Luận văn a new nodeling of the macpherson suspension system and its optimal pole placement control

A New Modeling of the Macpherson Suspension System and its Optimal Pale-Placement Control Pusan National University Pusan National University Hyun-Chul Sohn‘ Graduate College Pusan Na

A New Modeling of the Macpherson Suspension System

and its Optimal Pale-Placement Control

Pusan National University Pusan National University

Hyun-Chul Sohn‘

Graduate College Pusan National University Pusan, 609-735 Kerea

Abstract

In this paper a new model and an optimal polo-plicement control for the Macpherson suspension system arc invesligated ‘The fecus in this new modeling is the rotational mobon of the unsprung mass, ‘Lhe livo generalized coordinates selected in this new model are the vertical displacement of the sprung mass and the angular

displacement of the control arm The vertical acceleration of the sprimg, mase is measured, while the angular

displacement of the control am is estimated, It is shown that the conventioaal model is a special case of this new model since the transfer fnction af this new model coincides with that of the conventional one if the lower support point of the damper is located al the mass center of Une unsprung mass, i is also shown thai hề resonance frequencies of this new model agree better with the experimental revulls, ‘Uherefors, lhis now model

is more general in the sense that it provides an extra degree of freedom in determining a plant model for control system design An optimal pole-placement control which combines the LQ control and the pole-placement technique is investigated using this new model The control law derived for an active suspension system is applied to the system with a semi-active damper, and the performance degradation with a semi-active actuator is

evaluated Simulalions are proviđed

1 Introduction

In this paper, a new model of the suspension system of the Macpherson type for control system design and an optimal pole-placement control for the new model are investigated The roles of a suspension

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system are to support the weight of the vehicle, to isolate the vibrations trom the road, and to maintain

the traction between the tire and the road The suspension systems are classified inta passive and active systems acconding ta the existence of a control input The active suspension system is again

subdivided inlo (wo lypes: a full active and a gemi-active syslem based upon the generation method of (he control force ‘the semi-aclive suspension syslom produces the conizol force by changing the size

of an orifice, and therefore the contral force is a damping force ‘The fill active euspensian syste provides the control force with a separate hydraulic power source In addition, the suspension systems can be divided, by their control methods, into a variety of types: In particular, an adaptive suspension

system is the type of suspension system in which controller parameters are continuously adjusted by

adapting the time-varying characleristics of the sysicm, Adaptive sucthods include a gain scheduling scheme, a model zeference adaptive control, a sclf-uning control, etc

stem ix characterized by the tide quality, the drive stability, the

The performance of a suspension s

size of the rattle space, and the dynamic tire force The prime purpose of adopting an active

suspension system is to improve the ride quality and the drive stability To improve the ride quality,

it is important to isolate the vehicle body from road disturbances, and to decrease the resonance peak

at or near 1 Hz which is known to be a sensitive froqnency to the human body

Since the sky-hook conizol strategy, in which the damper is assumed to be dirceily connected toa stationary ceiling, was introduced in the 1970's, a number of irmovative control methodologies have

been proposed to realize this strategy Alleyne and Iledrick{3] investigated a nonlinear control technique which combines the adaptive control and the variable structure control with an experimental electro-hydrauilic suspension system In their research, the performance of the controlled system was

evalnated by the ability of the actuator outpat to track the specified skyhook force Kim and Yoon[4] investigated a scini-active control law that reproduces the control force of an optimally controlled aclive suspension system while de-emphasizing the damping coefficient variaion ‘Truscott and Wellstead[5] proposed a self-tuning regulator thai adapts the changed velticle conditions al slart-ap

and road disturbances for active suspension systems based on the generalized minimum variance control Teja and Srinivasa|s] investigated a stochastically optimized PTD controller far a linear

quarter car model

Compared with various control algorithms in the literature, research om models of the Macpherson stra wheel suspension ae rare Slemsson et al.[8] proposed three nomliwear models for the Macpherson strut wheel suspension for the analysis of motion, force and deformation Joasson[7]

conducted a fimite element analysis for evaluating the deformations of the suspension components

‘These madels would be appropriate for the analysis af mechanics, but are not adequate for control

system design Intthis paper, a new control-oriented model is investigated

Fig 1 shows a sketch of the Macpherson strut wheel suspension Fig 2 depicts the conventional quitter car maodel of the Macpherson strut wheel suspension for control system design, In the

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conventional model, only the up-down movements of fhe sprung and the mmeprnng masses are

incorporated As are shown in Fig 1 and Fig 3, however, the spring mass, which inchudes the axle and the wheel, is also Linked to the car body by a control am Therefore, the unsprung, mass can totale besides moving up md down, Considering (at belter conlzol performance iz being demanded

by the amlomotive industry, investigation of a now model Gial includes (he rotational motion of the

amspning mass and allows tor the variance of suspension types is justified

‘The Macpherson type snispension system has many merits, such as an independent usage a8 4

shock absorber and the capability of maintaining the wheel in the camber direction The control arm plays several important roles: it supports the suspension system as an additional link to the body, completes the suspension structure, and allows (he rolationl motion of the wsprung imass However, the function of the control wm is completely ignored in the conventional model

In this paper, a new model which incindes a sprang mass, an nnspring mass, a coil spring, 2

damper, and a contral arm is introduced for the first time The mass of the control arm is neglected

For this model, the equations of motion are derived by the Lagrangian mechanics The open loop characteristics of the new model is compared to that of the conventional one The frequency responses and the natural frequencies of the two models arc analyzed under the same conditions Then, it ix shown that the conventional 1/4 car model is a special casc of the new model An optimal pole-placement control, which combines the L( control and the pole-placement technique, is applied

to the new model ‘The clogert loop performance is analyzed Winally, the optimal pole-placement Jaw, derived for the active suspension system, is applied to the semi-active suspension system which is equipped with a continuously variable damper for the purpose of investigating the degradation of the

control performance

The results in this paper are summeuized as follows A new model for the Macpherson type suspension system thal incorporates the rolalional molion of the unsprung mass is suggested for the fiast time If the lower support point of the shock absorber is localed al Le mass center of the

umspning mass, the transfer function, irom road disturbance to the acceleration of the sprung mass, of

the new model coincides with that of the conventional one Therefore, the new model is more

general, fiom the point of view that it can provide an extra degree of fieedom in determining a plant model for control design purpose In the frequency response analysis, the natural frequencies of the new model agrce betlex with he experimenlal resulls An optimal pole-placement conlzol, which combines the LQ control and the pole-placerent Leclnique, is applied lo the new model The control

Yaw, derived for a full active suspension, is applied to the semi-active system with a continuously variable damper It is shown that a small degradation of control performance occurs with a

continnously variate damper

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sprung mass, s,=7,—2,, the tire deflection, x, =2, the velocity of the unspnmg mess[10] Then,

the state equation is

‘The schematic diagram of the Macpherson type suspension system is shown in Fig 3 It is composed

of a quarter car body, an axle and a tire, a coil spring, a damper, an axle, a load disturbance and a control am The car body is assumed to have only a vertical motion If the joint between the control am and Uke car body is assumed (a be @ bushing and the mass of the control am is nol neglecled, Lhe degrees of eedom of Lic whole system is four, ‘The generalized coordinates im [ủy case are z,, d, 6, and 0, However, if the mass of the control am is ignored and the bushing is

agsnmed to be a pin joint, then the degrees of freedom becomes two

‘As the mass of the control arm is much smaller than those of the sprang, mass and the unsprmg mass, it can be neglected Under the above assumption, a new model af the Macpherson type suspension system is introduced in Lig 4 ‘The veitical displacement z, of the sprung mass and the

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rotation angle 6 of the control arm are chosen as the generalized coordinates The assumptions

adopted in Fig, 4 are summarized as follows

1 The horizontal movement of the sprung, mass is neglected, i.e the sprung mass has only the

vertical displacement z„

2 The wisprung mass is linked Lo Lhe cat body im two ways One is via the damper and the other is via the control am, @ denotes the angular displacement of the control am,

3 The values of z, and 6 will be measured from their static equilibrium points

4, The sprung and the unsprung masses are assumed to be particles

$ The mass and the stiffiess of the control am are ignored

The coil spring deflection, the tire deflection and the damping forces are in the linear regions of their operaling ranges

Let (v2.23) (rez) and (¥o.z;) denote the coordinates of point A, B and C, respectively, when the suspension system is at an equilibrium poiat Let the sprung mass be translated by z, upward,

cape (-ĐẺ

(coset con) +? cosecon(er' OY}?

Xa, b, costo! @)

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~lz(sin —8,)—sin(-8,)) +}, h=ẽMg;

(Se)

where, a

3.1 Equations of Motion

‘The equations of motion of the new model are now derived by the L.aprangian mechanics Let 7,

# and D denote the kinetic energy the potential energy and the damping enerpy of the system, respectively Then, these are

Now, introduce the state variables as fy 3) x; x; =È, ý, 8 Of Then, (8)-(9) canbe

written in the state equation as follows

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In order to compare the characteristics of (10) with that of the conventional model, (10) is linearized at

Ue equilibriuim stale where, — Gig %aerFur%ee) — (0.0.0.0) Then, the resulting linear equation is

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4, 008-85) pelo % 6 Aighe | Bhyy sin’( @q)

cage 1

“4 1y fay By coser')

Now, let the output variables be »()=[z, @Ƒ Than the output equation is

wire,

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4 Comparison of Two Models

In the conventional model, where the rowd input is 2,, Une output variables wert assumed lo be the

accelerations of the sprung mass 2, and the nnsprung mass #, In (12), however, while the Toad

Fins, it is shown that the conventional model and the new model coincide if the lower support point of the shock absouber in the new model is located af the mass center of the wisprung mass Let Ig=l ty =t,c0sa and 6,=0' Then, equation (11) has the form

experimental data

For comparing the (wo models, the following parameter values of a typical Macpherson Lype

suspetusion system are used:

Table 1 Comparison of the two models for a typical suspension system

‘The frequency responses of the (wo models, with (he same road mpul, are compared in Vig, 5

‘There axe subslantial differences in the resonance frequencies and peaks belween the two models A tendency of the new model is that the smaller the [./l, is, the lower the resonance frequency is All

the ahove observations are summarized as follows:

(1) The conventional model is considered as a special case of the new model where /; =/„

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(2) The location of the first resonance frequency ix higher in the new model than it is in the conventional one This better agrees with the experimental results The damping ratio, however, is lower inthe new model

(3) For lhe second resonance frequency, bolh the location and ic damping ratio are lower in the new model

§ Optimal Pule-Placement Controk Active Case

In this section, an optimal pole-placement control which combines the LQ control and the pole- placement technique for the new model is presented, The closed Joop system is designed to have desired characteristics by means of Lhe pole-placement technique, while minimizing Uke cost fanction,

as defined by Lhe weightings of he inpul, stale and oulput of the system, as Dollows

‘The considered linear time-invariant system and the pertormance index are

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