Fault Detection in Automotive Semi-active

13AC-0123

Fault Detection in Automotive Semi-active Suspension: Experimental Results Author, co-author (Do NOT enter this information. It will be pulled from participant tab in MyTechZone) Affiliation (Do NOT enter this information. It will be pulled from participant tab in MyTechZone) Copyright © 2013 SAE International

sensors and / or actuators. The study case is a Quarter of Vehicle (QoV) with a suspension using a configurable shock absorber. The experimental testbed uses Hardware-in-theLoop (HiL) to validate the fault detection approach. The HiL is an automotive Electronic Control Unit (ECU), and the QoV model is embedded in a Field-Programmable Gate Array (FPGA). The proposed scheme is performed for discrete time systems. The LPV approach is designed to detect a sensor fault on a vertical acceleration dynamic system. A control system for a semi-active suspension can take into account that detection and be online modified to assure the best performance in comfort, as well as messages to driver of a malfunction. The main technical contribution is the capability of fault sensor detection using a differences equation of second order which uses a measurement-based scheduling parameter.

ABSTRACT A proposal of fault detection using Linear Parameter Varying (LPV) systems is experimentally validated on embedded systems. The fault detection system is oriented to faulty sensors and / or actuators. The study case is a Quarter of Vehicle (QoV) with suspension using a controllable shock absorber which uses a SkyHook controller. The experimental testbed uses Hardware-in-the-Loop (HiL) to validate the approach. The HiL is an automotive Electronic Control Unit (ECU), and the QoV model is embedded in a FieldProgrammable Gate Array (FPGA). Results exhibit the effectiveness of the approach which is consistent with numerical simulation. These results open the application of LPV approaches to commercial vehicles since it is easy of implementation for several features such as, low computation load, lumped parameter model and available for nonlinear dynamic systems.

This paper is organized as follows. Section 2 shows a literature review. Section 3 describes the automotive suspension system. Section 4 presents the proposed solution where the Hardware-in-the-Loop including the semi-active suspension control system testbed using a Controller Area Network (CAN) is described. Section 5 discusses results. Section 6 concludes the paper.

1.-INTRODUCTION A controlled automotive suspension in actual vehicles claims to improve comfort and road holding using semi-active shock absorbers. This control system uses measurements from vertical acceleration in sprung mass, unsprung mass and suspension deflection. Hence, if a fault is presented in sensors, the suspension performance will be deteriorated. The fault detection of sensors allows to reconfigurate the system to nominal performance and/or advice driver of the system fault for maintenance. Since these control systems are implemented in computation constrained embedded systems, fault detection methods with light computation are mandatory.

2.-LITERATURE REVIEW Fast fault detection of automotive suspensions in extreme applications or situations, i.e. racing (car, bike), multiple uses of vehicles (off-road to highway changes) has opportunity areas. The detection of fault in sensors in racing application will allows to always keep an optimum damping, while in vehicles will decrease the risks of loss of stability at high speeds / maneouvres after off-road suspension operation. Consumers actually expect safety, security, reliability, comfort, etc. from their vehicles as they do from their other electronic products, [1]. Advanced Fault Detection and Isolation (FDI) methods are classified into two major groups, those which do not assume any form of model information

A fault detection in an automotive semi-active suspension using Linear Parameter Varying (LPV) systems, [9], is experimental validated on automotive embedded systems. The fault detection system is oriented to faulty 1

(process history-based methods) and those which use accurate dynamic process models (model-based methods), [10]. Basically, the FDI selection depends on the availability of a reliable model or process measurements under the considered faulty modes of operation. Faults in engineering processes belong to one of the following categories, [4]: faults in sensor and actuator called soft faults or, process faults where some plant parameters change because of physical/mechanical problems.

linear viscous friction (𝐹𝑐 ), a spring force (𝐹𝑘 ) and an electrically variable force (𝐹𝑖 ) which is subject to saturations. It is herby proposed to model the effects of a damper failure as an arbitrary additive force, denoted by 𝐹𝑜 . Thus, the resulting model for the damper force (𝐹𝑀𝑅 ) can be stated as:

𝐹𝑀 = 𝐼𝑓 𝑐 tanh(𝑎1 𝑧̇𝑑𝑒𝑓 + 𝑎2 𝑧𝑑𝑒𝑓 ) + 𝑏 1 𝑧̇𝑑𝑒𝑓 + 𝑏 2 𝑧𝑑𝑒𝑓 + 𝐹𝑜 (2) ��� ��� ����������������� 𝐹𝑖

which can be rewritten:

FDI of automotive suspensions shows model based approaches have prohibitive computation [6, 12, 3, 2, 11]. In general, the approach methodology proposes the online register of data from the condition of the sensors / actuatos using mems, and then, through an online adapation of models and observers, the fault is detected. Fault detection that uses memory and online identification are prohibitive in automotive industry. Hence efficient computing and accurate methods to perform FDI in suspensions systems are an open issue.

