Optimization of damping in the passive automotive suspension system with using two quarter-car models

IOP Conference Series: Materials Science and Engineering PAPER OPEN ACCESS Optimization of damping in the passive automotive suspension system with using two quarter-car models To cite this article: Z Lozia and P Zdanowicz 2016 IOP Conf. Ser.: Mater. Sci. Eng. 148 012014 Related content - The dynamics of antilock brake systems Mark Denny - Hybrid sliding mode control of semi-active suspension systems Babak Assadsangabi, Mohammad Eghtesad, Farhang Daneshmand et al. - The braking performance of a vehicle antilock brake system featuring an electrorheologicalvalve pressure modulator Seung-Bok Choi, Kum-Gil Sung, Myung- Soo Cho et al. View the article online for updates and enhancements. This content was downloaded from IP address 148.251.232.83 on 18/06/2018 at 07:05

Optimization of damping in the passive automotive suspension system with using two quarter-car models Z Lozia 1,2 and P Zdanowicz 1 1 Warsaw University of Technology, Faculty of Transport, Poland 2 Automotive Industry Institute (PIMOT), Warsaw, Poland E-mail: lozia@wt.pw.edu.pl Abstract. The paper presents the optimization of damping in the passive suspension system of a motor vehicle moving rectilinearly with a constant speed on a road with rough surface of random irregularities, described according to the ISO classification. Two quarter-car 2DoF models, linear and non-linear, were used; in the latter, nonlinearities of spring characteristics of the suspension system and pneumatic tyres, sliding friction in the suspension system, and wheel lift-off were taken into account. The smoothing properties of vehicle tyres were represented in both models. The calculations were carried out for three roads of different quality, with simulating four vehicle speeds. Statistical measures of vertical vehicle body vibrations and of changes in the vertical tyre/road contact force were used as the criteria of system optimization and model comparison. The design suspension displacement limit was also taken into account. The optimum suspension damping coefficient was determined and the impact of undesirable sliding friction in the suspension system on the calculation results was estimated. The results obtained make it possible to evaluate the impact of the structure and complexity of the model used on the results of the optimization. 1. Introduction; review of the literature dealing with quarter-car models and their applications The quarter-car model (figure 2) is a system with two degrees of freedom (2DoF), describing the vertical vibrations of a sprung mass (a part of the vehicle body solid) and an unsprung mass (connected with the road wheel). The two masses are linked by a spring-damper system representing the spring-damping properties of the suspension of the specific wheel. A spring-damper (or springonly) element, situated between the unsprung mass and the ground, represents the road wheel properties in the radial direction. The models of this type appeared in 1970s (e.g. [11, 19, 27, 28]) and were employed in many publications of 1980s (e.g. [10, 15, 20, 30, 31]) and 1990s (e.g. [5, 12, 21]); they are useful even today, e.g. for the works that provide guides for vehicle designers (such as [2, 6, 7, 13, 17, 26, 29, 31, 32, 33, 34, 35]). They are both linear and non-linear, describing the passive, semi-active, and active suspension systems. In some of the works, the model analyses were supplemented with experimental examinations of systems of the quarter-car type [13, 26, 33, 36, 37]. The quarter-car models were used in sophisticated optimization algorithms, where Pareto-optimal solutions were sought [14], with random nature of selected model parameters (sprung mass determined by hardly-predictable vehicle load and tyre stiffness depending on inflation pressure, e.g. [6]) being taken into account, and at the assessment of designs of variable-damping, semi-active, and active suspension systems (e.g. [2, 4, 5, 6, 7, 29, 30, 31, 33, 34, 35]). Most authors of the publications where the optimization of suspension system parameters is addressed have enumerated three major assessment criteria, which are related to the minimization of measures of the driver and passengers ride discomfort and of changes in the normal reaction at the tyre/road contact. A reduction in the range Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1

