Faithful and Robust Reduced Order Models

Transcription

1 Faithful and Robust Reduced Order Models David B. Segala Naval Undersea Warfare Center, DIVNPT 76 Howell Street, Newport RI, 284 Tel: , David Chelidze Department of Mechanical Engineering University of Rhode Island, Kingston, RI 288 Tel: , There is a great importance for faithful reduced order models that are valid over a range of system parameters and initial conditions. In this paper, we demonstrate through two nonlinear dynamic models (pinned-pinned beam and thin plate) that are both randomly and periodically forced that smooth orthogonal decomposition (SOD)-based reduced order models (ROMs) are valid over a wide operating range of system parameters and initial conditions when compared to proper orthogonal decomposition (POD)-based ROMs. Two new concepts of subspace robustness the ROM is valid over a range of initial conditions, forcing functions, and system parameters and dynamical consistency the ROM embeds the nonlinear manifold are used to show SODs ability in developing robust faithful ROMs that embed the nonlinear manifold in a lower dimensional space than POD when the ROM is used to approximate a system other than the one that it was created from. Nomenclature ROMs Reduced Order Model. LNM Linear Normal Mode. NNM Nonlinear Normal Mode. POD Proper Orthogonal Decomposition. POM Proper Orthogonal Modes. POC Proper Orthogonal Coordinate. POV Proper Orthogonal Values. SOD Smooth Orthogonal Decomposition. SOM Smooth Orthogonal Modes. SOC Smooth Orthogonal Coordinate. SOV Smooth Orthogonal Values. SPM Smooth Projection Mode. Address all correspondence to this author. Introduction Reduced order models (ROM) have a greater impact with the increasing and advancing computing technologies. These advances in computing technologies allow one to numerically interrogate larger scale, more complex dynamical systems (i.e., structural dynamics and acoustics, computational oceanographic/atmospheric dynamics, molecular dynamics and rationale drug template design) for longer time periods and across multiple temporal and spatial scales. The ultimate goal is to develop a ROM for a dynamical system of interest that is valid over an appropriate range of initial conditions, system parameters, and forcing functions. If the ROM is robust enough to yield faithful results over these ranges then the ROM has successfully captured or embedded the nonlinear manifold. However, the following two questions still remain on the topic of ROMs: ) What is the lowest dimensional ROM? and 2) How well does the ROM capture the dynamics of the full scale system model or a perturbed version of that model? Using the newly developed concepts of subspace robustness and dynamical consistency the authors demonstrate through a periodically and randomly forced pinnedpinned beam and flat thin plate that smooth orthogonal decomposition (SOD)-based ROMs develop both dynamically consistent and robust ROMs as compared to proper orthogonal decomposition (POD)-based ROMs. 2 Background The approach for model reduction of linear and nonlinear systems is achieved in two different ways. For the sake of completeness the methodology for linear systems will be briefly mentioned. Given a set of equations describing a linear dynamical system, one projects the equations onto a set of basis vectors obtained through some empirical methods. These methods can include linear normal modes (LNMs),

2 Kyrlov subspaces, and POD [ 6] (also known as, singular value decomposition [2], principal component analysis [2,7], and Karhunen-Loe ve Decomposition [2, 8]). Model reduction for nonlinear systems is a much harder problem and currently the topic of many investigations. Some of these investigations have yielded techniques such as inertial manifold approximation, bilinearization about the equilibrium point, center manifold theory, nonlinear normal modes (NNMs), and POD. A thourough review can be found in [9 3]. Due to the scope of this paper, we will focus on the methodology which develops ROMs by taking the nonlinear system and projecting it onto a linear subspace spanned by a set of appropriate basis vectors. One such common set of basis vectors are proper orthogonal modes (POMs). Due to the orthogonally constraint on POMs, POD will extract a higher dimensional subspace