Harmonic Solutions Research Articles

Fischer decompositions for entire functions and the Dirichlet problem for parabolas

Let \(P_{2k}\) be a homogeneous polynomial of degree 2k and assume that there exist \(C>0\), \(D>0\) and \(\alpha \ge 0\) such that $$\begin{aligned} \left\langle P_{2k}f_{m},f_{m}\right\rangle _{L^2(\mathbb {S}^{d-1})}\ge \frac{1}{C\left( m+D\right) ^{\alpha }}\left\langle f_{m},f_{m}\right\rangle _{\mathbb {S}^{d-1}} \end{aligned}$$for all homogeneous polynomials \(f_{m}\) of degree m. Assume that \(P_{j}\) for \(j=0, \dots ,\beta <2k\) are homogeneous polynomials of degree j. The main result of the paper states that for any entire function f of order \( \rho <\left( 2k-\beta \right) /\alpha \) there exist entire functions q and h of order bounded by \(\rho \) such that $$\begin{aligned} f=\left( P_{2k}-P_{\beta }- \dots -P_{0}\right) q+h\text { and }\Delta ^k r=0. \end{aligned}$$This result is used to establish the existence of entire harmonic solutions of the Dirichlet problem for parabola-shaped domains on the plane, with data given by entire functions of order smaller than \(\frac{1}{2}\).

New results for periodic solution in Liebau phenomenon

This paper is devoted to the existence and bifurcations of positive periodic solutions of nonlinear differential equation related to the Liebau phenomenon in rigid pipe-tank flow configuration. We obtain some more general requirements on the existence of a positive harmonic solution which improves the results in literatures. It is revealed that the system undergoes saddle node bifurcation, period doubling bifurcation and Neimark–Sacker bifurcation generating various periodic solution of different stability types. The multiplicity of both the positive harmonic solution and positive second-order subharmonic solution is first detected in this equation by numerical bifurcation analysis. Moreover, this work reveals that the positive subharmonic solutions of the nonlinear differential equation will also lead to the Liebau phenomenon in the model.

Development of a generalized extended harmonic solution for analyzing the combination of chatter suppression techniques in milling

In this paper a comprehensive and structured analysis of the stabilizing properties linked to the combination of different chatter suppression techniques in milling was carried out. Specifically, the spindle speed variation, the stiffness variation and the usage of tools with variable pitch and variable helixangles were considered.For this purpose, a unique, general and efficient frequency domain formulation (extended harmonic solution) was developed. It is suitable for dealing with differential delayed equations characterized by multiple and distributed time dependent delays and time periodic dynamics. It was demonstrated that the developed solution converges to the reference cases, which were studied in literature through semi-discretization and its extensions. Moreover, by selecting a proper number of considered harmonics, the computation time was reduced of one order of magnitude, in the face of a slight reduction of accuracy with respect to time domain approaches.The computed stability lobe diagrams demonstrated that the combination of multi-chatter suppression techniques allows exploiting the best features of each methodology. The simultaneous use of the spindle speed variation and the stiffness variation was studied for the first time. The absolute limit of stability increased up to 204%, much more than by adopting the techniques in a separate manner. If the enlargement of the stability region is considered as the reference target, the best results were granted by the combination of all the three strategies. Indeed, the improvement with respect to the use of only two techniques was at least 10%.

Two-dimensional Harmonic Modelling for Electro-magnetic Solution in Cartesian Coordinates

This paper first presents a general two-dimension (2D) harmonic analytical solution for the magnetic field of electric machines in the Cartesian coordinates. In this solution, the relative permeance is directly considered in Laplace and Poisson’s equations, and the particular solutions in Cartesian coordinates are solved. By applying the complex Fourier separation method, with the boundary and interface conditions, the magnetic field in the inhomogeneous region is solved from system equations. Numerical examples validate the presented method and the obtained results have a satisfactory agreement with the finite element analysis. The proposed model in this paper has a significant value for modelling electric machines, such as linear permanent magnet (PM) machines and axial flux PM machines.