−𝐹𝑀𝑅 − 𝑘𝑠 𝑧𝑑𝑒𝑓 𝑘𝑠 𝑧𝑑𝑒𝑓 + 𝐹𝑀𝑅 − 𝑘𝑡 (𝑧𝑢𝑠 − 𝑧𝑟 )

(3)

where 𝜌1 = tanh(𝑎1 𝑧̇𝑑𝑒𝑓 + 𝑎2 𝑧𝑑𝑒𝑓 ). Therefore a state-space representation of the QoV model is given by: 𝑧̈𝑠 𝑧̇𝑠 � � 𝑧̈𝑢𝑠 � �𝑧̇� 𝑢𝑠 𝑥̇

𝑏1

=

𝑧̈𝑠 𝑧̈ � 𝑢𝑠 � 𝑧𝑑𝑒𝑓 ���

(1)

where 𝑚𝑠 and 𝑚𝑢𝑠 are the sprung and unsprung mass, respectively; 𝑧𝑑𝑒𝑓 is the suspension deflection; 𝑧𝑟 represents the road profile; 𝑘𝑠 and 𝑘𝑡 are respectively the stiffness of the suspension and the tire and 𝐹𝑀𝑅 the semiactive damping force.

𝑦

𝑐1

𝑓𝜌

=

𝑥̇ 𝑦

𝑏1

𝑐1

⎡− 𝑚𝑠 − 𝑚𝑠 𝑚𝑠 ⎤ 𝑧̇𝑠 𝑚𝑠 ⎢1 ⎥ 𝑧𝑠 0 0 0 ⎢ 𝑏1 𝑐1 𝑏1 𝑐1 +𝑘𝑡 ⎥ �𝑧̇ � 𝑢𝑠 − − ⎢𝑚𝑢𝑠 𝑚𝑢𝑠 𝑚𝑢𝑠 𝑚𝑢𝑠 ⎥ 𝑧 �� � 𝑢𝑠 ⎣0 ⎦ ��������������������� 0 1 0

The dynamic behavior of a QoV with a semiactive suspension, Figure 1, is described by the following system of ordinary differential equations: = =

𝐹𝑘

𝐹𝑀 = 𝐼𝑓𝑐 𝜌1 + 𝑏1 𝑧̇𝑑𝑒𝑓 + 𝑏2 𝑧𝑑𝑒𝑓 + 𝐹𝑜

3.-AUTOMOTIVE SUSPENSION SYSTEM

𝑚𝑠 𝑧̈𝑠 𝑚𝑢𝑠 𝑧̈𝑢𝑠

𝐹𝑐

𝑏1

𝐴

𝑐 1 0⎤ ⎡− 𝑚𝑠 0 𝐼 ⎢ 0 0⎥ �𝑧 � + ⎢𝑓0𝑐𝜌1 ⎥ 𝑟 𝑘𝑡 0⎥ � 𝐹𝑜 ⎢ 𝑚𝑢𝑠 𝑚𝑢𝑠 ⎣����������� ⎦ 0 0 0 𝑢

𝑐1

𝐵 𝑏1

𝑥

(4)

𝑐1

𝑧̇ ⎡− 𝑚𝑠 − 𝑚𝑠 𝑚𝑠 ⎤ 𝑠 𝑚𝑠 𝑧 ⎢ 𝑏1 𝑐1 𝑏 𝑐 +𝑘 ⎥ 𝑠 − 1 − 1 𝑡 ⎥ �𝑧̇𝑢𝑠 � ⎢𝑚𝑢𝑠 𝑚𝑢𝑠 𝑚𝑢𝑠 𝑚𝑢𝑠 ⎣��������������������� ⎦ 𝑧𝑢𝑠 0 1 0 −1 𝐶 𝑓𝑐 𝜌1

= =

1⎤ 𝐼 ⎡− 𝑚𝑠 0 ⎢ ⎥ 𝑧 𝑓 𝜌 𝑘 𝑡 + 𝑐 1 0⎥ � 𝑟 � ⎢ 𝑚𝑢𝑠 𝑚𝑢𝑠 𝐹𝑜 ⎣����������� 0 0 0⎦

𝐴𝑥 + 𝐵𝑢 𝐶𝑥 + 𝐷𝑢

𝐷

(5)

with 𝑐1 = 𝑘𝑠 + 𝑏2 .

