Structural Optimization of Vented Brake Rotors with a Fully Parameterized Model

Abstract

Vented brake rotors used in an automobile behave similarly to centrifugal fans, drawing cool air from the inboard side, passing it through the disc vents, and exhausting it from the periphery. A vented brake rotor with a better heat dispersing ability is often superior to a solid rotor, in both thermal performance and brake efficiency. In this research, a fully parameterized model for a ventilated brake rotor is created using the ANSYS Parametric Design Language, to uniquely define the rotor’s geometry. With this parameterized model, two structural optimization cases are studied in this paper. The first one investigated is a modal frequency separation problem: The frequency differences in a tangential mode sandwiched between two nodal diameter modes of the brake rotor model are maximized. An automatic identification scheme for extracting correct mode orders is implemented in the program to track the correct modes during optimization. The second case is a thermal deformation problem: The distortion on the frictional surfaces of the rotor loaded with heat flux generated during the braking process is minimized. The optimization results show that a brake rotor design with a thinner outboard disc and a thicker inboard disc provides a great choice for rotor coning reduction.

1. Introduction

Disc brakes are commonly installed in vehicles as one of the critical safety components. Substantial thermal and mechanical loads applied to the disc brake system to stop a moving vehicle can cause severe noise and vibration problems, which are categorized among the noise, vibration, and harshness (NVH) phenomena and are not only uncomfortable to the passengers but can sometimes damage the braking system and the vehicle as well. The braking performance of a vehicle can significantly be affected by the temperature rise in the brakes. The frictional heat generated on the interface of the brake rotors and the braking pads can lead to high temperatures. On some occasions, the temperature may exceed the critical value for a given material, which leads to undesirable effects, such as thermoelastic instability, premature wear, brake fluid vaporization, bearing failure, thermal cracks, and thermally excited vibration [1]. Solid brake rotors, though cheaper, disperse heat slowly. Therefore, ventilated rotors are often used in vehicle brake systems to improve cooling by enhancing air circulation. A vented brake rotor behaves similarly to a centrifugal fan, drawing cool air from the inboard side, passing it through the disc vents, and exhausting it from the circumference of the disc. With more air passing through the vents, more heat generated from braking is carried away, and consequently, the brake rotor is cooled down more rapidly.

In the literature, research studies on brake rotors mainly focus on thermomechanical and vibration-induced issues. Belhocine and Bouchetara [2,3] analyzed the thermal behavior of a solid brake rotor and a ventilated rotor using ANSYS/CFX in a fluid analysis to extract the heat transfer coefficients on the surfaces of the brake rotors, and then, the coefficients were applied to the rotors as the boundary condition in a transient thermal conduction simulation. Kim [4] also performed finite element analysis (FEA) to predict the temperature distribution and thermal deformation of a brake disc. Jung et al. [5,6] proposed an analysis technique to estimate the temperature rise and thermal deformation of ventilated discs with consideration of various braking conditions and the properties of the disc and pads. The analytical process of the braking power generation during braking was mathematically derived. Thilak et al. [7] performed a transient thermal and structural analysis on a brake rotor to evaluate its performance under severe braking conditions and to assist in disc rotor design. Very recently, Güleryüz and Karadeniz [8] conducted a series of CFD and transient thermal analyses on brake rotors with various types of ventilation vent geometries and rotor–wheel–hub arrangements, including a new integrated rotor–hub design. CFD analyses were performed to extract convective heat transfer coefficients for two assembly conditions: rotor–hub and rotor–hub–wheel assemblies. The extracted convective heat transfer coefficients were then added to the finite element models as boundary conditions, together with thermal conduction and radiation effects, to carry out a transient thermal analysis. To ensure the accuracy of the CFD results and keep the computational cost low, a grid independence study was also executed by generating 14 different meshes. Significant improvements in the average convective heat transfer coefficients, cooldown period, and the maximum wheel-bearing temperature were observed for the new integrated design over the baseline model. In a recent study, Zhang et al. [9] investigated the thermal and structural coupling effects of brake discs and pads for mega-watt wind turbine brakes during the braking process.