of working displacements of the suspension system has also been considered [4, 5, 6, 7, 11, 12, 17, 19, 20, 21, 26, 27, 28, 29, 30, 31, 33, 34, 35]. A contradiction between these requirements has been highlighted in [30]: increased damping in the suspension system impairs the comfort but improves the safety. It has been shown in [17] that optimum levels of damping in suspension systems can be determined as a compromise between the two assessment criteria. In most of the works reported in the literature, the excitation was assumed as a Gaussian stationary random process characterized by the power spectral density of road profile height. For the non-linear models, sinusoidal waveforms or random process realizations were applied as the input. The wavelength values ranged from 0.1 m to 100 m [19, 20, 27, 30]. The smoothing properties of vehicle tyres were taken into account [3, 4, 15, 16, 17]. The vehicle speed range was 10-50 m/s [7]. In the publications quoted, frequencies from the 0-80 Hz band were taken for the analyses [2, 19, 20, 22, 28, 33, 34, 35]. This range is also mentioned in the ISO standards applicable to the comfort of vibrations (e.g. [8]). 2. Objective The work was aimed at the optimization of damping in the passive suspension system of a motor vehicle moving rectilinearly with a constant speed on a road with rough surface. The impact of the structure and complexity of the quarter-car model used on the optimization results obtained was assessed within this work. 3. The quarter-car models and their equations of motion Figure 1a shows a linear quarter-car model. The model consists of a sprung mass m 1 and an unsprung mass m 2. The suspension stiffness and the radial stiffness of the pneumatic tyre have been denoted by k 1 and k 2, respectively. The symbols c 1 and c 2 have the meaning of the suspension damping coefficient and the coefficient of radial damping of the tyre, respectively. The symbol ζ represents the time-dependent kinematic excitation (input) from the uneven road surface, measured along the Oζ axis. The vertical displacements of individual model masses from the static equilibrium position for zero road profile height are denoted by z 1 and z 2, respectively. The motor vehicle, as well as the model under consideration, moves rectilinearly with a constant speed V [km/h] (or v [m/s]). Figures 1b and 1c show a non-linear quarter-car model. The symbols m 1, m 2, z 1, and z 2 have the same meanings as they have in the case of the linear model. Symbols ζ 1 and ζ 2 represent the vertical coordinates of the centres of model masses along the Oζ axis, which is fixed to the road. Fs 1 is the suspension spring force, approximated by degree 5 polynomials; Fts 1 is the sliding friction force in the suspension system, represented by a broken line [15, 16, 36]; Ft 1 is the viscous damping force in the suspension system, represented by a broken line [16]; Fs 2 is the radial force of tyre elasticity, approximated by a degree 3 polynomial [15, 16, 36]; Ft 2 is the damping force in the pneumatic tyre, represented by a linear function of the rate of radial deflection of the tyre [1, 10, 11, 12, 15, 16, 20, 27, 33]; g = 9.81 m/s 2 ; ζ is the excitation (input) from the uneven road surface. Figure 1. Linear (a) and non-linear (b, c) quarter-car model (for the notation used, see the text); a, b general structure of the models; c system of forces in the non-linear model. 2

The equations of motion for the linear model have been derived from the principle of dynamic force analysis. Their matrix form is shown in relation (1), where the symbols of the matrices of inertia M, viscous damping C, stiffness K, excitation influences transmitted by damping in the pneumatic tyre C ζ, and excitation influences transmitted by radial stiffness of the pneumatic tyre K ζ have been pointed out. The vectors of generalized coordinates (displacements), velocities, and accelerations have been denoted by q, q&, q&, respectively. This notation is used in the concise form (2) of this relation. (1) M q& + C q& + K q = C ζ ζ& + K ζ & (2) For equation (2), the Laplace transform has been formulated, at zero initial conditions. After transformations, a solution in the form of equation (3) has been obtained, where the domain s = r + i ω has a real part r and an imaginary part ω, while i 2 = 1 (ω [rad/s] is the radian frequency): q s = Hq s ζ s (3) ( ) ( ) ( ) The operational transmittance (transfer function) for displacements H q(s) has the form (4): Hq (s) 1 q(s) 2 1 H q( s) = = = ( M s + C s + K) ( Cζ s + Kζ ) (4) Hq (s) ζ( s) 2 We can easily pass from the Laplace transform to the Fourier transform. The operational transmittances will then become spectral transmittances. In formal terms, this is expressed in passing from domain s to argument i ω, by assuming the real part r of the expression s = r + i ω as zero. With such a substitution, the relations (3) and (4) still hold. The equations of motion for the non-linear model (figures 1b and 1c) have also been derived with the use of the principle of dynamic force analysis. They have the form presented by equations (5): Fs1+ Fts1+ Ft1 && z1 = g m1 (5) Fs Fts Ft + Fs + Ft && 1 1 1 2 2 z2 = g m2 Their solutions are obtained in the time domain with the use of approximation techniques, by numerical integration. The results presented herein have been obtained with the use of an authorial program built in the Matlab Simulink environment [36]. 