On the Controllability of a Slowly Rotating Timoshenko Beam

Abstract

We consider a slowly rotating Timoshenko beam in a horizontal plane whose movement is controlled by the angular acceleration of the disk of a driving motor into which the beam is clamped. The problem to be solved is to transfer the beam from a position of rest into a position of rest under a given angle within a given time. We show that this problem is solvable, if the time of rotation prescribed is large enough.

Solutions Stability of Initial Boundary Problem, Modeling of Dynamics of Some Discrete Continuum Mechanical System

“…Исследованию динами-ки механических систем, содержащих твердые тела и упругие стержни, посвящено большое количество работ. Упругие стержни рассматриваются как в рамках модели Эйлера-Бернулли [6][7][8][9], так и в рамках модели балки Тимошенко [10][11][12][13][14][15][16]. Изучают-ся задачи стабилизации и управления поведением таких механических систем.…”

Controllability of a slowly rotating Timoshenko beam

“…In [8], the case γ = 1 and L = 1 is considered and controllability from rest to rest is proved. The initial state of the beam is described by the conditions…”

“…The control is performed by the angular acceleration of the axis, to which the beam is clamped. In [8], Krabs and Sklyar have shown controllability from a position of rest to a position of rest for the Timoshenko beam for a special parameter value (namely γ = 1).…”

Construction of an Optimal Rest-to-Rest Control of a Rotating Beam

“…Note that entries of Cramer matrix of this system depend on the first N eigenvalues λ n , which also have to be found numerically, but using the analytic formulas from [2] that can be done with any required level of precision. Thus we reduced the infinite dimensional problem of optimal control (1)- (5) to solving a system of a finite number of linear equations.…”

“…A number of works concerning various controllability problems for the model were published. In particular, Krabs and Sklyar [2] analyzed the appropriate non-Fourier trigonometric moment problem in 1999, inspired by Russel (e.g., [3]). They showed that the system is rest-to-rest controllable, under some conditions on the physical properties of the materials of the beam, if the time of steering is strictly greater than the certain critical Communicated by Günther Leugering.…”