Brock’s approaching zero method improved as approaching fixed point frequency to solve optimum damping ratio of dynamic vibration absorber in machine tools and experimental confirmation

The design parameters of the dynamic vibration absorber significantly affect the motion performance of the main vibration system. The Brock's approaching zero method was improved as approaching the fixed point frequency method. A general method of obtaining the explicit exact solution to the optimum damping ratio was presented to improve the accuracy of calculating the dynamic vibration absorber's optimum parameter. Some exact closed-form solutions, for example displacement amplitude gain, fixed point coordinate, and optimum damping ratio, were deduced with the real number form of differential equation of load motion and employing L'Hospital first rule. Many computational parameters of the main vibration system were attained. The fixed point theory essentially computes the extreme large value, not the maximum value. The numerical simulation results of the present paper's absorber are closer to the vibrational experimental results than those of the Ormondroyd absorber and Lanchester absorber. Moreover, the present paper's absorber has larger band width than the Ormondroyd absorber and Lanchester absorber. The current answers may be applicable to realize and control the accurate dynamic performances of the main vibration system and dynamic vibration absorber in operation.