Brake squeal, which is a high-frequency noise produced during braking, has been one of the most difficult issues associated with automotive brake systems [10,11]. Brake rotors are often bolted to wheel hubs; however, finite element analyses on brake rotors with free–free boundary conditions can still provide valuable insights into the squeal phenomenon [12,13,14,15]. Hassan et al. [16] and Ouyang et al. [17] examined the thermal effects on the squealing problem in disc brakes using complex eigenvalue analysis and dynamic transient analysis, respectively, by employing the finite element approach. In another research, a validated model of a disc brake was used as the basis for parametrically studying and evaluating a set of design guidelines that were intended to minimize the possibility of squeal noise production in the brake system [18]. Hassan et al. [19] described a fully coupled transient thermomechanical model of a disc brake that took into account the so-called rotating-heat-source effect. The experimental results from a sequence of dynamometer tests replicating a family of drag braking events, based on the SAE J2521 drag braking schedule [20], were established to validate the model. A recent study reported by Gao et al. [21] studied the mechanism of high-frequency brake squeal through vibration energy.

Structural optimization is a natural extension to the analysis and design of brake rotors. In the literature, due to their geometrical complexity, structural optimization of brake rotors usually involves commercial software for optimization and repeated analyses, which either employ the finite element method to solve static, dynamic, or modal problems, or employ the finite volume method to solve CFD problems. Galindo-Lopez and Tirovic [22] aimed to maximize the convective heat transfer of brake discs mounted on railway vehicles and proposed the specific power dissipation, the product of the average convective heat transfer coefficient and the disc wetted area, to measure the effectiveness in convective cooling for various brake disc designs. The authors adopted fluid dynamic software ANSYS CFX to evaluate the performance of disc models and were able to generate a new disc design with an increase in convective cooling power of over 10% when compared with the existing disc. Roy and Bharatish [23] employed five geometric parameters of a ventilated brake disc to minimize the maximum total deformation and equivalent stress of the model using the ANSYS software and the response surface methodology and achieved approximately 10% reduction in both deformation and stress. Topouris and Tirovic [24] performed shape optimization to minimize the maximum principal stresses on finger structures of a cast-iron brake disc and then topology optimization to reduce the mass of the disc. Finite element analysis and optimization packages, Abaqus and OptiStruct, were used in their research. In another recent research, the shape and pattern of through-holes and slits on the friction surface of a brake rotor were optimized to minimize the maximum stress and temperature in the rotor model [25].

In this paper, a fully parameterized model for ventilated brake rotors is created using the ANSYS Parametric Design Language (APDL), with 22 parameters uniquely defining the rotor geometry. Then, an advanced, zero-order optimization procedure is applied iteratively on the model to maximize or minimize a predefined objective function with some constraints imposed. Two structural optimization cases are studied in this paper. The first one is a maximization problem for frequency separation between certain eigenmodes of the rotor structure. The second case minimizes the thermal deformation on both sides of the friction surfaces of the rotor with a thermal loading of heat flux generated during the braking process. A conceptual flowchart for this research is shown in Figure 1.

2. Research Methods

2.1. Brake Rotor Modes Description

A vented brake rotor is composed of two major portions—the hub and the disc, as shown in Figure 2. The vibration mode shapes of a brake rotor under a free–free boundary condition can be categorized as in-plane modes or out-of-plane modes according to their primary directions of motion. The out-of-plane modes mainly vibrate transversely out of the disc plane or the top surface plane of the hub, and they can be classified as diametrical bending modes or circumferential bending modes (Figure 3). The oscillations of the in-plane modes remain mostly on either or both of the planes and the modes are further characterized as radial modes or tangential modes (Figure 4). The tangential in-plane modes appear in two distinct patterns: shear and compression, as shown in Figure 4b,c. Two sets of coordinate systems are used in Figure 3 and Figure 4. In Figure 3, a Cartesian coordinate system is employed, and the color contour represents vibration amplitudes in the y-direction (normal to the disc plane). In Figure 4, a cylindrical coordinate system with normal z-axis in the disc plane is used, and the amplitudes in a radial direction and circumferential direction are shown in Figure 4a–c, respectively.

The tangential in-plane compression modes have been found as the most significant cause of brake squeal [11,13,14], and Chen et al. [11] stated that the coupling of a tangential in-plane compression mode and an adjacent diametrical bending mode constitutes the dominant cause of squeal noise. The diametrical bending modes behave similarly to the nodal diameter (ND) modes from the classical circular plate vibration theories, e.g., Leissa [26]. Hereafter, diametrical modes are named ND modes having nodal diameters across the rotor, e.g., one nodal diameter is defined as 1ND mode, two as 2ND mode, etc. Additionally, the tangential in-plane compression modes are referred to as T modes for short. T modes typically have much higher frequencies than ND modes.

