Harmonic Balance

COSMOS Pass 3 establishes a production Newton-Raphson solver for the static two-body sector of the DBI + Cubic Galileon modified-gravity framework. The solver employs sparse LU factorisation (SuperLU via SciPy), a 5×5 graph-coloured finite-difference Jacobian, and homotopy continuation in the nonlinearity parameter ε from 0 to 1. A formal verification and validation campaign confirms second-order spatial convergence (L² orders 1.999/2.000 at N=64/128/256), Jacobian directional-derivative consistency (order 1.000), nonlinear operator self-consistency at machine precision (sR init s ~ 10-15), positive kinetic eigenvalues (λ min = +1.023-+1.028, ELLIPTIC), and symmetry preservation to 10-23. Two critical implementation defects discovered during verification were corrected (Jacobian boundary exclusion; MMS source sign). The ε-continuation engine is validated. Nontrivial source seeding (OI-1b) is the active blocking item for the first screened-branch solution and subsequent Solar-System constraint analysis.

COSMO-S Pass 3 establishes a production Newton-Raphson solver for the static two-body sector of the DBI + Cubic Galileon modified-gravity framework. The solver employs sparse LU factorisation (SuperLU via SciPy), a 5×5 graph-coloured finite-difference Jacobian, and homotopy continuation in the nonlinearity parameter ε from 0 to 1. A formal verification and validation campaign confirms second-order spatial convergence (L² orders 1.999/2.000 at N=64/128/256), Jacobian directional-derivative consistency (order 1.000), nonlinear operator self-consistency at machine precision (sR init s ~ 10 -15 ), positive kinetic eigenvalues (λ min = +1.023-+1.028, ELLIPTIC), and symmetry preservation to 10 -23 . Two critical implementation defects discovered during verification were corrected (Jacobian boundary exclusion; MMS source sign). The ε-continuation engine is validated. Nontrivial source seeding (OI-1b) is the active blocking item for the first screened-branch solution and subsequent Solar-System constraint analysis.

In this work we predict the nonlinear behaviour of an eight-pole quasi-elliptic bandpass high temperature superconducting (HTS) filter with an equivalent circuit extracted from intermodulation measurements performed at the centre of the filter passband. We present measurements that show that the equivalent circuit is able to predict the intermodulation products produced by the filter when driven by two in-band or out-of-band sinusoidal signals. Numerical techniques based on harmonic balance are used to extract the elements of the equivalent circuit and to simulate its nonlinear performance.

This paper presents a new basic nonlinear track model consisting of an infinite Euler-Bernoulli beam (rail) resting on continuous foundation with two elastic layers (rail pad and ballast bed) and intermediate inertial layer (sleepers). The two elastic layers have bilinear elastic characteristic obtained from the load-displacement characteristic of the rail pad and ballast. A time-varying load with two components - time-constant one and harmonic other, representing the wheel/rail contact force is considered as the track model input. Rail deflection due to the time-constant component of the load is obtained solving the nonlinear equations of the balance position. Subsequently, the structure of the nonhomogeneous foundation is determined. Dynamic rail response in terms of receptance due to the harmonic component of the load is calculated using the linearised track model with nonhomogeneous elastic characteristic. Influence of the time-constant component and the reflected waves due to t...

Due to the extreme health concerns associated with asbestos-based brake linings, researchers are looking at using ecologically friendly bio-based biomaterials as reinforcing agents in composite materials used to make brake pad linings. The goal of this study is to see whether using sea shell (SS) powder in the production of asbestos-free brake lining materials is feasible. The powders were combined with the necessary fillers to make the brake lining. The reinforcing powders' compositions were varied between 20% and 35%, the resin binder's composition was changed between 58 and 43 percent, and the filler metal and curing agents' compositions were kept constant. These ingredients were weighed, prepared, combined, cured, and moulded. The composite materials were tested for water and oil absorption, compressive strength, hardness, and wear. According to the findings, increasing the quantity of reinforcing components increases the water and oil absorption of the samples. With the addition of up to 35% content, the composite's compressive strength increased proportionally. The tested coefficient of friction was found to be within acceptable limits. Furthermore, when mixed with other fillers and restricted, SS powder has shown tremendous promise in the production of brake linings.

