Modal Truncation Research Articles

SSPS Dynamic Modeling and the Flexible Vibration Suppression

Space solar power satellite (SSPS) as a very large flexible spacecraft structure with complex configuration, large size and number of units bring the difficulties to dynamics modeling and analysis. Considering SSPS structure characteristics, equivalent strain and kinetic energy theory is adopted to establish the equivalent beam model of SSPS truss structure. The assumed mode method is adopted to describe the flexible body. The modal truncation method implements the dynamics system order reduction. Mixed coordinates method is adopted to establish the rigid-flexible coupled dynamic model. The established dynamic model can reflect dynamic characteristics of SSPS, achieve control requirements for SSPS and decrease the workload of simulation calculation. The independent modal space control (IMSC) method is proposed to active control research view of the large displacement, nonlinearity, low and dense mode frequency, light damping of flexible structures. Simulation results on flexible solar array show the effectiveness of the control method.

Hybrid modal‐balanced truncation method based on power system transfer function energy concepts

The combined use of modal and balanced truncations methods is proposed for model order reductions. To efficiently combine these methods, a stopping criterion based on spectral energy concepts is also proposed. This criterion was implemented into the code of the widely known subspace accelerated dominant pole algorithm (SADPA), designed to compute a set of dominant poles and associated residues of transfer functions from large-scale, sparse, linear descriptor systems. The resulting enhanced SADPA code automatically stops once the computed set of dominant poles and associated residues is sufficient to build a modal reduced order model (ROM) whose energy content approaches that of the complete model within a specified tolerance and considering a frequency window of interest. The number of dominant poles in this set is much smaller than the number of poles of the full system model. Hence, their state-space realisation usually has a small enough dimension for the efficient application of the square root balanced truncation method. This new method, named hybrid modal-balanced truncation, produces ROMs whose order are much smaller than that of the modal ROMs and, most importantly, can also be applied to unstable models.

Exploring a newSU(4)symmetry of meson interpolators

In recent lattice calculations it has been discovered that mesons upon truncation of the quasizero modes of the Dirac operator obey a symmetry larger than the $SU(2{)}_{L}\ifmmode\times\else\texttimes\fi{}SU(2{)}_{R}\ifmmode\times\else\texttimes\fi{}U(1{)}_{A}$ symmetry of the QCD Lagrangian. This symmetry has been suggested to be $SU(4)\ensuremath{\supset}SU(2{)}_{L}\ifmmode\times\else\texttimes\fi{}SU(2{)}_{R}\ifmmode\times\else\texttimes\fi{}U(1{)}_{A}$ that mixes not only the u- and d-quarks of a given chirality, but also the left- and right-handed components. Here it is demonstrated that bilinear $\overline{q}q$ interpolating fields of a given spin $J\ensuremath{\ge}1$ transform into each other according to irreducible representations of $SU(4)$ or, in general, $SU(2{N}_{F})$. This fact together with the coincidence of the correlation functions establishes $SU(4)$ as a symmetry of the $J\ensuremath{\ge}1$ mesons upon quasizero mode reduction. It is shown that this symmetry is a symmetry of the confining instantaneous charge-charge interaction in QCD. Different subgroups of $SU(4)$ as well as the $SU(4)$ algebra are explored.

Improved Modal Truncation Error in the Directly Analytical Method for Damage Identification in Civil Engineering

Improved Modal Truncation Error in the Directly Analytical Method for Damage Identification in Civil Engineering

Evidence for a newSU(4)symmetry withJ=2mesons

Recently, a new symmetry of mesons has been found upon truncation of the quasizero modes of the overlap Dirac operator in lattice simulations. Namely, the $\ensuremath{\rho}$, ${\ensuremath{\rho}}^{\ensuremath{'}}$, $\ensuremath{\omega}$, ${\ensuremath{\omega}}^{\ensuremath{'}}$, ${a}_{1}$, ${b}_{1}$, ${h}_{1}$, and possibly ${f}_{1}$ $J=1$ mesons get degenerate after removal of the quasizero modes. This emergent symmetry has been established to be $SU(4)\ensuremath{\supset}SU(2{)}_{\mathrm{L}}\ifmmode\times\else\texttimes\fi{}SU(2{)}_{\mathrm{R}}\ifmmode\times\else\texttimes\fi{}U(1{)}_{\mathrm{A}}$. It is higher than the symmetry of the QCD Lagrangian and provides not only a mixing of quarks of given chirality in the isospin space, but also the mixing of left-handed and right-handed components. Here we study, with the overlap Dirac operator, the isovector $J=2$ mesons upon the quasizero mode reduction and observe a similar degeneracy. This result further supports the $SU(4)$ symmetry in mesons of given spin $J\ensuremath{\ge}1$.

