I even managed to giggle once or twice,much to the chagrin o

much to sb。'令某人。的是。,这是个短语介词；后面的全是这个介词的宾语；of the man sitting next to me holding the airsickness bag是chagrin（恼；委屈；气愤）的定语；sitting next to me holding the airsickness bag是现在分词短语,做the man的后置定语； next to me 是sitting的状语；holding the airsickness bag是sitting next to me的伴随状语。翻译为：我当时甚至设法咯咯地笑,这让紧挨着我坐的、手里拿着晕机呕吐袋的男士恼火不已。