Parametric Frequency Analysis of Mathieu–Duffing Equation

The classic linear Mathieu equation is one of the archetypical differential equations which has been studied frequently by employing different analytical and numerical methods. The Mathieu equation with cubic nonlinear term, also known as Mathieu–Duffing equation, is one of the many extensions of the classic Mathieu equation. Nonlinear characteristics of such equation have been investigated in many papers. Specifically, the method of multiple scale has been used to demonstrate the pitchfork bifurcation associated with stability change around the first unstable tongue and Lie transform has been used to demonstrate the subharmonic bifurcation for relatively small values of the undamped natural frequency. In these works, the resulting bifurcation diagram is represented in the parameter space of the undamped natural frequency where a constant value is allocated to the parametric frequency. Alternatively, this paper demonstrates how the Poincaré–Lindstedt method can be used to formulate pitchfork bifurcation around the first unstable tongue. Further, it is shown how higher order terms can be included in the perturbation analysis to formulate pitchfork bifurcation around the second tongue, and also subharmonic bifurcations for relatively high values of parametric frequency. This approach enables us to demonstrate the resulting global bifurcation diagram in the parameter space of parametric frequency, which is beneficial in the bifurcation analysis of systems with constant undamped natural frequency, when the frequency of the parametric force can vary. Finally, the analytical approximations are verified by employing the numerical integration along with Poincaré map and phase portraits.

An Innovative Aerial Manipulator with Tandem Ducted Fans: Modeling, Control, and Simulation

This paper proposes an innovative ducted fan aerial manipulator, which is particularly suitable for the tasks in confined environment, where traditional multirotors and helicopters would be inaccessible. The dynamic model of the aerial manipulator is established by comprehensive mechanism and parametric frequency-domain identification. On this basis, a composite controller of the aerial platform is proposed. A basic static robust controller is designed via H-infinity synthesis to achieve basic performance, and an adaptive auxiliary loop is designed to estimate and compensate for the effect acting on the vehicle from the manipulator. The computer simulation analyses show good stability of the aerial vehicle under the manipulator motion and good tracking performance of the manipulator end effector, which verify the feasibility of the proposed aerial manipulator design and the effectiveness of the proposed controller, indicating that the system can meet the requirements of high precision operation tasks well.