4. Qualitative differences between the linear and non-linear quarter-car models The differences between the two models arise, above all, from the form of the functions describing the forces in the suspension system and pneumatic tyre. The description of the suspension spring force in the non-linear model reflects the impact of suspension travel limiters in the rebound and compression phase (beginning and end of the characteristic curve), noticeable in experimental tests [15, 16, 36]. In the non-linear model, the sliding friction force is taken into account (while it is ignored in the linear model) and the asymmetry of shock absorber characteristics can be represented. The approximation of changes in the force of tyre elasticity by a degree 3 polynomial [15, 16, 36] can reflect the nonlinearities of this force at small and large tyre deflections and the possibility of wheel lift-off. 5. Random excitation from an uneven road surface An assumption has been made here that the road surface is undeformable and its vertical irregularities are a realization of a stationary Gaussian random process. In the relevant ISO standard [9], road surfaces have been classified from A to H, i.e. from very good (A) through good (B), average (C), and ζ 3

poor (D) to very poor (E and further F, G, and H). The road is described in the ISO standard [9] by the function of power spectral density (PSD) S d(ω) [m 3 /rad] of one track parallel to the road centreline: S d(ω) = S d(ω 0) (Ω/Ω 0) -w (6) where: Ω = 2π/L [rad/m] is angular frequency of the longitudinal road profile; L [m] is road roughness wavelength; Ω 0 [1/m] is reference angular frequency (in most cases, Ω 0 = 1.0); S d(ω 0) [m 3 /rad] is road roughness indicator, defining the general road surface condition (whether it is good or poor); w [ ] is road waviness indicator, informing whether long or short waves predominate in the road profile. Roads of different classes differ from each other in the S d(ω 0) values. The exponent w has a constant value of w = 2. The road roughness wavelength L values have been assumed as 0.1 100 m. This is the micro-profile of road surface irregularities [10, 19, 20]. Based on the known function S d(ω) and with the use of subprograms (taken from publication [10]), where the FFT (fast Fourier transform) techniques were employed, a discrete realization of the road profile was generated, which was then approximated by cubic splines [15, 16]. Such a form enables the use of realizations with high wavelength values, exceeding the maximum micro-profile wavelength (100 m), thanks to the periodical form of the approximating function, which has been described in publications [15, 16]. In the pneumatic tyre model, the point contact model is normally used for the radial direction of vibrations [3, 4, 15, 16]. Therefore, the road roughness wavelength that is taken into account must be bounded from below for unnatural overestimation of the damping forces in the tyre model to be avoided. To overcome this difficulty, the smoothing properties of vehicle tyres may be simulated by adopting a fixed footprint tyre model [3, 4, 15, 17] (with averaging the road profile height over the tyre/road contact patch, whose length is 2 l op) and by filtering the road roughness spectra. To do this, a filter was used with the absolute value of its transmittance being as specified in [15]. The inputs thus filtered make it possible to employ the one-point tyre/road contact model [3, 4, 15, 17]. 