In this application, it is proposed to implement fault detection and control algorithms for a quarter of vehicle system. The QoV model is implemented in an FPGA, National Instruments Compact RIO. The fault detection and control functionalities are implemented in a MPC5517 microcontroller. Both modules communicated via a CAN network. Figure 2 illustrates the components under consideration.

Figure 1. QoV model of a semi-active suspension

The semiactive suspension is represented by an MR damper, which can be modeled as the sum of various forces: a

The QoV model is implemented in the NI-FPGA inside the real-time module of the Compact RIO “cRIO-9004”. The 2

chassis includes a CAN module NI-9852 composed of two low-speed CAN.

Both fault detection and control algorithms are computed inside the Freescale microcontroller. The microcontroller runs at 18 𝑀ℎ𝑧. It is equipped with a CAN controller in order to achieve transmission with the cRIO FPGA. Thanks to the low computational load of the algorithms, real time computations are available at the constant rate specified by the system: 𝐹𝑠 = 100 𝐻𝑧. Computations are made in 32 bits floating point precision.

(a)

Once all data has been received, fault detection and control algorithms are computed. Then, those data are sent to cRIO. The CAN network ensures the communication between both modules. It has been chosen to use this type of network due to its great utilization in automotive industry. The CAN network is programmed in both components to work at a constant bit rate of 100 𝑘𝑏𝑖𝑡𝑠/𝑠. The National Instrument FPGA sends the 3 sensors data coded as 32 bits fixed point data. As the way of coding data is different between cRIO and Freescale, is has been chosen to multiply the decimal data to be sent by a constant 𝛼 = 10242 and transmit directly an integer. Then, division by 𝛼 is made in the receiving module. Each integer 𝐼 are split in 4 bytes 𝐼 = [𝐼1 𝐼2 𝐼3 𝐼4 ] as CAN network work with bytes.

(b)

After experiments, it has been shown that the transmission delay was roughly 1 ms for 1 integer. Transmitting the 3 data from cRIO to Freescale took 3 ms. Figure 3 illustrates the communication.

Figure 2. Experimental implementation (a) Photo, (b) Block diagram The model runs at the sampling frequency 𝐹𝑠 = 100𝐻𝑧 in order to achieve real-time computations. The state space model is implemented as : �

𝑋𝑘+1 𝐴 �=� 𝑌𝑘 𝐶

𝐵 𝑋𝑘 �� � 𝐷 𝑈𝑘

Figure 3. Communication between modules

(6)

4.-FAULT DETECTION APPROACH

where the numerical values are shown in the Appendix. It results as output the vectors of the next state 𝑋𝑘+1 and the outputs 𝑌𝑘 of the system. The inputs of the matrix are the states 𝑋𝑘 and the control input 𝑈𝑘 composed by the input current 𝐼𝑘 and the road profile 𝑧𝑟𝑘 .

This section presents the fault detection approach which is used to detect a fault on the sprung mass acceleration sensor. The aim is by making use of the system inputs and outputs, recover if the actuator is faulty or healthy. The approach is facing several difficulties such as :

Once all the outputs are computed, they are sent to the fault detector and controller in order to detect a fault and compute the input current in the system.

1. The non linearity of the inputs 2. The road profile which is unmeasured The approach presented in [8] is adapted for the system under consideration as it proposes a fault detector for LPV systems with uncertainties. Nevertheless, a simpler

All decimal numbers inside the FPGA are coded in fixed point since floating point computation is not available. 3

synthesis can considerations.

be

handled

according

to

the

The parity matrix 𝒲𝑜 exists if and only if the rank of [ℋ | 𝒢𝑢 ] is degenerated. However, it is known that the rank of the concatenation is given by :

further

The fault detection scheme is based on the parityspace approach. Since the nonlinearity of the system is only in the input matrices 𝐵 and 𝐷, it is proposed to consider the nonlinearity outside the matrices, directly inside the control input. For this purpose, it is considered the system : 𝑋𝑘+1 𝑌𝑘

= =

𝐴𝑋𝑘 + 𝐵𝐼 𝜌 � 𝑘 𝐼𝑘 + 𝐵𝑟 𝑧𝑘 𝐼𝜌𝑘

𝐴𝐶𝑘 + 𝐷𝐼 𝜌 � 𝑘 𝐼𝑘 + 𝐷𝑟 𝑧𝑘 + 𝐷𝑓 𝐹𝑜

rank([ℋ

(7)

