Search Results (3)

This study is about the dual spacelike curves lying on the dual lightlike cone, which can be either symmetric or asymmetric. We first establish the dual associated curve, which is related to the reference curve. Using these curves and the derivative of the reference curve, we derive the dual asymptotic orthonormal frame. Next, we define the dual structure function, curvature function, and Frenet formulae, and express the curvature function in terms of the dual structure function. This leads to a differential equation that characterizes the dual cone curve in relation to its curvature function. Since curves with constant curvature maintain the same curvature at every point, their geometry is more predictable. Therefore, we assume that the dual cone curvature function is constant and examine how this condition affects the behavior and geometric properties of the dual curves. As a result of this investigation, some new results and definitions are obtained. Full article

Simultaneous occurrence of Dirac-like cones at the center of the Brillouin zone (Γ) at two different energy states is termed Dual-Dirac-like cones (DDC) in this article. The occurrence of DDC is a rare phenomenon. Thus, the generation of multiple Dirac-like cones at the center of the Brillouin zone is usually non-manipulative and poses a challenge to achieve through traditional accidental degeneracy. However, if predictively created, DDC will have multiple engineering applications with acoustics and vibration. Thus, the possibilities of creating DDC have been identified herein using a simple square periodic array of tunable square phononic crystals (PnCs) in air media. It was found that antisymmetric deaf bands may play critical roles in tracking the DDC. Hence, pivoting on the deaf bands at two different energy states, an optimized tuning parameter was found to achieve Dirac-like cones at two distinct frequency states, simultaneously. Orthogonal wave transport identified as key Dirac phenomena was achieved at two frequencies, herein. It was identified that beyond the Dirac-like cone, the Dirac phenomena remain dominant when a doubly degenerated state created by a top band with positive curvature and a near-flat deaf band are lifted from a bottom band with negative curvature. Utilizing a mechanism of rotating the PnCs near a fixed deaf band, frequencies are tracked to form the DDC, and orthogonal wave transport is demonstrated. Exploiting the dispersion behavior, unique acoustic phenomena, such as ballistic wave transmission, pseudo diffusion and acoustic cloaking are also demonstrated at the Dirac frequencies using numerical simulation. The proposed tunable acoustic PnCs will have important applications in acoustic and ultrasonic imaging, waveguiding and even acoustic computing. Full article

Profile measurement is a key technical enabler in the manufacturing of highly curved freeform surfaces due to their complex geometrical shape. A current optical probe was used to measure nearly rotary freeform surfaces with the help of one rotation axis, because the probe needs to measure along the normal vector of the surface under the limitation of the numerical aperture (NA). This kind of measuring system generally has a high cost due to the high-precision, multi-axis platform. In this paper, we propose a low-cost, dual-axis rotation scanning method for a highly curved freeform surface with an arbitrary shape. The optical probe can scan the surface profile while always keeping consistent with the normal vector of the measuring points with the help of the double rotation axis. This method can adapt to the changes in curvature in any direction for a highly curved freeform surface. In addition, the proposed method provides a system error calibration technique for the rotation axis errors. This technique can be used to avoid the dependence of the measuring system on the high-precision platform. The three key system errors that affect the measurement accuracy such as installation error of the B-axis, A-axis, and XZ perpendicularity error are first analyzed through establishing an error model. Then, the real error values are obtained by the optimal calculation in the calibration process. Finally, the feasibility of the measurement method is verified by measuring one cone mirror and an F-theta mirror and comparing the results to those obtained using commercial equipment. The maximum measurable angle of the system is ±90°, the maximum measurable diameter is 100 mm, and the measurement accuracy of the system reaches the micron level in this paper. Full article