Abstract
Enhanced modal-based order reduction of forced structural dynamic systems with isolated nonlinearities has been performed using the updated LELSM (local equivalent linear stiffness method) modes and new Ritz vectors. The updated LELSM modes have been found via iteration of the modes of the mass normalized local equivalent linear stiffness matrix of the nonlinear systems. The optimal basis vector of principal orthogonal modes (POMs) is found via simulation and used for POD-based order reduction for comparison. Two new Ritz vectors are defined as static load vectors. One of them gives a static displacement to the mass connected to the periodic forcing load and the other gives a static displacement to the mass connected to the nonlinear element. It is found that the use of these vectors, which are augmented to the updated LELSM modes in the order reduction modal matrix, reduces the number of modes used in order reduction and considerably enhances the accuracy of the order reduction. The combination of the new Ritz vectors with the updated LELSM modes in the order reduction matrix yields more accurate reduced models than POD-based order reduction of the forced nonlinear systems. Hence, the LELSM modal-based order reduction is enhanced via new Ritz vectors when compared with POD-based and linear-based order reductions. In addition, the main advantage of using the updated LELSM modes for order reduction is that, unlike POMs, they do not require a priori simulation and thus they can be combined with new Ritz vectors and applied directly to the system.
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References
Guyan, R.J.: Reduction of stiffness and mass matrices. AIAA J. 2, 380 (1965)
Burton, T.D., Young, M.E.: Model reduction and nonlinear normal modes in structural dynamics. In: Nonlinear and Stochastic Dynamics Symposium. AMD, vol. 192. DE, vol. 78, pp. 9–16. ASME Winter Ann. Mtg., Chicago, IL, pp. 6–11 (1994)
Friswell, M.I., Penny, J.E.T., Garvey, S.D.: Using linear model reduction to investigate the dynamics of structures with local nonlinearities. Mech. Syst. Signal Process. 9(3), 317–328 (1995)
Burton, T.D., Rhee, W.: On the reduction of nonlinear structural dynamics models. J. Vib. Control 6, 531–556 (2000)
Kim, J., Burton, T.D.: Reduction of structural dynamics models having nonlinear damping. ASME Paper no. DETC2003/VIB-48435
Butcher, E.A., Lu, R.: Order reduction of structural dynamic systems with static piecewise linear nonlinearities. Nonlinear Dyn. 49, 375–399 (2007)
Shaw, S.W., Pierre, C.: Non-linear normal modes and invariant manifolds. J. Sound Vib. 150, 170–173 (1991)
Jiang, D., Pierre, C., Shaw, S.: Large amplitude nonlinear normal modes of piecewise linear systems. J. Sound Vib. 272, 869–891 (2004)
Peschek, E., Boivin, N., Pierre, C.: Nonlinear modal analysis of structural systems using multi-mode invariant manifolds. Nonlinear Dyn. 25, 183–205 (2001)
Burton, T.D.: Numerical calculation of nonlinear normal modes in structural systems. Nonlinear Dyn. 49, 425–441 (2007)
Shaw, S.W., Pierre, C., Pesheck, E.: Modal analysis-based reduced-order models for nonlinear structures—an invariant manifold approach. Shock Vib. Dig. 31, 3–16 (1999)
Pesheck, E., Pierre, C., Shaw, S.W.: Modal reduction of a nonlinear rotating beam through nonlinear modes. J. Vib. Acoust. 124, 229–236 (2002)
Sinha, S.C., Redkar, S., Butcher, E.A.: Order reduction of nonlinear systems with time periodic coefficients using invariant manifolds. J. Sound Vib. 284, 985–1002 (2005)
