Modeling Nonlinear Beam Vibrations: A Comparison Between Classical and Data-Driven Approaches

Vibrating slender structures often deform considerable, which trigger nonlinear behavior. For example, when the curvature is no longer small, the equations of motion contain nonlinear terms, which are often neglected assuming small vibration amplitudes. In this chapter, we experimentally observe nonlinear vibration behavior of a slender beam under harmonic excitation. We excite the bending modes of the beam and observe the forced response for different excitation levels. The frequency response shows a softening behavior with a jump phenomenon. The nonlinearity is more pronounced in higher bending modes but is already detected in the first clamped free bending motion. The experimental results are compared with an analytical approximation using the single nonlinear mode theory. The single-mode approach is appropriate for isolated modes and predicts a softening Duffing equation as a minimal nonlinear model of geometrically nonlinear beams. Finally, we obtain the governing equations directly from the measurements utilizing data-driven techniques. The underlying nonlinear differential equation is derived using test functions and sparse identification. The identified parameters are then compared to the analytical model.