2.2. The Fully Parameterized Brake Rotor Model

The geometry of the vented brake rotor under investigation was fully parameterized by 22 geometrical input parameters, excluding the bolt holes on top of the rotor hub, and these parameters can be seen in Figure 5 (shown in a quarter-sectioned model) and Figure 6. The parameter θ_RIB, shown in Figure 6, also determined the number of ribs sandwiched between the upper and lower discs. For example, θ_RIB = 10° leads to 360°/10° = 36 ribs and θ_RIB must be set in such a way that the division results in an integer. S_CH1, S_CH2, and S_CH3 represent the depths (or widths) of the respective 45° chamfers. These 22 parameters completely and uniquely defined the geometry of the brake rotor. With the help of ANSYS Parametric Design Language, assigning the values adopted from a commercial brake rotor to the input parameters (values given in

Table 1. Initial values of the 22 geometrical parameters for the baseline model ^§.

DCEN	DI1	DI2	DI3	DO1	DO2	DO3	R1	R2	R3	SCH1
61	123	142.6	147.6	135	230.6	222.6	2	2	2	1
SCH2	SCH3	SCHX	SCHY	TC	TF1	TF2	TRIB	WRIB	Y1	θRIB
3.5	2	6	2	6	6	6	5	5.4	27.5	10°

^§ All length units are in millimeters.

) rendered a solid model, as previously shown in Figure 2, which was defined as the baseline model. APDL is a programing language capable of parametrically creating solid models, meshing, building, and solving finite element models, manipulating FEA results, and even performing optimization. A finite element grid of this baseline model can be seen in Figure 7.

The brake rotor was made of grey cast iron, the material properties of which are listed in

Table 2. Material properties used in the finite element models.

Young’s Modulus (GPa)	Poison’s Ratio	Density (kg/m3)	Thermal Exp. Coef. (1/°C)	Thermal Conduct. (W/(m°C))	Specific Heat (J/(kg°C))
128.75	0.28	7220	1.1 × 10−5	55	500

. Based on the results of a preliminary study, modal analyses of the rotor model under a free–free boundary condition with 40 flexural modes were enough for the current study. Among the first 40 flexural modes, the ND modes and T modes were particularly of interest in brake rotor design. Analysis results showed that one nodal diameter (1ND mode) did not exist for the free–free rotor model. However, 2ND through 7ND modes and 1T mode of the baseline model were among the 40 flexural modes. Figure 8 shows these ND and T modes with their mode shapes, mode orders, and frequencies. In order to properly show their mode shapes, two different coordinate systems are used in Figure 8. For ND modes, vibration amplitudes in the y-direction (perpendicular to the rotor disc plane) are much more prominent, and therefore, a Cartesian coordinate system is used. For T modes, oscillations are mainly confined on the rotor disc surface and behave similarly to torsion motions, and therefore, a cylindrical coordinate system with its z-axis perpendicular to the disc surface is employed. The deep red and deep blue color contours on the mode shapes in Figure 8 represent two regions with amplitudes of opposite directions. For a nearly axially symmetric structure such as the brake rotor under investigation, ND modes are expected to appear in nearly orthogonal pairs with two consecutive modes phase-shifted 90° apart and very close in frequency, and so are the T modes. Figure 8 shows only the first modes of these orthogonal pairs, which are also the modes of interest in this research.

2.3. Convergence Test

Before performing further finite element analyses and optimizations, a convergence test was executed to ensure the credibility of the finite element model. Gradually reducing the element sizes, remeshing the solid model with higher-order 3D elements, and performing multiple modal analyses on the baseline model of the rotor under a free–free boundary condition revealed its first 40 natural frequencies and mode shapes, excluding the rigid body modes. A total of 12 finite analyses were performed, with element sizes reducing from 10 mm to 1.7 mm.

Table 3. Element sizes and numbers of nodes in the model.

