SYNTHESIS AND SIMULATION OF A MULTILINK SUSPENSION MECHANISM

Transcription

1 Journal Mechanisms and Manipulators Vol. 5, Nr. 2, 26. pp ARoTMM - IFToMM SYNTHESIS AND SIMULATION OF A MULTILINK SUSPENSION MECHANISM Giorgio FIGLIOLINI*, Pierluigi REA* DiMSAT, Universit of Cassino LARM (Laborator of Robotics and Mechatronics) G. Di Biasio 43, 343 Cassino (Fr), ITALY figliolini@unicas.it, rea@unicas.it Abstract: This paper deals with the dimensional snthesis and simulation of a multilink suspension mechanism. In particular, an optimal dimensional snthesis of a five-link suspension mechanism in terms of lengths of the five links and positions of the ten spherical joints, which are installed on the vehicle chassis and wheel carrier, is presented. This optimiation has been carried out b using the Matlab Optimiation Toolbox with the aim to obtain a bod motion of the wheel carrier with respect to the vehicle chassis as close as possible to a vertical translation. In order to test the kinematic performances according to the design specifications, the position analsis of the snthesied five-link suspension mechanism is formulated and suitable three-dimensional simulations obtained through the ADAMS software are shown. Finall, sketches of McPherson rear suspension mechanisms are reported to extend this snthesis method. 1. Introduction The suspension of modern vehicles has to satisf several specifications which can be in conflict between them because of different operating conditions (loaded/unloaded, acceleration/breaking, level/uneven road, straight running/cornering). The suspension sstem allows the roll and pitch motions of the vehicle chassis, while the aw motion is indispensable for turning. A basic stud of the vehicle roll motion is reported in [1]. Vehicle suspensions can be with rigid axles, semi-rigid axles and independent wheels in which the relative motion of each wheel carrier with respect to the vehicle chassis is independent. This characteristic gives several advantages as no mutual wheel influence, little space required and low weight. Thus, the tpe and dimensional kinematic snthesis of the spatial mechanism, which connects the wheel carrier to the vehicle chassis is fundamental to satisf the required specifications. Multilink and McPherson mechanisms are well know examples of independent wheel suspensions. The dimensional snthesis of a multilink suspension mechanism can be carried out b imposing seven arbitraril poses of the wheel carrier. This method can be considered an extension of the Burmester problem to spatial mechanisms, as proposed in [2] and extended in [3]. In particular, as proposed in [4], the planar kinematic mapping can be applied to the five-position Burmester problem in order to snthesie a four-bar linkage when five distinct poses of the coupler link are given. This snthesis method gives man solutions of the possible mechanisms, which ensure the rigid motion of the wheel carrier through the assigned seven poses. Nevertheless, the choice of the best mechanism, among those obtained, should be carried out b taking into account their practical feasibilit and the space required on both wheel carrier and chassis. Thus, this paper deals with the optimal dimensional snthesis of a multilink suspension mechanism. The optimiation has been carried out through the Matlab Optimiation Toolbox, while the kinematic performances of the snthesied five-link suspension mechanism have been tested through the results of the position analsis and b means of three-dimensional simulations, which have been obtained through the ADAMS software. A similar kinematic approach was proposed in [5], while the displacement and force analsis of a five-link suspension mechanism is reported in [6]. Finall, two McPherson rear suspension mechanisms are sketched in order to extend this snthesis method to the case of the Magneti Marelli suspension sstem, which is installed on the automobile of tpe Alfa Romeo 156. Some results about the kinematic analsis and snthesis of McPherson suspensions are reported in [7-9], while the brake noise was analed through the ADAMS software in [1]. 2. Kinematic Snthesis Referring to Fig.1, the vehicle chassis and the wheel carrier are sketched through the rigid bodies A and B respectivel. In the case of five-link suspension mechanism, the independent wheel suspension is obtained b connecting B to A through ten spherical joints A i and B i and with five links of lengths l i for i = 1 5. Reference frame F A (O x,, ) is fixed to A and moving frame F B (O x,, ) is attached to B. This mechanism was first introduced b Deimler-Ben on the W21 (Mercedes 19/19E) and W124 (2D/3E) series in order to obtain rear suspension sstems with high performances, as reported in [6]