A review of different mascon approaches for regional gravity field modelling since 1968

Abstract. The geodetic and geophysical literature shows an abundance of mascon approaches for modelling the gravity field of the Moon or Earth on global or regional scales. This article illustrates the differences and similarities between the methods, which are labelled as mascon approaches by their authors. Point mass mascons and planar disc mascons were developed for modelling the lunar gravity field from Doppler tracking data. These early models had to consider restrictions in observation geometry, computational resources or geographical pre-knowledge, which influenced the implementation. Mascon approaches were later adapted and applied for the analysis of GRACE observations of the Earth's gravity field, with the most recent methods based on the simple layer potential. Differences among the methods relate to the geometry of the mascon patches and to the implementation of the gradient and potential for field analysis and synthesis. Most mascon approaches provide a direct link between observation and mascon parameters – usually the surface density or the mass of an element – while some methods serve as a post-processing tool of spherical harmonic solutions. This article provides a historical overview of the different mascon approaches and sketches their properties from a theoretical perspective.

Photothermal excitation and Thomson impact in a semiconductor microelongated thermoelastic medium with microtemperatures in the gravity

AbstractThe heating of solids occurs due to the absorbed light from the heat source. Photothermal spectroscopy is a method of measuring the optical absorption and thermal properties of semiconductor materials using high‐sensitivity spectroscopic techniques. In the present paper, the harmonic solution as an analytical solution is provided to describe the semiconductor photothermal microelongated medium with microtemperatures in the gravity field. The proposed photothermal microelongated model is subjected to an induced magnetic field to observe the Thomson effect. The numerical calculations for the components of thermal stresses, displacement, temperature field, and carrier density are obtained using the harmonic solution approach. The propagation of heat, elastic, and plasma waves, as well as the distributions of each investigated field, were examined and explained. The comparison is also used to see how the gravity, Thomson effect, and the photo‐excitation have more observable effects on the distribution of the medium fields during their values changes. With the development of technologies, the semiconducting materials were used widely in modern engineering. The study of wave propagation in a semiconducting medium will have important academic significance and application value.

Modified Emden Type Oscillator Equations with Exact Harmonic Solutions

This paper is devoted to investigating the existence of exact harmonic solutions and limit cycles of certain modified Emden-type equations. The exact and general solutions obtained are in opposition to the predictions of classic existence theorems.

Analytic Matrix Method for Frequency Response Techniques Applied to Nonlinear Dynamical Systems II: Large Amplitude Oscillations

This work is the second in a series of articles that deal with analytical solutions of nonlinear dynamical systems under oscillatory input that may exhibit harmonic frequencies. Frequency response techniques of nonlinear dynamical systems are usually analyzed with numerical methods, because in most cases analytical solutions such as the harmonic balance series solution turn out to be difficult, if not impossible, as they are based on an infinite series of trigonometric functions with harmonic frequencies. The key contribution of the analytic matrix methods reported in the present series of articles is to work with the invariant submanifold of the problem and to propose the solution as infinite power series of the oscillatory input; this procedure is a direct one that speeds up the computations compared to traditional series solution methods. The method reported in the first contribution of this series allows for the computation of the analytical solution only for small and medium amplitudes of the oscillatory input, because these series may diverge when large amplitudes are applied. Therefore, the analytic matrix method reported here, which is a reconfiguration of the method proposed in the first contribution in this series, allows the solving of problems in the regime of large-amplitude oscillations where the contributions of the high order harmonics affect the amplitudes of the low order harmonics, leading to amplitude- and frequency-dependent coefficients for the infinite series of trigonometric function expansion.