𝑟 = 𝒲 ⋅ (𝑌𝑠 − 𝒢𝐼 𝑈𝐼𝑠 )

The parity space approach aims to express the output of the system 𝑌𝑘 along a horizon 𝑠. It leads to the following system :

𝐷𝑥 ⎡𝐶𝐵 ⎢ 𝑥 𝐶𝐴𝐵 ⎢ 2𝑥 where 𝒢𝑥 = ⎢𝐶𝐴 𝐵𝑥 ⎢ ⋮𝑥 ⎢𝐶𝐴𝑠−2 𝐵𝑥 ⎣ and

𝐶 ⎡𝐶𝐴 ⎤ ⎢ ⎥ ℋ = ⎢𝐶𝐴2 ⎥ ⎢⋮ ⎥ ⎣𝐶𝐴𝑠−1 ⎦

0 𝐷𝑥 𝐶𝐵𝑥 𝐶𝐴𝐵𝑥 ⋱ 𝐶𝐴𝑠−3 𝐵𝑥

⋯ 0 ⋱ ⋱ ⋱ ⋯

⋯ ⋯ ⋱ ⋱ ⋱ 𝐶𝐴𝐵𝑥

[5] proposed a controller to minimize the vertical motion of the sprung mass by connecting a virtual damper between the body and the sky, named Sky Hook (SH). It is a practical and feasible solution to improve comfort on primary ride driving. In practice, the adjustable damper is approximated to mimic the virtual damper. The needed damping force is:

(8)

⋯ ⋯ ⋱ ⋱ ⋱ 𝐶𝐵𝑥

0 0 ⎤ ⎥ ⋮ ⎥ ⋮ ⎥ 0 ⎥ 𝐷𝑥 ⎥ ⎦

𝑐 𝑧̇ 𝐹𝑆𝐻 = � 𝑆𝐻 𝑠 𝑐𝑚𝑖𝑛 (𝑧̇𝑠 − 𝑧̇𝑢𝑠 )

𝑧̇𝑠 (𝑧̇𝑠 − 𝑧̇𝑢𝑠 ) ≥ 0 (12) 𝑧̇𝑠 (𝑧̇𝑠 − 𝑧̇𝑢𝑠 ) < 0

The SH force was converted from Newtons to Amperes using a proportional relation of 5 A / 3000 N that is the maximum force that the identified model of the MR damper can achive in the span of 0 to 5 A, [7]. The electric current was saturated in the aforementioned span.

5.-EXPERIMENTAL RESULTS The road profile is composed of a sine wave. Table 1 presents the frequencies and corresponding normalized amplitudes. The real amplitudes has been multiplied by 𝐴𝑜 = 15 𝑚𝑚. First, the vehicle is in uncontrolled system, meaning the control input is 𝐼 = 0𝐴. Then, the SH controller is integrated. Several experiments corresponding to different excitation frequencies has been performed. The sprung mass

To fulfill the requirements of sensitivity to the fault and insensitivity to disturbances like the road profile, it can be synthesized a parity-matrix 𝒲𝑜 as : | 𝒢𝑢 ] = 0

𝑖𝑓 𝑖𝑓

where 𝐹𝑆𝐻 is the SH force, 𝑐𝑆𝐻 is the damping coefficient to achieve the comfort, 𝑐𝑚𝑖𝑛 is the minimum damping coefficient, 𝑧̇𝑠 , and 𝑧̇𝑢𝑠 are the vertical velocity of sprung and unsprung masses. This technique certainly improves the ride comfort. This controller is the reference in comfort oriented suspensions due to its practical implementation and design.

In order to detect the fault 𝐹𝑜 , a residual has to be synthesized. Two approaches has been carried out. First, an exact decoupling face to the state and the road profile has been handled. Then a sub-optimal optimization process face to road profile is presented, making use of lower computation time.

𝒲𝑜 ⋅ [ℋ

(11)

This application allows to reduce the horizon 𝑠, and so computations and computation time, [8].

𝑜

𝑈𝐼𝑠

(10)

The parity matrix is synthesized considering 𝑠 = 𝑛 + 1, where 𝑛 is the number of states of the system. It results a residual given by :

Now, the system under consideration is an LTI system. The aim is to synthesize a residual sensitive to the fault 𝐹𝑜 , and insensitive to the road profile 𝑧𝑟 .

