THE INFLUENCE OF FORCED STEERING VIBRATIONS ON A WHEEL AND DYNAMIC EFFECT OF A WHEEL WITH ABS BRAKING ON UNDULATED ROAD I

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1 The 3rd International Conference on Coputational Mechanics and Virtual Engineering COMEC OCTOBER 009, Brasov, Roania THE INFLUENCE OF FORCED STEERING VIBRATIONS ON A WHEEL AND DYNAMIC EFFECT OF A WHEEL WITH ABS BRAKING ON UNDULATED ROAD I Vasii Marian¹; Scutaru Maria Luinita ; Vlase Sorin Renault Technologie Rouanie, Departent Prestations Clients, arian.vasii-reneter@renaults.co Transilvania University of Brasov-Roania, Departent of Mechanics, luinitascutaru@yahoo.co, svlase@yahoo.co Abstract: A steering/suspension syste of an autoobile ehibits a rather cople configuration and possesses any degrees of freedo. A siplification is necessary to conduct a sensible analysis to gain insight into its general dynaic behavior and into the influence of iportant paraeters of the syste. Investigation of the steering ode of vibration requires at least the steering degree of freedo of the front wheel, possibly etended with the rotation degree of freedo of the steering wheel. For the sake of siplicity, one degree of freedo ay be suppressed by holding the steering spring claped at the node of the natural ode of vibration. Keyword: vibration, degrees of freedo, otion, wheel, brake. INTRODUCTION We will consider the influence of two ore degrees of freedo: the vertical ale otion and the longitudinal deflection of the suspension with respect to the steadily oving vehicle ass. The picture of Fig. shows the lay-out of syste. Due to the assued orthogonally of the syste (wheel ais, king-pin, road plane) the dynaically coupled horiontal otions ( ) are not coupled with the vertical ale otion when sall displaceents are considered and tyre contact forces are disregarded. First, we will eaine the dynaics of the free syste not touching the road. After that, the tyre is loaded and tyre transient odels for the in-plane and out-of-plane behavior are introduced and the syste response to wheel unbalance will be assessed and discussed. Figure : Configuration of a siple steering/suspension syste and the resulting natural frequencies and vibration odes. 807

2 . DYNAMICS OF THE UNLOADED SYSTEM EXCITED BY WHEEL UNBALANCE The siple syste depicted in Fig. possesses two horiontal degrees of freedo: the rotation about the vertical steering ais,, and the fore and aft suspension deflection,. The figure provides details about the geoetry, stiffnesses, daping and inertia. The ass represents the total ass of the horiontally oving parts. The length i denotes the radius of inertia: i = i /. The wheel ri that revolves with a speed Ω is provided with an unbalance ass un (in the wheel centre plane at a radius r un ). The centrifugal force has a coponent in forward direction: Fun, = unrunω sinωt () The equations of otion of this fourth-order syste read: & b && + k + & + c = F () un, ( i + b ) && b && + k & + c = lf un, & (3) With daping disregarded, the agnitude of the frequency response function becoes: Ω Ω { ( ) } & 0 = un run c Ω Ω { ( ) }{ ( ) } in which we have the ero frequency: 0 l l b = (5) and the two natural frequencies:, = 4β (+ β ){( + ) ± ( ) + } (6) + β ( ) where we have introduced the 'uncoupled' natural frequencies: = c c and = (7) ( i + b ) and the coupling factor: b β = (8) i (4) For the analysis, we are interested in the influence of the fore and aft copliance of the suspension. In the righthand diagra of Fig. the two natural frequencies have been plotted as a function of the longitudinal natural frequency ratio (squared), which is proportional to the longitudinal stiffness c, together with the constant vertical natural frequency and the ero frequency for three different values of f/b. In addition, the location of the two centers of rotation according to the two odes of the undraped vibration have been indicated. The two steer natural frequencies and increase with increasing longitudinal suspension stiffness. The lower natural frequency with a centre of rotation located at the inside of the king-pin, approaches the uncoupled natural frequency. Fro (5) it is seen that a ero does not occur if the unbalance ar length l lies in the range 0<l<b which does not represent a usual configuration. For the noral situation with b>0 the ero frequency line ay cross the second natural frequency curve if l is not too large. If the two frequencies coincide, the second resonance peak of the steer response to unbalance will be suppressed. In that case, the unbalance force line of action passes through the centre of rotation of the higher vibration ode. It ay be noted that the situation with contact between wheel and road can be siply odeled if the wheel is assued to be rigid. The sae equations apply with inertia paraeters adapted according to the altered syste with a point ass attached to the ale in the wheel plane. The point ass has the value l w / r where l w denotes the wheel polar oent of inertia and r the wheel radius. 808

3 3. DYNAMICS OF THE LOADED SYSTEM WITH TIRE PROPERTIES INCLUDED In a ore realistic odel the in-plane and out-of-plane slip, copliance and inertia paraeters should be taken into account. A possible iportant aspect is the interaction between vertical tyre deflection and longitudinal slip which ay cause the appearance of a third resonance peak near the vertical natural frequency of the wheel syste. The longitudinal carcass copliance gives rise to an additional natural frequency around 40 H of the wheel rotating against the foot print Due to daping, originating fro tangential slip of the tyre, a supercritical condition will arise beyond a certain forward velocity. This causes the additional natural frequency to disappear. For the etended syste with road contact, the following coplete set of linear equations apply: && & &&& (9) ( + ζ + ) b F = F un, & ζ & (0) ( + + ) = Fun, &&& + & & () l ( ζ + ) b+ lf M = lfun, l ( Ω& w + r0 + F = 0 () σ k F& + V 0 F = CFkVs (3) σ F& V F = C V (4) a V s y + 0 y Fa sy l r0 ( Ω Ω0) + Ω0 = & & η (5) V sy = V 0 (6) M = ta Fy k & CgyrV F& 0 y (7) V 0 Fun, = unrunω0 sinω0t (8) Fun, = unrunω0 cosω0t (9) V = r (0) 0 0Ω0 Table. Paraeter values of wheel suspension syste and type considered Note that echanical caster has not been considered so that the lateral slip speed is siply epressed by (6). The rolling resistance oent has been neglected in () and the average effective rolling radius has been taken equal to the average ale height or loaded radius r0 (in reality re is usually slightly larger than r0). 809

4 Figure : Steer vibration aplitude due to wheel unbalance as a function of wheel frequency of revolution values of the longitudinal suspension stiffness. =Ω 0 / π n for various In Table the set of paraeter values used in the coputations have been listed. The oent of inertia about the steering ais is denoted with i and equals ( b + i ). The aplitude of the steer angle that occurs as a response to a wheel unbalance ass of 0.kg has been plotted as a function of the wheel speed of revolution in Fig.. To eaine the influence of the longitudinal suspension copliance a series of values of the longitudinal stiffness c has been considered. Figure 3: Variation of the three natural frequencies with longitudinal suspension stiffness at different speeds together with the constant vertical natural frequency. The circles on the horiontal ais ark the stiffness cases of Figure. In figure 3 these values have been indicated by arks on the stiffness ais. Clearly, in agreeent with the variation of the natural frequencies assessed in this figure, the two resonance peaks ove to higher frequencies when the stiffness is raised. A third resonance peak ay show up belonging to the vertical natural frequency. This peak reains at the sae frequency. It is of interest to observe that when the lowest steer natural frequency n coincides with n the interaction between vertical and horiontal otions causes the peak to reach relatively high levels. The ero frequency closely follows the forula (5) of the free syste. At the lowest stiffness the ero frequency n 0 alost coincides with the second natural frequency n and suppresses the second peak. Figure 4: Influence the varies type paraeters on the steer angle aplitude response curve 4 CONCLUSION In Fig.4 the result of aking these paraeters equal to ero has been depicted for the stiffness case c = 3X0 N /. Neglecting the factor η (case ) eaning that the effective rolling radius would not change with load, appears to have a 80 5

