DYNAMIC BEHAVIOUR AND OPTIMISATION OF FRAMES FOR ROAD AND MOUNTAIN BIKES C. FERRARESI, L. GARIBALDI, D. PEROCCHIO, B.A.D. PIOMBO Dipartimento di Meccanica Politecnico di Torino Corso Duca degli Abruzzi, 24 10129 Torino, Italy ferraresi@polito.it ABSTRACT. The frames of a road bicycle and a mountain bike for downhill competitions have been analysed from the point of view of their structural dynamic behaviour. First, the numerical results, obtained by modal analysis performed on finite element models, were compared to experimental data and a good correspondence was found The analysed frames were then optimised from the structural point of view in order to minimise their mass without lowering their capacity to be resistant to mechanical stresses. Moreover, the influence of the geometrical design of the mountain bike frame on its mode shapes has been studied. Finally, the fatigue behaviour of the optimised model obtained has been analysed. INTRODUCTION This work presents the dynamic analysis of a few different bicycle frames, provided for riding on different paths, i.e. asphalt roads, mountains or downhill competitions (see figures la, lb). As it is well known, the frames designed for these purposes hold different characteristics. This is mainly due to the different needs required in their use: the road frame, for example, must be as lighter as possible but the stresses which are subjected to are often weak; as a counterpart, the frames designed for other applications as the mountain bike and the downhill bicycles can be heavier, providing an excellent stiffness and manoeuvrability are guaranteed [1]. These latest characteristics are usually met by adopting oversized frames, special designs, or by taking advantage from high tech materials as carbon fibres, titanium or aluminium alloys. Incidentally, it must be noted that, with the actual technology, these frames are provided with a suspension system on the front wheel, or with both the wheels suspended, especially for the case of downhill frames. Another relevant aspect concerning the off roads frames is the level and the quality of the excitation due to the wheel-terrain contact. For the case of a road bicycle, in fact, the excitation has a spectrum (depending upon the roughness of the road) which has frequency contents and amplitudes considerably different from those coming from the off road cases. The latest, very often give origin to violent vibrations of the frame, with a consequent lost of manoeuvrability and riding precision; this is why the constructors tend to design frames very stiff and, possibly, with a high fundamental resonant frequency. As already said, this is usually achieved by increasing the beams dimensions or their thickness, keeping their weight as lighter as possible. 387
In this the paper, seven different commercially available frames are considered for comparing these 'quality' parameters, and a structural modification approach (optimisation) is carried on for a single interesting case by means of a FEM routine. The results obtained are encouraging and effective to improve its behaviour by increasing the first natural frequency and lowering the weight by one third. FINITE ELEMENT MODEL DESIGN As a first step for the dynamic analysis, a road bike (figure 2a) and a downhill mountain bike (figure 2b) frames have been modelled using a Finite Element code Ansys 5.0 [2], based on two real frames available on the market. The mountain bike frame is provided with a rear suspension system usually defined as "rising rate type". As regards the road bike frame, shell type elements have been considered, due to the high tube slenderness, whereas for modelling the two dropouts, solid elements have been preferred. The overall number of nodes is 3820 whilst the number of elements is 4900. The mountain bike frame has been also modelled with the same shell elements, excluding the attachments for the rear triangle and the suspension swingarm, which have been modelled with solid elements; in this case, the total number of nodes resulted 3503, with 4617 elements. EXPERIMENTAL MODAL ANALYSIS The experimental modal analysis, to be compared with the FEM dynamic model, was performed on a road frame whose data where already available from a previous experimental test carried on a very similar model. The test was carried on by adopting a simply suspended configuration by means of soft elastics, as to reproduce the classic free-free conditions commonly used in modal analysis. Notwithstanding some differences with the FEM model already described, the agreement between the experimental and the numerical results appeared satisfying, showing a reasonable accordance in terms of eigenvalues (reported in the table 1 hereafter) and their corresponding shapes (figures 3a to 3t). Mode number FEM Experimental Table 1. 1 88.7 86.6 2 107.2 131.7 3 220.0 209.0 4 233.7 214.8 5 300.7 298.7 6 307.6 322.4 Road frame: comparison between FEM and experimental eigenvalues. As a comment on these modes, it should be noted that the first, the third and the fifth modes are given by out of plane flexural motions, which are often responsible for the low manoeuvrability of the bike; the second mode is mainly affecting the rear triangle only, whilst modes fourth and sixth are in plane motions strongly affected by the slenderness of the tubes of the front triangle. In table 2 shown hereafter are reported the six lower frequencies of the mountain bike frame; in this case a more rude approximation of the FEM values with the experimental results can be noted. Mode number Table 2. 1 2 3 4 5 6 FEM 485.0 667.2 686.0 745.7 1023.7 1187.3 Mountain bike between FEM eigenvalues. frame: and Experimental 450.3 533.8 624.0 732.7 774.7 903.0 comparison experimental This is certainly due to the actual differences existing between the numerical frame model and 388
