How are gyros used on a turn indicator?
The picture given is not complete, it is missing several components. Unlike the other two gyros, the turn coordinator is not allowed to remain rigid in space while the gimbal moves freely around it. The plane of rotation is forced to change with changes in the heading, because of this it does not use a full gimbal. It has one axis other than the gyro core. The housing applying force to the gyro causes precession. In other words, the gyro would like to remain fixed in space but the housing applies force as the aircraft changes heading and bank angle. The resulting precession is restricted by springs.
The higher the rate of change in heading the more torque is generated by the gyro attempting to precess. This force can be measured by the change in spring length, indicated as movement on the gauge. By tilting the free axis up from zero pitch, changes in bank angle will also cause precession. With 90 degrees or a vertical axis the instrument would only precess with changes in bank, angles between 0 and 90 provide a varying mix of the two signals.(Angling the axis down will cause bank to react in the opposite direction, usually this is not desired.)