Closed-form Relations Research Articles

Chaotic behaviour in the dynamical response of coaxial magnetic gears during acceleration

Slippage during acceleration has been a significant issue in coaxial magnetic and has been thoroughly investigated in the recent years. Depending on the applied outer load it is important to know the maximum acceleration that can be induced so that the system will not diverge. In the present work a non-dimensional analysis has been conducted in order to develop a stability criterion for the maximum acceleration that can be applied to the system as a function of the outer load. A closed-form relation for the period of the oscillations of the system that will be caused by the acceleration, has also been derived. Furthermore, the case of acceleration with ripple has been investigated. It was demonstrated that during acceleration with ripple under constant applied outer load, the system showcases a chaotic behaviour similar to the driven pendulum. In particular, a thorough analysis on the frequency of the ripple has been conducted. It was observed that when the ripple frequency was slightly lower than the frequency of the oscillation under steady acceleration, the system could potentially diverge even if the acceleration of the system was lower than the critical value. With the present work, a detailed non-dimensional model has been derived that could be a useful tool for determining the stability of coaxial magnetic gears without the requirement of numerical solution of the governing equations. In addition, some interesting observations are made regarding the chaotic behaviour of the dynamical response of coaxial magnetic gears during acceleration with ripple.

Characteristics of wave propagation in pre-stressed viscoelastic Timoshenko nanobeams with surface stress and magnetic field influences

The current investigation explores the behavior of pre-stressed viscoelastic Timoshenko nanobeams under the influence of surface effects and a longitudinal magnetic field. Utilizing a modified version of non-local strain gradient theory through the Kelvin–Voigt viscoelastic model, a closed-form dispersion relation using a suitable analytical approach has been derived. To account for surface stresses, Gurtin–Murdouch surface elasticity theory has been employed. Additionally, the study delves into the impact of a longitudinal magnetic field on a single-walled carbon nanotube, considering Lorentz magnetic forces. The validity of the findings is established by deriving results in the absence of surface effects and magnetic fields, aligning well with existing literature. The investigation indicates that pre-stress has marginal effects on flexural and shear waves, while surface effects, magnetic fields, non-locality, characteristic length, and nanotube diameter significantly influence the phase velocity. Additionally, the threshold velocity and blocking diameter are discussed for the model.

On the equivalence between steady-state Kalman filter and DPLL

Fundamental results in the literature showed that a class of Kalman filters (KF) converges to a digital phase lock loop (DPLL) structure. Results were proved for second- and third-order KFs. We extend these results to KFs of any order and derive in closed form the equivalent loop filter constants as a linear combination of the steady-state Kalman gains. We also give the inverse relation. Both closed-form relations involve Stirling numbers of the first or second kind. We implement both filters in a global navigation satellite system (GNSS) receiver and illustrate their equivalence with real-world data. These extended theoretical results provide a deeper understanding of the equivalence between KF and DPLL and may be of practical interest in highly dynamic scenarios to design and tune tracking filters.

A study of the effect of array geometry on improving channel hardening in mm-Wave massive MIMO systems

The impact of array geometry on enhancing channel hardening in mm-wave massive MIMO systems is investigated. As the number of physical radio links increases, small-scale fading is averaged out, leading to a hardened channel and increased communication capacity. A comprehensive analysis is conducted using the physical ray distribution model (3GPP) to investigate the impact of various array geometries on channel hardening. A clear insight is obtained through the derivation of a closed-form relation, providing an understanding of the corresponding effect. Additionally, a novel measure called Array Spread is introduced, offering a practical tool for designing array geometry that minimizes channel variations in every specific physical scenarios. The findings of this study contribute to the overall understanding and optimization of MIMO communication systems, particularly in the context of massive MIMO deployments in mm-wave frequencies.

AC Magnetic Susceptibility: Mathematical Modeling and Experimental Realization on Poly-Crystalline and Single-Crystalline High-Tc Superconductors YBa2Cu3O7-δ and Bi2-xPbxSr2Ca2Cu3O10+y.