Element Size (mm)	No. of Nodes	Max. Relative Error
10	23,953	5.49% (35th mode)
8	30,170	5.40% (35th mode)
6	47,723	4.45% (35th mode)
5	69,025	3.89% (35th mode)
4	135,872	1.94% (35th mode)
3	281,539	1.02% (35th mode)
2.5	429,193	0.57% (35th mode)
2.2	535,311	0.44% (35th mode)
2.1	662,995	0.33% (35th mode)
2	790,894	0.19% (35th mode)
1.8	1,096,105	0.04% (8th mode)
1.7	1,250,497	-

gives the element sizes and the resulting numbers of nodes in the finite element models, with more than 1.25 million nodes in the model of element size 1.7 mm. All 40 natural frequencies appeared to converge to certain values as the number of nodes increased. Figure 9a,b illustrate, respectively, the frequency converging trends for the 2ND mode and 1T mode.

Since the subsequent optimization tasks in this research require a great number of repeated finite element calculations, an accurate finite element model with a proper mesh size should be established before optimization to reduce the computational burden. A relative error in computed frequency for a particular mode (i) and with element size (s) is defined as

$${\mathsf{\epsilon }}_{i,s}=\frac{f_{i,s}-f_{i,1.7}}{f_{i,1.7}}$$

(1)

where f_i,s is the natural frequency for mode i of the finite element model with element size s, and f_i,1.7 is computed from the finite element model with the element size of 1.7 mm. In Table 3, the maximum relative errors and the corresponding modes for element sizes ranging from 10 mm to 1.7 mm are also listed. The smallest model (with the fewest nodes) having relative errors, defined in Equation (1), of less than 1% in all 40 natural frequencies was selected for the subsequent optimization problem. The resulting finite element model, having an element size of 2.5 mm, consisted of 275,226 higher-order elements and 429,193 nodes (shown in Figure 7) and produced frequency errors of less than 0.6% for all 40 modes. The 2ND through 7ND modes and the 1T mode predicted by the selected model are shown in Figure 8, in which, the 1T mode, for example, is the 30th mode, with a frequency of 9701 Hz, and 7ND mode is at 10,317 Hz (the 34th mode).

2.4. Automatic Identification of the ND and T Modes

To make the optimization process fully automatic, ND modes and T modes must be determined automatically. Based on the analysis results of the baseline model, the following facts were observed, according to which actions were taken for the brake rotor model under investigation: All ND and T modes appeared in nearly orthogonal pairs, and for each pair, only the mode with the lower frequency was used for optimization; the largest amplitudes of the ND and T modes were located on the outermost rim of the rotor disc; the number of regions on the rim containing the positive displacements was identical to that of the negative regions, and each region of positive/negative displacements occupied a similar arc length on the rim, as shown in Figure 8. The procedure to identify the ND and T modes is described below:

• During modeling, 360 hardpoints evenly distributed on the rim of the top surface of the rotor disc were first defined, and the finite element nodes at these hardpoints (defined as the key nodes) were extracted and renumbered sequentially;
• Modal analysis was performed to solve for 40 flexural modes;
• The Cartesian coordinate system was selected;
• The largest amplitude in the y-direction among all finite element nodes of each mode was identified, by which all nodal y displacements of the key nodes within the same mode were divided (normalization);
• Starting from the first mode (the lowest frequency), modes with normalized y displacements at the key nodes equal or very close to +1/−1 were identified, alternating behavior of positive and negative displacements at the key nodes was checked, and the numbers of alternating regions and the key nodes in each region were counted;
• Consecutively, 2ND through 7ND modes were identified;
• The cylindrical coordinate system was selected;
• Excluding the identified ND modes, the above procedure was repeated, with θ direction replacing the y-direction, and the 1T and 2T modes were identified, if necessary.

These steps, which are efficient and easy to implement, were realized and incorporated with finite element analysis and optimization schemes in APDL codes and proved very effective in identifying the ND and T modes.

3. Results and Discussion

3.1. Case 1: Frequency Separation Optimization

In brake rotor design, an important issue is to avoid the coupling of ND and T modes, i.e., to prevent T modes from growing too close to ND modes in frequency. Due to the manufacturing process, two brake rotors with the same design specification can yield a variation in natural frequencies as high as 7%, especially at high frequencies [27]. Other factors, e.g., wear, may also deviate the natural frequencies of a rotor from its design values. Therefore, separating T modes from ND modes as far as possible is a prudent decision.

