Citation: Casoli,P.; Vescovini, C.M.; Scolari, F.; Rundo, M. Theoretical Analysis of Active Flow Ripple Control in Positive Displacement Pumps. Energies 2022, 15, 4703. https://doi.org/10.3390/ en15134703 Academic Editor: Kamel Hooman Received: 9 June 2022 Accepted: 24 June 2022 Published: 27 June 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). energies Article Theoretical Analysis of Active Flow Ripple Control in Positive Displacement Pumps Paolo Casoli 1, * , Carlo Maria Vescovini 1 , Fabio Scolari 1 and Massimo Rundo 2 1 Department of Engineering and Architecture, University of Parma, 43124 Parma, Italy; carlomaria.vescovini@unipr.it (C.M.V.); fabio.scolari@unipr.it (F.S.) 2 Department of Energy, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129 Turin, Italy; massimo.rundo@polito.it * Correspondence: paolo.casoli@unipr.it Abstract: Positive displacement machines present a well-known major drawback that is the oscillation in delivered flow rate. This paper presents two active solutions for reducing the flow ripple generated by a pump with an external device actuated by means of a piezo-stack actuator. The work is focused on a theoretical analysis, with the aim of collecting information about the performance of the solutions proposed and their main advantages and drawbacks. The active methods proposed involve a cylindrical actuator connected to the delivery line of the pump. The piston could be actuated directly by a piezo-stack actuator or by a differential pressure modulated by a proportional piezo actuated valve. The actuators were modelled and a control algorithm based on Least Mean Square algorithm was used to achieve the adaptability for both systems at different operating conditions. The developed mathematical model permits to define the great potential of these solutions that can drastically reduce the flow ripple. The first architecture presented resulted as the best solution, while the second one allowed reduction of the production cost. Keywords: positive displacement pump; active control; flow ripple; pressure ripple 1. Introduction Positive displacement machines are commonly utilized in hydraulic circuits since they provide high power density together with robustness and reliability. However, one of the major drawbacks is the oscillation of the delivered flow rate, which in turn generates a pressure ripple and finally produces noise and vibrations that can cause stress to the components along the hydraulic circuit. This phenomenon can negatively affect durability and functionality of the hydraulic components [1]; furthermore, vibrations can affect the human being and environment [2]. Researchers dealing with the reduction in the flow and pressure ripple generated by positive displacement pumps have focused on different solutions, many of which contemplate the geometric optimization of components. In axial piston pumps, for example, a careful design of the port plate allows a smooth transition of the fluid pressure between suction and delivery phase and vice versa [3–5]. In gear pumps instead, side bushings are designed with particular grooves to improve the machine performance from this point of view [6–11]. Other methods involve devices external to the pump and could be classified as passive or active methods. Passive methods contemplate fitting external devices, such as resonators, outside the pump. One major drawback of passive devices connected on a pump delivery line is that they usually are designed and set to operate in particular conditions, being effective only at certain pressures or against particular frequencies [12–14]. This means, for example, that if the pump shaft speed changes, these systems lose their effectiveness, with the pressure ripple frequency being determined directly by the pump rotational speed. In [13], a passive Energies 2022, 15, 4703. https://doi.org/10.3390/en15134703 https://www.mdpi.com/journal/energies