Deformation equivalent reduction of the forcesexciting axle vibrations in the crank shafts Текст научной статьи по специальности «Электротехника, электронная техника, информационные технологии»

yflK 621.43

DEFORMATION EQUIVALENT REDUCTION OF THE FORCESEXCITING AXLE VIBRATIONS IN THE CRANK SHAFTS

N.S. Ivanov, Seniour Assistant, Dr-Eng., Z.D. Ivanov, Seniour Assistant, Dr-Eng., Technical University - Varna, Bulgaria

Summary. The problem of analysis of the axial force which is used to calculate the crankshaft axial vibrations is raised in this article. The crankshaft is considered as a space frame for precise calculation. It was shown that the dependence of inductive forces on crank turning angle can be developed into the Fourier series.

Key words: deformation, force, vibration, Fourier series, crankshaft.

Introduction

In the specialized literature are examined the problems, concerning the nature and excitation of the axle vibrations, determination of the mass and elastic parameters of the discrete dynamic model and the demphering of the resonance axle vibrations. The study shows, that these problems are not thoroughly solved in comparison with the similar problems twisting vibrations. The main reason for this conclusion is probably the significantly greater conventionality in the choice and building of the equivalent discrete dynamic model of axle-vibrating crank shaft line. This conclusion to a great extend is concerning the determination of the sources of vibrations excitation as well. In the proposed working out is examined the excitation of single joined axle vibrations in the crank shaft lines by the forces and the moments, acting on the crank shaft in a piston internal combustion engine.

It is convenient the axle displacements of a crank from the crank shaft to be determined by introduction of a force A, acting in the axis direction, which should be equivalent in relation to the displacements of the radial and tangential actually acting forces. In [2] the relation between the axial force A and radial force is given by the expression:

(1)

where a - length of the connecting rod neck;

1 m

Ym = —XYi ; li is the angle between each two legs;

m i=1

m - number of the cranks of the crank shaft; R -radius of the crank; k - coefficient of proportionality.

It is obvious that (1) does not take into consideration the constructional differences between the separate crank shafts, as the coefficient k must be experimentally determined for each one of them. Besides, (1) is not accounting the influence on the force A of the tangential force.

In [3] axial force A is determined by analysis with registering the moment forces in the crank under the influence of the radial force Z. The crank shaft is examined as a cut beam. By this way the influence of the neighboring cranks on the distribution of the bending moments in the examined crank is not calculated. An advantage of the method is the introduced coefficient kn, representing the overlapping of the connecting rod and the main crankpins.

In [4] by an experimental way is solved the problem between the relation of the radial deformation of the connecting rod crankpin and the axle deformation of the crank as well as the average angle between the examined crank and its neighboring cranks. Despite the accounting to a certain extent the influence of the neighboring cranks on the formation of the axle deformation of the examined crank, this method has the same disadvantages as the in [2] as well.

Targeting the experimental clearing up the influence of the load, applied on the neighboring cranks, the authors research the axle deformations in a real engine by a tensiometering of the crank shaft [7, 8]. On fig. 1 is given the axle vibrogram of the fifth crank for engine DP5800 on the outer frequency characteristic. The rate of 900 rpm is chosen due to the prevalent influence on the sum force Z5 of the gas force Fi5 and the smaller relative quantity of the inertia forces Fj5 of the piston group. The index 5 is for the fifth cylinder. With the thick lines are the experimental curves for the axle deformation of the

fifth crank S5 and the indicator force Fi2, registered for the second crank. With dotted line are drawn the indicator diagrams for the rest cranks and cylinders. The axle deformation S5 is registered with tensioresistors, reacting only on bending of the arms of the fifth crank in direction to the less strength, i.e. only in the plain of the crank, fig. 1 and [7]. Under the experimental vibrogram is given the graphic and the theoretically calculated sum force Z5. By the visual comparison of the curves S5 and Z5 follows the conclusion, that the reason for the axle deformation of the fifth crank are the sum force Z5 (biggest influence) and the forces applied on the neighboring cranks as well. The rate 900 rpm has no resonances of axle and twisting vibrations as well.