The modal analysis of the baseline model showed that, in the first 40 modes, 1T mode was the only T mode, and it was sandwiched by 6ND and 7ND modes. Therefore, in this optimization case, 1T mode was forced to be separated as far as possible from 6ND and 7ND modes by optimizing a few geometric parameters. The frequency differences between 6ND and 1T modes and 1T and 7ND modes were combined as one objective function and then maximized. The mathematical formulation to define the frequency separation optimization problem in this research can be stated as

$$\mathrm{Maximize}{f}_{obj}=f_{sep1}+f_{sep2}$$

(2)

$$\mathrm{subjected}\mathrm{to}{L}_{DV}\le DV\le U_{DV}$$

(3)

$$M_a-M_{base}\le {\epsilon }_M$$

(4)

$$f_{sep1}=f_{1T}-f_{6ND}>0$$

(5)

$$f_{sep2}=f_{7ND}-f_{1T}>0$$

(6)

$$\mathrm{and}DV={\left[\begin{array}{cccc}D{I}_1& T_{F1}& T_{F2}& T_{RIB}\end{array}\right]}^T$$

(7)

where f_obj is the objective function of this optimization problem; f_sep1 and f_sep2 are the separation frequencies between the 6ND (f_6ND) and 1T (f_1T) modes and 1T and 7ND (f_7ND) modes, respectively; DV contains the design variables and L_DV and U_DV denote the lower and upper bounds, respectively, of the design variables; M_base is the total mass of the baseline model, M_a represents the mass of current model; ε_M is a small positive value. Equation (4) ensures that the mass of the optimized model will not exceed the baseline model by more than a small amount. In this study, ε_M was set to 1% of the baseline mass. The weight issue is always an important concern when designing automotive parts. Four design variables DI₁, T_F1, T_F2, and T_RIB were chosen based on engineering judgment and sensitivity analysis that revealed their high sensitivities. The sensitivity of a frequency f_j to a design variable DV_i is defined as

$${\mathrm{s}}_{ij}=\frac{\partial f_j}{\partial D{V}_i}\cong \frac{f_j\left(D{V}_i+\Delta D{V}_i\right)-f_j\left(D{V}_i\right)}{\Delta D{V}_i}.$$

(8)

by assigning the baseline value and setting a 1% increment to one design variable at a time, and then repeating the finite element analysis, the sensitivities of 1T mode to DI₁, T_F1, T_F2, and T_RIB, for example, were found to be −11.96, 66.00, −13.33 and −75.6 Hz/mm, respectively. The upper and the lower bounds were set as ±5% from the baseline value for DI₁, and ±20% for T_F1, T_F2, and T_RIB. Over relaxed upper and/or lower bounds might cause numerical difficulties during finite element analysis in some situations.

Applying Equations (2)–(7) with an APDL code using the subproblem approximation method (SAM) [28], which is an advanced, zero-order optimization procedure using approximations to all dependent variables and can be applied efficiently to a broad spectrum of engineering problems, convergence was achieved in 12 iterations.

Table 4. Optimization results for Case 1.

DVs and Conditions	Baseline Model	Optimized Model
DV: DI1 (mm)	123.00	126.90
DV: TF1 (mm)	6.00	6.58
DV: TF2 (mm)	6.00	7.11
DV: TRIB (mm)	5.00	4.05
fsep1, 1T and 6ND (Hz)	1328.0	936.1
fsep2, 7ND and 1T (Hz)	616.0	1189.3
fobj (Hz)	1944.0	2125.4
Mass of the model (kg)	3.507	3.525 (+0.51%)

lists the optimization results for Case 1. Although f_sep1 decreased, f_sep2 increased by a large amount after optimization, resulting in a combined improvement of 181.4 Hz in the objective function. Figure 10 shows the 6ND, 1T, and 7ND mode shapes and frequencies for the optimized model. Notably, the mode orders of these three modes all changed. The mode orders of the 6ND, 1T, and 7ND modes were initially 26th, 30th, and 34th (baseline model), and after optimization, they were 28th, 32nd, and 38th (optimized model), respectively. In fact, during the optimization process, the mode orders of all analyzed modes were constantly changing as the design variables evolved. It is critical that, in each iteration, a new set of ND and T modes were determined based on the automatic identification scheme mentioned in the last section. If the modes of interest were not correctly tracked during optimization, the process would most likely fail to converge to an optimal solution.