6. Automotive suspension damping selection criteria (optimization criteria) 6.1. Basic optimization criteria As in publications [2, 4, 5, 6, 7, 11, 12, 17, 19, 20, 21, 27, 28, 29, 30, 31, 32, 33, 34, 35], three criteria have been adopted to assess the correctness of selection of suspension damping coefficient c 1: minimization of the measure of vehicle occupants discomfort, i.e. the standard deviation of sprung mass acceleration, σ a [m/s 2 ]; minimization of the safety hazard, measured by the standard deviation of the vertical component of the normal reaction at the tyre/road contact, σ F [N]; reduction in the working displacements (range of changes in the deflection) of the suspension system to a value lower than the suspension displacement limit r zg [m]. They may be described as functions of suspension damping coefficient c 1 and vehicle speed V: σ a(c 1, V) minimum σ F(c 1, V) minimum 6 σ uz(c 1, V) r zg (7) where: σ a(c 1, V) and σ F(c 1, V) [m/s 2 ] are the objective functions subject to minimization and σ uz(c 1, V) [m/s 2 ] is the standard deviation of suspension deflection; r zg [m] is the suspension displacement limit. The above criteria were adopted for each of all the road surface quality classes taken into consideration. Factor 6 in formula (7) comes from the Gaussian distribution of the suspension deflection, but for the linear model. For such a model, the standard deviations σ a, σ F, and σ uz were calculated from the known formulas [1, 10, 11, 12, 17, 18, 19, 20, 22, 23, 24, 25, 27, 35] as definite integrals of power spectral densities of sprung mass accelerations, vertical component of the normal road reaction, and suspension deflection. These densities were calculated as products of the power spectral density of the excitation (road profile height) and squared absolute values of the transmittances of the quantities mentioned above [1, 10, 11, 12, 17, 18, 19, 20, 22, 23, 24, 25, 27, 35]. In the case of the non-linear model, σ a, σ F, and σ uz were calculated from time histories of the vertical acceleration of the sprung mass, & z 1 (c1, V, t), vertical component of the normal road reaction, F(c 1, V, t), and suspension deflection, u 1(c 1, V, t), for the assumed values of parameters c 1 and V. 4

6.2. Modified optimization criteria The criterion that is easiest to use is the one related to reducing the range of working travel of the suspension system. The other two criteria (see (7)) give rise to more difficulties. Due to different units of measure, it is not easy to formulate a joint synthesized measure of discomfort and safety hazard. A way to solve this problem has been shown in [17]. The proposed modification of the criteria should be started with normalizing the criterial quantities, i.e. σ a and σ F, by dividing each of the values in an individual curve (plotted for a specific vehicle speed V) by the maximum value of the curve within the c 1 domain. The symbol max means here the highest value instead of a formal maximum. σ au(c 1, V) = 100 % σ a(c 1, V)/max {σ a(c 1, V)} (max. relative to c 1, for a specific V value) (8) σ Fu(c 1, V) = 100 % σ F(c 1, V)/max {σ F(c 1, V)} (max. relative to c 1, for a specific V value) (9) The minimums of the curves are better visible than they were before the normalization [17]. The next step in the modification process is to change the reference system, with maintaining the variability range from zero to 100 %. Thus, the comfort indicator WP1 and the safety indicator WP2 are determined [17]. Here, the symbol min has the meaning of a formal minimum. WP1(c 1) = σ au(c 1, V) + min {σ au(c 1, V)} + 100 % (min. relative to c 1, for all V values) (10) WP2(c 1) = σ Fu(c 1, V) + min {σ Fu(c 1, V)} + 100 % (min. relative to c 1, for all V values) (11) The modified criteria of selection of suspension damping coefficient c 1 (i.e. the criteria of optimizing the selection of suspension damping in respect of ride comfort and safety) have the form [17]: Q z(c 1) = w k WP1(c 1) + w b WP2(c 1) maximum (12) 6 σ uz(c 1, V) r zg (13) where the new symbols are defined as follows: Q z(c 1) [%] is the modified objective function subject to maximization; w k [ ], w b [ ] are weighting factors for comfort and safety, from within a range of <0, 1>; WP1 [%],WP2 [%] are comfort and safety indicator, respectively (see relations (10) and (11); r zg [m] is suspension displacement limit. In publication [17], attention has been directed to a change in the terminology used: the terms discomfort and safety hazard have been replaced with the notions comfort indicator WP1 and safety indicator WP2, because of their different monotonicity as functions of c 1. Now, the maximum of the new objective function Q z(c 1) is sought. 