你说的这种结构是很常用的一种,类似独立主格结构,即：
with + 名词或代词 + （现在分词、过去分词、不定式、介词短语、副词等）
本结构所起作用类似伴随状语,其中名词或代词和后面的成分是逻辑主谓关系,如果是主动关系就用doing （强调动作在进行）或 to do（强调过程或要去做的事）,被动关系就用过去分词,表示状态的可根据具体情况选用介词短语或副词.如：
The teacher came in with a book in his hand.
A man walked in with a dog running after him.
He often sleeps with his classes on.
The man was smoking with a magazine held in his hand.
I was very busy with a lot of work to do.