What you are feeling is likely a combination of things. As your jet
accelerates down the runway for takeoff the speed of the jet's tires
racing along the runway is accelerating as well. If the tires are out
of balance then the vibration associated with their being out of
balance will increase as the speed of the tire increases. That speed
is at its greatest the first few seconds after takeoff. Airplane
tires are big heavy things. If they are out of balance they are
capable of producing a lot of vibration.
There is usually a hesitation of a few seconds after liftoff before
the landing gear is retracted. This hesitation is to verify that a
positive rate of climb has been established before we retract the
gear. During the retraction phase it is common for the brakes to be
automatically applied to the main gear as it is being retracted into
the body of the airplane for aerodynamic efficiency. Once the tire
rotation has stopped the vibration should stop as well. But during
those few seconds the tires will be hanging in the slipstream spinning
away.
The nose gear tire is also a common offender that causes vibration.
It is common for the nose gear to shimmy, shake, rattle and bang away
as it is being retracted into the body of the airplane. The nose gear
tire has no breaking mechanism to stop its rotation; steering yes,
brakes no. The nose gear strut assembly takes a beating on a regular
basis and is often in need of maintenance. I probably write up at
least one nose gear strut a month for additional maintenance.
In many airplanes the landing gear is closed up inside the body of
the jet by aerodynamically shaped doors. In many cases these doors
must be opened to allow for the landing gear to be brought into the
body of the aircraft. As these doors cycle open it is possible for
them to induce airframe vibration if they aren't rigged properly. In
most cases your jet is accelerating to 250 knots (287 mph) (Federal
airspeed limit below 10,000') as expeditiously as practical after
takeoff. Anything dragging in the slipstream will produce vibration
and noise at those speeds. Vibration and noise are common byproducts
as those landing gear doors are being opened to allow for the
retraction and extension of the landing gear itself.
Additionally, the landing gear may not retract symmetrically and that
can cause vibration. Also, you may also notice a yawing tendency if
the gear does not come up in unison.
I am not sure what your fellow passenger meant by ground effect
causing vibration after takeoff. I suspect they were confusing
terms. Ground effect is an aerodynamic phenomenon that takes place on
takeoff and landing; its maximum effects are experienced at
approximately half the distance of the wingspan above the runway. As
the wing gets that close to the ground it becomes much more efficient
in producing lift. See the following link for a detailed explanation:
http://en.wikipedia.org/wiki/Ground_effect_in_aircraft
I don't see any connection between ground effect and vibration
associated shortly after takeoff. On takeoff your jet will climb out
of the region where it can encounter "ground effect" very rapidly.