In this paper, we develop an adaptation of the LuGre friction model so as to allow the development of the friction force and its application in any directions on systems subjected to varying normal forces. This is achieved by projecting a modified LuGre model adapted to varying normal forces in 3D along an arbitrary orthogonal system. Consequently, the direction of the friction force is automatically oriented in the correct direction, thus stick, stick-slip, and slip behavior can be represented in all directions. The projected friction model has the following friction features: stick-slip, presliding displacement, frictional lag, varying break-away force, viscous friction, Stribeck effect, and is adapted to varying normal forces. The equivalence of this projected LuGre model with the modified one is proven analytically. The friction model is then applied to simulate the friction on two mechanical systems. The first system consists of a cube sliding on a plane with a transition from stick to slip due to varying normal forces and with a pulling force oriented in multiples directions of the contact plane. The second one is a more complex system consisting of three turbine blades that uses friction to damp their resonance. The results obtained for both systems are consistent with literature.