Complex Normal Mode Research Articles

FFParam-v2.0: A Comprehensive Tool for CHARMM Additive and Drude Polarizable Force-Field Parameter Optimization and Validation.

Developing production quality CHARMM force-field (FF) parameters is a very detailed process involving a variety of calculations, many of which are specific for the molecule of interest. The first version of FFParam was developed as a standalone Python package designed for the optimization of electrostatic and bonded parameters of the CHARMM additive and polarizable Drude FFs by using quantum mechanical (QM) target data. The new version of FFParam has multiple new capabilities for FF parameter optimization and validation, with an emphasis on the ability to use condensed-phase target data in optimization. FFParam-v2 allows optimization of Lennard-Jones (LJ) parameters using potential energy scans of interactions between selected atoms in a molecule and noble gases, viz., He and Ne, and through condensed-phase calculations, from which experimental observables such as heats of vaporization and free energies of solvation may be obtained. This functionality serves as a gold standard for both optimizing parameters and validating the performance of the final parameters. A new bonded parameter optimization algorithm has been introduced to account for simultaneously optimizing multiple molecules sharing parameters. FFParam-v2 also supports the comparison of normal modes and the potential energy distribution of internal coordinates towards each normal mode obtained from QM and molecular mechanics calculations. Such comparison capability is vital to validate the balance among various bonded parameters that contribute to the complex normal modes of molecules. User interaction has been extended beyond the original graphical user interface to include command-line interface capabilities that allow for integration of FFParam in workflows, thereby facilitating the automation of parameter optimization. With these new functionalities, FFParam is a more comprehensive parameter optimization tool for both beginners and advanced users.

Low Energy Vibrations and Crystalline Structure of Ambroxol Hydrochloride by THz Spectroscopy and DFT Calculations

AbstractThe widely used drug ambroxol hydrochloride (A‐HCl), was subjected to analysis by terahertz time domain (THz‐TDS) spectroscopy and quantum chemistry calculations. The performed X‐ray measurement confirmed the C2/c crystallographic group for A‐HCl. On the contrary, the presented theoretical calculations, achieved by solid‐state density functional theory (ss‐DFT), determined the structure in the global minima on potential energy surface which belong to the Cc crystallographic group. In the frame of this symmetric group, the ss‐DFT calculations resulted in 40 vibrational normal modes up to wavenumber 110 cm−1. The three different functionals (B3LYP, PBE, and PW1PW) were applied and all of them provided similar results. In the same range of wavenumbers (10–110 cm−1) the THz spectrum was measured at two different temperatures (room and cryogenic). Thanks to the very good correspondence of the experiment and calculations in both position and intensities of the peaks, we were able to analyze each peak from percentage contributions of inter‐ and intra‐molecular movements point of view and trivial nomenclature for such complex normal modes was suggested as well.

Construction of nonlinear lattice with potential symmetry for smooth propagation of discrete breather

We construct a nonlinear lattice that has a particular symmetry in its potential function consisting of long-range pairwise interactions. The symmetry enhances smooth propagation of discrete breathers, and it is defined by an invariance of the potential function with respect to a map acting on the complex normal mode coordinates. Condition of the symmetry is given by a set of algebraic equations with respect to coefficients of the pairwise interactions. We prove that the set of algebraic equations has a unique solution, and moreover we solve it explicitly. We present an explicit Hamiltonian for the symmetric lattice, which has coefficients given by the solution. We demonstrate that the present symmetric lattice is useful for numerically computing traveling discrete breathers in various lattices. We propose an algorithm using it.

Effects of residual stress and viscous and hysteretic dampings on the stability of a spinning micro-shaft

This study examines the effects of the residual stress and viscous and hysteretic dampings on the vibrational behavior and stability of a spinning Timoshenko micro-shaft. A modified couple stress theory (MCST) is used to elucidate the size-dependency of the micro-shaft spinning stability, and the equations of motion are derived by employing Hamilton’s principle and a spatial beam for spinning micro-shafts. Moreover, a differential quadrature method (DQM) is presented, along with the exact solution for the forward and backward (FW-BW) complex frequencies and normal modes. The effects of the material length scale parameter (MLSP), the spinning speed, the viscous damping coefficient, the hysteretic damping, and the residual stress on the stability of the spinning micro-shafts are investigated. The results indicate that the MLSP, the internal dampings (viscous and hysteretic), and the residual stress have significant effects on the complex frequency and stability of the spinning micro-shafts. Therefore, it is crucial to take these factors into account while these systems are designed and analyzed. The results show that an increase in the MLSP leads to stiffening of the spinning micro-shaft, increases the FW-BW dimensionless complex frequencies of the system, and enhances the stability of the system. Additionally, a rise in the tensile residual stresses causes an increase in the FW-BW dimensionless complex frequencies and stability of the micro-shafts, while the opposite is true for the compressive residual stresses. The results of this research can be employed for designing spinning structures and controlling their vibrations, thus forestalling resonance.