The multifaceted inductive technique of AC magnetic susceptibility (ACMS) provides versatile and reliable means for the investigation of the respective properties of magnetic and superconducting materials. Here, we explore, both mathematically and experimentally, the ACMS set-up, based on four coaxial pick-up coils assembled in the second-derivative configuration, when employed in the investigation of differently shaped superconducting specimens of poly-crystalline YBa2Cu3O7-δ and Bi2-xPbxSr2Ca2Cu3O10+y and single-crystalline YBa2Cu3O7-δ. Through the mathematical modeling of both the ACMS set-up and of linearly responding superconducting specimens, we obtain a closed-form relation for the DC voltage output signal. The latter is translated directly to the so-called extrinsic ACMS of the studied specimen. By taking into account the specific characteristics of the studied high-Tc specimens (such as the shape and dimensions for the demagnetizing effect, porosity for the estimation of the superconducting volume fraction, etc.), we eventually draw the truly intrinsic ACMS of the parent material. Importantly, this is carried out without the need for any calibration specimen. The comparison of the mathematical modeling with the experimental data of the aforementioned superconducting specimens evidences fair agreement.

A closed-form structural stress-strain relation and implications on plastic design limits in codes and standards

The deformation of elastic-perfectly plastic beam or plate under combined through-thickness membrane and bending stresses (i.e., structural stress) is analyzed in this paper. Closed-form solutions of “plane-remains-plane” strain (i.e., structural strain) distribution as a function of applied stress and yield strength is derived. The relationship between structural stress and structural strain is also revealed for gaining insights on not only the plastic collapse conditions, but also through-thickness deformation behaviors leading to limit state. Based on the closed-form strain solutions, four deformation modes are identified: (1) elastic, (2) one-side yielding, (3) two-side yielding, and (4) plastic collapse. A membrane-bending stress interaction diagram is constructed to depict four zones that are associated with the four deformation modes in the normalized membrane and bending stress space. Closed-form solutions of residual stress, residual strain, and structural stress range-strain range relation after a loading-unloading cycle are also derived. Based on the parabolic plastic collapse stress limit curve, parabolic design stress limit curves with coherent safety margins for any combinations of membrane and bending stresses are constructed. The new insights gained in this study offer a more consistent definition of design stress limit than those used in the current codes and standards, e.g., ASME BP&V Code.

Design and analysis of an ideal scramjet flowpath

The current work presents a novel conceptual framework for the fluid and gasdynamics that govern the design and performance of an ideal scramjet flowpath. These include a theoretical comparison between ram and scram modes, the physics of thrust loss during inlet unstart, and the design of an optimal scramjet flowpath. We present a unique explicit, closed-form relation for the wall divergence of an ideal scramjet combustor. The accompanying derivations and discussions, which leverage this formulation, seek to address uncertainties and misconceptions regarding the dominant fluid processes present in these engines. It is shown that scram and ram modes exhibit theoretical similitude for maximum thrust potential at conditions beyond the one-dimensional Rayleigh choking limit but can diverge below the global choking threshold. Additionally, it is shown that even for an ideal scramjet heat engine cycle, thermodynamic efficiencies at various flight conditions deviate from those of the classical Brayton cycle. These insights and accompanying theoretical analyses are meant to establish a foundation for the thermodynamics and gasdynamics relevant to the performance of dual-mode scramjet engines. The resulting work offers an intuitive technical perspective on supersonic combustion and the fundamentals of dual-mode scramjet operation that can be applied across a wide range of scramjet-related experimental and computational studies and design efforts in the future.

Reassessing the transport properties of fluids: A symbolic regression approach.

The viscosity and thermal conductivity coefficients of the Lennard-Jones fluid are extracted through symbolic regression (SR) techniques from data derived from simulations at the atomic scale. This data-oriented approach provides closed form relations that achieve fine accuracy when compared to well-established theoretical, empirical, or approximate equations, fully transparent, with small complexity and high interpretability. The novelty is further outlined by suggesting analytical expressions for estimating fluid transport properties across the whole phase space, from a dilute gas to a dense liquid, by considering only two macroscopic properties (density and temperature). In such expressions, the underlying physical mechanisms are reflected, while, at the same time, it can be a computationally efficient alternative to costly in time and size first principle and/or molecular dynamics simulations.

