PARAMETRIC CONTROL OF RADIUS GRADIENT IN GAUSSIAN-MODIFIED CYCLOIDAL PROFILES

Abstract

The article is devoted to solving the urgent problem of geometric modeling of high-performance nose cones for strike systems and aerospace objects. The object of research is the mathematical process of forming modified cycloidal curves, in which the concept of parametric control of the radius gradient is implemented by integrating Gaussian distribution functions directly into the structure of the analytical description of the base generatrix. In this work, the mathematical apparatus of geometric modeling has been improved by introducing special normalization functions and scaling coefficients. The proposed approach ensures the constancy of the total radius difference regardless of variations in smoothness parameters. Primary attention is paid to the numerical study of the standard deviation parameter σ as a key factor determining the intensity and localization of the radius change. The research methodology is based on a comprehensive analysis of the differential characteristics of the synthesized surfaces. By constructing and visualizing 3D surfaces of the logarithmic curvature log|k (t)| and its gradient in the coordinates of «modification parameter – generatrix characteristic», physically grounded critical limits for the σ values have been established. During the numerical experiment, it was revealed that an excessive concentration of the radius gradient, observed at values of σ < 0.27, leads to the emergence of mathematical anomalies and a violation of the curvature monotonicity. From an aerodynamic perspective, such geometric effects are critical, as they initiate premature boundary layer separation and the transition from laminar to turbulent flow. It has been established that the range
σ ∈ [0.27, 0.95] is the most valid for the analytical synthesis of so-called «laminar» profiles. Such profiles are characterized by a stable distribution of geometric parameters and asymptotic dynamics of curvature change, which is confirmed by the absence of sharp peaks on the curvature gradient surface log|dk/dt|. The practical significance of the obtained results lies in the possibility of their implementation in modern computer-aided design systems for the precision synthesis of aerodynamic surfaces with predefined flow properties. The identified optimal parameters allow the designer to flexibly «dose» the sharpness of the nose cone while maintaining high geometric smoothness in the transition zones to the cylindrical body of the vehicle.

Keywords: geometric modeling, cycloidal profile, Gaussian function, radius gradient, surface curvature, aerodynamic nose cone, standard deviation parameter, surface synthesis.