中国观众在看京剧的时候，看得高兴时，就用喝彩，鼓掌的方式来表示自己高兴的心情，他们也用这种方式来赞扬京剧演员
What's the meaning of lai2 in this sentence? Is it a common meaning of this? What are other examples in which I can use it with this meaning? Is "fang1 shi4 lai2" a fixed expression? What does it mean?
It seems to me that it's [description of method] + fang1 shi4 lai2 + [verb] and it means "to do [verb] in a [description] way".
It is a common structure
[用 (object) 来 (verb)] = [use (object) to (Verb)]
[用 (这种方式) 来 (赞扬京剧演员)] = [use (this method) to (praise) Peking Opera actors]
方式 is the object; 赞扬 is the verb; 京剧演员 is the indirect object
Example:
[用(水) 来 (灭)火] = [use (water) to (put out) fire]
水 is the object; 灭 is the verb; 火 is the indirect object
"用这种方式赞扬京剧演员" and "用水灭火" mean the same as "用这种方式来赞扬京剧演员" and "用水来灭火". The difference is adding 来 strengthen the connection between the noun phrase and the verb phrase.
The connection between noun and verb in "use water put out fire" is not as strong as "use water to put out fire"
More example:
"用枪杀敌" (use gun kill enemy) might be broken into two unrelated phrases if you say it like "用枪， 杀敌" (use gun, kill enemy). While "用枪来杀敌" (use gun to kill enemy) cannot be broken into two unrelated phrase no matter how you say it
來 can be be omitted here, it don't have any meaning in this situation.
According to dictionary owned by Ministry of Education in Taiwan 來 have several meaning one of them is:
當口語中的襯字，無義。
(Used to adds to other words in verbal language without meaning)
I notice that even if there is no 來, the meaning of sentence is complete. And other meanings cannot be used here, so I think 來 is no meaning in this situation.
