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Optimization of Damper Top Mount Characteristics for Semi-Active Suspension System.

INTRODUCTION

The main task of the vehicle suspension system is to isolate the vehicle from the road irregularities while providing adequate road holding capability. The limitations of passive suspensions regarding the conflict between ride comfort and road holding have motivated the investigation of controlled suspension systems, both active and semi-active [1]. The foundation of controlled suspensions for the car mass-market can probably be dated back to the early 1960s, when Citroen introduced hydro-pneumatic active suspension in its luxury cars; at that time those suspensions were still untouched by electronics [2]. Given this tribute to Citroen, the 1980s was the real start of electronic suspension; analog electronics were already developed, and the magic of the fully active suspensions attracted both the Formula 1 competition and the car manufacturers [3]. During this decade the exceptional potential of replacing the conventional spring-damper system with a fully electronically controllable fast-reacting hydraulic actuator was demonstrated. However, the high costs, significant power absorption, bulky and unreliable hydraulic systems, and uncertain management of the safety issues prevented its impact on the mass-market of the vehicles [4].

Since the 1990s, it became increasingly clear that the best compromise of cost (component cost, weight, electronics and sensors, also power consumption, etc.) and performance (comfort, handling and safety) was to be found in another technology of electronically controllable suspensions: the semi-active suspension or the Continuous Variable Damping suspension. For example, the car manufacturers like Audi, VW, Mercedes, Ford, Volvo and GM cooperated together with the controlled dampers suppliers such as Sachs, Tenneco, Bilstein and Monroe to design, manufacture and develop different types of the semi-active suspension systems which are introduced in the mass-market of cars [4]. In last few years most of the luxury car manufactures have started offering semi-active suspension as standard equipment. In fact, now it is not uncommon to see semi-active suspension system as an option on non-luxury vehicles.

After a survey of the previous research studies on the semi-active suspension systems in the automobile market, it can be concluded that the semi-active dampers with solenoid valve are well developed and made commercially available, and therefore have already found their way to the vehicle market in comparison with the other types of the controlled dampers. The semi-active control algorithms have been developed by suppliers or OEM's to achieve the correct balance of comfort and road holding control [5], [6], [7], [8], [9], [10], [11]. On the other hand, the available research studies on the semi-active suspension systems have been applied only for the ride comfort frequency range up to 15 Hz. However, semi-active suspension systems might have a negative effect on the vehicle harshness performance, which was not considered yet in the available research studies [4].

The damper top mounts are used in the vehicles not only to provide ideal Noise-Vibration-Harshness (NVH) performance but also to improve ride comfort, driving safety, and handling. The ride comfort and harshness can be considered as the vibration response of the vehicle body at different frequency ranges. Vehicle ride comfort can be evaluated with vertical acceleration of the body up to 20 Hz while the harshness can be considered as the body vertical acceleration in the frequency range over 20 Hz until 100 Hz [1]. Another type of the vehicle harshness is the impact harshness (IH) which also affects the subjective impression of ride comfort. This type of harshness involves the vibration response of the vehicle which is referred to as IH events [12], [11], [14], [15].

It is relatively easy and cost effective to change the compliance of the damper top mounts compared with changing the characteristics of other suspension components to improve the vehicle ride comfort and harshness performance [16], [17], [13], [14]. For the adjustment of top mount characteristics to improve the ride comfort and harshness performance, the correlation between these has to be understood. For this purpose, in [18], a detailed mathematical damper top mount model based on restoring force mapping technique is presented which enables accurate simulation of the damper top mount force response. The presented damper top mount model consists of three elements, which are the nonlinear elastic element, the nonlinear friction element, and the nonlinear viscous element. The amplitude dependence of the top mount characteristics is modeled by using the friction element and the elastic element, while the frequency dependence of the top mount is modeled by using the restoring-force-mapping technique. Furthermore, a new procedure is proposed for the identification of the model parameters, which is based on a two-stage optimization routine where two sets of measurement data from the amplitude and frequency dependent tests are used. The mathematical model parameters of different commercial damper top mounts are identified using experimental data. The model is then validated by comparing the measured and simulated forces for three different top mounts. Good agreement between the measured and the simulated forces is obtained. Furthermore, the proposed model is found to be superior to the existing rubber isolator models. The damper top mount model proposed in [18] is used first to study the influence of the damper top mount characteristics on the ride comfort and harshness performance of a vehicle [19], and to optimize the damper top mount characteristics for the passive suspension systems to improve the harshness performance of the vehicle [4]. All top mount parameters investigated in this work have been shown to have visible effects on either one or both of the ride comfort and harshness characteristics of the vehicle. Therefore, it was concluded that, significant improvements can be achieved in both ride comfort and harshness performance of a vehicle through proper adjustment of the damper top mount characteristics.