A Transformada de Laplace é uma das ferramentas mais poderosas da matemática aplicada, utilizada para transformar equações diferenciais em equações algébricas, facilitando sua resolução. Tradicionalmente, ela é entendida como uma mudança do domínio do tempo para o domínio complexo. Entretanto, sob uma perspectiva vetorialética, essa transformação não é apenas uma mudança de domínio, mas uma projeção estrutural que revela a organização interna de sistemas dinâmicos.

This paper is concerned with the development and evaluation of a number of modeling techniques which improve Qucs Harmonic Balance simulation performance of RF compact device models. Although Qucs supports conventional SPICE semiconductor device models, whose static current/voltage and dynamic charge characteristics exhibit second and higher order derivatives may not be continuous, there is no guarantee that these will function without Harmonic Balance simulation convergence problems. The same comment also applies to a number of legacy compact semiconductor device models. The modeling of semiconductor devices centered on non-linear Equation-Defined Devices and blocks of Verilog-A code, combined with linear components, is introduced. These form a class of compact macromodel that has improved Harmonic Balance simulation performance. To illustrate the presented modeling techniques RF diode, BJT and MESFET macromodels are described and their Harmonic Balance performance simulated with Q...

Following the release of Qucs-0.0.18 in August 2014 the Qucs Development Team considered in detail a number of possible directions that future versions of the software could take. Spice4qucs is one of these routes. It addresses a number of problems observed with the current version of Qucs while attempting to combine some of the best features of other GPL circuit simulation packages. The project also aims to add additional model development tools to those currently available in Qucs-0.0.18. Qucs was originally written as an RF and microwave engineering design tool which provided features not found in SPICE, like S parameter simulation, two and multiport small signal AC circuit analysis and RF network synthesis. Since it was first release under the General Public License (GPL) in 2003 Qucs has provided users with a relatively stable, flexible and reasonably functional circuit simulation package which is particularly suited to high frequency circuit simulation. In the years following ...

This paper is concerned with the development and evaluation of a number of modelling techniques which improve Qucs Harmonic Balance simulation performance of RF compact device models. Although Qucs supports conventional SPICE semiconductor device models, whose static current/voltage and dynamic charge characteristics exhibit second and higher order derivatives may not be continuous, there is no guarantee that these will function without Harmonic Balance simulation convergence problems. The same comment also applies to a number of legacy compact semiconductor device models. The modelling of semiconductor devices centered on non-linear Equation-Defined Devices and blocks of Verilog-A code, combined with linear components, is introduced. These form a class of compact macromodel that has improved Harmonic Balance simulation performance. To illustrate the presented modelling techniques RF diode and bipolar junction transistor macromodels are described and their Harmonic Balance performance simulated with Qucs and Xyce c .

Passive and semipassive ultrawideband (UWB) radio-frequency identification (RFID) technology has been recently proposed to offer high-accuracy localization capabilities in next-generation RFID systems. This technology relies on the modulation of backscattered signals, i.e., backscatter modulation, from multiple tags present in the environment. The detection of multiple tags based on backscattered signals is challenging in harsh environments with nonideal conditions such as clutter, near-far interference effects, and clock drift. This paper analyzes the detection of multiple tags employing UWB backscatter modulation and proposes practical signaling, spreading codes, and detection schemes that are robust to nonideal conditions. A case study is presented to evaluate the performance of the proposed technique for the detection of multiple tags based on impulsive backscattered signals.

Duffing harmonic oscillator is a common model for nonlinear phenomena in science and engineering. This paper presents He´s Energy Balance Method (EBM) for solving nonlinear differential equations. Two strong nonlinear cases have been studied analytically. Analytical results of the EBM are compared with the solutions obtained by using He´s Frequency Amplitude Formulation (FAF) and numerical solutions using Runge-Kutta method. The results show the presented method is potentially to solve high nonlinear oscillator equations.