5 considerable effect indicating that, with η, the vertical otion does aplify the steering oscillation. Oitting the gyroscopic tyre oent (), while reinstating η, appears, as epected, to effectively decrease the steer daping. This is strengthened by additionally deleting the oent due to tread width (3). Oitting the relaation lengths (4) lowers the peaks, thus reoving the negative daping due to tyre copliance. Deleting, in addition, the side force and the aligning torque (5) raises the peaks again indicating that soe energy is lost through the side slip. Disregarding the horiontal tyre forces altogether (6) brings us back to the (horiontally) free syste. As predicted by the analysis, two sharp resonance peaks arise as well as the dip at the ero frequency. REFERENCES [] Cossalter, V., Da Lio, M., Lot, R., and Fabbri, L. (999): A General Method for the Evaluation of Vehicle Manoeuvrability with Special Ephasis on Motorcycles. Vehicle Syste Dynaics, 3, 999. [] Davis, D.C. (974): A radial-spring terrain-enveloping tire odel. Vehicle Syste Dynaics, 3, 974. [3] Dijks, A. (974): A Multifactor Eaination of Wet Skid Resistance of Car Tires. SAE Paper 7406, 974. [4] Dugoff, H., Fancher, P.S., and Segel, L. (970): An analysis of tire traction properties and their influence on vehicle dynaics perforance. In: Proceedings FISITA Int. Auto. Safety Conference, SAE Paper , 970. [5] Eldik Thiee, H.C.A. van (960): Eperiental and Theoretical Research on Mass- Spring Systes. In: Proceedings of FISITA Congress, The Hague 960, The Netherlands. [6] Fiala, E. (954): Seitenkr~ifte a rollenden Luftreifen. VDI Zeitschrift, 96, 954. [7] Frank, F. (965a): Grundlagen ur Berechnung der Seitenfiihrungskennlinien von Reifen. Kautchuk und Gui, 8, 8, 965. [8] Frank, F. (965b): Theorie des Reifenschrdglaufs. Dissertation, Braunschweig, 965. Freudenstein, G. (96): Luftreifen bei Schr~ig-und Kurvenlauf. Deutsche Krafifahreugforschung und Str. Verk. techn., 5, 96. [9] Frit, W. (977): Federhdrte yon Reifen und Frequengang der Reifenkr~ifte bei periodischer Vertikalbewegung der Felge. Dissertation, Karlsruhe, 977. [0] Fro, H. (94): Kurer Bericht iiber die geschichte der Theorie des Radflatterns. [] Bericht 40 der Lilienthal Gesellschaft, 94; NACA TM 365, 954. [] Gillespie, T.D. (99): Fundaentals of Vehicle Dynaics. SAE, 99. [3] Gipser, M. (987): DNS-Tire, a Dynaical Nonlinear Spatial Tire Model in Vehicle Dynaics. In: Proceedings of the nd Workshop on Road Vehicle Systes and Related Matheatics, ed. Neunert, ISI Torino, 987, Teubner Stuttgart, 989. [4] Gipser, M., Hofer, R., and Lugner, P. (997): Dynaical Tire Forces Response to Road Unevennesses. In: Proceedings of "d Colloquiu on Tyre Models for Vehicle Analysis, eds. F.Btih and H.P.Willueit, Berlin 997, Suppl. Vehicle Syste Dynaics, 7, 996. [5] Gipser, M. (999): Ftire, a New Fast Tire Model for Ride Cofort Siulations. International ADAMS User Conference, Berlin, 999. [6] Goncharenko, V.I., Lobas, L.S., and Nikitina, N.V. (98): Wobble in guide wheels. Soviet Applied Mechanics, 7, 8, 98. [7] Gong, S. (993): A Study fln-plane Dynaics of Tires. Dissertation, TU Delft, 993. [8] Gong, S., Savkoor, A.R., and Pacejka, H.B. (993): The influence of boundary conditions on,the vibration transission properties of tires. SAE Paper 9380,