Development and validation of the damped pendulum model for cantilever conveyor trough using Lagrangian method

Cantilever conveying trough has often caused the unbalanced swing in the rice combine harvester, leading to the high failure rate with the low operation efficiency. The purpose of this study is to establish a damped pendulum model for the cantilever conveyor trough of the combine harvester in the support of the hydraulic cylinder, in order to accurately analyze the unbalanced pendulum during operation. Taking the rice combine harvester as the research object, the force and dynamic models were established for the cantilever conveying trough using Lagrange equation. The hydraulic cylinder stiffness, damping and the total excitation were also determined in the damped pendulum model of the cantilever conveyor trough. Firstly, a static analysis of hydraulic cylinder was carried out using Workbench software. The hydraulic cylinder stiffness was solved to be 1.023 6×10⁸ N/m. Then, the modal analysis was performed on the hydraulic cylinder, where the main vibration mode was corresponded to the intrinsic frequency of 60.604 Hz. The vibration acquisition test was also carried out, where the measurement points were set up at the front end and the rear end of the hydraulic cylinder. The hydraulic cylinder amplitude was obtained to integrate the acceleration signal. The logarithmic attenuation rate of amplitude was ln2. The intrinsic frequency and logarithmic attenuation rate of amplitude were then substituted into the damping formula to calculate the damping c of the hydraulic cylinder, which was 228.58 N·s/m. Finally, the half- and full-width field harvesting tests were implemented for the external excitation of conveyor trough. Four points were measured at the connection between the cutting platform and the conveyor trough. The acceleration signals were collected during the field tests. The quadratic integration and the Fast Fourier Transform (FFT) were utilized to obtain the amplitude of cantilevered conveyor trough and the main frequency components of vibration. The excitation characteristics of cantilever conveyor trough were obtained to verify the subsequent differential equations of dynamics. All parameters were verified in the damped pendulum model of the cantilever conveyor trough. The ODE45 function in MATLAB was used to solve the equations of angular velocity and pendulum angle of damped pendulum model. The amplitudes of pendulums were compared between the theoretical solution and field test at the measurement points 3 and 4 in the initial and steady state, indicating the excellent performance. It was found that the theoretically solved pendulum amplitudes of measurement points 3 and 4 were very close to the test under stable conditions, with the errors of about 1.11% and 4.30%, respectively. Therefore, the accuracy and feasibility of damped pendulum model can be expected to serve as the cantilever conveyor trough.