Abstract
In this work, we study the vibration control of a flexible mechanical system. The dynamic of the problem is modeled as a viscoelastic nonlinear Euler–Bernoulli beam. To suppress the undesirable transversal vibrations of the beam, we adopt a control at the right boundary of the beam. This control law is simple to implement. We prove uniform stability of the system using a viscoelastic material, the multiplier method and some ideas introduced in [20]. It is shown that a large range of rates of decay of the energy can be achieved through a determined class of kernels. Unlike most of the existing classes in the market, ours are not necessarily strictly decreasing.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Flexible systems exert an increasing influence in different industries and fields. For instance, we may cite flexible manipulators, flexible robot arm and marine risers for oil and gas transportation. The vibration problem of flexible systems has become a crucial topic of research. It is a widespread phenomena in engineering. The origin of these vibrations and their nature might be different. They can cause numerous harmful effects on the production process, including the damage of the equipment with significant financial consequences. There are many approaches to deal with vibration and stabilize flexible systems. Boundary control is the most practical and efficient one. In reference [5], active boundary controls to reduce vibration of an Euler–Bernoulli beam systems in one dimension are considered. In [12], nonlinear vibrations and stability issues are studied. In [4], boundary controllers are used to reduce the vibration of a coupled nonlinear flexible marine riser. Reference [11] considered an adaptive boundary control for an axially moving belt system to eliminate the vibration. In [8] using the direct method of Lyapunov, the exponential stability of a closed-loop system is proven with the help of boundary controls. Kelleche and Tatar [9] designed a nonlinear boundary control for a viscoelastic flexible system. Park et al. [14] studied the Euler–Bernoulli beam equation with memory, they proved the existence and the exponential stability of solutions for the problem
under the boundary and initial conditions
where
They supposed that the kernel k verifies
for some \(c_{i}>0, i=1,...,3.\) Furthermore, a similar result in [15] was established under a boundary control
Seghour et al. [17] investigated the following system
where the positives coefficients \(d_{s}, M_{s}\) represents the vessel damping and the mass of the surface vessel. They showed an exponential decay result for solutions with the following conditions on the kernel:
for some positive function \(\zeta (t)\) and
Moreover, in [18], the authors considered a similar problem under \( u(t)=d_{s} v_{t}(L,t) \), for kernels k verifying
for some function \(\eta (t)\). Later, the authors in [1] established uniform stability of the same problem for kernels satisfying
In [3], the authors considered the vibrating flexible beam system
\(\left\{ \begin{array}{l} \rho A v _{tt}(x,t)+EI v _{xxxx}(x,t)-P_{0} v _{xx}(x,t)-\dfrac{3}{2}EA v _{xx}(x,t)v _{x}^{2}(x,t)=0, \\ \quad \text {in }(0,L)\times [0,\infty ), \\ v _{xx}(0,t)=v _{xx}(L,t)=v (0,t)=0, \forall t\ge 0 \\ -EI v _{xxx}(L,t)+P_{0} v _{x}(L,t)+\dfrac{1}{2} EA v _{x}^{3}(L,t)=-K_{1}v _{t}(L,t), \forall t\ge 0,\text { }K_{1}>0. \end{array} \right. \)
They imposed a linear control force at the boundary to achieve the exponential stability of the system. Motivated by this work [3], the objective of the present paper is to consider the nonlinear viscoelastic Euler–Bernoulli beam equation
under the boundary conditions
The initial conditions are
where
EI: the flexural rigidity
\(\rho A\): the mass per unit length
EA: the axial stiffness
v(x, t): the transverse displacement and
\(P_{0}\): the tension force
The variance length envisaged with the tension force will be assumed to be weak compared to the overall length of the beam. We show an arbitrary decay result for problem (1)–(3) with weaker hypotheses on the relaxation function \(\psi \) than the existing ones for similar problems. Namely, we do not limit ourselves to polynomially or exponentially decaying functions only. Relaxation functions that can have zero derivatives on certain subsets of \(\left( 0,\infty \right) \) are considered, see [20,21,22]. We assume that the zone where the kernel is flat and is small. Consequently, a wide range of materials with various viscoelastic properties can be used in modern engineering.
The rest of our paper is arranged as follows: In Section 2, we give some useful lemmas needed for our result. The arbitrary decay of the energy result is shown in Section 3.
