Optimization of Damping in a Semi-Active Car Suspension System with Various Locations of Masses

Abstract

The key request for a vehicle suspension system is vibration control and decreasing the actual inertia forces. This ensures ride comfort for the crew and influences the fatigue level of the driver and overall driving safety. Implementing semi-active damping control in the vehicle suspension allows for adjusting the damping process in the vehicle for minimum acceleration applied to the seats, driver, and passengers. In order to implement theoretical analysis, we used a mathematical full-car model in Simulink/MATLAB. As the load, we added simulations of various artificially generated road profiles. The damping coefficient of the semi-active suspension system was optimized for maximum comfort level for a driver only. Results from the full-car simulation process deliver a graph of the output accelerations showing kinematic excitation from road deformities under various locations of vehicle load positions.

1. Introduction

The wheels are connected to the vehicle through the suspension system. In real vehicle suspensions, the relative movement of the wheels in relation to the vehicle body is used for vibration control and for the transmission of force and torque between the wheels and body. Vibration control helps hold good and uninterrupted contact of the tires with the road and ensures driving safety and ride comfort regardless of the quality of the road surface [1]. Depending on vibration control, the suspension types are passive, semi-active, and fully active. Semi-active suspension can smoothly change the damping coefficient and can be nearly as effective as a fully-active suspension for increasing ride quality [2]. It combines the advantages of both the other suspension types. Various systems are discussed in reference [3] noting their strengths, weaknesses, and relative performance or equipment requirements. The semi-active suspension system becomes a suitable choice due to its small power consumption, low expense, and fewer parameters [4]. The main element of semi-active suspensions is a variable damper of a few possible types: the magnetorheological damper, the electrorheological damper, the electrohydraulic damper, and the electromagnetic damper [5,6].

Road conditions often alter during driving. Different emphasis is placed on ride comfort and handling stability under different road conditions or driving styles [7,8,9]. The properties of the suspension should also change depending on the road conditions [10,11,12]. When ride comfort is the priority, a semi-active suspension system using a variable damper can have an effectiveness close to that of an active suspension [3,13,14,15,16]. However, the variable damper in a semi-active suspension system can only change the rate of dissipating energy without receiving energy, and it has major restrictions when one requires controlling the height and attitude movement of a vehicle. The semi-active suspension system with variable stiffness can solve this problem [17,18,19]. Meanwhile, some authors use both a variable damper and stiffness [20,21,22,23]. Therefore, semi-active suspension systems are very helpful in enhancing ride comfort and vehicle handling capabilities. One of the main problems is to determine the control signal for the semi-active suspension. Various control strategies [6,24,25,26] have been developed in which the control signal is determined by the measurement of different parameters. Real-time control strategies such as skyhooks or groundhooks are common for semi-active vibration damping based on road statistical properties, i.e., an optimal damping ratio control [26]. In addition, a preview strategy was implemented to improve the behavior of semi-active systems. Therefore, the system needs information about road conditions.

At present, researchers have carried out many studies of models and methods of road recognition. Some of them are difficult to realize or are very expensive. Various methods and equipment for measurement of the road properties are briefly discussed in [27,28,29]. Different, inexpensive response-based road profile identification methods are reviewed and presented in [30].

Several vehicle models with semi-active suspension systems have been commercially developed in recent years [6]. The usage of road profile information before a wheel hits road unevenness, investigated in [13,31,32,33,34,35], has been applied to mass production cars as well. The early attempts were made by Nissan in 1990 using Super Sonic Suspension [36] and, recently, the Magic Body Control feature [37] used in Mercedes Benz. H. Eric Tseng and Davor Hrovat in [38] state, “the next advances in active and semi-active suspension design will mainly come from two thrust areas. The first is the increased efficiency in actuator design and implementation such as the usage of systematic control software algorithm design combined with the previously mentioned innovative hardware design measures such as inclusion of possibly variable in-series/parallel compliances and fast load levelling. The second is more comprehensive usage of preview information from camera, Global Positioning System (GPS), and electronic horizon such as vehicle-to-vehicle communication, vehicle-to-infrastructure communication, vehicle localization and real-time accessing of cloud information, and crowd sourcing leading to up-to-date road profile maps”. So, there are still many areas for investigation [23,39,40,41,42,43].

We planned to keep in check our developed (semi-)active suspension [44] using information acquired from a segment of traveled road profile [45,46]. The central vehicle safety block would receive the road characteristics and set damper damping values to minimize the vibration of the vehicle body. The needed step is to relate the obtained road characteristics to the required optimal damping values. For that purpose, we needed a gridded matrix of damping coefficients at certain fixed values of speed and road characteristics. We found only a few papers with the results needed for this purpose; only a few points needed for the whole matrix can be found in them, and they used different car models, road types and characteristics, and optimization methods (see

Table 1. Simulation of road profile impact to the vehicle body.

Road Type	Car Model	Remarks	Refs.
Asphalt, concrete, rough	Quarter-car	Modified skyhook control	[47]
Sinusoidal and random road	Full-car (7DOF)	LIDAR and diagonal recurrent neural network control model	[23]
ISO class B, C, D	Quarter-car	Optimized damping using three criteria or modified criteria; various speeds	[48,49]
Good and poor quality	Quarter-car	Optimized damping and stiffness	[50]
Random (max elevation 50 mm)	Full-car (12DOF)	Optimized damping and stiffness; separated engine	[21]
Random (standard deviation 10 mm)	Quarter-car	Genetic algorithm	[51]
Main road profile	3D model with driver	Optimized damping and stiffness for passive, semi-active, and active suspensions	[52]
Bump 50 mm	Half-car	C-K figures; genetic algorithm; various control methods	[53]

).

This paper provides a relationship between the detected w₁, w₂ waviness values for a wide range of road profiles and the optimized damping coefficient for a full-car model (12DOF) traveling them at various speeds. To minimize the number of possible choices, here we focus on instances when the damping coefficient was optimized for the maximum level of driver comfort. In addition, there are various locations of masses, and we used the same value of damping for all four wheels. This idea is similar to the developed method of minimizing the acceleration at a selected point on the vehicle body [54].

2. Methods

The random, artificial road profile can be generated through various methods, based on: (1) linear filtering, (2) superposition of harmonics, or (3) inverse fast Fourier transformation of discretized power spectra density (PSD) [55]. One of the most commonly implemented [56,57,58,59,60,61], the superposition of harmonics (or sinusoidal approximation), we also used. The longitudinal road profiles were generated using the original MATLAB code, implementing the method of superposition of harmonics in the spatial domain according to Equations (1) and (2):

$$z(x)={\displaystyle \sum _{i=1}^{N=\left({\Omega }_{\mathrm{U}}-{\Omega }_{\mathrm{L}}\right)/\Delta \Omega }{Z}_i\cos \left({\Omega }_{i}x+{\phi }_i\right)}\mathrm{with}$$

(1)

$$Z_i=\sqrt{2G_{\mathrm{d}}({\Omega }_i)\Delta \Omega }$$

(2)

where $N=\left({\Omega }_{\mathrm{U}}-{\Omega }_{\mathrm{L}}\right)/\Delta \Omega$—number of harmonic samples (in our case, N = 1000); ${\Omega }_{\mathrm{U}},{\Omega }_{\mathrm{L}}$—upper and lower angular spatial frequencies [rad/m] in the PSD spectrum; $\Delta \Omega$—the width of each frequency band; $Z_i$—amplitude of the ith harmonic; ${\Omega }_i={\Omega }_{\mathrm{L}}+\left(i-1\right)\Delta \Omega$—angular spatial frequency of the ith harmonic; ${\phi }_i$—phase angle of the ith harmonic uniformly distributed in the range from 0 to 2π; and $G_{\mathrm{d}}({\Omega }_i)$—displacement PSD at angular spatial frequency ${\Omega }_i$. The ISO 8608 standard recommends the lower and upper limits of angular spatial frequencies (=2πn) equal to 2π × 0.01 rad/m and 2π × 10 rad/m for general on-road measurements, respectively [62]. To obtain various road profiles that more closely resemble real roads, we changed the linear fitting of $G_{\mathrm{d}}(\Omega )$ proposed by ISO [62] to two split fittings offered by Andrén [63] in Equation (2), and we modified it by reducing the amplitude of the frequencies higher than Ω₂ and lower than 0.04 × 2π rad/m:

$$G_{\mathrm{d}}(\Omega )=\{\begin{array}{c}{G}_{\mathrm{d}}({\Omega }_0){\Omega }^{-1}\\ G_{\mathrm{d}}({\Omega }_0){\Omega }^{-w_1}\\ G_{\mathrm{d}}({\Omega }_0){\Omega }^{-w_2}\\ G_{\mathrm{d}}({\Omega }_0){\Omega }^{-w_3=5}\end{array}\begin{array}{c}\mathrm{for}\\ \mathrm{for}\\ \mathrm{for}\\ \mathrm{for}\end{array}\begin{array}{c}\Omega \le 0.04\times 2\pi \mathrm{rad}/\mathrm{m}\\ 0.04\times 2\pi \mathrm{rad}/\mathrm{m}\le \Omega \le {\Omega }_1\\ {\Omega }_1\le \Omega \le {\Omega }_2\\ {\Omega }_2\le \Omega \end{array},$$

(3)

where w_i is waviness. The reference frequency Ω₀ = 1 rad/m, the lower break frequency Ω₁ = 0.21 × 2π rad/m, and the higher break frequency Ω₂ = 1.22 × 2π rad/m produced a minimal error for the Swedish road network [63]. Additionally, a Welch-type windowing function with the value of the exponent equal to 10 was used to minimize the appearance of abrupt shifts at the connections between profile segments [64]. The profile generated in this way applied to the left side, and the profile for the right side was formed by randomly increasing or decreasing it by 20%. For comparison, identical sequences of random numbers were used to generate each profile.

The Range Rover Evoque dynamic response on road irregularities was analyzed on the full-car model. Our developed dynamic model included ten masses (the car’s mass; four wheels’ masses; and the masses of a driver, three passengers, and a baggage box) and moments of inertia about the X and Y axes [65]. This model is a dynamic system with twelve degrees of freedom: ten vertical displacements along the Z axis of the above-mentioned masses and rotation of the car body about the car’s center of mass around the X and Y axes. The model was built using Lagrange’s equation of the second type. These equations were solved analytically; after differentiating, the final system, consisting of twelve equations presented in this form, was obtained:

$$\begin{array}{l}{a}_{ii}{\dot{q}}_i+b_{i1}{\dot{q}}_1+b_{i2}{\dot{q}}_2+\mathrm{\dots }+b_{i\mathrm{n}}{\dot{q}}_{\mathrm{n}}+c_{i1}{q}_1+c_{i2}{q}_2+\mathrm{\dots }+c_{i\mathrm{n}}{q}_{\mathrm{n}}=\\ =d_{i1}{\eta }_1+d_{i2}{\eta }_2+d_{i3}{\eta }_3+d_{i4}{\eta }_4+d_{i1}^{*}{\dot{\eta }}_1+d_{i2}^{*}{\dot{\eta }}_2+d_{i3}^{*}{\dot{\eta }}_3+d_{i4}^{*}{\dot{\eta }}_4\end{array}$$

(4)

where a, b, c, d, and d* (with corresponding indexes) are coefficients of equations derived from matrixes of stiffness, dissipation, and inertia; $q_i$ is a generalized coordinate, in respect to which differentiation was applied during the building of a system of equations; n is the number of generalized coordinates; and ${\eta }_1,{\eta }_2,{\eta }_3,{\eta }_4$ are coordinates along which the system of the car is excited kinematically. More information about our model, its derivation, and the parameters used can be found in reference [65]. Road irregularities are assumed in the model as the artificial road profiles. Both rear wheels pass the corresponding profiles shifted by a distance equal to the length between the front and rear wheel centers.

