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In recent years, the demand for lighter and more compact equipment has been increasing due to reduce product manufacturing time and the viewpoint of environmental load. Therefore, further lightweighting is demanded with the ball screw. The ball screw has step shape on both ends of shaft. Therefore, the lightweighting method of drilling to center of ball screw shaft makes it difficult to significantly reduce weight due to the thicker wall thickness to secure the strength of the shaft end. The lightweighting method devised is to form the step shape of the shaft end by swaging process to have a structure with a large space inside, which enables a significant weight reduction. The appearance and surface roughness of the lightweight ball screw shaft does not differ from that of conventional shaft. The weight of 0.70 kg is 0.55 times that of the conventional shaft, which enables a significant weight reduction. In the rotary bending fatigue test, the fatigue limit decreases with the ratio of wall-thickness, but the allowable torsional stress of the conventional shaft is satisfied. The axial rigidity is approximately 0.76 times that of the conventional shaft, which is smaller due to the reduction of the area moment of inertia in the hollow structure. The maximum torque is 0.02 Nm, satisfying the specified value of ±0.41 Nm. The torque fluctuation ratio is ±5 %, satisfying the best accuracy grade of ±15 %. The quietness is equivalent to number of revolutions above 2100 and improve at number of revolutions below 1000. Vibration characteristic is on the high-frequency side of resonance frequency in bending vibration, which increases the allowable rotational speed by approximately 1.2 times.

Ball screws and ball bearings are important mechanical elements in automobiles and machine tools due to their high mechanical transmission efficiency. However, with the spread of electric vehicles, the designs that ensure low noise and high durability are required. Currently, numerical analysis is mainly used to clarify the inner balls motion, but it is necessary to tune simulation models based on knowledge obtained from experiments in order to reproduce the behavior of experimental systems. Moreover, while there are the quantification methods of contact states and rolling motion of steel balls inside the ball screw using mathematical methods, the contact state and the ball motion are only considered in limited circumstances. Therefore, this study aims to extend mathematical methods to clarify the contact state and rolling motion of steel balls and guide grooves inside ball screws in order to contribute to noise reduction and longevity of ball screws. Four arc shapes of a Gothic arc, considering the helix orbit of steel balls and the position and dimensional errors of the steel balls in the contact plane under non-driven and non-loaded conditions of the ball screw, are given as geometric shapes, and the balance and rolling motion of steel balls during ball screw drive are calculated. Next, numerical examples are presented to examine the effects of changes in the arc shape on the revolution and rotation of steel balls and the impact on noise, wear, and lost motion inside the ball screw. Finally, the validity of the mathematical model is demonstrated by comparing the measured values with the theoretical values using a device composed of intersecting straight grooves with a crossing angle.