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A New Mechatronic Solution for the Passive Piloted CRONE Suspension: A Magneto-Rheological Damper
Copyright @ IFAC Mechatronic Systems, Darmstadt, Germany, 2000
A NEW MECHATRONIC SOLUTION FOR THE PASSIVE PILOTED CRONE SUSPENSION: A MAGNETO-RHEOLOGICAL DAMPER
x. Moreau, C. Ramus-Serment, M. Nouillant and A. Oustaloup Laboratoire d'Automatique et de Productique EP 2026CNRS Universite Bordeaux I - ENSERB 351 cours de la Liberation, 33405 Talence cedex, France Tel : +33 (0)556 842 417 - Fax : +33 (0)556 846 644 [email protected]
Abstract: Semi-active control devices have received significant attention in recent years because they offer the adaptability of active control devices without requiring the associated large power sources. The passive piloted CRONE suspension, which uses a specific continuously controlled damper, is an example of a semi-active control device. Its design permits manufacture at the traditional automotive damper cost. Bench tests on a prototype have validated theoretical expectations. From magneto-rheological dampers, a new mechatronic solution can be developed for the CRONE suspension. Magnetorheological dampers are semi-active control devices that use magneto-rheological fluids to produce controllable dampers. They potentially offer highly reliable operation and can be viewed as fail-safe in that they become passive dampers should the control hardware malfunction. Copyright @2000 IFAC Keywords: vehicle dynamics, semi-active suspension, control, analysis
theory [Oustaloup, 1991] for the vehicle suspension.
1. INTRODUCTION The subject of ''vehicle dynamics" is concerned with the movements of vehicles on a road surfaces [GiIlespie, 1992]. The movements of interest are acceleration and braking, ride, and turning. Dynamic behavior is determined by the forces imposed on the vehicle from the tires, gravity, and aerodynamics. The vehicle and its components are studied to determine what forces will be produced by each of these sources at a particular maneuver and trim condition, and how the vehicle will respond to these forces. A traditional analysis method consists in decomposing vehicle dynamics in vertical dynamics (heave, pitch and roll motions) and horizontal dynamics (yaw, longitudinal and lateral motions). These two dynamics are linked [Pacejka and Sharp, 1991] by dynamic behavior of wheel/tire subsystem (Fig. 1). Generally, in numerous works [Brovat, 1997], design and analysis of vehicle chassis control (braking, steering and suspension) are developed in accordance with the decomposition in subsystems of Fig. 2. Here, the approach is concentrated on developing a robust control strategy based on CRONE control
133
lateral 1\
It
."-'''1\
pitch
roll
Fig. 1. Decomposition of vehicle dynamics The suspension device is composed of a traditional spring and a magneto-rheological damper which is a
semi-active control device that uses magnetorheological fluids to produce controllable dampers [Spencer, et al., 1996]. Magneto-Rheological Fluids (or simply "MR" fluids) are suspensions of micronsized, magnetizable particles in oil. Normally, MR fluids are free flowing liquids having a consistency similar to that of motor oil. However, when a magnetic field is applied their consistency changes, virtually instantly, to something that is more like cold peanut butter. The degree of change is proportional to the magnitude of the applied magnetic field. MR fluids actually develop yield strength and behave as Bingham solids when a magnetic field is applied. This change can appear as a very large change in effective viscosity and occurs in less than a few milliseconds. MR fluids are similar to ER, electro-rheological, fluids but much stronger, more stable and easier to use. This paper is organized as follows. In the next section mode ling of road, vehicle and controllable damper is presented. In the following section, the control strategies for the vehicle suspension system are derived. In section 4, performance is presented and fmally concluding remarks are given.
Fig. 2. Decomposition of vehicle in subsystems 2. MODELING 2.1. Road model
There are many possible ways to analytically describe the road inputs, which can classified as shock or vibration. Shocks are discrete events of relatively short duration and high intensity, as e.g. caused by a pronounced bump or pothole on an otherwise smooth road. Vibrations, on the other hand, are characterized by prolonged and consistent excitations that are felt on rough roads. Obviously, a well-designed suspension must perform adequately in a wide range of shock and vibration environments. In the context of vibrations, the road roughness is typically specified as a random process of a given displacement power spectral density (PSD). Many works [Dodds and Robson, 1973] have shown that the road input is a fractal signal. A way of synthesising the vertical road deflection 2o(t) consists in filtering a white gaussian noise wet), whose power spectral density is a constant C, by a non-integer (so called fractional) order filter H(s) whose frequency response is given by [Moreau, et al., 1999] :
H(jOJ)= Ho
in which 12: rl > r2 > 0 and (0 I is called corner frequency. Filtering a white noise wet) by a real noninteger filter provides an output signal whose the power spectral density Sw(w) is given by :
(2)
2.2. Vehicle model
For any vehicle, the suspension system must perform two main functions . Firstly, it provides a high degree of isolation for the vehicle body from the loads applied between the wheels and roads to ensure passenger comfort and secondly, it keeps the wheels in close contact with the road surface to ensure adequate adhesion when accelerating, braking or cornering. These functions must then be optiroized within several constraints: first, the minimum value of the relative body/wheel workspace usage; second, control of vehicle attitude in maneuvering; third minimum value of the power consumption. In fact, the conflict between these various aspects of vehicle behavior is the main problem in suspension system design [Hrovat, 1997]. Consider the two-degree-of-freedom model of a vehicle moving with the constant forward speed V shown in Fig. 3. The two degrees of freedom Z2(t) and ZI(t) are associated with vehicle's sprung and unsprung masses m2 and mb respectively. The model shown in Fig. 3 is often adequate in preliminary studies of vehicle ride dynamics, where the primary suspension (e.g. tire) is mode led via a linear spring with stiffness k 1• The investigation is concentrated on the proper choice of a secondary suspension that would result in the most comfortable ride. The secondary suspension, located between the sprung and unsprung masses, is composed of a traditional spring with stiffness k2 and a semi-active damper that develops a force u(t) .
Z2(t)
Fig. 3. Quarter-car model
134
, (1)
fO(t)
If it is assumed that the tire does not leave the ground and that ZI(t) and Z2(t) are measured from the static equilibrium position, then the application of the fundamental law of dynamics leads to the linearised equations of motion: (3) and in which Z21(t) is the suspension deflection given by (5)
fl(t) the vertical force developed by tire on unsprung mass and whose expression is given by (6) and fo(t) a force applied on sprung mass resulting from aerodynamic loading, braking, turning, accelerating ... The Laplace transform of equations (3) and (4), assuming zero initial conditions leads to expressions of ZI(S) and Z2(S), namely:
ZI(s)=~{fj(s)+k2 Z21(S)-U(s)}
(7)
mls
and
Z2(S )=_1_2 {Fo(s )-k2 Z21 (s )+U(s)} .
(8)
m2 s
To analyse road-holding ability, ride comfort and suspension stroke capability, three additional variables are used: - vertical force fl(t) defmed by relation (6); - vertical acceleration a2 (t )= z2 (t) ; - suspension deflection Z21(t) defmed by relation (5). From relations (7) and (8) it is possible to define a multivariable plant (Fig.4), its inputs and output signals being (Fo(s), Zo(s), U(s» and (FI(s), A 2(s), Z21(S», respectively, namely:
[;:~~J= Z21(S)
[p]
[2~:~] U(s)
,
(9)
the conventional passive system, but without the heavy penalties in cost, weight and complexity associated with fully active suspensions. The concept proposed by Crosby and Kamopp was to use a controllable damper in parallel with a spring. At the present time, there are two types of controllable dampers : - variable orifice hydraulic dampers [Abadie, 1998]; - dampers using magneto-rheological fluids . In the first type of damper, the damping coefficient is varied by regulating the orifice area, which is typically achieved using solenoid-actuated poppet valves or some other electro-mechanically driven valve arrangement. Dampers using MR fluids recently developed by Lord Corporation [Spencer, et aI., 1996] regulate the damping force by modulating the magnetic field through which the MR fluid flows . They have many attractive features, including high yield strength, low viscosity and stable behaviour over a broad temperature range. MR fluids consist of micron-sized, magnetically polarizable particles dispersed in a carrier medium such as mineral or silicone oil. When a magnetic field is applied to the fluids, particle chains form, and the fluid becomes a semi-solid, exhibiting plastic behaviour. MR fluids belong to the group of non-Newtonian fluids and their description as Bingham-plastics is generally recognised. Hence, the following relation between shear stress T and shear rate i is given by : T=TO
+TrY,
(12)
with 1'/ dynamic viscosity (in Pa.s) and TO field induced shear stress (in Pa) which depends on the magnetic field strength. Transition to rheological equilibrium can be achieved in a few milliseconds, providing devices with high bandwidth. Moreover, MR fluids can operate at temperatures from -40°C to 150°C with only slight variations in the yield stress. The MR fluid can be readily controlled with a low voltage (e.g., 12-24 V), current-driven power supply outputting only -1-2 A. A prototype MR damper is considered as shown in Fig. 5. The magnetic field produced in the device is generated by a small electromagnet.
with
and
Fig. 5. Schematic ofMR damper
Fig. 4. Block diagram of the plant
The MR damper is capable of providing a wide dynamic range of force control for very modest input power levels as shown in Fig. 6. The force u(t) developed by the MR damper can be described by an expression of the form : u{t )=U2 (t)+ usAt) , (13) with
2.3. Magneto-rheological damper model
The concept of semi-active suspension incorporating a controllable damper was developed by M. Crosby and D. Karnopp in the 1970's [Kamopp, et al., 1974]. The objective was to produce a suspension system that provided a significant performance improvement over 135
(14) where b2 is damping coefficient obtained with zero current (Fig. 6) and \lsa(t) the semi-active force which is a function of current i(t). fon:e extension 1-----.................
1 - - - - - ........ """'"
-
velocity
compression
Fig. 6. Force versus velocity envelope 3. AUTOMATIC CONTROL The proposed control structure is shown in Fig 7. It consists of feedforward and feedback strategies. Fig. 8 shows the block diagram representing the control strategy in the case of four independent suspension systems. ~-+--===-~~-,
f,
w
1+~
C2{s)=CO
m
:b
in which Co is static gain, ())b and ())h low and high break frequencies, and m a non-integer (so called fractional) order between 0 and 1. This strategy leads to a better robustness of stability degree of vehicle versus sprung mass variations, so a better bodywork holding ability without a diminution of road holding ability [Moreau, et al., 1999]. From this concept, two technological solutions have been developed in the automotive domain. The CRONE suspension can be manufactured from passive and semi-active devices. The first, called passive CRONE suspension [Oustaloup, et aI., 1996], has received the "Trophee AFCET'95" french national award. The second, called passive piloted CRONE suspension is a semi-active device which is composed of a traditional spring with stiffness k2 in parallel with controllable damper [Abadie, 1998]. The objective is to obtain, at each time, a force f2(t) developed by the suspension, namely:
h{t)=-{k2Z21{t)+U{t)),
S=.WIj.I...-=",.".,
(16)
[ 1+- ] OJh
(17)
Brake pressure
equal to the theoretical force fcrone(t) given by the inverse Laplace transform of relation (15). By introducing the relation (13) ofu(t), the expression ofusa(t) is given by: usAt)= fcronAt )-{k2Z21 (t )+b2:i 2l (t)) , (18)
Vehicle speed
~~~.~;..::.:;.~=
Fig. 7. Block diagram of the control strategy
Fig. 8. Block diagram of the global suspension control It frequently happens that one can estimate part or all
of disturbance fo(t) which is the force applied on sprung mass resulting from aerodynamic loading, braking, turning and accelerating. This estimation is obtained from measures of steering wheel position, brake pressure, vehicle speed end vehicle acceleration. This part of the control strategy will be developed in future paper. The development of a robust feedback control strategy based on CRONE control theory for vehicle suspension system is fully discussed in previous works [Moreau, et al., 1999]. With this strategy, the suspension system called CRONE suspension develops a theoretical force fcrone(t) which is a function of the suspension deflection Z21(t) and which obeys symbolically to the relation Fcrone{S)=-C2{S)Z21{S) , (15) where C2(s) is the suspension force-displacement transfer function defined by a non-integer expression of the form:
136
We have shown in previous paper [Moreau, et aI., 1998] that supply energy during activation phases of the CRONE suspension is negligible in comparison with dissipated energy during passive phases. That is why performance of the CRONE piloted suspension is in accordance with theoretical expectation. Finally, the current i(t) applied to electromagnet is deduced from a linearised inverse model of the MR damper, or from a look up table (L.V.T.) obtained after bench tests on the MR damper. The Fig. 9 summarises the control strategy.
Fig. 9. Block diagram representing the control structure of the MR damper 4. PERFORMANCE
The parameters of quarter-car-model used simulation are : m20 = 300 kg - nominal sprung mass: with 150kg ~ m2 ~ 450kg ; - unsprung mass: ml = 24 kg ; - stiffness of tire : k\ = 180000 N/m; - stiffness of suspension: k2 = 5625 N/m ;
for
- damping coefficient : b2= 1000 Ns/m ; - non-integer order: m = 0.8 ; - static gain: Co = 1194; (i)b=0.5rd/s and (i)b=5Ord/s. - break frequencies: Two types of road solicitation are considered: - harmonic solicitation for the nominal sprung mass at • bodywork resonance with an amplitude of 1 cm; • wheel hop resonance with an amplitude of 0.5 cm; - step solicitation with an amplitude of 1 cm for minimal, nominal and maximal values of the sprung mass. Fig. 10 and 11 show the variations with time of the force fl(t) developed by tire, the vertical acceleration a2(t) of the sprung mass and the suspension travel Z21(t), at bodywork resonance and at wheel hop resonance, with zero current and with CRONE control. When the continuous control strategy is activated, the effect on the suspension system is clearly evident, except in the case of the vertical acceleration at wheel hop resonance (Fig.l1.b).
f1(t)(N ) .~r----.--'---.---'----r---r--,
time Cs)
Cal
f1(t)(N)
z21(t) (an)
·20
~
·1001L---'---...i-.---'----'-----i.---" 2 8 10 12 1. time (s)
(a)
02 -
0.1 - -
'().1
Fig. 11. Harmonic solicitation for the nominal sprung mass at wheel hop resonance with an amplitude of 0.5 cm, with zero current before t = 2.5 s and with CRONE control after t = 2.5 s, for: (a) fl(t) ; (b) a2(t) and (c) Z21(t)
'()2
.().3l:----'--'--....L-...i......C"----'--_-'-_--'-_ 2 8 10 12 time (s)
__' 1.
(b) z21(t) (an)
1.5.---r----,----,---,-----.---, oroc.,,,....
------ ..:------
.
CRONE control -- -- -- --~-- -- --
Fig. 12 shows the step responses of the sprung and unsprung masses with zero current (Fig.12.a and c) and with CRONE control (Fig.12.b and d). With the continuous control strategy, it can be seen that the first overshoot remains constant, showing a better robustness in the time domain.
0.5 -- -
·1 .5L---'---...i......C--'---'--~---'
2
6
a
10
12
1.
_(I)
(c)
Fig. 10. Harmonic solicitation for the nominal sprung mass at bodywork resonance with an amplitude of 1 cm, with zero current before t=7 s and with CRONE control after t=7 s, for: (a) fl(t); (b) a2(t) and (c) Z21(t) 137
5. CONCLUSION
"(1)(=)
1.5 , - - - . . , . - - : - - - , - - - - - - r - - r - - - . . , . - - - ,
,
,
,
,
I
In this paper, the performance of a MR damper controlled with the CRONE strategy has been evaluated. The results presented demonstrate the effectiveness of the passive piloted CRONE suspension. This work is a part of a more important project on global chassis control as shown in Fig. 13. The purpose of this project is to investigate the interaction between the aforementioned active or semi-active subsystems in order to provide us with a better understanding of the global chassis control.
.
--- -;---- ---;--------:--------i--------,------
0.5
· ,
,
.,
oL--~-~-~--~-~-~
o
0.5
1.5
2.5
time (s)
(a) " (1)(= )
1.5 , - - - . . , . - - - - , - - - - - - r - - r - - - . . , . - - - ,
,
I
, ,
,
I
I
··
0
, ,
I I
, ,
I ,
,
. .....
,
,
.. ... .
···· ·· 0
,
- - -- ~- - -- --- ~ ----- - - ~- -------:- -- -- - -- :- -- -- --
0.5
I
Fig. 13. Block diagram representing the global chassis control structure
..
.
0.5
2.5
1.5 time (s)
(b) zl (t) (cm)
REFERENCES
1.5
GilIespie, T.D. (1992). Fundamentals of vehicle dynamics. Published by Society ofAutomotive Engineers, Inc. Pacejka, H. B. and R. S Sharp. (1991). Shear Force Development by Pneumatic Tires in Steady-State Conditions : A Review of Modeling Aspects. Vehicle Systems Dynamics, Vo1.20, pp.121-176. Hrovat, D. (1997). Survey of Advanced Suspension Developments and Related Optimal Control Applications. Automatica, Vol. 33, nO 10, pp. 17811817.
i
lA
.:
,
I
,
,
I
- - - - - - - ~ - . - - - - - ~ - - - - - - --:- - - - - - - -:-- - - - - - -~ - - - - --
0.5
··· ··· ,
.. ..
... . ,
I
,
.. .. ._ _
·· ·
oL--~-~-~
o
0.5
~_~_~
2.5
1.5 rime (5)
Oustaloup, A. (1991). La commande CRONE. Edition Hermes, Paris.
(c)
Spencer, B.F., S.l Dyke, M.K. Sain and lD. Carlson of a (1996). Phenomenological Model Magnetorheological Damper. ASCE Journal of Engineering Mechanics, March 10. Dodds C.l and 1.D. Robson (1973). The description of road surface roughness. Journal of Sound and Vibration, Vol. 31, nO 2, pp. 175-183. Moreau X., A. Oustaloup and M. Nouillant (1999). From Analysis to Synthesis of Vehicle Suspension : The CRONE Approach. Proceedings of ECC '99, Karlsruhe, Germany, 31 August-3 September. Karnopp, D., M.J. Crosby, and R.A. Harwood (1974). Vibration control using semi-active force generators. Journal of Engineering for Industry, Vol. 96, n02, pp. 619-626. Abadie V. (1998). Continuously-variable damping system. Patent n098400985.2, PSA Peugeot Citroen. Oustaloup A., X . Moreau and M. Nouillant (1996). The CRONE suspension. Journal of Control Eng. Practice, VolA, W8,pp. 1101-1108. Moreau X., A. Oustaloup and M. Nouillant (1998). Sky hook and CRONE suspensions : a comparison. Proceedings of IFAC Workshop ICV'98, Seville, Spain, March 23·24.
zt (t) (cm)
1.5 r - - . . , . - - - - , - - - - - - r - - r - - - . . , . - - - ,
I
,
,
- ----- -}----- --~-- -- - --
0.5
,
I I I
~-
•
,
- - - - - - -;-- - -- - --)- - - - -- -
, , ,
.
• I ,
.
I I I
.
oL--~-~-~--~-~-~
o
0.5
1.5
2.5
time (. )
(d)
Fig. 12. Step responses of the sprung mass (a) and (b), and of the unsprung mass (c) and (d), with zero current (a) and (c), and with CRONE control (b) and (d), for m2 = 150 kg, m2 = 300 kg, m2 = 450 kg
138
A NEW MECHATRONIC SOLUTION FOR THE PASSIVE PILOTED CRONE SUSPENSION: A MAGNETO-RHEOLOGICAL DAMPER
x. Moreau, C. Ramus-Serment, M. Nouillant and A. Oustaloup Laboratoire d'Automatique et de Productique EP 2026CNRS Universite Bordeaux I - ENSERB 351 cours de la Liberation, 33405 Talence cedex, France Tel : +33 (0)556 842 417 - Fax : +33 (0)556 846 644 [email protected]
Abstract: Semi-active control devices have received significant attention in recent years because they offer the adaptability of active control devices without requiring the associated large power sources. The passive piloted CRONE suspension, which uses a specific continuously controlled damper, is an example of a semi-active control device. Its design permits manufacture at the traditional automotive damper cost. Bench tests on a prototype have validated theoretical expectations. From magneto-rheological dampers, a new mechatronic solution can be developed for the CRONE suspension. Magnetorheological dampers are semi-active control devices that use magneto-rheological fluids to produce controllable dampers. They potentially offer highly reliable operation and can be viewed as fail-safe in that they become passive dampers should the control hardware malfunction. Copyright @2000 IFAC Keywords: vehicle dynamics, semi-active suspension, control, analysis
theory [Oustaloup, 1991] for the vehicle suspension.
1. INTRODUCTION The subject of ''vehicle dynamics" is concerned with the movements of vehicles on a road surfaces [GiIlespie, 1992]. The movements of interest are acceleration and braking, ride, and turning. Dynamic behavior is determined by the forces imposed on the vehicle from the tires, gravity, and aerodynamics. The vehicle and its components are studied to determine what forces will be produced by each of these sources at a particular maneuver and trim condition, and how the vehicle will respond to these forces. A traditional analysis method consists in decomposing vehicle dynamics in vertical dynamics (heave, pitch and roll motions) and horizontal dynamics (yaw, longitudinal and lateral motions). These two dynamics are linked [Pacejka and Sharp, 1991] by dynamic behavior of wheel/tire subsystem (Fig. 1). Generally, in numerous works [Brovat, 1997], design and analysis of vehicle chassis control (braking, steering and suspension) are developed in accordance with the decomposition in subsystems of Fig. 2. Here, the approach is concentrated on developing a robust control strategy based on CRONE control
133
lateral 1\
It
."-'''1\
pitch
roll
Fig. 1. Decomposition of vehicle dynamics The suspension device is composed of a traditional spring and a magneto-rheological damper which is a
semi-active control device that uses magnetorheological fluids to produce controllable dampers [Spencer, et al., 1996]. Magneto-Rheological Fluids (or simply "MR" fluids) are suspensions of micronsized, magnetizable particles in oil. Normally, MR fluids are free flowing liquids having a consistency similar to that of motor oil. However, when a magnetic field is applied their consistency changes, virtually instantly, to something that is more like cold peanut butter. The degree of change is proportional to the magnitude of the applied magnetic field. MR fluids actually develop yield strength and behave as Bingham solids when a magnetic field is applied. This change can appear as a very large change in effective viscosity and occurs in less than a few milliseconds. MR fluids are similar to ER, electro-rheological, fluids but much stronger, more stable and easier to use. This paper is organized as follows. In the next section mode ling of road, vehicle and controllable damper is presented. In the following section, the control strategies for the vehicle suspension system are derived. In section 4, performance is presented and fmally concluding remarks are given.
Fig. 2. Decomposition of vehicle in subsystems 2. MODELING 2.1. Road model
There are many possible ways to analytically describe the road inputs, which can classified as shock or vibration. Shocks are discrete events of relatively short duration and high intensity, as e.g. caused by a pronounced bump or pothole on an otherwise smooth road. Vibrations, on the other hand, are characterized by prolonged and consistent excitations that are felt on rough roads. Obviously, a well-designed suspension must perform adequately in a wide range of shock and vibration environments. In the context of vibrations, the road roughness is typically specified as a random process of a given displacement power spectral density (PSD). Many works [Dodds and Robson, 1973] have shown that the road input is a fractal signal. A way of synthesising the vertical road deflection 2o(t) consists in filtering a white gaussian noise wet), whose power spectral density is a constant C, by a non-integer (so called fractional) order filter H(s) whose frequency response is given by [Moreau, et al., 1999] :
H(jOJ)= Ho
in which 12: rl > r2 > 0 and (0 I is called corner frequency. Filtering a white noise wet) by a real noninteger filter provides an output signal whose the power spectral density Sw(w) is given by :
(2)
2.2. Vehicle model
For any vehicle, the suspension system must perform two main functions . Firstly, it provides a high degree of isolation for the vehicle body from the loads applied between the wheels and roads to ensure passenger comfort and secondly, it keeps the wheels in close contact with the road surface to ensure adequate adhesion when accelerating, braking or cornering. These functions must then be optiroized within several constraints: first, the minimum value of the relative body/wheel workspace usage; second, control of vehicle attitude in maneuvering; third minimum value of the power consumption. In fact, the conflict between these various aspects of vehicle behavior is the main problem in suspension system design [Hrovat, 1997]. Consider the two-degree-of-freedom model of a vehicle moving with the constant forward speed V shown in Fig. 3. The two degrees of freedom Z2(t) and ZI(t) are associated with vehicle's sprung and unsprung masses m2 and mb respectively. The model shown in Fig. 3 is often adequate in preliminary studies of vehicle ride dynamics, where the primary suspension (e.g. tire) is mode led via a linear spring with stiffness k 1• The investigation is concentrated on the proper choice of a secondary suspension that would result in the most comfortable ride. The secondary suspension, located between the sprung and unsprung masses, is composed of a traditional spring with stiffness k2 and a semi-active damper that develops a force u(t) .
Z2(t)
Fig. 3. Quarter-car model
134
, (1)
fO(t)
If it is assumed that the tire does not leave the ground and that ZI(t) and Z2(t) are measured from the static equilibrium position, then the application of the fundamental law of dynamics leads to the linearised equations of motion: (3) and in which Z21(t) is the suspension deflection given by (5)
fl(t) the vertical force developed by tire on unsprung mass and whose expression is given by (6) and fo(t) a force applied on sprung mass resulting from aerodynamic loading, braking, turning, accelerating ... The Laplace transform of equations (3) and (4), assuming zero initial conditions leads to expressions of ZI(S) and Z2(S), namely:
ZI(s)=~{fj(s)+k2 Z21(S)-U(s)}
(7)
mls
and
Z2(S )=_1_2 {Fo(s )-k2 Z21 (s )+U(s)} .
(8)
m2 s
To analyse road-holding ability, ride comfort and suspension stroke capability, three additional variables are used: - vertical force fl(t) defmed by relation (6); - vertical acceleration a2 (t )= z2 (t) ; - suspension deflection Z21(t) defmed by relation (5). From relations (7) and (8) it is possible to define a multivariable plant (Fig.4), its inputs and output signals being (Fo(s), Zo(s), U(s» and (FI(s), A 2(s), Z21(S», respectively, namely:
[;:~~J= Z21(S)
[p]
[2~:~] U(s)
,
(9)
the conventional passive system, but without the heavy penalties in cost, weight and complexity associated with fully active suspensions. The concept proposed by Crosby and Kamopp was to use a controllable damper in parallel with a spring. At the present time, there are two types of controllable dampers : - variable orifice hydraulic dampers [Abadie, 1998]; - dampers using magneto-rheological fluids . In the first type of damper, the damping coefficient is varied by regulating the orifice area, which is typically achieved using solenoid-actuated poppet valves or some other electro-mechanically driven valve arrangement. Dampers using MR fluids recently developed by Lord Corporation [Spencer, et aI., 1996] regulate the damping force by modulating the magnetic field through which the MR fluid flows . They have many attractive features, including high yield strength, low viscosity and stable behaviour over a broad temperature range. MR fluids consist of micron-sized, magnetically polarizable particles dispersed in a carrier medium such as mineral or silicone oil. When a magnetic field is applied to the fluids, particle chains form, and the fluid becomes a semi-solid, exhibiting plastic behaviour. MR fluids belong to the group of non-Newtonian fluids and their description as Bingham-plastics is generally recognised. Hence, the following relation between shear stress T and shear rate i is given by : T=TO
+TrY,
(12)
with 1'/ dynamic viscosity (in Pa.s) and TO field induced shear stress (in Pa) which depends on the magnetic field strength. Transition to rheological equilibrium can be achieved in a few milliseconds, providing devices with high bandwidth. Moreover, MR fluids can operate at temperatures from -40°C to 150°C with only slight variations in the yield stress. The MR fluid can be readily controlled with a low voltage (e.g., 12-24 V), current-driven power supply outputting only -1-2 A. A prototype MR damper is considered as shown in Fig. 5. The magnetic field produced in the device is generated by a small electromagnet.
with
and
Fig. 5. Schematic ofMR damper
Fig. 4. Block diagram of the plant
The MR damper is capable of providing a wide dynamic range of force control for very modest input power levels as shown in Fig. 6. The force u(t) developed by the MR damper can be described by an expression of the form : u{t )=U2 (t)+ usAt) , (13) with
2.3. Magneto-rheological damper model
The concept of semi-active suspension incorporating a controllable damper was developed by M. Crosby and D. Karnopp in the 1970's [Kamopp, et al., 1974]. The objective was to produce a suspension system that provided a significant performance improvement over 135
(14) where b2 is damping coefficient obtained with zero current (Fig. 6) and \lsa(t) the semi-active force which is a function of current i(t). fon:e extension 1-----.................
1 - - - - - ........ """'"
-
velocity
compression
Fig. 6. Force versus velocity envelope 3. AUTOMATIC CONTROL The proposed control structure is shown in Fig 7. It consists of feedforward and feedback strategies. Fig. 8 shows the block diagram representing the control strategy in the case of four independent suspension systems. ~-+--===-~~-,
f,
w
1+~
C2{s)=CO
m
:b
in which Co is static gain, ())b and ())h low and high break frequencies, and m a non-integer (so called fractional) order between 0 and 1. This strategy leads to a better robustness of stability degree of vehicle versus sprung mass variations, so a better bodywork holding ability without a diminution of road holding ability [Moreau, et al., 1999]. From this concept, two technological solutions have been developed in the automotive domain. The CRONE suspension can be manufactured from passive and semi-active devices. The first, called passive CRONE suspension [Oustaloup, et aI., 1996], has received the "Trophee AFCET'95" french national award. The second, called passive piloted CRONE suspension is a semi-active device which is composed of a traditional spring with stiffness k2 in parallel with controllable damper [Abadie, 1998]. The objective is to obtain, at each time, a force f2(t) developed by the suspension, namely:
h{t)=-{k2Z21{t)+U{t)),
S=.WIj.I...-=",.".,
(16)
[ 1+- ] OJh
(17)
Brake pressure
equal to the theoretical force fcrone(t) given by the inverse Laplace transform of relation (15). By introducing the relation (13) ofu(t), the expression ofusa(t) is given by: usAt)= fcronAt )-{k2Z21 (t )+b2:i 2l (t)) , (18)
Vehicle speed
~~~.~;..::.:;.~=
Fig. 7. Block diagram of the control strategy
Fig. 8. Block diagram of the global suspension control It frequently happens that one can estimate part or all
of disturbance fo(t) which is the force applied on sprung mass resulting from aerodynamic loading, braking, turning and accelerating. This estimation is obtained from measures of steering wheel position, brake pressure, vehicle speed end vehicle acceleration. This part of the control strategy will be developed in future paper. The development of a robust feedback control strategy based on CRONE control theory for vehicle suspension system is fully discussed in previous works [Moreau, et al., 1999]. With this strategy, the suspension system called CRONE suspension develops a theoretical force fcrone(t) which is a function of the suspension deflection Z21(t) and which obeys symbolically to the relation Fcrone{S)=-C2{S)Z21{S) , (15) where C2(s) is the suspension force-displacement transfer function defined by a non-integer expression of the form:
136
We have shown in previous paper [Moreau, et aI., 1998] that supply energy during activation phases of the CRONE suspension is negligible in comparison with dissipated energy during passive phases. That is why performance of the CRONE piloted suspension is in accordance with theoretical expectation. Finally, the current i(t) applied to electromagnet is deduced from a linearised inverse model of the MR damper, or from a look up table (L.V.T.) obtained after bench tests on the MR damper. The Fig. 9 summarises the control strategy.
Fig. 9. Block diagram representing the control structure of the MR damper 4. PERFORMANCE
The parameters of quarter-car-model used simulation are : m20 = 300 kg - nominal sprung mass: with 150kg ~ m2 ~ 450kg ; - unsprung mass: ml = 24 kg ; - stiffness of tire : k\ = 180000 N/m; - stiffness of suspension: k2 = 5625 N/m ;
for
- damping coefficient : b2= 1000 Ns/m ; - non-integer order: m = 0.8 ; - static gain: Co = 1194; (i)b=0.5rd/s and (i)b=5Ord/s. - break frequencies: Two types of road solicitation are considered: - harmonic solicitation for the nominal sprung mass at • bodywork resonance with an amplitude of 1 cm; • wheel hop resonance with an amplitude of 0.5 cm; - step solicitation with an amplitude of 1 cm for minimal, nominal and maximal values of the sprung mass. Fig. 10 and 11 show the variations with time of the force fl(t) developed by tire, the vertical acceleration a2(t) of the sprung mass and the suspension travel Z21(t), at bodywork resonance and at wheel hop resonance, with zero current and with CRONE control. When the continuous control strategy is activated, the effect on the suspension system is clearly evident, except in the case of the vertical acceleration at wheel hop resonance (Fig.l1.b).
f1(t)(N ) .~r----.--'---.---'----r---r--,
time Cs)
Cal
f1(t)(N)
z21(t) (an)
·20
~
·1001L---'---...i-.---'----'-----i.---" 2 8 10 12 1. time (s)
(a)
02 -
0.1 - -
'().1
Fig. 11. Harmonic solicitation for the nominal sprung mass at wheel hop resonance with an amplitude of 0.5 cm, with zero current before t = 2.5 s and with CRONE control after t = 2.5 s, for: (a) fl(t) ; (b) a2(t) and (c) Z21(t)
'()2
.().3l:----'--'--....L-...i......C"----'--_-'-_--'-_ 2 8 10 12 time (s)
__' 1.
(b) z21(t) (an)
1.5.---r----,----,---,-----.---, oroc.,,,....
------ ..:------
.
CRONE control -- -- -- --~-- -- --
Fig. 12 shows the step responses of the sprung and unsprung masses with zero current (Fig.12.a and c) and with CRONE control (Fig.12.b and d). With the continuous control strategy, it can be seen that the first overshoot remains constant, showing a better robustness in the time domain.
0.5 -- -
·1 .5L---'---...i......C--'---'--~---'
2
6
a
10
12
1.
_(I)
(c)
Fig. 10. Harmonic solicitation for the nominal sprung mass at bodywork resonance with an amplitude of 1 cm, with zero current before t=7 s and with CRONE control after t=7 s, for: (a) fl(t); (b) a2(t) and (c) Z21(t) 137
5. CONCLUSION
"(1)(=)
1.5 , - - - . . , . - - : - - - , - - - - - - r - - r - - - . . , . - - - ,
,
,
,
,
I
In this paper, the performance of a MR damper controlled with the CRONE strategy has been evaluated. The results presented demonstrate the effectiveness of the passive piloted CRONE suspension. This work is a part of a more important project on global chassis control as shown in Fig. 13. The purpose of this project is to investigate the interaction between the aforementioned active or semi-active subsystems in order to provide us with a better understanding of the global chassis control.
.
--- -;---- ---;--------:--------i--------,------
0.5
· ,
,
.,
oL--~-~-~--~-~-~
o
0.5
1.5
2.5
time (s)
(a) " (1)(= )
1.5 , - - - . . , . - - - - , - - - - - - r - - r - - - . . , . - - - ,
,
I
, ,
,
I
I
··
0
, ,
I I
, ,
I ,
,
. .....
,
,
.. ... .
···· ·· 0
,
- - -- ~- - -- --- ~ ----- - - ~- -------:- -- -- - -- :- -- -- --
0.5
I
Fig. 13. Block diagram representing the global chassis control structure
..
.
0.5
2.5
1.5 time (s)
(b) zl (t) (cm)
REFERENCES
1.5
GilIespie, T.D. (1992). Fundamentals of vehicle dynamics. Published by Society ofAutomotive Engineers, Inc. Pacejka, H. B. and R. S Sharp. (1991). Shear Force Development by Pneumatic Tires in Steady-State Conditions : A Review of Modeling Aspects. Vehicle Systems Dynamics, Vo1.20, pp.121-176. Hrovat, D. (1997). Survey of Advanced Suspension Developments and Related Optimal Control Applications. Automatica, Vol. 33, nO 10, pp. 17811817.
i
lA
.:
,
I
,
,
I
- - - - - - - ~ - . - - - - - ~ - - - - - - --:- - - - - - - -:-- - - - - - -~ - - - - --
0.5
··· ··· ,
.. ..
... . ,
I
,
.. .. ._ _
·· ·
oL--~-~-~
o
0.5
~_~_~
2.5
1.5 rime (5)
Oustaloup, A. (1991). La commande CRONE. Edition Hermes, Paris.
(c)
Spencer, B.F., S.l Dyke, M.K. Sain and lD. Carlson of a (1996). Phenomenological Model Magnetorheological Damper. ASCE Journal of Engineering Mechanics, March 10. Dodds C.l and 1.D. Robson (1973). The description of road surface roughness. Journal of Sound and Vibration, Vol. 31, nO 2, pp. 175-183. Moreau X., A. Oustaloup and M. Nouillant (1999). From Analysis to Synthesis of Vehicle Suspension : The CRONE Approach. Proceedings of ECC '99, Karlsruhe, Germany, 31 August-3 September. Karnopp, D., M.J. Crosby, and R.A. Harwood (1974). Vibration control using semi-active force generators. Journal of Engineering for Industry, Vol. 96, n02, pp. 619-626. Abadie V. (1998). Continuously-variable damping system. Patent n098400985.2, PSA Peugeot Citroen. Oustaloup A., X . Moreau and M. Nouillant (1996). The CRONE suspension. Journal of Control Eng. Practice, VolA, W8,pp. 1101-1108. Moreau X., A. Oustaloup and M. Nouillant (1998). Sky hook and CRONE suspensions : a comparison. Proceedings of IFAC Workshop ICV'98, Seville, Spain, March 23·24.
zt (t) (cm)
1.5 r - - . . , . - - - - , - - - - - - r - - r - - - . . , . - - - ,
I
,
,
- ----- -}----- --~-- -- - --
0.5
,
I I I
~-
•
,
- - - - - - -;-- - -- - --)- - - - -- -
, , ,
.
• I ,
.
I I I
.
oL--~-~-~--~-~-~
o
0.5
1.5
2.5
time (. )
(d)
Fig. 12. Step responses of the sprung mass (a) and (b), and of the unsprung mass (c) and (d), with zero current (a) and (c), and with CRONE control (b) and (d), for m2 = 150 kg, m2 = 300 kg, m2 = 450 kg
138