7. Calculation data: parameters of the model and of the test conditions The model parameters taken as an example corresponded to the data of the front suspension system of the Isuzu D-max motor vehicle: m 1 = 578 kg, m 2 = 69.5 kg. The coefficients of stiffness for the linear model were k 1 = 42 520 N/m, k 2 = 220 000 N/m. The coefficient of viscous damping of the tyre was c 2 = 150 N s/m and c 1 was a variable in the calculations. The length of the tyre/road contact patch in static conditions was 2 l op = 0.185 m. Three roads were chosen for the calculations, i.e. road of class B (good), with S d (Ω 0) = 0.000004, Ω 0 = 1.0, and w = 2; road of class C (average), with S d(ω 0) = 0.000016, Ω 0 = 1.0, and w = 2; and road of class D (poor), S d(ω 0) = 0.000064, Ω 0 = 1.0, and w = 2 (see [9]). The wavelength values varied between 0.1 m and 100 m. In consideration of the ISO standard provisions that concern the vibration comfort (e.g. [8]), the frequency band of 0-80 Hz (0-502.65 rad/s) was adopted for the analyses. The analyses were carried out for 4 constant vehicle speeds V [km/h], i.e. 30 km/h, 60 km/h, 90 km/h, and 120 km/h. The vehicle speed values expressed as v [m/s] and V [km/h] are connected with each other by the generally known relation v = V/3.6. The damping coefficient c 1 [N s/m] was changed and its values were indirectly expressed by means of a relative damping coefficient [1], i.e. by values γ [ ] defined as follows for the linear model: γ = c 1/c 1kr, where: c 1kr = 2 m 1 ω 01; (14) c 1kr [N s/m] is critical damping coefficient; ω 01 = 2 π f 01 [rad/s] is the first (lower) natural radian frequency of the undamped system; f 01 [Hz] is the first (lower) natural Hertz frequency of the undamped system. For the system under analysis, these values were ω 01 = 7.839 rad/s, f 01 = 1.248 Hz, c 1kr = 9 062 N s/m. The analyses were carried out for 26 values of the relative damping coefficient γ [ ], changed from 0.1 to 0.6 in steps of 0.02. The damping coefficient in the suspension system was: 5

c 1 = γ c 1kr = γ 2 m 1 ω 01 [N s/m] (15) The working displacements of the suspension system were limited to r zg = 0.12 m, corresponding to the real linear range of operation of the suspension system of the vehicle under analysis. 8. Calculation results For the model parameters and characteristics and the test conditions as specified, the quantities taken as the criteria of selection of damping in passive automotive suspension systems (optimization criteria) were calculated in accordance with the algorithm presented in a previous part of this paper. 8.1. Determining the optimum values of the relative suspension damping coefficient γ The optimization criterion related to reduction in the working displacements of the suspension system can be most easily applied. In figure 2, standard deviations of suspension deflection σ uz have been presented as functions of relative damping coefficient γ for the linear and non-linear model, for roads of class B, C, and D, and for vehicle speed values of 30 km/h, 60 km/h, 90 km/h, and 120 km/h. In qualitative terms, the curves are similar to each other, but their values are considerably lower for the non-linear model. For the linear model, the requirements expressed by (7) and (13) were met for the road of class B (good). For the road of class C (average), they were also met, but only for the relative damping coefficient γ higher than 0.159. On the road of class D (poor), the requirements of relations (7) and (13) could only be met when the vehicle speed did not exceed 60 km/h and the relative damping coefficient γ was higher than 0.30. For the non-linear model, the requirements expressed by (7) and (13) were met for the road of class B (good) and C (average) for all the vehicle speeds and the whole range of changes in the relative damping coefficient under analysis. For the requirements to be met on the road of class D (poor), the vehicle speed had to be limited to 60 km/h and the relative damping coefficient γ had to be higher than 0.25. Actually, it is hard to expect that any driver would decide to travel with a high speed on a road being in poor technical condition (of class D). It should be stressed that the nonlinearities of the real vehicle suspension system and tyres could be reproduced in the latter model and this explains the fact that the criterion related to reduction in the working displacements of the suspension system, expressed by relations (7) and (13), could be easier satisfied. Figure 2. Standard deviations of suspension deflection σ uz for the linear model (solid lines) and nonlinear model (dashed lines) vs. relative damping coefficient γ, for roads of class B (good, figure a), C (average, figure b), and D (poor, figure c) according to ISO [9] and for four vehicle speed values. Figure 3 shows standard deviations of sprung mass acceleration σ a and of the vertical component of the normal road reaction σ F as functions of relative damping coefficient γ for the linear and non-linear 6

