تجزیه و تحلیل حساسیت حرکتی و بهینه سازی از ارتباطات راهبری مسطح دندانه دار و پینیون

In this paper, the combined kinematic and sensitivity optimization of a rack-and-pinion steering linkage is performed. This steering linkage is the most common steering system used in passenger cars. Although, the steering linkage has received a lot of attention for the minimization of the steering errors, no attempt has been made so far to investigate the sensitivities of optimum dimensions relative to variation of link lengths. The kinematic optimization of the linkage is carried out using three homogenous design parameters. The objective of the proposed optimization is to minimize maximum steering error during cornering. This is followed by a sensitivity analysis to predict how the steering error is affected by manufacturing tolerances, assembly errors, and clearances resulting due to wear. Since the optimized kinematic error is very sensitive to the variations of the linkage parameters, the kinematic and post-optimal sensitivity optimization of the steering linkage is performed in an integrated manner. The methodology proposed in this work helps the designers of rack-and-pinion steering linkage to choose the linkage parameters whose maximum steering error (MSE) and sensitivity are minimum.

Among the steering linkages, rack-and-pinion steering linkage is the most widely used in passenger cars. It consists of two steering arms, two tie rods, and a rack. The linkage has two common configurations, namely, central take-off and side take-off, as shown in Fig. 1. In central take-off (CTO) configuration, the tie rods and the rack are connected at the middle of the rack as depicted in Fig. 1a, while in side take-off (STO) configuration, these connections are at the rack ends as shown in Fig. 1b. Each of the above configurations can be either trailing or leading type, as shown in Fig. 2a and b, respectively [1]. Full-size image (28 K) Fig. 1. Rack-and-pinion steering linkage and its configurations. Figure options Full-size image (21 K) Fig. 2. Trailing and leading type of rack-and-pinion steering linkages. Figure options In order to provide pure rolling to the road wheels and to reduce wear and tear of the tires, a steering linkage must handle the vehicle so that it follows Ackermann principle (see Fig. 3). This principle states that during low speed cornering when free from lateral inertia forces, the verticals drawn from the centers of the wheels should meet at the center of bend, i.e., point O of Fig. 3. For a two-wheel steering vehicle, this point must lie on the common axis of the rear wheels [1]. Referring to Fig. 3, the relation between the inner wheel angle, θI, and the outer wheel angle according to Ackermann principle, θOA, is given as equation(1) View the MathML sourceθOA(θI)=tan-11cotθI+WtWb=tan-11cotθI+1/wb Turn MathJax on where wb = Wb/Wt is the normalized expression of the wheel base, Wb, with respect to wheel track, Wt. In reality, Eq. (1) is never satisfied for every radius of orientations. Hence, there are efforts to synthesize the linkage so that Ackermann principle is satisfied for any orientation of the wheels as closely as possible. In order to do that, it is necessary to obtain the angle θO for a given value of θI. Hence, an appropriate kinematic model of the steering linkage is essential. In addition, the kingpin inclination and caster angles that provide compliance to the steering linkage with the suspension system have little influence on the motion transmission of the steering linkage. As a result, the real rack-and-pinion steering linkage, which is spatial in nature, can be modeled as a planar linkage for the investigation of Ackermann condition. Such a simplification of the steering system has been also used by other researchers, e.g. [2] and [3]. Full-size image (6 K) Fig. 3. Ackermann condition for a vehicle when turning. Figure options Error optimization studies in steering linkages have been attempted by many researchers. Zarak and Townsend [3] optimized the STO configuration as shown in Fig. 1b, where they considered the distance between the inner and outer wheel turning centers as the steering error and minimized it for different rack travels. During optimization, they used four parameters that were normalized with respect to the wheel track. Felzien and Cronin [4] investigated the minimization of steering errors. They considered an integrated McPherson suspension and steering linkage model, and minimized the weighted sum of the squares of steering errors. Simionescu and Smith [5] discussed Watt II function generating cognates, and showed that STO configuration of the steering linkage has infinite number of cognates, of which one is the CTO configuration. Simionescu and Smith [2] used three parameters, namely, a normalized link length/a link length ratio and two angles, in the case of CTO/STO configuration to optimize the steering errors of the linkages. The choice of the design parameters used by them, some of which are in terms of angles, is not appropriate however for link-length sensitivity analysis. Hence, the kinematic optimization carried out in this paper uses three homogeneous design parameters, namely, those which have the units of length. Moreover, manufacturing tolerances, assembly errors, and clearances resulting due to wear, which are inherent to any real steering linkage, may affect the objective function value significantly. Hence, it is important to perform a post-optimal sensitivity analysis, in addition to kinematic optimization. Furthermore, a method based on rack-and-pinion steering linkage cognates given in [5] is used to generalize the steering optimization. Using this methodology, the steering linkage can be optimized once and the results are extended to desired rack-and-pinion steering linkage whether CTO or STO with any rack length. To the best of the authors’ knowledge, no work concerning the sensitivity of the error for the optimized dimensions of a rack-and-pinion steering system has been reported so far. In addition, the sensitivity minimization, which is carried out in this paper as a part of optimization process, is also new in the context of rack-and-pinion steering linkage design. Such minimization of both the objective function and the sensitivity has been considered important in the literature of robust optimal design [6] and [7]. In summary, the contributions of the paper are • A simple generalized methodology for the optimization of a rack-and-pinion linkage for both CTO and STO is proposed. • Post-optimal sensitivity of the steering linkage is performed. • The steering linkage is optimized for minimum steering error as well as for minimum sensitivity. • A methodology for multi-objective optimization problem is also presented, where the number of design parameters is reduced by one, thus increasing the efficiency and speed of the optimization. This paper is organized as follows: Section 2 shows that the optimized results for a CTO linkage can be used to find the optimized parameters of a corresponding STO linkage, consequence that any STO configuration has a CTO cognate configuration. Section 3 presents a generalized kinematic modeling of a planar six-bar rack-and-pinion steering linkage, which would then be used in Section 4 to optimize the steering linkage under study. Post-optimal sensitivity analysis is then carried out in Section 5, which is further used in Section 6 to optimize the steering linkage that has minimum sensitivity in addition to minimum steering error. The proposed methodologies are illustrated in Section 7, followed by the conclusions in Section 8.

In this paper a combined kinematic and sensitivity optimization of a rack-and-pinion steering linkage are presented for the first time. Since the variations of the linkage parameters cannot not be avoided in practice, attention to the sensitivity analysis and optimization is considered essential. The kinematic optimization of the steering linkage is carried out using three homogenous design parameters all having the unit of length as a requirement for post-optimal sensitivity analysis. A simple but generalized methodology based on rack-and-pinion steering linkage cognates is proposed to analyze the central take-off configurations. The results are applicable to side take-off configuration of the steering linkage as well. The analysis of the linkage shows that the system is very sensitive with respect to variations of the links lengths close to optimum and are different left and right of the optimum point. This has implications in assigning tolerances to the links lengths. Next observation is that the sensitivities of the steering error with respect to the variations of different design parameters vary very widely. Hence, it should be taken into account that some alignments during maintenance, namely toe angle by changing the tie rod length, can increase the steering error extensively. For optimization of both the kinematic steering error and sensitivity to link length variation, a two-level multi-objective optimization was proposed. The advantage of the proposed method is that it reduces the number of design variables by one, thus increasing the stability and efficiency of the optimization. The proposed methods were successfully applied to the rack-and-pinion steering linkage of a real vehicle, and the numerical results provided in the paper.