The role of the shoulder movement from the takeback to the contraction position is so important and subtle that we go over it here in more detail. 

It is possible to pull the racquet forward in several ways, most of which are not stable at high speeds. We illustrate below a method that is stable, versatile, and adaptable. These two rollover illustrations focus only on the area from the shoulder to the elbow.
 

On the left, the takeback position shifts to a "partial" contraction position. This is an important reference point to illustrate how the elbow is pulled forward to the rotation position. This movement is like a pendulum swinging from left to right. The shoulder is the suspension point of the pendulum, and the elbow is the end point. You can study this pendulum motion by moving the cursor over the image on the right. This movement takes some study and experimentation. In fact, it feels most unnatural at first. 

Also note, in the illustration on the right, how the shoulder is pulled forward to advance the racquet. As this happens, the upper arm automatically begins to rotate forward. As a result, if you consciously rotate the upper arm as well, you will likely overrotate the racquet and shear or hook the ball to some degree. In any case, overrotation is an unpleasant feeling.

Proceed carefully until you develop the flexibility to do this easily. 

The basic movement is the same for all grips. What changes is the angle between the forearm and the upper arm. 

Also note that the wrist is cocked back as far as possible and that this configuration is maintained until the ball has been struck.