model, for road of class C (average), and for the four vehicle speed values. The curves show similar monotonicity, but their minimums for both models occurred at different locations. It can be noticed that for the non-linear model, the minimum (i.e. optimum) values are distinctly shifted towards lower γ values. The results obtained for roads of class B and D are similar, in qualitative terms. Figure 3. Standard deviations of sprung mass acceleration σ a (figure a) and of the vertical component of the normal road reaction σ F (figure b) vs. relative damping coefficient γ for the linear model (solid lines) and non-linear model (dashed lines), for road of class C (average) according to ISO [9], and for the four vehicle speed values. In result of the procedure described by relations (8)-(21), the criteria of assessment were modified, i.e. they were normalized and the reference system was changed, with their variability range from zero to 100 % having been maintained. Thus, a modified objective function Q z(c 1) has been created, which describes the quality of tuning the suspension damping, c 1, in respect of comfort and safety of vehicle ride. The independent variable is now the relative damping coefficient γ of the linear model, related to c 1 according to equation (15). The motor vehicle ride comfort and safety may be treated equally (w k = w b) or with different preferences (w k w b). Figure 4 shows modified objective function Q z as a function of relative damping coefficient γ for the linear and non-linear model, for roads of class B, C, and D, for four vehicle speed values, and for two sets of the weighting factors for comfort and safety: set 1 (solid lines in the graph) with w k = w b = 0.5 and set 2 (dashed lines in the graph) with w k = 0.4 and w b = 0.6. Set 2 represents the case that safety is deemed 1.5 times as important as comfort (1.5 = 0.6/0.4). For this set, the optimum values of the relative damping coefficient γ (at the maximums of Q z) are somewhat higher than those for comfort and safety being equally treated (set 1), whether the linear or non-linear model is used. The results obtained from the linear model are similar for different roads in terms of curve shape and monotonicity. They show identical optimum damping values and similar Q z function values for the corresponding damping. For the non-linear model, the results largely depend on the road type. The worse the road (changing from B to C and D), the higher the optimum damping is for both sets of the weighting factors. In the extreme case, the optimum damping coefficient value is even doubled (from 0.12 for road B to 0.24 for road D, at a vehicle speed of 30 km/h). For each of the roads and vehicle speeds considered, the optimum damping determined with the use of the non-linear model is lower than that obtained from the linear model. The better the road, the bigger differences are observed. 8.2. Discussion on the reasons for the differences in the optimum suspension damping values for the non-linear model compared with the linear model The authors have formulated a hypothesis that the shift of the optimum suspension damping value obtained from the nonlinear model in relation to that determined with the use of the linear model chiefly arises from the fact that sliding friction in the suspension system is represented in the nonlinear model. Figure 5 shows results analogous to those presented in figure 4, but obtained without taking into account the sliding friction in the non-linear model. As it can be seen, the optimum values of the suspension damping values came considerably nearer to those obtained from the linear model. An exception was the case where the vehicle moved on road D (poor) with a speed of 120 km/h, but this may only serve information purposes, as a vehicle drive in such conditions is hardly imaginable. 7

The results presented confirm the hypothesis having been formulated and highlight the significant importance of sliding friction for vibration damping and safety measures. Although friction, which is inevitable in the construction of motor vehicle suspension systems, is connected with the known disadvantageous properties of suspension systems [36, 37], it softens the requirements for the necessary level of viscous damping in shock absorbers. Figure 4. Objective function Q z vs. relative damping coefficient γ for the linear model (left) and non-linear model (right), for a road of class B (good, figure a), C (average, figure b), and D (poor, figure c), for four vehicle speed values, and for two sets of the weighting factors for comfort and safety: set 1 (solid lines) with w k = w b = 0.5 and set 2 (dashed lines) with w k = 0.4 and w b = 0.6. Figure 5. Objective function Q z vs. relative damping coefficient γ for the linear model (left) and non-linear model (right), with the sliding friction being disregarded in the latter model. Graphs plotted for road of class B (good, figure a), C (average, figure b), and D (poor, figure c), for four vehicle speed values, and for two sets of the weighting factors for comfort and safety: set 1 (solid lines) with w k = w b = 0.5 and set 2 (dashed lines) with w k = 0.4 and w b = 0.6. 8

