A Surrogate Reduced Order Free Vibration Model of Linear and Non-Linear Beams using Modified Modal Coefficients and HOSVD Approaches
Subject Areas : Mechanical Engineering
Mohammad Kazem
Moayyedi
1
(Department of Mechanical Engineering,
University of Qom, Iran)
Keywords: Structural Dynamics, Proper Orthogonal Decomposition, Low-Dimensional Model, Free vibration, High Order Singular Value Decomposition,
Abstract :
In the present work, two low-dimensional models are presented and used for vibration simulation of the linear and non-linear beam models. These models help to compute the dynamical responses of the beam with fast computation speed and under the effects of different conditions. Also the obtained results can be used in the conceptual and detailed design stages of an engineering system overall design. First, a finite element analysis based on Euler-Bernoulli beam elements with two primary variables (deflection and slope) at each node is used to find static and dynamic responses of the considered linear and non-linear beams. Responses to three different static load cases are obtained and applying them as initial conditions, the time responses of the beam are calculated by the Newmark's time approximation scheme. A low-dimensional POD model which was extracted from the ensemble under the effect of an arbitrary loading is reconstructed. To apply the model to simulate the response of beam under the effect of other loads, POD modal coefficients are updated due to change of initial condition. This modification is performed based on the recalculation of the eigenvalues due to a new initial condition. Also, another low-dimensional model is constructed which is developed based on an ensemble under the effect of several parameters. To apply the model to simulate the response of the beam under the effect of other loads and variations of beam thickness, POD-HOSVD modal coefficients are updated due to the change of desired parameters. The results obtained from the low-dimensional model are showing good agreement to the benchmark data and proving high level accuracy of the model.
[1] Lucia, D. J., Beranb, P. S., and Silva, W. A., Reduced Order Modeling: New Approaches for Computational Physics, Progress in Aerospace Sciences, Vol. 40, 2004, pp. 51-117.
[2] Holmes, P., Lumley, J. L., and Berkooz, G., Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge Monographs on Mechanics, Cambridge University Press, 1996.
[3] Taeibi-Rahni, M., Sabetghadam, F., and Moayyedi, M. K., Low-Dimensional Proper Orthogonal Decomposition Modeling as a Fast Approach of Aerodynamic Data Estimation, Journal of Aerospace Engineering, Vol. 23, Issue 1, 2010, pp. 44-54.
[4] Sirovich, L., Kirby, M., Low-Dimensional Procedure for the Characterization of Human Faces, J. Optical Society of America, Vol. 4, No. 3, 1987, pp. 519–24.
[5] Shinde S., Pandy M., Modelling Fluid Structure Interaction Using One-way Coupling and Proper Orthogonal Decomposition, Proceedings of the 11th International Conference on Engineering Sciences (AFM 2016), At Ancona, Italy, 2016.
[6] Tissot, G., Cordier, L., Noack, B. R., Feedback Stabilization of an Oscillating Vertical Cylinder by Pod Reduced-Order Model, Journal of Physics: Conference Series, Vol. 574, 2015, pp. 012137.
[7] Feeny, B. F., Kappagantu, R., On the Physical Interpretation of Proper Orthogonal Modes in Vibration, Journal of Sound and Vibration, Vol. 211, No. 4, 1998, pp. 607-616.
[8] Han, S., Linking Proper Orthogonal Decomposition Modes and Normal Modes of the Structure, Proceedings of the 18th International Modal Analysis Conference (IMAC), 2000.
[9] Kereschen, G., Golinval, J. C., Physical Interpretation of the Proper Orthogonal Modes using the Singular Value Decomposition, Journal of Sound and Vibration, Vol. 249, No. 5, 2002, pp. 849-865.
[10] Han, S., Feeny, B., Application of Proper Orthogonal Decomposition to Structural Vibration Analysis, Mechanical Systems and Signal Processing, Vol. 17, No. 5, 2003, pp. 989-1001.
[11] Amabili, M., Sarkar, A., and Paı̈doussis, M. P., Reduced-Order Models for Nonlinear Vibrations of Cylindrical Shells Via the Proper Orthogonal Decomposition Method, Journal of Fluids and Structures, Vol. 18, No. 2, 2003, pp. 227-250.
[12] Lieu, T., Farhat, C., Adaptation of POD-Based Aeroelastic ROMs for Varying Mach Number and Angle of Attack: Application to a Complete F-16 Configuration, AIAA Paper 2005-7666, U.S. Air force T&E Days, Tennessee, 2005.
[13] Gonçalves, P. B., Prado, Z. J. G. N. D., Low-Dimensional Galerkin Models for Nonlinear Vibration and Instability Analysis of Cylindrical Shells, Nonlinear Dynamics, Vol. 41, No. 1, 2005, pp. 129-145.
[14] Georgiou, I., Advanced Proper Orthogonal Decomposition Tools: Using Reduced Order Models to Identify Normal Modes of Vibration and Slow Invariant Manifolds in the Dynamics of Planar Nonlinear Rods, Nonlinear Dynamics, Vol. 41, No. 1, 2005, pp. 69-110.
[15] Allison, T. C., System Identification via the Proper Orthogonal Decomposition, PhD Dissertation, Dep't Mechanical Engineering, Virginia Polytechnic Institute and State University, 2007.
[16] Gonçalves, P. B., Silva, F. M. A., Del Prado, Z. J. G. N., Low-dimensional Models for the Nonlinear Vibration Analysis of Cylindrical Shells Based on a Perturbation Procedure and Proper Orthogonal Decomposition, Journal of Sound and Vibration, Vol. 315, No. 3, 2008, pp. 641-663.
[17] Gonçalves, P. B., Silva, F. M. A., Del Prado, Z. J. G. N., Reduced Order Models for the Nonlinear Dynamic Analysis of Shells, Procedia IUTAM, Vol. 19, 2016, pp. 118-125.
[18] Speet, J., Parametric Rreduced Order Modeling of Structural Models by Manifold Interpolation Techniques, M.Sc. Thesis, Department of Mechanical Engineering at Delft University of Technology, 2017.
[19] Ilbeigi, S., A New Framework for Model Reduction of Complex Nonlinear Dynamical Systems, PhD Dissertation, Department of Mechanical Engineering at University of Rhode Island, 2017.
[20] Guo, M., Hesthaven, J. S., Reduced Order Modeling for Nonlinear Structural AnalysisU Gaussian Process Regression, Computer Methods in Applied Mechanics and Engineering, Vol. 341, 2018, pp. 807-826.
[21] Reddy, J. N., An Introduction to Finite Element Method, 3rd Edition, McGraw Hill, 2006.