Traditionally, the models of the unsteady aero- dynamic loads needed for aircraft flight simula- tions have been estimated using the aerodynamic derivatives approach, which, using linear aerody- namic models, provides the influence of the air- craft motion rates on the aerodynamic forces and moments. With increasing aircraft maneuverabil- ity resulting in nonlinear unsteady flow regimes, however, the linearity assumption of the conven- tional aerodynamic derivatives approach makes the **method** questionable. Methods with higher reliability have been show to be achievable by using knowledge of the aircraft aerodynamic re- sponse to **harmonic** excitations. Prompted by the need of rapidly and accurately estimating such response, this study demonstrates the applicabil- ity of the nonlinear frequency–domain Navier– Stokes **Harmonic** **Balance** **method** for predicting periodic aircraft flows with low and high levels of nonlinearity. Using the NASA Common Re- search Model aircraft case study, it is found that the **Harmonic** **Balance** technology yields esti- mates of the unsteady forces differing negligibly from those of the standard time–domain Navier– Stokes **method** with a runtime analysis reduced by at least one order of magnitude over that of the time–domain approach.

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Recently, Neild & Wagg [8] have extended the work of Jezequel & Lamarque [7] to deal directly with second-order systems of the form (1.1); see §1 for the details. One purpose of this paper is to survey their **method** and to demonstrate its power through applications. Primarily though, we shall show how the second-order **method** can be recast using the general theory of normal forms, where forcing and damping terms can be treated as specific forms of unfoldings. One does not need to make the additional approximations inherent in the **harmonic** **balance** **method** when deriving the normal form itself. Application of the **harmonic** **balance** **method** to the derived normal form, though, is shown to lead to remarkably powerful predictions for resonant responses, leading to so-called backbone curves in structural analysis, for mode switching in complex multimodal responses and for analytically finding NNMs.

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The approximate solution of **harmonic** **balance** **method** and the multiple scales **method** from the perturbation techniques are chosen to perform the analytical analysis of a base excited hardening nonlinear system. **Harmonic** **balance** **method** is assumed to have the solution in terms of single mode as written in equation (4) while multiple scales **method** is solved up to first order expansion. Both **method** were expressed in the amplitude - frequency relationship for comparison.

Again these expressions can also be obtained through the lowest order **harmonic** **balance** **method** for (6) with (31). They solve exactly the coefficients of the fundamental and third order harmonics in the residual of this eq- uation when x ( ) τ = A cos ( ) τ . Consequently, the higher order approximation to the solution is taken once again as in (19). It is obvious here that the determination of the Fourier expansion of f x ( ( ) τ ) is untractable. One can first envisage to expand f x ( ( ) τ ) in power series of d prior to the computation of the Fourier coefficients. However, upon substituting (19) in (6) with (31), it is simple to reduce it to the same denominator and consider only its numerator. Then by considering the coefficients of cos ( ) τ , cos 3 ( ) τ and cos 5 ( ) τ we find after some algebraic manipulations that

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21. Hu, H., “Solution of a quadratic nonlinear oscillator by the **method** of **harmonic** **balance**,” J. Sound Vib., Vol. 293, 462, 2006. 22. Itovich, G. R. and J. L. Moiola, “On period doubling bifurcations of cycles and the **harmonic** **balance** **method**,” Chaos Solitons Fractals, Vol. 27, 647, 2005.

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Abstract: The Van der Pol oscillator is a nonlinear damping and non-conservative oscillator. Energy is generated at low amplitude and dissipated at high amplitude. This nonlinear oscillator was first introduced by Dutch electrical engineer and physicist B. Van der Pol and it was originally used to investigate vacuum tubes. Nowadays, it is used in both physical and biological sciences. It is also used in sociology and even in economics. It has a limit cycle and in earlier it was determined by the classical perturbation methods when the nonlinear term is small. Then the **harmonic** **balance** **method** was used to determine the limit cycle for stronger nonlinear case. Moreover, many researchers have been analyzed this oscillator by various numerical approaches. In this article, a new analytical approach based on **harmonic** **balance** **method** is presented to determine the limit cycle as well as approximate solutions of this nonlinear oscillator. The frequency as well as the limit cycle obtained by new approach has been compared with those obtained by other existing methods. The present **method** gives better result than other existing results and also close to the corresponding numerical result (considered to the exact result). Moreover, the present **method** is simpler than the existing **harmonic** **balance** **method**.

The electrical characteristics of harmonics are reflected by not only the voltage (or electric field) but also the current (or magnetic field). It is the power (there is no any other quantity) that involves both of them (voltage and current). So only the quantitative liability division **method** for **harmonic** pollution based on **harmonic** power is the most complete and explicit one. In this paper, employing our achievements, two quantitative liability divi- sion methods for **harmonic** pollution based on the **harmonic** power are proposed. Simulation and analysis show that the quantitative liability division **method** for **harmonic** pollution based on line-transferred active power component is the most desirable one.

Figure 4 shows probability density functions (PDFs) for res- idence times. Theoretically, the global PDFs of evaporation and precipitation residence time should be identical. How- ever, they slightly differ, which can be attributed to inconsis- tencies in the ERA-I forcing data as well as the assumptions made in the tracking model. Regarding the forcing data, it seems from Fig. 1 that the lifetime of atmospheric moisture is slightly too short, a common issue in all models (Tren- berth et al., 2003), indicating that our results may be slightly skewed towards lower residence times. According to Tren- berth et al. (2003), precipitation also falls too early in the day in all models, thus the amplitude of residence times over land could also be affected, but it is unclear to what extent. Regarding the modeling assumptions, in 3D-T we assume a humidity-weighted well-mixed atmosphere during precipita- tion and the starting location of the trajectories is randomized over the grid cell. These assumption may lead to an underes- timation of the number of water particles that undergo a very fast cycle, and may, thus, slightly skew our results towards higher residence times. By definition of mass **balance**, how- ever, the actual mean of the distribution should not change. When more water particles undergo a faster(slower) cycle, as a logical consequence, also more water particles undergo a slower(faster) cycle. Adding age tracers to online tracking methods (Wei et al., 2016), but then applied to new global methods (e.g., Singh et al., 2016), would allow to check the validity and consequences of these assumptions in more de- tail, however, would still depend on the model world.

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In this case the reference signal to be amplified will be the voltage at the point of common coupling. So the voltage calculated in the real time simulation is transferred to the Power Amplifier control unit after an ADC and DAC respectively. Once the signal is in the control unit of the power amplifier, it is processed through the DFT transforms. The DFT algorithm used allows for a variable frequency of the processed signal, so being accurate during frequency changes. The DFT output values, in the frequency domain, will be phase-shifted **harmonic** by **harmonic** and phase by phase according to the measured loop delay, hence compensating the time delay. Then, the reconstruction of the signal into the time domain will be carried out. The reconstructed phase-shifted waveform is then introduced as reference for the PI-Resonant controller type of the Power Amplifier that finally will amplify the voltage to that of the reference and applies it to the device under test as shown in Fig 2. In this manner the time delay compensation will not affect to the system topology and therefore the dynamic behaviour of the original system will stay as it originally was in terms of power angles and V-I phase relationships for all the harmonics processed. The main limitation of this algorithm is that it is not appropriate for the accurate reproduction of fast transients, i.e. sub-cycle step- changes to the fundamental or **harmonic** amplitudes. The DFT window length is finite, and in this paper a 2-cycle triangular window, adaptive to fundamental frequency, is used. So, any step change in fundamental or **harmonic** content or voltage within the simulation will be represented as a smoothed “ramp” of the component’s amplitude/phase over 2 cycles in the PHIL environment. This is the drawback of the proposed approach. On the other hand, the benefit is that ALL the voltage-to-voltage and voltage-to-current amplitude and phase relationships for the fundamental and harmonics should be maintained accurately for quasi-steady-state operation, i.e. for all dynamic cases except the most abrupt “sub-2-cycle” step changes which will be “smoothed” over 2 cycles.

Approximate analytical methods have been used extensively for finding approximate solutions to nonlinear ordinary differential equations. In this paper we compare the recently developed direct normal form transformation with two other very well known and long standing methods, **harmonic** **balance** and the **method** of multiple scales. We will show that the direct normal form **method** combines some of the key advantages of **harmonic** **balance** and multiple scales whilst reducing some of the limitations.

In this paper, an improved maximum power point (MPP) tracking (MPPT) with better performance based on voltage-oriented control (VOC) is proposed to solve a fast-changing irradiation problem. In VOC, a cascaded control structure with an outer dc link voltage control loop and an inner current control loop is used. The currents are controlled in a synchronous orthogonal d , q frame using a decoupled feedback control. The reference current of proportional-integral (PI) d -axis controller is extracted from the dc-side voltage regulator by applying the energy-balancing control. Furthermore, in order to achieve a unity power factor, the q -axis reference is set to zero. The MPPT controller is applied to the reference of the outer loop control dc voltage photovoltaic (PV). Without PV array power measurement, the proposed MPPT identifies the correct direction of the MPP by processing the d -axis current reflecting the power grid side and the signal error of the PI outer loop designed to only represent the change in power due to the changing atmospheric conditions. Simulations and experimental results demonstrate that the proposed **method** provides effective, fast, and perfect tracking.

their controlled turn-on and turn off. In this project the concept of HEPWM technique is described with performance analysis of waveform. Total **Harmonic** Distortion (THD) is helped to proof the calculation by showing the reduction value of THD. Finally, the simulation result with all analysis for the strategy applies in differential evolution and for three phase inverter using HEPWM is presented.

