth condition and Equation (105) describes the asymptote of , for any small there are only finitely many eigenvalues of in the following half-plane:

Following Theorem 2.4 in the study of Gao [10], all the properties of operator found above, allow us to claim that for the semigroup generated by the spectrum-determined growth condition holds the following:with

By using Theorem 4.1, the exponential stability of system (1) can be concluded.

Theorem 4.2. Consider that and . If , then the system (1) is exponential stable for any and . That is, there are constants and such as the energyof system (1) satisfies , for any initial condition

Proof. The operator is dissipative. In effect, For We also know that generates a semigroup of contractions on . Moreover, the spectrum of the operator has an asymptote as follows:Then, the study of the exponential stability is equivalent to verifying that . Let with be an eigenvalue of operator on the imaginary axis and be the corresponding eigenfunction, then .
, thenSince and are continuous with , we obtain the following:
andThen,Moreover , then from Equations (74) and (124)the following differential equation is satisfied by We show with the help of Rolle’s Theorem that the null function is the unique solution of Equation (125). For this, the reader is invited to follow a method used by Jean-Marc et al. [6].
From Theorem 4.1 and the spectrum-determined growth condition, the system is exponentially stable for any and with

Now, using an idea of Wang [7], we study the situation where is continuous and indefinite in [0,1]. We have the following theorem.

Theorem 4.3. Note and letHence, the operator can be written as . Note . Ifthen the system (1) is exponentially stable.

Proof. According to Section 2 of this paper, the operator is bounded and symmetric, so it is a self-adjoint operator (see Brezis [15]) andBy Theorem 4.2 and the definition of operator , is a contraction semigroup and . Applying the perturbation theory of linear operator’s semigroup (see Pazy [12]), we have whenever . Furthermore, Theorem 4.1 ensures thatTherefore, the system (1) is exponentially stable if

5. Conclusion

During our analysis, after showing the basic properties of the operator, we established an asymptotic expression of the operator, which generalizes several other results. Then, using the Riesz basis property, we have shown that the system is exponentially stable in the general case following the sign of . In addition, our results can also be extended to other Euler–Bernoulli beam problems.

Data Availability

No underlying data were collected or produced in this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

References

M. D. Aouragh and N. Yebari, “Stabilisation exponentielle d’une équation de poutres d’Euler-Bernoulli à coefficients variables,” Annales Mathématiques Blaise Pascal, vol. 16, no. 2, pp. 483–510, 2009.
View at: Publisher Site | Google Scholar
B.-Z. Guo, “Riesz basis property and exponential stability of controlled Euler–Bernoulli beam equation with variable coefficients,” SIAM Journal on Control and Optimization, vol. 40, no. 6, pp. 1905–1923, 2002.
View at: Publisher Site | Google Scholar
C. J. J. Koffi, G. J.-M. Bomisso, K. A. Touré, and A. Coulibaly, “Riesz basis property and exponential stability for a damped system and controlled dynamically,” International Journal of Numerical Methods and Applications, vol. 22, 2022.
View at: Google Scholar
K. A. Touré, A. Coulibaly, and A. A. Hermith Kouassi, “Riesz basis, exponential stability of variable coefficients Euler-Bernoulli beams with an indefinite damping under a force control in position and velocity,” Electronic Journal of Differential Equations, vol. 2015, no. 54, pp. 1–20, 2015.
View at: Google Scholar
J.-M. Wang, G.-Q. Xu, and S.-P. Yung, “Riesz basis property, exponential stability of variable coefficient Euler–Bernoulli beams with indefinite damping,” IMA Journal of Applied Mathematics, vol. 70, no. 3, pp. 459–477, 2005.
View at: Publisher Site | Google Scholar
B. G. Jean-Marc, T. K. Augustin, and Y. Gozo, “Stabilization of variable coefficients euler-bernoulli beam with viscous damping under a force control in rotation and velocity rotation,” Journal of Mathematics Research, vol. 9, no. 6, pp. 1–13, 2017.
View at: Publisher Site | Google Scholar
J.-M. Wang, “Riesz basis property of some infinite-dimensional control problems and its applications,” The University of Hong Kong, 2004, Ph.D. Thesis.
View at: Google Scholar
G. D. Birkhoff, “On the asymptotic character of the solutions of certain linear differential equations containing a parameter,” Transactions of the American Mathematical Society, vol. 9, no. 2, pp. 219–231, 1908.
View at: Publisher Site | Google Scholar
G. D. Birkhoff, “Boundary value and expansion problems of ordinary linear differential equations,” Transactions of the American Mathematical Society, vol. 9, no. 4, pp. 373–395, 1908.
View at: Publisher Site | Google Scholar
B. Z. Guo, “On the boundary control of a hybrid system with variable coefficients,” Journal of optimization Theory and Applications, vol. 114, no. 2, pp. 373–395, 2002.
View at: Publisher Site | Google Scholar
B.-Z. Guo and J.-M. Wang, “Riesz basis generation of abstract second order partial differential equation systems with general non-separated boundary conditions,” Numerical Functional Analysis and Optimization, vol. 27, no. 3-4, pp. 291–328, 2006.
View at: Publisher Site | Google Scholar
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44, Springer-Verlag, New York, 1983.
M. A. Naimark, Linear Differential Operators, vol. 1, F. Ungar, New York, 1967.
R. F. Curtain and H. Zwart, “An introduction to infinite dimensional linear system theory,” in Texts in Applied Mathematics, vol. 21, Springer-Verlag, Berlin, 1995.
View at: Google Scholar
H. Brezis, “Analyse fonctionnelle,” in Théorie et Applications, Masson, Paris, 1983.
View at: Google Scholar

Copyright

Copyright © 2023 Teya Kouakou Kra Isaac et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

435

Downloads

272

Citations