Will an unbalanced wave be able to induce resonance?

[Kingson]

Hi guys, this is the first time I post a thread here.

I have a problem that maybe be simple for your guys. If the force as the attached image shows is applied to an object and the frequency of the force equals to the resonant frequency of the object, will resonance happen? If will, will the object vibrate in a similar unbalance way?

 

[mfb]

Your period is then the time for a half-wave. Yes resonance can occur, but the required frequency is different from a sine wave with the same "look".
A good resonance means your external force is tiny compared to the internal forces, so the oscillation can be very close to the undisturbed way.

 

[Khashishi]

Your driving force looks like a rectified sine wave. I did a quick search for fft rectified sine.
http://www.mathworks.com/matlabcentral/fileexchange/42506-fft-analysis-of-rectified-sine-waves
You can think of any wave as a superposition of pure sine waves of various frequencies. In this case, there is plenty of energy in the pure wave component at twice the frequency of your rectified sine wave. So, it will resonate with a natural frequency at this higher frequency.