## 1 Introduction
[20] explored the effects of the flame equivalence ratio and fuel flow rate on nonlinear thermoacoustic instability in a swirl combustor, and mode switching between higher harmonic and fundamental oscillation was observed. It has also been found that the fuel type and air/fuel supply method affect the triggering and growth of thermoacoustic oscillations [21; 22; 23; 24]. The influence of geometric parameters on thermoacoustic oscillations cannot be ignored. [PERSON] et al. [25] evaluated the damping capacity according to geometrics and the number of resonators using a Rijke tube, and the decay time of thermoacoustic oscillations was measured to quantify the damping capability in the time domain. Therefore, by illuminating the influence of parameter settings on the triggering time and climb rate of thermoacoustic oscillations, combined with using low-order models to construct a model-based control system [26; 27; 28], a better design for active control systems for thermoacoustic oscillations can be achieved for real burners. The analytical models that have been used by researchers to study the thermoacoustic oscillation phenomenon contain the G-equation method [29; 30] or the \(n\)-\(\tau\) model [31]. In this study, only linear acoustic vibrations were expected in the combustor chamber; the nonlinear effects were supposed to be related to the flame heat-release rate, which exhibited saturation characteristics. To predict limit cycle thermoacoustic oscillations, scholars establish the flame's G-equation by linking the flame surface with flame heat release. The empirical approach is to use a parameter-dependent \(n\)-\(\tau\) model. Therefore, this study intended to construct a low-order model of thermoacoustic oscillations with the \(n\)-\(\tau\) model, aiming to explore the temporal signal characteristics of thermoacoustic oscillation under different key parameter settings. Through analyzing the differences in the triggering time and climb rate of thermoacoustic oscillations, this study will provide references to better construct a model-based control system for thermoacoustic oscillation feedback control. In this work, the triggering times and climb rates of thermoacoustic oscillations are theoretically studied. In order to present the impact of changes in the combustion parameters, three main factors of the combustion system were researched--burner length, inlet air temperature and thermal power. Five burner lengths were measured in this study. The inlet air temperature for combustion ranged from 0 to 100 \({}^{\circ}\)C. The thermal power of the burner was also set to five different values. Under these parameters, the triggering times and climb rates of thermoacoustic oscillations were recorded and studied. A low-order model of thermoacoustic oscillation was constructed in MATLAB software 2021R1. This research aimed to reveal the parameter settings of a lean-premixed combustor to trigger and grow thermoacoustic oscillations to better achieve the feedback control effect of thermoacoustic oscillations in modern gas turbine engines. This study found that the triggering time and climb rate affect the safety of a combustion system; therefore, the present investigation examines the effect of different parameter settings on thermoacoustic oscillations to better understand the evolution properties of triggering time and climb rate. This study provides a theoretical basis for the prevention of combustion thermoacoustic oscillation. At the same time, it can also be seen that an incorrect burner parameter design will lead to difficulties in suppressing thermoacoustic oscillations.