Coupling nonlinearities and dynamics between the hybrid tri-stable piezoelectric energy harvester and nonlinear interfaced circuit

Tri-stable piezoelectric vibration energy harvester (T-PVEH) recently has been widely investigated due to its good electro-mechanical performance. However, most studies emphasizing the mechanical configuration usually only regard the interfaced circuit as equivalent to a linear resistance load. The coupling nonlinearities and dynamics between the T-PVEH and the nonlinear interfaced circuit are yet to be well understood. Considering a hybrid T-PVEH interfacing with a nonlinear AC-DC rectifying circuit, this paper aims to derive its electro-mechanical equations to characterize the mechanical and energetic dynamics, as well as the coupling nonlinearities between the mechanical terminal and electrical terminal, which can be helpful to optimize the T-PVEH configuration and the circuit topology, so as to enhance the energy harvesting efficiency and effective broadband. The general harmonic balance solutions and the Jacobian matrix used to estimate the solution stability are presented based on the derived electro-mechanical equations. The influences of the magnetic distance, coupling constant and load resistance on the mechanical and energetic dynamics are simulated. The contradiction between the bandwidth and efficiency caused by the coupling nonlinearities is also analyzed. The results show that proper selection of the magnetic distance, load resistance and coupling constant in their suitable range is beneficial to enhance the effective bandwidth and harvested power of the global inter-well motions. It also shows that strong coupling constant with large load resistance will introduce additional mechanical damping and stiffness, attenuating the response amplitude and effective bandwidth of the system. Experimental results show reasonable agreement with the simulations. The rectified voltage obtained by experiment is 2.6 V, which meets the power supplying demand of the low-powered electronic devices.

Modelling and Control Development of a Cascaded NPC-Based MVDC Converter for Harmonic Analysis Studies in Power Distribution Networks

Today’s power distribution networks are predicted to incorporate more Power Electronics (PE)-based power conversion systems, widely acknowledged as harmonic sources. Concerns about power harmonic severity in the distribution networks can arise, especially with the growing numbers of Medium Voltage Direct Current (MVDC) systems, which are also facilitated by such power converters. Yet, an accurate harmonic model of the MVDC converter is required to investigate its harmonic emissions, propagations, effects, and solutions in today’s distribution networks. This article is devoted to the development of a detailed model of a cascaded Neutral Point-Clamped (NPC)-based MVDC converter for accurate harmonic analysis studies. An appropriate control system with a simple Proportional Integral (PI) controller tuned using the loop-shaping technique is developed. An interleaved Sinusoidal Pulse-Width Modulation (SPWM) scheme aiming to improve the harmonic performance of such an application is introduced. The detailed model of the MVDC system was developed using the Simulink/MATLAB simulation environment, for which the concept of operation was validated, control system performance was investigated, and the effectiveness of the harmonic reduction method was analysed. The key finding is that the interleaved SPWM technique has significantly reduced the Total Harmonic Distortion (THD) to 2% with no significant even-order harmonic components in comparison to the reported models.

Inverse Mean Value Property of Metaharmonic Functions

A new analytical characterization of balls in the Euclidean space ℝm is obtained. Previous results of this kind involved either harmonic functions or solutions to the modified Helmholtz equation (both have positive fundamental solutions), whereas solutions to the Helmholtz equation are used here. This is achieved at the expense of a restriction imposed on the size of admissible domains—a feature absent in the inverse mean value properties known previously.

Transmutation Operators and Liouville Normal Form, Leading to Harmonic Solutions with Variable Components

Transmutation Operators and Liouville Normal Form, Leading to Harmonic Solutions with Variable Components

A broadband flow energy harvester induced by the wake of a bluff body

Wake-galloping energy harvesting has been extensively developed to scavenge flow energy from vortex-induced oscillations. Hence, the wake-galloping harvester only has a natural frequency which leads to a very narrow bandwidth. Therefore, it does not operate well under the wide region of shedding frequencies in variable wind speed. To overcome the vital issue, this paper we explored a novel two-degree-of-freedom nonlinear flow energy harvester to collect flow energy induced by the wake of a bluff body. The nonlinear restoring force is realized by using a repulsive magnetic force between two cuboid-shaped permanent magnets, and the electromechanical coupling equations are presented. Based on the method of harmonic balance, the electromechanical governing equations are decoupled, and the first-order harmonic solutions are implemented. The modulation equations are established, and the amplitude–frequency figures of displacement and voltage are depicted with different detuning parameters. The superiority of the presented energy harvester is contrasted with the single-degree-of-freedom linear and nonlinear cases, and the results revealed that the two-degree-of-freedom nonlinear scheme can enhance the bandwidth of flow energy capture. The effect of physical parameters on the scavenged power is discussed. The accuracy and efficiency of the approximate analytical data are examined by numerical simulations.

