This technical note proposes a clapping vibration energy harvesting system (CVEH system) installed in a rotating system. This device includes a rotating wheel, a drive shaft that rotates the wheel, and a double elastic steel sheet fixed on the drive shaft. One of the free ends of the steel is fixed with a magnet, and the free end of the other elastic steel is fixed with a PZT patch. We also install an array of magnets on the periphery (rim) of the wheel. The rim magnets repulse the magnet on the elastic steel sheet of the transmission shaft, causing the elastic steel to oscillate periodically, and slap the piezoelectric patch installed on the other elastic steel sheet to generate electricity. In this study, the authors’ previous study on the voltage output was improved, and the accurate nonlinear natural frequency of the elastic steel was obtained by the dimensional analysis method. By adjusting the rotation speed of the wheel, the precise frequency was controlled to accurately excite the energy harvesting system and obtain the best output voltage. A simple experiment was also performed to correlate with the theoretical model. The voltage and power output efficiencies of the nonlinear frequency to linear frequency excitation of the CVEH system can reach 15.7% and 33.5%, respectively. This study confirms that the clapping VEH system has practical power generation benefits, and verifies that nonlinear frequencies are more effective than linear frequencies to excite the CVEH system to generate electricity.

The problems of steady, inviscid, isentropic, irrotational supersonic plane flow passing a body with a small thickness ratio was solved by the linearized theory, which is a first approximation at and near the surface but fails at far fields from the body. Such a problem with far fields was solved by W.D. Hayes’ “pseudo-transonic” nonlinear theory in 1954. This far field small disturbance theory is reexamined in this study first by using asymptotic expansion theory. A systematic approach is adopted to obtain the nonlinear Burgers’ equation for supersonic far fields. We also use the similarity method to solve this boundary value problem (BVP) of the inviscid Burgers’ equation and obtain the nonlinear flow patterns, including the jump condition for the shock wave. Secondly, the transonic and hypersonic far field equations were obtained from the supersonic Burgers’ equation by stretching the coordinate in the y direction and considering an expansion of the freestream Mach number in terms of the transonic and hypersonic similarity parameters. The mathematical structures of the far fields of the supersonic, transonic, and hypersonic flows are unified to be the same. The similar far field flow patterns including the shock positions of a parabolic airfoil for the supersonic, transonic, and hypersonic flow regimes are exemplified and discussed.

The infinite series solution to the boundary-value problems of Laplace’s equation with discontinuous Dirichlet boundary conditions was found by using the basic method of separation of variables. The merit of this paper is that the closed-form solution, or the singular similarity solution in the semi-infinite strip domain and the first-quadrant domain, can be generated from the basic infinite series solution in the rectangular domain. Moreover, based on the superposition principle, the infinite series solution in the rectangular domain can be related to the singular similarity solution in the semi-infinite strip domain. It is proven that the analytical source-type singular behavior in the infinite series solution near certain singular points in the rectangular domain can be revealed from the singular similarity solution in the semi-infinite strip domain. By extending the boundary of the rectangular domain, the infinite series solution to Laplace’s equation in the first-quadrant domain can be derived to obtain the analytical singular similarity solution in a direct and much easier way than by using the methods of Fourier transform, images, and conformal mapping.