The focus of this contribution is set on the derivation of the scheme and the illustration of its superior features by application to a number of **structural** **dynamics** problems.
References
[1] N.M. Newmark. A method of computation for **structural** **dynamics**. Journal of Engineering Mechanics, ASCE, 85(EM3):67–94, 1959.

Chapter 5
Building a flexible docking receptor library
The utility of **structural** **dynamics** in drug design is introduced in chapter 2. While in chapter 4 we presented a practical case of rational drug design targeting alternative conformation of a specific residue, an in-depth characterization of the conforma- tions accessible to side chain is needed for more general cases. In this chapter we will present how information from molecular **dynamics** simulations can be used to design a library of conformers. Simulations were ran both on the human AChE and on TcAChE, for which many X-ray structures already exist. In this chapter we will discuss about simulation of native hAChE and TcAChE. Comparing different simulation strategies, one long simulation versus a combination of multiple shorter simulations, we will argue that combining multiple short simulations is preferable for sample the populations of side chains rotamers. With the multiple simulations approach, we compared the **dynamics** of key residues in the gorge of TcAChE and hAChE, revealing differences in their dynamical behavior. Flexible docking meth- ods are usually not specific to the **dynamics** of the receptor, but are rather based on statistical properties of a particular amino acid. We developed a method that takes advantage of MD simulations to generate a receptor library that reflects the motions proper to specific residues of AChE.

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period, namely the empirical mode decomposition (EMD) [126] and the Hilbert vibration decomposition (HVD) [127, 128].
The basic idea of EMD is to decompose the original signal into a sum of elemental components, the intrinsic mode functions (IMFs). The extraction process, termed sifting process, relies on a spline approximation of the lower and upper envelopes of the signal based on its extrema. To be amenable to the Hilbert trans- form, each IMF must satisfy two properties, i.e. the number of extrema and zero-crossings can differ by no more than one, and, at any point, the mean value of the envelopes defined by the local maxima and minima should be zero. It follows that an IMF is a monochromatic signal, the amplitude and frequency of which can be modulated. Taken collectively, the Hilbert spectra of the IMFs give a complete characterisa- tion of a multicomponent signal in terms of amplitudes and instantaneous frequencies. A first effort to gain fundamental understanding of EMD in nonlinear **structural** **dynamics** was made in Refs. [129, 130]. More specifically, a one-to-one relationship between the analytically-realised slow-flow **dynamics** of a nonlinear system and the IMFs derived from measured time series was demonstrated. Based on this theoretical link, the slow-flow model identification method, a linear-in-the-parameters identification approach applicable to multi-degree-of-freedom nonlinear systems, was developed [129]. As discussed in Section 3.5, the corre- spondence between theoretical and empirical slow flow analyses was further utilised for modal identification using the concept of intrinsic modal oscillators (IMOs). EMD was also used in conjunction with perturbation analysis for nonlinear system identification in Ref. [131].

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On the one hand, some of these points can be fulfilled a priori by choices at the development or selection stage of the time integration procedure, notably of the principle underlying the discretization of the equations of motion. For instance, item 1. precludes explicit algorithms that are only conditionally stable [2]; item 5. rules out the use of linear multistep solvers for time integration in **structural** **dynamics**; the use of the generalized framework for one-step three-level schemes (the number of levels is defined as the ratio between the state-vector dimension and the number of degrees of freedom in the problem [3]) proposed by Zhou and Tamma [4] enables the translation of these criteria into constraints on the generalized scheme parameter sets and, thus, their verification at the design level. On the other hand, the stability and accuracy properties of integration schemes, items 1. and 3., are typically evaluated at a subsequent stage of the development of an integration procedure. The importance of their correct assessment stems from two purposes: ( i ) the establishment of the procedure convergence that is guaranteed by its consistency and stability (Lax-Richtmyer equivalence theorem [5]), and ( ii ) the selection criteria for choosing an integration scheme for a specific application partly rely on these properties.

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The strength of the method lies in its ability to deal with nonlinear and nonstationary data (see, e.g., Refs. 14 and 15), despite the absence of a serious analytical foundation under those assumptions. In this paper, we attempt to provide a fundamental understanding of the HHT in nonlinear **structural** **dynamics** by linking its outcome to the slow-flow **dynamics**. The slow-flow model is established by performing a partition between slow and fast **dynamics** using the complexification-averaging technique, and a dynamical system described by slowly-varying amplitudes and phases is obtained. These variables can also be extracted directly from the experimental measurements using the Hilbert transform coupled with the EMD. The comparison between the experimental and analytical results forms the basis of a nonlinear system identification method, termed the slow-flow model identification (SFMI) method. The SFMI method can be viewed as a generalization of Feldman’s FREEVIB approach to MDOF systems. In addition, it identifies the parameters of the equations of motion, something which was not considered in Refs. 4 and 5.

