As a new efficient rock‐breaking technology, harmonic impact drilling technology has received great attention, but the research on its rock‐breaking mechanism is insufficient, which limits its development. Based on the theory of vibration, a simplified model of high‐frequency harmonic vibration impact system is established in this paper. The numerical model was solved by Matlab and the motion equations of drill bit and rock at different stages of motion are obtained, respectively. Based on the factor analysis of the mathematical model, the dynamic characteristics of the impact system under harmonic excitation are studied. Finally, numerical simulations are carried out to further analyze the drilling effect of harmonic impact drilling and verify the correctness of the simplified model. The results show that when the excitation frequency equals the resonance frequency of rock, the vibration displacement of rock reaches the peak value. The drilling speed is greatly increased by harmonic impact drilling compared with conventional drilling.

The large fluctuation of net torque and the existence of negative torque on the crank output shaft of the beam pumping unit are the decisive factors leading to its low efficiency and high energy consumption. The conventional pumping unit CYJ10-4.2-53HF was selected as the study object on the basis of the fixed shaft secondary balance principle and the positive torque modulation scheme was first proposed according to the following secondary balance principle based on the linkage. The kinematics analysis of the suspension point and the secondary incremental velocity mechanism were carried out using the theory of rigid body plane kinematics. The force analysis of each moving part of the pumping unit was carried out, the net torque expression of the crank output shaft was obtained, and an example calculation was performed. The positive torque beam pumping unit was developed and tested in a field test. The tests showed that the positive torque beam pumping unit was able to fully realize positive torque operation under field well conditions, with a power saving rate of 23.73% and a 14.5% increase in system efficiency, and that the reliability of the pumping unit meets the requirements for field application.

Beam pumping units have numerous energy transfer links, significant net torque fluctuations and negative net torque in gear boxes, which lead to a high installed power and low system efficiency. In order to improve the efficiency of the pumping unit, in this paper, an eccentric balanced positive torque pump unit was designed based on the principle of eccentric balance that is small in size and adopts a flexible energy transfer system instead of a rigid structure to significantly reduce the energy transfer link. The kinematics and dynamics of the eccentric balanced positive torque pumping unit were analyzed and the ability to operate with positive torque over the entire cycle was verified using theoretical computations. And the results of the theoretical calculation were verified by using the virtual prototype simulation and analysis software ADAMS (Automatic Dynamic Analysis of Mechanical Systems); the pumping unit was optimized for relevant parameters such as the balance crank angle λ and the counterweight center of mass offset angle δ. Indoor testing and comparative field application have demonstrated that the installed power has been reduced by 80%, the weight of the entire prototype has decreased by 25% and the system efficiency has reached 24.3%. Additionally, electricity savings of over 50% have been achieved. These results indicate that the pumping unit has been designed correctly, has high reliability and has significantly improved system efficiency while also providing an obvious energy-saving effect.

The large net torque fluctuations in the reducer output shafts of conventional beam pumping units and the existence of negative torque are the decisive factors that lead to their low efficiency and high energy consumption. This study developed a positive torque modulation scheme for conventional beam pumping units, which was based on the principle of the follow-up secondary balance of the connecting rod. The CYJ10-4.2-53HF conventional beam pumping unit was selected as the research object. The kinematic and dynamic simulation analysis of the modified pumping unit was carried out using ADAMS software. The results showed that secondary balance torque curves could realize the function of “peak cutting and valley filling” for the curves after the primary balance and that the modified pumping unit could achieve a full-cycle positive value for the reducer output shaft and verify the feasibility of our modulation scheme. A secondary balance offset angle of 315° was the best choice as the amplitude of the torque curve clearly increased and the phase remained basically the same when the radius of the mass center of the secondary balance increased. Therefore, when the offset angle value of the secondary balance weight was determined, the radius of the mass center could be changed by adjusting the position of the secondary balance weight to achieve the balance adjustment.

In the light of the problems of multi-transmission segments, poor movement performance, big volume and weight, large fluctuating range of payload, negative power and torque of beam pumping units, the principle of eccentric balance was suggested, eccentric rotating mechanism is adopted to change the interval of upstroke and down stroke, positive torque with alternating uniformly, energy saving and high efficiency were realized. Basing on the setting up motion and dynamic models of eccentric flexible pumping unit, the motion and dynamic characteristic was analyzed. Field test demonstrated that positive torque can be realized completely, the installed power is decreased 80% and the weight of whole prototype is decreased 25%, system efficiency is improved from 14% to 24.3%, electricity saving reached to more than 50%.

Counterexamples are given to show that some results concerning a weighted ${L^2}$ projection presented earlier by Bramble and the author are sharp, i.e., that certain error and stability estimates are impossible in some cases.