In this study, an optimization technique to optimize the damper top mount characteristics of the semi-active suspension system is developed. The proposed optimization technique employs a new combined objective function based on ride comfort and harshness evaluation. A detailed and accurate damper top mount mathematical model is implemented inside a validated full vehicle model to constitute the simulation environment required for the optimization study. The damper top mount characteristics are optimized for the semi-active suspension system based on the Rule-Optimized Fuzzy-Logic controller published by the authors in previous papers [20], [21], [22], [23]. The ride comfort and harshness of the full vehicle are evaluated by analyzing the body acceleration in different frequency ranges. While, the dynamic stiffness of the damper top mount is used to describe the optimum damper top mount characteristics for different optimization case studies.

VEHICLE MATHEMATICAL MODEL

In this study, a generic full vehicle suspension model with non-linear suspension component characteristics has been used. The vehicle model contains seven masses which represent the body, four wheels, engine, and driver. The model has eleven degrees of freedom which include the bounce, pitch and roll motions of the body and engine, and the bounce motion of the wheels and driver. The suspension between the vehicle body and wheel is modeled using a linear spring and a viscous damper in series with a rubber top mount at each corner. The damper and top mount masses are neglected. The nonlinear characteristics of the damper and the top mount are taken into account by using detailed damper top mount model presented in [18] and non-linear Force-Velocity damper characteristics. Point contact is assumed between the road and the tire at each corner and tire-road separation is taken into account. The non-linear force characteristics of the tires are considered within the model by using an experimentally identified non-parametric Force-Displacement-Velocity map for the tires. Furthermore, the anti roll bars at front and rear axles are also -included in the vehicle model. The engine is assumed to have bounce, pitch and roll degrees of freedom with respect to the vehicle body. The engine body is supported by three equivalent vertical mounts, which are modeled using linear springs in parallel with viscous dampers. The driver is included inside the full vehicle model as a one degree of freedom system with the vertical motion only, and it is modeled as a mass supported by a linear spring and a linear damper. The axis convention and the notations used for the vehicle body, wheels and road profile are shown in Figure 2.

The equations of motion for the bounce, pitch and roll responses of the vehicle body are written as follows:

[M.sub.b][Z.sub.b] = -[F.sub.sb] - [F.sub.sb2] - [F.sub.sb3] - [F.sub.sb4] + [F.sub.el]+[F.sub.e2]+[F.sub.e3] + [F.sub.st] (1)

[mathematical expression not reproducible] (2)

[mathematical expression not reproducible] (3)

The vehicle is front-engine front-drive type and the engine-gearbox block is supported by three mounts. First mount is situated at the left side while the second one installed at the right edge of the engine. The third engine mount is the sub-frame mount and installed at the middle and near to the left side. The equations of motion for the engine vertical, pitch and roll responses are:

[M.sub.e][Z.sub.e] = -[F.sub.e1]-[F.sub.e2]-[F.sub.e3] (4)

[I.sub.e][phi]e= [Le.sub.1] [F.sub.e1] - [L.sub.e2][F.sub.e2] - [L.sub.e3][F.sub.e3] (5)

[J.sub.e][x.sub.e] = [B.sub.e1][F.sub.e1]-[B.sub.e2][F.sub.e2]-[B.sub.e3][F.sub.e3] (6)

The equations of motion for vertical response of the driver can be written as:

[M.sub.st][Z.sub.st] = -[F.sub.st] (7)

The equations of motion for the bounce motion of the wheels can be written as:

[M.sub.wi][Z.sub.wi] = -[F.sub.wi] + [F.sub.swi] (8)

The equation of motion for the damper top mount dynamic characteristics can be written as the following, and more details about it is shown in [18].