Unbiased construction of constitutive relations for soft materials from experiments via rheology-informed neural networks.

The ability to concisely describe the dynamical behavior of soft materials through closed-form constitutive relations holds the key to accelerated and informed design of materials and processes. The conventional approach is to construct constitutive relations through simplifying assumptions and approximating the time- and rate-dependent stress response of a complex fluid to an imposed deformation. While traditional frameworks have been foundational to our current understanding of soft materials, they often face a twofold existential limitation: i) Constructed on ideal and generalized assumptions, precise recovery of material-specific details is usually serendipitous, if possible, and ii) inherent biases that are involved by making those assumptions commonly come at the cost of new physical insight. This work introduces an approach by leveraging recent advances in scientific machine learning methodologies to discover the governing constitutive equation from experimental data for complex fluids. Our rheology-informed neural network framework is found capable of learning the hidden rheology of a complex fluid through a limited number of experiments. This is followed by construction of an unbiased material-specific constitutive relation that accurately describes a wide range of bulk dynamical behavior of the material. While extremely efficient in closed-form model discovery for a real-world complex system, the model also provides insight into the underpinning physics of the material.

Electrohydrodynamics of diffuse porous colloids.

The present article deals with the electrohydrodynamic motion of diffuse porous particles governed by an applied DC electric field. The spatial distribution of monomers as well as the charge distribution across the particle are considered to follow sigmoidal distribution involving decay length. Such a parameter measures the degree of inhomogeneity of the monomer distribution across the particle. The diffuse porous particles resemble several colloidal entities which are often seen in the environment as well as in biological and pharmaceutical industries. Considering the impact of bulk pH and ion steric effects, we modelled the electrohydrodynamics of such porous particulates based on the modified Boltzmann distribution for the spatial distribution of electrolyte ions and the Poisson equation for electric potential as well as the conservation of mass and momentum principles. We adopt regular perturbation analysis with weak field assumption and the perturbed equations are solved numerically to calculate the electrophoretic mobility and neutralization fraction of the particle charge during its motion as well as fluid collection efficiency. We further deduced the closed form relation between the drag force experienced by the charged porous particle and the fluid collection efficiency. In addition to the numerical results, we further derived the closed form analytical results for all the intrinsic parameters indicated above derived within the Debye-Hückel electrostatic framework and homogeneous distribution of monomers within the particle for which the decay length vanishes. The deduced mathematical results as indicated above will be useful to analyze several electrostatic and hydrodynamic features of a wide class of porous particulate and environmental entities.

Boundary-layer approach to the linear karren instability

The present paper concerns the linear fate of transverse perturbations in a gravity-driven, thin-film flow over a soluble substrate. We propose a reduced-order model, based on a boundary-layer treatment of the solute transport and a depth-integration of the Stokes equations, using two extended lubrication methodologies found in the literature. We obtain a closed-form dispersion relation, which we compare to a previous, fully resolved analytical investigation (Bertagni and Camporeale, J. Fluid Mech., vol. 913, 2021, A34). The results allow us to distil the essential physical mechanisms behind the instability.

Is the out-of-plane shear strength an independent property? A micromechanical perspective about a macromechanical question

For unidirectional laminae, five elastic properties and six strengths are usually required. The goal of this paper is to evaluate if the out-of-plane shear strength is really an independent property or if it is somehow related to the transversal tensile and compressive strengths. The analytical VSPKc micromechanical model is used to derive closed-form relations. Fiber-matrix interface failure due to normal and shear interface tractions is evaluated considering a perfectly brittle behavior. The proposed model is compared with four analytical models from the literature and their estimations are verified against six experimental data. The relations proposed obtained the best results compared with the experimental data, with an average error equal to 8.84%, while the error using the second-ranked model is 17.40%, indicating with good reliability that the out-of-plane shear strength is not an independent property.

