MECHANICAL MATHEMATICAL MODELS DESCRIBING THE DYNAMIC BEHAVIOR OF VEHICLE WITH ARM SUSPENSION

Assoc. prof. PhD Eng. Kunchev L ., M. Eng. Pavlov N.

Technical University – Sofia, Bulgaria

Abstract: In this paper dynamic behavior of different types of suspensions for a car are investigated. For describing the mechanical system is used mathematical model based on vector-matrix algebra. The results for the all types of suspensions are compared.

Keywords: MATHEMATICAL MODELS, VIBRATIONS, NATURAL FREQUENCIES, RIDE COMFORT

1. Introduction The aim of this work is to present an approach through which to

formalize the process of searching the system differential equations describing the dynamic behavior of cars with different kind of suspension and different number and method of disposal of the masses over suspended. Formalization of the design process based on mathematical modeling allows you to automate finding the matrices of mass, elasticity and Damping of mechanical-mathematical model of the car.

In the paper is considered mechanical-mathematical models of cars with different schemes of suspension. Scheme of the model are shown in Figures 1 to 6. The models take into account the kinematics of the levers mechanisms in different types of suspension. This allows for more accurate description of the dynamic behavior compared to models in which non suspended masses are presented like the masses with only vertical movement.

The scheme shown in Fig. 1 is a car with front transverse arms and rear longitudinal arms. Longitudinal position of the arms was taken back almost all modern cars from small and medium-sized front-wheel drive and independent rear suspension. This allows a flat floor of the car, on the other hand between the suspension arms can be embed the fuel tank and spare wheel. When working with longitudinal suspension arms, wheels move parallel to one another. When driving in a road with a curve, the wheels slope with angle equal to the slope of the car body. This reduces the forces of adhesion between the road surface end wheels.

The scheme in Fig. 2 is a transverse-arm in the suspension and both axles of the vehicle. Rear suspension lateral arms is mainly used in cars with rear-wheel drive and four-wheel drive, but can be seen on cars with front-wheel drive. It has the following advantages:

- Little change the track of the car and stable linear motion, even on the road with bumps;

- Opportunity for greater lateral forces when the car moved in the curves;

- Small amount of lowering the mass center after loading the car.

Fig. Scheme 3 shows a car with front transverse-arm suspension and rear suspension dependent rigid beam, which are jointed left and right wheel. Moving the left ore right wheel of one dependent suspension movement causes the other. The dependant suspension They lead to an increase in non suspended masses, but have advantages such as constant ground clearance and high reliability.

The models shown in Fig. 4. and Fig. 5 are designed of the suspension as model of Fig. 1. Here is taken account the influence of the mass power unit (engine and gearbox) (Fig. 4) and the masses of seats end passengers on them (Fig.5).

On Fig. 6 is shown front and rear axles of the vehicle with transverse-arm suspension for both axes (as Fig. 2), with additional influences of the rubber-metal shock absorbers in the arms.

The systems looked above are composed of suspended masses and non suspended masses. In the systems of Fig. 4 and 5 are included the masses over suspended (additional) masses. The

suspended masses incorporate the masses of the car body, passengers and cargo. In the mass center the suspended masses attached local coordinate system O0x0y0z0. Non suspended masses include the wheels, beam of Fig. 3, differential axles and connecting elements in an element that is attached to the sprung masses.

Each of these elements is fixed local coordinate system, respectively O1x1y1z1, O2x2y2z2, O3x3y3z3, O4x4y4z4. Masses over suspended include the mass of the power unit (Fig. 4) and tables and seating passengers on them (Fig. 5). They are fixed local coordinate systems Odxdydzd, Od1xd1yd1zd1, Od2xd2yd2zd2, Od3xd3yd3zd3. In the equilibrium condition of the system all the axes of coordinate systems are parallel. All movements of local coordinate systems are calculated after take account to the absolute coordinate system OАxАyАzА.

On the systems from fig. 1 to фиг.6 are applied the next conditions:

- system elements are rigid bodies; - antiroll bar(stabilizer) link are massless and their elasticity is

considered as equivalent springs connected to a lower control arm at a distance Lsf for the front axle end Lsb for the rear axle;

- here takes into account the elastic end damping elements crf, crb, βrf, βrb, , and the elasticity of the front tires cgf and the rear tires cgb;

- here take into account the elastic and damping properties of elastic elements supporting the power unit cde, βde (for Fig. 4);

- here take into account the elastic and damping properties of seats cd, βd (Fig. 5);

-here take into account the elastic properties of the shock absorbers in the arms ct (Fig. 6).

- elastic and damping elements are with linear characterization; - the system is placed in equilibrium by the mass centers of the

wheels resting on a horizontal axis. O1y1 axis coincides with the axis O2y2, and coincides with the axis O3y3 O4y4.

Generalized coordinate systems are:

- z0 - linear displacement of the local coordinate system O0x0y0z0 axis Oz;

- φ0, ψ0 - angular displacement of the local coordinate system O0x0y0z0 respectively around the axes Ox and Oy;

- θ0 - angular rotation of the local coordinate system O0x0y0z0 axis Oz (for Fig. 6);

- φ1 - angular rotation about the axis O1x1 of the coordinate system O1x1y1z1;

- φ2 - angular rotation about the axis O2x2 of the coordinate system O2x2y2z2;

- φ3 - angular rotation about the axis O3x3 of the coordinate system O3x3y3z3 (Fig. 2, 3 and 6);

- φ4 - angular rotation about the axis O4x4 of the coordinate system O4x4y4z4 (Fig. 2 and 6);

- ψ3 - angular rotation about the axis O3y3 of the coordinate system O3x3y3z3 (Fig. 1, 4 and 5);

- ψ4 - angular movement about the axis O4y4 of the coordinate system O4x4y4z4 (Fig. 1, 4 and 5);

57

xA

yA

zA

x1

x0

x2

y1

y2

y0

z1

z0

z2

OA

O2

O1

O0

Lcf

Lmpf

bkf

crf

csf

Lsf

bf

bf

bb

mp

mp

mp

Rkf

cgf

cgf

cgb

cgb

Lf

Lb

bkb

hcf

m0

bf+Lcf

Ag21

H

011'

Ag11

Ag12

qf1

x3

y3

z3

O3

x4

y4

z4

O4

mp

Rkb

csbLsb

hcb

Lkb

Lcb

Lmpb

bb

qb1

qf2

qb2

Brf

Brb

crb

Brf

crf

crbBrb

Fig. 1 Kinematical scheme of a car with front transverse arms and rear longitudinal arms

xA

yA

zA

x1

x3

x4

x0

x2

y1

y2

y3

y4

y0

z1

z0

z3

z4

z2

OA

O2

O1

O4

O3

O0

Lcf

Lmpf

bkf

Lcb

Lmpb

bkb

crb

crb

crf

crf

csf

csb

Lsb

Lsf

bf

bf

bb

bb

mp

mp

mp

mp

Rkf

cgf

cgf

cgb

cgb

Lf

Lb

hcb

hcf

m0

bb+Lcb

bf+Lcf

Ag21

H

011'

Ag11

Ag12

Rkb

q

Brb

Brb

Brf

Brf

Fig. 2 Kinematical scheme of the car with front and rear transverse arms

xA

yA

zA

x1

x0

x2

y1

y2

y0

z1

z0

z2

OA

O2

O1

O4

O0

Lcf

Lmpf

bkf

crb

crb

crf

crf

csf

csb

Lsf

bf

bf

mp

mp

Rkf

cgf

cgf

cgb

cgb

Lf

Lb

hcb

hcf

m0

Lcb

bf+Lcf

Ag21

H

011'

Ag11

Ag12

Rkb

q

x3

y3

O3

z3

mgr

bkb

Brf

Brf

Brb

Brb

Fig. 3 Kinematical scheme of the car with front transverse arms and rear dependent suspension

58

xA

yA

zA

x1

x0

x2

y1

y2

y0

z1

z0

z2

OA

O2

O1

O0

Lcf

Lmpf

bkf

crf

csf

Lsf

bf

bf

bb

mp

mp

mp

Rkf

cgf

cgf

cgb

cgb

Lb

bkb

m0

Ag21

H

011'

Ag11

Ag12

x3

y3

z3

O3

x4

y4

z4

O4

mp

Rkb

csb Lsb

hcb

Lkb

Lcb

Lmpb

bb

Brb

crb

Brf

crf

crbBrb

bd

Brf

Bde

Bde

cde

xd

yd

zd

d

Hd

d

ld

Lf

md

Od

Ld

Fig. 4 Kinematical scheme of the car with front transverse arms and rear longitudinal arms, end suspended power unit on four springs

xA

yA

zA

x1

x0

x2

y1

y2

y0

z1

z0

z2

OA

O2

O1

O0

Lcf

Lmpf

bkf

crf

csf

Lsf

bf

bf

bb

mp

mp

mp

Rkf

cgf

cgf

cgb

cgb

Lf

Lb

bkb

hcf

m0

bf+Lcf

Ag21

H

011'

Ag11

Ag12

q

x3

y3

z3

O3

x4

y4

z4

O4

mp

Rkb

csb Lsb

hcb

Lkb

Lcb

Lmpb

bb

q

q

q

Brf

Brb

crb

Brf

crf

crbBrb

bd1

bd2

bd3

md1

md2

md3

cd1

cd2

cd3

Bd1Ldb

Ldf

Od1

Od2

Od3

Hd1

zd1

zd2

zd3

Fig. 5 Kinematical scheme of the car with front transverse arms and rear longitudinal arms end three additional masses

odresoreni tables.

Lcf

qf1qf2

Lmpf

bf

bkf

mp Y1Z1

H

hcf

Lsf

Y4Z2

Brf Brf crfcrf

m0

mp

z0

y0

z1 z2

y1ct

Lcb

qb1qb2

Lmpb

bb

bkb

mp Y3Z3

H

hcb

Lsb

Y4Z4

Brb Brb crbcrb

m0

mp

z0

y0

z3 z4

y3

ct

Fig. 6 Vehicle's scheme of front and real axles with added shock absorbers

ct

y

y

z

z ct

x

Fig. 7 Rubber shock absorbers

59

- z3- linear displacement of the local coordinate system O3x3y3z3 to absolute OАxАyАzА for axis Oz (Fig. 3);

- zd - linear displacement of the local coordinate system Odxdydzd to absolute OАxАyАzА for axis Oz (Fig. 4);

- φd, ψd - angular rotation of the local coordinate system Odxdydzd to absolute OАxАyАzА respectively around the axes Ox end Oy (Fig. 4);

- zd1, zd2, zd3 - linear displacement of the local coordinate system Od1xd1yd1zd1, Od2xd2yd2zd2 end Od3xd3yd3zd3 for axis Oz (Fig. 5);

- z1, z2, z3, z4 - vertical displacements of the arms at the points of attachment to the body (Fig. 6);

- y1, y2, y3, y4 - horizontal displacements of the arms at the points of attachment to the body (Fig. 6);

Longitudinal elasticity in shock absorbers don’t take account. All displacements are counted against the absolute

coordinate system OАxАyАzА.

List of names: m0 –suspended masses ; mp, mgr- non suspended masses; J0x , J0y , J0z - moment of inertia of the suspended masses in

relation to the axes Ох, Оy, Оz; Jpxf , Jpyf , Jpzf - moments of inertia of the non suspended

masses the front axle in relation to the axes Ох, Оy, Оz; Jpxb, Jpyb, Jpzb - moments of inertia of the non suspended

masses the rear axle in relation to the axes Ох, Оy, Оz; Jgrx, Jgry - moments of inertia of the non suspended masses the

rear axle in relation to the axes Ох, Оy (only for Fig. 3); H - vertical coordinate of center of the non suspended masses

in relation to the arms; bf, bb - horizontal coordinate of mass center of the non

suspended masses to the articulation point of the front and rear arms;

Ldf, Ldb - distance from the mass center to the front axle and the rear axle

bkf , bkb length of the front and rear transverse arms; Lkb length of the longitudinal rear arms (Fig. 1); Lmpf, Lmpb - distance from the mass centers of the front and

rear arms to their articulation point; Lcf , Lcb - distances from the points of attachment of the front

and rear elastic elements to the arms articulation points. For Fig. 3 Lcb is the distance from the point of attachment of the rear elastic element to the longitudinal axis of the vehicle;

Lsf, Lsb - distance from the points of attachment of the front and rear transverse rollbar link to the articulation points;

Rkf, Rkb - radiusis of front and rear wheels; crf, crb - coefficient of elasticity of the front and rear elasticity

elements; cgz - coefficient of elasticity of tires; csf, csb - coefficient of elasticity of the front and rear rollbar

link; βrf, βrb - damping coefficients of front and rear shock

absorbers; md - mass power unit (Fig. 4); Jd - moment of inertia of the power unit along the axes Ox

and Oy (Fig. 4) Hd - vertical coordinate of mass center of the power unit to

the mass center of the suspended masses (Fig. 4); Ld - distance from the mass center of the suspended masses to

the power unit center (Fig. 4); ld, bd - distances from the mass center of the power unit to the

point of attachment on its axes Ox and Oy (Fig. 4); Ldf, Ldb - distances from the mass center of the suspended

masses to the mass centers of the front and rear seats to axel Ох (Fig. 5);

bd1, bd2, bd3 - distances from the longitudinal axis of the vehicle to the mass centers of the seats on the Oy axis (Fig. 5);

Hd1, Hd2, Hd3 - vertical coordinates of mass centers of the seats to the mass center the suspended mass (Fig. 5);

cdе, βde –coefficients of elasticity and damping of the power unit supports (Fig. 4);

cd, βd - coefficients of elasticity and damping of the seat supports (Fig. 5);

ct - coefficient of elasticity of rubber-metal shock absorbers (Fig. 6).

2.Mechanical Mathematical models of arm suspension in various locations and types of arms in the rear axle(Fig. 1, 2 and 3)

In this part is modeled three types of suspension in the rear axle. The consider models are with transverse arms in front axle and the three most common suspension in the rear axle. In the rear axle the type of arms are with longitudinal position, transverse position and suspension-type dependent "whole beam."

2.1.Mechanical mathematical model describing the

dynamic behavior of suspension-type figure. 1., Work [1]. Generalized coordinates for the system are:

{ }

=

4

3

2

1

0

0

0

ψψϕϕψϕz

q }{

=

4

3

2

1

0

0

0

ψψϕϕψϕ

z

q }{

=

4

3

2

1

0

0

0

ψψϕϕψϕ

z

q

(1)

To find the laws of motion in the absolute coordinate system OАxАyАzА is necessary to define the transition matrix of each local coordinate system to the absolute.

- matrix of transition from O0x0y0z0 to OАxАyАzА is:

−−

−

=

1000cos.cossinsin.cos

0cos.sincossin.sin0sin0cos

000000

00000

00

0 zТ А

ψϕϕψϕψϕϕψϕ

ψψ

(2)

- x0 and y0 are zero because it is considered only linear vibration on axis Oz;

- matrix of transition from one coordinate system to O0x0y0z0 can present as:

- matrix of transition from coordinate systems O1x1y1z1, O2x2y2z2, O3x3y3z3, O4x4y4z4, to O0x0y0z0, are:

−−−

=

1000cossin0sincos0001

11

1101 H

bL

T f

f

ϕϕϕϕ

(3)

−−

=

1000cossin0sincos0

001

22

2202 H

bL

T f

f

ϕϕϕϕ

(4)

−−−−

=

1000cos0sin

010sin0cos

33

33

03 H

bL

T b

b

ψψ

ψψ

(5)

−

−−

=

1000cos0sin

010sin0cos

44

44

04 H

bL

T b

b

ψψ

ψψ

(6)

60

- matrix of transition from coordinate system O1x1y1z1 to OАxАyАzА is:

+−−+−++−−−−−−

+−−

==

1000coscossinsincoscoscoscossinsinsincoscoscossinsincos

cossincossinsincoscossinsincossincossincoscossinsinsincoscossinsinsincos

.000000100101001000

00000100101001000

0010100

0101 zHbL

HbLHL

TTTff

ff

f

AA

ψϕϕψϕϕψϕϕϕϕψϕϕϕψϕψϕϕψϕϕψϕϕϕϕψϕϕϕψϕ

ψψϕψϕψψ

(7)

- the transition matrices of coordinate system O2x2y2z2,

O3x3y3z3, O4x4y4z4 to OАxАyАzА, AT2 ,

AT3 , AT4 are similar to

AT1 . The full form of all matrices is presented in [1].

Determination of the radius vectors of mass centers: - radius vector determining mass center of the suspended

mass m0: [ ]Tmo z 1,,0,0 0=ρ (8)

[ ]Tmo z 1,,0,0 0 =ρ (9)

- radius vector determining the mass center of the front right arm mp1:

[ ]Tmpfmp L 1,0,,011 −=ρ (10)

+−−+−−+−−+−

++

==

1coscossinsincossincoscoscossin

cossincossinsinsincossincoscossincossinsin

00000010010

0000010010

0010

1111 zHbLLL

HbLLLHLL

Tffmpfmpf

ffmpfmpf

fmpf

mpAA

mp ψϕϕψϕϕψϕϕϕψϕϕψϕϕψϕϕϕ

ψψϕψ

ρρ

(11)

Working with small angles, for easier calculation is applied sinφ = φ, cosφ = 1. After changing the radius-vector is:

+−−+−−+−−+−

++

=

100010

00010

010

1 zHbLLLHbLLL

HLL

ffmpfmpf

ffmpfmpf

fmpf

Amp ϕψϕϕ

ϕψϕϕϕψϕψ

ρ

(12)

Later on the terms will apply the rules that the magnitudes of higher order will be neglected. The terms like φ.φ sinφ.sinφ etc. don’t takes account.

After differentiation and remove members of a high order is obtained:

−++−=

1)( 1000

0

0

1 ϕψϕϕψ

ρ

mpfffmpf

Amp LLbLz

HH

(13)

- radius - vector determining the mass center of the front left arm mp2 is:

[ ]Tmpfmp L 1,0,,022 =ρ (14)

+−+++++−−

++−

==

1coscossinsincossincoscoscossin

cossincossinsinsincossincoscossincossinsin

00000020020

0000020020

0020

2222 zHbLLL

HbLLLHLL

Tffmpfmpf

ffmpfmpf

fmpf

mpAA

mp ψϕϕψϕϕψϕϕϕψϕϕψϕϕψϕϕϕ

ψψϕψ

ρρ

(15)

Appling the rules for small angles sinφ = φ, cosφ = 1 is obtained:

+−+++++−−

++−

=

100020

00020

020

2 zHbLLLHbLLL

HLL

ffmpfmpf

ffmpfmpf

fmpf

Amp ϕψϕϕ

ϕψϕϕϕψϕψ

ρ

(16)

After differentiation and remove members of a high order is obtained:

++++=

1)( 2000

0

0

2 ϕψϕϕψ

ρ

mpfffmpf

Amp LLbLz

HH

(17)

- radius-vector determining the mass center of the rear right arm mp3 is: [ ]Tmpbmp L 1,0,0,3

3 −=ρ (18)

+−−−−−+−++

+−+−

==

1coscossinsincossincoscoscossincoscossincossinsinsincossincossinsin

sincossinsincoscos

000000300300

00000300300

003030

3333 zHbLLL

HbLLLHLLL

Tbbmpbmpb

bbmpbmpb

bmpbmpb

mpAA

mp ψϕϕψϕψψϕψψϕψϕϕψϕψψϕψψϕ

ψψψψψψ

ρρ

(19)

After take account for angles sinφ = φ, cosφ = 1 is obtained:

+−−−−−+−++

+−+−

=

100030

0003000

030

3 zHbLLLHbLLL

HLLL

bbmpbmpb

bbmpbmpb

bmpbmpb

Amp ϕψψψ

ϕψϕψϕψϕψψψ

ρ

(20)

After differentiation and remove members of a high order is obtained:

−+−−=

1)( 3000

0

0

3 ψψϕϕψ

ρ

mpbbmpbb

Amp LLLbz

HH

(20)

- radius-vector determining the mass center of the rear left arm mp4 is:

[ ]Tmpbmp L 1,0,0,44 −=ρ (21)

+−+−−−++++

+−+−

==

1coscossinsincossincoscoscossincoscossincossinsinsincossincossinsin

sincossinsincoscos

000000400400

00000400400

004040

4444 zHbLLL

HbLLLHLLL

Tbbmpbmpb

bbmpbmpb

bmpbmpb

mpAA

mp ψϕϕψϕψψϕψψϕψϕϕψϕψψϕψψϕ

ψψψψψψ

ρρ

(22)

After take account for angles sinφ = φ, cosφ = 1 is obtained:

+−+−−−++++

+−+−

=

100040

0004000

040

4 zHbLLLHbLLL

HLLL

bbmpbmpb

bbmpbmpb

bmpbmpb

Amp ϕψψψ

ϕψϕψϕψϕψψψ

ρ

(23)

After differentiation and remove members of a high order is obtained:

−+−+=

1)( 4000

0

0

4 ψψϕϕψ

ρ

mpbbmpbb

Amp LLLbz

HH

(24)

If it is take account only linear displacements z, a compilation of the kinetic energy equation using only those components of the radius vectors which refer to the axis Oz.

Determination of the angular velocity of the elements. The components of the angular velocity of the suspended

masses are preset like: 00

ϕω =Ax

00 ψω =Ay (25)

00 =Azω

The angular velocity of the arms where there is relative movement can determine taking account the transition matrices. The angular velocity ωi of the i-th unit to the absolute coordinate system is equal to:

)(00)(000 Tii

Tiii TTTT =−=∗ω (26)

Where ∗0

iω is matrix of the projection of vector ωi to the main coordinate system:

−−

−=∗

00

00

ixiy

ixiz

iyiz

i

ωωωωωω

ω

(27)

To obtain the angular velocity of arms (26) we have to define matrices )(

1TAT , )(

2TAT ,

)(3

TAT , )(4

TAT and their derivatives AT1 , AT2

, AT3 , AT4

. Type of these matrices is presented in [1].

61

After multiplying the matrices and simplify the resulting expressions for the components of angular velocity in three axes are obtained:

0101 cosψϕϕω +=Ax

001001 sinsincos ϕψϕϕψω +=Ay (28)

000011 sincossin ϕψϕψϕω −=Az

After removal of members of a higher order of angular speeds of the front right side is obtained:

101 ϕϕω +=Ax

01 ψω =Ay (29)

01 =Azω

Тhe angular speeds of the other arms are determined as:

- for the front left arm

202 ϕϕω +=Ax

02 ψω =Ay (30)

02 =Azω

- for the rear right arm

03 ϕω =Ax

303 ψψω +=Ay (31)

03 =Azω

- for the rear left arm

04 ϕω =Ax

404 ψψω +=Ay (32)

04 =Azω

Determination of the elastic deformation elements:

Top of the main elastic elements is rigidly fixed to the suspended mass and therefore the study of change in length can be held to coordinate system O0x0y0z0 instead seek to coordinate the absolute coordinate system OАxАyАzА..

- radius vectors the points of joints of the right front spring respect to coordinate system O0x0y0z0 is:

[ ]Tcfcfff hLbL 1,),(,011 +−=′ρ (33)

[ ]TcfL 1,0,,0111 −=′′ρ (34)

−−−−

=

−−−−

=

=

−

−−−

=′′=′′

11sincos

10

0

1000cossin0sincos0001

11

1

11

11111

01

011

HLbL

L

HLbL

L

LHb

L

T

cf

fcf

f

cf

fcf

f

cff

f

ϕϕϕ

ϕϕϕϕ

ρρ

(35)

After removal of members of a higher order and take account that sinφ = φ, cosφ = 1, the change in the length of the elastic element are:

[ ]Tcfcf HLh 0,,0,0 10

110

11 ++=′′−′ ϕρρ (36)

From the radius vector (36) will use a component on axel Oz in the equation of potential energy.

- radius vectors the points of joint of the left front spring respect to coordinate system O0x0y0z0 is:

[ ]Tcfcfff hLbL 1,,,012 +=′ρ (37)

[ ]TcfL 1,0,,0212 =′′ρ (38)

−+

=

−+

=

=

−−

=′′=′′

11sincos

10

0

1000cossin0sincos0

001

22

2

22

22212

02

012

HLbL

L

HLbL

L

LH

bL

T

cf

fcf

f

cf

fcf

f

cff

f

ϕϕϕ

ϕϕϕϕ

ρρ (39)

[ ]Tcfcf HLh 0,,0,0 20

120

12 +−=′′−′ ϕρρ (40)

- radius vectors the points of joints of the right rear spring respect to coordinate system O0x0y0z0 is:

[ ]Tcbbcbb hbLL 1,,),(021 −+−=′ρ (41)

[ ]TcbL 1,0,0,321 −=′′ρ (42)

−−−−−

=

−−−

−−

=

=

−

−−−−

=′′=′′

11sin

cos

100

1000cos0sin

010sin0cos

33

3

33

33

321

03

021

HLb

LL

HLb

LL

L

HbL

T

cb

b

bcb

cb

b

bcb

cb

b

b

ψψ

ψ

ψψ

ψψ

ρρ (43)

[ ]Tcbcb HLh 0,,0,0 30

210

21 ++=′′−′ ψρρ (44)

- radius vectors the points of joints of the left rear spring respect to coordinate system O0x0y0z0 is:

[ ]Tcbbcbb hbLL 1,,),(022 +−=′ρ (45)

[ ]TcbL 1,0,0,422 −=′′ρ (46)

−−

−−

=

−−

−−

=

=

−

−

−−

=′′=′′

11sin

cos

100

1000cos0sin

010sin0cos

44

4

44

44

422

04

022

HLb

LL

HLb

LL

L

HbL

T

cb

b

bcb

cb

b

bcb

cb

b

b

ψψ

ψ

ψψ

ψψ

ρρ (47)

[ ]Tcbcb HLh 0,,0,0 40

220

22 ++=′′−′ ψρρ (48)

- radius vector the contact patch of the front right wheel is: T

kfkfg Rb ]1,,,0[111 −−=ρ (49)

+−−++−−−+−−+++−

+++

==

10001010

0000110

0010

111111 zHbLRRbb

HbLRRbbHLRb

Tffkfkfkfkf

ffkfkfkfkf

fkfkf

gAA

g ϕψϕϕϕϕϕψϕϕϕϕϕ

ψψϕψ

ρρ

(50)

From the radius vector (50) will using a component on axel Oz in the equation of potential energy

100011 )( ϕψϕρ kffkffA

zg bLbbz −++−= (51)

- radius vector of the contact patch of the front left wheel is: T

kfkfg Rb ]1,,,0[212 −=ρ (52)

62

+−+++−+++−++−

+++−

==

10002020

0000220

0020

212212 zHbLRRbb

HbLRRbbHLRb

Tffkfkfkfkf

ffkfkfkfkf

fkfkf

gAA

g ϕψϕϕϕϕϕψϕϕϕϕϕ

ψψϕψ

ρρ

(53)

From the radius vector (53) will using a component on axel Oz in the equation of potential energy

200012 )( ϕψϕρ kffkffA

zg bLbbz ++++= (54)

- radius vector of the contact patch of rear right wheel is: T

kbkbkbg RbL ]1,,,[321 −−−=ρ (55)

+−−−+−−−−+−++−−+

+−+++−

==

100030030

00003003000

00330

321321 zHbLRRbLL

HbLRRbLLHLRRLL

Tbbkbkbkbkbkb

bbkbkbkbkbkb

bkbkbkbkb

gAA

g ϕψψψϕψψϕψϕϕψψϕψϕψϕ

ψψψψψ

ρρ

(56)

After mathematical transformation the component on axel Oz is:

300021 )()( ψψϕρ kbkbbkbbA

zg LLLbbz −+−+−= (57)

- radius vector the contact patch of rear left wheel is: T

kbkbkbg RbL ]1,,,[422 −−=ρ (58)

+−+−+−+−−++++−++

+−+++−

==

100040040

00004004000

00440

422422 zHbLRRbLL

HbLRRbLLHLRRLL

Tbbkbkbkbkbkb

bbkbkbkbkbkb

bkbkbkbkb

gAA

g ϕψψψϕψψϕψϕϕψψϕψϕψϕ

ψψψψψ

ρρ

(59)

After mathematical transformation the component on axel Oz is:

400022 )()( ψψϕρ kbkbbkbbA

zg LLLbbz −+−++= (60)

To find the deformation of the rollbar link should be

determined radius-vectors of the two its joints respect to coordinate system attached to the sprung mass O0x0y0z0.

- radius-vectors the joint points of the front rollbar link is:

The coordinates of the right joint are: T

sfsf L ]1,0,,0[11 −=ρ (61)

−−−−

=

−

−−−

==

1sincos

10

0

1000cossin0sincos0001

1

1

11

1111

01

01 HL

bLL

LHb

L

Tsf

fsf

f

sff

f

sfsf ϕϕ

ϕϕϕϕ

ρρ

(62)

The coordinates of the left joint are: T

sfsf L ]1,0,,0[22 =ρ

−+

=

−−

==

1sincos

10

0

1000cossin0sincos0

001

2

2

22

2222

02

02 HL

bLL

LH

bL

Tsf

fsf

f

sff

f

sfsf ϕϕ

ϕϕϕϕ

ρρ

(63)

The rollbar link deflection respect to Oz is: 21

02

01 ϕϕρρ sfsfsfsf LL −−=− (64)

- radius-vectors the joint points of the rear rollbar link is: - the coordinates of the right side are:

Tsbsb L ]1,0,0,[3

1 −=ρ (65)

−−−

−−

=

−

−−−−

==

1sin

cos

100

1000cos0sin

010sin0cos

3

3

33

33

31

03

01 HL

bLLL

HbL

Tsb

b

bsbsb

b

b

sbsb ψ

ψ

ψψ

ψψ

ρρ

(66)

- the coordinates of the left joint are: T

sbsb L ]1,0,0,[42 −=ρ

−−

−−

=

=

−

−

−−

==

1sin

cos

100

1000cos0sin

010sin0cos

4

4

44

44

42

04

02

HLb

LL

L

HbbL

T

sb

b

sb

sbb

sbsb

ψ

ψ

ψψ

ψψ

ρρ (67)

The rollbar link deflection respect to Oz is: 43

02

01 ψψρρ sbsbsbsb LL +−=− (68)

The kinetic energy of the system is:

24000

23000

22000

21000

20

20

240

230

220

210

200

200

200

))((21

))((21

))((21

))((21

)21(2)

21(2

)(21)(

21)(

21

)(21

21

21

21

ψψϕ

ψψϕ

ϕψϕ

ϕψϕ

ϕψ

ψψψψϕϕ

ϕϕψϕ

mpbbmpbbp

mpbbmpbbp

mpfffmpfp

mpfffmpfp

pxbpyf

pybpybpxf

pxfyx

LLLbzm

LLLbzm

LLbLzm

LLbLzm

JJ

JJJ

JJJzm

−+−++

+−+−−+

++++++

+−++−+

+++

+++++++

+++++=Τ

(69)

Potential energy of the system is:

243

221

224000

213000

222000

211000

24

23

22

21

)(21)(

21

))()((21

))()((21

))((21

))((21)(

21

)(21)(

21)(

21

ψψϕϕ

ψψϕ

ψψϕ

ϕψϕ

ϕψϕψ

ψϕϕ

sbsbsbsfsfsf

bkbkbbkbbgz

bkbkbbkbbgz

fkffkffgz

fkffkffgzcbrb

cbrbcfrfcfrf

LLcLLc

qLLLbbzc

qLLLbbzc

qbLbbzc

qbLbbzcLc

LcLcLc

+−+−−+

+−−+−+++

+−−+−+−+

+−+++++

+−−++−++

++−+=Π

(70)

The function of Relay, taking account only of the damping resistance is:

24

23

22

21 )(

21)(

21)(

21)(

21 ψβψβϕβϕβ cbrbcbrbcfrfcfrf LLLLR ++−+= (71)

After applying the Lagrange equation for the dynamic equation can obtained:

∂∂

−

∂Π∂

−=

∂Τ∂

−

∂Τ∂

qR

qqqdtd

, (72)

for equations describing the laws of motion of the system of Fig. 1:

[ ] [ ] [ ] [ ]FqCqq =+Β+Μ (73)

To analyze how a change in the matrix models with different

suspension and taking into account the additional mass and elastic parameters of rubber-metal vibroizolatori sequentially will be considered the other five models shown in the figure above.

63

[M1] is the inertia matrix that is symmetric about main diagonal with dimension 7x7 and has the following form:

m0+4mp 0 2mpLf-2mp(Lmpb+Lb)

-mpLmpf mpLmpf -mpLmpb -mpLmpb

0

J0x+2Jpxf+ 2Jpxb+

2mpbb2+2mp

(Lmpf+bf)2

0 Jpxf+mpLm

pf (Lmpf+bf)

Jpxf+

mpLmpf

(Lmpf+bf)

mpbbLmp

b -

mpbbLmpb

2mpLf-2mp

(Lmpb+Lb) 0

J0y+2Jpyb+

2Jpyf+2mpLf2

+2mp

(Lmpb+Lb)2

-mpLfLmpf mpLfLmp

f

Jpyb+

mpLmpb

(Lmpb+Lb)

Jpyb+

mpLmpb

(Lmpb+Lb)

-mpLmpf Jpxf+mpLm

pf (Lmpf+bf) -mpLfLmpf

Jpxf+

mpLmpf2

0 0 0

mpLmpf Jpxf+mpLm

pf (Lmpf+bf) mpLfLmpf 0

Jpxf+

mpLmpf2

0 0

-mpLmpb mpbbLmpb Jpyb+mpLmpb

(Lmpb+Lb) 0 0 Jpyb+mp

Lmpb2 0

-mpLmpb -mpbbLmpb Jpyb+mpLmpb

(Lmpb+Lb) 0 0 0 Jpyb+mp

Lmpb2

[C1] is the matrix of elasticity, which is also symmetric and has dimension 7x7:

4cgz 0 2cgzLf -2cgz

(Lb-Lkb) -cgzbkf cgzbkf -cgzLkb -cgzLkb

0 2cgz(bf+bkf

)2+2cgz

(bb+bkb)2 0

cgzbkf

(bf+bkf) cgzbkf

(bf+bkf) cgzLkb

(bb+ bkb)

-cgzLkb

(bb+ bkb)

2cgzLf -2cgz

(Lb+Lkb) 0

2cgzLf2+2cgz

(Lb+Lkb)2 -cgzbkfLf cgzbkfLf

cgzLkb

(Lb+ Lkb)

cgzLkb

(Lb+ Lkb)

-cgzbkf cgzbkf

(bf+bkf) -cgzbkfLf

crfLcf2+

cgzbkf2

+csfLsf2

csfLsf2 0 0

cgzbkf cgzbkf(bf+b

kf) cgzbkfLf csfLsf

2

crfLcf2+

cgzbkf2+

csfLsf2

0 0

-cgzLkb cgzLkb

(bb+ bkb) cgzLkb(Lb+ Lkb)

0 0

crbLcb2+c

gz

Lkb2+

csbLsb2

-csbLsb2

-cgzLkb -cgzLkb

(bb+ bkb) cgzLkb(Lb+ Lkb)

0 0 -csbLsb2

crbLcb2+c

gz

Lkb2+

csbLsb2

64

[B1] is the matrix of dissipative forces, showing the influence of dampers - with symmetrical dimensions 7x7:

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 βrf.L2cf 0 0 0

0 0 0 0 βrf.L2cf 0 0

0 0 0 0 0 βrb.L2cb 0

0 0 0 0 0 0 βrb.L2cb

2.2.Mechanical mathematical models describing the

dynamic behavior of suspension-type figure. 2 and 3, [2]. Generalized coordinates and their derivatives of these models

are: - for Fig. 2:

{ }

=

4

3

2

1

0

0

0

ϕϕϕϕψϕz

q

}{

=

4

3

2

1

0

0

0

ϕϕϕϕψϕ

z

q

}{

=

4

3

2

1

0

0

0

ϕϕϕϕψϕ

z

q

(74)

for fig. 3:

{ }

=

3

3

2

1

0

0

0

z

z

q

ϕϕϕψϕ

}{

=

4

3

2

1

0

0

0

z

z

q

ϕϕϕψϕ

}{

=

4

3

2

1

0

0

0

z

z

q

ϕϕϕψϕ

(75)

Transition matrices - the matrix of transition from coordinate system O0x0y0z0 to

OАxАyАzА..For Fig. 2 and 3 is the same as AT0 for Fig. 1

(equation (2)). - the transition matrices of coordinate systems O1x1y1z1 and

O2x2y2z2 to O0x0y0z0 for Fig. 2 and 3 are the same as 01T , 0

2T for Fig. 1 (equations (3) and (4)).

- differences in the transition matrices of coordinate systems O3x3y3z3 and O4x4y4z4 to O0x0y0z0 is obligated to differences in the type of rear suspension:

-for the suspension type Fig. 2 the transition matrices are:

−−−−

=

1000cos0sin

010sin0cos

33

33

03 H

bL

T b

b

ψψ

ψψ (76)

−

−−

=

1000cos0sin

010sin0cos

44

44

04 H

bL

T b

b

ψψ

ψψ , (77)

-for the suspension type Fig. 3 the transition matrices are:

+−−−−

=

1000cossin0sincos0

001

333

3303 zH

bL

T b

b

ϕϕϕϕ , (78)

for the suspension type Fig. 3 no transition matrice T04

Angular velocity

The components of the angular velocity of the suspended masses are preset and are same as the model of Fig. 1 (equation 25).

Angular velocity of the shoulder: - for Fig. 2 differ only rotational speed of the rear shoulders: - rear right side:

303 ϕϕω +=Ax

03 ψω =Ay

(79)

03 =Azω

- rear left side: 404 ϕϕω +=A

x

04 ψω =Ay

(80)

04 =Azω

- for Fig. 3 differ only angular velocity of the beam rear axle: 303 ϕϕω +=A

x

03 ψω =Ay

(81)

03 =Azω

After expression of kinetic and potential energies and function of Relay for both systems apply the Lagrange equation of second kind (72). For the equations describing the laws of motion of the system is in force (73):

[ ] [ ] [ ] [ ]FqCqq =+Β+Μ ,

65

Where: [M1, 2.3] is the inertia matrix, including changes in patterns Fig. 1,2 and 3, which is symmetric about main diagonal with

dimension 7x7 and has the following form: m0+4mp

I, II 0 2mpLf-

2mp(Lmpb+Lb) I -mpLmpf mpL

mpf -mpLmpb

I,II -mpLmpb

I

m0+2.mp

+mgr III

2mpLf-2mpLb II

0 III

mpLmpb II

2.mp.Lf

-mgrLb III

mgr III

-mpLmpb I, J0x+2Jpxf+ 2Jpxb+ 2mpbb

2+2mp (Lmpf+bf)2 I

0 Jpxf+mpLmpf (Lmpf+bf)

Jpxf+

mpLmpf (Lmpf+bf)

mpbbLmpb I

-mpbbLmpb I

0 III

J0x+2Jpxf+ 2Jpxb+2mp (Lmpf+bf)2

+2.mp.

(Lmpb+bb)2

Jpxb+mp.Lmpb.(Lmpb+bb) II

Jpxb+mp.Lmpb.(Lmpb+bb) II

J0x+2Jpx+Jgrx+2.mp.(Lmpf+bf)2 III

Jgrx III

0 III

2mpLf-2mp (Lmpb+Lb) I

0 J0y+2Jpyb+

2Jpyf+2mpLf2

+2mp (Lmpb+Lb)2 I

-mpLfLmpf

mpLfLmpf

Jpyb+

mpLmpb

(Lmpb+Lb) I

Jpyb+

mpLmpb

(Lmpb+Lb) I

2mpLf-

2mp Lb II

J0y+2.Jpyb+2.Jpyf +2.mp.Lf

2+2.mp. Lb2

II

mp.Lb.Lmpb I -mp.Lb.Lmpb

II

2.mp.Lf –

mgr.Lb III

J0y+2.Jpy+Jgry+2.mLf

2+mgr. Lb2 III

0 III

-mgr.Lb III

-mpLmpf

Jpxf+mpLmpf (Lmpf+bf)

-mpLfLmpf Jpxf+

mpLmpf2

0 0 0

mpLmpf Jpxf+mpLmpf (Lmpf+bf)

mpLfLmpf 0 Jpxf+

mpLmpf

2

0 0

-mpLmpb

II mpbbLmpb

I Jpyb+mpLmpb

(Lmpb+Lb) I 0 0 Jpyb+mp Lmp

I 0

0 Jpxb+mp.Lmpb.(Lm

pb+bb) II mp.Lb.Lmpb Jpxb+mp.Lmpb

2 II

Jgrx III

0 I Jgrx III

-mpLmpb I

-mpbbLmpb I

Jpyb+mpLmpb (Lmpb+Lb) I

0 0 0 Jpyb+mp Lmpb2

I

mp.Lmpb II

Jpxb+mp.Lmpb.(Lm

pb+bb) II -mp.Lb.Lmpb Jpxb+mp.Lmpb

2

II

mgr III

0 III

-mgr.Lb I mgr III

- [C1, 2,3] is the matrix of elasticity, taking into account changes in the patterns Fig. 1,2 and 3, which is also symmetric and has dimension 7x7:

4cgz 0

2cgzLf -2cgz

(Lb+Lkb) I -cgzbkf cgzbkf

2.cgz.Lf -2.cgz.Lb II

2.cgz.Lf -2.cgz.Lb III

0

2cgz(bf+bkf)2+2cgz

(bb+bkb)2 I, II 0

cgzbkf

(bf+bkf) cgzbkf (bf+bkf)

2.cgz.(bf+ bkf)2+2.cgz.bkb III

2cgzLf -2cgz

(Lb+Lkb) I 0

2cgzLf2+2cgz

(Lb+Lkb)2 I -cgzbkfLf cgzbkfLf

2.cgz.Lf 2.cgz.Lf2+2.cgz.Lb

2 II

66

-2.cgz.Lb II

2.cgz.Lf

-2.cgz.Lb III 2.cgz.Lf

2+2.cgz.Lb III

-cgzbkf cgzbkf

(bf+bkf) -cgzbkfLf

crfLcf2+

cgzbkf2

+csfLsf2

csfLsf2

cgzbkf cgzbkf(bf+bkf) cgzbkfLf csfLsf 2

crfLcf2+

cgzbkf2+

csfLsf2

-cgzLkb I cgzLkb

(bb+ bkb) I cgzLkb(Lb+ Lkb) I

0 0 -cgzbkb II cgz.bkb.(bb+ bkb) II cgz.bkb.Lb II

0 III 2cgz.bkb2 III 0 III

-cgzLkb I -cgzLkb

(bb+ bkb) I cgzLkb(Lb+ Lkb) I 0 0

- [B1, 2,3] is the matrix of dissipative forces, including changes in patterns Fig. 1,2 and 3 and showing the influence of dampers - with symmetrical dimensions 7x7:

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 βrf.L2cf 0 0 0

0 0 0 0 βrf.L2cf 0 0

0 0 0 0 0 βrb.L2

cb 0 2βrb.Lcb

2 III

0 0 0 0 0 0 βrb.L2

cb

2βrb III

Cells colored in orange (index I) refer to the system of Figure 1, those in blue (index II) of Figure 2 and those in green on Fig. 3 (index III). The white cells without index are common to the three matrix schemes.

3.Mechano-mathematical models of the arm suspension with different location and type of masses over suspended and take into account rubber shock absorbers (Fig. 4, 5 and 6).

3.1. Mechanical mathematical model describing the dynamic behavior of the car for the effect of the power unit (Fig. 4), work [3].

For the basic scheme of the arm suspension is considered the model of Fig. 1. Generalized coordinates and their derivatives for the system of Fig. 4 are:

{ }

=

d

d

dz

z

q

ψϕ

ψψϕϕψϕ

4

3

2

1

0

0

0

}{

=

d

d

dz

z

q

ψϕ

ψψϕϕψϕ

4

3

2

1

0

0

0

}{

=

d

d

dz

z

q

ψϕ

ψψϕϕψϕ

4

3

2

1

0

0

0

(82)

- matrix of transition from O0x0y0z0 to OAxAyAzA is the

same as AT0 for models Fig. 1, 2 and 3, equation (2);

- matrix of transition from O1x1y1z1, O2x2y2z2, O3x3y3z3, O4x4y4z4 to OAxAyAzA are the same as 0

1T , 02T , 0

3T , 04T

for models Fig. 1, 2 and 3, equation (3), (4), (5) and (6); - matrix of transition from Odxdydzd to O0x0y0z0 for small

angles has the form:

−−−−

=

10001

01.01

0

dddd

ddd

dd

d zH

L

Tϕψ

ϕψϕψ

(83)

The components of the angular velocity of the suspended and non suspended masses are identical to those of the model in Fig. 1 - equations (25), (29), (30), (31) and (32).

Angular velocity of the power unit is obtained after multiplying the matrices )(0 T

dT and 0dT and simplifying the

resulting expressions: d

Adx ϕϕω += 0

dAdy ψψω += 0

(84)

04 =Azω

After expression of kinetic and potential energy function of Relay system applies Lagrange equation (72). For the equations describing the laws of motion for the system can write:

[ ] [ ] [ ] [ ]FqCqq =+Β+Μ

67

Matrices [M4], [B4] and [C4] for the model of Fig. 4, which further includes the mass of power unit, located on the suspended masses to amend these models in Fig. 1, 2 and 3 as follows:

- the matrix [M4] change is in rows 1 to 3 - M4(1,1), M4 (1.3), M4(2.2), M4(3.1) and M4(3.3 ) supplemented with shown below articles showing the correlation between the suspended mass and

masses over suspended; appear new rows 8 ÷ 10 , and new columns 8 ÷ 10 which are colored in gray;

- the matrix [C4] appears only new rows 8 ÷ 10 , and new columns 8 ÷ 10 which are colored in gray

- The matrix [B4] also appears only new rows 8 ÷ 10 , and new columns 8 ÷ 10 which are colored in gray.

Inertia matrix [M4]:

М1+md

М1+0

М1+mdLd

М1 М1 М1 М

1 М

1-md

М1

М1

М1 М1+Jd

М1+0 М1 М1 М1 М

1 М

1 М

1+Jd

М1

М1+mdLd

М1 М1+J

d+mdLd2 М1 М1 М1

М1

М1-mdLd

М1

М1+Jd

М1 М1 М1 М1 М1 М1 М

1 М

1 М

1 М

1

М1 М1 М1 М1 М1 М1 М

1 М

1 М

1 М

1

М1 М1 М1 М1 М1 М1 М

1 М

1 М

1 М

1

М1 М1 М1 М1 М1 М1 М

1 М

1 М

1 М

1

М1-md

М1 М1-

mdLd М1 М1 М1

М1

М1+md

М1

М1

М1 М1+Jd

М1 М1 М1 М1 М

1 М

1 М

1+Jd

М1

М1 М1 М1+Jd

М1 М1 М1 М

1 М

1 М

1 М

1+Jd

Elastic matrix [C4]:

C1 C1 C1 C1 C1 C1 C1 C1 C1 C1

C1 C1 C1 C1 C1 C1 C1 C1 C1 C1

C1 C1 C1 C1 C1 C1 C1 C1 C1 C1

C1 C1 C1 C1 C1 C1 C1 C1 C1 C1

C1 C1 C1 C1 C1 C1 C1 C1 C1 C1

C1 C1 C1 C1 C1 C1 C1 C1 C1 C1

C1 C1 C1 C1 C1 C1 C1 C1 C1 C1

C1 C1 C1 C1 C1 C1 C1 C1

+4cde C1 C1

C1 C1 C1 C1 C1 C1 C1 C1 C1

+4cdbde2 C1

C1 C1 C1 C1 C1 C1 C1 C1 C1 C1

+4cdeld2

Damping matrix [B4]:

B1 B1 B1 B1 B1 B1 B1 B1 B1 B1

B1 B1 B1 B1 B1 B1 B1 B1 B1 B1

B1 B1 B1 B1 B1 B1 B1 B1 B1 B1

B1 B1 B1 B1 B1 B1 B1 B1 B1 B1

B1 B1 B1 B1 B1 B1 B1 B1 B1 B1

B1 B1 B1 B1 B1 B1 B1 B1 B1 B1

B1 B1 B1 B1 B1 B1 B1 B1 B1 B1

B1 B1 B1 B1 B1 B1 B1 B1

+4βde B1 B1

B1 B1 B1 B1 B1 B1 B1 B1 B1

+4βdebd2 B1

B1 B1 B1 B1 B1 B1 B1 B1 B1 B1

+4βdeld2

In matrix [M4], [C4], [B4] elements labeled M1, C1, B1

show that the same elements are equal numbers of matrix elements [M1]. In this way, do not repeat recordings.

For example, elements М4(1,1)=М1+md, which means that the М4(1,1)=M1(1,1)+md=m0+4mp+md.

68

This method of approach will be used in subsequent expression of matrix.

3.2.Mechanical mathematical model describing the

dynamic behavior of the vehicle and takes into account elastic suspended passenger seats (Fig. 5), work [4].

In the case, comparing the model takes into account the masses over suspended masses (Fig. 5) with a model fig. 1 generalized coordinates and their derivatives are:

{ }

=

3

2

1

4

3

2

1

0

0

0

d

d

d

zzz

z

q

ψψϕϕψϕ

}{

=

3

2

1

4

3

2

1

0

0

0

d

d

d

zzz

z

q

ψψϕϕψϕ

}{

=

3

2

4

3

2

1

0

0

0

d

d

d

zzz

z

q

ψψϕϕψϕ

(85)

- matrix of transition from O0x0y0z0 to OАxАyАzА is the same

as AT0 for models Fig. 1, 2 and 3, equation (2); - matrix of transition from O1x1y1z1, O2x2y2z2, O3x3y3z3,

O4x4y4z4 to OАxАyАzА are the same as 01T , 0

2T , 03T , 0

4T for models Fig. 1, 2 and 3, equation (3), (4), (5) and (6);

- the transition matrices of coordinate systems Od1xd1yd1zd1, Od2xd2yd2zd2, Od3xd3yd3zd3 to O0x0y0z0 have to type:

−=

1000100010001

11

101

dd

d

df

d zHbL

T (86)

−−

=

1000100010001

22

202

dd

d

df

d zHb

L

T (87)

−−

=

1000100010001

33

303

dd

d

db

d zHb

L

T (88)

The components of the angular velocity of the suspended and non suspended masses are identical to those of the model in Fig. 1 - Equations (25) (29) (30) (31) (32).

Angular velocity of the masses over suspended masses obtained after multiplication of matrices )(

1TA

dT , )(2

TAdT , )(

3TA

dT

and their derivatives AdT 1 , A

dT 2 , A

dT 3 by formula (26):

0221 ϕωωω === Axd

Axd

Axd

0321 ψωωω === Ayd

Ayd

Ayd

(92)

0321 === Azd

Azd

Azd ωωω

After determining the kinetic and potential energy, and

function of Relay and applying Lagrange’s equation for the equations of motion of the system can write:

[ ] [ ] [ ] [ ]FqCqq =+Β+Μ

Inertia matrix [M5]:

М1+md

1+md2+md3 М1+md1bd1-

md2bd2-md3bd3 М1+md1Ld1+md2Ld1+md3Ld2

М1 М1 М1 М1 М1-md1 М1-

md2 М1-md3

М1+md

1bd1-md2bd2-md3bd3

М1+Jd1+md1bd1

2+Jd2+md2bd22+

Jd3+md3bd32

М1+md1Ld1bd1-md2Ld1bd2+md3Ld2bd3

М1 М1 М1 М1 М1-

md1bd1 М1+

md2bd2 М1+md3bd3

М1+md

1Ld1+md2Ld1-md3Ld2

М1+md1Ld1bd1-

md2Ld1bd2+md3Ld2bd3

М1+Jd1+md1Ld12+

Jd2+md2Ld12+Jd3+md3L

d22

М1 М1 М1 М1 М1-

md1Ld1 М1-

md2Ld1 М1+md3Ld2

М1 М1 М1 М1 М1 М1 М1 М1 М1 М1

М1 М1 М1 М1 М1 М1 М1 М1 М1 М1

М1 М1 М1 М1 М1 М1 М1 М1 М1 М1

М1 М1 М1 М1 М1 М1 М1 М1 М1 М1

М1-md1 М1-md1bd1 М1-md1Ld1 М1 М1 М1 М1 М1+md

1 М1 М1

М1-md2 М1+md2bd2 М1-md2Ld1 М1 М1 М1 М1 М1 М1+

md2 М1

М1-md3 М1+md3bd3 М1+md3Ld2 М1 М1 М1 М1 М1 М1 М1+md3

Elastic matrix [C5]:

C1 C1 C1 C1 C1 C1 C1 C1 C1 C1

C1 C1 C1 C1 C1 C1 C1 C1 C1 C1

C1 C1 C1 C1 C1 C1 C1 C1 C1 C1

C1 C1 C1 C1 C1 C1 C1 C1 C1 C1

C1 C1 C1 C1 C1 C1 C1 C1 C1 C1

C1 C1 C1 C1 C1 C1 C1 C1 C1 C1

69

C1 C1 C1 C1 C1 C1 C1 C1 C1 C1

C1 C1 C1 C1 C1 C1 C1 C1+cd1 C1 C1

C1 C1 C1 C1 C1 C1 C1 C1 C1+

cd2 C1

C1 C1 C1 C1 C1 C1 C1 C1 C1 C1+

cd3

Damping matrix [B5]:

B1 B1 B1 B1 B1 B1 B1 B1 B1 B1

B1 B1 B1 B1 B1 B1 B1 B1 B1 B1

B1 B1 B1 B1 B1 B1 B1 B1 B1 B1

B1 B1 B1 B1 B1 B1 B1 B1 B1 B1

B1 B1 B1 B1 B1 B1 B1 B1 B1 B1

B1 B1 B1 B1 B1 B1 B1 B1 B1 B1

B1 B1 B1 B1 B1 B1 B1 B1 B1 B1

B1 B1 B1 B1 B1 B1 B1 B1+βd1 B1 B1

B1 B1 B1 B1 B1 B1 B1 B1 B1+βd2 B1

B1 B1 B1 B1 B1 B1 B1 B1 B1 B1+βd3

Matrices [M5], [B5] and [C5] in model of Fig. 5, which

included additional seats on the suspended masses have some differences with models in Fig. 1, 2 and 3 as follows:

- in the matrix M change is in rows 1 to 3 where elements M(1,1) ÷ M(1,3), M(2,1) ÷ M(2,3) and M(3,1) ÷ M(3.3) have additional members showing the correlation between the suspended mass and the masses over suspended. Here are appeared new rows 8 ÷ 10 and new columns 8 ÷ 10 colored in gray.

- in the matrix C are appeared only new rows 8 ÷ 10 and new columns 8 ÷ 10 colored in gray.

- in the matrix B are appeared new rows 8 ÷ 10 and new columns 8 ÷ 10 colored in gray.

Indices in the new elements of matrices [M5], [C5], [B5] show the serial number of the seat included in the mechanical model. By increasing the number of additional seats on the suspended masses, number of rows and columns of the matrix increases by one for each additional mass (passenger seat). New rows and columns in the matrix show the effect of additional masses on the behavior of the entire vibrating system. With the inclusion of more passenger seats matrices will be increased by one row and one column.

3.3.Mechanical mathematical model describing the

dynamic behavior of the car and taking into account the influence of rubber-metal shock absorbers.

Scheme is the same model as the model shown in Fig. 2

with the difference that the points of attachment of the arms are added elastic elements, taking into account the elastic properties of rubber-metal shock absorbers (Fig. 7). Detailed diagrams of front and rear axles are shown in Fig. 6.

The task considered vibration of the suspended mass and non suspended masses taking into account the movement of the axes Oz, Oy and rotations around the axes Ox, Oz, Oy.

The generalized coordinates of the model and their derivatives are:

{ }

=

4

3

2

1

4

3

2

1

0

0

4

3

2

1

0

0

0

yyyyzzzz

y

z

q θ

ϕϕϕϕψϕ

{ }

=

4

3

2

1

4

3

2

1

0

0

4

3

2

1

0

0

0

yyyyzzzz

y

z

q

θ

ϕϕϕϕψϕ

{ }

=

4

3

2

1

4

3

2

1

0

0

4

3

2

1

0

0

0

yyyyzzzz

y

z

q

θ

ϕϕϕϕψϕ

(93)

Matrix of transition from O0x0y0z0 to OAxAyAzA

+−−+−−−

−

=

1000coscoscossinsinsincossinsincossincoscossincoscossinsinsinsincoscossinsin

0sinsincoscoscos

0000000000000

0000000000000

00000

0 zy

Т А

ψϕθϕθψϕθϕθψϕψϕθϕθψϕθϕθψϕ

ψθψθψ

(94) The transition matrices of coordinate systems Od1xd1yd1zd1,

Od2xd2yd2zd2, Od3xd3yd3zd3 to O0x0y0z0 have to type:

+−−−−

=

1000cossin0sincos0001

111

11101 zH

ybL

T f

f

ϕϕϕϕ

(95)

70

+−+−

=

1000cossin0sincos0

001

222

22202 zH

ybL

T f

f

ϕϕϕϕ (96)

+−−−−

−

=

1000cossin0sincos0

001

333

33303 zH

ybL

T b

b

ϕϕϕϕ (97)

+−+−−

=

1000cossin0sincos0

001

444

44404 zH

ybL

T b

b

ϕϕϕϕ (98)

The components of the angular velocity of the suspended masses are preset as:

00ϕω =A

x

00 ψω =Ay

(99)

00 θω =Az

For the components of the angular velocity of the arms and

the three axes is obtained: 101 ϕϕω +=A

x

01 ψω =Ay

(100)

01 θω =Az

- the front left side:

202 ϕϕω +=Ax

02 ψω =Ay

(101)

02 θω =Az

- rear right side:

303 ϕϕω +=Ax

03 ψω =Ay

(102)

03 θω =Az

- rear left side: 404 ϕϕω +=A

x

04 ψω =Ay

(103)

04 θω =Az

The kinetic energy of the system:

240040

20040

230030

20030

220020

20020

210010

20010

20

20

20

20

240

230

220

210

200

200

200

200

200

))((21

)(21))((

21

)(21))((

21

)(21))((

21

)(21)

21(2)

21(2)

21(2)

21(2)(

21

)(21)(

21)(

21

21

21

21

21

21

ϕψϕ

θϕϕψϕ

θϕϕψϕ

θϕϕψϕ

θϕθθψψϕϕ

ϕϕϕϕϕϕθψϕ

mpbbbmpbp

bpmpbbbmpbp

bpmpfffmpfp

fpmpfffmpfp

fppzbpzfpybpyfpxb

pxbpxfpxfzyx

LLbLzzm

LHyymLLbLzzm

LHyymLLbLzzm

LHyymLLbLzzm

LHyymJJJJJ

JJJJJJymzm

+−++++

+++++−−+−++

+++−+++++++

+−+++−++−++

+−+−+++++++

+++++++++++=Τ(104)

Potential energy of the system:

24

23

22

21

24

23

22

21

24343

22121

2244000

22440

002

1330002

1330

002

2220002

2220

002

1110002

1110

002

442

332

222

11

)(21)(

21)(

21)(

21)(

21)(

21)(

21)(

21)(

21

)(21))((

21)

)((21))((

21)

)((21))((

21)

)((21))((

21)

)((21)(

21)(

21)(

21)(

21

zczczczcycycycyczzLLc

zzLLcqzbLbbzcqyRL

RHycqzbLbbzcqyRL

RHycqzbLbbzcqyRL

RHycqzbLbbzcqyRL

RHyczLczLczLczLc

ttttttttsbsbsb

sfsfsfbkbbkbbgzbykbb

kbgybkbbkbbgzbykbb

kbgyfkffkffgzfykff

kfgyfkffkffgzfykff

kfgycbrbcbrbcfrfcfrf

+++++−++−+−+−−+

+−+−−+−++−+++−+++

++++−+−−+−+−−++

++++−++++++−++−

−+++−+−++−+−−+−

−+++−−+−+−−+−=Π

ϕϕ

ϕϕϕψϕϕθ

ϕϕψϕϕθ

ϕϕψϕϕθ

ϕϕψϕϕθ

ϕϕϕϕϕ (105)

The function of Relay, taking account only of the damping

resistance is:

244

233

222

211 )(

21)(

21)(

21)(

21 zLzLzLzLR cbrbcbrbcfrfcfrf −−+−+−−+−= ϕβϕβϕβϕβ

(106)

Matrices [M6], [B6] and [C6] model of Fig. 6, which includes

additional rubber-metal shock absorbers in the arms, and further consideration of the movement along the axes Oz, Oy and rotations around the axes Ox, Oz, Oy, to amend these models in Fig. 1, 2 and 3 as follows:

- the matrix [M6], are appeared new rows 8 ÷ 17 and new columns 8 ÷ 17 colored in gray;

- the matrix [C6], are appeared new rows 8 ÷ 17 and new columns 8 ÷ 17 colored in gray;

- the matrix [B6] are appeared new rows 8 ÷ 17 and new columns 8 ÷ 17 colored in gray.

Inertia matrix [M6]: М1 М1 М1 М1

М1

М1 М1 М

1 М

1 М

1+mp М

1+mp М

1+mp М

1+mp М

1 1 1 М

1

М1 М1 М1 М1 М

1 М1 М1

М1+4mpH

М1-

2mpHLf+2mpH

Lb

М1-

mp(Lmpf+bf)

М1+mp(Lmpf+bf)

М1-

mp(Lmpb+b

b)

М1+mp(Lmpb+bb)

М1-

mpH 1+mpH

1-mpH

М1+mp

H

М1 М1 М1 М1 М

1 М1 М1

М1

М1

М1+mpLf

М1+mp

Lf

М1-

mpLb

М1-

mpLb

М1 1 1

М1

М1 М1 М1 М1 М

1 М1 М1

М1

М1

М1-

mpLmpf

М1

М1

М1

М1 1 1

М1

М1 М1 М1 М1 М

1 М1 М1

М1

М1

М1

М1+mpLmpf

М1

М1

М1 1 1

М1

М1 М1 М1 М1 М

1 М1 М1

М1

М1

М1

М1

М1-

mpLm

pb

М1

М1 1 1

М1

М1 М1 М1 М1 М М1 М1 М М М М М М1+mp

М М

71

1 1 1 1 1 1 Lmpb 1 1 1 1

М1 М1+4

mpH М1 М1 М

1 М1 М1

М1+m0+4

mp

М1-

2mpLf+2mpLb

М1

М1

М1

М1

М1-mp 1+

mp 1-mp

М1+mp

М1 М1-

2mpHLf+2mpHLb

М1 М1 М

1 М1 М1

М1-

2mpLf+2mpLb

М1+J0z+2Jpzf+2Jpz

b+2mpLf2+2mpLb

2

М1

М1

М1

М1

М1+mp

Lf 1-mpLf

1-mpLb

М1+mp

Lb

М1+mp

М1-mp(Lmpf+bf)

М1+mpLf

М1-mpLmpf

М1

М1 М1 М

1 М

1 М

1+mp М

1 М

1 М

1 М

1 1 1 М

1

М1+mp

М1+mp(Lmpf+bf)

М1+mpLf

М1 М

1+mpLm

pf М1 М1

М1

М1

М1

М1+mp

М1

М1

М1 1 1

М1

М1+mp

М1-mp(Lmpb+bb)

М1-mpLb

М1 М

1 М1-

mpLmpb М1

М1

М1

М1

М1

М1+mp

М1

М1 1 1

М1

М1+mp

М1+mp(Lmpb+bb)

М1-mpLb

М1 М

1 М1

М1+mpLmpb

М1

М1

М1

М1

М1

М1+mp

М1 1 1

М1

М1 М1-

mpH М1 М1 М

1 М1 М1

М1-mp

М1+mpLf

М1

М1

М1

М1

М1+mp 1 1

М1

М1 М1+mpH М1 М1

М1

М1 М1 М

1+mp М

1-mpLf М

1 М

1 М

1 М

1 М

1 1+mp 1

М1

М1 М1-

mpH М1 М1 М

1 М1 М1

М1-mp

М1+mpLb

М1

М1

М1

М1

М1 1 1+

mp

М1

М1 М1+mpH М1 М1

М1

М1 М1 М

1+mp М

1+mpLb М

1 М

1+ М

1 М

1 М

1 1 1 М

1+mp

Elastic matrix [C6]:

C1 C1 C1 C1 C1 C1 C1 C

1 C

1

C1+cg

z

C1+cgz

C1+cgz

C1+cgz 1 1 1

C1

C1 C1 C1 C1 C1 C1 C1

C1+2cgy(H+Rkf)+2cgy(H+Rkb

)

C1-

2cgyLf(H+Rkf)+2cgyLb(H+R

kb)

C1-

cgz(bf+bkf

)

C1+cgz(bf+bkf)

C1+-

cgz(bb+bkb

)

C1+cgz(bb+bkb)

1-cgy(H+Rkf)

1+cgy(H+Rkf)

1-cgy(H+Rkb)

C1+cgy(H+Rkb)

C1 C1 C1 C1 C1 C1 C1 C

1 C

1

C1+cgz

Lf

C1+cgzLf

C1-cg

zLb

C1-cg

zLb

1 1 1 C

1

C1 C1 C1 C1 C1 C1 C1 C

1+cg

yRkf

C1-

cgyLfRkf

C1-

crfLc

f-cgzbk

f-csfLs

f

C1+csfLsf

C1

C1

1-cgyRkf

1 1 C

1

C1 C1 C1 C1 C1 C1 C1 C

1+cg

yRkf

C1-

cgyLfRkf

C1-

csfLs

f

C1+crfLcf+cgzbkf+csfLsf

C1

C1 1

1+cgyR

kf 1

C1

C1 C1 C1 C1 C1 C1 C1 C

1+cg

yRkb

C1+cg

yLbRkb

C1

C1

C1-

crbLc

b-cgzbk

b-csbLs

b

C1+csbLs

b 1 1 1-cgyRkb

C1

C1 C1 C1 C1 C1 C1 C1 C

1+cg

yRkb

C1+cg

yLbRkb

C1

C1

C1-

csbLs

b

C1+crbLc

b+cgzbk

b+csbLs

b 1 1 1

C1+cgyR

kb

C1 C1+2cgy(H+Rkf)

+2cgy(H+Rkb) C1

C1+cgyRkf

C1+cgyRkf

C1+cgyRkb

C1+cgyRkb

C1+4c

gy

C1-

2cgyLf+2cgyL

b

C1

C1

C1

C1 1-

cgy 1cg

y 1-cgy

C1+c

gy

C1 C1-

2cgyLf(H+Rkf)+2cgyLb( H+Rkb)

C1 C1-

cgyLfRkf C1-

cgyLfRkf

C1+cgyLbRk

b

C1+cgyLbRk

b

C1-

2cgy

C1-

2cgy

C1

C1

C1

C1 1+c

gyL1-cgy

1-cgy

C1+cgyL

72

Lf+2cgyL

b

Lf2+

2cgyLb

2

f Lf Lb b

C1+cgz

C1-cgz(bf+bkf) C1

+cgzLf

C1-crfLcf-cgzbkf-

csfLsf

C1+-csfLsf

C1 C1 C

1 C

1

C1+crf+cgz+csf+ct

C1-csf

C1

C1 1 1 1

C1

C1+cgz

C1+cgz(bf+bkf) C1

+cgzLf C1+csf

Lsf

C1+crfLcf+cgzbkf+c

sfLsf C1 C1

C1

C1

C1-csf

C1+crf+cgz+csf+

ct

C1

C1 1 1 1

C1

C1+cgz C1-cgz(bb+bkb)

C1- cgzLb

C1 C1

C1-crbLcb-cgzbkb-csbLsb

C1-csbLsb

C1

C1

C1

C1

C1+crb+cgz+csb+ct

C1-csb 1 1 1

C1

C1+cgz

C1+cgz(bb+bkb) C1-

cgzLb C1 C1

C1+csbLsb

C1+crbLcb+cgzbkb+cs

bLsb

C1

C1

C1

C1

C1-csb

C1+crb+cgz+csb

+ct 1 1 1

C1

C1 C1-cgy(H+Rkf) C1 C1-

cgyRkf C1 C1 C1

C1-cgy

C1+cg

yLf

C1

C1

C1

C1

1+cgy+ct

1 1 C

1

C1 C1+cgy(H+Rkf) C1 C1 C1+cgy

Rkf C1 C1

C1+cg

y

C1-

cgyLf

C1

C1

C1

C1 1

1+cgy+ct

1 C

1

C1 C1-cgy(H+Rkb) C1 C1 C1 C1-

cgyRkb 0 C

1-cgy

C1-

cgyLb

C1

C1

C1

C1 1 1

1+cgy+ct

C1

C1 C1+cgy(H+Rkb) C1 C1 C1 C1 C1

+cgyRkb

C1+cg

y

C1+cg

yLb

C1

C1

C1

C1 1 1 1

C1+cgy+ct

Damping matrix [B6]: B1 B1 B1 B1 B1

B1

B1

B1

B1

B1

B1

B1

B1 1 1 1 1

B1 B1 B1 B1 B1 B

1 B

1 B

1 B

1 B

1 B

1 B

1 B

1 1 1 1 1

B1 B1 B1 B1 B1 B

1 B

1 B

1 B

1 B

1 B

1 B

1 B

1 1 1 1 1

B1 B1 B1 B1 B1 B

1 B

1 B

1 B

1 B

1-βrfLc B

1 B

1 B

1 1 1 1 1

B1 B1 B1 B1 B1 B

1 B

1 B

1 B

1 B

1

B1+βrfLc

f

B1

B1 1 1 1 1

B1 B1 B1 B1 B1 B

1 B

1 B

1 B

1 B

1 B

1

B1-

βrbLc

b

B1 1 1 1 1

B1 B1 B1 B1 B1 B

1 B

1 B

1 B

1 B

1 B

1 B

1

B1+βrbL

cb 1 1 1 1

B1 B1 B1 B1 B1 B

1 B

1 B

1 B

1 B

1 B

1 B

1 B

1 1 1 1 1

B1 B1 B1 B1 B1 B

1 B

1 B

1 B

1 B

1 B

1 B

1 B

1 1 1 1 1

B1 B1 B1 B1-

βrfLcf B1

B1

B1

B1

B1

B1+βrf

B1

B1

B1 1 1 1 1

B1 B1 B1 B1 B1+βrf

Lcf B

1 B

1 B

1 B

1 B

1 B

1+βrf B

1 B

1 1 1 1 1

B1 B1 B1 B1 B1 B

1-βrbLcb B

1 B

1 B

1 B

1 B

1 B

1+βrb B

1 1 1 1 1

B1 B1 B1 B1 B1 B

1

B1+βrbL

cb

B1

B1

B1

B1

B1

B1+βrb 1 1 1 1

B1 B1 B1 B1 B1 B

1 B

1 B

1 B

1 B

1 B

1 B

1 B

1 1 1 1 1

B1 B1 B1 B1 B1 B

1 B

1 B

1 B

1 B

1 B

1 B

1 B

1 1 1 1 1

B1 B1 B1 B1 B1 B

1 B

1 B

1 B

1 B

1 B

1 B

1 B

1 1 1 1 1

B1 B1 B1 B1 B1 B

1 B

1 B

1 B

1 B

1 B

1 B

1 B

1 1 1 1 1

73

New rows and columns in the matrix show the influence of the elasticity of rubber-metal shock absorbers on the behavior of the vibration system.

4.CONCLUSION

Presented mechanical mathematical models of cars with

arm suspension to allow doing the following conclusions: 1. The models have the identical mathematical structure

allowing their comparison. 2. Following the sequence of arm constructions and

forming matrices [M], [B], [C] and [Q] is used the method of mathematical induction. The structure of matrices shows that the inclusion of new masses and new generalized coordinates is manifested in the creation of new lines and columns in the matrix.

3. The models are tested numerically and it is shown in previous papers [1], [2].

4. The models allow to obtain frequencies and their own forms of vibrations. With this models can to investigate the dynamic behavior the vehicle driven on the road roughness.

5.ACKNOWLEDGEMENTS

This work is a part of the project DO 02-

47/10.12.2008 funded by "National Science Fund – Ministry of Education, Youth and Science".

6.REFERENCES 1. Павлов, Н., Л. Кунчев. Механо – математично

моделиране на независимо раменно окачване. ЕКОВАРНА'2009 – Сборник доклади, ТУ-Варна, 2009, 97-114.

2. Kunchev, L., N. Pavlov. Comparative Analysis on Mathematical Models Describing Vibrations of Road Vehicles with Different Types Suspensions. JUMV International Automotive Conference with Exhibition SCIENCE AND MOTOR VEHICLES 2011, Belgrade, Serbia, 487-498.

3. Pavlov, N., L. Kunchev. Determination of Natural Frequencies of a Car Taking Into Consideration Influence of a Power Unit. - Machines, Technologies, Materials, 2010, № 10-11, 30-35.

4. Kunchev, L., N. Pavlov. Modeling Drive Comfort of a Car With Full Arm Suspension Taking Account Masses Over Suspended. - 15th International Scientific Conference - Quality and Reliability of Technical Systems, Nitra, Slovakia, 2010, 365-371.

5. Kunchev, L., N. Pavlov, G. Yanachkov. Mechanic-mathematical Modeling of an Independent Suspension as Takes Account of Elasticity of Silent Blocks. - Machines, Technologies, Materials, 2010, № 3, 29-35.

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