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a weak correlation between PB variations in the NMR ensemble and PB substitutions at structurally equivalent positions in its homologues of phospholipase A2.
Weak correlations at majority of the loop region structurally equivalent positions in NMR ensembles and their homologues inform us about the different local structures adopted at these positions but it does not inform us on how different these local structures are. To quantify the differences in adopted local structures, we calculated ‘average PB variation scores’ and ‘average PB substitution scores’ respectively from NMR ensembles and their homologues at structurally equivalent positions in loop regions (see Materials and Methods). We compared the distributions of ‘average PB variation scores’ and ‘average PB substitution scores’ (fig. 6) for the structurally equivalent positions in loop regions. The significance of difference between these distributions was assessed by paired t- test. ‘Average PB substitution scores’ were significantly lower than ‘average PB variation scores’ (t=38.36, df=1280, p-value=2.2e-16), suggesting that observed local **structural** variations in NMR ensembles are quantitatively different than the local **structural** differences observed in their homologues at loop region structurally equivalent positions. Lack of negative scores in the score distributions in figure 6 also confirm that inherent **structural** **dynamics** as well as evolutionary **structural** **dynamics** follow the rules defined for local **structural** changes in global PB variation and substitution matrices.

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6. Nonparametric Methods
In computational mechanics, random uncertainties in model predictions are due to data uncertainties and to model uncertainties. Data uncertainties concern the parameters of the mathematical–mechanical model and can be taken into account by parametric probabilistic approaches, such as the ones previously presented. However, for complex mechanical systems, the con- structed model cannot be considered representative due to the introduction of approximations and because some details are not accurately known, such as in joining areas. Clearly, model uncertainties should be introduced but can- not be taken into account by the parameters of the mathematical–mechanical model under consideration. A description of the nonparametric probabilistic approach for dynamical systems can be found in [18, 19]. Model uncertainties are considered in a global way when considering the matrices of the dynam- ical model as random matrices, built using the maximum entropy approach. Developments for taking into account data and model uncertainties in linear dynamical systems are presented in [47]. Recently, a generalized probabilis- tic approach that models both model-parameter uncertainties and modeling errors in **structural** **dynamics**, for linear and nonlinear problems, has been introduced and validated [48].

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[3] J. Argyris, H.P. Mlejnek, **Dynamics** of Structures. North-Holland, 1991.
[4] R.R. Craig, A.J. Kurdila, Fundamentals of **Structural** **Dynamics**. Wiley, 2nd Edi- tion, John Wiley and Sons, 2006.
[5] C. Soize, A. Batou, Stochastic reduced-order model in low-frequency **dynamics** in presence of numerous local elastic modes. Journal of Applied Mechanics - Transactions of the ASME, 78(6), 061003-1 to 9, 2011.

reactions. This opened the way for developing various types of pump-probe techniques. To name a few, we can mention two- dimensional electronic spectroscopy, [4] able to probe quantum coherences and electronic couplings between different states, broadband impulsive vibrational spectroscopies [5] , impulsive stimulated Raman [6] and also optical absorption [7] , which are able to follow molecular vibrations during non-equilibrium dynamical processes. However, the fs optical pulses used in the IR, visible and ultraviolet regime deliver a very limited **structural** information. Therefore, femtosecond probes using electron pulses, [8] high harmonic generation [9] and X-ray [10] were developed. Compared to table-top plasma sources or synchrotron, [11] able to generate 100 fs X-ray pulses, X-FELs represent an ultimate source, generating much more intense X-ray pulses as short as few fs and with tunable wavelength. [12] A recent review detailed how time- resolved X-ray methods can map photoinduced electronic and **structural** **dynamics** of molecular systems. [13] These experimental inputs are very important for developing relevant theoretical models, with multi-dimensional potentials focussing on identified electronic states and key nuclear degrees of freedom. We will illustrate here how combined ultrafast optical and X-ray methods can deliver detailed view of the subtle and coupled changes of electronic and nuclear structures during molecular transformations, and provide a physical picture necessary for developing light-activated functions. Here we will focus our attention on transition metal complexes, which are of importance in biochemistry, catalysis, solar energy conversion, and photomagnetic materials for example. Understanding their light- activated processes, and the role of intermediate states, is

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1 INTRODUCTION
The idea of using slow-flow **dynamics** for nonlinear system identification dates back to Feldman [1,2] who exploited the Hilbert transform. The proposed procedure is one of the most successful approaches to tracking the varying nature of vibration of a large class of nonlinear systems thanks to the extraction of backbone curves from experimental data. Alternative approaches for slow flow-based identification were developed, in particular the Wigner-Ville approach [3] and the wavelet transform [4,5] Using the Gabor transform, Bellizzi et al.[6] related the slow-flow **dynamics** to the concept of coupled nonlinear modes. Because multicomponent signals do not admit a well-behaved Hilbert transform, the Hilbert-Huang transform (HHT) was introduced in [7]. It has been shown to be effective for characterizing a wide range of signals in terms of elemental components, termed intrinsic mode functions (IMFs), through what has been called the empirical mode decomposition (EMD). Several applications of this technique to **structural** **dynamics** recently appeared [8-10].

