Harmonic Truncation Research Articles

The Influence of Coordinate Systems on the Stability Analysis of Lateral–Torsional Coupled Vibration

Stability analysis of lateral–torsional coupled vibration is obligatory for rotating machinery, such as aero-engines. However, the state-of-the-art method may lead to stability misjudgment under different coordinate systems. The cause of this misjudgment has not yet been well explored. The purpose of this paper is to clarify the error source of the stability analysis in a more comprehensive manner. A vertical Jeffcott rotor model including torsion vibration is built, and the Lagrange approach is applied to establish the motion equations. The coordinate transformation matrix is used to transfer the motion equations into the rotating coordinate system, making the coefficients of the motion equation constants. The differences in the unstable speed regions in the two coordinate systems are captured. The limitations of the Floquet theory and Hill’s determinant analysis in the stability estimation of the lateral–torsional coupled vibration are explained. It is found that, for Hill’s method, increasing the number of the harmonic truncation cannot correct the misjudgment, and the matrix truncation is the fundamental error source. The above research provides more accurate theoretical support for the analysis of the lateral–torsional coupling instability of rotors.

An Extended Generalized Average Modeling Framework for Power Converters

The Generalized Averaged Modeling (GAM) technique is traditionally employed to capture the dynamic performance of power electronic converters. This paper proposes an improved version of it, named the Extended-GAM (EGAM) technique, which supports the multiplication of two Double Fourier Series (DFS) signals in the time domain. Multiplication of DFS signals in the time domain translates to the 2D-convolution of coefficients of the DFS terms of their equivalent Discrete Fourier Image (DFI) representations. Thus, the proposed EGAM technique, capable of capturing many harmonics present in the output of a power converter, effectively captures the dynamic behavior of power converters excited by two distinct frequencies. The proposed technique is then converted into an algorithm suitable for numerical platforms, which typically use Ordinary Differential Equation (ODE) solvers. The proposed algorithm is validated based on the observations of the effects of harmonic truncation. The efficacy of the proposed technique is assessed through a case study, wherein a single-phase inverter employs LC filters on both the dc-link and the ac-side. Finally, it is shown that the results obtained with the proposed method show an excellent congruence between simulation and hardware experimental models. Additionally, the proposed algorithm is packaged into a MATLAB toolbox and shared for future implementations.

An extended Vlasov–Fokker–Planck approach for kinetic simulations of laser plasmas

Vlasov–Fokker–Planck simulation codes occupy an important niche in modeling laser-produced plasmas, since they are well suited to studying the effect of collisions on electron kinetic phenomena, especially energy transport. One of the most important elements of energy transport is the absorption of laser light by the plasma; however, simulating this in detail requires resolving oscillations of the laser light, whose characteristic timescale is orders of magnitude shorter than the simulation time needed to study transport physics. For this reason, most Vlasov–Fokker–Planck codes used to study electron transport in laser plasmas rely on simplified models of the laser–plasma coupling. Their underlying assumptions nominally preclude their use for modeling laser light having short-scale structure in space or time, such as broadband lasers. In this work, we derive a more general computational framework suitable for arbitrarily structured laser fields. Our approach is based on an extended set of Vlasov–Fokker–Planck equations that separately solve for the low- and high-frequency plasma response. We implement these extended Vlasov–Fokker–Planck equations in the spherical harmonic code K2 and demonstrate the performance of the method on several laser absorption test problems, with particular attention to the judicious selection of time steps, time integrators, and spherical harmonic truncation, according to the intensity and spectrum of the laser light under consideration. Comparison with the widely used Langdon absorption operator shows the Langdon operator performs remarkably well for predicting laser heating in the simple cases considered here, even in situations that would seem to violate its underlying assumptions.

Are Chebyshev-based stability analysis and Urabe’s error bound useful features for Harmonic Balance?