2 The optimal dimensional snthesis of this one d.o.f. (degree of freedom) five-link suspension mechanism is formulated in terms of link lengths l i and positions A i and B i of the ten spherical joints. The design specifications are aimed to avoid excessive sidewas thrust and minimie the variation of both toe and camber angles during the vertical motion of the wheel carrier with respect to the vehicle chassis. In other words, a rigid motion as close as possible to a vertical translation is required. F A (O x,,) x A 2 O A 3 A 1 l 1 l 2 l 3 l 4 x' B 2 B 1 ' O' B 3 B 4 B5 ' B F B (O x,, ) A 4 l 5 wheel rotation axis A A 5 Fig. 1. Sketch of a five-link suspension mechanism Referring to the real five-link suspension mechanism of the Mercedes 19, the starting position of the moving frame F B is obtained through a pure translation with respect to F A and its origin O = (, 75 mm, 32 mm) is chosen on the wheel rotation axis that coincides with the -axis. Moreover, the starting coordinates of the spherical joints A i and B i, and consequentl the starting link lengths l i for i = 1 5, are reported in Tab.1. Thus, supposing to remove the hpothesis of rigidit of the five links and imposing a pure vertical translation of B along the -axis, the starting link lengths change in order to allow the imposed rigid motion. The optimiation problem is formulated through the following objective function 5 n 2 Ai Ai Ai Bi Bi Bi i 1,...,5 i i i j i 1 j 1 F ( x,,, x,, ) [ l ( A B ) ] (1) where (A i B i ) is the variable distance between the spherical joints A i and B i for i = 1 5 and n is the number of steps between the minimum and maximum vertical displacement of the wheel carrier for j = 1 to n. The variable distances (A i B i ) for i = 1 5 at step j -th are given b (Ai B i ) j = (xaij - x Bij ) +( Aij - Bij ) + (Aij - Bij ) (2) Table 1. Starting coordinates of the spherical joints Joint Coordinates in frame O x,, Link length Joint Coordinates in frame O x,, A 1 12, 25, 231 l 1 = 461 B 1 43, 662, 28 A 2 321, 496, 251 l 2 = 32 B 2 44, 632, 163 A 3 29, 44, 296 l 3 = 272 B 3 141, 666, 269 A 4 29, 437, 42 l 4 = 26 B 4 78, 662, 397 A 5 5, 356, 416 l 5 = 31 B 5 5, 666, 427 Referring to Fig.2, a suitable variation range of the Cartesian coordinates of each spherical joint A i and B i is assumed during the optimiation procedure in order to minimie the objective function and obtain a practical and feasible five-link suspension mechanism. These variation ranges for the i-th link A i B i are sketched in Fig.2 through two parallelepipeds with edges Δx Ai, Δ Ai, Δ Ai and Δx Bi, Δ Bi, Δ Bi around the starting positions of the spherical joints A i and B i respectivel

