Can you detect Kate's motivation? Was she just a shrew, was she too smart to be content to be subservient, or perhaps something else?
Despite the fighting and the screaming and the blustering, Kate is a fairly simple character. She wants to be loved, first by her father then by her husband, Petruchio. Her motivation is simple, then--love.
She doesn't stand a chance of being loved by her father, and not because he's bad but because he's weak. He's been duped (taken in) by the rather shallow and simpering love Bianca shows him, and there's just no room for Kate. In order to get some attention from him, she has become a violent, unpredictable shrew. She can't "out-nice" her younger sister, so she turns mean and vindictive in hopes of capturing her father's interest. The plan backfires, of course, as he is not a strong man and is actually afraid of her. When she does meet a strong man worthy of loving, though, her tactics do work.
When she is faced with the whirlwind that is Petruchio, she only has those behaviors as her defenses. She barks and blusters and bites, assuming he'll shy away from her like all the other people in her life have done. Instead, he stays--admittedly, for all the wrong reasons for 99% of the play, but he stays. Eventually she realizes her husband is someone she can love and respect--if she "trains" him correctly. It is this give and take, this tension, that motivates her to win him. And she does. Her shrewishness eventually looks like subservience, but it's all part of her plan to win her husband's love.
Kate is not content to be subservient; however, she is content to let Petruchio think she is in order to gain his love. Her motivation grows out of desperation in the beginning, but she understands this is a man worthy of her affection and is willing to do what it takes--even make herself look foolish--to attain that goal.
I like Kate, though not everyone does. Perhaps if I didn't I'd say she was a shrew; instead, though, I say she acted shrewish because it was the only way she could win the attention and affection of her father and the love of her husband.