Boundary Stabilization of Euler-Bernoulli and Rayleigh Beam Vibrations

In this Demonstration, we apply the finite-difference method to the stabilization of the vibrations of the Rayleigh beam equation, modeled as the bending profile of beams. The beam is hinged both on the left and on the right. The Rayleigh beam equation retains the effect of rotational inertia of the cross-sectional area if . This provides some improvement to the Euler–Bernoulli beam model, with .
Consider two types of damping to suppress vibrations: a distributed (weak) viscous damping and the boundary damping injected through the right end . The solutions are filtered by adding a viscosity term (structural damping type) to the Rayleigh beam equation.
The relevant partial differential equation (PDE) is:
where represents the beam deflection at any point and time. The moment of the tip of the beam is controlled by the velocity of the shear angle of the tip . In other words, the rate of change of the shear angle in time is measured and fed back to the system. This makes the given PDE a closed-loop control system.
The discretized version using the filtered semi-discretized finite differences is:
where is the number of discrete nodes in the discretization of the interval ,
,
,
and
.
The following values can be set with the controls:
• type of beam: Rayleigh or Euler–Bernoulli
• filtering: whether or not the system has a filtering term
: number of nodes
and : non-negative controller gains
: the controller for filtering in the numerical scheme
and : frequency of initial normal-mode displacement and velocity
For simplicity, we take .

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In infinite-dimensional control theory, vibrational dynamics are mostly governed by PDEs. To demonstrate the corresponding complex dynamics, model reductions are unavoidable, as they rely heavily on the finite element or finite difference-based approximations. For control-free dynamics, these approximations yield reliable and robust results. However, in controlled systems, where the controllers change the dynamics through the boundary of the region, known approximations fail to provide reliable results [1].
Consider the boundary-controlled one-dimensional PDE for a hinged vibrating beam of length 1:
,
,
,
,
,
where is the deflection of the beam, and is the material constant for the moment of inertia term of the cross-sectional area. If , this reduces to the Rayleigh beam model; if , it becomes the so-called Euler–Bernoulli beam model. Considering the boundary control term as a state feedback controller, the given model is known to have exponentially stable solutions, whereas its approximations by the known numerical approaches, including the finite difference method, finite element method and the Galerkin method, fail to mimic this behavior.
Applying the finite difference method on the space variable , let the mesh parameter be , where is the number of nodes in the discretization:
.
We introduce two fictitious nodes and to approximate the boundary conditions and by the central differences. The approximated solution to at each node is represented by . Therefore, the semi-discretize finite difference–approximated model is:
,
,
,
,
,
.
This problem is extensively treated in [4]. The PDE model has infinitely many eigenvalues; the semi-discrete finite difference–approximated model has only a finite number. Infinitely many eigenvalues of the PDE model satisfy a uniform gap condition, whereas, for a finite number, eigenvalues of the semi-discrete finite difference–approximated model do not. In fact, for high-frequency eigenvalues, the gap between two consecutive eigenvalues approaches zero as the mesh parameter tends to zero:
as or .
This is particularly bad for the boundary observability of the high-frequency eigenfunctions (signals) of the corresponding control-free system, since unobservable systems are also uncontrollable [2]. This is simply because the observer cannot distinguish one signal from another.
To remedy this, an indirect filtering technique was first proposed in [5] for the Euler–Bernoulli equation. A viscosity term
is added to the discretized equation above to obtain:
,
,
,
,
,
.
This convergence of the solutions of the filtered model to those of the unfiltered model is shown as the mesh parameter . The gap condition holds true, and the uniform observability of the filtered model is also shown as the mesh parameter [4, 5]. Unlike the technique considered in this project, in [4], a direct filtering technique is used instead. More importantly, this approach is recently applied in a demonstration project for a boundary-controlled wave equation [6].
References
[1] H. T. Banks, K. Ito and C. Wang, "Exponentially Stable Approximations of Weakly Damped Wave Equations," Estimation and Control of Distributed Parameter Systems (W. Desch, F. Kappel and K. Kunisch, eds.), Basel: Birkhäuser, 1991 pp. 1–33. doi:10.1007/978-3-0348-6418-3_1.
[2] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Basel: Birkhauser, 2009.
[3] L. T. Tebou and E. Zuazua, "Uniform Boundary Stabilization of the Finite Difference Space Discretization of the 1-D Wave Equation," Advances in Computational Mathematics, 26(1), 2007 pp. 337–365. doi:10.1007/s10444-004-7629-9.
[4] A. Ö. Özer, "Uniform Boundary Observability of Semi-discrete Finite Difference Approximations of a Rayleigh Beam Equation with Only One Boundary Observation," in 2019 IEEE 58th Conference on Decision and Control (CDC), Nice, France, 2019, IEEE, 2019 pp. 7708–7713. doi:10.1109/CDC40024.2019.9028954.
[5] E. Zuazua and L. Leon, "Boundary Controllability of the Finite-Difference Space Semi-discretizations of the Beam Equation," ESAIM: Control, Optimisation and Calculus of Variations, 8, 2002 pp. 827–862. doi:10.1051/cocv:2002025.
[6] D. Price, E. Moore and A. Ö. Özer. "Boundary Control of a 1D Wave Equation by the Filtered Finite-Difference Method" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/BoundaryControlOfA1DWaveEquationByTheFilteredFiniteDifferenc.
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