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2.DISC BRAKE DYNAMICS AND DRAG PERFORMANCE
2.1.Dynamic System of a Disc Brake
A disc brake system in braking is a dynamic system in two directions, i.e., rotor axial direction and rotor rotating direction. The schematic diagram of a disc brake system can be simplified as shown in Fig. 2. Along the rotor axial direction, the governing equation of the dynamic system can be expressed as
In the direction of the rotor rotation, the governing equation can be expressed as
In general, equations (1) and (2) are coupled. The braking torque is directly related to the normal pad force generated by the applying pressure. Since our primary interest is on the caliper drag performance which is mostly affected by the dynamics along the rotor axial direction, we focus on Eq.(1) in the present study.
2.2. Drag Performance
Drag is the residual torque on the rotor after brake release. The caliper drag performance depends on how much a piston can retract from its displaced position.
The more retraction of the piston, the less likely the drag will occur or the lower the drag will be. A systematic view of the drag behavior is shown in Fig. 3. Among all the contributing factors, housing stiffness, lining compliance, and seal/groove are three major factors to the drag problem. The stiffness of the housing and lining dictates the piston travel during a brake apply. A softer caliper system will result in a larger piston travel and possibly dislocate the piston/seal relative position. Contrarily, a stiffer caliper system would limit the piston travel, and provide larger spring back force to recover the relative piston/seal position. The friction in the system and the hysteresis of the lining and rubber materials cause energy dissipation and thus reduce the ability for the system to retract the piston. Seal/groove design is crucial to the piston dynamic characteristic. It determines squeezed state of rubber seal and subsequently impacts the stick-slide behavior between the seal and piston. Additionally, the slide force of the suspension (guide) pin is also a contributing factor of the drag performance. In certain situations, the suspension pin slide force could be quite high. Finally, rotor runout, which is not directly related to piston retraction, also negatively affects the drag performance.
Retraction can be related to drag performance. Figure 4 illustrates the dimension changes in a caliper under hydraulic pressure application. Equation (3) shows the relation between the dimension changes. The left-hand side represents the sum of housing compliance (∆H ) and lining compliance (2∆H ), and the right-hand side is the measured relative position change ( ∆H ) between the housing and piston. When the left-hand side is greater than the right-hand side, there is some amount of dislocation between the housing and piston from the neutral state, which consequently causes the caliper residual drag torque. We use the term "residual retraction" to represent this dislocation, and it can be quantified by equation (4). As discussed later, we have found it is a reasonable drag performance index for our caliper simulation.
3.ONE-DIMENSIONAL BRAKE SYSTEM MODEL
3.1. Model Description
Figure 5 shows the one-dimensional model of a disc brake system for drag performance simulation. The model includes inboard and outboard lining pads, piston, housing, and rubber seal. Each component is idealized as a spring-mass-damper unit. In caliper assembly, frictional pads are held in position by retention springs to prevent falling off from and rattling against the bracket. Retention springs provide clamping effect on the pads, which generally induces friction on the pad in the direction perpendicular to rotor surface. On the other hand, the caliper housing is held in position by two suspension (guide) pins that slide along the two anchor bores on the bracket. The friction force between the sliding pins and anchor bores also affects the dynamics of the brake assembly. Therefor, the stick-slide behavior at these locations is modeled. In all the areas that friction is modeled, friction is assumed to follow the Coulomb friction law. Friction lining materials, semi-metallic or non-asbestos organic (NAO), usually exhibit a significant amount of hysteresis （Fig. 6). Therefore hysteresis behavior of the lining is included in the simulation. The interfaces between the rotor and lining, the piston and the inboard lining pad, as well as the outboard flange of the housing and the outboard lining pad are modeled as "gap" elements that allow the separation but prevent the penetration.