When we pronounce the vowel [ы] the tongue and the mouth are in the same position as for pronouncing [у] and the lips stretch as for pronouncing [и].
We suggest to students to first pronounce the sound [y] and after that pronouncing of the sound [и] while changing only the position of the lips: уу-ыыы. Also Russian sound [ы] can be learnt using the sound [и], it is necessary to move the tongue back so that its tip leaves the bottom teeth. To fix the position of the tongue you can use a pen, a pencil or a spoon.
To understand the position of the tongue and lips the following exercises can help:
1. Round the lips so that they are slightly stretched forward (pic. 6)
2. Stretch the lips forward as when kissing while the mouth is nearly closed (pic. 7)
3. Smile, your lips will stretch (pic. 8)
4. Smile so that your lips are stretched and tensed, the teeth slightly can be seen (pic. 9)
Note! These articulations are meant for pronouncing the following vowels:
1 – o, 2 – y, 3 – и, 4 – ы
«Learn how to move your tongue forward and back»
1. Move your tongue forward, try to though your chin with the tip of your tongue, while the front part of your mouth is narrow and tensed.
Страницы из пособия А. А. Акишиной "Учимся учить"
А вот как шутят иностранцы по поводу правильного произношения звука Ы: