The expression of azimuth of radial vibration antinode on the hemispherical resonator has been derived for the study on the operating principle of hemispherical resonator gyro(HRG) firstly.Then the state equation about vibration displacement and velocity was established and the analytic solutions of the state equation were obtained.The steady state solution of antinode azimuth andthe solution error of angular rate at the angular rate input and the density distribution nonuniformityuneven distribution have been given.Finally, the tolerance of fourth harmonic density was determined to ensure the accuracy of HRG under open loop positional excitation mode.