Modal Truncation Research Articles

Parameterization of temperature and spectral distortions in future CMB experiments

CMB spectral distortions are induced by Compton collisions with electrons. We review the various schemes to characterize the anisotropic CMB with a non-Planckian spectrum. We advocate using logarithmically averaged temperature moments as the preferred language to describe these spectral distortions, both for theoretical modeling and observations. Numerical modeling is simpler, the moments are frame-independent, and in terms of scattering the mode truncation is exact.

Nonlinear Galerkin method for low-dimensional modeling of fluid dynamic system using POD modes

Truncation of higher POD modes may describe the behaviors of the system inaccurately. In this context, nonlinear Galerkin method considering effects of higher modes via approximate inertial manifolds is presented to approximate the nonlinear partial differential equations for fluids. The system with infinite dimension is reduced to lower dimension using POD modes. The space spanned by the modes satisfying the boundary condition of the flow field is split into two subspaces, a finite-dimensional one spanned by lower modes and its complement spanned by higher modes. Furthermore, the interaction between the lower modes and higher modes is considered via approximate inertial manifolds. Then, a numerical example of the flow past NACA0012 airfoil for low dimensional modeling analysis is presented to prove the efficiency of the proposed method. The results show that the presented method can give an accurate description for the dynamic behaviors of the system because of considering the effect of higher modes and preserve the topological structure of the system with less computational time, compared with traditional POD method.

Vibration of vehicle–pavement coupled system based on a Timoshenko beam on a nonlinear foundation

This paper focuses on the coupled nonlinear vibration of vehicle–pavement system. The pavement is modeled as a Timoshenko beam resting on a six-parameter foundation. The vehicle is simplified as a spring–mass–damper oscillator. For the first time, the dynamic response of vehicle–pavement coupled system is studied by modeling the pavement as a Timoshenko beam resting on a nonlinear foundation. Consequently, the shear effects and the rotational inertia of the pavement are included in the modeling process. The pavement model is assumed to be a linear-plus-cubic Pasternak-type foundation. Furthermore, the convergent Galerkin truncation is used to obtain approximate solutions to the coupled vibratory response of the vehicle–pavement coupled system. The dynamic responses of the vehicle–pavement system with the asphalt pavement on soft soil foundation are investigated via the numerical examples. The numerical results show that the calculation for the coupled vibratory response needs high-order modes. Moreover, the coupling effects between the pavement and the vehicle are numerically examined by using the convergent modal truncation. The physical parameters of the vehicle–pavement system such as the shear modulus are compared for determining their influences on the coupled vibratory response.

Model reduction of a multibody system including a very flexible beam element

An optimum method to reduce the system size of a multibody system including a very flexible beam element is presented. In this paper, the absolute nodal coordinate formulation (ANCF) is used to describe the motion of the flexible beam in this system. Based on contin- uum mechanics approach, the bending stiffness matrix of the system is assumed to be a constant matrix when the axial strain is small. Thus, free-interface method, in which modal coordinate vector is free from the boundary region, is used to reduce the degrees of freedom of a flexible beam with sliding joints. Finally, two numerical examples are presented to verify the efficiency and accuracy of this method. The results show that the present method can reduce the computing time with precision. Meanwhile, a deformation energy criterion of mode truncation is established.

Generalized mode acceleration and modal truncation augmentation methods for the harmonic response analysis of nonviscously damped systems

The modal truncation problem is frequently encountered in nonviscously (viscoelastically) damped systems since only the modes of interest are usually considered in the dynamic analysis of engineering problems. This study aims at accurately calculating the steady-state responses of nonviscously damped systems by only considering the modes of interest. Based on the Neumann expansion theorem and the frequency shifting technique, a property obtained from the first-order terms of the Neumann expansion of the frequency response function (FRF) matrix of nonviscously damped systems is given. However, this procedure cannot be extended to consider the further higher-order terms. It means a truncation expansion problem exists for nonviscously damped systems. By considering the first-order terms of the Neumann expansion, a generalized mode acceleration method (GMAM) is presented to handle the modal truncation problem. The GMAM can overcome the singular problem of the stiffness matrix. The modal truncation augmentation method (MTAM) is also presented to handle the modal truncation problem by making the equilibrium equations into a subspace equation spanned in terms of the columns of a projection basis given in the GMAM. Several conclusions concerning the implementation of the presented methods are formulated on the basis of the results of three examples.

