My 1996 jeep cherokee headlamps turn off and on on 5 second intervals. what could be the solution

Bright lights wont shut off ? Assume you mean high beams ? Reading how they work might thing high beam relay has stuck contacts inside .

Headlamps
The headlamps may be turned ON 2 different ways. First, when the driver places the headlamp switch in the HEADLAMP position, for normal operation. Second, with the headlamp switch placed in the AUTO position, for automatic lamp control (ALC). During ALC, the headlamps will be OFF in daylight conditions, or low beam operation in low light conditions. Battery positive voltage is supplied to the low beam relay and the high beam relay in the underhood fuse block. When low beam headlamps are requested, the body control module (BCM) supplies ground to the coil side of the low beam relay. This activates the relay and the relay supplies voltage to the left hand low beam headlamp through the LH LOW BEAM fuse and to the right hand low beam headlamp through the RH LOW BEAM fuse. Ground for the left hand low beam headlamp is provided at G101. Ground for the right hand low beam headlamp is provided at G102. When high beam headlamps are requested, the body control module (BCM) supplies ground to the coil side of the high beam relay. This activates the relay and the relay supplies voltage to the left hand high beam headlamp through the LH HIGH BEAM fuse and to the right hand high beam headlamp through the RH HIGH BEAM fuse. Ground for the left hand high beam headlamp is provided at G101. Ground for the right hand high beam headlamp is provided at G102.

Ive changed the relay and the headlight switch on the dash .. still wont turn off headlights ? Just changing parts won't fix it , need to test . Hooking up a professional scan tool , one that can read codes in the BCM - body control module is the way to start to find the problem . Then familiarize yourself with how they work.

Headlamps
Major components of this system are the body control module (BCM), headlamp relay, beam select relay, ambient light sensor, headlamp and panel dimmer switch, turn signal/multifunction switch, park brake switch, and the low and high beam headlamps.
The headlamps may be turned ON in 2 different ways. First, when the driver places the headlamp switch in the ON position, normal operation occurs. Second, when the headlamp switch is placed in the AUTO position, automatic lamp control (ALC) operation occurs. During ALC control, the headlamps will be OFF in daylight conditions or in low/high beam operation in low light conditions.
The headlamp relay receives battery positive voltage directly from the battery. The BCM supplies a ground signal to the headlamp relay for normal headlight operation. The BCM also supplies voltage to the beam select relay coil for high beam headlight operation. When the driver places the headlamp switch in the ON position and the dimmer switch is in the low beam position, the headlamp relay supplies current flow through the left and right low beam headlamp fuses to both low and high beam headlamps. Both low beam headlights are grounded through the beam select relay. When activated, the dimmer switch sends a ground signal to the BCM in the high beam position and a momentary ground signal in the flash-to-pass (FTP) position from G106. The BCM then energizes the beam select relay which switches the headlamp grounds from the low beam headlight bulbs to the high beam headlight bulbs illuminating the high beam headlights. With the headlights in the low beam position, the high beams may be momentarily turned ON or flashed by activating the FTP portion of the switch.
The instrument panel cluster (IPC) illuminates the high beam indicator when the BCM detects that the high beams are requested. The IPC receives a class 2 message from the BCM requesting illumination.
The Headlamps On at Exit and Perimeter Lighting modes can be adjusted using the steering wheel controls to select the function as displayed in the driver information center (DIC). The Headlamps On at Exit selection will allow the driver to customize the period of time that the headlamps remain illuminated upon exiting the vehicle. This period can be adjusted from 0-180 seconds. The Perimeter Lighting feature allows the driver to select if they want the headlamps and backup lamps to illuminate when the door UNLOCK button on the key fob is activated. This feature can be turned ON or OFF by using the DIC steering wheel control buttons.

DTC B2575 Headlamp Control Circuit

Circuit Description
When the body control module (BCM) senses a request for headlamp or daytime running lamp (DRL) illumination, the BCM sends a ground signal to the headlamp relay through the low beam headlamp relay control circuit. The headlamp relay will illuminate the low beam headlamps.

Install a scan tool.
Turn ON the ignition, with the engine OFF.
With a scan tool, command the headlamps ON and OFF.
Does the headlamp relay turn ON and OFF with each command?

Do you know what bi-directional control means ? Need a professional scan tool to do these type of test .

DTC B2602: Daytime Running Lamp Control Circuit Low or Open
DTC B2603: Daytime Running Lamp Control Circuit High

Without this type of info. from service repair manual an using a scan tool your missing the big picture .

Your best bet , take your vehicle to a qualified repair shop . Before you want to replace the BCM . . Professional tech would check for DTC'S - diagnostic trouble code's in the BCM before replacing part's that are not needed .

No signal at signal wire on relay ???? Which circuit wire mite that be ? Hooking up a factory or professional scan tool to view input data from the switch would also be a good step .