Adaptive POD‐based low‐dimensional modeling supported by residual estimates

SummaryAn adaptive low‐dimensional model is considered to simulate time‐dependent dynamics in nonlinear dissipative systems governed by PDEs. The method combines an inexpensive POD‐based Galerkin system with short runs of a standard numerical solver that provides the snapshots necessary to first construct and then update the POD modes. Switching between the numerical solver and the Galerkin system is decided ‘on the fly’ by monitoring (i) a truncation error estimate and (ii) a residual estimate. The latter estimate is used to control the mode truncation instability and highly improves former adaptive strategies that detected this instability by monitoring consistency with a second instrumental Galerkin system based on a larger number of POD modes. The most computationally expensive run of the numerical solver occurs at the outset, when the whole set of POD modes is calculated. This step is improved by using mode libraries, which may either be generic or result from former applications of the method. The outcome is a flexible, robust, computationally inexpensive procedure that adapts itself to the local dynamics by using the faster Galerkin system for the majority of the time and few, on demand, short runs of a numerical solver. The method is illustrated considering the complex Ginzburg–Landau equation in one and two space dimensions. Copyright © 2015 John Wiley & Sons, Ltd.

Polarization properties of a broadband multi-moded concentrator.

We present the design and performance of a non-imaging concentrator for use in broadband polarimetry at millimeter through submillimeter wavelengths. A rectangular geometry preserves the input polarization state as the concentrator couples f/2 incident optics to a 2π sr detector. Measurements of the co-polar and cross-polar beams in both the few-mode and highly over-moded limits agree with a simple model based on mode truncation. The measured co-polar beam pattern is nearly independent of frequency in both linear polarizations. The cross-polar beam pattern is dominated by a uniform term corresponding to polarization efficiency of 94%. After correcting for efficiency, the remaining cross-polar response is -18 dB.

English

This work is about cultural hegemony and Africa’s development process. It examines the backwardness associated with the truncation of African cultural modes of development and the unsuccessful adaptation to western systems introduced by colonial powers. Among the Africa’s lost cultural values, as stated in this work, include the African languages for thinking and planning development processes; sense of community life for effective community organization and development which also would have culminated in the evolution of an ideology for state development in Africa. The paper points out the challenges posed by the destruction of Africa’s cultural values especially language and conceptualization; traditional religion; sense of community life; social capital development; and, democratic system. The paper accentuates the point that Africa should desist from what could be regarded as the ‘colonial overstretch’ and finally, argues that because of the adjustment failures to western traditional systems by African states, the revival or equipment of African traditional values for development, constitute a way forward.    Key words: Africa, culture, development, hegemony, values, trajectories.

Comparison between balanced truncation and modal truncation techniques for linear state‐space symmetric systems

In this study, two well-known model reduction techniques: balanced truncation and modal truncation are compared. The comparison is based on their error bound formulas for a special class of systems derived in this study. The comparison is illustrated by a numerical example.

High order modal approximation techniques in the frequency domain

This paper presents high order modal methods that increase the accuracy of the standard modal truncation scheme with an emphasis on frequency-domain accuracy. The existing Mode Acceleration and Modal Truncation Augmentation methods are compared alongside a new interpretation utilizing the interpolating polynomial. This technique is motivated by taking limits of the modal sum, introducing a residual that appears as a Laurent polynomial. The results presented focus on the case when the frequency range of interest is in the middle of the system’s spectrum of eigenvalues, meaning that at least some of the system’s natural frequencies lie below the lower bound of the frequency range. This case allows for a possible error in the Modal Truncation Augmentation method that is demonstrated here. Also, convergence properties of the methods are discussed and demonstrated.