Harmonic Balance is one of the most popular methods for computing periodic solutions of nonlinear dynamical systems. In this work, we address two of its major shortcomings: First, we investigate to what extent the computational burden of stability analysis can be reduced by consistent use of Chebyshev polynomials. Second, we address the problem of a rigorous error bound, which, to the authors’ knowledge, has been ignored in all engineering applications so far. Here, we rely on Urabe’s error bound and, again, use Chebyshev polynomials for the computationally involved operations. We use the error estimate to automatically adjust the harmonic truncation order during numerical continuation, and confront the algorithm with a state-of-the-art adaptive Harmonic Balance implementation. Further, we rigorously prove, for the first time, the existence of some isolated periodic solutions of the forced-damped Duffing oscillator with softening characteristic. We find that the effort for obtaining a rigorous error bound, in its present form, may be too high to be useful for many engineering problems. Based on the results obtained for a sequence of numerical examples, we conclude that Chebyshev-based stability analysis indeed permits a substantial speedup. Like Harmonic Balance itself, however, this method becomes inefficient when an extremely high truncation order is needed as, e.g., in the presence of (sharply regularized) discontinuities.

Semianalytic stability algorithm in the frequency domain for interrupted milling

The construction of stability diagrams of interrupted milling cases is generally carried out by means of time-consuming numerical methods in either frequency or time domain, since the period doubling lobes arising under interrupted cutting are omitted by the time-averaged parametric scan of the traditional zeroth order approximation. This paper presents a novel seminalytic method in the frequency domain for interrupted milling. Taking advantage of their analytical chatter frequency distribution, this method adds the period doubling lobes calculated in a single frequency scan to the existing Hopf limits of the zeroth order solution. This, together with intelligent selection of scanning frequency ranges and truncation to the minimum set of harmonics, allows very fast calculation of the stability charts of interrupted milling cases while retaining the analytical basis and advantages of the zeroth order algorithm. The method accurately describes the stability limits at period doubling dominated zones and improves the existing zeroth order solution, but exhibits slight inaccuracies in the prediction of Hopf boundaries due to mode interaction and harmonic truncation effects.

Quasi-3-D Nonlinear Analytical Modeling of Magnetic Field in Axial-Flux Switched Reluctance Motors

Analytical modeling (AM) of the magnetic field for axial-flux switched reluctance motors (AFSRMs), considering three-dimensional (3-D) features and nonlinear saturation effects, remains to be a knotty problem. This paper proposes a quasi-3-D nonlinear AM method to predict magnetic field distributions and electromagnetic performances of an AFSRM. Firstly, the harmonic modeling technique is applied to obtain the spatiotemporal distribution characteristics of the axial and tangential magnetic flux density at the mean radius. Secondly, the nonlinearity of the soft magnetic material in teeth is considered by an adaptive convergence rate algorithm. Finally, a radial dependence function is formulated to extend the solution of the annular section to any point in the 3-D space of the AFSRM. To confirm the accuracy of the proposed method, the analytical solutions of magnetic flux density and electromagnetic torque have been compared with FEM simulation results. Meanwhile, an additional validation has been done for the stability of this nonlinear AM method under severe saturation conditions. At last, the reasonable selection of the harmonic truncation order number has been discussed to reach a good balance between the accuracy and time cost. The quasi-3-D nonlinear AM method can consider the curvature and edge effect as well as the saturation effect effectively, which is suitable for performance predictions and parametric studies of AFSRMs.

GRACE Fast Mascons From Spherical Harmonics and a Regularization Design Trade Study

AbstractMass concentration (mascon) solutions have become a prominent medium for investigating time‐variable gravity recovered by GRACE and GRACE Follow‐On. While GRACE Level‐2 spherical harmonic products require various postprocessing techniques to eliminate correlated noise, mascon formulations employ spatial regularization strategies during the estimation step to improve signal recovery. However, mascon estimation has traditionally required large computing resources and GRACE Level‐1B processing capabilities. In this study, we show that a typical mascon estimation system can be reformulated to allow for the estimation of regularized mascons from Level‐2 spherical harmonics. Provided that spherical harmonic solution covariances are available, the computed mascons will be mathematically equivalent to similar mascons estimated from Level‐1B observations. This method is computationally efficient, better leverages GRACE spherical harmonics than past methods, and matches the performance of typical mascon solutions without locking scientists into predetermined regularization designs. We develop a proof‐of‐concept solution using ITSG‐Grace2018 and compare results with traditional mascons from the Jet Propulsion Lab (JPL) and NASA Goddard Space Flight Center (GSFC). We then assess the effects of spherical harmonic truncation and use of regularization correlations on basin signal recovery. We find that spherical expansions to degree and order 60 provide the minimum expansion necessary to study most basins, while larger expansions help further localize signals. We also find that diagonal regularizations (i.e., regularizations that do not contain inter‐mascon correlations) are adversely affected by leakage, especially across boundaries such as coastlines where signals are not highly correlated, whereas including inter‐mascon correlations and regional boundaries in the regularization greatly improves signal recovery.

