One of the most "beloved" letters in the Russian alphabet for foreigners is the letter "Ы" [Y]. I've used inverted commas for a reason. Those students who start learning the Russian phonetics sooner or later meet this sound. Sometimes they think it’s physically impossible to pronounce the sound "Ы" [Y].
With regard to the letter "Ы" and difficulties of its pronunciation there are a lot of jokes. "To pronounce sound "Ы", imagine that someone punched you in the stomach" or "What resembles sound "Ы"? Howl of injured seal". Interestingly, sometimes in the Russian-speaking society little children and some adults have difficulties with pronunciation of several sounds such as "Р" [R], "Л" [L], "Ш" [SH], "Щ" [SHCH] (mostly it's consonants and hushing sounds) while pronunciation of "Ы" doesn’t cause any trouble. So we can understand your emotions but they are groundless. Russians prove – the speech apparatus is capable of pronouncing this nightmarish sound, so eventually you will definitely manage.
Actually, it is better to copy with phonetics and positioning of difficult sounds with a real teacher. If you have any opportunity, spend a couple of hours on face-to-face lesson or a lesson via Skype. Of course, there is not always an opportunity to find a Russian-speaking interlocutor or teacher, so any advice would be a great help. Hope, I can help you with that, too.
So, let's start our fight against the sound "Ы" [Y]:
First thing you should do – feel the difference between the sounds "И" [I] and "Ы" [Y] and work at it. Almost all the students who have difficulties with this sound automatically pronounce it like "И" [I]. Probably, you also notice that you pronounce "ми" [mi] instead of "мы" [my], "ви" [vi] instead of "вы" [vy]. Even if you don't notice it, actually the fact remains. Your sound "Ы" [Y] resembles any of the sounds "И" [I], "Э" [EH] or "О" [O].
To transform the sound "И" [I] into "Ы" [Y], make some exercises on articulation.
Pronounce the sound "И" [I]. Pronounce it slowly, feel the position of your tongue. Feel that the tongue is set against your lower teeth and lips are stretched like in smile. So it's ok with pronunciation of the sound "И" [I]. Try to change the position of your tongue by relaxing it. Just move it back from your teeth. You feel that the sound "И" disappears, and then any other sounds disappear too. Air stream goes through the extended gap between palate and tongue, sound becomes voiceless, air starts "whistling".
It is all right. Let's turn to the next step.
Pronounce the sound "И" [I] again. Then move your tongue back from teeth so that it doesn’t touch them. If you can't do it, practise more. You should pay your attention to two things:
– your lips are stretched in smile
– your tongue is relaxed and doesn't touch teeth
Try to keep your jaw slightly strained. Check this strain on. Try to pronounce the sound "Ы" [Y].
If you pronounce something like "И" again, try to move your low jaw slightly forward. Now when the sound becomes better, return to previous position (lips are stretched less than during pronunciation of sound "И" [I], jaw in its natural position) and try to "catch" the sound "Ы" [Y].
Consonant sounds can help you to produce the sound "Ы". Best friends – letters "К" [K] and "Г" [G].
Try to pronounce the syllables and words:
One more exercise. At first repeat the syllables slowly, working on clearness of sound. Then add speed. Pronounce last words in a row as clear and loud as you can.
The exercise with a pencil can be effective too. Gripe a pencil with your teeth and try to pronounce the sound "И". The pencil will not allow your tongue to touch the teeth and you won't be able to clench them. Thus, you will pronounce the sound "Ы" better and better. Working at it, try to pronounce the sound "Ы" [Y] without any pencil, memorize the articulation.