SOURCE: need torque spec for 2003 F150 9.75 differential
77 ftlbs
I am assuming you have the run out and backlash all ready set.
SOURCE: 1991 ford explorer 4wd wont work
No, mostlikely its the "transfer case actuator motor" failing to engage the 4WD mode. its located on the outside of the transfer case @ the back of the unit and mount by 2 or 3 screws usually. You may need to replace it, but if you up to it, call a few trans. shops to see if they can test it if you were to remove it and bring the motor to them for testing. Its an electric motor that in most cases auto parts stores WONT take back for return if that doesnt solve the problem, thats why i recommend talking to a trans shop about having yours tested 1st.
SOURCE: My 2002 Ford Explorer 4WD
Usually, the rear pinion bearing will make a higher pitched whining noise under a slight load. As soon as the throttle pedal is released, the noise stops. The rear axle shaft bearings will make a lower growling rumble when they are bad.
Another thing to check are the Emergency brakeshoes inside the hub of the rear rotors. The shoe material frequently separates from the rusty shoe plates and the bonded material gets pulverized. This would be similar to a "rocks crushing" sound and can cause vibration in the drivetrain.
The U-joints can cause this problem too. Check for rust and oil streaks which appear shortly before the U-joints fall apart. You should find telltale streaks on the undercarriage of your vehicle.
The front driveshaft can cause noise without being powered. There is just enough sympathetic rotation when the 4 wheel drive is Off, to cause sounds and vibration.
Testing can be aided by using a Commercial hoist and having the drivetrain engaged with the engine running. You would observe the shaking or use a probe "long broom handle" to the stationary parts like the axle tubes, differential housing, pinion bearing area, and so on. This would require a helper to sit in the hoisted vehicle and a second person to stand under a running machine. Not for the faint of heart. No baggy or loose fitting clothes, no long hair, etc.
Now there is a specialized machine which can Electronically investigate noise and vibration. There are sensors which are placed under the vehicle and it listens to vibrations. I saw one machine one time. It helps in balancing driveshafts too. A driveshaft balancing shop may be able to fix you up.
I hope my solutions are very helpful to you. If you do not have the exotic testing equipment you may have to drop the driveshafts to check your U-joints.
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Disassembly
Make sure that you have all the
parts and tools you will need. The extent of disassembly
depends on the job being done and the inspection
findings. Lift the vehicle using an appropriate lift or a
jack and safe jack stands. Always make certain that the
vehicle is safely supported before working underneath.
Unbolt the driveshaft from the yoke. Remove the
differential cover or unbolt the third member. Let the
oil drain into a suitable container. Please recycle your
waste oil. Remove c-clip axles by removing the
differential cross pin bolt and cross pin shaft, pushing
the axles in and pulling the c-clips. Full float axles
are unbolted at the hubs. Punch both carrier caps with
identification marks so that you will be able to
re-install them on the same side and in the same
direction. Most carriers can be pried out of the housing
with a pry bar. Further disassembly depends on the job
being done. If you're changing the ring and pinion
or the pinion bearings, remove the pinion nut with an air
gun while holding the yoke, or use a long breaker bar and
brace the yoke (bolt it to a long board) so that it
can't move. Knock the pinion gear out to the rear
with a brass punch, taking care not to damage the
threads. Keep
track of the location and thickness of all of the
original shims.
Pinion bearings must be pressed off. Carrier bearings can
be pulled using a bearing puller. Internal parts (inside
the carrier) can be removed as necessary.
Inspect all bearings and races for
pitting or uneven wear. The inner carrier bearing races
should not spin on the carrier journals. The carrier
races should have a snug fit in the housing. Inspect the
carrier race bores for grooves from spinning races. The
side gear bores inside the carrier should not have any
abnormal wear. All gear teeth (including the spider
gears) should be smooth but not excessively shiny.
Inspect all gear teeth for pitting, chips, breaks, and
for signs of uneven wear and overheating. Inspect
positraction clutches for scoring and wear. Inspect the
axles for pitted, grooved, or dull and rough bearing
surfaces. Check for worn axle splines. All questionable
parts should be replace
Assembly
Differential Adjustments
The four essential differential
adjustments are pinion depth, pinion bearing preload,
backlash and carrier bearing preload. The tables at the
back can be used to write down shim combinations and
results.
Preparing Parts
CAUTION: Turning the differential case (4204) or an axle shaft (4234) with the differential pinion shaft (4211) removed will cause the differential pinion gears (4215) to fall out of the assembly and damage the components. CAUTION: Do not damage the rubber O-ring in the U-washer groove. CAUTION: Do not damage the wheel bearing oil seal (1177). CAUTION: Never remove the wheel bearing oil seal (1177) by itself. Always remove the rear wheel bearing (1225) and the inner wheel bearing oil seal at the same time. 
Installation 
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