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Because multicomponent signals do not admit a well-behaved Hilbert transform, the Hilbert-Huang transform (HHT) was intro- duced in [11]. It has been shown to be effective for characterizing a wide range of signals in terms of elemental components, termed intrinsic mode functions (IMFs), through what has been called the empirical mode decomposition (EMD). Several appli- cations of this technique to **structural** **dynamics** recently appeared. For instance, Yang et al. [12,13] used it for modal analysis and were able to relate the IMFs to the modal properties, giving a clear interpretation of the outcome of HHT in linear **dynamics**. The strength of the method lies in its ability to deal with nonlinear and nonstationary data (see, e.g., [14-15]), despite the absence of a serious analytical foundation under those assumptions. In this paper, we attempt to provide a fundamental understanding of the HHT in nonlinear **structural** **dynamics** by linking its outcome to the slow-flow **dynamics**. The slow-flow model is established by performing a partition between slow and fast **dynamics** using the complexification-averaging technique, and a dynamical system described by slowly-varying amplitudes and phases is obtained. These variables can also be extracted directly from the experimental measurements using the Hilbert transform coupled with the EMD. The comparison between the experimental and analytical results forms the basis of a nonlinear system identification method, termed the slow-flow model identification (SFMI) method. The SFMI method can be viewed as a generalization of Feldman’s FREEVIB approach to MDOF systems. In addition, it identifies the parameters of the equations of motion, something which was not considered in [4,5].

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6 Conclusions
There is no need for lengthy conclusions. The WG2 benchmarks were selected in a pragmatic fashion which limited their coverage of Engineering applica- tion domains. Within the domain of interest - large steel and concrete structures - the analysis shows that SHM techniques are adequate for the detection and localisation of large damages. Given that the subject of the action is **structural** **dynamics**, this is all that one could have hoped for, as SHM for smaller structures: aircraft, ground transportation and rotating machinery is likely to require the use of higher-frequency tools. The only other conclusion one should draw is that the COST F3 action has been of inestimable value in al- lowing this benchmarking exercise - much has been learnt.

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Abstract
This paper discusses procedures for nonlinearity detection, localisation and identification in structures from time domain vibration measurements. The detection and the localisation techniques use pattern recognition tools and are based on a dissimilarity measure between the signals coming from a linear structure and the corresponding nonlinear one. The detection procedure distinguishes between linear structures and structures with a nonlinearity employing nearest neighbour techniques. The localisation procedure combines substructuring with a nonlinearity detection procedure. This technique is useful for cases of local nonlinearity, when its localisation can be of value for the consequent understanding and modelling of the structure. The identification procedure makes use of the Karhunen-Loeve transform, known also as Proper Orthogonal Decomposition (POD). It is a powerful tool for solving inverse problems in nonlinear **structural** **dynamics**. The identification procedure works on the basis of the minimisation of a difference function between the experimental and the simulated proper orthogonal modes (POM). The proposed techniques are demonstrated on a beam test case with a local damping type nonlinearity.

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joseph.morlier@isae.fr
Abstract
This paper presents a framework and one pedagogical application of motion tracking algorithms applied to **structural** **dynamics**. The aim of this work is to show the ability of high speed camera to study the dynamic characteristics of simple mechanical systems using a marker less and simultaneous Single Input Multiple Output (SIMO) broadband analysis. KLT (Kanade-Lucas-Tomasi) trackers are used as virtual sensors on mechanical systems video. First we introduce the paradigm of virtual sensors in the field of modal analysis using video processing. Then we present a pedagogical example of flexible beam (Fishing rod) video. From KLT tracking we extracted displacements data (virtual sensors) which are then enhanced using filtering and smoothing and then we can identify natural frequency and damping ratio from classical modal analysis. The experimental results (mode shapes) are compared to an analytical flexible beam model showing high correlation but also showing the limitation of linear analysis. The main interest of this paper is that displacements are simply measured using only video at FPS (Frame Per Second) that respects the Nyquist frequency. There is no target needed on the structure only few critical pixels that are good features to track and which become virtual sensors.

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Conclusion
The main goal of this paper is to show a pedagogical image processing tool to understand **structural** **dynamics**. OpenCV framework can easily be used for displacemet measurement on a video of a vibrating system according the speed of camera respect the Nyquist criteria. KLT trackers are simply used as virtual sensors to measure displacement on video choosing good features to track on the image. We succeed to estimate the first three main modes of a flexible beam (cantilever composites fishing rod) under broad band excitation. For educational purposes, this simple application can also be used with the help of less expensive tools than high speed camera, e.g. with a classical camera (frequency max is 12.5Hz at image resolution of 1024*768 pixels). Finally it will be interesting to develop a 3D framework with several synchronised camera to continuously monitor an important structure like a bridge.

