This exercise is designed to help you gain more control over your vibrato, so you can change the width and rate of the vibrato depending on what you are trying to express musically. It is important that you have a solid understanding of how vibrato works as well as its function in music as an expressive technique before you begin this exercise:
Set your metronome to 52.
Start with a finger on a string and in a position with which you feel most comfortable.
Using the swinging motion of your forearm, alternately shade the pitch sharp and then flat at the rate of a quarter note. Keep the swinging motion fluid so that the vibrato is “round” in character. Be careful not to use a sharp or abrupt motion, and do not exaggerate how sharp or flat you shade the note. For the more advanced player who is used to vibrating for expression, this exercise will seem very cold. Remember though, the point of this exercise is to develop control so that you are able to be more expressive with your vibrato.
Once you become comfortable vibrating at a quarter note speed, it is time to move to eighth notes, then triplets, then sixteenths, et cetera. At some point the motion will turn into a pure vibrato, with no discernible rhythmic pulse, which of course is what vibrato should sound like. However, at this stage you should have more control over the speed and width of the vibrato, giving you more ways of expressing yourself musically.
Use this same procedure to improve your vibrato with other fingers, strings, and positions. Over time you will develop a very beautiful and mature vibrato.
It is also a good idea to incorporate this exercise with your long-tone scale studies. As you move from note to note, vibrate at the same rhythmic pulse, such as quarters or eighths. Go through the scale again at a faster rate, perhaps as triplets, sixteenths, et cetera.