This article analyzes a strongly nonlinear oscillator with cubic and harmonic restoring force and proposes an efficient analytical technique based on the modified energy balance method (MEBM). The proposed method incorporates higher-order approximations. After applying the proposed MEBM, a set of complicated higher-order nonlinear algebraic equations are obtained. Higher-order nonlinear algebraic equations are cumbersome to investigate especially in the case of a large initial oscillation amplitude. This limitation is overcome in the proposed method by using the iterative procedure based on the homotopy perturbation method. The approximated results agree well with the numerically obtained exact solutions. These third-order approximate solutions are found to be almost the same as exact solutions, which cannot be found using the existing energy balance method. Highly accurate result and simple solution procedure are advantages of this proposed method, which could be applied to other n...

We propose a new method, namely, the modified harmonic balance method. This paper also analyses and offers the high-order approximate periodic solutions to the strongly nonlinear oscillator with cubic and harmonic restoring force. The existing harmonic balance method cannot be applied directly to such kind of nonlinear oscillators in the presence of forcing term. It is possible if we rewrite the original form of the nonlinear oscillators. If we do so, the results are valid only for small values of amplitude of the oscillation. Moreover, after applying the existing harmonic balance method, a set of complicated higher-order nonlinear algebraic equations are obtained. Analytical investigation of these equations is cumbersome especially when the amplitude of the oscillation is large. These limitations are removed in the proposed method. In addition, a suitable truncation principle has also been used in which the solution achieves better results than existing solutions. The approximate results agree well with numerically obtained exact solutions. Highly accurate results and a simple solution procedure are the advantages of this proposed method, which could be applied to other nonlinear oscillatory problems arising in nonlinear science and engineering.

In this paper, we implement a new approach coupled with the iteration method. This procedure is obtained by combining He's frequency-amplitude formulation and He's energy balance method into a new iteration procedure such that excellent approximate analytical solutions, valid for small as well as large values of amplitude, can be determined for nonlinear oscillators. This study has clarified the motion equation of nonlinear oscillators by the iteration method to obtain the relationship between amplitude and angular frequency. We compare the approximate periods obtained by our procedure with the numerical solution and with other methods like energy balance method and variational iteration method. The results show that the approximations are of extreme accuracy.

Report presenting a flight and wind-tunnel investigation to determine the aerodynamic vibration characteristics of ailerons for the Douglas D-558-2 airplane. Testing was conducted to investigate the possibilities of a single-degree-of-freedom flutter known as aileron "buzz" or aileron compressibility flutter. Results regarding the three types of aileron vibration observed and vibration amplitudes are provided.

The assimilation of discrete higher fidelity data points with model predictions can be used to achieve a reduction in the uncertainty of the model input parameters which generate accurate predictions. The problem investigated here involves the prediction of limit-cycle oscillations using a High-Dimensional Harmonic Balance method (HDHB). The efficiency of the HDHB method is exploited to enable calibration of structural input parameters using a Bayesian inference technique. Markov-chain Monte Carlo is employed to sample the posterior distributions. Parameter estimation is carried out on both a pitch/plunge aerofoil and Goland wing configuration. In both cases significant refinement was achieved in the distribution of possible structural parameters allowing better predictions of their true deterministic values.

This work proposes a extends a novel approach to compute transonic Limit Cycle Oscillations using high fidelity analysis. CFD based Harmonic Balance methods have proven to be efficient tools to predict periodic phenomena. This paper’s contribution is to present a methodology to determine the unknown frequency of oscillations using an implicit formulation of the HB method to accurately capture Limit Cycle Oscillations (LCOs); this is achieved by defining a frequency updating procedure based on a coupled CFD/CSD Harmonic Balance formulation to find the LCO condition. A pitch/plunge aerofoil and respective linear structural models is used to exercise the new method. Results show consistent agreement between the proposed and time-marching methods for both LCO amplitude and frequency.

The dynamic behavior of a physical pendulum system of which the support is subjected to both rotation and vertical vibration are studied in this paper. Both analytical and computational results are employed to obtain the characteristics of the system. By using Lyapunov's direct method the conditions of stability of the relative equilibrium position can be determined. Melnikov's method is applied to identify the existence of chaotic motion. The incremental harmonic balance method is used to "nd the stable and unstable periodic solutions for the strong non-linear system. By applying various numerical results such as phase portrait, PioncareH map, time history and power spectrum analysis, a variety of the periodic solutions and the phenomena of the chaotic motion can be presented. The e!ects of the changes of parameters in the system could be found in the bifurcation and parametric diagrams. Further, chaotic motion can be veri"ed by using Lyapunov exponent and Lyapunov dimension. The global analysis of basin boundary and fractal structure are observed by the modi"ed interpolated cell mapping method. Besides, non-feedback control, delayed feedback control, adaptive control, and variable structure control are used to control the chaos e!ectively.