to capture the nonlinear manifold as compared to SOD. Therefore, POD does determine an embedding space but it cannot provide the most efficient embedding space which will lead to a higher dimensional ROM that may contain unwanted time scales. The multivariate multiscale data analysis method of SOD is proposed to find linear subspaces that fully embed the nonlinear manifold in an efficient low dimensional subspace. SOD can be thought of as an extension of POD [4 2]. In addition to considering spatial (i.e., statistical) characteristics of the data set as in POD, SOD considers temporal (i.e., dynamical) characteristics of the data set as well. In particular, SOD identifies coordinates (smooth orthogonal coordinates- SOCs) that have both minimal temporal roughness and maximal spatial variance. Therefore, SOD obtains the smoothest low dimensional approximation of a high dimensional system. Proper and Smooth Orthogonal Decomposition Consider a dynamical system of which n state variables are measured. In addition, each n variable is sampled m times, which are arranged into a m n ensemble matrix Y. If the n state variables are n spatial locations, then the Y i j element represents the i th time instant value for the j th spatial location. Next, an ensemble of each time-derivatives of the state variable is formed by Ẏ DY where D is some discrete differential operator (e.g., based on forward difference). Provided that Y and Ẏ are zero mean, the autocovariance matrices can be formed by Σ yy = m Y T Y, Σẏẏ = mẏ T Ẏ. () The solution for POD, is achieved by solving the eigenvalue problem of auto-covariance matrix Σ yy in Eq. (), Σ yy φ k = λ k φ k, (2) where λ k are proper orthogonal values (POVs), φ k are proper orthogonal modes (POMs), and proper orthogonal coordinates (POCs) are q = Φ T Y where Φ = [φ,φ 2,...,φ n ] R n. POVs are ordered such that λ λ 2...λ n. In practice, the solution for solving POD is achieved by using singular value decomposition on the matrix Y. The solution for SOD, is achieved by solving the generalized eigenvalue problem of the matrix pair Σ yy and Σẏẏ in Eq. () Σ yy ψ k = λ k Σẏẏ ψ k, (3) where λ k are now smooth orthogonal values (SOVs), ψ k R n are smooth projection modes (SPMs), smooth orthogonal modes (SOMs) are Φ = Ψ T, and smooth orthogonal coordinates (SOCs) are given by q = ΨY where Ψ = [ψ,ψ 2,...,ψ n ] R n. It should be noted that if we were to replace Σẏẏ with the identity matrix, then the formulation would yield the proper orthogonal decomposition. In practice, if Y is ill-conditioned, the solution for solving SOD is achieved by using generalized singular value decomposition on the matrix pair Y and DY. 3 Nonlinear Model Reduction Consider the general class of nonlinear ODEs described as, Mẍ +Cẋ + Kx = F(ẋ,x,t), (4) where x R n is a dynamic state variable, t is time, M, C, K R nxn are the global mass, damping, and stiffness matrices, respectively, and F R n describes any nonlinear forcing components. Using a coordinate transformation of x = Pq where q R k where k << n is a reduced state variable and P = [e,e 2,...,e n ] is an appropriate set of basis vectors obtained through some empirical methods (e.g., POD or SOD), the corresponding ROM is P T MP q + P T CP q + P T KPq = P T F(P q,pq,t). (5) 4 Model Reduction Subspace Selection The purpose of this paper is show that using SOD one can develop robust ROMs that embed the active nonlinear manifold as compared to POD. This is demonstrated using the newly developed concepts of subspace robustness and dynamical consistency. In choosing the appropriate subspace of the full scale dynamical system for model order reduction, the selected subspace needs to ) embed or capture the active NNM manifold and (2) be insensitive to a varying set of initial conditions, system parameters, and forcing functions. Refer to [2, 22] for a complete description. 4. Subspace Robustness The key idea behind subspace robustness is that the subspace spanned by the basis vectors need to be insensitive

