Engine failure on a twin

[Old Smokey]

If you consider a cross-wind in flight, we simply alter the Heading of the aircraft to compensate for the drift, and rudder input is not necessary. On the ground, however, we cannot allow drift, and must use rudder to off-set the yawing moment caused by the cross-wind in order to keep the aircraft tracking down the runway centre-line.
If there was. for example, a strong cross-wind from the left, the aircraft would tend to "weather cock" towards the left, and right rudder would be required to keep the aircraft straight. Some of the rudder authority to the right has already been used, leaving less rudder authority in that direction to compensate for an engine failure of the left engine/s (i.e. the up-wind engine/s). Thus, more rudder input will be needed for a failure of the up-wind engine/s, and conversely, less rudder input required for a failure of the down-wind engine/s.

All of this applies between Vef (just below V1) and the point where the aircraft leaves the ground, just after Vr.

Once airborne, drift is compensated for with Heading change, and fairly equal rudder input should be required for an engine failure on either side of the aircraft. (The fore-going is true for a jet aircraft, there may be assymetric propeller thrust considerations at high angles of attack for a propeller aircraft).

Regards,

Old Smokey