Abstract Summary
Large amplitude vibrations in slender structures tend to introduce nonlinearities in the structural response due to coupled bending and membrane stretching effect. This is generally observed as a hardening or a softening type nonlinearity which results in amplitude dependent shifts in modal frequencies [1]. These effects cannot be captured using the traditional eigenvalue analyses approach due to linearization approximations. Therefore, new methods are required to be developed in order to account for the nonlinear effects. For generality in application to various types of structures, it is convenient to have finite element (FE)-based formulations. Recently, an approach for model order reduction technique using a momentum subspace formulation [2] has been developed which is applicable to FE models. The geometric nonlinearity is introduced in the system using the von Karman kinematic model where higher order displacement derivatives are accounted for. The stiffness tensors are subsequently computed as up to fourth order derivatives of the total strain energy in the system. This results in an equation of motion (EOM) with quadratic and cubic nonlinearities. To obtain the reduced order model (ROM), an adaptation of the Koiter-Newton method [3] for post-buckling analyses is utilized. A quadratic mapping is used to correlate the real displacements to the reduced subspace. Eigenmode shapes obtained using the linear eigenvalue analysis are utilized in the reduction basis matrix to obtain the ROM. The EOM in the reduction subspace is transformed to a set of first order differential equations using the Hamiltonian formulation. A solution in the time domain is subsequently obtainable using numerical integration techniques. Alternatively, a frequency response curve is obtained using continuation algorithms [4]. Studies conducted using this approach on simple structures, such as rectangular plates, have demonstrated high accuracy and efficiency. Furthermore, experiments conducted on a stiffened plate to measure the nonlinear frequency response have been used to validate the numerical results. Comparisons show that for the chosen test cases, accurate results are obtainable using up to 2-DOF reduced order models. References: [1] Alijani, F., & Amabili, M. (2014). Non-linear vibrations of shells: A literature review from 2003 to 2013. International journal of non-linear mechanics, 58, 233-257. [2] Sinha, K., Singh, N. K., Abdalla, M. M., De Breuker, R., & Alijani, F. (2020). A momentum subspace method for the model-order reduction in nonlinear structural dynamics: Theory and experiments. International Journal of Non-Linear Mechanics, 119, 103314. [3] Liang, K., Abdalla, M., & Gürdal, Z. (2013). A Koiter‐Newton approach for nonlinear structural analysis. International journal for numerical methods in engineering, 96(12), 763-786. [4] Krauskopf, B., Osinga, H. M., & Galán-Vioque, J. (2007). Numerical continuation methods for dynamical systems (Vol. 2). Berlin: Springer.