3 to various forcing functions, initial conditions, and system parameters. Conventional choices like POMs would be a good choice but they can vary with initial conditions, system parameters, and forcing functions. Therefore, a decomposition identifies a robust subspace if all subspaces estimated from trajectories starting from different initial conditions and/or perturbed system parameters are mutually and nearly linearly dependent. Fig.. A simply supported beam with two nonlinear springs positioned left and right of center. Given a matrix S R n ks whose columns contain a set of basis vectors {Pi k}s i= for each k-dimensional subspace, the corresponding subspace robustness γ k s is given by the following: γ k s = 4 π arctan n i=k+ σ2 i. (6) k i= σ2 i From Eq. (6), if γ k s = then the individual subspace realizations Pi k (i =,...,s) are mutually linearly dependent. However, if γ k s = then the individual subspace realizations are mutually linearly independent. Therefore, the subspace identified through some empirical procedure can only be used for model reduction if its subspace robustness γ k s is close to unity. Otherwise, that subspace may not capture all the needed system dynamics for particular sets of parameters. 4.2 Dynamical Consistency The idea is to find subspaces whose resulting reduced trajectories are deterministic and smooth (there is no folding or singularities). Dynamical consistency is determined by estimating a ratio of the number of false nearest neighbors over the total number of nearest neighbor pairs in a particular k-dimensional subspace: ζ k = Nk f nm N nm, (7) where N k f nm is the estimated number of false nearest neighbors in a k-dimensional subspace and Nnm k is the total number of nearest neighbor pairs used in the estimation. If ζ k is close to unity, then that k-dimensional subspace is dynamically consistent. N k f nm is estimated by comparing the distance between the temporally uncorrelated nearest neighbors in a k-dimensional space to the same distance in the k + - dimensional space. If the change in the distances is one order of magnitude larger than the original k-dimensional distance, then these points are denoted as false nearest neighbors in k- dimensional space. 5 Nonlinear Dynamical Systems 5. Simplify Supported Pinned-Pinned Beam The eighteen degree-of-freedom pinned-pinned beam is shown in Fig.. The length of the beam is.6 m, height Fig. 2. A simply supported pate with a nonlinear spring positioned at the center node.. m, width.5 m, Young s modulus is e9 N/m, and density is 785 kg/m 3. The beam is discretized into nine Euler Bernoulli beam elements with two (displacement and rotation) degrees-of-freedom per node. The first and last node experience pinned boundary conditions. The steady state simulation trajectory consisted of, points with the time step of.7. The equations of motion describing the dynamics of the beam are given by an 8-dimensional second-order equation which is described by Eq. (4) where M R 8 8 is the global mass matrix, K R 8 8 is the global stiffness matrix, C R 8 8 is the global proportional damping matrix, and F(x,ẋ,t) can be decomposed into two terms: mainly, f e (t) and f n (x). Two nonlinear springs whose nonlinear force f n (x) is described by Eq. (8) are positioned left and right of center of the center of the beam. f n (x) = 3x 8x 3. (8) A sine wave with frequency 2π was chosen for the periodic forcing and the random forcing was generated by interpolated random sequences. To generate the data for the subspace robustness, the model was simulated for four different forcing amplitudes,.,.7,.3, and 2., which were each was simulated using ten different initial conditions. This resulted in 4 different simulations. 5.2 Simplify Supported Plate The seventy-eight degree-of-freedom plate is shown in Fig. 2. The length of the plate is. m, thickens. m, width. m, Young s modulus is 3e6 N/m, shear correction factor is 5/6, Poissons ratio is.3, and density is 7 kg/m 3. The plate is discretized into sixteen shear deformable plate elements with four nodes per element. Two point integration for the bending term while one point integration is used for the shear term. All the nodes on the perimeter of plate are simply. The steady state simulation trajectory consisted of, points with the time step of.2. The