Bayesian Active Learning for Optimization and Uncertainty Quantification in Protein Docking.

Ab initio protein docking represents a major challenge for optimizing a noisy and costly "black box"-like function in a high-dimensional space. Despite progress in this field, there is a lack of rigorous uncertainty quantification (UQ). To fill the gap, we introduce a novel algorithm, Bayesian active learning (BAL), for optimization and UQ of such black-box functions with applications to flexible protein docking. BAL directly models the posterior distribution of the global optimum (i.e., native structures) with active sampling and posterior estimation iteratively feeding each other. Furthermore, it uses complex normal modes to span a homogeneous, Euclidean conformation space suitable for high-dimensional optimization and constructs funnel-like energy models for quality estimation of encounter complexes. Over a protein-docking benchmark set and a CAPRI set including homology docking, we establish that BAL significantly improves against starting points from rigid docking and refinements by particle swarm optimization, providing a top-3 near-native prediction for one third targets. Quality assessment empowered with UQ leads to tight quality intervals with half range around 25% of the actual interface root-mean-square deviation and confidence level at 85%. BAL's estimated probability of a prediction being near-native achieves binary classification AUROC at 0.93 and area under the precision recall curve over 0.60 (compared to 0.50 and 0.14, respectively, by chance), which also improves ranking predictions. This study represents the first UQ solution for protein docking, with rigorous theoretical frameworks and comprehensive empirical assessments.

Exact diagonal representation of normal mode energy, occupation number, and heat current for phonon-dominated thermal transport

Collective excitations of crystal vibrations or normal modes are customarily described using complex normal mode coordinates. While appropriate for calculating phonon dispersion, the mixed representation involving the complex conjugates does not allow the construction of equivalent phonon occupation number or modal dynamical quantities such as the energy or heat current specific to a wave-vector direction (q). Starting from a canonical solution that includes waves going to the left and right directions, we cast the Hamiltonian, normal mode population, and heat current in an exactly diagonalizable representation using real normal mode amplitudes. We show that the use of real amplitudes obviates the need for a complex modal heat current while making the passage to second quantization more apparent. Using nonequilibrium molecular dynamics simulations, we then compute the net modal energy, heat current, and equivalent phonon population in a linear lattice subjected to a thermal gradient. Our analysis paves a tractable path for probing and computing the direction-dependent thermal-phononic modal properties of dielectric lattices using atomistic simulations.

Complex Modal Analysis of a Nonmodally Damped Continuous Beam

This work represents an investigation of the complex modes of continuous vibration systems with nonmodal damping. As an example, a cantilevered beam with damping at the free end is studied. Assumed modes are applied to discretize the eigenvalue problem in state-variable form and then to obtain estimates of the true complex normal modes and frequencies. The finite element method (FEM) is also used to get the mass, stiffness, and damping matrices and further to solve a state-variable eigenvalue problem. A comparison between the complex modes and eigenvalues obtained from the assumed-mode analysis and the finite element analysis shows that the methods produce consistent results. The convergence behavior when using different assumed mode functions is investigated. The assumed-mode method is then used to study the effects of the end-damping coefficient on the estimated normal modes and modal damping. Most modes remain underdamped regardless of the end-damping coefficient. There is an optimal end-damping coefficient for vibration decay, which correlates with the maximum modal nonsynchronicity.

An Investigation of the Temperature Influence on a Shift of Natural Frequencies Using Digital Image Correlation

The contribution describes an original optical measuring technique serving for the investigation of temperature influence on the shift of investigated objects natural frequencies. The mentioned method uses a modified high-speed digital image correlation system with original program Modan 3D created in Matlab, by which it is possible to non-contactly determine the natural frequencies using complex mode indicator function (CMIF) or normal mode indicator function (NMIF), the mode shapes as well as the damping ratio. Influence of the temperature on the shift of natural frequencies was analyzed at the planar steel specimen of circular shape with three holes. To the excitation a modal hammer was used. The specimen heating was realized by a hot-air gun, whereby the influence of the elevated temperature on the shift of natural frequencies was investigated by three temperatures distinguishing the room temperature 50° C, 75° C and 100° C.