The same damping coefficient for all wheels, which defines the behavior of the suspension, was optimized to reach the minimal root-mean-square (RMS) value of driver vertical acceleration; in other words, the suspension system was adjusted for maximum driver comfort level. The mathematical solution of Equation (4) and the optimization process were processed using Simulink/MATLAB software and its response optimization.

3. Results and Discussion

Initially, the road profiles with various waviness w₁ values of 1, 2, 4, and 6; w₂ values of 0.5, 1, 2, and 3; and displacement PSD value $G_{\mathrm{d}}({\Omega }_0)=4\times {10}^{-6}{\mathrm{m}}^3$ [62] (ISO road class B) were generated. The length of the artificial road profile was 200 m. Finally, our dynamic full-car model simulated a vehicle passing these profiles at speeds v = 20, 50, 70, 90, and 130 km/h. Damping coefficient values can vary in the range from 1000 Ns/m up to 15,000 Ns/m. Most research authors studying suspension system optimization highlight three basic assessment touchstones related to minimization of the crew ride discomfort and changes in the tire contact with the road: (1) minimization of the standard deviation of sprung mass (body) acceleration, (2) minimization of the standard deviation of the normal reaction at the tire/road contact, and (3) reduction in the displacements of the suspension system [49,66,67]. In order to evaluate the ride comfort, statistics such as the suspension travel and sprung mass (body) acceleration [68], the standard deviation (RMS) of vertical acceleration [69,70], and the vertical jerk [71,72] are considered. The intuitive relationship between safety and comfort is contradictory [3], and a concurrent enhancement of both of them is not viable. In ref. [48] it is showed that the dependence of these parameters in some ranges of damping consistently grows and shrinks. In addition, a combined performance index with different weighting for each performance measure was used. The weightings depend on the intent of the design; for example, in the case of a luxury car, the highest weight is given to ride comfort. Thus, in this work, we considered only the vertical acceleration of the driver as a criterion for evaluation of the ride comfort, as in ref. [54]. All these optimizations were repeated for the seven masses: (1) m₁ = m₂ = m₃ = m₄ = 80 kg; (2) m₁ = 80 kg; (3) m₁ = m₂ = 80 kg; (4) m₁ = m₃ = 80 kg; (5) m₁ = m₄ = 80 kg; (6) m₁ = 160 kg; and (7) m₁ = 100 kg, m₂ = 60 kg, m₃ = 50 kg, and m₄ = 40 kg. The optimized suspension’s damping coefficient h_s values for generated profiles and for all masses 80 kg are shown in

Table 2. The optimized damping coefficients when all masses m₁ = m₂ = m₃ = m₄ = 80 kg.

	hs, Ns/m
Waviness	20 km/h	50 km/h	70 km/h	90 km/h	130 km/h
w1 = 1; w2 = 0.5	1131	1153	1141	1106	1291
w1 = 2; w2 = 0.5	1129	1191	1410	1378	1658
w1 = 4; w2 = 0.5	1122	1390	2433	2687	2701
w1 = 6; w2 = 0.5	1116	1953	4729	5739	4148
w1 = 1; w2 = 1	1326	1351	1280	1170	1319
w1 = 2; w2 = 1	1319	1408	1598	1478	1690
w1 = 4; w2 = 1	1312	1688	2735	2881	2726
w1 = 6; w2 = 1	1299	2382	5267	5944	4160
w1 = 1; w2 = 2	1856	1844	1581	1295	1357
w1 = 2; w2 = 2	1839	1941	1973	1663	1725
w1 = 4; w2 = 2	1808	2375	3324	3197	2752
w1 = 6; w2 = 2	1776	3375	6297	6272	4175
w1 = 1; w2 = 3	2747	2412	1858	1402	1378
w1 = 2; w2 = 3	2689	2565	2317	1811	1748
w1 = 4; w2 = 3	2596	3170	3867	3422	2767
w1 = 6; w2 = 3	2512	4585	7280	6508	4180

, Figure 1a.

