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Description
This paper presents a method for integrating base excitations into the harmonic balance method within the context of path continuation. Footpoint excitation can be applied via linear or non-linear elements to the mechanical system. First, a brief introduction into the harmonic balance method and the continuation procedure are given. Then, different types of base excitations are discussed: displacement, velocity or acceleration driven base excitation. Also, the relevant connections in the time and the frequency domain as well as the required derivatives are derived. Afterwards, the handling of the linear coupling case is shown as it is straightforward and only requires the usage of the derivatives declared beforehand. The equations of motion can be rearranged such that the base excitation appears exclusively on the right-hand side as an external force which may depend on the excitation frequency. Following this, two methods are presented to handle the nonlinear coupling case: The first method uses the definition of relative coordinates to achieve the base excitation to be included into the external forces again as in the linear case. The downside of this approach is that it is not generally applicable, which will be discussed. A second more general approach includes the base excitation into the nonlinear force term which then depends on the excitation frequency. Again, the derivatives needed for continuation are derived. This approach allows to retain the original set of coordinates. In the following, the proposed method is applied to a single degree of freedom system. The system shows a behavior where for certain choices of the excitation amplitude and the nonlinearity there is no turning point near resonance and the FRF splits up into two separate curves. This behavior is further investigated by using an analytic approach which allows reformulating the problem into a third order polynomial. Several parameter studies are shown, including the locus curves of the local maximum and the turning point under variation of the base excitation amplitude. Additionally, frequency sweep data is shown to verify the effect. The proposed method is then applied onto a multi degree of freedom system which additionally has nonlinear couplings within its physical coordinates. The FRF under variation of the base excitation amplitude shows the same behavior of having no turning point. Moreover, the applicability to arbitrary NDOF systems is proven.