4 Subspace Coherence POD R POD P SOD P SOD R Dynamical Consistency, ζ(k) POD: Harmonic POD: Random SOD: Harmonic SOD: Random Subspace Coherence SOD P POD R POD P SOD R Dynamical Consistency, ζ(k) POD: Harmonic POD: Random SOD: Harmonic SOD: Random 5 5 Subspace Dimension (# of modes) Subspace Dimension, k Subspace Dimension (# of modes) Subspace Dimension, k Fig. 3. (Left) Subspace coherence and (Right) dynamical consistency for both periodically and randomly forced POD ( and, respectively) and SOD ( and, respectively)-based ROMs for the beam. Fig. 4. (Left) Subspace coherence and (Right) dynamical consistency for both periodically and randomly forced POD ( and, respectively) and SOD ( and, respectively)-based ROMs for the plate. equations of motion describing the dynamics of the plate are given by a 39-dimensional second-order equation which is described by Eq. (4). Using the same matrices as the beam, M R 39 39, K R 39 39, C R 39 39, and F(x,ẋ,t) can be decomposed into two terms: mainly, f e (t) and f n (x). A nonlinear springs whose nonlinear force f n (x) is described by Eq. (9) is positioned at the center of the plate. f n (x) = 4x 8x 2. (9) A sine wave with frequency 4π/5 was chosen for the periodic forcing and the random forcing was generated by interpolated random sequences. To generate the data for the subspace robustness, the model was simulated for four different forcing amplitudes, 5,, 5, and 2, which were each was simulated using ten different initial conditions. This resulted in 4 different simulations. 6 Results The subspace robustness is depicted for the beam Fig. 3 (Left) and for the plate in Fig. 4 (Left). In both the periodically and randomly forced beam and plate models, the subspace robustness for SOD-based ROMs approaches unity monotonically; resulting in the ROM being robust in five and ten dimensions for the beam and plate, respectively. For both forcing cases in the plate, there is a slight decrease in subspace robustness in the first few dimensions before the values begin to increase. On the other hand, for both periodically and randomly forced beam and plate models, the subspace robustness for POD-based ROMs never approaches unity, although is close in value at certain points. In the plate model, dimensions ten through twelve and nineteen are close to unity but then drop off rapidly. The dynamical consistency is depicted for the beam Fig. 3 (Right) and for the plate in Fig. 4 (Right). For both forcing cases, the POD and SOD-based ROM are similar in results. For the beam model, POD and SOD-based ROMs are dynamically consistent in four dimensions or greater. In the plate model, they are consistent in three dimensions or greater. Figure 5 depicts the results for the POD (Top) and SODbased (Bottom) ROMs for periodic forcing with f =.. The blue curve represents the phase space trajectory for degree-of-freedom (DOF) eight. Based on visual inspection, POD-based ROM requires a four-dimensional subspace to reconstruct or track the phase space trajectories (red curves) whereas SOD-based ROM requires five dimensions. Similar results can be drawn for the periodically forced plate as shown in Fig. 6. Here the plate is forced with an amplitude of f =.. The blue curve represents the phase space trajectory for DOF twenty one. Both a twodimensional POD and SOD-based ROM tracks the full scale dynamics quite well but a three-dimensional ROM fully captures the dynamics. If we consider a one-dimensional ROM, SOD captures the general structure whereas POD does not. The results for the randomly forced models parallel that of the periodically forced models. For the beam whose forcing amplitude is f =.6, a three dimensional ROM captures the dynamics for both SOD and POD, Fig. 7. Interestingly, a one and two-dimensional SOD-based ROM captures the general structure and better tracks the dynamics at the origin better than the equivalent POD-based ROM. The results from the randomly forced plate with forcing amplitude f =. are shown in Fig. 8 where POD outperforms SOD-based ROMs. POD can capture the full scale dynamics in only three dimensions and can track the dynamics quite well in only two dimensions. Likewise, SOD tracks the dynamics in two dimensions with the same accuracy but in order to fully capture the dynamics, a six-dimensional ROM is required. Now, we can investigate how well the subspace extracted for model reduction from one set of forcing values and initial conditions approximates the full scale dynamics for a full scale dynamic model with different forcing values, systems parameters, and initial conditions. For the periodically forced beam, the SOD and POD-based subspaces were developed from the full scale dynamical system which are excited with a forcing amplitude of f = 2. and a random initial condition. These extracted subspaces are now used to approximate the full scale dynamic model with forcing amplitude f =.3 and zeros as the initial condition. The results are depicted in Fig. 9 where the blue curve is the full scale trajectory and the red curve is the POD-based (Top) and SOD-based (Bottom) ROM. Using only a one dimensional subspace, SOD-based ROM is able to track the general nature of full scale dynamic model s response (blue curve). By

