Nonlinear Dynamics

, Volume 84, Issue 2, pp 977–990 | Cite as

Effect compensation of the presence of a time-dependent interior delay on the stabilization of the rotating disk-beam system

Original Paper


In this article, we investigate the compensation problem of the effect of the interior time-dependent delay on the stabilization of a rotating disk-beam system. The physical system consists of a flexible beam free at one end, and attached to the center of the rotating disk whose angular velocity is time-varying. Assuming that a time-dependent interior delay is present in the system, we introduce a dynamic boundary force control at the free end of the beam and a torque control on the disk. Then, we show the destabilizing effect of the delay is compensated. Specifically, it is shown that the presence of such proposed controls assures the exponential stability of the system, provided that some reasonable conditions on the angular velocity of the disk and delay are fulfilled. Numerical examples in the case of constant delay are also provided to highlight the stability result.


Rotating disk-beam Interior control Time-dependent delay Dissipative boundary force control Torque control Stability 

List of symbols

\(\ell \)

Length of the beam

\(\rho \)

Mass per unit length of the beam


Flexural rigidity of the beam

\(I_\mathrm{d} \)

Disk’s moment of inertia


Beam’s displacement at time t with respect to the spatial variable x

\(\omega (t)\)

Angular velocity of the disk at time t

\(\mathcal{U}_\mathrm{I} (t)\)

Interior control

\(\mathcal{U}_\mathrm{F} (t)\)

Boundary force control


Torque control

\(\alpha \ge 0\)

Feedback gain of the interior control

\(\beta >0\)

Force control feedback gain

\(\gamma >0\)

Torque control feedback gain



The author would like to thank the Associate Editor and the anonymous referees for their constructive comments and valuable suggestions which have led to an improved version of this paper. Also, the author is grateful to one of the referees for having mentioned the recent articles [14, 15, 16, 17, 18]. Finally, the author acknowledges the support of Sultan Qaboos University.


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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsSultan Qaboos UniversityAl-Khod, MuscatOman

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