[mathematical expression not reproducible] (9)

[mathematical expression not reproducible] (10)

[mathematical expression not reproducible] (11)

[mathematical expression not reproducible] (12)

[F.sub.ej] = [K.sub.ej]([Z.sub.ej] - [Z.sub.bej])+[C.sub.ej]([Z.sub.ej] - [Z.sub.bej]) (13)

[F.sub.st] = [K.sub.st]([Z.sub.ej] - [Z.sub.bs]) + [C.sub.st]([Z.sub.st] - [Z.sub.bs]) (14)

[F.sub.wi] = [K.sub.ti]([Z.sub.wi] - [Z.sub.oi]) (15)

[F.sub.swi] = [K.sub.si]([Z.sub.bi] - [Z.sub.wi]) + [F.sub.di] (16)

[mathematical expression not reproducible] (17)

[mathematical expression not reproducible] (18)

[mathematical expression not reproducible] (19)

[mathematical expression not reproducible] (20)

[T.sub.stf] = [2B.sub.s1][K.sub.stf](([Z.sub.b1] - [Z.sub.w1]) - ([Z.sub.b3] - [Z.sub.w3])) (21)

[T.sub.str]= [2B.sub.s2][K.sub.str](([Z.sub.b2] - [Z.sub.w2]) - ([Z.sub.b4] - [Z.sub.w4])) (22)

With i = 1,2,3,4, j = 1,2,3

The damper of the semi-active suspension system used in this study can be controlled continuously between upper and lower limits of the Force-Velocity characteristic curves. Therefore, it is theoretically capable of tracking a damping force demand independently of the damper velocity. The demanded damper force [F.sub.sci] is generated by the controller according to defined control laws and strategies. Since the controller output cannot be always realized because of the non-linear restriction in the damper, i.e. it can only dissipate energy and not input it, the system cannot work with the output control signal only [2], [1], [22]. Therefore, an additional control law is required to keep the control force in the controllable damper limits. In this study, the Rule Optimized Fuzzy Logic controller is used to generate the demanded suspension control force [F.sub.sci] [22]. The additional control law defined in Equation 23 is used to keep the actual semi-active damper force [F.sub.svi] always within the allowable force limits of the damper.

[mathematical expression not reproducible] (23)

The damping characteristics of the semi-active damper used in this research study are shown in Figure 3. The presented force characteristics are obtained through experimental identification of the dampers available in the market. The semi-active damper can yield a damping force within the range between the soft and the hard curves shown in the figure. In order to control the semi-active damper, a controlled solenoid valve which can be used to change the damping characteristics is employed inside the semi-active damper.

The dynamic response of the semi-active damper has also significant influence on the performance of the semi-active suspension system. Therefore, the overall dynamics of the semi-active damper and its solenoid valve is simulated using a first order transfer function (Equation 24) which models the delay behavior of the damper.

[mathematical expression not reproducible] (24)

In Equation 24. [mathematical expression not reproducible] is the Laplace transform of the desired damping force [mathematical expression not reproducible], while [F.sub.svi](s) is the actual damping force whose time domain form is [F.sub.svi](f) shown in Equation 23. and [t.sub.d] is the time delay constant.

The system equations of the vehicle model presented above are solved by using the Simulink. More details about the vehicle model and its experimental validation analysis are presented in [4] and [20].

Semi-Active Controller

The Rule-Optimized Fuzzy-Logic controller developed by the authors in [20], [22], [23] is used in the study to demonstrate effectiveness of the top mount optimization process for the semi-active suspension system.

The Fuzzy-Logic controller uses the body vertical velocity and acceleration as controller input and generates the demand damper forces as the controller output. The controller surface map is shown in Figure 4. More details about the design and optimization of the Rule-Optimized Fuzzy-Logic controller can be found in [20] and [22].

The control process of the semi-active suspension starts by providing the body velocity and acceleration to the Rule-Optimized Fuzzy Logic controller. The controller uses this information to generate the demanded damper force [F.sub.sci]. The additional semi-active damper control law given with Equation 28 is used to calculate the value of the damping force [F.sub.svi] that is inside the controllable damper limits, according to the demanded control signal [F.sub.sci] and the damper velocity. In order to obtain the force output from damper [F.sub.di], the damper control signal is filtered by using a first order transfer function that models the delay behavior of the voltage controller and the damper valve. The vehicle model is used to calculate the body acceleration and velocity, and the damper velocity which are then fed back to the control system.

DAMPER TOP MOUNT OPTIMIZATION METHODOLOGY

As concluded from the damper top mount parametric study for passive suspension system presented in [4], the damper top mount has a significant influence on the vehicle harshness performance, while the influence on the vehicle ride comfort is very small. On the other hand, improving the vehicle ride comfort performance using the semi-active suspension might have a negative effect on the harshness performance if the damper top mount is not properly tuned. Therefore, the optimization of the damper top mounts for the semi-active suspension system is crucial.