5 2 POD Dim 2 2. POD Dim SOD Dim 2 2. SOD Dim Fig. 5. (Blue) Phase portrait of degree-of-freedom 8 full scale trajectory with periodic forcing with amplitude f =. for the beam. (Red) One through six dimensional ROM trajectory with periodic forcing with amplitude f =. for (Top) POD and (Bottom) SOD-based ROMs. increasing to a two-dimensional subspace, the SOD-based ROM is able to track the dynamics almost perfectly whereas the POD-based ROM does not track well. Both SOD and POD capture the dynamics in a three-dimensional subspace. The periodically forced plate is shown in Fig. whose full scale dynamics are forced at a frequency of ω = 4π/5 with a random initial condition (blue curve). The ROM is developed for the model with frequency of ω = 3π/5 and zeros as the initial condition. SOD is able to fully capture the dynamics in six dimensions where the ROM is continually improved as dimensions are increased. Therefore, the ROM improves with each new added dimension. On the other hand, POD does not capture the dynamics in fewer than seven dimensions. The random forcing cases depict the same conclusions. In the beam model, a forcing amplitude f = 2. with zeros as the initial condition for both the full scale model and the ROM is used. However, the coefficient in front of the nonlinear spring term is changed from β = to β = 5. SOD is able to track the dynamics in a four dimensional space whereas it takes POD five dimensions. Again, as each dimension is added, SOD increases its performance. Lastly, for the plate, the full scale dynamical system which is excited with a random initial condition and a nonlinear spring coefficient β = 6. The extracted subspaces are Fig. 6. (Blue) Phase portrait of degree-of-freedom 2 full scale trajectory with periodic forcing with amplitude f =. for the plate. (Red) One through six dimensional ROM trajectory with periodic forcing with amplitude f =. for (Top) POD and (Bottom) SOD-based ROMs. now used to approximate the full scale dynamic model with zeros as the initial condition and nonlinear spring coefficient β = 8. With continuous improvement as dimensions are increased, SOD outperforms POD needing three dimensions to capture the dynamics. POD needs at least four dimensions to capture the dynamics. 7 Discussion The purpose of this work was to show SODs ability to embed the nonlinear manifold of a nonlinear dynamical system which will therefore produce robust ROMs that are insensitive to changes in system parameters and/or initial conditions. This is demonstrated using an eighteen degree-offreedom pinned-pinned beam supported by two nonlinear springs left and right of center and a seventy-eight degreeof-freedom plate with a nonlinear spring in the center. For both the periodically and randomly forced beam and plate models, when the ROMs are created and compared against the same full scale dynamic models POD always outperforms SOD. By definition, POD tries to minimize the error in the projections in the least squares sense. Therefore, the subspace that POD captures is going to be the lowest dimensional subspace needed to capture the nonlinear mani-

