![]() Here, the BPs are defined along with the saddle node relevance by estimating the angles of the arc-length from the unit vector. Thus, the specific objectives of this study are as follows: First is to suggest a method to determine the bifurcation point (BP) of the HBM solutions with respect to the arc-length vector. In addition, various prior studies have been conducted to examine the nonlinear dynamic responses in a torsional system with clearance-type nonlinearities by employing a multiterm HBM 19, 20, 21, 22, 23, 24.īased on prior studies of various nonlinear analysis problems, this study suggests a method to investigate the bifurcation characteristics using both the HBM and Hill’s method. To capture the sub-harmonic effects, we artificially modified the input conditions with respect to the relevant sub-harmonic input terms. In addition, an excitation perturbation method to trigger a sub-harmonic resonance has been suggested 18. Here, the strong nonlinear cubic stiffness subject to non-probability was estimated using the IHB method. The incremental harmonic (IHB) method was used to investigate the limit cycle oscillation of a two-dimensional airfoil with parameter variability in an incompressible flow 11. In this study, NOFRFs in strong nonlinear equations were implemented by employing the Volterra series to extend the classic FRF to nonlinear cases. For example, nonlinear output frequency response functions (NOFRFs) as a type of HBM have been suggested to examine the nonlinear motions of the Duffing oscillator 10. Many studies have been conducted to resolve these problems. Thus, it is difficult to recognize the practical dynamic behaviors that occur in the physical system, especially regarding the unstable regimes determined by Hill’s method. In addition, the employed Hill’s method can only project the information of the stability conditions. However, the incremental HBM employed in this study has some limitations in examining complex dynamic behaviors because the basic mathematical model is constructed based on integer valued harmonic terms 5, 6. In addition, Hill’s method can be employed to reveal the stability conditions, while the system analysis is conducted under the frequency up- or down-sweeping conditions 5, 6, 17. The harmonic balance method (HBM) is efficiently used to investigate nonlinear dynamic behaviors in a torsional system induced by piecewise-type nonlinearities under sinusoidal input conditions, especially to examine multiple numbers of periodic responses 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16. Thus, this study suggests fundamental method to understand the bifurcation phenomena using only the HBM with Hill’s method. The methods employed in this study successfully explain the basic ways to examine the bifurcation phenomena when the HBM is implemented. Finally, the real parts of the eigenvalues are analyzed to examine the practical dynamic behaviors, specifically in the unstable regimes, which reflect the relevance of various bifurcation types on the real part of eigenvalues. Then, their points are defined for various bifurcation types. Second, the bifurcation points are determined by tracking the stability variational locations on the arc-length continuation scheme. To analyze the bifurcation phenomena, the HBM is first implemented utilizing Hill’s method where various local unstable areas are found. Thus, the main goal of this study is to suggest mathematical and numerical approaches to determine the complex dynamic behaviors regarding the bifurcation characteristics. However, it is difficult to understand the practical dynamic behaviors with only their stability conditions, especially with respect to unstable regimes. In addition, the stability conditions can be examined by employing Hill’s method. The results of the harmonic balance method (HBM) for a nonlinear system generally show nonlinear response curves with primary, super-, and sub-harmonic resonances.
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