In this study, an optimization routine with a combined objective function is developed. The ride comfort and the harshness performance of the vehicle have been considered in the proposed objective function. As shown in Equation 25, the objective function is the sum of the cost function of Ride Comfort and Harshness, Road Holding and Suspension Working Space in the frequency domain. Suspension Working Space represents the suspension travel resulting from the change of relative distance between the vehicle body and wheels. The cost function for the Ride Comfort and Harshness performance (Equation 26) includes the weighted sum of the PSD of the vehicle body vertical acceleration at each corner in the ride comfort frequency range (R) from 0 Hz to 20 Hz and in the harshness frequency range (H) from 20 Hz to 100 Hz. Furthermore, the cost function of the Road Holding and Suspension Working Space shown in Equations 27 and 28 includes the weighted sum of the PSD of the dynamic tire load and suspension deflection at each corner of the vehicle in the overall ride comfort and harshness frequency range from 0 Hz to 100 Hz.

Cos[t.sub.total] = Cos[t.sub.RCH] + Cos[t.sub.RH] + Cos[t.sub.SWS] (25)

[mathematical expression not reproducible] (26)

[mathematical expression not reproducible] (27)

[mathematical expression not reproducible] (28)

In Equation 26 the parameters from [q*.sub.1] to [q*.sub.8] can be used to emphasize the relative importance between the ride comfort and harshness performance, while the other weighting parameters from [q*.sub.9] to [q*.sub.16] in Equation 27 and 28 are employed to specify other design objectives. It should be noted that, the parameters q/1..16 include both the weighting parameters [q.sub.i] which describe the relative importance of a term and the normalization factors [mathematical expression not reproducible] which account for the nominal values of the terms used in the objective function. The normalization factors [mathematical expression not reproducible] are obtained through the simulations of the vehicle model with a set of experimentally identified rubber mount parameters.

[mathematical expression not reproducible] (29)

In the optimization analysis presented in this paper, all weighting parameters of the proposed objective function [q.sub.1]..[q.sub.16] are selected as 1 in order to achieve a balanced ride comfort, harshness and road holding performance. The proposed optimization routine is presented schematically in Figure 5. During the optimization process, the full vehicle model is simulated using a measured road unevenness profile.

Figure 6 shows the power spectral density of the used road profile in logarithmic scale.

The simulation outputs of interest; the vehicle body vertical acceleration at each corner, dynamic tire loads and suspension working space, are used to generate the power spectral densities (PSD) required by the proposed objective function. The Genetic Algorithm (GA) is employed as an optimum search routine to obtain the optimum front and rear damper top mount characteristics which minimize the combined objective function. Detailed information about the design variables of the damper top mount can be found in [19]. The dynamic stiffness of the damper top mount is used to present the optimal top mount characteristics together with the displacement-velocity-force characteristics of the damper top mount.

OPTIMIZATION RESULTS

Figure 7 presents a comparison between the front and rear damper top mount dynamic stiffness curves in the overall ride comfort and harshness frequency range for the nominal top mount characteristics with semi-active suspension system and optimal top mount characteristics with passive and semi-active suspension system. In the damper top mount optimization for semi-active suspension system case, it is obvious that, the optimization process ends up with softer damper top mount characteristics in order to improve the harshness performance. As already mentioned, the harshness performance considers only the frequencies above 20 Hz. Therefore, the optimization routine tries keeping the dynamic stiffness low at all frequencies over 20 Hz. In this case, the decrease in the dynamic stiffness of the top mount in the harshness frequency range from 20 Hz to 100 Hz is accompanied by a reduction in the dynamic stiffness in the ride comfort frequency range, but this reduction has no visible effect on the ride comfort of the vehicle. The results imply that, for the considered vehicle configuration, a softer top mount should be used in the semi-active suspension compared to the passive system.

Figure 8 shows the Force-Velocity-Displacement characteristics for the nominal top mount characteristics of the semi-active suspension system and optimal top mount characteristics of the passive and semi-active suspension systems. The Force-velocity-displacement characteristics prove the conclusions derived from the dynamic stiffness curves shown in Figure 7. Optimization process for the damper top mount model parameters leads to softer damper top mount characteristics in all operation points of the top mount.

