The problem with most friction models is that they concentrate on dynamic conditions and ignore static friction. The best they might do is include coulombic and viscous friction (both dynamic conditions) using
F = V Fv +SIGN(V)Fc
Where F = total friction force; V= velocity;Fv= dynamic friction;Fc= coulombic (static) friction; and SIGN(V) is programming command that returns either -1 or +1.
The limitations of a model that includes only dynamic friction comes clearer when considering the problem of pulling a block just 1 in. on a surface through a compliant system such as a spring. The spring may stretch 3 in. before reaching a breakaway force. Then the block jumps more than the needed distance. A recently developed model of mechanical friction improves computerized simulations of mechanical systems because it includes both static and dynamic friction.
An example where dual-friction models are valuable is positioning systems operating at near-zero velocity. Gimbals and low-speed grinders, for instance, include mechanisms that must pass through or operate in the static-friction region where surfaces behave as though welded. Rejecting or zeroing out base motion is a big problem with pointing or tracking scenarios, such as those in inertial-stabilized targeting systems aboard aircraft. These must make small corrections to account for the motion of their mounting platform (the airframe) and illustrate the need for a robust friction model.
The accompanying initial graph is typical for most friction models. They return a required coulombic plus viscous output for non-zero velocities. But at zero speed in the regime of static friction, the old model can only return zero, or positive and negative coulomb friction (Fc). Such limited models perform adequately for systems such as speed regulators because during normal operation components are always moving and they don’t see static friction effects. Motion control or positioning systems, on the other hand, perform in both static and dynamic-friction regions. Thus, dynamic-only friction models cannot accurately predict realistic performance when used as a simulation tool for motion-control systems. A better model must accurately return values for both static and dynamic friction.
In the static region — the vertical force line in the graphs — a model should return reactions equal to the algebraic sum of applied forces. But the static reaction force must be limited by breakaway friction. This is the value of static friction which when exceeded produces initial motion and a transfer to the dynamic-friction region.
The New friction model shows the required composite performance. In the dynamic or non-zero speed regions, the model returns the same frictional force as the graph above it. In the static region, however, the model returns any value of static friction required to maintain the system in static equilibrium.
A more complete friction model with dynamic and static performance has been derived and tested for dynamic system-simulation software. To demonstrate the robustness of the model, a test was devised in which four stacked blocks are positioned on an immovable base. Masses, friction values, and applied forces are selected to make each bearing boundary operate and transfer between static and dynamic friction regimes. A force applied to one of the blocks for 2 sec gets the system moving.
The subsequent distribution of forces in the stack moves the blocks together at first and then relative to each other after forces overcome breakaway friction. Removing the force lets the blocks slow from coulomb friction (static friction) and reseize until they come to rest. Reducing viscous friction in the model to a minuscule value lets the static and coulomb friction effects dominate the results. The distribution of frictional forces and algebraic summation with applied forces are shown integrated with the system kinematics in A diagram for sliding blocks with friction.
For the simulation, only force P2 pushes on Block 2. One way to view relative motion depicts states of block activity at crucial times in the simulation when block surfaces transition between static and coulomb friction. Opposition forces produced by the friction model and velocities of each block are graphed in additional illustrations.
A demonstration in detail
The four stacked blocks provide one test for the recently developed static-dynamic friction model. Although the graph The friction model in motion describes the interaction between blocks, it can be difficult to interpret. A more detailed reading follows.
Between time t=0 and t1, friction values F23 (friction between Blocks 2 and 3), F34, and F4B rise with the applied force P2. Friction value F23 remains at zero exactly as required. At time t1, the force applied to Block 2 reaches the static breakaway value between Block 4 and the base. F4B makes the transition from static to dynamic friction and remains there until t8. Also at t1, F12, F23, and F34 abruptly change from static friction to correct values in conjunction with P2 and F4B, causing all four blocks to accelerate as one welded unit. At t2, the static force between Blocks 3 and 4 reaches its breakaway friction value, abruptly transfers to its assigned dynamic friction value, and Blocks 1 through 3 separate from Block 4. Block 4 continues to accelerate as a function of its mass and algebraic summation of the dynamic values of F34 and F4B.
At the same time, there is the required abrupt change in static values of F12 and F23. From t2 to t3, Blocks 1, 2, and 3 accelerate as a unit under the influence of P2, the dynamic value of F34, and the sum of their masses. The dynamic friction values of F12 and F23 continually adjust to assure common acceleration of the three individual masses. At t3, the static value of F12 reaches its breakaway limit and Block 1 separates from Block 2. Blocks 2 and 3 accelerate as a single unit and static force F23 readjusts. At t4, the static value of friction reaches its breakaway value and the final separation occurs. Block 2 continues to accelerate under the influence of the applied force P2 minus the dynamic friction values at its two interfaces, F12 and F23. After reaching point tp, the acceleration of Block 2 turns negative as it begins to slow due to the removal of the applied force P2. Blocks 2 and 3 reach matching speed at t5 and seize together by static friction. The velocity of Block 1 matches that of Blocks 2 and 3 and enters seizure at t6. At t7, all blocks seize and move together slowly due to coulomb friction between Block 4 and the base. All blocks are at rest at t8 and all four static friction forces become zero.