Numerical simulations of vehicle platform stabilization

ELSEVIER

Available online at www.sciencedirect.com MATHEMATICAL AND

S C I E N C E ~____DIRECT" COMPUTER MODELLING

Mathematical and Computer Modelling 41 (2005) 1389-1402 www.elsevier.com/locate/mcm

N u m e r i c a l S i m u l a t i o n s of Veh ic l e P l a t f o r m S t a b i l i z a t i o n

W. n. LINDSEY, A. M. SPAGNUOLO, J. C. CHIPMAN AND M. SHILLOR Department of Mathematics and Statistics, Oakland University

Rochester, MI 48309, U.S.A. <walindse> <spagnuol>< chipman><shillor>~oakland, edu

(Received and accepted October 2003)

Abstrac t - -A new model for low-power active control of automotive suspension is described and numerically simulated. It is based on the vertically compressed horizontally sliding spring (VeCHSS) and is in the form of a nonlinear coupled system of ordinary differential equations. The numerical simulations indicate that when the system has one steady state, it is stable; when there are three steady states two are stable and the middle one is unstable. However, when there are two steady states, both are unstable, although the system trajectory is shown to be bounded. The simulations also indicate that the system, without any controls, can follow a periodic road profile well. (~) 2005 Elsevier Ltd. All rights reserved.

K e y w o r d s - - V e h i c l e platform stabilization, Nonlinear dynamical system, VeCHSS, Simulations.

1. I N T R O D U C T I O N

In this work we continue the investigation of a new model for the stabilization of an automotive

platform with a low-power active suspension, begun in [I]. The topic is of considerable importance

for the automotive industry, especially so in racing cars, where up to 40~ of the engine power

may be taken by the active suspension mechanism, placing a considerable burden on the engine.

In the current climate of environmental awareness that places higher value on fuel-efficient cars,

a high-power active control that increases the fuel consumption of the car is not very practical.

Much work has been done with the goal of increasing the performance of active controls for

vehicle suspensions, see, e.g., [2,3] and the references therein.

A novel approach to low-power active suspension has been proposed in [4] and some of its

aspects were studied in [2,3,5-8]. It is based on a nonlinear spring, the so-called VeCHSS (pro-

nounced 'vex') vertically compressed horizontally sliding spring, the central steady state of which

is unstable. A detailed description of the VeCHSS can be found in [9]. The low-power consump-

tion is achieved by the interaction between linear springs and the VeCHSS, that are connected

in series. The operating displacement range of the suspension system is near the unstable steady

state, where the VeCHSS behaves as a spring with negative stiffness, while supporting a posi-

tive load. Using this instability for the active control of the suspension leads to a considerable

decrease in the power needed, as was shown in [1,5].

0 8 9 5 - 7 1 7 7 / 0 5 / $ - see front matter (~) 2005 Elsevier Ltd. All rights reserved. doi: 10.1016/j.mcm.2003.10.054

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1390 W . A . LINDSEY et al.

In this work, we are concerned with the stability of the steady states of the system and its numerical simulations. We do not address the control issues, yet. Tha t will be done in the future. Once gravity is taken into account, the model can have one, two, or three steady states, and their stability is investigated numerically. We also simulate the way the system follows a periodic road.

The model for this system, described in Section 2, is in the form of three coupled nonlinear ordinary differential equations. The system is shown, in Section 3, to have one, two, or three

steady states, depending on the parameters. Numerical simulations are set up and the results are shown in Section 4. The paper concludes in Section 5, where some open control problems are also described.

2. T H E M O D E L

In this section, we follow [6] and describe a model of a vehicle suspension, and as is customary in the industry, we consider a 'quarter vehicle', that is, the system is connected to one of the wheels. During the vehicle's motion the suspension reacts to the contact forces acting on the wheels, and this influences the behavior of the vehicle and determines the comfort of the passengers. The design engineer's goal is to make the ride as smooth as possible. Our aim is to model this behavior and to investigate the conditions for quick and efficient damping of the resulting platform oscillations, thus providing the designer with an effective design tool which allows for the determination of the conditions for a rough ride, and the ability to address them via the

actuator.

The physical setting, depicted in Figure 1, consists of three ordinary (linear) springs and a nonlinear spring, the VeCHSS [6], which is the heart of the system, and will be described below. The mathematical model for the system consists of a nonlinear coupled system of three ordinary differential equations.

X2

Actuator

FA

x 1 l m l

I FD

m 2

k s -Jc s

I R(t)

Figure 1. The setting.

Platform Stabilization 1391

To describe the system's reaction to the road, we assume that the road's profile is given by R = R(t), which has to be determined from measurements. The effects of the rest of the vehicle on the suspension are denoted by FD = FD (t) and the reaction of the force actuator is described by FA = FA(t). This work, however, does not deal with the actuator, so it is always assumed to be zero.

The system, depicted in Figure 1, consists of three masses ml, m2, and m3 connected with springs and dampers and an actuator for the platform control. Mass ml represents the wheel and the suspension itself; m2 represents the part of the vehicle above the wheel (the 'quarter vehicle') including the driver, passengers, and any other paraphernalia that may be in the car. The mass ma is small compared to the other two and represents the active control mechanism.

The tire is characterized by a stiffness kT and a damping coefficient CT; the suspension by ks and cs respectively; and the control mechanism by kR. The nonlinear spring, the VeCHSS, is characterized by the force-displacement relation ~ which will be described below. A detailed investigation of it can be found in [9], and an additional description along with analysis can be found in [6].

Let xl = zl( t ) , x2 = x2(t), and xa = xa(t) be the respective displacements of the masses from their reference positions, positive when upward. The reference position is taken to be the state of the system when gravity is not considered and all the springs, except the VeCHSS, have their natural length. Then, the motion of the system is described in the following model [5],

m l x ' ~ = ] ~ T ( R - - Xl) + -~- CT(R / - - x l ) -.~ k s ( x 2 - - Xl) -~- c s ( x l 2 - - x l ) 4;- kR(x 3 -- Xl) -- m l g , (2.1)

- ~ 2 2 = x '1 k s (x1 - x2) + c s ( x l - ~ ; ) - ~ ( ~ a - ~2) + F ~ + FA - -~2g, (2.2)

, ~ : 4 = k R ( x , - x~) + ¢,(x~ - x~) - -PA - ~ g . (2.3)

Together with the initial conditions,

X 1 (0) = Xl0 , X 2 (0 ) : X20 , X 3 (0 ) = X30 , ( 2 . 4 )

' x ; (0) x~ (0) (2.5) z l (0) = Vl0, = v20, = v30.

Here, the prime indicates the time derivative, the xio are the initial positions of the masses and the rio are their initial velocities, for i = 1, 2, 3.

Gravity force (mg) is taken into account for each mass. Also, by ( f )+, we mean max{0, f}, the positive part of the function f . It is used in the term (R - xl)+ in (2.1) to ensure that the road generates only compressive force on the tire, and cannot cause tension.

The nonlinear spring ~ that is used in equations (2.2) and (2.3) is the VeCHSS [1,6,9]. This spring is positioned vertically, held fixed at the top at O ~, while the lower end is restricted to movement on a horizontal rail. Let Lo be the natural length of the spring, L, the compressed length (the distance from O ~ to O, where the latter is the point on the rail nearest to O/), k, the spring constant and x = x(t), the horizontal distance of the movable end from the origin O. Then, a simple argument shows that the force generated by the VeCHSS is given by

L o (2.6)

This restoring force • = ~(x) is depicted in Figure 2. Clearly, for this choice of • the following relation holds, (g,)2 + L 2 = L02, where x = 4-g* are the two stable equilibrium positions of the VeCHSS. The details and analysis of such a system can be found in [1,9].

Returning to system (2.1)-(2.3), it is noted that since the nonlinear spring • and the func- tion (f)+ are Lipschitz continuous, and the other terms on the right-hand sides are linear, it follows from standard results for ordinary differential equations that the system has a unique local solution for each choice of the initial conditions (2.4) and (2.5).

1392 W . A . LINDSEY et al.

\ \ \ \ \

\

-6

20000

10000

\

\ ~ -'z /Ioo~oo \\~......//"

-20000

2 ,~ \ X

\ \ \ Figure 2. T h e r e s to r ing force ~ vs. d i s p l a c e m e n t of t h e VeCHSS.

3. S T E A D Y S T A T E S A N D E N E R G Y C O N S E R V A T I O N

Since the system is nonlinear, its steady states and their stability provide considerable insight

into the types of motions that it may predict. In this short section, we describe the steady states

of system (2.1)-(2.3). Since the VeCHSS may have one or three steady states, the system itself

may have either one, two, or three steady states, depending on the system parameters.

The steady states of a system are found by setting the time derivatives of equations (2.1)-(2.3)

to zero. Assuming no external forces or inputs besides gravity, we obtain

0 : ]gT ( - - :~1) . I- @ ]~S (2~2 - - :El) n t- kR (X3 -- X l ) - - ? n l g ,

0 : ]gS (:El - - :g2) - - • (X3 -- :E2) - - 77~2g,

0 = k R (:~i - - :~3) - - kIJ (:E3 - - :E2) - - m3g.

(3.1) (3.2) (3.3)

Let 7/* be a solution of the equation,

¢ ( r ] ) - - k R + k S rl + g k R k s " (3.4)

Then, each steady state is determined by

ml ÷ m 2 + m 3 ) :~1 : - - g kT ,

2.2 - k s + kRrl + x l - g --~R + kS ) '

k s rt* + ~21 - g X3 -- kS __ kR jr. kS

(3.5)

(3.6)

(3.7)

A representation of the solutions of (3.4), when the system has only one steady state, is depicted in Figure 3 and is given by the intersection of the curve with the straight line. The right-hand side of the equation is a straight line, and both its slope and the y-intercept are governed by the system parameters, so different cases in which the system has one, two or three steady states can be obtained by altering the slope or the intercept, by changing appropriately the parameters.

We note that by including gravity in the system, the line does not, generally, pass through the origin, as opposed to the case without gravity, when the line always passes through the origin. Without gravity, these equations are those found in [6], and there, there were either one or three

Platform Stabilization 1393

/ r

60000 , ~ / .

40000 ..... "

,

~ \ 20000 ,../

. . . . . . . . . i . . . . . . , < , , , i . . . . I ~ , \ , i

\ \ ~" - 2 0 0 0 0

~ - 4 0 0 0 0

. "' - 6 0 0 0 0 //'

Figure 3. Equation (3.4) has only one solution.

solutions for the system, with a rather standard bifurcation diagram. The bifurcation diagram here is somewhat more complicated since two roots can occur.

For the sake of completeness, we show that the solutions of the system are bounded. This means that even if all the steady states are unstable (when there are two of them) the solutions do not approach infinity, and the trajectories remain in a bounded set.

The system kinetic energy is

1 , 2 lm2(x~(t))2 + ~ma(xa(t)) , EK(t) = -~ml(xl(t)) + 1 , 2 (3.s)

and the potential energy is

1 2 1 Ep (t) : -~kT (Xl (t))2~ - ~:s (x2 (t) -- x I (t)) + ~ k R (x 3 (t) - x I (t))2-~UA (x3 (t) -- x2 (t)). (3.9)

Here, UA is the VeCHSS potential energy,

1 .2"~ UA(X)= l-k2 (x(t)) 2 - k L o ~ / ( x ( t ) ) 2 + L 2 + k L ~ - ~ g ) .

The total energy of the system is given by

E(t ) = EK( t ) + E e ( t ) . (3.10)

Consider now system (2.1)-(2.5). Multiplying equation (2.1) by zi( t) , (2.2) by z~2(t), and (2.3) by x~3(t), integrating over 0 _< s _< t and adding up the resulting expressions, it follows, after integration by parts and rearranging, that

/0 t /0 t E (t) -- E (0) - CT (x i (s)) 2 ds - cs (x i (s) - x i @))2 ds

[ + _ (krR + cTR') xl (s) ds + F~x; (s) ds + f a (x; (~) - ~'~ (~)) ds. 3o

(3.11)

The two integrals with eT and Cs represent the energy dissipated by the two dampers over the time interval [0, t]; the other three integrals on the right-hand side of (3.11) represent the energy input due to the work of external forces.

1394 W . A . LINDSEY et al.

Consequently, when system (2.1)-(2.5) is considered without external inputs, i.e., when R = FD = FA = 0, and without dissipation, i.e., CT = cs = 0, then the energy of the system is conserved. Thus,

E ( t ) : E(O),

for all 0 < t. This estimate, in addition, guarantees tha t the system has a global solution, i.e.,

the solution exists on any time interval [0, T]. On the other hand, when the system includes dissipation but without external inputs, it follows from (3.11) that the energy is decreasing with time. The system, eventually, settles down at one of its steady states, which are discussed above.

Actually, it follows from (3.11), with zero inputs, cs = 0 and CT > 0 that

d E

d - 7 : ,

that is, the system energy decays, and therefore, so do the system oscillations. However, the rate of convergence of xl , x2, and x3 to their steady states is left open.

In particular, it follows from the Cauchy inequality with s applied to (3.11) when FA = FD = O,

that

/o E (t) <__ E (0) + C (kTR + CTR') ds,

where C is a positive constant independent of time, and therefore,

(3C~ ( t ) ) 2 -[- (X 1 ( t ) ) 2 -[- (X~ ( t ) ) 2 -~- (39 2 ( t ) ) 2 -[- (X~ ( t ) ) 2 -~- (X 3 ( t ) ) 2 ~ C ( R , R ' , T ) , (3 .12)

where C is another constant that depends on R, R ~, and T for 0 < t < T. We conclude that the system remains in a bounded set, the sphere with radius v/C, in the six-dimensional phase space for 0 < t < T. When R = 0, the only inputs are the initial data, and then the system will stay confined in a ball with a radius that does not depend on time, but only on the initial conditions. In the absence of dissipation, the system's t rajectory in phase space will forever wander in this ball.

In the numerical simulations presented below, the stability of the steady states is investigated, as well as some types of system behavior. It is found that when there is exactly one steady state,

it is stable; tha t is, small perturbations in the initial positions do not lead to large oscillations in the system. When three steady states are present, the one nearest the origin is unstable while the two other states are stable. The interesting finding is that in the case where there are two steady states, they both appear to be unstable, but, the discussion above indicates that for a road profile R, the solution remains bounded for 0 < t < T. The phase space structure seems to be complicated and this case warrants further study.

4. N U M E R I C A L S I M U L A T I O N S

In this section, we describe the numerical algorithm that was used to simulate the system and

then present the results of some simulations. We note that the simulations described here are for the system without any controls, i.e., FA = FD = 0. As a first step, since each equation contains second derivatives, we convert the system into a first-order system with six equations. This is

done in the usual way. Let ul = xl , u2 = x~, U3 : X2, U4 = X~, U5 ----- X3, and u6 = x~. Then, system (2.1)-(2.3) can be rewritten as

' (4 .1 ) U 1 ~ U2,

Tt%lUt2 = ]gT (1~ -- U l ) j _ ~ - C r ( i~ ' - - %t2)J-~ S (U 3 -- %tl) -J-C S (%L 4 -- U2) -~-k R (~5 -- U l ) - - m l . q , (4 .2)

' (4 .3 ) 11'3 ~ U4,

m 2 u l 4 -~- k S (i t 1 - u3) Jr- e S ( u 2 - u4) - t I / ( u 5 - u3) - m 2 g , (4 .4)

u~ = u6, (4.5)

mau 6' = kR (Ul - us) + ~ (us - u3) - mag. (4.6)

P l a t fo rm Stabi l izat ion 1395

This system can now be solved numerically by any one of the s tandard methods. The one used

to obtain the following simulations was the predictor-corrector scheme in which the fourth-order

Adams-Bashforth predictor method was coupled with the third-order Adams-Moulton eorrector.

Four simulations are shown below. The first three describe the three different situations tha t

can arise for the steady states. The simulations suggest tha t when there is one steady state, it is

stable. When there are two steady states, it seems tha t neither one is stable. When the system has three steady states, the middle one is unstable and the outer two are stable. This is likely to be the case used in automotive applications, and then the design and the actuator will be needed to control the system.

The fourth simulation describes the system response to traveling on a periodic road.

The initial conditions used for all simulations are xl(0) = x2(0) = x3(0) = 0 [m] and x~(0) = I 0 x2( ) = x~(0) = 0 [m/s]. The model parameters used are ml = 50, m2 = 400, m3 = 5 [kg];

kT = 18000, ks = 3000, kR = 20000 [kg/s2]; CT = 100, cs = 100 [kg/s], unless otherwise noted. The parameters used for the VeCHSS, (2.6), are k = 2000 [kg/s2], L = 4 [m], and L0 = 6 [m].

Gravi ty was taken to be 9.8 [m/s2]. Time is measured in seconds, [s].

The computat ions were done on a Pentium 4, 1.6 GHz machine, and a typical run took 180

seconds of CPU time.

We now describe the simulations.

-0 .1 ¸

- 0 . 2

- 0 . 3

- 0 . 4

- 0 . 5

x l [ t ] : dt = .1 e -2 , t = [0., 10 .000 ] , S teps /P t = 10

, 2 . . . . ,6 , 1,o

itit

!//i !i ;x / t l

ii / , i \,J 'v' t,/

x l Phase Plane: dt = . l e - 2 , t = [0., 1 0 . 0 0 0 ] , S teps /P t = 10

i.5 f [ ~.~ --q.3 -q,.2 ~,~i t~

t t / J /U ,

\ = . . . . . . .

Figure 4. T h e g raph of x l vs. t; t h e phase plane, in Case (I).

0.5

-0.5

-1

1396 W.A. LINDSEY et al.

x2[t]: dt = .1 e -2 , t = [0., 10 .000 ] , Steps/Pt = 10 2 . . . . 4, . . . . ,6 , ,

ti / ' i '°'"t I t I

t l i I l i / i t

- ° ° l t ' I I " ~ I ! i

t ~ I t . l

t,/,t / lO

x2 Phase Plane: dt = . l e - 2 , t = [0., 10 .000 ] , Steps/Pt = 10

! t-i(({ - o . , . . . . ~ . ~ . . . . ~ , . . . ) . j~;) ~. Jo t \ \ t . . . . ,/

Figure 5. The graph of x2 vs. t; the phase plane, in Case (I).

ONE STEADY STATE. The first case (1) is when the system has exactly one steady state. The

graph of equation (3.4) is given in Figure 3. The graphs of xl, x2 [m], as functions of time t Is],

and their respective phase planes are depicted in Figures 4 and 5, respectively.

Both variables are seen to oscillate with a slow decay, and from the phase portraits, we see

that xt stabilizes quickly into a periodic oscillation from its initial value.

Two STEADY STATES. The second case (II) is when there are two steady states. From the

applied point of view this case is artificial, since it can be obtained only when the line on the

right-hand side of (3.4) is tangent to the curve. This tangency is structurally unstable, since any

small change in the parameters will modify it to either intersecting the curve at two different

points or not touching. Therefore, it is unlikely to happen in practice. Indeed, it took some

experimentation to obtain this case, and the parameters that were needed, in addition to the

above, are m3 = 362.6089 [kg], ~T = 180000, ks = 30000, kR = 9809.3729 [kg/s2], and k = 20000 [kg/s2]. These parameters lead to the graph of equat ion (3.4) as in Figure 6. The graphs

of Xl and xa as functions of t ime can be seen in Figure 7, the graph of x2 and the phase plane of x2 are given in Figure 8.

P l a t f o r m S t a b i l i z a t i o n 1 3 9 7

\

0.04 :

0.02 :

0

-0.02 :

-0 .04 -

-0.06 :

-0.08 -

-0.1-

-0,12

- 0 . 5 i

--1"

-1 .5

- 2

- 2 . 5

40000

2 0 0 0 0

"" -20000

• . z z

-40000

, J

. x

x J

/ J t

X

Figure 6. The graph of equation (3.4) with two steady states.

xl [t]: dt = .1e-2 , t = [0., 10.000], Steps/Pt = 10

!i ,,!j 19 r /I

i,; lJ \/ l v i t i ? v

? x3[t]: clt = .1 e-2, t = [0., 10.000], Steps/Pt = 10

+ . . . . ,6 . . . . B 10

l

' i

f , i

..... ! i i ~ ........ i

?

:+I ........... I+ i ...... l+j ....

Figure 7. The graphs of Xl and x3 vs. t in Case (II).

1398 W . A. LINDSEY et aL

x2[t]: dt = .le-2, t = [0., 10.000], Steps]Pt = 10

o.o A ' " il !,i i I A /~ o.~ i fi i. ,!,

~ i . . . . . A , ........ i l / i i

o,, i:,;~il l ' ..... I ' ' ~" ............./~'~" ~ ........ i l i i ~ t / i / t l t

-0.2 ,q

x2. Phase Plane: dt = . l e -2 , t = [0., 10.000] , Steps/Pt = 10

\ t-°,~tt t ~ \ . ~ ) ! ) ~ ) i ) °~lji ,

Figure 8. The graph of x2 vs. t and the phase p]ane in Case (If).

\ \ 10000 \

5000

, \ , ~ . . . . --2 \

\ , > ........... • , . , , " " - 15000

/

.,'/+ ~ \

\

\

F i g u r e 9. T h e g r a p h of e q u a t i o n (3.4) in C a s e ( I I I ) .

Plat form Stab i l i za t ion 1399

xl[t]: dt = . l e - 2 , t = [0., 10 .000] , Steps]Pt = 10

o ~ ~ I ' ! ! % ~ i ,' ~ 1,o

I i 1 1 I I i ! I , ; ~ / ~ / w.5. i i I I i i ] I ] t /

/ i [ ] i ] 1 ] i ] I i ] I ! i ~ I , l t / I i t I t , ! ; I t i \/ v

- , t I ! I ~ i U " t t f k/ v

x l P h a s e Plane: dt = .le.~2, t = [0., 1 0 . 0 0 0 ] , Ste! ~s/Pt = 10

/ / # . - /

, , - 0 , 5 , ,

Figure 10. The g raph of x l vs. t and the phase p lane in Case (III) .

x 2 [ t ] : d t = . le--2, t = [0., 10 .000] , Steps/Pt = 10 2 4 6 8 10

I . . . . I . . . . I . . . . I

0

--1"

- 2

-3 ̧

-4

-5 ̧

-6.

'\

........... ~ ................ ~ ! .................... I t. , ] i ' ~ L. ' J i ................ ] ................. i ................. J ........ i .............. / .................... i ........... ;

J i i I ! ~ / i I / i / ~ / i J \ I

t J , , i ! f V v

V v Figure 1 1 . The g raph of x2 vs. t and the phase p lane in Case (III) .

1400 W. A. LINDSEY et al,

x2 Phase Plane: dt = .le-2, t = [0., "i 0.000], Steps/Pt = t 0

/ / J / / / \ \ \\",, \ I ! i ~ k ' . . . . . , ; ~ / l ' l , \\\%, / / / / / /

. . . .

..... ----C~CL ...... 11

F i g u r e 11. (cont . )

xl[t'l: dt = le-1, t = [0, 100.00], Steps/Pt = 10 ---0.0:

10

"5

0

-5

- 1 0

-0.

-0.1,=

-0.:=

--0.25

-0.3

-0.35

-0.4

0 20 40 6o 8o lOO

x l Phase Plane: dt = .1 e - l , t = [0., 100.00], Steps/Pt = I0

0.6 - ~ " - ~ ' I " ~ ~ ' ~ " ~ - ~

/ , , / / . . . . . ~" .......................... -...._., ~ - - . . ' ~

o ~ 4 . . . . ~.3 --o; --o. --o:

-0.2-

Figure 12. T h e g r a p h of x l a n d i t s p h a s e p l a n e w i t h one s t e a d y s t a t e a n d p e r i o d i c roo~d.

-0.3

P l a t fo rm Stabi l izat ion

x2[t]: dt = .le-1, t = [0., 100.00], Steps]Pt = 10

1401

-0.4

--0,5 . i I I

-0.6 I

-0.7 t

-0.8 0 20 40 60 BO 1 O0

X2 Phase Plane: dt = . l e - 1 , t = [0., 100.00] , StepsdPt = 10

0"5 I

~.5

\

Figure 13. T h e g raph of x2 and its phase p lane wi th one s t eady s t a t e and periodic road.

The numerical simulations seem to indicate that the oscillations are erratic, possibly chaotic, and the system may be locally unstable. However, as was shown above (see also [6]) the system variables are bounded, and there may be a locM attractor in the system. This case certainly is interesting from the mathematical point of view.

THREE STEADY STATES. The last case (III) is when three steady states exist. The only param-

eter tha t needed to be changed was kR = 40000 [kg/s2]. This leads to a graph of equation (3.4) depicted in Figure 9. The graphs of Xl, x2, and their phase planes are given in Figures 10 and 11. It is seen that the oscillations are regular and decaying.

P E R I O D I C R O A D . N o w , leaving all of the parameters the same, a periodic road was introduced. The road was assumed to have the equation R(t) = B sin(wt). For the following simulations, we took B = 0.1 [m] and ~ = 3 [rad/s]. Thus, it was Case (I) with a periodic source input. The output is shown in Figures 12 and 13. It is seen that the system settles quickly into periodic oscillations in tandem with the road profile.

5. C O N C L U S I O N S

We described shortly the model for a low-power active automotive suspension, following [6]. Gravity was added to the system presented there, and the resulting system was briefly analyzed

1402 W.A. LINDSEY et al.

and numerically simulated. With the addition of gravity, the system was shown to have either one, two or three steady states. The first and the last cases follow the usual pattern. However, there is numerical evidence that in the case of two steady states both are unstable, while the solution is bounded.

In the numerical simulations presented, the behavior of the steady states was checked. When the simulations started with the initial positions of the masses at their steady states, there was no movement in the system, as was to be expected. When there was exactly one steady state,

this position was found to be stable; that is, small perturbations in the initial positions did not lead to large oscillations in the system. When three steady states were present, the one nearest the origin seemed to be unstable while the other two states were stable. The simulations in Cases (I) and (III) show that the system presented with reasonable parameters behaves as was

to be expected. The system oscillations decayed steadily in the presence of damping. When a periodic road was introduced, the oscillations followed the road after an initial adjusting period of time. Furthermore, the phase planes are mostly smooth after the initial few seconds.

From the point of view of the automotive applications, it seems that Case (III), with three steady states is the one suited for the suspension control of [4-6], since the displacements would be confined to be near the unstable central steady state. Case (II) with two steady states is structurally unstable and is unlikely to be actualized. Case (I) with one steady state is similar to the case of using an ordinary spring, and would not be useful for the low-energy active suspension.

We conclude that the model and the numerical algorithm may be used as an effective tool for the design engineer for such active automotive suspensions. The interesting finding was that in Case (II) where there are two steady states, they both appear to be unstable. Although it is not of interest to the automotive applications, it seems to be of mathematical interest and its mathematical analysis, and further numerical investigations are warranted.

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