Application of the Heun functions to rotor vibratory gyroscope, their numerical computation and accuracy estimation

For an asymmetric rotor vibratory gyroscope that is oscillating in an elastic suspension means, the equation of motion is derived from the Eurler–Lagrange equation and the exact solution is obtained as Heun functions (Hfs). A fast and effective method for calculating Heun functions by direct calculation of solutions of the Heun differential equation (Hde) using standard numerical integration methods is developed. Three methods of accuracy check are employed in this case. The accuracy of the numerical solutions deteriorated in the vicinity of the singularity. To overcome this difficulty, an optimised method for calculating the Hfs is developed, which give a uniform accuracy of the calculated values on the interval. The optimised method for calculating the numerical Hfs by means of the solution of the governing initial value problem gave acceptable accuracy in modelling the behaviour of an asymmetric rotor gyroscope.