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Handle ID: . http://hdl.handle.net/10985/13277
To cite this version :
Malik MUHAMMAD HARIS, Domenico BORZACCHIELLO, José Vicente AGUADO, Francisco CHINESTA - Advanced parametric space-frequency separated representations in **structural** **dynamics**: A harmonic–modal hybrid approach - Comptes Rendus - Mecanique - Vol. 346(7), p.590-602. - 2018

through Fe–L elongation was limited by the decay of the MLCT towards the T state, where antibonding orbitals were populated since the initial 1 MLCT state had on equally bonding character as the initial LS state. The molecular expansion only began once the T state was reached, which took about 120(10) fs, before decaying towards the nal HS state within 70(10) fs. The d–d excitation process instantaneously populated anti- bonding orbitals, which launched the Fe–L expansion and moved the system towards the HS potential. The HS state was then reached within 70(10) fs. Herein, we could accurately study the coherent **structural** **dynamics** during LIESST through the intense d–d HS band in the FeN 4 O 2 system, strongly modulated by the Fe–L distance. By reducing the lifetime of intermediates compared to MLCT excitation, d–d excitation allowed a faster LIESST and preserved **structural** breathing coherence since this

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Photo-induced **structural** **dynamics**
Measurements of transient changes to both the re ﬂectivity and birefringence, using pump pulses at a photon energy tuned in resonance with the infrared-active phonon modes, hν ≃ 85 meV (21 THz), reveal multiple oscillatory responses at frequencies signi ﬁcantly below the one of the pump (see Fig. 2 a, b). The highest-frequency oscillation is centered at 1.1 THz (4.3 meV) and is assigned to the Raman-active E g soft mode of LaAlO 3 24 , 28 associated with a rhombohedral instability of the R3c lattice structure (see Supplementary Note 1). This mode comprises rotations of the oxygen octahedra around an axis perpendicular to the [111] pseudocubic direction as shown in Fig. 2 c. The longer time delay further reveals oscillatory components at two discrete frequencies f TA and f LA in the GHz frequency range. Such a pattern originates from interference between light pulses re ﬂected at the crystal surface and re ﬂections from an acoustic wave front propagating into the bulk (Fig. 2 c). In transparent materials, the frequency of the oscillations f is related to the refractive index n of the material at the probe wavelength 27 , the speed of sound v s , the angle θ w.r.t. the sample normal and the wavelength λ of the probe by the relation 29 f ¼ 2nv s cosðθÞ=λ. In our experiments we

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Figures 6a–c depict the three key dynamical groups as a function of time, representing the ultrafast **structural** **dynamics**. The time constant of these key modes is 2.3 ps. The time-dependent changes of ξ are summarized in Figure 6d. The reaction coordinates of three motions represent relative atomic motions from the initial LS state at ξ = (0, 0, 0) to towards the final HS state at ξ = (1, 1, 1). Previous picosecond X-ray diffraction studies on later processes associated with SCO indicate a local excited HS molecule within a LT unit cell, which subsequently changes to a HT unit cell through molecular rearrangement. [21,22] Here, we find that the photoexcited HS structure, with ξ = (0.9, 0.8, 0.75), is similar but distinct from the thermal HS structure since the photoexcited HS molecule is constrained inside the smaller LT unit cell. We experimentally detect a local unit cell expansion on the ultrafast timescale, which is induced by the Fe–N bond elongation and ligand motion through intermolecular elastic interaction, which was not observed in previous studies. This is of key interest to understand how IVR and light-induced intermolecular forces within the unit cell drive lattice expansion that is subsequently responsible for cooperative elastic spin-state switching. [21]

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Biasin et al 214 studied the **structural** **dynamics** upon SCO in photoexcited [Co II (terpy)
2 ] in an aqueous solution with
femtosecond X-ray scattering at LCLS. Their analysis showed that photoexcitation leads to elongation of the Co-N bonds, followed by coherent Co-N bond length oscillations arising from the displacive excitation of a vibrational mode dominated by the symmetrical stretch of all six Co-N bonds (Figure 13). The mode has a period of 0.33 ps and decays on a sub-picosecond time scale. It was found that the equilibrium bond-elongated structure of the high spin state is established on a single-picosecond time scale and that this state has a lifetime of ∼7 ps. This first demonstration of vibrational wave packets by fs XRS in solution opens the way to more systematic studies on molecular systems, as the tools are refined and the data treatment becomes more routine. Among these is the possibility to exploit the polarization properties of the pump and probe pulses as explored in a simulation by Penfold

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