Derivation Of Analytical Equations Research Articles

Derivation of analytical equations with experimental verification for working mechanism of triboelectric nanogenerators in contact-separation mode

Abstract Since the invention of triboelectric nanogenerator (TENG) in 2012, various works have attempted to address the basic physical principles of TENG operation such as Maxwell displacement current or air breakdown effect. Nevertheless, physical mechanisms incorporating the measurement system to give a better comprehension of the testing process have not been reported yet. Here, as evidenced by the measurement results of a ZnO-PDMS TENG, we investigate the fundamental mechanisms of short-circuit current and open-circuit voltage output of TENGs in contact-separation mode by analyzing analytical equations based on a capacitor-like and charged-plate model for better comprehension. The analysis can provide deeper insights into the output waveforms of TENGs during contact and separation operations. Moreover, the impact of the testing conditions of a measurement system on the results can be elucidated. The output waveforms could be disguised and misleading to some degree depending on the sampling rate in data collection. More importantly, the inherent resistor associated with the voltmeter dramatically governs the voltage output of TENGs, which is especially sensitive to transfer charge distribution.

Photobleaching statistics in single-molecule on-/off-time distributions.

The on- and and off-time distributions from fluorescence single-molecule experiments are widely used to extract kinetics parameters with the goal to provide a quantitative description for the molecule's behavior on the ensemble level. Such experiments are inevitably influenced by photobleaching, where the fluorescent probe transitions to a nonemissive state. Yet, it appears that few reports went beyond acknowledging this unavoidable complication; in fact, it has so far been ignored when evaluating off-time distributions. Here, we present a theoretical framework that allows the derivation of analytical equations in which photobleaching kinetics are rigorously incorporated. Unexpectedly, our results indicate that the off-time distribution should be nonexponential even when all the rate processes are single exponential. With the analytical theory understood and demonstrated as easy to implement, such ubiquitous photochemical processes can now be readily included in routine experimental analyses.

Standard deviation of line objects in geographic information science

Standard deviation of points is regarded as an effective precision indicator and has been used widely for over 100 years. However, to date, no standard deviation for line objects exists, despite lines being the most fundamental geometric objects in geographic information science. This paper proposes a new theory: the measurement of random line precision using standard deviation. The new theory involves: (1) standard deviation presented graphically, in a band-shape: termed the standard deviation band; (2) the rigorous derivation of analytical equations for the standard deviation band; (3) the probability that a line falls within the standard deviation band. The main contributions of this research include: (1) the derivation of an analytical equation of the standard deviation band; (2) a method to estimate the probability of a line falling within a standard deviation band. These contributions form a foundation for the proposition of further control measures for spatial data quality.

Diffusion orientation transform revisited

Diffusion orientation transform (DOT) is a powerful imaging technique that allows the reconstruction of the microgeometry of fibrous tissues based on diffusion MRI data. The three main error sources involving this methodology are the finite sampling of the q-space, the practical truncation of the series of spherical harmonics and the use of a mono-exponential model for the attenuation of the measured signal. In this work, a detailed mathematical description that provides an extension to the DOT methodology is presented. In particular, the limitations implied by the use of measurements with a finite support in q-space are investigated and clarified as well as the impact of the harmonic series truncation. Near- and far-field analytical patterns for the diffusion propagator are examined. The near-field pattern makes available the direct computation of the probability of return to the origin. The far-field pattern allows probing the limitations of the mono-exponential model, which suggests the existence of a limit of validity for DOT. In the regimen from moderate to large displacement lengths the isosurfaces of the diffusion propagator reveal aberrations in form of artifactual peaks. Finally, the major contribution of this work is the derivation of analytical equations that facilitate the accurate reconstruction of some orientational distribution functions (ODFs) and skewness ODFs that are relatively immune to these artifacts. The new formalism was tested using synthetic and real data from a phantom of intersecting capillaries. The results support the hypothesis that the revisited DOT methodology could enhance the estimation of the microgeometry of fiber tissues.

Strong principle of corresponding states: Reduction of a p-v-T surface to a line

Abstract Traditional principles of corresponding states collapse the p-v-T surfaces of whole families of substances onto a single reduced p ★ -v ★ -T ★ surface. The recent derivation of analytical equations of state based on statistical-mechanical perturbation theory now allows us to further collapse such a surface to a single curve. This is accomplished by using the theory to extract a quantity that is a function only of a reduced density. This quantity, G ( b ϱ), corresponds physically to an effective pair distribution function for hard convex bodies at contact, where b is the analogue of the van der Waals covolume and ϱ is the density. The information needed can be obtained from the intermolecular potential, or more conveniently from just the second virial coefficient as a function of temperature. Plots of G −1 vs. b ϱ are found empirically to be essentially straight lines whose slope depends on the particular substance, covering the entire range from the low-density gas to the compressed liquid, and including both subcritical and supercritical regions. Thus the final result permits the construction of an entire p-v-T surface from as little information as the second virial coefficient plus a few liquid densities (to determine the slope of G −1 vs. b ϱ). The corresponding equation of state is only cubic in the density and has an accuracy amounting to parts per thousand in the density.

An analysis of the stress-strain hysteresis loop and the cyclic stress-strain response of steels, based on dislocation dynamics (1st report, Derivation of analytical equations and some analyzed results)

Cyclic internal stress was determined through a strain rate change test, and cyclic hardening and softening properties were measured during low cycle fatigue tests on structural steels greatly different in strength and ductility levels. The internal stress and the mechanical quantities on cyclic deformation were empirically formulated as functions of the yield stress and the work hardening exponent in a static tension, and the dislocation density. These formulated quantities were introduced into the Johnston-Gilman's equation modified by the authors, in which the macroscopic plastic strain rate was correlated with the dislocation motion. Then, the stress-strain hysteresis loops and the cyclic stress-strain response of the structural steels for machine components were analyzed in terms of dislocation dynamics. The analyses were in quite good agreement with the present test results. Consequently, it was shown that the cyclic stress-strain response of the steels could be analytically determined using the static mechanical properties.