Utilizing a blending function to improve nonlinear frequency response predictions for a pendulum system
This paper investigates the use of blending functions to improve nonlinear frequency response predictions for a pendulum system. The harmonic balance method is used to generate approximate analytical solutions for the amplitude-dependent natural frequency of an unforced pendulum, as well as the frequency response behavior of a forced pendulum. For both cases, separate results are obtained and compared from two different approximations for the sine function: Taylor series expansion (evaluated at the stable equilibrium position of the pendulum) and Legendre polynomial series expansion (evaluated for the pendulum's entire range of motion). The solutions obtained from these two approximations feature a classical trade-off, with the Taylor series predictions offering superior accuracy at small to moderate amplitudes of oscillation, while the Legendre polynomial series predictions offer superior accuracy at moderate to large amplitudes of oscillation. We propose a method for blending these two results in order to obtain a hybrid analytical expression that captures the nonlinear amplitude-dependent characteristics of the pendulum system across its entire range of motion, and we validate these results with numerical studies and experimentation. We believe that this blending approach could also be applied to more complex systems whose dynamics exhibit similar nonlinear behavior to the pendulum.