Multiple‐support response spectrum analysis using load‐dependent Ritz vectors

SUMMARYLoad‐dependent Ritz (LDR) vectors are used in conjunction with the multiple‐support response spectrum (MSRS) combination rule for analysis of structures subjected to spatially varying earthquake ground motions. The LDR vector approach for MSRS analysis is motivated by the fact that LDR vectors in general are more accurate and computationally simpler than eigenvectors in mode superposition analysis, and because many researchers and engineers are using LDR vectors in linear structural dynamic analysis. Mode truncation rules for the original MSRS method are modified to apply to LDR vectors. Two methods for selecting LDR vectors for multicomponent MSRS analysis are introduced. Idealized models of two real bridges with differing structural characteristics are used to investigate the accuracy and efficiency of the two LDR‐MSRS methods in comparison with results obtained by the original MSRS method as well as an extended version that accounts for the static contribution of truncated modes. The results show that the LDR‐MSRS method is generally more accurate than the original MSRS method and at least as accurate as the extended MSRS method. Copyright © 2014 John Wiley & Sons, Ltd.

Improved modal truncation method for eigensensitivity analysis of asymmetric matrix with repeated eigenvalues

An improved modal truncation method with arbitrarily high order accuracy is developed for calculating the second- and third-order eigenvalue derivatives and the first- and second-order eigenvector derivatives of an asymmetric and non-defective matrix with repeated eigenvalues. If the different eigenvalues λ1, λ2, …, λr of the matrix satisfy |λ1| ⩽ … ⩽ |λr| and |λs| < |λs+1| (s ⩽ r−1), then associated with any eigenvalue λi (i ⩽ s), the errors of the eigenvalue and eigenvector derivatives obtained by the qth-order approximate method are proportional to |λi/λs+1|q+1, where the approximate method only uses the eigenpairs corresponding to λ1, λ2, …, λs. A numerical example shows the validity of the approximate method. The numerical example also shows that in order to get the approximate solutions with the same order accuracy, a higher order method should be used for higher order eigenvalue and eigenvector derivatives.

Numerical computation of 3D heat transfer in complex parallel heat exchangers using generalized Graetz modes

We propose and develop a variational formulation dedicated to the simulation of parallel convective heat exchangers that handles possibly complex input/output conditions as well as connection between pipes. It is based on a spectral method that allows to re-cast three-dimensional heat exchangers into a two-dimensional eigenvalue problem, named the generalized Graetz problem. Our formulation handles either convective, adiabatic, or prescribed temperature at the entrance or at the exit of the exchanger. This formulation is robust to mode truncation, offering a huge reduction in computational cost, and providing insights into the most contributing structure to exchanges and transfer. Several examples of heat exchangers are analyzed, their numerical convergence is tested and the numerical efficiency of the approach is illustrated in the case of Poiseuille flow in tubes.

Quantum signaling in cavity QED

We consider quantum signaling between two-level quantum systems in a cavity in the perturbative regime of the earliest possible arrival times of the signal. We present two main results: First, we find that, perhaps surprisingly, the analog of amplitude modulated signaling (Alice using her energy eigenstates $\left|g\right\ensuremath{\rangle},\left|e\right\ensuremath{\rangle}$, as in the Fermi problem) is generally suboptimal for communication, namely, e.g., phase-modulated signaling (Alice using, e.g., ${\left|+\right\ensuremath{\rangle},\left|\ensuremath{-}\right\ensuremath{\rangle}}$ states) overcomes the quantum noise already at a lower order in perturbation theory. Second, we study the effect of mode truncations that are commonly used in cavity QED on the modeling of the communication between two-level atoms. We show that, on general grounds, namely for causality to be preserved, the UV cutoff must scale at least polynomially with the desired accuracy of the predictions.

Vertical random vibration analysis of vehicle–track coupled system using Green's function method

A vertical vehicle–track coupled dynamic model, consisting of a high-speed train on a continuously supported rail, is established in the frequency-domain. The solution is obtained efficiently by use of the Green's function method, which can determine the vibration response over a wide range of frequency without any limitations due to modal truncation. Moreover, real track irregularity spectra can be used conveniently as input. The effect of the flexibility of both track and car body on the entire vehicle–track coupled dynamic response is investigated. A multi-body model of a vehicle with either rigid or flexible car body is defined running on three kinds of track: a rigid rail, a track stiffness model and a Timoshenko beam model. The results show that neglecting the track flexibility leads to an overestimation of both the contact force and the whole vehicle vibration response. The car body flexibility affects the ride quality of the vehicle and the coupling through the track and can be significant in certain frequency ranges. Finally, the effect of railpad and ballast stiffness on the vehicle–track coupled vibration is analysed, indicating that the stiffness of the railpad has an influence on the system in a higher frequency range than the ballast.