Anomalous eddy viscosity for two-dimensional turbulence

We study eddy viscosity for generalized two-dimensional (2D) fluids. The governing equation for generalized 2D fluids is the advection equation of an active scalar q by an incompressible velocity. The relation between q and the stream function ψ is given by q = − (−∇2)α/2ψ. Here, α is a real number. When the evolution equation for the generalized enstrophy spectrum Qα(k) is truncated at a wavenumber kc, the effect of the truncation of modes with larger wavenumbers than kc on the dynamics of the generalized enstrophy spectrum with smaller wavenumbers than kc is investigated. Here, we refer to the effect of the truncation on the dynamics of Qα(k) with k < kc as eddy viscosity. Our motivation is to examine whether the eddy viscosity can be represented by normal diffusion. Using an asymptotic analysis of an eddy-damped quasi-normal Markovian (EDQNM) closure approximation equation for the enstrophy spectrum, we show that even if the wavenumbers of interest k are sufficiently smaller than kc, the eddy viscosity is not asymptotically proportional to k2Qα(k), i.e., a normal diffusion, but to k4−αQα(k) for α > 0 and k4Qα(k) for α < 0, i.e., an anomalous diffusion. This indicates that the eddy viscosity as normal diffusion is asymptotically realized only for α = 2 (Navier–Stokes system). The proportionality constant, the eddy viscosity coefficient, is asymptotically negative. These results are confirmed by numerical calculations of the EDQNM closure approximation equation and direct numerical simulations of the governing equation for forced and dissipative generalized 2D fluids. The negative eddy viscosity coefficient is explained using Fjørtoft’s theorem and a spreading hypothesis for the spectrum.

Optimal Spectral Decomposition (OSD) for Ocean Data Assimilation

AbstractOptimal spectral decomposition (OSD) is applied to ocean data assimilation with variable (temperature, salinity, or velocity) anomalies (relative to background or modeled values) decomposed into generalized Fourier series, such that any anomaly is represented by a linear combination of products of basis functions and corresponding spectral coefficients. It has three steps: 1) determination of the basis functions, 2) optimal mode truncation, and 3) update of the spectral coefficients from innovation (observational increment). The basis functions, depending only on the topography of the ocean basin, are the eigenvectors of the Laplacian operator with the same lateral boundary conditions as the assimilated variable anomalies. The Vapnik–Chervonkis dimension is used to determine the optimal mode truncation. After that, the model field updates due to innovation through solving a set of a linear algebraic equations of the spectral coefficients. The strength and weakness of the OSD method are demonstrated through a twin experiment using the Parallel Ocean Program (POP) model.

Reduced order structural models for the calculation of wet contact forces due to impacts in hydraulic valves

Impacts in hydraulic valves generate structure-borne sound. Numerical methods are applied to calculate the contact forces, which are related to the acoustic noise generation. The fluid between the moving valve stem and the end stop creates a squeeze film force. A model order reduction technique in two steps is applied to capture the flexibility of the impact bodies in a multi-physics system simulation. The first reduction step is based on modal truncation using normal and attachment modes. The second step aims to select the important states or modes within the model bandwidth based on Hankel singular values, where two selection procedures are compared. The squeeze film effect is represented by an analytical formula which is closely incorporated to the compliant contact model. This contact deformation relation for the wet impact has to account for the transition between the pure squeeze motion and the structural collision. Measurements are used to support the calculation results. The coupled calculation of elastic and hydraulic effects leads to improved models for predicting the noise design and the functionality of hydraulic valves.

Operational modal analysis with uncertainty quantification for SDDLV-based damage localization

The Stochastic Dynamic Damage Locating Vector (SDDLV) approach is a vibration-based damage localization method based on a finite element model of a structure in a reference state and output-only measurements in both reference and damaged states. A stress field is computed for loads in the null space of a surrogate of the change in the transfer matrix at the sensor positions, where the null space is obtained based on the identified modal parameters in both structural states. Then, the damage location is related to positions where the stress is close to zero. The localization results of this generic approach are perturbed by mainly two sources: modal truncation (not all modes of the structure are available) and modal parameter identification errors (estimation is subject to statistical uncertainties). In this paper, we show how damage localization with the SDDLV approach is improved by taking into account the estimation uncertainties of the underlying identified modal parameters.