The results of all other cases are given as supplementary material Tables S1–S6 or are presented as the difference between optimized and non-optimized values in Figure 2a, Figure 3a, Figures S3a, S4a, S5a and S6a. As seen in the results, as the values of w₁ and w₂ increase, the values of optimal damping increase, except in situations of speed = 20 km/h and increasing w₁. Analogic dependencies occur for all seven mass cases. This could be explained in two ways. First, using Equation (3), we have a small decrease in the $G_{\mathrm{d}}(\Omega )$ values in the range from 0.16 × 2π rad/m up to 0.21 × 2π rad/m increasing w₁. Second, due to the relationships between angular spatial frequency Ω, frequency f, and velocity v $\Omega =\frac{2\mathsf{\pi }f}{v}$, we need higher angular spatial frequencies at decreasing speeds to excite the same minimal frequency. For example, to excite 1 Hz vibrations driving at 20 km/h, we need an angular spatial frequency of 0.18 × 2π rad/m. The highest damping values occur when w₁ = 6 and speed varies from 70 km/h up to 90 km/h. Comparing the optimized damping with its value in a passive suspension system 2700 Ns/m [65], one can say that all values are smaller, except in several situations (m₁ = m₂ = 80 kg, w₁ = 1, w₂ = 3; m₁ = 160 kg, w₁ = 1, 2, 4, 6, w₂ = 3; m₁ = 100 kg, m₂ = 60 kg, m₃ = 50 kg, m₄ = 40 kg, w₁ = 1, 2, w₂ = 3). The surface area of the bigger values increases with increasing car speed and reaches maximum damping values when the speed is 70 km/h or 90 km/h and w₁ = 6. This was observed as well for other locations of masses (see Figure 2a, Figure 3a, Figures S3a, S4a, S5a and S6a). The highest damping is needed when mass m₁ = 160 kg (from 1.1 up to 1.43 times bigger compared with m₁ = 80 kg or 1.06–1.47 times compared with m₁ = m₂ = m₃ = m₄ = 80 kg). Comparing differences in the optimized damping of various locations of masses of 80 kg, the variations are less than 100 except in situations when w₁ = 6 or v = 90 km/h. Graphic representation of the original and optimized values is presented in Figure 1b–d.

The dependencies of the optimized damping coefficient on vehicle speed (see Figures S1 and S2) show that:

• When fixed waviness index = w₁:
  • Higher damping values are needed, increasing values of w₂ at low speed (differences amount to several hundreds);
  • This difference decreases with increasing speed (down to several tens);
  • Higher w₁ values significantly increase damping values when the speed is more than 50 km/h;
  • It reaches the maximum when w₁ = 6 and the speed is 70 km/h or 90 km/h.
• When fixed waviness index = w₂:
  • The differences of damping amount to several tens at 20 km/h speed;
  • This difference increases with increasing speed and reaches the maximum at 70 km/h or 90 km/h speed.

The dependence of RMS values of the optimized vertical acceleration of the driver, the difference between the RMS of the vertical accelerations of the driver when optimized and non-optimized, and the RMS of the non-optimized vertical accelerations of the driver on the road profile and speed when all masses are 80 kg are shown in Figure 1b–d, respectively. Other differences between the optimized and non-optimized RMS of the vertical accelerations of the driver are presented in Figure 2b and Figure 3b. One can observe the maximum RMS of the driver acceleration increasing as the speed shifts from lower w₂ values to higher (see Figure 1b,d). We obtained all smaller driver acceleration values after optimization for all cases (Figure 1c, Figure 2b, Figure 3b, Figures S3b, S4b, S5b and S6b). For example, the change varies from 0 m/s² up to 0.66 m/s² when all masses are 80 kg, and up to 1 m/s² when mass is 160 kg. This means that the acceleration of the driver could be reduced by up to 33% in comparison with non-optimized cases, depending on driving speed and road roughness. According to the ISO 2361 standard, the comfort level is very low when the total values of overall vibration magnitudes are from 1.25 m/s² up to 2.5 m/s² and becomes extremely uncomfortable when they are greater than 2.5 m/s² [73]. When optimized acceleration is still high, we could recommend additionally a lower speed value. Thus, combining information about speed, masses and their locations from car sensors, optimal damping, RMS of vertical acceleration stored in a microcomputer or the cloud, and road information, we could control damping and recommend or reduce the driving speed [40,74].