Statistical modal analysis for variation characterization and application in manufacturing quality control

A Statistical Modal Analysis (SMA) methodology is developed for geometric variation characterization, modeling, and applications in manufacturing quality monitoring and control. The SMA decomposes a variation (spatial) signal into modes, revealing the fingerprints engraved on the feature in manufacturing with a few truncated modes. A discrete cosine transformation approach is adopted for mode decomposition. Statistical methods are used for model estimation, mode truncation, and determining sample strategy. The emphasis is on implementation and application aspects, including quality monitoring, diagnosis, and process capability study in manufacturing. Case studies are conducted to demonstrate application examples in modeling, diagnosis, and process capability analysis.

Modal Modification of Damped Asymmetric Systems without Using the Left Eigenvectors

This study focuses on the prediction of complex eigenpairs of modified asymmetric systems. A method, which combines the reduced basis and the projection technique, is presented to predict the eigenpairs of modified asymmetric systems. The reduced basis obtained using the homotopy perturbation technique only requires the eigensolution of interest such that the modal truncation problem can be avoided. More importantly, the method can be carried out without using the left eigenvectors. The accuracy of the reduced basis approximation can be improved by introducing the projection technique. A rotor dynamic system is used to illustrate the capacity of predicting the changes of eigensolutions.

Application of Krylov Reduction Technique for a Machine Tool Multibody Modelling

Quick calculation of machine tool dynamic response represents one of the major requirements for machine tool virtual modelling and virtual machining, aiming at simulating the machining process performance, quality, and precision of a workpiece. Enhanced time effectiveness in machine tool dynamic simulations may be achieved by employing model order reduction (MOR) techniques of the full finite element (FE) models. The paper provides a case study aimed at comparison of Krylov subspace base and mode truncation technique. Application of both of the reduction techniques for creating a machine tool multibody model is evaluated. The Krylov subspace reduction technique shows high quality in terms of both dynamic properties of the reduced multibody model and very low time demands at the same time.

Complex eigenvector scaling from mass perturbations

This paper presents an approach to normalize experimentally extracted complex eigenvectors so that their outer product gives transfer function residues. The approach, an implementation of the mass perturbation strategy, is exact for arbitrary perturbation magnitudes and number of sensors when the modal space is complete and is robust against modal truncation. It is shown that improvements over a sensitivity solution are significant when the relation between the eigenvalue and the perturbation magnitude is strongly nonlinear.

Fixed-base poles and eigenvectors from transmission zeros

An approach to identify building eigenproperties for fixed base conditions, applicable to cases where measurements are from seismic excitation is presented. The approach extracts the fixed base poles from a state space realization where the input is the free field and the output is the base motion, which is assumed rigid. The formulation holds for arbitrary damping distributions and provides, in addition to the poles, the fixed base eigenvectors at any point where the structural response is measured. The approach is built on the fact that the transmission zeros of the foundation transfer matrix can be extracted from measurements and that these zeros are the poles of the fixed base system. Under the premise of linearity and finite dimensionality the method is exact when the map from the free field to the foundation coordinates is itself exact. In practice modal truncation is inevitable and approximation from this source is mitigated using a pseudo-static correction that for a half-space proves a function of the foundation geometry but not of the shear wave velocity. Performance of the approach is examined in simulations and an application in a real structure is illustrated.

Eliminating the modal truncation problem encountered in frequency responses of viscoelastic systems

Increasing the number of degrees of freedom used in finite element analysis for mechanical and structural systems with viscoelastic damping, the need to consider the modal truncation problem of viscoelastic systems is more than ever before. The higher modes may be unnecessary to obtain in dynamic analysis for engineering applications. For viscoelastic systems, the modal truncation problem may be more frequently encountered since the nonviscous modes are difficult or even impossible to be found accurately even if a small-scaled problem is considered for some eigensolution methods. This study aims at eliminating the influence of the higher modes on the frequency responses of viscoelastically damped systems. A method is presented by making the equilibrium equations of motion into a subspace equation spanned in terms of the columns of a projection basis obtained by considering the use of the contribution of the lower modes and the first two terms of the Neumann expansion of the contribution of the unavailable modes. Finally, three example studies are provided to illustrate the effectiveness of the derived results. It is shown that the proposed method can reduce the modal truncation error significantly.

Investigation of the effectiveness of using an experiment to validate experimental substructure models

Experimental-to-analytical substructuring has been investigated as a way to avoid costly tests and analyses on large systems by experimentally characterizing certain subcomponents and then coupling them to an analytical model for the rest of the structure. Unfortunately, substructuring has sometimes proven difficult to implement because the substructuring calculations can be sensitive to modal truncation and experimental noise and often require the measurement of rotational motions at the interface. These issues have led to decreased confidence in substructuring calculations, causing it to remain underutilized in industry. This work proposes performing a small, inexpensive validation test to give confidence or reveal flaws in an experimentally derived model. This is only meaningful if the error in the validation test is characteristic of the error in an application of interest; this work explores this assumption. The validation test is evaluated in two scenarios which represent substructuring realities. The first scenario investigates measurement errors by applying normally distributed noise to a simple beam structure and evaluating how well the validation test compares to the application of interest. The second scenario simulates substructuring of a complicated engine–generator system with a statically indeterminate connection, where the effect of modal truncation is not easy to anticipate.

