# Analysis of Dynamic Characteristics of Attached High Rise Risers
## 3 Dynamic Analysis of VNA System under Combined Excitation
### Vibration Response of the VNA System under First Order Frequency Locking
Figure 7: Relationship between vibration amplitude of VNA system and excitation frequency under different constraint types. (**a**) Amplitude frequency response of I, II and III; (**b**) Amplitude frequency response of I, IV and V; (**c**) Amplitude frequency response of I, VI, and VII. As shown in Figure 7a, when the nonlinear stiffness of the elastic constraint remains unchanged, the resonance response amplitude of the system decreases slightly with the increase in the linear stiffness, and the resonance region also shifts to the right, all as a consequence of the fact that the increase in the linear stiffness makes the linearized natural frequency of the structure enlarged and the resonance sensitive region to the excitation frequency shift to the right. From Figure 7b, it can be seen that as the linearized stiffness remains constant and the nonlinear stiffness gradually increases, the frequency drift of the system becomes more and more obvious, and the rightward shift exhibits a significant \"hard\" characteristic. With the emergence of the drift phenomenon, the resonance response amplitude of the structural system decreases significantly. In summary, as the nonlinear phenomenon gradually becomes prominent, the multivalued response region of the VNA system under a single excitation frequency becomes wider, and the amplitude jump phenomenon becomes more obvious. As can be seen from Figure 7c, when the linearized stiffness of elastic constraints changes synchronously with the nonlinear stiffness, the resonance response amplitude of the VNA system also decreases significantly. However, due to the increase in linearized stiffness, the multivalued response region of the structure at a single excitation frequency does not widen as much as in Figure 7b, especially under Type VII constraints. In order to study the impact of external input energy on the response of the VNA system, fundamental excitations of different amplitudes are applied to the VNA system. Under the frequency-locked condition of vortex-induced vibration (i.e., \(\sigma_{2}=0\)), the system response amplitudes of different elastic foundation constraint stiffnesses are qualitatively studied. The constraint type is selected from the previous section, that is,  
* Linear stiffness \(10^{3}\) N/m, nonlinear stiffness \(10^{3}\) N/m\({}^{3}\);
* Linear stiffness \(10^{4}\) N/m, nonlinear stiffness \(10^{4}\) N/m\({}^{3}\);
* Linear stiffness \(10^{5}\) N/m, nonlinear stiffness \(10^{5}\) N/m\({}^{3}\). The relationship between the steady-state response amplitude of the VNA pipeline and the fundamental excitation amplitude is studied when the main resonance is fully tuned (\(\sigma_{1}=0\)) and the main resonance is detuned (\(\sigma_{1}=490.33\)). Results are shown in Figure 8, where \(\eta_{A}\) is the dimensionless amplitude at the top of the VNA system pipeline. As shown in Figure 8a, when the main resonance parameters of the VNA system are fully tuned with the gradual increase in the foundation acceleration excitation amplitude, the vibration response amplitudes at the top of the VNA pipeline are all unique, determined values, and the system exhibits stable periodic motion. furthermore, the amplitudes grow nonlinearly with the increase in the foundation acceleration excitation, and the growth rate gradually decreases when the foundation acceleration excitation amplitude is large. In Figure 8b, the main resonance parameters are completely detuned, and the steady-state  
Figure 8: Relationship between vibration amplitude of VNA system and excitation amplitude under different constraint types. (**a**) resonance frequency conditions; (**b**) non resonant frequency conditions. amplitude of the VNA pipeline simultaneously exhibits multiple steady-state solutions within a certain range. Two of the three solutions are stable and the other is unstable, and the specific real response is determined by the initial conditions. There is only one stable solution for the system outside the multistable solution region.