8.3. Determining the optimum damping values for the two sets of the weighting factors for comfort and safety The results obtained from the linear model (figure 4) show that the optimum value of the relative damping coefficient γ for set 1 of the weighting factors for comfort and safety is 0.31; for set 2, it is 0.32. Based on equation (15) and the model data, the optimum values of the suspension damping coefficient c 1 may be calculated: for set 1: c 1 = 0.31 c 1kr = 0.31 9 062 N s/m 2 810 N s/m (16) for set 2: c 1 = 0.32 c 1kr = 0.32 9 062 N s/m 2 900 N s/m (17) The difference between the c 1 values is not too big, in spite of significant differences between the weighting factors adopted. The results obtained from the non-linear model, but with disregarding the sliding friction (see figure 5), show that the optimum value of the relative damping coefficient γ for set 1 of the weighting factors for comfort and safety is about 0.29 and for set 2, it is about 0.31 (except for the case with the vehicle moving on road D with a speed of 120 km/h). These values are close to those obtained from the linear model (compare with (16) and (17): the differences are about 6.5 % and 3.1 %, respectively). Therefore, it is reasonable to adopt the results obtained from the linear model and represented by relations (16) and (17). In spite of efforts made to reduce the sliding friction in suspension systems, this phenomenon will always occur in real systems, which will result in stronger suspension damping. The values of the friction impact are not only small; they will also lower the said percentage differences between the calculation results obtained from the non-linear and linear models. 9. Conclusion The optimum values of the suspension damping coefficient determined by means of the non-linear model are markedly lower than those obtained from the linear one. For the road of class B (good) and the vehicle speed being low enough (e.g. 30 km/h), the optimum values of coefficient γ calculated from the more complex model are lower by even more than a half than those obtained with the use of the linear representation (γ falling in the ranges of 0.12-0.14 and 0.31-0.32, respectively). This is chiefly due to the fact that sliding friction, always taking place in suspension systems but not taken into account in the linear approximation, is represented in the non-linear model. The disregarding of this undesirable phenomenon leads to a very important conclusion that the calculation results obtained from the linear and non-linear model are close to each other (γ of about 0.31-0.32 and 0.29-0.31, respectively, if the unrealistic case of the vehicle being driven on the road of class D with a speed of 120 km/h is ignored). The next stage of this work will be dedicated to optimization of the shock absorber asymmetry coefficient. Afterwards, the authors will focus on the verifying calculations, carried out with the use of a three-dimensional vehicle motion model, which most accurately represents the properties of a real motor vehicle. Acknowledgements This work was done within Research Project PBS3/B6/27/2015, sponsored by the National Centre for Research and Development. Part carried out by the team from the Warsaw University of Technology. References [1] Arczyński S 1993 Mechanika Ruchu Samochodu (Warszawa: WNT) [2] Cao D, Song X and Ahmadian M 2011 Editors perspectives: road vehicle suspension design, dynamics, and control Vehicle System Dynamics 49 (1-2) pp 3-28 [3] Captain K M, Boghani A B and Wormley D N 1979 Analytical tire models for dynamic vehicle simulation Vehicle System Dynamics 8 pp 1 32 [4] Crolla D A 1996 Vehicle dynamics theory into practice J. of Autom. Eng. 210 pp 83 94 [5] Firth G R June 1991 The Performance of Vehicle Suspensions Fitted with Controllable Dampers (The University of Leeds, Department of Mechanical Engineering, Ph.D. thesis) [6] Gobbi M, Levi F and Mastinu G 2006 Multi-objective stochastic optimization of the suspension system of road vehicles J. of Sound and Vibration 298 pp 1055 72 9

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