Modal Selection Through Effective Interface Mass With Application to Flexible Multibody Cranktrain Dynamics

The development of a multibody model of a motorbike L-twin engine cranktrain is presented in this work. The need for an accurate evaluation of the loads acting on the main engine components at high rotational speed makes it necessary to take element flexibility into account in order to capture elastodynamic effects, which might have a major impact on the dynamics of the system. Starting from finite element descriptions of both the crankshaft and the connecting rod, the classical Craig–Bampton (CB) technique is employed to obtain reduced models, which are suitable for the subsequent multibody analysis. A particular component mode selection procedure is implemented based on the concept of effective interface mass, allowing an assessment of the accuracy of the reduced model prior to the nonlinear simulation phase. Bearing dynamics also plays an important role in such a high-speed engine application: angular contact ball bearings are modeled according to a 5DOF nonlinear scheme in order to grasp the main bearings behavior while an impedance-based hydrodynamic bearing model is implemented providing an enhanced operation prediction at big end locations. The assembled cranktrain model is simulated using a commercial multibody software platform. Numerical results demonstrate the effectiveness of the procedure implemented for the flexible component model reduction. The advantages of this technique over the traditional mode truncation approach are discussed.

Virtual collocation method for a flexible rotor supported by active magnetic bearings

Local vibration control systems of mechanical structures can be collocated or non-collocated. If a sensor is placed at the same location as an actuator the system is said to be collocated. Otherwise, the system is non-collocated. It is not always possible to collocate the vibration control actuator at the location of interest, e.g. in active magnetic bearings (AMBs) eddy current sensors are used to measure displacements of the supported shaft. Due to inevitable electromagnetic interferences the sensors cannot be placed exactly at locations of electromagnetic coils of the AMB. Non-collocation complicates the control problem because the dynamics of the structure between the control actuator and sensor disturbs the performance of the vibration control system. Non-collocated systems have non-minimum-phase zeros at the right half-plane, which may even destabilize the control system. Non-collocation effects were suppressed by several methods including phase shifting, time delay or passive vibration absorber methods. However, none of these methods ensured the required stability, nor a good performance. The present paper recommends the virtual (recalculated) collocation method for local control loops of the AMB. The idea is to calculate displacements at actuators' locations given the displacements at sensors' locations. The approach is illustrated with numerical results of the flexible rotor supported by AMBs and controlled with four proportional integral derivative (PID) controllers. The finite element model of the rotor is developed and transformed to the state-space. The model is reduced by the modal truncation technique. Then, an observer for the reduced system is developed. This way the shaft displacements at any discretization node along the shaft axis are simply estimated and introduced to the inputs of PID controllers. Calculation results demonstrate a good performance of the collocated PID control systems, confirming the potential of the method and giving a rationale for its further development.

Modeling and experimental verification of a flexible rotor/AMB system

PurposeIn this paper, the aim is to present a modeling strategy for a flexible rotor/active magnetic bearing (AMB) system with non‐collocation. Special attention is paid to the vibration reduction and the stable passage through the first critical speed.Design/methodology/approachThe finite element method based on Euller‐Bernoulli beam theory is applied in the formulation of the rotor model. Since rotor/AMB systems are complex mechatronic systems, reduced order approach is used in the control system design. This study applies the modal decomposition method and the modal truncation method, thus retaining the lower order bending modes. The obtained numerical results are compared with the measured open loop frequency responses and the existing differences are compensated in order to obtain accurate numerical model.FindingsFrequency response of the entire system model (flexible shaft, actuators, power amplifiers and sensors) with amplitudes expressed in rotor lateral displacements can be verified by the measured frequency responses. The deviations in the amplitude and phase diagrams are then successfully corrected using the appropriate model modifications.Practical implicationsThe results of this research find direct applications in flexible rotors supported by AMBs, e.g. high speed spindles, turbo molecular pumps, flywheel energy storage systems, etc. The presented procedure can be especially valuable in the design of model based controllers.Originality/valueAn AMB system model is developed and presented in this paper, in conjunction with a systematic description of an efficient procedure for the elimination of the typical mismatches between the simulation and experiment. Firstly, rotor/AMB test rig is stabilized with an appropriately tuned PID controller and an open loop frequency response is obtained for such a system. This response is then compared to corresponding simulation results for which mismatches are identified and eliminated thus yielding an accurate model of the system.