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Switching strategy aims to calculate the firing angles in such a way that, firstly, the voltage amplitude of the fundamental frequency be generated and secondly, excessive harmonics not be produced or the total **harmonic** distortion of the output voltage be small. One of the strategies used in the multilevel inverters is SHEPWM. In other words, in addition to the functionality of multilevel inverters to reduce the total **harmonic** content, by use of the SHEPWM strategy, some harmonics are selected and as much as possible closed to zero. Therefore, the main problem to control the multilevel inverters is to find the firing angles. This **method** is used to eliminate the low order harmonics, provided that the fundamental harmonics are satisfying.

Very interestingly, the **harmonic** oscillator comprises one of the most important examples of elementary Quantum Mechanics. There are several reasons for its pivotal role. The linear **harmonic** oscillator describes vibrations in molecules and their counterparts in solids, the phonons. Many more physical systems can, at least approximately, be described in terms of linear **harmonic** oscillator models. However, the most eminent role of this oscillator is its linkage to the electron, one of the conceptual building blocks of microscopic physics. For example, electron describe the modes of the radiation field, providing the basis for its quantization [17].

ABSTRACT: This paper proposes a design and simulation of Solid State Power Amplifier (SSPA) using ADS which will be useful for pulsed radar applications. The selection of proper device for the desired application plays a vital role. There are various steps involved in the present design such as Direct Current (DC) simulation in order to get the proper quiescent operating point under the platform. Subsequently, Stability simulations are to be carried out to make sure that the amplifier can be unconditionally stable. Load pull and Source pull simulations are required to match the load and source impedances, and finally **Harmonic** **Balance** (HB) simulations are to be carried out to verify the **harmonic** levels of the signals other than designed frequency of operation.

increasing, lead to power system **harmonic** distortion is more and more serious, precise analysis and detection of the **harmonic** in power system become more and more important.Different types of harmonics in the grid produce different types of harmonics.At present, for the **harmonic** detection is mainly based on fast Fourier transform and its improved algorithm, as well as time series analysis and neural network **method** [1], because of the Fourier transform analysis of **harmonic** is simple and effective, and can accurately every **harmonic** analysis of stationary signal, so it has been the main methods of power system **harmonic** analysis.However, with the increase of the **harmonic** content of the power grid, the components become more and more complex, the transient mutation and other unsteady components become more and more prominent, and it becomes very difficult to analyze the harmonics by FFT only.

The drawback of using NS CFD for analyzing yawed flows is its high computational cost. Time–domain (T D) NS simulation of HAWT periodic flows require long runtimes as several rotor revolutions need to be simulated before a periodic state is achieved. This runtime can be significantly reduced by solving the governing equations in the frequency–domain. A widespread **method** of this type is the **harmonic** **balance** (HB) NS technology, initially introduced for turbomachinery blade aeroelasticity [14] and successively also used for multi–stage turbomachinery aerodynamics [15, 16], several vibratory motion modes of aircraft configurations [17, 18, 19], and recently also for wind turbine aerodynamics [20, 21] and aeroelasticity [22]. The use of the HB NS **method** for the simulation of this type of periodic flows has been shown to reduce by one to two orders of magnitude CFD runtimes with respect to conventional T D NS analyses. Several other nonlinear frequency–domain NS methods exist and have been applied in the abovesaid areas, as more extensively reported in [20].

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The quantification of surface water -groundwater interactions using isotope and geochemical tracers is achieved by applying the isotopic or geochemical mass **balance** methods [17]. Interaction of river and the groundwater components in the Huaisha River basin of China [18], the Huasco and Limari river basins in the North Central Chile [19], the Hillsborough River watershed of West-Central Florida [20] etc. was estimated by applying the isotope mass **balance** and the conductivity mass **balance** **method** with other conventional methods like groundwater level monitoring and stream flow measurements.

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Abstract The extended homogeneous **balance** **method** is used to construct exact traveling wave solutions of the Maccari system, in which the homogeneous **balance** **method** is applied to solve the Riccati equation and the reduced nonlinear ordinary differential equation. Many exact traveling wave solutions of the Maccari system equation are successfully obtained.