the real one, showing a similar geometry, but belonging to different producers [3]. The mode sequence is anyway respected; modes one, two and four are out of plane modes, whilst modes three, five and six are in plane bending motions. As it can be evinced from the two tables, the mountain bike shows a considerably higher stiffness, which demonstrates fundamental to improve its behaviour on off road terrain. FRAME OPTIMISATION The stress analysis of the latter frame (figure 4), under the same forces as computed from an off road simulations performed by ADAMS [4][5)[6], demonstrated a considerably low level of stresses under operating conditions. With the aim of recreating the on field boundary conditions, the frame was modelled as clamped at the bottom of the steerer tube, and simply pinned in its connection points with the swingarm and the rear triangle. The maximum stress area was found near the previously mentioned points, reaching the value of 0.86E8 N/m 2, ten times lower than the maximum stress compatible with the material adopted for its construction (titanium). These results have suggested a further analysis to achieve a reduction of the frame weight, providing a reasonable stress distribution being preserved. To this target, a routine of design optimisation has been employed, adopting the following sequence of operations: 1) the design is initialised, i.e. the actual main dimensions of tubes and the materials properties are set up, as well as other characteristic distances, and a first attempt analysis is carried on. 2) the maximum value of allowed stress is settled and an object function is defined (in this case the volume, and hence the tube thickness, must be minimised, since the material and the frame geometry have been already chosen) 3) an iterative procedure can be carried on, leading to the best design, with a quasi uniform stress distribution. For the frame analysed, a consistent weight lowering has been obtained, by adopting thinner pipes as reported in the following table (3): Original design Optimised design (jmax [N/m 2 ] 0.863E8 0.207E9 tube thickness [ mm] 1.3 0.8 volume [m 3 ].329E-3.207E-3 mass [kg] 1.467 0.92 Table 3. Mountain bike optimisation: original design and optimised parameters. For the sake of completeness it should be noted that this result has partially lowered all the eigenfrequencies by less than 1 0%, reducing the first one from 485.0 Hz to 4 51.5 Hz. BEST SHAPE FOR A DOWNHILL FRAME The existence of an optimal design for a downhill frame is evaluated in this paragraph, by comparing different frame geometry of downhill bicycles available on the market. A design optimisation seeking for a solution which takes into account a simply weight reduction of the tubes can be unsatisfactory: in fact, a complete re-design of the frame, particularly regarding his own geometry, can be the only right solution to accomplish the stiffness requirements and the weight reduction. A number of frame geometry is then considered, by assuming some simplifications to limit the computer time consuming. For these cases, only the beam and pipe elements have been adopted, giving a first approximation of frequencies and dynamic mode shapes. Due to the roughness of the models, the optimisation cannot be carried out 389
in a reasonable manner on these models, which are conversely useful to describe the mode sequence and to evaluate the related eigenvalues. The latter being a consistent parameter to describe the comfort and the handling qualities of the bicycle. Two main categories of frames can be distinguished: the first being made by welded pipes with circular cross section, the second adopting at least one or two main tubes with rectangular cross sections. All the geometries considered are supposed to be designed by using the titanium as material for the frame. The table 4 reports, per each frame geometry as shown in figures 5 (a to g), the mass, the first mode frequency and a parameter, said "Quality Factor (QF)", somehow indicating the ratio between the manoeuvrability and the mass per each frame. Frame First frequency [Hz] Mass [kg] "QF" A 426.3 1.59 268 B 445.3 1.20 371 c 467.0 1.54 303 D 406.6 1.12 363 E 354.8 1.10 323 F 328.3 0.98 335 G 367.6 1.05 350 Table 4. CONCLUSIONS Mass and first eigenfrequency for seven different geometries on the market. A few typical concerns of the bicycles users are met in this paper, which is focused on the dynamics of different type of bicycle frames, for road and off road riding. One of the most important parameter emerging from the analysis done is the stiffness of the frame, which is by far higher for the mountain bikes, such as to give a first natural frequency 4-5 times the corresponding for the road frame. This has demonstrated very important in terms of manoeuvrability, precision and riding comfort for these kind of bicycles, the downhill bikes, which are subject to severe excitations transmitted by the terrain. Obviously, a right compromise between the weight and the stiffness of the frame is the challenge to produce a successful bicycle, but different designs can improve and stiffen the frame better than other ones. A first analysis has been done trying to reduce the frame weight, contemporary maintaining other characteristics as the maximum stress admitted and its main dimensions, giving a 35% reduction in weight with an extremely limited reduction of eigenfrequencies. The optimal design aspect, which can be reached by improving the geometry, has been highlighted by comparing seven different frames currently available on the market: the results demonstrate how these modifications can improve the global performances of the bicycle, which is becoming, over the times, a more and more sophisticated product of the technology. BIBLIOGRAPHY [1] Catania D., "Arnica bicicletta", Edizioni Gruppo Abele, Torino,Italy, 1995 [2] ANSYS- User's Manual for Revision 5.0. [3] Filaferro F., "Analisi numerica e sperimentale della dinamica di biciclette biammortizzate", Tesi di Laurea, Politecnico di Torino, Italy, 1995. [4] Petrone N., Tessari A., Tovo R., "Acquisition and analysis of service load histories in mountain-bike", International Conference on Material Engineering, Gallipoli (Leece), Italy, 4-7 Sept. 1996. [5] Vinassa M, "Studio dinamico con tecniche multi body di biciclette dotate di 390
sospensioni", Tesi di Laurea, Politecnico di Torino, Italy, 1996. [6) Catelani D., "ADAMS: programma per la simulazione dinamica dei sistemi multicorpo", M.D.I., Italia. Figure 1 a. Example of road bicycle Figure 1 b. Example of downhill bicycle Figure 2a. FE model ofthe road frame. Figure 2b. FE model ofthe downhill frame. 391
Figure 3a. Mode 1: 88.7 Hz Figure 3b. Mode 2: 107.2 Hz Figure 3c. Mode 3: 220.0 Hz Figure 3d. Mode 4: 233.7 Hz Figure 3e. Mode 5: 300.7 Hz Figure 3f. Mode 6: 307.6 Hz 392
Figure 4. The optimised downhill frame Figure 5a. Model A geometry Figure 5(b,c,d). Model B geometry Model C geometry Model D geometry Figure 5( e,f,g). Model E geometry Model F geometry Model G geometry 393