Thermoelastic interactions on the propagation of surface waves in a piezoelectric half-space: A comparative analysis of GN-III type and three-phase-lag model

This study elucidates the propagation of surface (Rayleigh type) waves in a homogeneous, transversely isotropic, piezothermoelastic half-space subjected to stress free, electrically open or shorted, and thermally insulated or isothermal boundary conditions, based on GN-III type and three-phase-lag thermoelastic models. Plane harmonic wave solutions are employed to find the mechanical displacements, electrical potential, and temperature change. With the aid of these expressions, stresses, electrical displacement, and temperature gradient are derived. Based on different boundary conditions, four secular equations are derived in the considered half-space. Path of the surface particles traces an elliptic path in vertical plane parallel to the direction of wave propagation and the eccentricity of the ellipse is calculated. Particle path degenerates a straight line when there is no phase difference between vertical and horizontal components of displacements. A pre-established analysis is discussed as a particular case of this study. Effect of various characteristics of waves like phase velocity, attenuation coefficient, and specific loss is demonstrated graphically for the GN-III type and three-phase-lag thermoelastic models engaging cadmium selenide (6 mm class) material of hexagonal symmetry. This mathematical framework may be utilized to design and develop temperature sensors, and other piezoelectric surface acoustic wave (SAW) devices.

Study on nonlinear force transmissibility of flywheel rotor system considering periodic base motions

This paper reports on the study of the dynamic force transmissibility (DFT) of flywheel rotor system (FRS), considering periodic base motions and nonlinear support stiffness of angular contact ball bearings (ACBBs). The energy method and Lagrange equation are adopted to deduce the base induced additional excitations, including the internal (gyroscopic and stiffness) and external excitations. The influence of combined loads and contact angle variation is introduced into the Jones formula to accurately solve for the internal load distribution and nonlinear stiffness of the ACBBs. The lateral vibration model of FRS considering base motions and ACBB nonlinear support stiffness is then established. The DFT of the system is obtained by combining the harmonic balance method and alternate frequency/time domain method. The arc length continuation method is employed to track the multi-solution region owing to bifurcation. In addition, the stability of harmonic balance solutions is determined through eigenvalue analysis in the frequency domain. Both numerical integration and comparison with literature results are used to verify the accuracy of the proposed DFT model and solution method. On the basis of these results, the effects of base motion amplitude, ACBB axial preload, and rotor damping on the DFT of the system are discussed. With the increase of radial deformation, the support stiffness of ACBB under axial preload shows “soft” and “stiff” characteristics in turn. The additional excitation caused by the base motions is the key factor determining the nonlinear characteristics of FRS. These results support designing a reasonable vibration isolation system and reducing the transmission of FRS dynamic response to the spacecraft platform.

Harmonic balance analysis of magnetically coupled two-degree-of-freedom bistable energy harvesters

Because a magnetically coupled two-degree-of-freedom bistable energy harvester (2-DOF MCBEH) shows the rich, complicated nonlinear behaviors caused by its coupled cubic nonlinearities, understanding the dynamics remains challenging. This paper reports and investigates the important nonlinear dynamical phenomena of the 2-DOF MCBEHs by performing the harmonic balance analysis (HBA). All periodic solution branches are identified in order to study and comprehend the complicated dynamics of the 2-DOF MCBEHs. This end requires care when truncating the harmonic balance solution. For a 1-DOF MCBEH, which is the conventional type, the fundamental harmonic is able to approximately describe the steady-state periodic response. However, high-order harmonics are significant for the 2-DOF MCBEH. This paper demonstrates that the harmonic balance solution should involve the high-order terms instead of using the oversimplified single-harmonic solution. By performing the proposed HBA, important solution branches are reported, and their dynamical behaviors are studied. Moreover, the complete architecture of the frequency response of the 2-DOF MCBEH is disclosed across the entire frequency range. The HBA also reveals the underlying physics of building a bridge between the first and second primary resonant areas under a strong excitation. In the future, the findings in the present report can be utilized in the design process of the 2-DOF MCBEHs.

