Figure 9(a) shows the simulation result of the residual retraction as a function of housing and lining stiffness in a three-dimensional surface plot. We have observed that, while maintaining housing stiffness, a stiffer lining results in a smaller residual retraction, and conversely, a softer lining would result in a greater residual retraction. A similar trend is observed in Fig. 9(c) with housing stiffness.

3.3.2. Piston Slide Force vs. Housing Stiffness

Figure 10 shows a three-dimensional surface that signifies the residual retraction as a function of piston slide force and housing stiffness. We have observed that, while maintaining piston slide force, a stiffer housing results in a smaller residual retraction, and conversely, a softer housing would result in a greater predicted residual retraction. Similarly, a higher piston slide force would result in a greater residual retraction.

3.3.3. Piston Slide Force vs. Lining Stiffness 

Figure 11 shows the residual retraction as a function of piston slide force and lining stiffness in a three-dimensional surface plot. Similar to what are observed in Piston Force vs. Housing Stiffness, while maintaining piston slide force, a stiffer lining results in a smaller residual retraction, and a softer lining would result in a greater predicted residual retraction. While maintaining lining stiffness, higher piston slide force would result in greater residual retraction.  

The three -dimensional surface plots shown in Figs. 9(a), 10, and 11 provide helpful guidelines for caliper system designs. They not only point out the general trend of the drag performance resulting from the variation of each design parameter, but also indicate the trade-off between the design parameters to achieve certain drag performance. Using this tool, caliper engineers can perform trade-off study as the design requirement changes. For instance, in Fig. 9(a), point A indicates the residual retraction at initial design with baseline lining stiffness and a 0.6 housing stiffness. When a softer lining is selected, the lining stiffness is reduced to 0.8 time of the baseline stiffness. Simulation results reveal that the housing stiffness has to be increased accordingly to 0.9 to achieve the same level of residual retraction, as shown at point B.

3.4. Case Study

A case study was conducted to demonstrate the merit of the one-dimensional model as an effective analytical design tool to guide the brake engineers in their disc brake system development.

In the study, the initial design contained semi-metallic linings and a phenolic piston. Case 1in Fig.12 is the residual retraction for the caliper design with the initial configuration. It had a relatively smaller residual retraction, implying a relatively lower drag torque. At later stage of the development, NAO lining were used to replace the semi-metallic linings to satisfy other caliper performance requirements. The increased lining compliance caused a much greater residual retraction, and consequently, a higher drag. The case study continued to investigate the possibility to recover some of the loss in drag performance without invoking changes in seal groove configuration. To reduce the residual retraction, a steel piston was then considered  to replace the phenolic piston. The use of steel piston usually results in a lower piston slide force and hence reducing caliper residual retraction. To further reduce the residual retraction, a modification to the housing was implemented that increases the housing stiffness by a significant percentage. As a result of this change, a lower residual retraction was achieved.

 This study shows that, with the change of lining material from semi-metallic to NAO, the loss of drag performance cannot be completely compensated from the changes in the other two components, piston and housing stiffness. The model is very effective in assisting engineers to understand the trade-off among the three components, i.e. Housing, lining and piston, and evaluate design alternatives to meet the performance specification.