Response optimization of underactuated vibration generators through dynamic structural modification and shaping of the e
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ORIGINAL ARTICLE
Response optimization of underactuated vibration generators through dynamic structural modification and shaping of the excitation forces Roberto Belotti 1 & Dario Richiedei 2
&
Iacopo Tamellin 2 & Alberto Trevisani 2
Received: 6 April 2020 / Accepted: 14 September 2020 # The Author(s) 2020
Abstract Resonant vibration generators, such as vibratory feeders or ultrasonic sonotrodes, are often employed in manufacturing to generate harmonic vibrations with suitable amplitude, spatial shape, and frequency, in order to meet the process requirements. These underactuated systems are usually excited in open loop by few actuators, and therefore, it is not ensured that the desired response is correctly achieved, since the feasible motions should belong to the subset of the allowable motions. To achieve the closest approximation of the desired vibrations, some new solutions are here proposed. The first strategy is the optimal shaping of the harmonic forces exerted by the actuators, by solving an inverse dynamic problem through a coordinate transformation and the projection of the desired response onto the subspace of the allowable motion. By exploiting the formulation of such a subspace, a second approach that involves concurrently both the force shaping and the modification of the inertial and elastic system parameters is proposed. The idea of this approach is to exploit the modification of the elastic and inertial parameters to properly shape the allowable subspace in such a way that it spans the desired response. A solution method is developed, and analytical sensitivity analysis is proposed to choose the design variables. Validation is proposed through a linear vibratory feeder with a long flexible tray, taken from the literature. The results show the effectiveness of the proposed strategies that lead to a very precise approximation of the desired response. Keywords Vibration generators . Vibratory feeders . Underactuated systems . Feedforward control . Dynamic structural modification . Concurrent design
Nomenclature bn B
BA Approximation of the nth bilinear term Force distribution matrix
* Dario Richiedei [email protected]
D EJ f, f0 f 0i , f 0M
Roberto Belotti [email protected]
fc
Iacopo Tamellin [email protected]
Fext
Alberto Trevisani [email protected]
GAA, GAU, GUU
1
Faculty of Science and Technology, University of Bozen-Bolzano, Piazza Università 5, 39100 Bolzano, Italy
2
Department of Management and Engineering, University of Padova, Stradella S. Nicola 3, 36100 Vicenza, Italy
GAA , GAU , GUU
I
Upper-triangular matrix obtained from the QR decomposition of B Auxiliary matrix for the sensitivity analysis Flexural stiffness of the tray Generalized force vector and force amplitude vector ith excitation force amplitude and maximum excitation force amplitude Convex approximation of the nonconvex function to optimize Generalized forces for the Lagrange equation of motion Inverse of the receptance matrix partitions in the transformed system Inverse of t
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