Harmonic solutions and weak solutions of two-dimensional rotational incompressible Euler equations

In this paper, two families of exact solutions to two-dimensional incompressible rotational Euler equations are constructed by connecting the Euler equations with the Laplace equation via a stream function. The constituent solutions in the first family are smooth, orthogonal, and conjugate harmonic solutions, while their constituent velocities are nonlinear with respect to the spatial variables. The second family are weak solutions in the distribution sense.

Influence of time-delayed feedback on the dynamics of temporal localized structures in passively mode-locked semiconductor lasers.

In this paper, we analyze the effect of optical feedback on the dynamics of a passively mode-locked ring laser operating in the regime of temporal localized structures. This laser system is modeled by a set of delay differential equations, which include delay terms associated with the laser cavity and the feedback loop. Using a combination of direct numerical simulations and path-continuation techniques, we show that the feedback loop creates echoes of the main pulse whose position and size strongly depend on the feedback parameters. We demonstrate that in the long-cavity regime, these echoes can successively replace the main pulses, which defines their lifetime. This pulse instability mechanism originates from a global bifurcation of the saddle-node infinite-period type. In addition, we show that, under the influence of noise, the stable pulses exhibit forms of a behavior characteristic of excitable systems. Furthermore, for the harmonic solutions consisting of multiple equispaced pulses per round-trip, we show that if the location of the pulses coincides with the echo of another, the range of stability of these solutions is increased. Finally, it is shown that around these resonances, branches of different solutions are connected by period-doubling bifurcations.

Determination of Weak Terrestrial Water Storage Changes from GRACE in the Interior of the Tibetan Plateau

Time series of the Gravity Recovery and Climate Experiment (GRACE) satellite mission have been successfully used to reveal changes in terrestrial water storage (TWS) in many parts of the world. This has been hindered in the interior of the Tibetan Plateau since the derived TWS changes there are very sensitive to the selections of different available GRACE solutions, and filters to remove north-south-oriented (N-S) stripe features in the observations. This has resulted in controversial distributions of the TWS changes in previous studies. In this paper, we produce aggregated hydrology signals (AHS) of TWS changes from 2003 to 2009 in the Tibetan Plateau and test a large set of GRACE solution-filter combinations and mascon models to identify the best combination or mascon model whose filtered results match our AHS. We find that the application of a destriping filter is indispensable to remove correlated errors shown as N-S stripes. Three best-performing destriping filters are identified and, combined with two best-performing solutions, they represent the most reliable solution-filter combinations for determination of weak terrestrial water storage changes in the interior of the Tibetan Plateau from GRACE. In turn, more than 100 other tested solution-filter combinations and mascon solutions lead to very different distributions of the TWS changes inside and outside the plateau that partly disagree largely with the AHS. This is mainly attributed to less effective suppression of N-S stripe noises. Our results also show that the most effective destriping is performed within a maximum degree and order of 60 for GRACE spherical harmonic solutions. The results inside the plateau show one single anomaly in the TWS trend when additional smoothing with a 340-km-radius Gaussian filter is applied. We suggest using our identified best solution-filter combinations for the determination of TWS changes in the Tibetan Plateau and adjacent areas during the whole GRACE operation time span from 2002 to 2017 as well as the succeeding GRACE-FO mission.

Exact and Sinusoidal Periodic Solutions of Lienard Equation Without Restoring Force

In this paper we study a Lienard equation without restoring force. Although this equation does not satisfy the classical existence theorems, we show, for the first time, that such an equation can exhibit harmonic periodic solutions. As such the usual existence theorems are not entirely adequate and satisfactory to predict the existence of periodic solutions.