𝑌𝑠

𝒢𝑢 ]) ≤ rank([ℋ]) + rank([𝒢𝑢 ])

Finally, the condition of existence of the parity matrix leads to a long horizon time 𝑠, so large computations. The next part makes use of an optimization approach in order to guarantee the orthogonality (decoupling face) to the states combines with an optimization approach in order to minimize the effect of the road profile.

𝐼𝜌𝑘

𝐹𝑜 𝑧𝑟 𝑦 𝐼 ⎡ ̇ ⎤ ⎡𝑧̇𝑟 ⎤ ⎡𝑦̇ ⎤ ⎡𝐼 ̇ ⎤ 𝐹 ⎢ 𝑜 ⎥ ⎢𝑧̈ ⎥ ⎢𝑦̈ ⎥ ⎢ ⎥ ⎢ ⎥ − 𝒢𝐼 ⎢𝐼 ̈ ⎥ = ℋ𝑥 + 𝒢𝐹 ⎢𝐹𝑜̈ ⎥ + 𝒢𝑢 ⎢ 𝑟 ⎥ ⎢⋮ ⎥ ⎢⋮ ⎥ ⎢ ⋮ ⎥ ⎢⋮ ⎥ (𝑠) ⎣� ⎣𝑦 (𝑠) ⎦ ⎣𝑧𝑟 (𝑠) ⎦ ��� 𝐼 (𝑠) ⎦ ⎣𝐹 ⎦

|

(9)

4

accelerometer is considered as faulty and its fault has to be estimated. Table 1.Frequencies and relative amplitude for each sine wave tested. F(Hz) A(mm) F(Hz) A(mm)

0.5 1

0.57 1

0.65 1

0.76 1

0.85 1

0.97 1

1.10 0.98

1.26 0.95

1.44 0.91

1.64 0.87

1.87 0.83

2.14 0.79

F(Hz) A(mm)

2.44 0.76

2.78 0.72

3.17 0.67

3.62 0.57

4.13 0.47

4.72 0.36

F(Hz) A(mm)

5.38 0.29

6.14 0.26

7.01 0.23

8.00 0.20

9.13 0.14

10.42 0.10

F(Hz) A(mm)

11.89 0.09

13.53 0.08

15.48 0.07

17.66 0.06

20.16 0.05

23.00 0.04 Figure 5. Faulty outputs: left plots are for uncontrolled systems and right plots are for SH control system

Figure 4 compares the uncontrolled system and the SH control system. The SH controller has been implemented. Several experiments for different frequencies between 𝑓𝑖 = 0.5 𝐻𝑧 to 𝑓𝑒 = 23 𝐻𝑧 has been carried out. A pseudoBode diagram shows the benefits of a SH control system against uncontrolled system for the resonance frequenct. At this frequency the sprung mass has a high peak when the suspension is soft (I = 0A), and low peak for the SH control system.

The fault estimation procedures estimate presented fault in Figure 6. This figure shows the effectiveness of the approach. The road profile effect is well attenuated, even in the suboptimal approach. The proposal approach is robust since the fault estimation is good no matter the changes in the road profile and the SH control compensation. This shows the exact decoupling face to the road profile and the manipulation of the semi-active damper.

The residual as proposed in (11) is implemented. However, only the most significant resonating frequency 𝑓𝑐 ≃ 1.88 𝐻𝑧 is presented.

The proposed approach has been proved to be feasible for commercial implementation given its light computing time and high efficiency.

6.-CONCLUSIONS This paper presents an applicative fault detection combined with a Sky Hook Controller for a class of semiactive controllers. The proposed test-bed aims to highlight the performances of a fault detector and a controller for practical industrial applications. Two theoretical approaches has been presented for fault detection. For industrial implementation reasons, a simpler form based on the combination of a parityspace approach and optimization face to road profile is taken for study. The performances of the methodology highlight the interest of this approach. On the other hand, a SkyHook controller has been implemented. The combination of those low computational load algorithms allows for real time implementation, even for industrial applications.

Figure 4. Comparison of sprung mass acceleration for uncontrolled and SH control systems

Acknoledgement.

A fault of 𝐹𝑜 = +1𝑚. 𝑠 −2 is present on the output all the time, except between 𝑡𝑖 = 2.4𝑠 and 𝑡𝑒 = 6.6𝑠 in uncontrolled system and between 𝑡𝑖 = 6.2𝑠 and 𝑡𝑒 = 12.1 in SH control system. The outputs are presented in Figure 5.