Headlamps
The headlamps may be turned ON 2 different ways. First, when the driver places the headlamp switch in the HEADLAMP position, for normal operation. Second, with the headlamp switch placed in the AUTO position, for automatic lamp control (ALC). During ALC, the headlamps will be OFF in daylight conditions, or low beam operation in low light conditions. Battery positive voltage is supplied to the low beam relay and the high beam relay in the underhood fuse block. When low beam headlamps are requested, the body control module (BCM) supplies ground to the coil side of the low beam relay. This activates the relay and the relay supplies voltage to the left hand low beam headlamp through the LH LOW BEAM fuse and to the right hand low beam headlamp through the RH LOW BEAM fuse. Ground for the left hand low beam headlamp is provided at G101. Ground for the right hand low beam headlamp is provided at G102. When high beam headlamps are requested, the body control module (BCM) supplies ground to the coil side of the high beam relay. This activates the relay and the relay supplies voltage to the left hand high beam headlamp through the LH HIGH BEAM fuse and to the right hand high beam headlamp through the RH HIGH BEAM fuse. Ground for the left hand high beam headlamp is provided at G101. Ground for the right hand high beam headlamp is provided at G102.

Headlamps Inoperative - Low Beams
Step
Action
Yes
No
Schematic Reference: Headlights/Daytime Running Lights (DRL) Schematics
Connector End View Reference: Master Electrical Component List
1
Did you perform the Diagnostic System Check - Vehicle?
YES - Go to Step 2
NO - Go to Diagnostic System Check - Vehicle
2
Place the headlamp switch in the HEADLAMP position.
Place the headlamp dimmer switch in the LOW position.
Do the low beam headlamps illuminate?
YES - Go to Testing for Intermittent Conditions and Poor Connections
NO - Go to Step 3
3
Disconnect the turn/headlamp switch connector C2.
Connect a 15-amp fused jumper between the headlamp low beam signal circuit and a good ground.
Do the low beam headlamps illuminate?
YES - Go to Step 4
NO - Go to Step 5
4
Inspect for poor connections at the harness connector of the turn/headlamp switch. Refer to Testing for Intermittent Conditions and Poor Connections and Connector Repairs .
Did you find and correct the condition?
YES - Go to Step 7
NO - Go to Step 6
5
Repair an open or high resistance in the headlamp low beam signal circuit. Refer to Circuit Testing and Wiring Repairs .
YES - Did you complete the repair?
NO - Go to Step 7
--
6
Replace the turn/headlamp switch. Refer to Turn Signal Multifunction Switch Replacement .
YES - Did you complete the replacement?
NO - Go to Step 7
--
7
Operate the system in order to verify the repair.
Did you correct the condition?
YES - System OK
No - Go to Step 2

It's more then likely the dimmer switch .

DTC B2575
Circuit Description
The body control module (BCM) turns on the LOW BEAM relay when the ambient light sensor senses a dark condition or the low beam headlamps are requested ON. Battery positive voltage is supplied to the LOW BEAM relay at all times. The BCM grounds the LOW BEAM relay control circuit in order to activate the relay.
DTC Descriptor
This diagnostic procedure supports the following DTC:
DTC B2575 Headlamp Control Circuit

B2575.01
Will set if there is a short to voltage in the low beam relay control circuit.
B2575.06
Will set if there is an open or short to ground in the low beam relay control circuit.

Looks like the dimmer switch is built into the combo turn signal wiper switch on the column.
I would unplug that switch and see if the indicator light goes off.
If it does the problem may be inside the combo switch.

The Low Beam relay is located in the fuse block behind the panel at the left knee panel below the instrument panel.

1. Turn the headlamps ON and select the LOW beams for vehicles in all markets except Japan. Vehicles for the Japanese market should have the headlamp HIGH beams selected.
2. Insert a screwdriver through the outboard clearance hole (1) in the front lamp unit cover (2) to rotate the headlamp vertical adjustment screw on each front lamp (3) to adjust the beam height as required.
3. For export markets only, insert a screwdriver through the inboard clearance hole in the front lamp unit cover to rotate the headlamp horizontal adjustment screw on each front lamp to adjust the beam right or left as required.

For Audi A4 the fuse box is located on the left side of dashboard behind a plastic cover (for left hand steering wheel vehicle). On inner face of the plastic cover you will find the fuse diagram.
First fuse on first row from left upper corner is Fuse Nr.20 = F20 (15A) - RH headlamp low beam, headlamp adjustment.
First fuse on second row from left upper corner is Fuse Nr.21 = F21 (15A) - LH headlamp low beam, headlamp adjustment.

you need to replace the multifunction switch also

try taking your tilt steering(steeringwheel) and check which setting its on if the tilt steering is all the way down click it up to the next setting and then try your low beams if that still doesn't work try it one more click up you might have a short or pinched wire in your wheel colum

had the same problem on my 92....check under the hood on the drivers side,towards the front,there is a "junction box" with all of the lighting wires connecting from the switches to the lights.My wires were all corroded inside,so I cleaned the ends and bought a new box from the local electronic supply store.The problem has not come back.

Cheers