In the following we consider a 2-dimensional system of ODE's containing quasiperiodic terms. The system is proposed as an extension of Mathieu-type equations to higher dimensions, with emphasis on how resonance between the internal frequencies leads to a loss of stability. The 2-d system has two 'natural' frequencies when the time dependent terms are switched off, and it is internally driven by quasiperiodic terms in the same frequencies. Stability charts in the parameter space are generated first using numerical simulations and Floquet theory. While some instability regions are easy to anticipate, there are some surprises: within instability zones small islands of stability develop, and unusual 'arcs' of instability arise also. The transition curves are analyzed using the method of harmonic balance, and we find we can use this method to easily predict the 'resonance curves' from which bands of instability emanate. In addition, the method of multiple scales is used to examine the islands of stability near the 1:1 resonance.

In this study, the parallel inverted dual time stepping (PIDTS) method has been investigated for analyzing turbomachinery unsteady flows. This is the first known effort in exploiting temporal parallelization to reduce wall-clock time for analyzing turbomachinery unsteady flows in all time scales. This method relaxes the sequential time dependency of solutions at different time instants in the dual time stepping method to achieve parallel solutions at the expense of an increased number of pseudo-time iterations. To demonstrate its parallel scalability and solution stability and accuracy, a one-dimensional De Laval nozzle with a time-periodic back pressure disturbance has been used as a test case. Study indicates that the higher number of pseudo-time iterations is inevitable with a larger number of time instants marching together. A hybrid explicit and implicit method for accelerating solution convergence significantly mitigates the need for an increased number of pseudo-time iterations. Further verification and application have been conducted using a case of rotor-stator interaction of a transonic compressor stage. The obtained parallel efficiency is about 96%, 90%, 83%, 77%, and 72% with 2, 4, 8, 12, and 16 time instants marching together, respectively. The recommended number of time instants for parallelization is 2-10, which is a result of balancing parallel benefits, time consumption of additional pseudo-time iterations, and additional memory consumption. The obtained wall-clock speedup compared with the dual time stepping method is 1.9 to 3.5. To reduce the wall-clock time in an unsteady flow analysis, there has been research effort on two fronts. One is parallel computing by making use of available computing hardware. The domain decomposition-based approach is widely used in daily life. The computational domain is decomposed into many sub-domains and each sub-domain is computed by one CPU core. This can reduce the analysis wall-clock time to a large extent. The other front is the development of efficient reduced order methods by exploiting turbomachinery flow and domain features. In recent years, many efficient reduced-order methods for turbomachinery unsteady flows have been developed, by exploiting the temporal and spatial periodicity in the flow field resulting from the cyclic symmetry of turbomachinery structures. The representative reduced order methods are the harmonic balance method (Hall et al., 2013; Sicot et al., 2013 and the nonlinear harmonic method . It can be considered that the two methods transform the original initial value problem into a boundary value problem. With one fundamental frequency, the time domain size is the reciprocal of the fundamental frequency. The time domain is discretized using uniformly distributed points. At the time boundary, a periodic boundary condition is applied. The approximation of the time derivative is achieved using spectral methods in the harmonic balance method. Nonetheless, other methods can also be used, such as the cubic spline approximation in the time collocation method (Du and Ning, 2014, 2017) and the finite difference method in the time vector marching method (Zhang and Wang, 2021). The transformation of an initial value problem to a boundary value problem makes it possible to parallelize analysis in time, leading to a further reduction of wall-clock time. Nevertheless, these reduced-order methods have their limitations. For example, the computational efficiency of the methods diminishes with the increase of the number of frequencies retained in an analysis; they cannot handle unsteady flows for which the frequencies of unsteadiness are not known a priori; the methods have convergence issues for near stall operating points. The time domain dual time stepping method is general and applicable to all scenarios. However, as mentioned earlier, it takes a long time to filter the influence of solution initialization. To benefit from the domain decomposition based parallelization, the size of each sub-domain should not be too small, otherwise the incurred overhead will offset any benefit. Therefore, there is a limit to further reduce wall-clock time using the domain decomposition