Figure 9 shows a comparison between the RMS values of the body vertical accelerations in the ride comfort and harshness frequency ranges for the nominal and optimal top mount characteristics of the passive and semi-active suspension systems, respectively. The results indicate that, worthwhile reduction in body vertical accelerations can be obtained in the ride comfort range by using the semi-active suspension system instead of the passive suspension system. In the ride comfort frequency range, the RMS values of the body vertical accelerations attained by using the semi-active suspension system with nominal and optimal damper top mounts are close to each other. On the other hand, the lowest body vertical acceleration level in the harshness range is obtained by using the optimal top mount with passive suspension system. The semi-active suspension with nominal damper top mount generates the highest RMS body vertical accelerations. When optimal damper top mount is used with the semi-active suspension system the body vertical accelerations in the harshness range are significantly improved.

Figure 10 shows a comparison between the RMS values of the dynamic tire load in the ride comfort frequency range for the nominal and optimal top mount characteristics with passive and semi-active suspension systems, respectively. The RMS values indicate that the semi-active suspension system with nominal or optimal damper top mounts improve the dynamic tire load at the front and rear axles compared with the passive systems. The use of optimal damper top mount with the semi-active dampers provides relatively small change in the dynamic tire load compared with the nominal top mounts.

The results presented in Figure 9 and Figure 10 prove that the influence of the damper top mount characteristics on the body vertical accelerations and dynamic tire loads in the ride comfort frequency range is very small for both passive and semi-active suspension systems. On the other hand, the body vertical accelerations and dynamic tire loads are improve in the ride comfort range as results of using the semi-active suspension system by 12-15% relative to the passive system. The body vertical accelerations are increased by 12 -16% in the harshness frequency range when the semi-active suspension is used. Optimization of the damper top mount characteristics in passive and semi-active suspension systems improves the body vertical accelerations in the harshness frequency range by 4% and 6-10%, respectively.

CONCLUSIONS

In order to improve the vehicle harshness performance for the semi-active suspension system, the optimization of the damper top mounts in the harshness frequency range is proposed in this research work. The RMS values of the vehicle body vertical acceleration at each corner at the harshness frequency range from 20 to 100 Hz are considered as evaluation criterion for the harshness performance. On the other hand, the dynamic tire loads and the suspension working space are included in the objective function in order to prevent degradation of these aspects during top mount optimization. The simulation results indicate that the influence of the damper top mount characteristics on the body vertical accelerations and dynamic tire loads in the ride comfort frequency range with both passive and semi-active suspension systems is not significant. On the other hand, the body vertical accelerations and dynamic tire loads improve by 12 -15%in the ride comfort range as results of using the semi-active suspension system instead of the passive system. In addition to this, the body vertical accelerations increase by 12-16% in the harshness frequency range when the semi-active suspension is used. The optimization of the damper top mount characteristics in the harshness frequency range for the passive suspension system improve the body vertical accelerations by 4%, while the body vertical accelerations improve in the harshness frequency range by 6-10% as results of damper top mount optimization for the semi-active suspension system. The results presented in this paper show that, top-mount optimization is crucial to realize the benefit for ride comfort, harshness and road-holding for semi-active system. It should be that, this work focuses on the analysis of the vertical dynamics of the vehicle. Since, the semi-active suspension system and its top mount characteristics have significant effect on the handling performance of the vehicle, the analysis should be extended to cover the lateral dynamics of the vehicle.

REFERENCES

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[2.] Crolla D.A., Firth G., and Horton D., An Introduction to Vehicle Dynamics. UK: Leeds University, Mechanical Engineering Department. 1992.

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[4.] Kaldas M.M.S., Improvement of Semi-Active Suspensions through Fuzzy-Logic and Top Mount Optimization, 1st ed. Germany: Shaker Verlag, 2015.

[5.] Ford-Motors-Company-Ltd, "Ford Technology Newsbrief," Ford Motors Company, 2010.

[6.] Murphy Tom, "CDC Puts Astra in Big Leagues," www.wardsauto.com. 2004.

[7.] Tenneco-Inc, ""Electronic Damper Technologies: Ride control innovation,"" www.tenneco.com. 2011.

[8.] Tenneco-Inc, "Tenneco Equips All-New Volvo XC90 First Edition with Monroe Intelligent Suspension," http://www.tenneco.com. 2014.

[9.] ThyssenKrupp-Bilstein-GmbH, "DampTronic Sky - Introduction and Function," http://www.thvssenkrupp-bilstein.de. 2013.