2 Notation and main results
We introduce the following notation
For the kernel \(\psi \) we assume:
- (H1):
-
\(\psi : {\mathbb {R}}_{+}\rightarrow {\mathbb {R}}_{+}\) is a differentiable function satisfying
$$\begin{aligned} 0<k=\int \limits _{0}^{+\infty }\psi (s)\mathrm{d}s<1 . \end{aligned}$$ - (H2):
-
\(\psi ^{\prime }(t)\le 0\) for almost all \(t\ge 0\).
- (H3):
-
There exists a positive increasing function \(\theta \left( t\right) \) such that \(\dfrac{\theta ^{\prime }\left( t\right) }{\theta \left( t\right) }=u(t)\) is a decreasing function and \(\int \limits _{0}^{+ \infty }\psi (s)\theta \left( s\right) \mathrm{d}s<+\infty .\)
We denote
and (., .), \(\left\| .\right\| \) the inner product and the norm of the space \(L^{2}\left( 0,L\right) \), respectively. The existence result for our problem (1)-(3) can be proved by Faedo–Galerkin method, the reader may consult [14].
Theorem 1
Suppose that (H1)-(H3) are satisfied. If \(( v_{0}, v_{1}) \in {\mathcal {H}} \times L^{2}(0,L)\), then there exists a unique solution v of problem (1)-(3), in the sense that for \(T> 0\),\(v \in L^{\infty }([0, T ), {\mathcal {H}}), v_{t} \in L^{\infty }([0, T ), {\mathcal {V}}), v_{tt} \in L^{2}([0, T ), L^{2}(0,L)).\) Moreover, we have \(v \in C([0, T ), {\mathcal {V}}), v_{t} \in C([0, T ),L^{2}(0,L))\).
We define the (classical) energy of problem (1)–(3) by
Then, the time derivative of energy is equal to
It is easy to see that
Then, we consider the modified energy
By differentiation, we obtain
If our non-negative relaxation function satisfies \(\psi ^{\prime }\le 0\), it follows that e(t) is nonincreasing and uniformly bounded above by \( e(0)=E(0)\). Next, we introduce the functionals
where \(\zeta \) is a positive constant to be determined later,
and \(\theta \left( t\right) \) is specified below. We define the second modified functional by
for \(\lambda _{i}>0\), \(i=1,2,3\) to be specified later. Our first result shows that this functional is an appropriate one to consider.
Proposition 2
There exist \(q_{i}>0, i=1,2\) such that
Proof
It is easy to see, from the above definitions, that
where \(c_{1}=\max (1+\zeta L, \dfrac{\rho A\zeta L}{P_{0}}, \dfrac{\rho Ac_{p}}{P_{0}})\). With these in mind, we have
and
Therefore, \(q_{1}\left( e(t)+\varphi _{2}\left( t\right) +\varphi _{3}(t)\right) \le L(t)\le q_{2}\left( e(t)+\varphi _{2}\left( t\right) +\varphi _{3}(t)\right) \) for some constant \(q_{i}>0\) and \(\lambda _{1}\) such that
\(\square \)
The next result [21] gives a better estimate for
Lemma 3
We have for a continuous function \(\psi \) on \([0, \infty )\) and \(v\in H^{1}(0,L)\)
3 Asymptotic behavior
In this section we state and show our result. To this end we require some notation. For every measurable set \({\mathcal {A}}\) \(\subset {\mathbb {R}}^{+}\), we define the probability measure
where \(k=\int _{0}^{\infty }\psi (s)\mathrm{d}s.\) The flatness set and the flatness rate of \(\psi \) are \(\left( \text {respectively}\right) \) defined by
and
Let \(t^{\star }>0\) and \(\int \limits _{0}^{t^{\star }}\psi (s)\mathrm{d}s=\psi _{\star }>0 \).