Comparing the differences in robustness between interaural time delay calculation methods

The interaural time delay (ITD) is a primary cue to lateral sound source direction and can be calculated from a head-related impulse response (HRIR). Although many ITD calculation techniques utilize cross-correlation, recent work suggests that for horizontal plane HRIRs, ITD calculation based on HRIR onset threshold may better match human perception of source location. The goal of the current study is to more comprehensively compare ITD calculation methods across a large sample of HRIRs. 2448 ITD calculation methods were compared using 42 240 HRIRs from the HUTUBS database (440 directions from 96 individuals). Within each method and individual, the calculated ITD function was compared to a spatially smoothed ITD function using a 10th-order spherical harmonic truncation. When large deviations from the smoothed version occurred (>250 μs), the method was tagged as being non-robust to calculation artifacts. Although the robustness of the onset threshold technique was found to be highly dependent upon threshold level and low-pass filter parameters, proper selection of these parameters resulted in little to no calculation artifacts and superior performance to other calculation methods. Improvements in the robustness of ITD calculation may help to optimize virtual auditory displays or HRIR personalization procedures using the minimum-phase plus ITD model.

Flutter analysis of a one-and-a-half-stage fan at low speed using nonlinear harmonic method

Multirow effects on flutter stability of a 1.5-stage fan at low speed are investigated using nonlinear harmonic method in a one-way coupled fashion. In the first part, the mesh-independence verification and validation of nonlinear harmonic method to simulate multirow effects are performed. In the second part, multirow effects are separated into two parts including acoustic reflection and rotor–stator interaction induced by relative motion between rotor and stators with each part investigated individually. Effect of acoustic reflection from upstream and downstream blade rows is investigated separately using a harmonic truncation method to avoid the change of time-mean flow. The results show that acoustic reflection can have a large effect on flutter stability of rotor blade. The simulation of the rotor–stator interaction effect indicates that the rotor–stator interaction does not significantly affect the flutter stability of rotor blade in this case. Lastly, the variation of aerodynamic modal damping ratio with the size of gap between inlet guide vane and rotor is investigated. Aerodynamic modal damping ratio at a nodal diameter whose fundamental mode is cut-on varies periodically with gap size. Wave splitting method is employed to further investigate the relation between the phase difference between incoming and outgoing wave and aero damping, which can be used to improve the flutter stability at the design stage.

Harmonic Modeling of a Diode-Clamped Multilevel Voltage Source Converter for Predicting Uncharacteristic Harmonics

In response to the growing demand for medium- and high-power trends, multilevel voltage source converters (VSCs) have been attracting growing considerations. One of the widely used VSCs are the diode-clamped multilevel VSC (DCM-VSC). As these converters proliferate, their harmonic impact may become significant. Nevertheless, a harmonic model for the DCM-VSC is currently lacking in the literature. In this paper, the ABCD matrix, mapping the input harmonics to the output harmonics of DCM-VSC, is derived. As the matrix is formulated in the time-domain, the output harmonics are exact and do not suffer from harmonic truncation errors. As the paper will demonstrate, the derived ABCD matrix can be easily applied to a microgrid system and users can easily predict all the uncharacteristic harmonics when a microgrid is subjected to various conditions of imbalance. In addition to all the results being validated with those of PSCAD/EMTDC, the computation time of the proposed method is in contrast much shorter.

Can the comprehensive model phase 4 (CM4) predict the geomagnetic diurnal field for days away from quiet time?