A normal mode projection technique for array response synthesis in range-dependent environments

A method is presented to examine the problem of simulating the beam time series response for an array of arbitrary shape in a range-dependent environment using a combination of parabolic equation (PE) forward modeling and local normal mode analysis. The procedure involves computing the acoustic pressure field as a function of depth, using the PE, at the nominal center location of the array. The field is then decomposed into local complex normal mode amplitudes. These mode amplitudes are used to compute the field response on each array element, via range-independent normal mode theory. Conventional plane-wave beamforming is then applied. It is shown that a single matrix computation can be used to map the field as a function of depth to the beam response as a function of angle. The method is applied to two broadband range-independent examples to demonstrate its accuracy. It is then applied to a shallow-water range-dependent experiment from off the Florida coast and a deep-water range-dependent experiment from sound scattering off a seamount in the open ocean. For both range-dependent examples, the model simulation results reproduce the qualitative features observed in the data.

Stability, complex modes, and nonseparability in rotating quadratic potentials

We examine the dynamics of a particle in a general rotating quadratic potential, not necessarily stable or isotropic, using a general complex mode formalism. The problem is equivalent to that of a charged particle in a quadratic potential in the presence of a uniform magnetic field. It is shown that the unstable system exhibits a rich structure, with complex normal modes as well as non-standard modes of evolution characterized by equations of motion which cannot be decoupled (non-separable cases). It is also shown that in some unstable cases the dynamics can be stabilized by increasing the magnetic field or tuning the rotational frequency, giving rise to dynamical stability or instability windows. The evolution in general non-diagonalizable cases is as well discussed.

Responses of discretized systems under narrow band nonstationary random excitations. Part 1: linear problems

The extended stochastic central difference (ESCD) method is proposed as a viable alternative for computing linear responses of discretized multi-degrees-of-freedom (mdof) systems under narrow band stationary and nonstationary random disturbances. The method provides a means of controlling the center frequencies and bandwidths of narrow band stationary and nonstationary random excitation processes. It is suitable for larger-scale random response analysis of complicated structures idealized by the finite element method. Its additional important feature is that application of normal mode or complex normal mode analysis or direct numerical integration algorithms such as the fourth-order Runge–Kutta scheme is not required. Examples, including one of flow-induced vibration of a pipe containing a moving fluid are included to demonstrate: (1) the capability of the proposed method and difference between responses of discretized systems under narrow band and wide band random excitations, and (2) its accuracy and efficiency by way of comparison to the Monte Carlo simulation data. Generalization of the ESCD method for computation of responses of nonlinear mdof systems is presented in a companion paper.

Quest for IR-Pumped Reactions in Dihydrogen-Bonded Complexes

As possible substrates for one-photon infrared-pumped reaction (IRPR), the structures and decomposition paths of complexes between AlH4- and three proton donors, H2O, HF, and HCl, have been studied by ab initio methods. In each case only one transition state was found for the proton-transfer and H2-loss process. For each cluster, the geometry and energy characteristics of reactants, complex, transition state, and products were analyzed with [AlH4···HCl]- emerging as the best IRPR candidate. The MP2//6-311++G**-calculated intrinsic reaction coordinate (IRC) confirmed the one-step proton transfer and H2 loss with no intermediate. Classical trajectories were calculated on the ab initio potential-energy surface, beginning from a large number of initial conditions. With zero-point vibrational energies (ZPVE = 1/2hνi) assigned to all normal modes, based on their calculated harmonic frequencies, νi, one or more additional excitation quanta were added to modes associated with Cl−H and Al−H stretching. Proton transfer from HCl and loss of H2 were calculated to occur on the femtosecond time scale when stretching modes involving the dihydrogen-bonded hydrogens were excited. However, many vibrational oscillations take place before H2 release. Analysis of the dynamics in terms of the complex's normal modes indicates that the excitation in the reaction-relevant modes remains localized on a time scale >1 ps.

State Space Implementation of the Algorithm of Mode Isolation

The Algorithm of Mode Isolation (AMI) extracts modal properties from complex frequency response data. Previous work used classic undamped modes as the analytical framework for the algorithm. The present work extends the algorithm to implement damped modal analysis, in which the eigenvalues and eigenvectors are complex. In order to assess how well this reformulation performs when natural frequencies are close and drive point mobilities are low, a prototypical system consisting of a cantilever beam with attached subsystems is introduced. One of these subsystems is selected to be a tuned vibration absorber for the isolated beam, so the system features a combination of modes whose natural frequencies are close and modes whose drive point mobility is low. The time domain response of this system is evaluated, contaminated with substantial white noise, and then FFT processed in order to obtain synthetic complex frequency response data. The performance of AMI is evaluated by comparing extracted values for natural frequency, modal damping ratio, and complex normal mode vectors to the analytical values. The results reveal that the pair of modes having proximite natural frequencies are accurately identified. Natural frequencies and damping ratios for those modes whose drive mobility is low are identified by processing the ensemble of frequency response functions, but identification of normal mode coefficients for such modes remains problematic.