[10.] ZF-Sachs-AG, "Electronically Controlled Adaptive Damping Systems," www.zf.com. 2014.

[11.] Ohlins-CES, "CES - Continuously controlled Electronic Suspension," www.ohlinsces.com. 2015.

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[14.] Li, L., Zhang, Y., Wu, S., and Chen, L., "Optimization of Suspension Elastomeric Bushing Compliance Under Constraints of Handling, Ride and Durability," SAE Int. J. Mater. Manuf. 3(1):468-483, 2010. doi :10.4271/2010-01-0721.

[15.] Aydin M. and Unliisoy Y. S., "Optimization of Suspension Parameters to Improve Impact Harshness of Road Vehicles," Int. J. Advanced Manufacturing Technology, vol. 60, no. 5, pp. 743-754, 2012.

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[23.] Caliskan, K, Kaldas, M., Henze, R., and Kucukay, F, "Performance Analysis of the Rule-Optimized Fuzzy-Logic Controller for Semi-Active Suspension," SAE Int. J. Mater. Manf. 9(3):707-717, 2016, doi :10.4271/2016-01-0444.

Mina M.S. Kaldas

Minia University

Kemal Caliskan, Roman Henze, and Ferit Kucukay

IAE, TU Braunschweig

CONTACT INFORMATION

Dr.-Ing. Mina M.S. Kaldas

Assistant Professor

Automotive Engineering Department, Faculty of Engineering, Minia

University, Egypt

m.kaldas@mu.edu.eg

Dr.-Ing. Kemal Caliskan

Team Leader Vibration and Acoustics

Institute of Automotive Engineering, TU Braunschweig, Germany

k.caliskan@tu-bs.de

Dr.-Ing. Roman Henze

Chief Engineer

Manager Chassis, ADAS + Automated Driving, Institute of

Automotive Engineering, TU Braunschweig, Germany

r.henze@tu-bs.de

Prof. Dr.-Ing. Ferit Kucukay

Director of the Institute of Automotive Engineering

TU Braunschweig, Germany

f.kuecuekay@tu-bs.de
NOTIONS

Symbol             Definition

[B.sub.bej]        Lateral distance from the vehicle body CG to
                   each mounting point of the engine
[B.sub.bs]         Lateral distance from the vehicle body CG to the
                   seat mounting point
                   Lateral distance from the engine CG to each
                   mounting point
[2B.sub.S,1],
[2B.sub.s,2]       Wheel track at front and rear axles
[C.sub.ej]         Engine mount damping coefficients
[C.sub.s]          Seat damping coefficient
[F.sub.di]         Damper force at each corner
[F.sub.ej]         The engine mount forces at each mounting point
[F.sub.elastici]   Damper top mount elastic force at each corner
[F.sub.frictioni]  Damper top mount friction force at each corner
[F.sub.frmaxi]     Damper top mount maximum friction force at
                   each corner
[F.sub.maxi]       Maximum force of the semi-active damper at
                   each corner
[F.sub.mini]       Minimum force of the semi-active damper at
                   each corner
[F.sub.mi]         Damper top mount force at each corner
                   Demanded suspension control force at each
[F.sub.sai]        corner
                   Total suspension force acting on the vehicle body
[F.sub.sbi]        at each vehicle corner
                   Overall demanded suspension control force at
[F.sub.sci]        each vehicle corner
                   Actual semi-active damper force at each vehicle
[F.sub.svi]        corner
[F.sub.swi]        Total suspension force acting to the each wheel
[F.sub.viscousi]   Damper top mount viscous force at each corner
                   1 = FL= Front Left
i = 1, 2, 3, 4     2 = RL= Rear Left
                   3 = FR= Front Right
                   4 = RR= Rear Right
[I.sub.b]          Body pitch moment of inertia
[I.sub.e]          Engine body pitch moment of inertia
                   1=Left
j =1,2,3           2 = Right
                   3 = Middle
[J.sub.b]          Body roll moment of inertia
[J.sub.e]          Engine body roll moment of inertia
[K.sub.e1i],
[K.sub.e2i],
[K.sub.e3i]        Damper top mount model elastic force
                   parameters
[K.sub.ej]         Engine mount stiffnesses
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Author:Kaldas, Mina M.S.; Caliskan, Kemal; Henze, Roman; Kucukay, Ferit
Publication:SAE International Journal of Commercial Vehicles
Article Type:Technical report
Date:May 1, 2017
Words:4958
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