Abstract. The most recent comprehensive model (CM4) of the geomagnetic field (Sabaka et al., 2004) has been used in conjunction with geomagnetic ground observatory station data to analyse and study the geomagnetic diurnal variation field for days away from quiet time and the CM4 prediction for these times. Even though much has been learnt about many components of the geomagnetic field, the diurnal variation field behaviour for days away from quiet time (moderately disturbed time) has not been intensively studied. Consequently, we analyse these, and the predictive ability of the CM4 for ground variations, and whether the CM4 prediction of the diurnal variation (whether at quiet time or away from quiet time) is valid outside the period of reference that from which the data were used in modelling. In carrying out the study, we compared the observatory station data and the CM4 prediction directly. Using the CM4 code, well-characterised internal and magnetospheric components were subtracted from the data, plots and global maps of the residual field generated and then compared with the CM4 to see how well the model performed in predicting the data at moderately disturbed time (Kp ≤ 5). The results show that the CM4 is valid and produces useful predictions outside the period covering the timespan of the model and during moderately disturbed time, despite the lack of active data in the original model dataset. The model predictability of the data increases as we move to higher spherical harmonic degree truncation, as the model–data misfit is reduced, but with increased roughness as a result of small-scale features incorporated. The observed results show that this relationship between the increase in spherical harmonic degree truncation and reduction in misfit can be restricted by data quality or quantity and global coverage or spread.

Effect of river discharge and geometry on tides and net water transport in an estuarine network, an idealized model applied to the Yangtze Estuary

Tidal propagation in, and division of net water transport over different channels in an estuarine network are analyzed using a newly developed idealized model. The water motion in this model is governed by the cross-sectionally averaged shallow water equations and is forced by tides at the seaward boundaries and by river discharge. Approximate analytical solutions are constructed by means of a harmonic truncation and a perturbation expansion in a small parameter, being the ratio of tidal amplitude and depth. The net water transport results from an imposed river discharge and from residual water transport generated by nonlinear tidal rectification. Two new drivers are identified that contribute to the net water transport in tidal estuarine networks, viz. the generation of residual water transport due to gradients in dynamic pressure and due to a coupling between the tidally averaged and quarter diurnal currents through the quadratic bottom stress. The model is applied in a case study on the Yangtze Estuary, to investigate tides and division of net water transport over its multiple channels during the wet and dry season, as well as before and after the construction of the Deepwater Navigation Channel. Model results agree fairly well with observations. Process analysis reveals that the decrease in tides from dry to wet season is due to enhanced bottom stress generated by river-tide interactions. Also, the seasonal variations in net water transport are explained. It is furthermore shown and explained that due to the Deepwater Navigation Channel tidal currents have increased and net water transport has decreased in the North Passage. These changes have profound implications for net sediment transport and salinity intrusion.

Improved source parameter constraints for five undersea earthquakes from north component of GRACE gravity and gravity gradient change measurements

Abstract The innovative processing of Gravity Recovery And Climate Experiment (GRACE) data using only the north component of gravity change and its corresponding gravity gradient changes allows the enhancement of the spatial resolution for coseismic deformation signals. Here, we report the study of five undersea earthquakes using this technique: the 2004 Sumatra–Andaman earthquake, the 2007 Bengkulu earthquake, the 2010 Maule, Chile earthquake, the 2011 Tohoku earthquake, and the 2012 Indian Ocean earthquakes. By using the high spherical harmonic degree (up to degree 96) data products and the associated GRACE data processing techniques, the retrieved north component of gravity change is up to − 34 ± 1.4 μ Gal for the 2004 Sumatra–Andaman earthquake, which illustrates by far the highest amplitude of the coseismic signal retrieved from satellite gravimetry among previous studies. We creatively apply the localized spectral analysis as an efficient method to empirically determine the practical spherical harmonic truncation degree. By combining least squares adjustment with the simulated annealing algorithm, point source parameters are estimated, which demonstrates the unique constraint on source model from GRACE data compared to other data sources. For the 2004 Sumatra–Andaman earthquake, GRACE data produce a shallower centroid depth (9.1 km), as compared to the depth (28.3 km) from GPS data. For the 2011 Tohoku earthquake, the GRACE-estimated centroid location is southwest of the GPS/seismic solutions, and the slip orientation is about 10° clockwise from the published GPS/seismic slip models. We concluded that these differences demonstrate the additional and critical offshore constraint by GRACE on source parameters, as compared to GPS/seismic data.