3 x Bi F A (O x,,) Ai x Ai (A i B i ) B i F B (O x,, ) A Ai x O A i Bi Bi O x B Fig. 2. Sketch showing the variation ranges of the coordinates of A i and B i The algorithm for the kinematic snthesis of the five-link rear suspension mechanism has been implemented in a Matlab program and the optimiation of the objective function in thirt variables has been carried out b using the Optimiation Matlab Toolbox. The numerical results of the optimiation procedure are reported in Tab.2 and the final link lengths are calculated consequentl. Of course, the overall sie of the snthesied mechanism depends b the total vertical displacement between the minimum and maximum -coordinates that are imposed at the wheel carrier with respect to the vehicle chassis, but the suspension mechanism has to be congruent and compatible with the vehicle design in order to occup the minimum volume into the passengers and luggage accommodation. Table 2. Coordinates of the spherical joints after the optimiation Joint Coordinates in frame O x,, Link length Joint Coordinates in frame O x,, A 1 87., 191., 238. l 1 = B 1 44., 671., 212. A , 482.4, 236. l 2 = B , 642.6, 173. A 3 21., 42., 289. l 3 = B , 673., 259. A , 422., 411. l 4 = B 4 79.,658., 388. A 5 3.4, 345., 431. l 5 = B 5 3.4, 637., Position Analsis A position analsis of the snthesied five-link rear suspension mechanism is formulated in order to test the kinematic performances in terms of the required design specifications, which should give a pure translation of the wheel carrier along the -axis. Thus, the position of each moving spherical joint B i for i = 1 5 with respect to the fixed frame F A can be expressed as the sum of the rotation and translation terms, in the form x B i x B i x O ' B i = R B i + O ' B i B i O ' (3) where the rotation matrix R is given b R c c s c s c s c s s c c c s s s c s s c s c s c s s s c c (4) where, and are the rotation angles around the x, and axes respectivel. Figure 3 shows the displacements of the wheel carrier with respect to the vehicle chassis for a given displacement Δ O of the origin O of F B, where the minimum and maximum values are ±1 mm. In particular, the displacements along the x and axes are diagrammed versus Δ O in Figs.3a and 3b, while the toe and camber angles are diagrammed versus Δ O in Figs.3c and 3d. Toe and camber angles represent the rotations of the wheel carrier around the -axis and x -axis respectivel

4 Z O' Z O' Z O' Z O' x O' O' Camber [deg] Toe [deg] a) b) c) d) Fig. 3. Displacements of the wheel carrier with respect to the vehicle chassis for a given displacements Δ O of the origin O of the moving frame : a) Δx O displacement; b) Δ O displacement; c) camber rotation around the x-axis; d) toe rotation around the -axis. All diagrams of Fig.3 pass through the point (, ) because corresponding to the starting position of the wheel carrier, while the maximum deviation of the curves with respect to the required design specifications are obtained in correspondence of the maximum -displacement. In particular, the maximum displacements are Δx O = 48 mm and Δ O = 36 mm, which are obtained for Δ O = 1 mm, while ΔC and ΔT are less than 4 and 2 deg respectivel. The toe angle changes also its versus of rotation. A three-dimensional sketch of the numerical results of Fig.3 are shown in Fig.4, where the starting position of the moving frame F B (O x,, ) and both final positions (O 1 x 1, 1, 1 ) and (O 2 x 2, 2, 2 ) of F B are referred to the fixed frame F A (O x,, ). Camber and toe angles are also indicated. Therefore, the kinematic performances of the snthesied five-link rear suspension mechanism can be considered good b taking into account the imposed limitations in terms of variation range of the Cartesian coordinates of the spherical joints, which also means overall occupied volume. F A (O x,,) x O ' 1 ΔT 1 ' 1 x' 1 ' O' 1 1 ΔC 1 ' ' 1 x' 1 O' x' ' ' 2 F B (O x,, ) ' 2 ΔT 2 ΔC 2 O'2 x' 2 '2 '2 x' 2 Fig. 4. Displacements of the moving frame F B (O x,, ) with respect to F A (O x,, ). 4. Simulations In order to better understand and test the kinematic performances of the snthesied five-link rear suspension mechanism, suitable three-dimensional simulations have been carried out b using the ADAMS software. Figure 5 shows the main configurations of the suspension mechanism b showing also the full 58 58

