## 4 Conclusions  
An investigation is conducted into the effect of variant flapping frequency during a flapping cycle on the aerodynamic force generation of a flapping wing rotor driven by an ultrasonic motor (USR30) through experiment and CFD simulation. In the experiment, a couple of optical sensors and a laser velocimeter are used to measure the FWR motion including flapping frequencies in up-stroke \(f_{u}\) and down-stroke \(f_{d}\) and rotational speed \(\dot{\psi}\). The lift force generated by the FWR is obtained from the difference of the total force and inertia force measured by a load cell. A CFD model is created for the FWR with the spar beam twist but without the wing skin deformation taken into account. The CFD model is validated before used for FWR aerodynamic simulation. As described in Sections 2.2 and 2.4, it is possible to vary the flapping frequency in up-stroke and down-stroke separately by employing the USM. The frequency and corresponding rotational speed of the FWR were measured by using laser velocimeter as described in Section 2.2. In this study, the flapping frequency was altered from 1 Hz to 3 Hz to set a combination of flapping motion cases including the constant flapping frequency mode in case 1, 2 and 3 and variant flapping frequency mode in case 4 and 5 as listed in Table 2. According to Table 2, it can be found that higher average flapping frequency usually lead to higher rotational speed \(\dot{\psi}\). However, given a specified power output, the lift generated by the FWR in VFF cases is significantly larger than the usual CFF cases. This is due to the reduced negative lift of the FWR during the slower up-stroke motion in the VFF cases. In the VFF case 4 (3 & 1 Hz), the lift is more than doubled comparing with that the CFF case having equivalent flapping frequency (1.3 Hz) and power. Due to slower up-stroke motion, a positive inertia force about 10% of the average lift is produced by the FWR in a flapping cycle although it does not contribute to lifting the model during flight. The CFD simulation reveals the aerodynamic mechanism and provides insight of the evolution and different flow characteristics of the LEV and TEV in the CFF and VFF cases. From the flow pressure and velocity results, it is found that the LEV and TEV growth on the FWR upper surface is much slower during up-stroke (takes over 60% of flapping period) in the VFF case 4 (3 & 1 Hz) hence a smaller negative lift (about 10-15% peak positive lift) than the CFF case 1 (3 Hz). Therefore, a kinematics of motion with an optimal VFF is superior comparing with the usual CFF subject to the same power. The installation angle \(\theta_{0}\) of the wings is a key factor that influences the FWR performance and lift. From the parametric study for a range of installation angles \(\theta_{0}\) = 15\({}^{\circ}\)-60\({}^{\circ}\), it is found that the larger the installation angle, the greater the lift force at a rotational speed. In practice however, a large installation angle results in a large drag and reduced rotational speed. When \(\theta_{0}\) \(\geq\) 45\({}^{\circ}\), the rotational speed is decreased to nearly zero. Without rotary motion, the FWR works like a conventional flapping wing with the lift reduced dramatically as shown in Figure 18. For this FWR model, an optimal \(\theta_{0}\) is found  
Figure 19: The lift efficiency and \(S_{t}\) number for the cases of installation angles 15\({}^{\circ}\)–30\({}^{\circ}\). around 30\({}^{\circ}\) for both CFF and VFF cases while the VFF normally results in a higher power efficiency and propulsion efficiency. With the optimal installation angle \(\theta_{0}=30^{\circ}\), the Strouhal number for the VFF cases is in the range of high propulsion efficiency 0.3-0.36 and higher \(P_{f}\) values than the CFF cases of the same power and equivalent flapping frequency. The ultrasonic motor provides a feasible solution for the FWR to obtain VFF and generate higher lift under a specified input power. Conceptualization, S.C. and L.W. ; methodology, S.C. and L.W.