IDENTIFICATION OF NON-PROPORTIONAL MODAL DAMPING MATRIX AND REAL NORMAL MODES

Equations of motion for non-proportionally damped structures cannot be decoupled by using the real normal modes. For such structures, the complex normal modes are in common use for this purpose, but for the validation of finite element mass and stiffness matrices where physical damping matrices are not available, the related experimental real normal modes must be known. In previous publications, an identification theory using the real normal modes and the non-diagonal modal damping matrix for the non-proportionally damped system and some applications with the computer code ISSPA were presented. However, the theory cannot assure the symmetry of the identified modal damping matrix, which must be theoretically symmetric. In this paper, a method for identifying the symmetric non-proportional modal damping matrix using undamped modal parameters obtained from ISSPA is presented and the validity of the method is demon-strated through both numerical and experimental examples.

On the approximate solution of aircraft landing gear under nonstationary random excitations

The motion of an aircraft landing gear over a rough runway can be modeled by a nonclassically damped system subject to nonstationary random excitations. In this paper, the approximate analysis methods based on either the real or complex normal modes for the computation of nonstationary response covariances are proposed. It has been found by simulation involving a realistic example that, for the nonclassically damped random vibrational systems, the approximate solution method based on the complex normal mode is superior to other approaches with respect to the accuracy and computation time.

Non-linear vibration of a travelling beam

Free and forced responses of a travelling slender beam including the non-linear terms are studied. A simple method of calculating the non-linear, non-stationary, complex normal mode is presented. The hardening characteristic and intercoupling of linear modes during the non-linear modal vibration are clearly brought out. The response of the beam, excited by a point harmonic load, is calculated using the non-linear normal modes. A stability analysis is provided to delineate the stable and unstable roots of the response.

Free Vibration and Stability Analysis of Internally Damped Rotating Shafts With General Boundary Conditions

Vibration analysis of an internally damped rotating shaft, modeled using Timoshenko beam theory, with general boundary conditions is performed analytically. The equations of motion including the effects of internal viscous and hysteretic damping are derived. Exact solutions for the complex natural frequencies and complex normal modes are provided for each of the six classical boundary conditions. Numerical simulations show the effect of the internal damping on the stability of the rotor system.

THE FORCED VIBRATION OF A BEAM WITH VISCOELASTIC BOUNDARY SUPPORTS

A method of analysis for the forced vibration of a beam with viscoelastic boundary supports is proposed based on complex normal mode analysis. The viscoelastic support regions are first described in terms of equivalent complex stiffness coefficients, and then using the complex modes of the beam system with complex stiffness at the boundary points, the equations of motion are completely uncoupled. The modal equation has precisely the same form as the equation of a structurally damped single-degree-of-freedom system. Effects of the viscoelastic supports on the forced vibration response are discussed as an example.

Parametrically Excited Non-Linear Traveling Beams with and without External Forcing

The effects of parametric excitation on a traveling beam, both with and without an external harmonic excitation, have been studied including the non-linear terms. Non-linear, complex normal modes have been used for the response analysis. Detailed numerical results are presented to show the effects of non-linearity on the stability of the parametrically excited system. In the presence of both parametric and external harmonic excitations, the response characteristics are found to be similar to that of a Duffing oscillator. The results are sensitive to the relative strengths of and the phase difference between the two forms of excitations.

The Structural Dynamic Formulation of Non‐Rotating and Rotating Systems Using Real and Complex Normal Modes

AbstractExperimentelle Modalanalyse‐Verfahren beschreiben das strukturdynamische Verhalten elastischer Strukturen teilweise mit komplexen Eigenschwingungen. Zur Steifigkeit nichtproportionale Dämpfungsverteilung wird allgemein als Ursache dieses Verhaltens angenommen. Andere Untersuchungen zeigen, daß komplexe Eigenschwingungen auch aus Unzulänglichkeiten im Versuch resultieren können. Die Extraktion reeller Eigenschwingungen aus komplexen Eigenschwingungskenngrößen wird deshalb auf rotierende Systeme als eine praktisch interessante Anwendung erweitert. Das Beispiel zeigt, daß diese Berechnung der reellen Eigenschwingungen eine Rück transformation in die Ausgangsgleichungen darstellt.