Object of the article

In the present article the authors are raising the problem of calculating by analysis the axial force A under the following assumptions: The crank shaft is examined as statically indefinable frame spatial beam on solid supports. The displacements only by the moment forces are taken into consideration and the same from the normal and cutting ones are neglected. For revealing of the statically indefiniteness is used the method of the upper-supports moments. This method, applied and further developed here is known in application for receiving the consecutive twisting moments [1]. The acceptance of the theoretical frame form of the crank shaft allows for their precise calculation. The assumption, that the supports of the crank shaft are ideally solid is acceptable as in most of the cases in practice the strength of the supports is

significantly bigger than the axle strength of the crankshaft.

i'h support

p

»port

/i

|x 1 7 transmission of

r power

d. T,^ ^2

| m| 1 '.SK oi i-,h crank ' (i+l>* support yji m|+i

b, a -b2 T;R

Fig.2 -

Ideally solid supports in comparison with the pliability of the crank shaft are the supports (roller rolling main bearings for the engines with tunnel type crankcase) [9]. In some engines with a purpose of increasing the strength of the crank shaft at supercharging and inter cooling of the supercharged air additionally are strengthened with extra ribs and suitable construction the supports of the main sliding bearings.

Solution of the presented problem

On the i_th crank (Fig.2) act the forces Zi, T and the twisting moment, transferred from the crank M. The second half of the crank in direction to transmission of the power is loaded additionally by the twisting moment TR, drawn on fig.2 by a vector. They are periodic functions of the angle of rotations of the crank shaft, which comply with the conditions of Dirihle and can be decomposed in a Furie line:

«Pi

1 3J,

Ef2

r, /

1 bl[

E(2 hi

bid,

(1,5^ - b,) + ^1^2 _ b,) + -1. [l,5^d2 (df - bf )- d2 (df - bf)+ d, (d| - b| ^

Jw

,s

3J

, Rbjd, | bjd.

Jw2x 3J2

U¡2^

1 3

Jwlxv 17 J 3 Jw,v 3J

' 2,x

E£

(U5e _ b,)+(e- b,)+± [o,5*(* + b, + b2 ) - (b? - b, b2 + b2 )

3J,

Jw

,x

3J

-i^-(<-b2)+^-(l,5<-b2)

2,x 2

3E r

bfd

- b,)+- d2)+i. [i^df - b?)- (d? - b?)]+ i(d| - bi)-J j J vV | y J J

R3d, b2d, 3ER h----------L + ^—1L +

Jw2,y J2

B*d2 (£-bl)+^- + -b2dl

Jw

1.0

2J Jw

2,0

O

1 bi ( K2> f2 bP+ 1 , R3 , (a + b2)3-b5| R3 | ER r^-b,)2

El2 Jl L/i C T 3 ^ J J 3Jw, y 3J 3Jw2 y G Jw1.0

■ +--------------------+ ‘

Table I

<p!+l(zi)=-<p!(z>

<p; 1 (m|)= <p;(m|11)

(mi*’ )= v! (m!)°

<•>;*’ (Ti )= -<I>! (T, >0+R0.; (SR,)

R f R R E a ^ o|+1 (m| )=o|(m

2E f. ^Jwl,y Jw2,y G J, /

<t>

3E^

Note: The sumbo

2 _ 3

(l,5£ - b,)- + * [o,5^ + b, + b2)- (b? - b,b2 + b^)

Jwly J

R3 b2

Jw2,y J2

2J Jw20 + ~(l)5f-b2)+

'[>!*' (m|+i )= )

®rl(SRi)=o!(SBil)

3ER

b,(<-b,) aR b2(f-b2)

Jw

1,0

2J Jw

2,0

oo means, that in the formulas for cpj and Ojmust be replaced the places of the indexes 1 and 2.

Z (a+Vu ) = -T1 + E Zv • sin (vot + vyu + ev)

2 v=p

T, (a+vu) = Hr + E Tv sin (va +vv i,/ +^v)

2 v=p

Mi (a+vu) = E Tj (a+vu)) =

j=1

(2)

'-1 T 0 "

= R E j+R E

j = 1 2 v= p

TvEsin (a + vVlj + ^v) j=1 v 7

where y1,, y 1 j - dephasing between the working

processes in the first and the /_th (j) cylinders; p = 0,5 for four-stroke engines ; p = 1 for two-stroke engines; v - engine order of accordion; ev, ^v - start phase of accordion with order v for first cylinder.