Mechanical modelling of viscoelastic hierarchical suture joints and their optimisation and auxeticity

It has been widely recognised that suture joints play a fundamental role in the exceptional mechanical properties of many biological structures like the cranium and ammonite fossil shells. Up to date, many efforts have been devoted to investigate suture interfaces to predict the relations between the characteristics of its two phases, the teeth and the interface layer, and the structure effective properties. However, in very few works the viscoelasticity of the suture components has been taken into account. To provide a contribution in this limitedly explored research area, this paper describes the mathematical formulation and modelling technique leading to explicit expressions for the effective properties of viscoelastic suture joints with a general trapezoidal waveform. It emerges a strong influence of the suture geometric and mechanical characteristics on the effective properties and a parametric analysis reveals that an auxetic behavior can be obtained simply by tailoring the suture parameters. The effects of adding hierarchy into the above system are also explored and closed-form relations for the effective moduli are derived. Optimal levels of hierarchy can be identified and, similarly to the non-hierarchical case, an auxetic behavior emerges for particular values of the suture parameters. Finally, an extension of the theory to the sinusoidal sutures in biology is reported. By considering the examples of the cranium and the woodpecker beak, it emerges that the first is optimised to obtain high stiffness, while the second to obtain high energy dissipation levels.

Theoretical Study and Systematic Design of Multiport Wire Antenna

This work presents a systematic study of multi-port wire antennas (MPWA) as a step toward reconfigurable radiation apertures, an essential component in modern wireless systems. We predict the surface current distribution (SCD) of the multi-port structure and derive a closed-form relation for the radiation pattern of the aperture as a function of its input port excitations. Based on this, we present a design procedure for optimal selection of port locations and their respective input excitations with a desired radiation pattern in mind. Finally, we develop a computationally efficient method to derive the impedance matrix of the multi-port aperture all without the need for full-wave simulation. As an example, we use this method to design a 15-port mm-wave antenna. We demonstrate that for a selected aperture size and port locations and only by adjusting the phases and amplitudes of excitation, we can reconfigure the MPWA to synthesize a desired beam at any frequency within the 20-40GHz range. There is good agreement between results predicted by this method compared to those from EM-based simulations, enabling an efficient and scalable approach for designing multi-port radiators.

Effective electromagnetic wave properties of disordered stealthy hyperuniform layered media beyond the quasistatic regime

Disordered stealthy hyperuniform dielectric composites exhibit novel electromagnetic wave transport properties in two and three dimensions. Here, we carry out the first study of the electromagnetic properties of one-dimensional 1D) disordered stealthy hyperuniform layered media. From an exact nonlocal theory, we derive an approximation formula for the effective dynamic dielectric constant tensor ε e (k q ,ω) of general 1D media that is valid well beyond the quasistatic regime and apply it to 1D stealthy hyperuniform systems. We consider incident waves of transverse polarization, frequency ω, and wavenumber k q . Our formula for ε e (k q ,ω), which is given in terms of the spectral density, leads to a closed-form relation for the transmittance T. Our theoretical predictions are in excellent agreement with finite-difference time-domain (FDTD) simulations. Stealthy hyperuniform layered media have perfect transparency intervals up to a finite wavenumber, implying no Anderson localization, but non-stealthy hyperuniform media are not perfectly transparent. Our predictive theory provides a new path for the inverse design of the wave characteristics of disordered layered media, which are readily fabricated, by engineering their spectral densities.

Thermoelastic resonance analysis of BNNT-reinforced nanocomposite beams heated by internal thermal source

The nanocomposite structures due to excellent chemical and thermal resistant have considerable potential for use in an aggressive environment with high corrosion and temperature gradient. This study analyzes subharmonic (Ω/ωL=2 and 3) and superharmonic (Ω/ωL=1/2 and 1/3) resonance behavior of boron nitride nanotube-reinforced composite (BNNT-RC) beams under harmonic excitation and internal thermal source. The thermal analysis is carried out for determining the temperature change using the two-dimensional heat conduction equation with the assumption of an arbitrary variation in the thickness direction. In the framework of adjacent equilibrium configurations and Galerkin method, the modified couple stress Timoshenko beam equations are reduced to one nonlinear decoupled equation. Next, closed-form relations of vibration amplitude (a) and detuning parameter (σ∼) are developed for subharmonic and superharmonic excitations using the multiple-scale perturbation technique. The nonlinear frequency and resonance responses are verified by comparison with the existing ones in the literature. Finally, parametric studies are presented to discuss the effects of length scale parameter, temperature change, energy rate of thermal source, detuning parameter, and geometric ratios. The snap-through buckling behavior is observed in the curves of ωNL/ω∗-ΔT and different unstable paths exist after the occurrence of instability. The thermoelastic effect can increase the stability range of harmonic load amplitude for various values of detuning parameter and it is more pronounced for the temperature variation induced by an internal thermal source.