Theorem 4
Let us suppose that \(\psi \) and \(\theta \) satisfy the hypotheses (H1)–(H3) and \({\mathcal {R}}_{\psi } < \dfrac{1}{4}\). Then, there exist positive constants C and \(\nu \) such that
Proof
A differentiation of \(\varphi _{1}\left( t\right) , \) with respect to t along the solution of (1)-(4), gives
We decompose the first integral into
Clearly
and
Moreover,
that is
and
Using Young and Cauchy–Schwartz inequality we estimate the integral
Next, Lemma 2 yields
Then
and
or
Taking into account (14)–(18), we have
The boundary control gives us
Moreover, by Young’s inequality
and, therefore,
Notice that
After substitution of \(-\rho A v_{tt}\) from (1) and integrating by part, we obtain
Again utilizing Young’s inequality, we get
For the second and the third term in (21), we have
Hence,
For \(\delta _{2}> 0\), we can write
and
Now we proceed to estimate \(J_{3}.\) For all measurable sets \({\mathcal {A}}\) and \({\mathcal {F}}\) such that \(\mathcal {A= {\mathbb {R}} }^{+}\backslash {\mathcal {F}}\), we see that
We denote \({\mathcal {Q}}_{t}={\mathcal {Q}}\mathcal {\cap }\left[ 0,t\right] \). Using Lemma 2, we obtain for \(\delta _{4}>0\)
clearly
and
Thus,
where \({\widehat{\psi }}\) is defined in (11). We end up with
For \(\delta _{5}>0\), we have
Taking into account (19)-(22) and the above estimations of \(J_{1},\) \(J_{2},\) \(J_{3}\), \(J_{4}\), we obtain
Further, a differentiation of \(\varphi _{2}\left( t\right) \) yields
Regarding \(\varphi _{3}^{\prime }\left( t\right) \) it appears that
that is
Collecting the estimations (7), (23)–(25), we find for \(t\ge t_{\star }\)
For \(n\in {\mathbb {N}}\), we introduce the sets [16]
Notice that
where \({\mathcal {N}}_{\psi }\) is the null set in which \(\psi ^{\prime }\) is not defined. The complement of \({\mathcal {A}}_{n}\) in \({\mathbb {R}}^{+}\) is denoted by \({\mathcal {F}}_{n}= {\mathbb {R}} ^{+}\backslash {\mathcal {A}}_{n}\). It appears that \(\underset{n\rightarrow \infty }{\lim {\widehat{\psi }}\left( {\mathcal {F}}_{n}\right) =}{\widehat{\psi }} \left( {\mathcal {F}}_{\psi }\right) \) since \({\mathcal {F}}_{n+1}\subset {\mathcal {F}} _{n}\) for all n and \(\underset{n}{\cap }{\mathcal {F}}_{n}={\mathcal {F}} _{\psi }\cup {\mathcal {N}}_{\psi }\). Then, we take \({\mathcal {A}}_{n}={\mathcal {A}}\) , \({\mathcal {F}}_{n}={\mathcal {F}}\), and we select \(\lambda _{1}\le \dfrac{ \delta _{1}}{c_{p}\psi (0)},\) so that
Choosing
we may write
we choose \(\zeta =1+2k\) so that
Also, we need \(K_{\theta }\left( 0\right) \le \min \left\{ 1-k+\dfrac{1}{k}, \text { }4\right\} \) and \(\delta _{1}=\dfrac{\psi _{\star }+1+2k}{2}.\) For small \(\delta _{5}\) and \(t^{\star }, n \) large enough, we see that if \( {\widehat{\psi }}\left( {\mathcal {F}}_{n}\right) <\dfrac{1}{4}\) then
and
with \(\sigma =\dfrac{3k\left( 1-\psi _{\star }\right) }{4\left( 1-k\right) \left( 1+2k\right) }\). For the relation
to hold, it suffices that
with \(\delta _{4}\) small enough. Taking \(K_{1} \le min \left\{ P_{0}, \dfrac{P_{0}}{30L}\right\} \), we get
This is possible if \(\psi _{\star }>\dfrac{8k^{2}-4-4k/5}{3k}.\) We also need \(\lambda _{1}\) so small that
As a consequence of the above consideration,
for some positive constants \(c_{i}\) \(i=1,...,5\). For \(\lambda _{1}\) even smaller if necessary, we get
where \(C_{1}\) is some positive constant. As u(t) is nonincreasing, we have u(t) \(\le u(0)\) for all \(t\ge t_{\star }\). Then (27) becomes
By Proposition 1, we obtain
for some positive constant \(C_{2}\). Integrating (28) over \(\left[ t_{\star },t\right] \) yields
Then using inequality (9) of Proposition 1, we find
The continuity of E(t) over the interval \(\left[ 0,t_{\star }\right] \) makes it possible to deduce
for some positive constants C and \(\nu \). \(\square \)
References
Berkani, A., Tatar, N.-E., Seghour, L.: Stabilisation of a viscoelastic flexible marine riser under unknown spatiotemporally varying disturbance. Int. J. Control. https://doi.org/10.1080/00207179.2018.1518596.