Sinha, S.C., Butcher, E.A., Dávid, A.: Construction of dynamically equivalent time invariant forms for time periodic systems. Nonlinear Dyn. 16, 203–221 (1998)
Feeny, B.F., Kappagantu, R.: On the physical interpretation of proper orthogonal modes in vibrations. J. Sound Vib. 211(4), 607–616 (1998)
Feeny, B.F.: On proper orthogonal co-ordinates as indicators of modal activity. J. Sound Vib. 255(5), 805–817 (2002)
Han, S., Feeny, B.F.: Enhanced proper orthogonal decomposition for the modal analysis of homogeneous structures. J. Vib. Control 8(1), 19–40 (2002)
Lenaerts, V., Kerschen, G., Golinval, J.C.: Proper orthogonal decomposition for model updating of nonlinear mechanical systems. Mech. Syst. Signal Process. 15(1), 31–43 (2001)
Kappagantu, R., Feeny, B.F.: An “optimal” modal reduction of a system with frictional excitation. J. Sound Vib. 224(5), 863–877 (1999)
Kerschen, G., Golival, J., Vakakis, A., Bergman, L.: The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: an overview. Nonlinear Dyn. 41, 147–169 (2005)
Kumar, N., Burton, T.D.: Use of random excitation to develop POD based reduced order models for nonlinear structural dynamics. In: Proceedings of the ASME IDETC, Paper DETC2007/VIB-35539, Las Vegas, NV, September 4–7 (2007)
Kumar, N., Burton, T.D.: On combined use of POD modes and Ritz vectors for model reduction in nonlinear structural dynamics. In: Proceedings of the ASME IDETC, Paper DETC2009-87416, San Diego, CA, September (2009)
Segalman, J.D.: Model reduction of systems with localized nonlinearities. J. Comput. Nonlinear Dyn. 2, 249–266 (2007)
Kline, K.A.: Dynamic analysis using a reduced basis of exact modes and Ritz vectors. AIAA J. 24(12), 2022–2029 (1986)
Balmès, E.: Optimal Ritz vectors for component mode synthesis using the singular value decomposition. AIAA J. 34(6), 1256–1260 (1996)
Wilson, E.L., Yuan, M.W., Dickens, J.M.: Dynamic analysis by direct superposition of Ritz vectors. Earthquake Eng. Struct. Dyn. 10, 813–821 (1982)
Lèger, P.: Application of load dependent vectors bases for dynamic substructure analysis. AIAA J. 28(1), 177–179 (1990)
Hurty, W.C.: Dynamic analysis of structural systems using component modes. AIAA J. 3(4), 678–685 (1965)
Craig, R.R., Bampton, M.C.C.: Coupling of substructures for dynamic analyses. AIAA J. 6(7), 1313–1319 (1968)
Apiwattanalunggarn, P., Shaw, S.W., Pierre, C.: Component mode synthesis using nonlinear normal nodes. Nonlinear Dyn. 41, 17–46 (2005)
Suy, H.M.R., Fey, R.H.B., Galanti, F.M.B., Nijmeijer, H.: Nonlinear dynamic analysis of a structure with a friction-based seismic base isolation system. Nonlinear Dyn. 50, 523–538 (2007)
Butcher, E.A.: Clearance effects on bilinear normal mode frequencies. J. Sound Vib. 224, 305–328 (1999)
Al-Shudeifat, M.A., Butcher, E.A., Burton, T.D.: Comparison of order reduction methodologies and identification of NNMs in structural dynamic systems with isolated nonlinearities. In: Proceedings of the International Modal Analysis Conference, Orlando, FL, February 9–12 (2009)
Belendez, A., Hernandez, A., Marquez, A., Neip, C.: Analytical approximation for the period of a nonlinear pendulum. Eur. J. Phys. 27, 539–551 (2006)
Milman, H., Chu, C.: Eigenvalue error analysis of viscously damped structures using Ritz reduction method. AIAA J. 30(12), 2935–2945 (1992)
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AL-Shudeifat, M.A., Butcher, E.A. Order reduction of forced nonlinear systems using updated LELSM modes with new Ritz vectors. Nonlinear Dyn 62, 821–840 (2010). https://doi.org/10.1007/s11071-010-9765-8
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DOI: https://doi.org/10.1007/s11071-010-9765-8