A numerical approach to directly compute nonlinear normal modes of geometrically nonlinear finite element models

• Extended an existing pseudo-arclength continuation algorithm to compute nonlinear normal modes of finite element models within the native code.
• Initial conditions for periodic solutions are based on a subset of the underlying linear modes, drastically reducing the number of variables to iterate on in the continuation algorithm.
• Demonstrated the approach on a beam model to highlight the performance of the algorithm.
• Demonstrated the approach on a realistic FEA model showing that this is applicable to large scale models.

The nonlinear normal modes of a dynamical system provide a modal framework in which the dynamics of a structure can be readily understood. Current numerical approaches use continuation to find a nonlinear normal mode branch that initiates at a low energy, linearized mode. The predictor-corrector based approach follows the periodic solutions as the response amplitude increases, forming the nonlinear normal mode. This method uses the Jacobian of the shooting function in a Newton–Raphson algorithm to find the initial conditions and integration period that result in a periodic response of the conservative equations of motion. Large scale finite element models require that the Jacobian be computed using finite differences since the closed form equations are not explicitly available. The Jacobian must be computed with respect to all of the states, making the algorithm prohibitively expensive for models with many degrees-of-freedom. In this paper, the initial conditions of each periodic solution are determined based on a subset of the linear modes of a geometrically nonlinear finite element model. The first approach, termed enforced modal displacement, sets the initial conditions as a linear combination of linear mode shapes. The second approach, here called the applied modal force method, applies a static load to the structure in a combination of applied forces that would excite a single linear mode, computes the static response to that load, and uses that to set the initial conditions. Both of these algorithms greatly reduce the number of variables that are iterated on during continuation. As a result, the cost of computing each solution along the nonlinear normal mode is only on the order of ten times the cost required to integrate the finite element model over one period of the response. The algorithm is initiated with only one linear mode and additional modes are added in a systematic way as they become important to the periodic solutions along the nonlinear mode branch. The approach is demonstrated on two geometrically nonlinear finite element models, showing a dramatic reduction in the computational cost required to obtain the nonlinear normal mode.