3.2. Case 2: Thermal Deformation Optimization

Coning of a brake rotor is caused by the thermal distortion of frictional surfaces due to uneven temperature distribution near the intersection of the hub and disc [29]. Severe brake rotor coning could result in brake system instability and is the main cause of brake judder [30]. As the second optimization case of this study, thermal deformation on the frictional surfaces of the rotor was investigated. By slicing through the baseline model while avoiding the holes on top of the hub, and taking advantage of the symmetrical feature of the rotor, a sliced model containing only one rib was created, and this reduced model with its boundary and loading conditions is demonstrated in Figure 11. The fixed BC applied on the areas on both sides of the rotor hub represents a bolted condition, which also reduced the adversity effects of the bolt holes that could hamper the symmetry assumption of the rotor. With heat flux loaded on the sliced model, deformations in y-direction at four measurement points (PT1 through PT4), which are named U_y1 through U_y4, respectively, were obtained, to calculate thermal distortion of the friction surfaces. Coning of the rotor can be represented by the differences between U_y1 and U_y2, and U_y3 and U_y4. To achieve this goal, the following steps were taken:

• A complete solid model of the brake rotor was first created using the parameterized model;
• The complete model was sliced through while avoiding the holes on top of the hub, to construct a sliced model containing only one rib;
• A transient thermal analysis was performed with a heat flux loading applied on the friction surfaces of the sliced rotor model;
• A series of temperature distributions as functions of time on the model were obtained;
• Then, with the temperature distributions as multistep loadings, a structural analysis was executed to acquire the thermal deformation of the model;
• U_y1 through U_y4 were extracted at four measurement points, and the brake rotor coning was calculated.

The heat flux loading for this study can be seen in Figure 12, with the peak heat flux of 1.2 W/mm² at 1.0 s and then linearly decreasing to 0 at 5.0 s. The heat flux data were acquired during a previous project in which the authors were involved. The simulation duration lasted 5.5 s, and the measuring time was also set at 5.5 s. Since the sliced model was only a 1/36 model of the complete rotor, a much finer mesh could be used. Figure 13 shows a finite element grid of the sliced baseline model that contains 38,917 elements and 98,100 nodes. Performing a transient thermal analysis and a structural analysis on the sliced baseline model with the material properties listed in Table 2 yielded the baseline results that were compared with those after optimization. Figure 14 shows temperature distributions (°C) of the sliced baseline model at various time steps, and Figure 15 illustrates deformations in the y-direction (meters) at various times for the same model. Large temperature gradients were observed near the junction of the hub and the outboard disc throughout different time steps, as shown in Figure 14. At time = 5.5 s, U_y1 through U_y4 were identified to be 0.1122, 0.0481, 0.0924 and 0.0344 mm, respectively, and coning was calculated on the outboard disc as U_y1 − U_y2 = 0.0641 mm and on the inboard disc as U_y3 − U_y4 = 0.0580 mm.

In optimization Case 2, thermal distortion of the frictional surfaces, i.e., the brake rotor coning, was minimized. The mathematical formulation to define this problem can be stated as

$$\mathrm{Minimize}{f}_{obj}=\left({\delta }_1+{\delta }_2\right)/2$$

(9)

$$\mathrm{subjected}\mathrm{to}{L}_{DV}\le DV\le U_{DV}$$

(10)

$$M_a-M_{base}\le {ϵ}_M$$

(11)

$${\delta }_1=U_{y1}-U_{y2}\ge 0$$

(12)

$${\delta }_2=U_{y3}-U_{y4}\ge 0$$

(13)

$$U_{y1}\ge 0$$

(14)

$$U_{y2}\ge 0$$

(15)

$$U_{y3}\ge 0$$

(16)

$$U_{y4}\ge 0$$

(17)

$$\mathrm{and}DV={\left[\begin{array}{cccc}D{I}_1& T_{F1}& T_{F2}& T_{RIB}\end{array}\right]}^T$$

(18)

where the objective function of this optimization problem is defined as the average coning calculated on the inboard and the outboard discs; δ₁ is the deformation difference between U_y1 and U_y2 at time = 5.5 s, and δ₂ is the difference between U_y3 and U_y4 at the same time. During optimization, we sought to ensure that the structural deformations in friction surfaces achieve a coning effect by forcing U_y1 through U_y4, with δ₁ and δ₂ all positive, i.e., Equations (12)–(17); nevertheless, we wanted this effect to be as small as possible.

