A Validation Study for the Computation of Nonlinear Modal Frequency using a Hamiltonian Reduced Order Model

A common structural design verification is to conduct modal analysis and to ensure that the vibration modes of the structure do not get harmonically excited during operation. Since the modal frequencies are dependent on the mass and stiffness properties of the structure, they are an influencing factor in the design optimization. Modal frequencies are obtainable using eigenvalue analysis after linearization approximations in the restoring force terms of the governing equation of motion. While this approximation is valid for lower loads, it is not acceptable for strong nonlinear vibrations. A measurable shift in the modal frequency is observable when the amplitude of vibrations is in the order of the thickness of the vibrating structure. This phenomenon is more pronounced in thin walled and flexible structures where the vibration amplitudes can exceed the linear threshold relatively more easily. A credible approach in computation of nonlinear modes is the utilization of parametric continuation schemes for generating nonlinear frequency-amplitude response. However, utilization of this scheme with a large degree of freedom model is a computationally intensive approach which necessitate the development of reduced order models. A model reduction method in the finite element framework, termed as Hamiltonian Reduced Order Model (ROM), has been recently developed at Delft University of Technology. The present study is conducted for the experimental validation of the ROM. Governing equations of motion of a stiffened plate have been derived using the Hamiltonian ROM and the nonlinear frequency response has been generated using a continuation scheme. For validation, experiments have been conducted using the Laser Doppler Vibrometer to obtain similar frequency response curves. A comparison of the numerical and the experimental results shows excellent agreement. Furthermore, accurate numerical response is obtainable using only a single degree of freedom in the reduced order model which proves the effectiveness of the Hamiltonian ROM. The results also demonstrate the necessity of a nonlinear damping model for obtaining comparable results.Aerospace Engineerin

A momentum subspace method for the model-order reduction in nonlinear structural dynamics: Theory and experiments

The article proposes a method developed for model order reduction in a Finite Element (FE) framework that is capable of computing higher order stiffness tensors. In the method, a truncated third order asymptotic expansion is used for transformation of an FE model to a reduced system. The basis matrix in the formulation of the reduced-order model (ROM) is derived from linear mode shapes of the structure. The governing equations are derived using Hamilton's principle and the method is applied to geometrically nonlinear vibration problems to test its effectiveness. An initial validation of the numerical formulation is obtained by comparison of results from time domain nonlinear vibration analyses of a rectangular plate using Abaqus. Subsequently, a stiffened plate is modeled to test a more complex structure and a continuation algorithm is used in combination with the ROM to compute nonlinear frequency response curves. The validation of the stiffened plate has been performed through comparisons with the results of nonlinear vibration experiments. The experiments are conducted with Polytec Laser Doppler Vibrometer and PAK MK-II measurement systems for large amplitude vibrations to validate the nonlinear vibration analyses.Aerospace Structures & Computational MechanicsDynamics of Micro and Nano System