The Responses of Rising or Falling Spherical Wind Sensors to Atmospheric Wind Perturbations

Abstract

The responses of rising or falling spherical wind sensors to atmospheric wind perturbations are analyzed using Fourier transform techniques. The linearized equations of motion of a sensor that is subject to drag and gravitational body forces are developed by perturbing a sensor about an equilibrium uniform motion with wind fluctuations which have vertical variations. The wind environment and sensor velocities are decomposed with stochastic Fourier-Stieltjes integrals, and the linearized equations of motion are used to derive the response functions and phase angles of the sensor motions. The results of the analysis are used to analyze the response properties of the Jimsphere balloon wind sensor. The transfer functions and phase angles are functions of the wind perturbation frequency ?(=?|w|, ? and w? being the wavenumber of the perturbation and the mean rise or fall rate of the sensor) and two parameters T and α which are functions of sensor mass, apparent mass of the sensor, the mass of air displaced by the sensor, the mean ascent or descent rate of the sensor as the case may be, and the acceleration of gravity. The quantity T is a time constant of the system and α is the ratio of the apparent mass to the mass of the system. Once these parameters are specified, the response properties of the sensor are completely specified in a linear context. In general, the system becomes more responsive as T and α approach zero and unity, respectively. As ?T ? 0, the transfer functions and phase angles approach unity and zero, respectively; so that at sufficiently low frequencies, the sensor essentially measures 100% of the wind perturbation Fourier amplitudes with no phase shifts. As ?T ? ∞, the transfer functions and phase angles approach α2 and zero, respectively; so that at sufficiently large frequencies, the sensor is capable of detecting 100a% of the wind perturbation Fourier amplitudes, again with no phase shifts. If apparent mass effects were not present (α = 0), the sensor transfer functions would approach zero as ?T?∞. Thus, the apparent mass effects make the sensor more responsive. At ?T = α?½ for the horizontal fluctuations and ?T=2α?½ for the vertical velocity fluctuations, the phase angles of the sensor Fourier amplitudes take on minimum values. In general, the transfer functions associated with the horizontal sensor motions are smaller than the transfer functions associated with the vertical sensor motions in the domain 0 ?T < ∞; thus, the sensor is more responsive to vertical than to horizontal air motions.