On magnetic estimation of Earth's core angular momentum variation

AbstractWe study systematically the estimation of Earth's core angular momentum (CAM) variation between 1962.0 and 2008.0 by using core surface flow models derived from the recent geomagnetic field model C3FM2. Various flow models are derived by changing four parameters that control the least squares flow inversion. The parameters include the spherical harmonic (SH) truncation degree of the flow models and two Lagrange multipliers that control the weights of two additional constraints. The first constraint forces the energy spectrum of the flow solution to follow a power law , where l is the SH degree and p is the fourth parameter. The second allows to modulate the solution continuously between the dynamical states of tangential geostrophy (TG) and tangential magnetostrophy (TM). The calculated CAM variations are examined in reference to two features of the observed length‐of‐day (LOD) variation, namely, its secular trend and 6 year oscillation. We find flow models in either TG or TM state for which the estimated CAM trends agree with the LOD trend. It is necessary for TM models to have their flows dominate at planetary scales, whereas TG models should not be of this scale; otherwise, their CAM trends are too steep. These two distinct types of flow model appear to correspond to the separate regimes of previous numerical dynamos that are thought to be applicable to the Earth's core. The phase of the subdecadal CAM variation is coherently determined from flow models obtained with extensively varying inversion settings. Multiple sources of model ambiguity need to be allowed for in discussing whether these phase estimates properly represent that of Earth's CAM as an origin of the observed 6 year LOD oscillation.

Interfacial Structural Transition in Hydration Shells of a Polarizable Solute

Electrostatics of polar solvation is typically described by harmonic free energy functionals. Polarizability contributes a negative polarization term that can make the harmonic free energy negative. The harmonic truncation fails in this regime. Simulations of polarizable ideal dipoles in water show that water's susceptibility passes through a maximum in the range of polarizabilities zeroing the harmonic term out. The microscopic origin of the nonmonotonic behavior is an interfacial structural transition involving the density collapse of the first hydration layer and enhanced number of dangling OH bonds.

Ocean contribution to seismic gravity changes: the sea level equation for seismic perturbations revisited

During megathrust earthquakes, great ruptures are accompanied by large scale mass redistribution inside the solid Earth and by ocean mass redistribution due to bathymetry changes. These large scale mass displacements can be detected using the monthly gravity maps of the GRACE satellite mission. In recent years it has become increasingly common to use the long wavelength changes in the Earth's gravity field observed by GRACE to infer seismic source properties for large megathrust earthquakes. An important advantage of space gravimetry is that it is independent from the availability of land for its measurements. This is relevant for observation of megathrust earthquakes, which occur mostly offshore, such as the Mw?9 2004 Sumatra–Andaman, 2010 Maule (Chile) and 2011 Tohoku-Oki (Japan) events. In Broerse et al., we examined the effect of the presence of an ocean above the rupture on long wavelength gravity changes and showed it to be of the first order. Here we revisit the implementation of an ocean layer through the sea level equation and compare the results with approximated methods that have been used in the literature. One of the simplifications usually lies in the assumption of a globally uniform ocean layer. We show that especially in the case of the 2010 Maule earthquake, due to the closeness of the South American continent, the uniform ocean assumption is not valid and causes errors up to 57?per?cent for modelled peak geoid height changes (expressed at a spherical harmonic truncation degree of 40). In addition, we show that when a large amount of slip occurs close to the trench, horizontal motions of the ocean floor play a mayor role in the ocean contribution to gravity changes. Using a slip model of the 2011 Tohoku-Oki earthquake that places the majority of slip close to the surface, the peak value in geoid height change increases by 50?per?cent due to horizontal ocean floor motion. Furthermore, we test the influence of the maximum spherical harmonic degree at which the sea level equation is performed for sea level changes occurring along coastlines, which shows to be important for relative sea level changes occurring along the shore. Finally, we demonstrate that ocean floor loading, self-gravitation of water and conservation of water mass are of second order importance for coseismic gravity changes. When GRACE observations are used to determine earthquake parameters such as seismic moment or source depth, the uniform ocean layer method introduces large biases, depending on the location of the rupture with respect to the continent. The same holds for interpreting shallow slip when horizontal motions are not properly accounted for in the ocean contribution. In both cases the depth at which slip occurs will be underestimated.

Analysis of Chaotic Characteristics of the Strong Nonlinear Vibration Isolation System Based on Harmonic Balance Method

The key of studying the nonlinear system of the stationary period solutions by the HBM is solving a group of higher order multi-degree-of-freedom nonlinear equations. Aiming at this difficulty, this paper can easily get a higher harmonic balance truncation order expression of solution using a powerful symbolic computation software, and avoid the tedious derivation process. And the main subharmonic response range was accurately obtained by the EACM for solving the nonlinear equations. Meanwhile, the chaotic region was estimated through subharmonic cascade regions combined with the bifurcation of the Feigenbaum rule by solving the system. And, numerical results calculated by the Runge-Kutta method were given to verify the results of the period doubling bifurcations and chaotic region obtained by this method. It has been shown that they are in good agreement.