6 5 POD Dim POD Dim Fig SOD Dim (Blue) Phase portrait of degree-of-freedom 8 full scale trajectory with random forcing with amplitude f =.6 for the beam. (Red) One through six dimensional ROM trajectory with random forcing with amplitude f =.6 for (Top) POD and (Bottom) SOD-based ROMs. Fig SOD Dim (Blue) Phase portrait of degree-of-freedom 2 full scale trajectory with random forcing with amplitude f =. for the plate. (Red) One through six dimensional ROM trajectory with random forcing with amplitude f =. for (Top) POD and (Bottom) SOD-based ROMs. fold. In some instances, for example the periodically forced plate in Fig. 6, SODs performance is comparable to POD. However, the ultimate goal of developing a ROM is to have it valid over a range of operating conditions. If the ROM is only valid for the set of parameters and initial conditions that were used to create the ROM, a new ROM will have to be created for each set of parameters and initial conditions. This negates the reason for the ROM since the full scale model will have to be simulated for at least a brief amount of time to create the projection modes. The amount of time is dependent on the nonlinearity, time step needed to capture the fundamental mode, etc. The subspace robustness for the beam and plate indicates that SOD-based ROMs are robust across a set of initial conditions and forcing functions. This can be observed by noting that for both the periodically and randomly forced cases, the robustness curve increase monotonically until reaching unity for increasing number of modes. For PODbased ROM in both the periodically and randomly forced cases, the subspace robustness never reaches a value of unity. With increasing modes, the robustness just oscillates. However, POD and SOD-based ROM for the beam and plate are dynamically consistent, in four dimensions or greater and three dimensions or great, respectively. This suggests that the ROM will embed the active nonlinear normal mode and the dynamics will be captured if the ROM is developed using at least a four-dimensional subspace or the beam and a three-dimensional subspace for the plate. In all the examples when the projection modes are generated from a full scale model with a set of parameters but is then compared against a full scale model that has different system parameter values, initial conditions, or forcing functions, SOD outperforms POD. This further supports the results from the subspace coherence. Furthermore, SOD adds continual improvement in its ROMs. As the dimension of the subspace is increased, the ROM captures the dynamics with improving accuracy. This can be seen by two different ways. First, by visual inspection of the phase portraits and by the subspace coherence which increases monotonically. The reason for this is due to the orthogonally of the projections modes (POMs and SPMs). POD requires that the POMs are orthogonal to each other whereas SPMs are orthogonal with respect to a set of covariance matrices (Eq. ()). If the nonlinear manifold is extended slightly into a particular higher dimension it may take POD several added dimensions to capture this. However, SOD can orient its modes to align with this extension in a fewer dimensions since the modes do not have to be orthogonal to each other. Therefore, SOD may require a lower dimensional subspace to adequately capture the nonlinear manifold than compared to

7 2 POD Dim POD Dim SOD Dim SOD Dim Fig. 9. (Blue) Phase portrait of degree-of-freedom 8 full scale dynamical system which are excited with a forcing amplitude of f = 2. and a random initial condition. (Red) Extracted subspaces are used to approximate the full scale dynamic model with forcing amplitude f =.3 and zeros as the initial condition for (Top) POD and (Bottom) SOD-based ROMs. Fig.. (Blue) Phase portrait of degree-of-freedom 2 full scale dynamical system which are excited with a forcing frequency of ω = 4π/5 and a random initial condition. (Red) Extracted subspaces are used to approximate the full scale dynamic model with a forcing frequency of ω = 3π/5 and zeros as the initial condition for (Top) POD and (Bottom) SOD-based ROMs. POD. 8 Conclusion A simply supported pinned-pinned beam with two nonlinear springs left and right of center and a simply supported plate with a nonlinear spring at the center were used to demonstrate SODs ability over POD to develop robust, faithful ROMs. Using the two newly developed concepts of subspace robustness and dynamical consistency, it is shown that SOD-based ROMs are valid over a wide range of system parameters, forcing functions, and initial conditions as compared to POD-based ROMs. Therefore, SOD is able to embed the nonlinear manifold of the dynamical system in a lower dimensional space than POD. Furthermore, SOD offers continual improvement of its ROM as the number of dimensions is increased until the ROM fully tracks the full scale dynamical trajectories. Acknowledgements