Do, K.D.; Pan, J.: Boundary control of transverse motion of marine risers with actuator dynamics. J. Sound Vib. 318(4/5), 768–791 (2008)
Fard, M. P., Sagatun, S. I.: Boundary control of a transversely vibrating beam via lyapunov method, Proceedings of 5th IFAC Conference on Manoeuvring and Control of Marine Craft. Aalborg, Denmark, pp. 263–268, (2000a).
Ge, S. S., He, W., B. V. E, How, Choo, Y. S.: Boundary control of a coupled nonlinear flexible marine riser. IEEE Trans. Control Syst. Technol. 18(5): 1080–1091 (2010)
Guo, B.-Z.; Jin, F.-F.: The active disturbance rejection and sliding mode control approach to the stabilization of the Euler–Bernoulli beam equation with boundary input disturbance. Automatica 49(9), 2911–2918 (2013)
He, W.; Zhang, S.: Control design for nonlinear flexible wings of a robotic aircraft. IEEE Trans. Control Syst. Technol 25(1), 351–357 (2017)
He, W.; Zhang, S.; Ge, S.S.: Adaptive boundary control of a nonlinear flexible string system. IEEE Trans. Control Syst. Technol. 22(3), 1088–1093 (2014)
He, W.; Nie, S.X.; Meng, T.T.; et al.: Modeling and vibration control for a moving beam with application in a drilling riser. IEEE Trans. Control Syst. Technol 25(3), 1036–1043 (2017)
Kelleche, A.; Tatar, N.-E.; Khemmoudj, A.: Stability of an axially moving viscoelastic beam. J Dyn. Control Syst. (2016). https://doi.org/10.1007/s10883-016-9317-8
Kelleche, A.; Berkani, A.; Tatar, N..-E.: Uniform stabilization of a nonlinear axially moving string by a boundary control of memory type. J. Dyn. Control Syst. 24, 313–323 (2020)
Liu, Y.; Zhao, Z.J.; He, W.: Stabilization of an axially moving accelerated/ decelerated system via an adaptive boundary control. ISA Trans. 64, 394–404 (2016)
Nayfeh, A., Mook, D.: Nonlinear oscillations. Wiley-Interscience, New York (1979)
Nguyen, T.L.; Do, K.D.: Boundary control of two-dimensional marine risers with bending couplings. J. Sound Vib. 332(16), 3605–3622 (2013)
Park, J.Y.; Kim, J.A.: Existence and uniform decay for Euler–Bernoulli beam equation with memory term. Math. Method. Appl. Sci. 27(14), 1629–1640 (2004). https://doi.org/10.1002/mma.512
Park, J.Y.; Kang, Y.H.; Kim, J.A.: Existence and exponential stability for a Euler–Bernoulli beam equation with memory and boundary out- put feedback control term. Acta. Appl. Math. 104(3), 287–301 (2008). https://doi.org/10.1007/s10440-008-9257-8
Pata, V.: Exponential stability in linear viscoelasticity. Q. Appl. Math. 64(3), 499 (2006)
Seghour, L., Khemmoudj, A., Tatar, N.-E.: Control of a riser through the dynamic of a vessel. Appl. Anal. 1957-1973 (2015)
Seghour, L.; Berkani, A.; Tatar, N.-E.; Saedpanah, F.: Vibration control of a viscoelastic flexible marine riser with vessel dynamics. Math. Model. Anal. 23(3), 433–452 (2018)
Shahruz, S.M.; Narasimha, C.A.: Suppression of vibration in stretched strings by the boundary control. J. Sound Vib. 204(5), 835–840 (1997)
Tatar, N.-E.: On a large class of kernels yielding exponential stability in viscoelasticity. Appl. Math. Comp. 215(6), 2298–2306 (2009)
Tatar, N.-E.: Arbitrary decays in linear viscoelasticity. J. Math. Phys. 52(1), 013502 (2011)
Tatar, N.-E.: A new class of kernels leading to an arbitrary decay in viscoelasticity. Meditter. J. Math. 10, 213–22 (2013)
Acknowledgements
The second author is very grateful to King Fahd University of Petroleum and Minerals for its continuous support.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Benterki, D., Tatar, NE. Stabilization of a nonlinear Euler–Bernoulli beam. Arab. J. Math. 11, 479–496 (2022). https://doi.org/10.1007/s40065-022-00368-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40065-022-00368-y