# Chaotic Characteristic Analysis of Spillway Radial Gate Vibration under Discharge Excitation
## 1 Introduction  
A spillway radial gate (SRG) is frequently used as the working gate for reservoir surface spillways owing to the advantages of its small opening and closingforce, good flow pattern, and easy operation. Therefore, its safe operation is of great significance to the whole hydropower station. However, the SRG is affected by the fluctuating loads of high-speed-flow water in the processes of opening and discharging, resulting in different modes of vibration [1]. Non-linear vibration refers to the motion that cannot be described by linear differential equations. For the gate vibration system, when the SRG vibrates under a discharge excitation condition, the supporting arm can create a dynamic unstable region due to the parametric vibration that is strongly non-linear, which can cause chaos. While assembled by welding and bolt connections [2], the vibrations weaken these junctions, which can finally lead to the failure of gate structures. As a result, further investigations into the characteristics of the vibration of SRGs are necessary to avoid these issues.  
To date, many scholars have investigated the fluid-induced vibration (FIV) of SRGs through prototype tests [3], numerical structural calculations [4; 5], hydroelastic modeling experiments [6], and principle analyses [7; 8]. The analyses showed that the supporting arms were the weakest component of the SRG, which could finally lead to the failure of the gate. On 17 July 1995, the rotation vibration around the supporting hinge of the No. 3 SRG of the Folsom dam in the United States and the low-frequency panel bending vibration were superimposed [9], which induced the strong self-excited vibration of the gate and led to the instability of the supporting arm. Furthermore, on 2 July 1967, the SRG of the Wachi dam in Japan was damaged owing to the out-of-plane bucking instability of the supporting arms [10]. Gate vibrations are often caused by forced, self-excited, and parametric vibrations occurring at the same time in practical engineering [11].  
According to the above-mentioned works, the theory of the stability analysis of supporting arms is gradually developing; nevertheless, the complexity of their vibrations havenot been analyzed. [PERSON] et al. [12] used the improved variational mode decomposition (IVMD) technique to conduct a multi-scale chaotic analysis of pipeline vibration responses. [PERSON] et al. [13] collected the time series of the fluctuating pressures in a slit-type energy dissipater of the discharge flow and analyzed the chaotic characteristics of the fluctuating pressure. [PERSON] et al. [14] analyzed the chaotic characteristics of the flow-induced vibration response of the plate gate. However, the method of the hydroelastic test that they adopted could be somewhat different from the actual operation of the gate. [PERSON] et al. [15] used the chaos theory to analyze the prototype test data of the supporting arm during the flood discharge. Both studies adopted the five-point cubic smoothing method to reduce the noise in the data, which can lead to the loss of critical vibration characteristics. On the other hand, the chaotic characteristics of the main beams of the SRG still remain unknown, and the causes of the chaotic phenomena in the gate vibration system are not clear. Hence, it is essential to thoroughly investigate the chaotic characteristics of SRGs.  
In this paper, the chaos theory is introduced to process the vibration time series of the supporting arm and main beams of an SRG, and the saturation correlation dimension and maximum Lyapunov exponent are used to quantitatively reverse analyze the chaotic characteristics of gate vibrations. On this basis, we highlight the vibration excitation source of the SRG. This work provides theoretical solutions for avoiding the operation of the gate under complex and small opening conditions in practical engineering, and also increases the use value of the measured data.