This work is partially supported by the PCP 03/10 CONACYT México and the Réseau Universitaire pour l'Innovation Industrielle from CNRS France, the Autotronics 5

Research chair at Tecnológico de Monterrey, and the French National Project INOVE / ANR 2010 BLAN 0308.

6.

Kim, J., Lee, H., 2011. “Sensor Fault Detection and Isolation Algorithm for a Continuous Damping Control System”. J. of Automobile Eng 225, 1347–1364 . 7. Lozoya-Santos, J., Tudón-Martínez, J.C., MoralesMenendez, R., Ramirez-Mendoza, R., Sename, O., Dugard, L., 2011. “Control Strategies for an Automotive Suspension with a MR Damper” . 18 𝑡ℎ IFAC World Congress, Italy , pp:1820–1825. 8. Varrier, S., D.Koenig, Martinez, J., 2012. “Robust Fault Detection for Vehicle Lateral Dynamics”, 51st IEEE Conf on Decision and Control, USA. 9. Varrier, S., Koenig, D., Martinez, J., 2012. “A parity Space-Based Fault Detection On LPV Systems: Approach For Vehicle Lateral Dynamics Control System”, 8 𝑡ℎ IFAC-SAFEPROCESS, pp:1191-1196. 10. Venkatasubramanian, V., Rengaswamy, R., Kavuri, S., Yin, K., 2003. “A Review of Process Fault Detection and Diagnosis Part I Quantitative Model-Based Methods”. Computers and Chemical Eng 27, 293–311 . 11. Vidal, Y., Acho, L., Pozo, F., Rodellar, J., 2010. “Fault Detection in Base-Isolation Systems Via a Restoring Force Observer”, Conf. on Control and Fault Tolerant Systems, France , pp. 94–99 . 12. Wang, H., Song, G., 2011. “Fault Detection and Fault Tolerant Control of a Smart Base Isolation System with Magneto-Rheological Damper”. Smart Mater. Struct. 20, 1–9 .

Figure 6. Fault estimation: upper plot shows the uncontrolled response, and lower plot, the SH control system.

Appendix 𝑧 𝑧̇ 𝑧̈ 𝑘𝑝 𝑐𝑝 𝑐𝑀𝑅 𝐼

REFERENCES 1.

2.

3.

4. 5.

Bertram, T., Bekes, F., Greul, R., Hanke, O., Hab, C., Hilgert, J., Hiller, M., 𝑂̈ttgen, O., Opgen-Rhein, P., Torlo, M., Ward, D., 2003. “Modelling and Simulation for Mechatronic Design in Automotive Systems”. Control Eng. Practice 11, 179–190 . Fischer, D., Isermann, R., 2004. “Mechatronic Semiactive and Active Vehicle Suspensions”. Control Eng. Practice 12 , 1353–1367 . Gáspár, P, Szabó, Z, Bokor, J, 2010. “LPV Design of Fault-Tolerant Control for Road Vehicles”, Conf on Control and Fault Tolerant Systems, France, pp: 807–812. Gertler, J., 1998. “Fault Detection and Diagnosis in Engineering Systems” . CRS Press, 1 𝑠𝑡 Ed. Karnopp, D, Crosby, MJ, Harwood, R, 1974 . “Vibration Control Using Semi-Active Force Generators”. Trans of ASME, J of Eng for Industry 96 , 619–626 .

Variable

𝑔 𝑚𝑠 𝑚𝑢𝑠 𝑘𝑠 𝑘𝑡 𝑓𝑐 𝑎1 𝑎2 𝑏1 𝑏2

6

deflection in meters deflection velocity in meters/sec deflection acceleration in meters/sec² passive stiffness constant in 𝑁/𝑚, passive damping constant in 𝑁𝑠/𝑚, MR damping constant in (𝑁 ⋅ 𝐴)/𝑚, applied current in A Value 9.81 500 100 118,000 170,000 600.95 37.85 22.15 2831 -7897

Unit 𝑚. 𝑠 −2 𝑘𝑔 𝑘𝑔 𝑁. 𝑚−1 𝑁. 𝑚−1 𝑁 𝑁. 𝑠. 𝑚−1 𝑁. 𝑚−1 𝑁. 𝑠. 𝑚−1 𝑁. 𝑚−1

Description gravity acceleration constant sprung mass unsprung mass spring rigidity constant tire rigidity constant Damper non-linear force Suspension non-linear damping Damper non-linear stifness Suspension passive damping Suspension passive stiffness