The calculated differences between the sums of the RMS of the vertical accelerations of the crew both when optimized for the driver and non-optimized, as presented in Figure 4a, show that it varies from 0.3 m/s² down to −4.4 m/s² (or in the range −35–6%). More detailed analysis showed that although driver acceleration was reduced, in a few situations the acceleration of the front-right passenger increased less than 0.1%. Meanwhile, the acceleration of the rear passengers decreased to 41% or increased to 31%. The biggest reduction of the rear passengers’ acceleration was observed when the sum of the acceleration decreased more than 10%. Moreover, the positive values of the differences between the sums of the RMS of the vertical accelerations are related to the bigger increase of the acceleration of the rear passengers. The question is, what is the most suitable to ensure the best driving comfort for all passengers and the driver. Usually, suspension systems have not one, but multiple different damping values in the front and rear of the vehicle.

The combined RMS vertical accelerations of the driver and rear-left passenger were chosen to optimize the suspension settings for ride comfort [75]. In addition, we optimized for the driver and front passenger or for the whole crew. The differences between the sum of the RMS of the vertical accelerations when optimized for the driver and front passenger or the whole crew, and when optimized for the driver, are presented in Figure 4b,c. The optimization for the driver only or for driver and front-right passenger give nearly the same results (the differences are less than 1%). In addition, it is true not only for the sum of the accelerations but also for the individual passenger’s acceleration. Consequently, both these optimizations could result in unacceptably high RMS vertical acceleration values for the rear passengers if left unchecked. Meanwhile, the sum of the RMS of the vertical accelerations is up to 9% smaller when optimizing for the whole crew than for only the driver. Simultaneously, the RMS of the vertical acceleration for the driver or front-right passenger increases up to 11% and 9%, respectively, and it decreases to 24% for rear passengers. Uys et al. [75] summarized the results of a simulation of an unmanned vehicle driving a measured road profile at 60 km/h: (1) when optimizing the RMS vertical acceleration of the driver, lower RMS vertical acceleration values for the driver are obtained; however, the RMS vertical acceleration of the rear-left passenger is 14% greater than that obtained by optimizing the combined driver and passenger RMS as an optimization objective; (2) when optimizing for combined RMS vertical acceleration, the lowest RMS vertical acceleration values for the passenger are obtained, but the driver RMS vertical acceleration is 13% higher than when optimizing for the driver only. One can state that these tendencies are true for all our cases when optimizing for the driver or the driver with a front passenger and when optimizing for the whole crew. So, further investigation will be required to discover how optimization is influenced if it will be used with different damping values for the front and rear wheels or for all wheels and various optimization objective functions.

4. Conclusions

The relationship between the detected road waviness values w₁, w₂ and the optimized damping coefficient under various locations of vehicle load positions could be used to control damping and recommend or reduce the driving speed. The needed optimized damping value could be interpreted from a 3D gridded matrix of damping coefficients calculated at certain fixed values of speed and waviness indexes w₁, w₂. The surface area of the bigger values increases in the case of increasing vehicle speed and reaches maximum damping values when the speed is 70 km/h or 90 km/h and w₁ = 6. Up to 1.5 times higher damping values are required when only the driver mass is doubled than in other cases under study.

When optimizing the full-car model for the driver RMS vertical acceleration, the acceleration of the driver could be reduced up to 33% in comparison with non-optimized cases depending on driving speed and road roughness. The optimization for the driver and combined optimization for the driver and rear passenger provides results with differences less than 1%. Nevertheless, both these optimizations could result in unacceptably high values of rear-passenger RMS vertical acceleration if they are not set with a limitation. When optimizing for the whole crew, the sum of the RMS of the vertical accelerations is up to 9% smaller than when optimizing for the driver. Simultaneously, the RMS of vertical acceleration for the front passengers increases up to 10%, and it decreases to 24% for the rear passengers.

The output of this research provides a reason for the new design of damping devices, which can ensure higher comfort for the vehicle crew without changing the suspension design.