A Study of Forced Vibrations with Nonlinear Springs and Dry Friction: Application to a Mechanical Oscillator with Very Large Vibrating Blades for Soil Cutting

To cut a clod of soil containing the roots of trees in nurseries, a semi-circular vibrating blade digging machine with diameters up to 1.2 m is increasingly used. The heart of the machine is the mechanical oscillator that produces an excitation torque supplied to the blade together with the cutting torque of the soil. The advantage of the vibrating blade is a reduction in the cutting torque up to 70%. This advantage led to the present study of the extension to blades of even 1.8 m for the digging of very large trees. To build an oscillator suitable for all blade sizes (from 0.6 to 1.8 m), it was necessary to equip it with nonlinear (quadratic) springs, since with traditional linear springs, it would not be versatile. The design and simulation of its operation required the development of a new mathematical model. Therefore, an approximate solution of the differential equation of the forced vibration with quadratic springs and dry friction between the blade and soil was developed, aimed at calculating the maximum blade displacement and the phase lag. These quantities, together with the cutting time, had to satisfy certain values to ensure the maximum reduction in the cutting torque (−70%). After the construction of the oscillator, it was coupled with all the blades (0.6, 0.9, 1.2, and 1.8 m) for experimental tests. For all diameters, the oscillator was able to optimally vibrate the blades, preventing the springs from reaching the end of the stroke when cutting the soil. Measuring the maximum blade displacement compared with the calculated one provided a good accuracy of the mathematical modeling, resulting in a mean error of 5.6% and a maximum error of 7.2%.