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The first thing to do is turn the chair over, and look at all hardware. Get a screwdriver or wrench and tighten any loose ones. Even try to tighten the parts that don't appear loose. Over time, it is easy for screws and bolts to become loose, causing certain chair parts to rub together irregularly and create a squeaky noise. Apply a lubricating oil on all of the nuts, screws and bolts to help loosen the joints. Simply spray oil directly on the the chair mechanisms, and pat them dry. You can also spray oil on a soft cotton cloth, and rub the oil onto your problem areas to have more control over where the oil goes.  Moisture in the air and air conditioners cause rust. Regularly applying oil prevents rust from occurring and building up. If the chair still squeaks after adding lubricant and tightening all bolts and screws, take them all out, and lubricate them with a light machine oil before putting them back in. To help you locate the area of the chair that is squeaking, have someone sit in it, and swivel from side to side. By applying weight to the chair to make it squeak, it will allow you to more easily locate the source of the noise to more accurately apply oil. Each time you apply more oil, ask your friend to swivel the chair so you can see if you put oil in the right spot. A chair may only squeak when you lean back, which is usually caused by too much tension where the spring's ends rub on the housing ends. To fix this, apply oil to the seat tension spring located inside the turn-knob housing. Simply loosen the seat tension turn-knob and remove the turn-knob to spray oil inside the housing. Desk chairs are often on wheels that may move a lot, so it is common for the axles in the wheels to need some silicone spray over time. Turn the chair over and spray the wheels. Then, flip the chair back over, and roll the chair to spread the silicone around the entire wheel. Collapsing into a chair will eventually cause it to squeak. Chairs go through a lot of wear and tear, so to keep your chair silent, be cautious when you sit down as to not speed up loosening joints.
Check the nuts, bolts, and screws. Grease the mechanisms. Remove the bolts and screws completely before adding any lubricant. Have a friend sit in the chair as you apply oil. Fix the springs in the back of the chair. Roll the chair back and forth to check out the wheels. Sit down gently.