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On the Calculation of Differential Parametrizations for the Feedforward Control of an Euler–Bernoulli Beam

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Dynamics and Control of Advanced Structures and Machines

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 156))

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Abstract

We are concerned with the motion planning for underactuated Euler–Bernoulli beams. The design of the feedforward control is based on a differential parametrization of the beam, where all system variables are expressed in terms of a free time function and its infinitely many derivatives. We derive an advantageous representation of the set of all formal differential parametrizations of the beam. Based on this representation, we identify a well-known parametrization, for the first time without the use of operational calculus. This parametrization is a flat one, as the corresponding series representations of the system variables converge. Furthermore, we discuss a formal differential parametrization where the free time function allows a physical interpretation as the bending moment at the unactuated boundary. Even though the corresponding series do not converge, a numerical simulation using the least term summation illustrates the usefulness of this formal differential parametrization for motion planning.

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Notes

  1. 1.

    The general constant-coefficient case

    $$ \mu \partial _{t}^{2}w(z,t)=-EI\partial _{z}^{4}w(z,t)\,,\quad 0\le z\le L\,,\,t\ge 0 $$

    with linear mass density \(\mu >0\), flexural rigidity \(EI>0\), and a spatial domain [0, L] can always be traced back to the normalized case (11.1) by transformations of the independent variables z and t.

  2. 2.

    The series (11.2a) and (11.2b) are formal solutions if they satisfy (11.1a) and (11.1b) after formally interchanging differentiation and summation, even if they do not converge.

  3. 3.

    In contrast to our double sum representation (11.28a), in [11], the parametrization of the deflection w(zt) is expressed by a single sum of real and imaginary parts of powers of complex numbers. Also, the input used in [11] has the opposite sign as compared to our input u(t).

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Acknowledgements

This work has been supported by the Austrian Science Fund (FWF) under Grant number P 29964-N32 and the Pro\(^2\)Future competence center in the framework of the Austrian COMET-K1 program under contract no. 854184.

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Correspondence to Markus Schöberl .

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Kolar, B., Gehring, N., Schöberl, M. (2022). On the Calculation of Differential Parametrizations for the Feedforward Control of an Euler–Bernoulli Beam. In: Irschik, H., Krommer, M., Matveenko, V.P., Belyaev, A.K. (eds) Dynamics and Control of Advanced Structures and Machines. Advanced Structured Materials, vol 156. Springer, Cham. https://doi.org/10.1007/978-3-030-79325-8_11

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  • DOI: https://doi.org/10.1007/978-3-030-79325-8_11

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  • Online ISBN: 978-3-030-79325-8

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