Apparatus for ultrasonic vibration-assisted machining, the apparatus comprising an ultrasonic transducer for generating ultrasonic waves in a vibration horn; first clamp on the vibration horn at a static node of the ultrasonic waves; and a second clamp between the first clamp and a lowermost end of the vibration horn. The second clamp comprises a linear bearing for reducing vibration of the vibration horn in a direction laterally of the vibration horn, and allowing vibration of the vibration horn in the direction of a longitudinal axial of the vibration horn.