David Segala would like to thank the Naval Undersea Warfare Center Division Newport Internal Investments and David Chelidze would like to thank the National Science Foundation under Grant No. 3. References [] Chatterjee, A., 2. An introduction to the proper orthogonal decomposition. Current Science, 78(7), pp [2] Liang, Y. C., Lee, H. P., Lim, S. P., Lin, W. Z., Lee, K. H., and Wu, C. G., 22. Proper orthogonal decomposition and its applications part i: Theory. Journal of Sound and Vibration, 252, pp [3] Feeny, B. F., and Kappagantu, R., 998. On the physical interpretation of proper orthogonal modes in vibrations. Journal of Sound and Vibration, 2(4), pp [4] Volkwein, S., 28. Model reduction using proper orthogonal decomposition. lecture notes,. Tech. rep., Institute of Mathematics and Scientific Computing, University of Graz. [5] Kershen, G., F, P., and Golinval, J., 27. Physical interpretation of independent component analysis in

8 2 POD Dim 2 2 POD Dim SOD Dim 2 2 SOD Dim Fig.. (Blue) Phase portrait of degree-of-freedom 8 full scale dynamical system which are excited with zeros as the initial condition and nonlinear spring coefficient β =. (Red) Extracted subspaces are used to approximate the full scale dynamic model with zeros as the initial condition and nonlinear spring coefficient β = 5 for (Top) POD and (Bottom) SOD-based ROMs. Fig. 2. (Blue) Phase portrait of degree-of-freedom 8 full scale dynamical system which are excited with random initial condition and nonlinear spring coefficient β = 6. (Red) Extracted subspaces are used to approximate the full scale dynamic model with zeros as the initial condition and nonlinear spring coefficient β = 8 for (Top) POD and (Bottom) SOD-based ROMs. structural dynamics. Mechanical Systems and Signal Processing, 2, pp [6] Feeny, B., 22. On the proper orthogonal modes and normal modes of a continuous vibration system. Journal of Sound and Vibration, 24(), pp [7] Shlens, J., 25. A tutorial on principal component analysis. Tech. rep., Institute for nonlinear science, University of California, San Diego, La Jolla, CA [8] Berkooz, G., Holmes, P., and Lumley, Jonh, L., 993. The proper orthogonal decomposition in the analysis of turbulent flows. Annuals Review of Fluid Mechanics, 25, pp [9] Rega, G., and Troger, H., 25. Dimension reduction of dynamical systems: Methods, models, applications. Nonlinear Dynamics, 4, pp. 5. [] Shaw, S., and Pierre, C., 994. Normal modes of vibration for non-linear continuous systems. Journal of Sound and Vibration, 69(3), pp [] Shaw, S., and Pierre, C., 99. Non-linear normal modes and invariant manifolds. Journal of Sound and Vibration, 5(), pp [2] Shaw, S., and Pierre, C., 993. Normal modes for non-linear vibratory systems. Journal of Sound and Vibration, 64(), pp [3] Vakakis, A., 997. Non-linear normal modes (nnms) and their application in vibration theory: An overview. Mechanical Systems and Signal Processing, (), pp [4] Chelidze, D., and Liu, M., 25. Dynamical systems approach to fatigue damage identification. Journal of Sound and Vibration, 28, pp [5] Chelidze, D., and Liu, M., 28. Reconstructing slowtime dynamics from fast-time measurements. Philosophical Transaction of the Royal Society A, 366, pp [6] Chelidze, D., and Cusumano, J. P., 26. Phase space warping: nonlinear time series analysis for slowly drifting systems. Philosophical Transactions of the Royal Society A, 364, pp [7] Chelidze, D., 24. Identifying multidimensional damage in a hierarchical dynamical system. Nonlinear Dynamics, 3, pp [8] Chelidze, D., and Zhou, W., 26. Smooth orthogonal decomposition based modal analysis. Journal of Sound and Vibration, 292(3 5), pp [9] Farooq, U., and Feeny, B., 28. Smooth orthogonal decomposition for modal analysis of ramdomly ex-

9 cited systems. Journal of Sound and Vibration, 36, pp [2] Chelidze, D., 23. Smooth robust subspace based model reduction. Proceedings of the ASME 23 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference. [2] Chelidze, D., 29. Nonlinear normal mode embedding for model reduction. Euromech Colloquium 53: Nonlinear Normal Modes, Model Reduction and Localization. [22] Segala, D.B,., and Chelidze, D., 23. Robust and dynamically consistent reduced order models. Proceedings of the ASME 23 International Mechanical Engineering Congress & Exposition.