Time‐variable gravity observations of ice sheet mass balance: Precision and limitations of the GRACE satellite data

Time‐variable gravity data from the Gravity Recovery and Climate Experiment (GRACE) mission have been available since 2002 to estimate the mass balance of the Greenland and Antarctic Ice Sheets. We analyze current progress and uncertainties in GRACE estimates of ice sheet mass balance. We discuss the impacts of errors associated with spherical harmonic truncation, spatial averaging, temporal sampling, and leakage from other time‐dependent signals (e.g., glacial isostatic adjustment (GIA)). The largest sources of error for Antarctica are the GIA correction, the omission of l=1 terms, nontidal changes in ocean mass, and measurement errors. For Greenland, the errors come mostly from the uncertainty in the scaling factor. Using Release 5.0 (RL05) GRACE fields for January 2003 through November 2012, we find a mass change of −258 ± 41 Gt/yr for Greenland, with an acceleration of −31 ± 6 Gt/yr2, and a loss that migrated clockwise around the ice sheet margin to progressively affect the entire periphery. For Antarctica, we report changes of −83 ± 49 and −147 ± 80 Gt/yr for two GIA models, with an acceleration of −12 ± 9 Gt/yr2 and a dominance from the southeast pacific sector of West Antarctica and the Antarctic Peninsula.

An improved method for head-related transfer function interpolation and grid matching

Today, there exists a growing number of HRTF datasets available with each set often proposing a unique variation on the spatial discretization measurement grid. These differing grids, typically determined by the mechanical system employed, result in datasets, which are not directly comparable or exploitable. To alleviate the limitation of incompatible grids and assist in the adaptation of measurements performed on one grid to another, facilitating the inter-exchange of HRTF sets, a fixed radius HRTF interpolation method is proposed. The approach is based on decomposition of the sound field using spherical harmonics, allowing for a global spatial recomposition. The frequency domain HRTF is separated into its complex components and interpolations are performed independently before reconstitution. Spherical harmonic truncation order is chosen to provide a system which is roughly square, improving matrix inversion with Tikhonov regularization. A high spatial density HRTF was used as a test case for evaluating the interpolation method. A series of measures are employed to quantitatively compare the quality of the interpolation as compared to traditional interpolation methods in both the time and frequency domains.

Recovery of the Earth’s gravity field from formation-flying satellites: Temporal aliasing issues

Temporal and mean gravity field models derived from the twin-satellite, leader–follower mission GRACE have provided a new type of information for Earth sciences. In this contribution, we study the potential of various alternative satellite formations for gravity field determination in the post-GRACE era in a simulation environment. In particular, the effects of spherical harmonic truncation and of temporal aliasing in the processing of gravity products from such future formations are investigated.Temporal aliasing is studied based on simulated observations of satellite formation flight (SFF) configurations and simulated errors in short-term background models for the non-tidal atmosphere, ocean masses and the continental water storage. We study the potential improvement of SFF gravity field solutions in terms of spherical harmonics truncation of up to degree/order 100. Furthermore, we simulate the recovery of the hydrological signal from various SFF configurations with and without the contamination of the aliasing errors, which were derived from model differences.We implement full-scale simulations for all scenarios to assess the quality of future gravity products in the form of spherical harmonic models. To this end, we create simulated measurements for four satellite configurations (known as pendulum, the radial Cartwheel, the LISA and the pendulum-3S) besides the GRACE-type configuration as a reference. All simulations are restricted to a time span of one month to limit computation costs.As expected, the gravity field products derived from future SFF configurations may surpass those obtained from GRACE in terms of isotropy of the gravity field. Our study provides detailed results for many scenarios ranging in terms of spherical harmonic truncation to up to 100 for mean and temporal solutions. Furthermore, we show that specific SFF configurations involving radial and cross-track information strengthen the recovery of the gravitational signal; in addition they are able to significantly reduce the aliasing errors appearing in the temporal GRACE gravity field monthly solutions. Following this, we suggest to fly a future mission in pendulum configuration.