A tutorial on a multi-mode identification procedure based on the complex-curve fitting method

Modal identification is a key step in modal analysis. It enables the researcher to extract modal parameters, such as natural frequency, amplitude, and damping from a given structure. There are a considerable number of techniques in the state of the art aiming to address this problem, where multi-mode approaches arise as an appealing choice due to their ability to deal with mode coupling. This tutorial paper focuses on the complex-curve fitting technique, originally conceived for an application distinct from modal analysis. It aims at guiding other researchers by providing a tutorial-like and in-depth analysis of this important method, associated with a nonlinear weighting procedure for improved precision. Additionally, this paper fills a gap on the original technique, which is limited to the ratio of two polynomials, by proposing an automatic parameter extraction technique. The original and improved methods are applied on both simulated and experimental data, highlighting the effectiveness of the proposed changes. The proposed procedure is also compared with the rational fraction polynomial method.

First Results on RR Lyrae Stars with the TESS Space Telescope: Untangling the Connections between Mode Content, Colors, and Distances

The Transiting Exoplanet Survey Satellite (TESS) space telescope is collecting continuous, high-precision optical photometry of stars throughout the sky, including thousands of RR Lyrae stars. In this paper, we present results for an initial sample of 118 nearby RR Lyrae stars observed in TESS Sectors 1 and 2. We use differential image photometry to generate light curves and analyze their mode content and modulation properties. We combine accurate light-curve parameters from TESS with parallax and color information from the Gaia mission to create a comprehensive classification scheme. We build a clean sample, preserving RR Lyrae stars with unusual light-curve shapes, while separating other types of pulsating stars. We find that a large fraction of RR Lyrae stars exhibit various low-amplitude modes, but the distribution of those modes is markedly different from those of the bulge stars. This suggests that differences in physical parameters have an observable effect on the excitation of extra modes, potentially offering a way to uncover the origins of these signals. However, mode identification is hindered by uncertainties when identifying the true pulsation frequencies of the extra modes. We compare mode amplitude ratios in classical double-mode stars to stars with extra modes at low amplitudes and find that they separate into two distinct groups. Finally, we find a high percentage of modulated stars among the fundamental mode pulsators, but also find that at least 28% of them do not exhibit modulation, confirming that a significant fraction of stars lack the Blazhko effect.

Real-Time Exercise Mode Identification with an Inertial Measurement Unit for Smart Dumbbells

Exercise is good for health, quality of life, and maintenance of human muscles. Dumbbells are popular indoor exercise equipment with several benefits such as low cost, high flexibility in space and time, easy operation, and suitability for people of all ages. Facilitated by advances in the Internet of Things, smart dumbbells that provide automatic counting and motion monitoring functions have been developed. To perform these tasks, the key process is identification of exercise mode. This study proposes a method to identify essential muscle groups’ (biceps, triceps, and deltoids) exercise modes of a dumbbell using an inertial measurement unit to provide three-axis angular velocities and accelerations. The motion angles were estimated from the axial acceleration and angular velocity. Phase diagrams and time plots of the axial angle, angular velocity, and acceleration were used to extract significant features of each exercise. Machine Learning and weighting functions were developed to combine these features into an identification index value for accurate identification and classification of the exercise modes. An algorithm was developed to verify the exercise mode identification. The results show that the proposed method and weighting function can successfully identify the six exercise modes. The identification algorithm was 99.5% accurate. The exercise mode identification of the dumbbell is confirmed.

Discovery, TESS Characterization, and Modeling of Pulsations in the Extremely Low-mass White Dwarf GD 278

We report the discovery of pulsations in the extremely low-mass (ELM), likely helium-core white dwarf GD 278 via ground- and space-based photometry. GD 278 was observed by the Transiting Exoplanet Survey Satellite (TESS) in Sector 18 at a 2 minute cadence for roughly 24 days. The TESS data reveal at least 19 significant periodicities between 2447 and 6729 s, one of which is the longest pulsation period ever detected in a white dwarf. Previous spectroscopy found that this white dwarf is in a 4.61 hr orbit with an unseen >0.4 M ⊙ companion and has T eff = 9230 ± 100 K and log g = 6.627 ± 0.056 , which corresponds to a mass of 0.191 ± 0.013 M ⊙. Patterns in the TESS pulsation frequencies from rotational splittings appear to reveal a stellar rotation period of roughly 10 hr, making GD 278 the first ELM white dwarf with a measured rotation rate. The patterns inform our mode identification for asteroseismic fits, which, unfortunately, do not reveal a global best-fit solution. Asteroseismology reveals two main solutions roughly consistent with the spectroscopic parameters of this ELM white dwarf, but with vastly different hydrogen-layer masses; future seismic fits could be further improved by using the stellar parallax. GD 278 is now the tenth known pulsating ELM white dwarf; it is only the fifth known to be in a short-period binary, but is the first with extended, space-based photometry.

Asteroseismology of a High-amplitude δ Scuti Star GSC 4552-1498: Mode Identification and Model Fitting

In this paper, the pulsation behavior of high-amplitude δ Scuti star GSC 4552-1498 was analyzed. Using the high-precision photometric data from the Transiting Exoplanet Survey Satellite, two new independent frequencies F1 = 22.6424(1) day−1 and F2 = 28.6803(5) day−1 were identified for this source, along with the fundamental one F = 17.9176(7) day−1, which was previously known. In addition, the classical O − C analysis was conducted to give a new ephemeris formula of BJDmax = T 0 + P × E = 2453321.534716(4) + 0.055811(0) × E. The O − C diagram reveals a continuous period increase, but the rate of (1/P)(dP/dt) = 1.11(3) × 10−7 yr−1 seems much larger (about hundreds) than predicted by evolution theories, which is long been noticed but not well understood, possibly related to nonlinear mode interaction. Based on frequency parameters (i.e., F, F1, and F2), a series of theoretical models were conducted by employing the stellar evolution code. It turns out that F1 should be a non-radial mode and F2 is the second overtone radial mode. Due to the mass–metallicity degeneracy, the stellar parameter of the star can however not be determined conclusively. We suggest high-resolution spectral observation is highly desired in the future to further constrain models. We note GSC 4552-1498 is located on the main sequence in the H-R diagram.