A bike seat that moves when you ride can mean only one thing -- a loose fitting. The fix will depend on the style saddle you have and how the seat moves. A saddle that swivels from left to the right requires a different approach than one that changes tilt. With the right tool, this is an easy fix that you can do in just a few moments.
If you notice the saddle changes pitch, the nut under the seat is loose. This means when you ride, only the saddle portion shifts up or down. Look directly under the saddle. On one side, probably the right, you will see a nut. This controls the horizontal alignment of the seat. A properly positioned saddle will have a slight tilt at the nose. Move the seat so the back end is just a tad lower than the front. Use a pair of pliers or an adjustable wrench to tighten the nut. You may need to insert a screwdriver into the screw head to hold it. Don't tighten the nut all the way just yet. Climb on the bike and make sure the seat is comfortable. If necessary, loosen the nut and make an adjustment before tightening it fully.
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A seat that swivels back and forth or that moves up and down probably has a loose stem. The stem is the long piece of the seat that slides into the frame. Bikes have one of two connections at the intersection of the stem and frame. Newer styles may have a lever on the body that holds the stem in place. Some models will have a nut and screw connection. Once you determine the style of connection for the stem, you will know what tool you need. A lever requires an Allen key, whereas a nut and screw system needs pliers or an adjustable wrench plus a screwdriver.
To fix a loose seat connected with a lever, place the Allen key in the hex nut on the opposite side of the lever handle. Turn the handle clockwise to tighten the connection. Pull back on the lever handle until it is straight. This will allow you to move the seat to the proper height. When you ride, the leg on the downward arc of the pedal stroke should be straight. The leg at the top of the arc should be slightly bent. Push down on the lever to close it. If it will not fold, the nut is too tight. Hold the nut in place with the Allen key and turn the handle slightly counterclockwise. Do this until the lever closes fully. Climb on the bike and check the height. If you need to make an adjustment, open the lever and move the seat.
Nut and Screw
This type of connection will require a pair of pliers or an adjustable wrench. Hold the bolt in place with a screwdriver and turn the nut counterclockwise. Move the seat to the proper height and tighten the nut. Avoid tightening it fully until you evaluate the seat height. Once you are comfortable with the adjustment, tighten the nut fully. This should hold the seat in place when you ride.