based approach. With the availability of computing resources, it is desired to further reduce analysis wall-clock time through parallel computing. This can only be achieved from time parallelization. Because of the one-sided difference scheme for time discretization in the dual time stepping method resulting from the invariable forward time direction, a temporal parallelization strategy cannot be the same as a domain decomposition strategy. Nievergelt (1964) first proposed the concept of temporal parallelization and pointed out that it can reduce wallclock time for integrating ordinary differential equations at the cost of redundant calculations. After that, many temporal parallel methods for time dependent simulations have been proposed in other fields. classified temporal parallel methods into four categories: 1) methods based on multiple shooting; 2) methods based on domain decomposition and waveform relaxation; 3) methods based on multigrid; and 4) direct time parallel methods. Methods based on multiple shooting decompose the whole time domain into several sub-domains and perform simulations separately in parallel to get an approximation solution for the entire time domain. The continuity of the solution on different time intervals is very important. A representative and widely used method is the parareal method . This method involves two time steps, a large one and a small one, corresponding to a coarse solver and a fine solver. In this method, the coarse solver is used to perform the solution over the global time interval, while the fine solver parallelizes the solution with a higher accuracy in each sub-time interval. The alternate iterative use of these two time steps ensures the solution accuracy over the entire time domain. However, its parallel efficiency is limited because of the sequential computing in the coarse solver applied this method to solving the Navier-Stokes equations and pointed out that a coarser discretization in space should be considered to match the large time step in the coarse solver to preserve stability and obtain a higher parallel speedup. Different from the methods above, methods based on domain decomposition and waveform relaxation (Gander, 2015; Lelarasmee et al., 1982; Crow and Ilic, 1990) are proposed to decompose the whole domain into several sub-domains that are relatively loosely coupled to neighbors along the time direction. Then an appropriate method is adopted to solve equations for each sub-domain considering its own characteristics. The dependency of each sub-domain on other ones is achieved through the solution obtained by the previous iteration, so each sub-domain is decoupled and can be solved in parallel. These methods are known as waveform relaxation (WR) methods. The advantage of WR methods is that different temporal discretization methods and time steps can be used for sub-domains with different time scales. However, there is no general strategy to decompose the entire domain so that the coupling of sub-domains is as loose as possible while the number of decomposed sub-domains is as many as possible. Methods based on multigrid solve equations in the entire time domain, which involve the time coarsening problem. The parallelism comes from the multi-level grid structure instead of the domain decomposition in the previous two methods. The parareal method can also be regarded as a variant of two-level multigrid method due to the utilization of two time steps . proposed a space-time multigrid method for the solution of parabolic partial differential equations (PDEs), which addresses the time coarsening problem firstly. A multigridreduction-in-time (MGRIT) algorithm can achieve significant speedup compared with the sequential time marching method

The objective of this work is to study the dynamical behavior of vehicle suspension systems employing asymmetrical viscous damping, with a focus on improving passenger comfort. Previous studies have shown that the use of asymmetrical dampers in these types of systems can be advantageous with regard to comfort of the passengers. The modeling and the behavior of a quarter-car model with asymmetrical viscous damping under harmonic excitation is presented. The response is obtained with an analytical approximation via the method of Harmonic Balance. The choice of the asymmetry ratio diminishes the effects that the uneven road causes on the displacement and acceleration of the sprung mass. Although current systems usually adopt larger damping during the expansion phase, it is shown in this work that, for lower frequencies, smaller damping in this phase results in better comfort.

Two different field solver tools have been developed in order to facilitate fast thermal and electro-static simulation of microsystem elements. The µS-THERMANAL program is capable for the fast steadystate and dynamic simulation of suspended multilayered microsystem structures. The 2D-SUNRED program is the first version of a general field solver based on an original method, the successive node reduction. SUNRED offers a very fast and accurate substitute of FEM programs for the solution of the Poisson equation. Steady-state and dynamic simulation examples demonstrate the usability of the novel tool.