Bio-convection flow of fractionalized second grade fluid through a vertical channel with Fourier’s and Fick’s laws

This study is conducted to discuss the thermal transport of second grade nanofluid flowing through a vertical channel. The nanoparticles, alumina (Al2O3), and titanium (TiO2), are dispersed into the second grade base fluid to form a hybrid nanofluid. The governing equation prescribed model is constructed by applying the laws of conservation of mass and energy. It can be modified by converting the non-dimensional classical model to the fractional derivative model using Fick’s and Fourier’s laws. With the help of the Laplace transforms, the resulting dimensionless form of the governing equations and the closed form relations of temperature, concentration, and velocity are obtained. The physical behavior of the obtained result is judged graphically. The nanoparticle’s volume percentage increases the concentration and temperature distribution, but it has the opposite impact on the velocity profile.

An efficient derivation of spatial Green's function of open-radiating rectangular cavity using CGF-CI technique

ABSTRACT A new technique for efficient calculation of spatial Green's function of open-radiating rectangular cavity is presented. The method is based on separability assumption of the original structure and characteristic Green's function method. Using complex image (CI) technique in spectral domain and well-known Weyl's identity, a closed-form relation for spatial Green's function can be derived. To obtain more accurate closed-form Green's function, before applying CI, discrete spectrum contribution part of the derived relation is modified by replacing more accurate reflection coefficients of surface wave modes from walls of the open-radiating cavity. Verification of the derivations of the proposed method is shown for dielectric loaded metallic box which is a separable structure. Method of moment + multilevel fast multipole method solution and also exact numerical integration approach for open-radiating rectangular cavity are considered to search the accuracy as well as speed of the proposed method. It is shown that the main advantage of the proposed method lies in its speed as well as its simple implementation while the accuracy is preserved.

Closed-form and technology independent phase noise relation for LC oscillators

PurposeThe purpose of this paper is to find a closed relation for the phase noise of LC oscillators.Design/methodology/approachThe governing equation of oscillators is generally a stochastic nonlinear differential equation. In this paper, a closed relation for the phase noise of LC oscillators was obtained by approximating the I–V characteristic of the oscillator with third-degree polynomials and analyzing its differential equation.FindingsThis relation expresses phase noise directly in terms of circuit parameters, including the sizes of the transistors and the bias. Next, for evaluation, the phase noise of the cross-coupled oscillator without tail current was calculated with the proposed model. In this approach, the obtained equations are expressed independently of technology by combining the obtained phase noise relation and gm/ID method.Originality/valueA technology-independent method using the gm/ID method and the closed relationship is provided to calculate phase noise.

Relationship between Hanbury Brown-Twiss effect and spectral degree of polarization in light scattering from a collection of particles of $\mathcal {L}$ types.

We consider the vectorial extension of the recently developed matrix theory for the correlation between intensity fluctuations (CIF) of the scattered field generated by a collection of particles of $\mathcal {L}$ types [Y. Ding and D. M. Zhao, Opt. Express 30 46460, 2022]. In the spherical polar coordinate system, we establish a closed-form relation that connects the normalized CIF of the electromagnetic scattered field with the pair-potential matrix (PPM), the pair-structure matrix (PSM), and the spectral degree of polarization $\mathcal {P}$ of the incident field. Based on this, we pay much attention to the dependence of the normalized CIF of the scattered field on $\mathcal {P}$. It is found that the normalized CIF can be monotonically increasing or be nonmonotonic with $\mathcal {P}$ in the region [0, 1], determined by the polar angle θ and the azimuthal angle ϕ. Also, the distributions of the normalized CIF with $\mathcal {P}$ at polar angles and azimuthal angles are greatly different. These findings are explained mathematically as well as physically, and may be of interest to some related fields, especially where the CIF of the electromagnetic scattered field plays important roles.