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I pinched myself to stop myself from falling asleep.

我掐自己以防自己睡着。

《牛津词典》

They often have trouble falling asleep.

他们经常难以入睡。

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Other products warn drivers if they are falling asleep or in danger of an accident.

VOA: special.2010.01.08

You start off with a transition period when you're falling asleep. We call that stage I.

一开始是过渡期,你开始进入睡眠状态，这是第一阶段。

耶鲁公开课 - 心理学导论课程节选

so I actually was like almost falling asleep in the second half, so you know.

我后半程几乎都快睡着了。

来伦敦旅游的 - SpeakingMax英语口语达人

The most common sleep problems in children are difficulty falling asleep and staying asleep and overall insufficient shut-eye.

WSJ: More Children With Sleep Problems Seek Overnight Tests

Sleep studies don't usually need to be done on children who have trouble falling asleep or staying asleep.

WSJ: More Children With Sleep Problems Seek Overnight Tests

By the third night, the babies were crying for a shorter period of time and falling asleep faster.

CNN: Should babies be allowed to 'cry it out'?