## 1 Introduction
[12], they suggested that a bi-rotor pump with indexed rotors had the potential to attenuate the flow ripple. [PERSON] et al. [14,15] conducted a comparison between the lumped parameter and the CFD approach for a unique insight into the pump's performance (volumetric efficiency, flow and pressure ripples, etc. ), with a specific focus on the rotors' radial micro-motions. With respect to the crescent pumps, [PERSON] [16] addressed the mathematical expressions for the ideal delivery from a kinematic aspect, focusing on the variations of the trapped volume. [PERSON] et al. [17] studied the theoretical flow characteristics based on the theory of gearing, and presented a set of conjugated involute gears that enabled better fluid delivery capacity. Additionally, the same research group in Reference [18] extended the study to the trapped volume performances via a discretization approach, with a specific focus on the gears' geometric parameters. [PERSON] [19] investigated the theoretical flowrate of the crescent pumps with respect to different combinations of the gear pair's tooth numbers (the gear shaft and the ring gear), suggesting that the increase of the gear shaft's tooth number helped attenuate the flow ripple. Moreover, the same research group in Reference [20] extended their research to cover the internal leakage and the outlet pressure ripple via a lumped parameter model built in LMS Amesim, validating the model through a comparison to experimental results on the steady-state flowrate and the outlet pressure ripples. Hence, it can be seen that few attempts have been made to study the flow ripple with respect to the tandem crescent pump design. Concerning the tandem pump design, it has been previously applied in other hydraulic pumps, particularly in axial piston pumps and in external gear pumps. [PERSON] et al. [21,22] addressed the theoretical torque ripple and flow ripple of a tandem axial piston pump with two identical rotating groups under different index angles from a kinematic aspect, suggesting that an index angle of 10\({}^{\circ}\) resulted in the greatest reduction of the flow ripple (roughly 75%) with respect to a nine-piston rotating group. [PERSON] et al. [23] studied the tandem piston pump's flow ripple via a lumped parameter model using LMS Amesim, suggesting that an index angle of 20\({}^{\circ}\) resulted in the greatest reduction of the flow ripple. In [PERSON] et al. 's work, they suggest that the reason for the difference in index angle from that in [PERSON] et al. 's work was due to the flow through the triangular grooves on the valve plate, which has been shown to have a great influence on the flow ripple (which was not considered in the work of [PERSON] et al.). [PERSON] et al. [24] presented a tandem external gear pump comprised of a set of spur gears and a set of helical gears, sharing the same driving and driven shafts. However, the presented tandem external gear pump did not share the same outlet, thus the flow produced by the two gear pairs was delivered to two different hydraulic systems, which is different to the aforementioned tandem axial piston pump and the tandem crescent pump presented in this work. Hence, few published works can be found on tandem crescent pump design. This paper focused on the flow ripple of the tandem crescent pump via a lumped parameter model built in Matlab. It was expected that the tandem crescent pump could lead to a decrease in the flow ripple through the proper selection of the design parameters in terms of the index angle and the displacement ratio.