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Sliding mode vibration control of an Euler–Bernoulli beam with unknown external disturbances

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Abstract

This paper addresses the stabilization problem of an Euler–Bernoulli beam system subject to an unknown time-varying distributed load and boundary disturbance. Based on Lagrangian–Hamiltonian mechanics, the model of the beam system is derived as a partial differential equation. Based on Lyapunov functions, a sliding surface is designed, on which the system exhibits exponential bounded stability and robustness against the external disturbances. A sliding mode controller which only uses boundary information is further proposed to drive the system to reach the sliding surface in finite time. Numerical simulations are shown to illustrate the validity of the proposed boundary control.

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Data availability

All data included in this study are available upon request by contact with the corresponding author.

Abbreviations

L :

Length of the beam

m :

Mass of the payload

\(E_{\text {I}}\) :

Bending stiffness of the beam

T :

Tension of the beam

\(\rho \) :

Uniform mass per unit length of the beam

w(x, t):

Displacement of the beam at the position x for the time t

f(x, t):

Time-varying distributed load on the beam except end point

u(t):

Boundary control force at the end of the beam

d(t):

Boundary input disturbance force at the end of the beam

\(C^1\) :

Continuously differentiable function space

\(R^+\) :

The sets of positive real numbers

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Authors and Affiliations

  1. School of IoT Engineering, Jiangnan University, Wuxi, 214122, China

    Zhan Wang, Wei Wu & Xuyang Lou

  2. German Research Center for Artificial Intelligence (DFKI), 67663, Kaiserslautern, Germany

    Daniel Görges

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  1. Zhan Wang
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  2. Wei Wu
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  3. Daniel Görges
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  4. Xuyang Lou
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Correspondence to Wei Wu.

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Wang, Z., Wu, W., Görges, D. et al. Sliding mode vibration control of an Euler–Bernoulli beam with unknown external disturbances. Nonlinear Dyn 110, 1393–1404 (2022). https://doi.org/10.1007/s11071-021-06921-2

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  • DOI: https://doi.org/10.1007/s11071-021-06921-2

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