EQUIVALENCE CLASSES AND OPTIMIZATION OF VEHICLE SWINGING SYSTEMS

(1)

EQUIVALENCE CLASSES AND OPTIMIZATION OF VEHICLE SWINGING SYSTEMS

T. PETER

Department of Transport Engineering ::IIathematics. Technical University. Budapest Received August 3, 1982

Presented by Prof. Dr. L. ILOSVAI

Introduction

Vehicles will be treated as swinging systems. In this case any particular swinging system ⁽⁽¹can be considered as au element of the set V of all vehicle swinging systems.

Let Q be an equivalence relation defined on the pairs of elements of V.

As it is well known, every equivalence relation generates a division of the elements of V into disjoint classes (Fig. 1).

Two vehicle swinging systems, say ((1 and ((2 are equivalent if and only if any pair of input and output belonging to a] helongs also to ((2' and con- versely (Fig. 2).

~:

J

^o,.oz.b€V

I

0, g °z (o,=oz(q))

, 0, 9

^b ⁽

0,

^;>^b^(~))

---

~090 ^¥ ^o€V

i 2. 0, ~ 02 ~ 0z q 0, ,

I 3. (011l 02)1\

(OZIl03)~01~021

Fig. 1

~ o,~

A,={g:.~}

Fig. 2

®

^v

{~.fl."t

^}cv

Fig. 3

(2)

From every equivalence class due to the equivalence relation it is sufficient to pick out one representatiye element making up a set {7., /3, jI, ••. }

c

V (Fig. 3).

I. Basic assumptions

The following assumptions 'will he made on the inyestigated vehicle swinging systems:

I. The car hody is rigid.

H. The springs and shock absorbers have nonlinear characteristics.

HI. The yehicle is excited only by the stochastic road unevennesses.

For the sake of simplicity only plane models will be considered neglect- ing transversal oscillations and studying only yertical ones (Figs 4 and 5).

The systems of differential equations describing such systems are either well known from the literature or can bp set up without particular difficulties.

The applicability of such models for vehicles having not too long frames has been justified by measurements by ROBSO::\",

J.

D., DODDS, C.

J.,

l\IITSCHKE,

11., ILOSVAY, L. etc. but exclusiydy for particular vehicle types i.e. nonlinear systems with given nunwrical characteristics and parameters ([1], [2], [3]).

Z 3 (t) '--'CFE--i---''iir-'

9,(1) '--":~~~~i7?~;'/'7;"";-:,*,7:'7:';:m;;;::;;;~~m!

Fig. 4-

Fig. 5

(3)

OPTDIlZATJO-, OF VEHICLE SWLYGI-,G SYSTEJIS 127

2. The transformations and the structure of systems

Essentially, our equivalence investigation is an expedient transformation of the nonlinear differential equation system describing the vehicle oscillations. Thereby equivalent systems are described by (numerically) identical sets of differential equations. However, the formal identity of the sets of differential equations of two vehicle swinging systems having t"WO or more inputs is only a necessary but not always sufficient condition of equivalence.

::\" ameiy, if thc sets of input functions of the two systems have no common element then no equivalence between the systems (as defined in the Introduc- tion) can be sho"wn. This is the case of two yehiclt' swinging systems equivalent up to tht'ir St'ts of differential equations but of different gauges. (This is the ca!3e of qnasi-('qniyalt'ncp and note that if tllP inputs of the two sY!3tems comprise identical ('xcitation then the systems will satisfy the strict conditions of the definition of t'quivalence.) Consequently, in the case of vehicles with two or more axles a sufficit'nt condition of tht' equivalence is the gauge identity between the two systems.

In this analysis, each equivalence class will comprise systems equivalent up to their sets of diffen>ntial equations; we choose representative elements from them, making up a representative system to be studit'd in the optimization procedurf'. In most practical cases the elements of the system have a constant paramt'ter [1. and freely chosen parameters [ff' 11'] wlwre:

tJ. matrix of generalized mass proportion factors,

[q,

li'J

Yector of nonlincar spring and shock absorber characteristics refer-

ring to unit generalized masses.

Thus, the problem is to determine the optimal value of [ff' 1p

J

for a given value of ^[1.(There are, however, favolIrabli~ cases where ^[.1.is not fixed either.) It should be mentioned that this method suits transformation of existing systems "with excellent properties to new systems with different dimensions and masses but with theoretically identical vibrational properties.

The coordinates of the swinging system are adyjsahly chosen according to Fig. 4,. N amcly then it is easy to see if the front and rear parts of our swinging system can be decomposed into two subsystems with two degrees of freedom vibrating independently of each other.

Similar advantages are offered by choosing the coordinates as shown in Fig . . S where:

Si nonlinear spring characteristics;

K_j nonlinear shock absorber characteristics;

Jl, m - car body masses;

m_i axle masses;

e

i moments of inertia of car bodies about their centroids.

(4)

The system of differential equations describing the vertical vibrations of the single vehicle in Fig. 4 is:

1nl 1nZl

0 0

where

1n12

⁰

1112 0

0

1n3

0 0

, -;-

CoO

e

(1- = - :

ilT

(I

W' ^~ l ^[(,{t, ^t,} I ^~

o Zz K2{tZ - t 4}

o Z:; K3{t3 - gl} -

K₁{t_{1 -}

t 3}

TIll

Zt.

^K1{tl -

g2} - Kz{t z - i

l }

(1)

Sl{Zl - Z3} =0

SZ{ZZ Z4}

[s,{z,

gl} Sl{Zl Zd S\{ZI g2y~S2{Z2 Z,}

l2 H2

m = J;f --=1,--_

2

U

The condition that the system decomposes to two independently vibrating subsystems i",

The system of differential equations of tllP vt>hicle-trailer pair in Fig. 5 is:

1nl m12

^{111 13} ⁰

m21

¹ⁿz m23 ⁰ m3l ^U132 ¹¹¹³ ⁰

0 0 0 m_l

0 0 0 0 0 0 0 0

+ 0 0 0 0 0 0 0 0 m5 ⁰ 0

1n6

- t 1 t z 23

21 K 4{t 4

t5

^K5{ts

_t6 K6{.t6

Sl{Zl - Z\}

Sz{Zz - Zs}

S3{Z3 - Z6}

S4{Z4 - gl} - Sl{Zl S5{Z5 gz} Sz{Zz So{Zo - g3} - S3{Z3

K l{t^{1 -}

t l}

K

₂

{t2 ts}

K3{t3 t 6}

gl} -- Kl{t1 - t 4}

- g2}

Kz{tz t 5}

g3} K3{t3 ts}

(2)

o

(5)

OPTIilUZATIOI> OF VEHICLE SWINGI!"G SYSTEMS 129 where

1 a²

m -_{1 -} L2 (~'''12 1Y.l! I 'e) 1 I ^I [2 (m15 I ² 'e)· 2 '

1 ^'> (1

+

a)2

m. _-= -_L2(NIli

+

^e1 )

+ _P '

^(ml~

+

^eo):

u - .

a(l a) •

e

--'----'- (m1s -L - 2);

[2

The condition that the system decomposes to three vibrating subsystems with two degrees of freedom is:

I

^ml2^m^{m 23}¹³⁼

⁼

⁼ ^m^m3l^m32²¹⁼

⁼

⁼ ⁰⁰

^o.

⁽³⁾

Hence the decomposition has the necessary condition l415 = D~. A sufficient condition is that the parameters of the s"'~nging system satisfy

13= - -L

2 ⁽⁴⁾

where 13 is the distance between the hinge and the rear axle of the vehicle [4]0 Replace the third and the fourth equations of system (1) by sums of the first and third and of the second and fourth equations, respectively. Again, replace the fourth, fifth and sixth equations of system (2) by sums of the first and fourth, the second and fifth, and finally, the third and sixth equations, respectively. Dividing now the i-th row of both systems by the main diagonal element mi of the mass matrix of reduced masses leads to the representative systems (5) and (6) defined by spring and damping characteristics

f{Ji and lPi and reduced mass proportion factors [lij referring to unit reduced masses:

[ .. ^[l3l

'~

^{.. l.2}^{.. l.}

[lu

5

[L12 0 1 0

(5)

(6)

where

u., =

mu :

, l) _

mi

m32 m 12

m3 m3

1n41 m 21

m.1 m.)

1 "u" ^,1.(J13 :0 0 0 2,

"u,1

,

_"un

_:0

₀ 0 2,

}J31 }J 32 1 :0 0 0 23

____________ 1 _ _ _ _ _ _ _ _

~---'"'It

0 Z.

t'ir-J!.4' "u43 :: 1 0

:;J51 .. }J~2'}JS3 _, _, :: _I0 0 Zs

2:5

~ }J ... }J'''' 110 0

U-~~ _ ~2_"_

,!3 ..

J:

.

q!1(Z,-Z•

!V,(Z,-Zs)

1!i3(Z3-;) '!'4(2:4 -91) 4'5(Z5-~) 4-'6(Z5-~)

(i = 1, 2, 3,4) (j = 1,2,3,4)

• '11(Z,-Z4) =Q '1p,-Zs) '13(Z3 -Z6) '14 (Z4-91 ) '£S(Z5-g2) 'l'f(Z6-g3)

s ,()

⁽ⁱ

111i (j

1,2,3, ... ,6);

L 2, 3, ... ,6) (6)

Consider the lower left sub matrix of the partitioned reduced mass proportion matrix [J. of systems (5) and (6). The non-diagonal elements of this matrix (framed in dashed lines) can be obtained as the product of the element of the upper left submatrix on the same place and the diagonal element in the same row of the lower left submatrix.

All in all, the considered systems with four and six degrees of freedom are determined by 12 and 21 independent parameters, respectively (reduced mass proportions and nonlinear characteristics referring to unit reduced masses).

Functioning of the systems is easier to understand by analyzing them decomposed to suitable subsystems. For the sake of simplicity, let the spring and shock absorber characteristics be linear.

Using Laplace transforms on the systems of differential equations per- mits to determine the transfer functions between each input and output. In occurrence of the deduced characteristics, the relations between the subsystems presenting the structure of the system are as shown in Fig. 6.

The possibility of decomposing the swinging vehicle systems to independently vibrating subsystems of t'wo degrees of freedom and reckoning with vehicle seats and passengers as biological vibrating systems justifies the analy-

(7)

OPTIMIZATION OF VEHICLE SWI.YGING SYSTE.1fS

VI, ( p) , - : - - - ' - - .. k) p.,. si

(i = 1.2. 3. {, i

{kj ; 4Jj (v) =ki -v;

Fig. 6

Si '

~i

^(Z)⁼^{Si .}^Z}

I

sis of so-called chain models of vehicle vihrating systems [5]. Fig. 7 shows a chain model of n degrees of freedom; its transformed system of differential equations is:

(J. Z(t)

+

1jJ(Z(t), g(t)) q;(Z(t), g(t)) = 0 (7) where

f.1.12' f.1.12fi23' f^t12,U'}2f.L34' ."

.

^., f.L12f.1.23.U34 ••• f.L(n-l).1l

1 _(.123' (.123(.134' "" ., f.1.23(.134 ••• f.1.(n-l).n

0 1 _f.1.34' "" '" f.L34 ••• f.L(n-l), n

0 0 1,

...

, f.L45 ••• f.L(n-l),n

_ .. - -----~ ~~-------_. --

0 0 0, ." ., 1

5*

(8)

g) = Sl(Zl - g) m1

A link chain model with n degrees of freedom is seen to have 3n - 1 independent parameteres.

j., }Jij = ^11'1•• " }JI,I.' ;

li<j)

Fig. 8

W j ( P ) = - - ' - - ; W,j(p)=kiP+Sj p 2.ki P'Sj

Fig. 8 shows the subsystems, their transfer functions and couplings in the case of a linear system.

3. The optimization

The optimization procedure 'will be illustrated on a plane model 'with 4 degrees of freedom. The original nonlinear spring and shock absorbed characteristics of the plane model are shown in Fig. 9 (full line). Fig. 10 shows the masses and geometrical dimensions of the model.

The example refers to a system such that 1112 # f}2. In the optimization, however, the couplings between the vibrating systems above the front and the rear axle are neglected.

(9)

&

K I ...j

i

30 -i

I

20 ~

I

I I

l

i

10

I

OPTUfIZATION OF T"EHICLE SWDiGL\-G SYSTEMS

Q

b

Fig. 9

M = 1260 ( kg]

60 80

[kg J [kg J

~ = 15000 [cm2 ] M

1₁=120,3 [cm]

[ cmJ

J

I Zl(tJLpE.s=::::::::T::::::=~T==~:=~~

o

b:3(t)L-\:'R::i~~,..::J. ~t-~====:::::=:::=:;f

gl(tJL.,;~~~~m~~,..m""",;~~~&,:;.:!

Fig. 10

133

(10)

The results show the vibration characteristics to be much improved by replacing the spring and shock absorber characteristics of the original system by the optimized ones.

As for the mathematical details, the problem is to optimize two swinging systems with two degrees of freedom, made according to the following steps [6]:

CD

A road section spectral density function Sg(v, w) is chosen, fitting hest to the expected stresses [7]. Beside the angular frequency w[rads] this function depends also on the vehicle speed v[ rn's].

eD

Amplitude transfer characteristics I Wz(fl, kl' kz, 8 1, 8 2, iw) ¹ of impor- tance for these investigations are determined. .

CD

Optimization is made for different speed values v by minimizing obj ec- tive functions for linear combinations of variances of output signah most important for evaluating the 'vibrations:

where

0) kl0PT(V), k20PT(V), 81OPT(V), 820PT(v)

Dz,-Z"OPT(V), Dz,_g,oPT(v), Dz.-Z"OPT(V), Dz,-g,OPT(V)

nz, ~ V ^~ /IWi-,(",k"k""""i

w)

I'S,(v,

w) dw

D z.-, ~ V ^~ /IWz.-,(",k" kg, '" 'g. iw) I' ^S,(v,

^w)

^dm

Di.-t,

= ~ f

¹^W^t.-t,(fl,k^{l ,}^k^{2 ,}^{8 1,8 2,}^ico)¹²^Sg(^V,^{w) dw.}

o

(8)

(9)

CD

The optimum nonlinear characteristics are determined hy generaliz- ing the statisticallinearization method hy BOOTON, R. C. and KAZAKOV, 1. E.

[8], [9]: nonlinear characteristics CfJi' "Pi' are to he found, which, statistically linearized at various speeds, hest approximate the linear optimum parameters computed in step

CD

^[10].

(11)

OPTDIIZATIO_, OF VEHICLE SWIXGISG SYSTEMS

r r

[klOPT(V) -

S

x!l(Dz,-i"OPT(V), x) . 1fl(a, x) dX]2 dv ^-->- Min!

V=

S

[~OPT(V) -

S

x!2(Dt,-z"OPT(V),X) 'IP2(b,x)dx]2dv-->-:Nlin!

o

Vmr,;:;

J

[SlOPT(V) -

J

^x!3(D

z

^,_g,^OPT(V),x) . Cfl(C, x) dX]2 dv -+ Min!

o

Vm"~

J

^[S20PT(V)

o

135

(10)

The functions

!J

(j = 1,2,3,4) in (10) are density functions of the input signals (depending on speed v) of the nonlinear characteristics ?Pi' CfJi

(i = 1,2) divided by the variances of these input signals.

In (10), nonlinear characteristics 71'i' and (Pi are sought for in some defined function form. Our calculations showed the most suitable function forms to be:

1/)I(a, x)

=

a l sign (x). [1 - exp ( - _{a 2 ,}x i)]; a

=Ha

l , a 2]

If'2(b, x) = b₁sign (x). [1 - exp ( - bzl x I)]; b = [bl' bz]

n

(FI(C, x) =

J:

Cjxj; C = [Cl' CZ' ••• , Cn]

j=l n

Cfz(d,x) =

J:

djxj; d = [d_{l ,}d2, •• ·, dn]·

j=l

In final account Eqs (10) serve for the determination of vectors a, b, c, d.

I a, b, c, d = ?

I

4. Results of the optimization

In Fig. 9, nonlinear characteristics of the O1'iginal system A have been plotted in full line, optimized damping characteristics of the system B obtained by optimizing only the shock absorbers in dash line, and optimized spring and damping characteristics of system C in dotted line. An analysis of nonlinear characteristics points out a slight softening of springs and a consider- able increase in damping due to optimization.

Fig. 10 shows swinging comfort factors K approximately proportional to the variance of vertical accelerations at various points of the car body for driving speed v = 50 [km h] on an asphalt road. The comfort factor im-

(12)

P [wl

Fig. 11

A 100 O(Z3-g11 (.,,]

20 ZSTATl

to o ,

20

10

o 50 A 100 o (Z4-g21 ('f,]

ZSTAT2

v (km/h]

100 150

v [km/h I 0~0~---S'b---~1OO---1T50~

Fig. 12

v [km/hi o+---~----~----~+

o 50 100 150

v [km/h]

040~----50~'---I~OO---IT50~~

Fig. 13

(13)

OPTIJIIZATIO_Y OF I-EHICLE STrTYGLYG SYSTEJIS 137

proved by 20 to 25% upon optimlzmg the dampings; combined spring and damping optimization resulted in an improvement of 30 to 45%.

Fig. 11 shows the specific power losses (in W) versus driving speed of the swinging systems. The optimized system shows an average power loss by 5-10%, lower than the original system.

The stability index characterizing the road :;:ection tracking ability of the -wheel versus the driving speed has been plotted in Fig. 12 hy referring the variance of the relative displacement between the axle and the road profile to the static sinking of tl1(' geon1(~trical centre of the wheel. (Cpper diagram referE to the front axle and the lower onE' to the real' axle.)

The optimization improyed the stability indices in front by 25 - .30°"

and in the rear by 30-45~".

Finally, Fig. 13 ~hows the effeetiy(, m{'ans of tllP sppcific dynamic streEses in the hearing springs. A non-negligihle result of the optimization is an improvement of the average dynamic spring loads by 35 - 50o~, in front and by 25--100 ;, in rear. compared to tllt' original sy:;:telll.

Summary

Systems equivalent up to their systems of differential equations have been studied·

Plane models ,,-ith four and six deu:rees of freedom have been found to have 12 and 21 independent parameters, resp.; link mo~dels ,,-ith n degrees of freedom to have 3n-l independent paralueters.

Optimization referred to linear systems for various speeds. An inverse statistical linear- ization method has been applied to find optilhnm nonlinear characteristics which_ statistically linearized for each driving speed_ provide the best quadratic approximation of the optimum characteristics.

The results of the optimization are shown on a plane model of four degrees of freedom corresponding to a medium category passenger car. Optimization of the springs and shock absorbers of this vehicle showed a swinging comfort increase by 30 -'t5%, a 5 -10% diminu- tion of power losses through absorbers_ an improvement by 25 - 50% (in terms of the stability index defined in this paper) of the road section tracking capability of the wheels and finally

a 25 to 500 {) decrease of dynamic stresses in the bearing springs.

References

1. ROBSO", J. D.-DoDD5, C. J.: The response of yehicle components to random road-surface undulations, XIII. FISITA Congress, 1970 17. 2, D.

2. MITSCHKE, ;'.1.: Dynamik der Kraftfahrzeuu:c. Berlin-Xew York. 1972.

3. ILOSVAI, L.: Gepj~rmiivek lengeskenyelme ~s kerek-talaj kapcsolata, ~ITA Doktori erte- kezes, 1978.

4. IC~D_~R. L.-PETER, T.: The effect of trailer on car vibratiom. Periodica Polytechnica.

1982. (In press.) . - -

5. FIALA. E.-CHE::-;CHA""A.;'.1. E.: Untersuchungen an dem linearisierten Schwingungsmodell ein·es Strassenfahrzeuges. ATZ. 1967. Xo.~ 4-5. ~ ~ 6. PETER, T_: Optimization~ of swinging vehicle systems in the presence of stochastic road

excitation. GA;'.nITAGUXG, 1982. Sektion 7.

7. PEYZ::-;ER. J. ~I.-TIKIIO:\"OY. A. A.: Avtomobilllaya promlishlellnosty. :\"0. 1. pp. 15-18, 1964_

(14)

8, BOOTol", R, c,: ~olllillear control systems with random inputs, Trans, IRE, POTe 1954 ..

pp. 9-18.

9. K.AZAKOV, 1. E.: Priblizhenniy metod statisticheskava isledovanija nelineinich sistem.

Trudi. \vIA. im. ~. E. ZHUKOYSKAYA, vip. 294, 195't,

10. PETER, T.: Examination of the linearizability of car vibration models described by nonlinear stoch lstic differential equation systems. Doct. Techn. Thesis, Budapest, 1977.

Dr. Tamas PETER H-1521, Budapest

EQUIVALENCE CLASSES AND OPTIMIZATION OF VEHICLE SWINGING SYSTEMS