Study on Spatial Nonlinear Vibration Characteristics of Cables with a Particular Constraint

In order to study the nonlinear vibration problems of cable with a particular constraint, the mathematical model for the object of study should be established firstly with the elastic mechanics method. In order to obtain the mathematical model of differential form, the Galerkin method is used to make the mathematical model discrete and modal truncation, so the infinite dimensional solution space of the mathematical model can be transformed into the finite dimensional one. Main factors influencing the vibration of the cable system are analyzed, and the simulation analysis about the effects of various parameters on frequency is carried out. The analysis results show that the internal resonance modes may exist, and provide a theoretical basis for improving the safety and reliability of laying cables in actual conditions.

Dynamic Analysis and Modal Truncation of High-speed Cam Mechanism

Dynamic Analysis and Modal Truncation of High-speed Cam Mechanism

Transverse Free Vibration of Axially Moving Stepped Beam with Different Length and Tip Mass

Axially moving stepped beam (AMSB) with different length and tip mass is represented by adopting Euler-Bernoulli beam theory, and its characteristics and displacements of transverse free vibration are calculated by using semianalytical method. Firstly, the governing equation of the transverse free vibration is established based on Hamilton’s principle. The equation is cast into eigenvalue equation through the complex modal analysis. Then, a scheme is proposed to derive the continuous condition accordingly as the displacement, rotation, bending moment, and shear force are all equal at the connections of any two segments. Another scheme is to derive frequency equation from the given boundary conditions which contain a tip mass in the last segment. Finally, the natural frequency and modal function are calculated by using numerical method according to the eigenvalue equation and frequency equation. Due to the introduction of modal truncation, displacement and, the free vibration solution can be obtained by adopting modal superposition after Hilbert transform. The numerical examples illustrate that length, velocity, mass, and geometry affect characteristics and displacements significantly; the series of methods are effective and accurate to investigate the vibration of the AMSB with different length and tip mass after comparing several results.

Application of model reduction techniques to jacket structures

Offshore structures like jackets are exposed to various kinds of external loading conditions and examination of broader range of modes and natural frequencies are required for their dynamic analysis. In this study, four model reduction techniques — two modal truncation methods and two balanced truncation methods — are introduced and applied to a jacket structure. The reduction results are compared with those of the conventional Paz condensation methods. Four new methods show better fidelity, especially in high frequency ranges than the conventional methods.

An integrated prediction model including the cutting process for virtual product development of machine tools

Recently activities for shortening the development time of machine tools and demands for high-value products are consistently being increased. As a result, many studies for predicting the performance in design stage have been conducted to solve these issues. So far, most manufacturers of machine tools have verified designs excluding the cutting process, even though the main performance is the machining quality of the workpiece. Therefore, it has been required to consider together the cutting process model for analyzing the machining quality before the manufacturing of the prototype. This paper presents an integrated dynamic model considering the all interaction of the machine structure, the control system, and the cutting process. The modal truncation technique using the steady-state dynamic analysis in a selected location and frequency range of interest is applied to increase the efficiency of mechatronics simulations including the cutting process. In particular, this proposed prediction model allows accurate design verifications for high-speed machines because it reflects the influences of the high inertia forces and the contour errors caused by high feed rate. Several peripheral milling applications were demonstrated in order to show the feasibility of the proposed prediction model, and we confirmed the superiority of prediction by comparing the simulated and measured results in high feed milling.

Efficient and accurate calculation of sensitivity of damped eigensystems

The exact methods (the algebraic method and Nelson’s method) to calculate eigensensitivities need a matrix decomposition for each eigensensitivity, and therefore are time-consuming. The modal method may be the most efficient method for eigensensitivity analysis, but suffers from modal truncation error. A method is presented by exactly expressing the modal truncation error as a sum of available modes and system matrices. The proposed method maintains original space without having to use the state-space form. It is shown that the proposed method yields good trade-off between accuracy and computational complexity in terms of the theoretical analysis and numerical studies.