5 wheel and the spring-damper device, which is alwas included in the suspension sstem in order to report the wheel carrier in the starting configuration and damp the relative motion. In particular, Figs.5a, 5b and 5c show the -view, Fig.5d, 5e and 5f show the x-view and Fig.5g, 5h and 5i show the x-view of the overall suspension sstem. Moreover, Figs.5a, 5d and 5g are referred to the starting configuration, Figs.5b, 5e and 5h are referred to the lowest position and Figs.5c, 5f and 5i to the highest position of the wheel along the -axis. a) b) c) d) e) f) g) h) i) Fig. 5. ADAMS simulation of the snthesied five-link suspension mechanism. Front view: a), b) and c); Top view: d), e) and f); Lateral view: g), h) and i). Starting configuration: a), d) and g). Minimum vertical displacement: b), e) and h). Maximum vertical displacement: c), f ) and i ). 5. McPHERSON Rear Suspension Figure 6a shows the sketch of a conventional McPherson rear suspension mechanism, while Figs.6b shows the kinematic sketch of the modified McPherson suspension mechanism and Fig.6c shows a view of the correspondent real suspension sstem. In particular, this mechanism is composed b the wheel carrier

6 (1), which is connected to the vehicle chassis through the strut link (2) and the SS links (3), (4) and (5) a) b) c) Fig. 6. McPherson rear suspension: a) conventional kinematic sketch; b) kinematic sketch of the modified structure; c) left-hand side of the suspension sstem of the Alfa Romeo Conclusions The optimal dimensional snthesis of a five-link rear suspension mechanism has been carried out through the formulation of a suitable algorithm, which has been implemented in a Matlab program in order to obtain the optimal mechanism according to the imposed design specifications. The Matlab Optimiation Toolbox has been used to minimie the objective function. The snthesied suspension mechanism has been tested b analing the results of the position analsis and through three-dimensional simulations, which have been carried out with the ADAMS software. Finall, two McPherson rear suspension mechanisms are sketched in order to extend this snthesis method to the case of the Magneti Marelli suspension sstem of the Alfa Romeo 156. Acknowledgments: The authors wish to thank the COFAP Automotive Suspension of the Magneti Marelli of Turin to have kindl given the rear suspension sstem of the Alfa Romeo 156 for research purposes. References [1] Oaki A., Basic stud of vehicle roll motion and possibilit of inward roll: examination b a mechanical model of rigid axle suspension, JSAE Societ of Automotive Engineers of Japan, 23, 22, pp [2] Innocenti C., Polnomial solution of the spatial Burmester problem, ASME Journal of Mechanical Design, 117 (1), pp , [3] Liao Q., McCarth J.M., On the seven position snthesis of a 5-SS platform linkage, ASME Journal of Mechanical Design, 123 (1), pp , 21. [4] Haes M.J.D., Zsombor-Murra P.J., Solving the Burmester problem using kinematic mapping, ASME Design Engineering Technical Conferences, Montreal, Canada, 22, paper: DETC22/MECH [5] Simionescu P.A., Beale D., Snthesis and analsis of the five-link rear suspension sstem used in automobiles, Mechanism and Machine Theor, 37, pp , 22. [6] Knapck J., Dierek S., Displacement and force analsis of five-rod suspension with flexible joints, ASME Journal of Mechanical Design, 117 (4), pp , [7] Kang H.Y., Suh C.H., Snthesis and analsis of spherical-clindrical (SC) link in the McPherson strut suspension mechanism, ASME Journal of Mechanical Design, 116 (2), pp , [8] Mántaras D.A., Luque P., Vera C., Development and validation of a three-dimensional kinematic model for the McPherson steering and suspension mechanisms, Mechanism and Machine Theor, 39, pp , 24. [9] Makita M., An application of suspension kinematics for intermediate level vehicle handling simulation, JSAE Societ of Automotive Engineers of Japan, 2, 1999, pp [1] Donle M., Riesland D., Brake Groan simulation for a McPherson Strut tpe suspension, SAE Noise & Vibration Conference and Exhibition, MI (U.S.A.), 23, paper: