otto engine dynamics - petrescu

7/31/2019 Otto Engine Dynamics - PETRESCU

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OTTO ENGINE DYNAMICS

PETRESCU Florian Ion*, PETRESCU Relly Victoria**, GRECU Barbu***

*,**,***Polytechnic University of Bucharest Romania

Abstract: Otto engine dynamics are similar in almost all common internal com-

bustion engines. We can speak so about dynamics of engines: Lenoir, Otto, and

Diesel. The dynamic presented model is simple and original. The first thing nec-

essary in the calculation of Otto engine dynamics, is to determine the inertialmass reduced at the piston. One uses then the Lagrange equation.

Key words:Lagrange equation, dynamic model

1. INTRODUCTION

The first thing necessary in the calculation of Otto engine dynamics, is to determine the in-

ertial mass reduced at the piston (1).

+

++=

++++=

+++=

22

2

2

22

1

22

2

2

222

2

1

2

22

2

2

2

1

2

2

)cos(cossin

cos)sin1(

)cos(cossin

1]cos)sin1()[(

cos

cos

'''

*

mmmM

l

J

r

JmmM

s

J

s

J

s

rmmMM

t

bAt

bAt

(1)

Then it derives the reduced mass to the crank position angle (2). Were used for piston the

next kinematics parameters (4). Lagrange equation is written in the form (3).

2

21

2

)cos(cossin

)(cos2)

cos

sin

sin

cos()2()(

+

+

=

mmmM

d

dMt (2)

pFxskxd

dMxM =+ )('

2

1'' 22

(3)

=

+

=

+=

3

223

cos

cossin

cos

)2cos(cos''

)cos(coscos

sin'

coscos

rrrs

rs

llrs

(4)


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2. DYNAMIC EQUATIONS

The dynamic equation of motion of the piston, obtained by integrating the Lagrange equa-tion (3), takes the form 5.

cos

)cos(coscos

cos43

32

++

= ccmk

ksx

t

(5)

Dynamic reduced velocity (6) and dynamic reduced acceleration (7) are obtained by deri-

vation:

sin

)cos(coscos

sin'' 433 2 +

+

= ccmk

ksx

t

(6)

cos

)cos(coscos

cos'''' 433 2 +

+

= ccmk

ksx

t

(7)

Angular velocity* is obtained through kinetic energy conservation (8-12).

2*2*

2

1*

2

1DDJJ = (8)

)sin1()sin1(cos)(cos 22222 ===== mmmmmD D (9)22

1

* 'smrmJJ tbA ++= (10)

22

1

* 'xmrmJJ tbAD ++= (11)

)sin1(30'

' 2222

1

22

1*

++

++=

n

smrmJ

xmrmJ

tbA

tbA (12)

Dynamic velocity (13) and kinematics velocity (14) are written:*'=xx& (13)

30''

nsss m

==

& (14)

Dynamic acceleration (15) and kinematics acceleration (16) are written:2*'' =xx&& (15)

900''''

222 n

sssm

==

&& (16)

3. NOTATIONS

In the picture number 1 one presents the crank shaft.

])2(

)(6.184.4.)[(sin96)(4

)(3

34444

44223

44

mp

mp

mm

mm

pp

pp

mmm

mm

DDrb

DDr

dD

Dl

dD

DldDErblG

dDGEk

++

++

++

+++

=

(17)

The relation (17) determines the elastic constant of the crank shaft, k. For the masses one

uses the notations (18); see the picture two.


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the ratio between lengths of crank and rod;l

r=

pm the mass of the piston, with piston bolt and segments;

bm

the mass of the rod;

rh

dm

Dm

b lmlp/2

dpDp

Fig. 1 Crank Shaft

=

+=

+=

=+=+==

2

22

2

11

''''''

l

Jm

r

Jmm

mmm

mmmllll

lmm

l

lmm

bA

bBpt

bbBbAbbBbbA

(18)

The parameters c1-c4 take the forms (19):

=

=

+=

=

][

][

][

][

14

213

21

2

2

2

1

mmcc

mccc

kgmmc

kg

m

k

rc

t

(19)

The moment of inertia 1J can be determined with the relation (20).


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]}8)()[()2()()2{(32

2224444

1 rdDdDbldDblJ mmmmmppp ++++

=

(20)

The crank length, r, and the length of the connecting-rod, l, can be seen in the kinematics

schema of an Otto mechanism (fig. 2).

l

ll

G

A

B

mbA

mbB

O

mb

rJ1

J2

mp

Fig. 2 Otto mechanism kinematics schema

4. DYNAMIC ANALYSIS OF THE MECHANISM AND

CONCLUSIONS

When increases the mechanism dynamics is deteriorating.

r=0.25 [m] l=0.3 [m] )3(8.0= For n=8000 [r/m] the mechanism is working normally (see the accelerations diagram from

the picture 3):

-400000

-300000

-200000

-100000

0

100000

200000

300000

0 100 200 300 400x2p [ms-2]

s2p [ms-2]

Fig. 3 Dynamic and kinematics accelerations; n=8000 [r/m]; 83.0=


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r=0.25 [m] l=0.3 [m] )3(8.0= At n=9000 [r/m] the mechanism work abnormally (see the accelerations diagram from the

picture 4):

-600000

-400000

-200000

0

200000

400000

600000

800000

0 100 200 300 400

x2p [ms-2]

s2p [ms-2]

Fig. 4 Dynamic and kinematics accelerations; n=9000 [r/m]; 83.0= r=0.25[m];l=0.3[m]

For a proper operation is necessary reduction of the ratio, especially if we want to in-crease the engine speed (see the next diagrams).

-800000

-600000

-400000

-200000

0

200000

400000

600000

0 100 200 300 400x2p [ms-2]

s2p [ms-2]

Fig. 5 Dynamic and kinematics accelerations; n=12000 [r/m];

r=0.25[m];l=0.6[m] 42.0=

-800000

-600000

-400000

-200000

0

200000

400000

600000

800000

0 100 200 300 400

x2p [ms-2]

s2p [ms-2]

Fig. 6 Dynamic and kinematics accelerations; n=14000 [r/m];

r=0.25[m];l=0.9[m] 27.0=


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otto engine dynamics - petrescu

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