By applying Equations (9)–(18) and incorporating them with the SAM optimization scheme in precisely the same way as the previous case in another APDL program, convergence was observed in 15 iterations. In each iteration, the six steps to acquire brake rotor coning at time = 5.5 s, including a transient thermal analysis and a structural analysis, were realized and repeated.

Table 5. Optimization results for Case 2.

DVs and Conditions	Baseline Model	Optimized Model
DV: DI1 (mm)	123.00	127.68
DV: TF1 (mm)	6.00	4.81
DV: TF2 (mm)	6.00	7.20
DV: TRIB (mm)	5.00	5.86
Uy1 (mm)	0.1122	0.0257
Uy2 (mm)	0.0481	0.0211
Uy3 (mm)	0.0924	0.0054
Uy4 (mm)	0.0344	0.0027
Uy1 − Uy2 (mm)	0.0641	0.0046
Uy3 − Uy4 (mm)	0.0580	0.0027
Objective function (mm)	0.0611	0.0037
Mass, complete model (kg)	3.507	3.264 (−6.9%)

lists the optimization results for Case 2. After optimization, a very significant reduction in thermal distortion was obtained, with the objective function values reduced from 0.0611 mm to 0.0037 mm, a great improvement of 94%. As an added bonus, the total mass of the optimized, complete model (before slicing) also decreased by 6.9%, compared with that of the baseline model. Figure 16 shows the temperature distribution (°C) and deformation in the y-direction (meters) of the optimized model at 5.5 s for Case 2.

Notably, the thickness of the outboard disc, T_F1, reduced from 6.00 mm to 4.81 mm, and the inboard disc, T_F2, increased to 7.20 mm, resulting in a pattern of deformation distribution after optimization, shown in Figure 16b, that was very different from that of the baseline model at 5.5 s, shown in Figure 15. Comparing Figure 14 and Figure 16a reveals that at time = 5.5 s, the temperature distribution was more uniform between the outboard and inboard discs for the baseline model, while the inboard disc assumed a much lower temperature than the outboard disc for the optimized model. This uneven temperature distribution in both discs might have offset, to some degree, the coning effects of abrupt temperature variation near the intersection of the hub and outboard disc.

4. Conclusions

A fully parameterized model for ventilated brake rotors was created using the ANSYS Parametric Design Language, with 22 parameters uniquely defining the rotor geometry. To ensure the credibility of the finite element model and to ease the computational burden during optimization, a convergence test was performed, resulting in an accurate finite element model with a moderate grid size. With this parameterized model, two optimization cases were investigated. In Case 1, frequency differences between the 6ND and 1T modes and 1T and 7ND modes were combined and maximized. Frequency separation optimization was achieved, resulting in a combined improvement of 181.4 Hz in the objective function. Care had to be taken to ensure correct modes were identified and tracked during optimization because mode orders indeed changed as the geometrical parameters of the rotor model varied in the process. In Case 2, a complete analysis involved a transient thermal analysis and a structural analysis on a sliced rotor model. A transient thermal analysis was first performed, with a heat flux loading applied on the friction surfaces, and a series of temperature distributions on the model were obtained. Then, a structural analysis was executed with the temperature variations as a multistep loading, to finally acquire the thermal deformation of the model. Minimization on the deformation was performed to accomplish a great reduction in thermal distortion, with the objective function values reduced from 0.0611 mm to 0.0037 mm. Since wear and the manufacturing process cause the geometrical values of a brake rotor to deviate from its design values, all calculations are valid only for a new braking system in an ideal condition. Some general conclusions from the present study can be summarized as follows:

• Structural optimization, including frequency separation maximization and thermal distortion minimization, based on a fully parameterized model for vented brake rotors can be efficient and very effective;
• During an optimization process involving modal analysis, mode orders alter as the geometry changes. Therefore, it is of the utmost importance that the code be able to automatically identify and track correct modes, and consequently, that the modal frequencies not be misused for calculating constraints or the objective function;
• Abrupt temperature variation near the intersection of the hub and disc caused brake rotor coning. However, uneven temperature distribution between the inboard and outboard discs could offset the coning effects to some extent. A brake rotor design with a thinner outboard disc and a thicker inboard disc provides a great choice for rotor coning reduction.