We examine the excitation of the forces and moments with definite order. Then on fig.2 follows:

= Zv sin (va + v yu +ev)

T = T v sin(v0t+v yu +§v) (3)

Mi = RTv E sin (a + vy1, + ) •

j=2

The over-support moments m‘, m‘+1, Mi and M,'+1

are drawn by a vector on fig. 2, Perpendicular to the plain, in which act in such a way, that if we look

from the top of the vector the moment rotates the body counterclockwise. Their determination is executed by the following way:

For each crank are determined the angles for rotation of the over-support section in the plain xz of the crank <p‘ and 9J+1 in the plain yz, perpendicular to

the plain of the crank - O', OJ+1. The upper index shows the number of the support. The rotation of the over-support in the plain xz is executed under the influence of the force Zt and the moments m,, m,+1. The rotation of the over-support sections in the field yz is done under the influence of the force T, and the moments M,, Mi, Mf1. The angles of rotation are determined by the integrals of Mor for each load separately, taken for one. In tables 1 and 2 are stated the formulas for calculation of the rotation angles of the over-support sections under the influence of single forces (moments). The type of the force (moment) is indicated in brackets. The lineal measurements are stated on fig.2. Besides we have: J1, J2 - axle inertia moments of the cross section of main crankpins, respectively for the i"th and (,+1)-th; J - the same for the connecting rod crankpin; Jw1,x, Jw2,x - inertia moments of the cross section of the arm in direction to the smaller strength; Jw1,y, Jw2,y - the same in direction to the bigger

strength; Jw10, Jw2 0 - inertia moments at twisting

the arms. The full angles of rotation of the oversupport sections of the i _th crank will be:

9; = Z i 9; (Zi ) + mfâ ( m\ ) + m'+y (+1 )

moments mi and Mi must be equal to the resultant

9+1 = Z9+1 (Zi) + mi9'+1 (mi)+ mf1^1 ( ) (4)

®■ = rio‘ (t ) + Mt®i (m; ) + m‘®i (m; ) + Mf1®; (M/+1 ) ®'+1 = T ®i+1 (T )+ Mi OÎ+1 (M; )+

+m; ®i+1 (m; )+Mi+1®;+1 (m;+1 ).

For determination of the over-support moments are used the following relations. We examine the f* support, limited by the (,-1)-th and i"th crank. The coordinate system (fig. 1) is considered connected to each crank, i.e. between the coordinate axes (plains) respectively for the (i-1)-th crank and the i"th crank, will have angle y,_u equal to the angle between the

cranks. We calculate it by the following way: if we look from the positive direction to the axe z, to see the axe x for the ,-th crank to rotate counterclockwise to axe x for the (i-1)-th crank. The resultant of the

of the moments m'_1 and M‘_1 with opposite directions - (fig.3-a). If we project the equality on the coordinate axes of the (i-1)-th crank we get the following relations between the over-support moments:

m'_1 + m; sin y + micos y,._u = 0

Mi_1 +M; cos Y;_1,i _ m‘i sin Yi_1,i = 0

(5)

If the angles of rotation of the ,-th over-support section are examined as vectors, received by the rules for the vectors of the moments (fig.3-b), then after projection on the coordinate axes of the (,-1) crank, we acquire the following relations:

9;_1 =9; cos Y;_1,i +®; sin Y;_1,i

_1 =_9i sin Y i _1,i +®; cos Y i _1,i

(6)

The equations (5) and (6) are build for each support as (4) is replaced in (6). By this way is received a system of algebraic equations, from which can be determined all the over-support moments. As according to (2) and (3) the forces and moments with given order, applied on the different cranks and on one crank have different starting phases, the oversupport moments from each force (moment) separately and applied consecutively on all the cranks should be determined for each crank. Following this pattern we can examine the multicrank shaft as build of a line of simple one-crank shafts, which allows comparatively easy with the help of the integrals of Mor to determine the axle displacements - fig.4.

The axle displacement 8, of the i"th support in relation to the (,+1)-th of the i-th crank is due only to the over-support moments (m,) , (mi+1) and the

V >qj V >qj

force Z,. The index defines the type of the load (Z, T, M), from which is determined the corresponding over-support moment , and the index j is the number of the crank, on which has been applied the load q. The formulas for calculation of the axle

displacements (8,+ , derived by the integrals of Mor on fig. 4 are the following:

From the over-support moments for axle displacement we receive:

The full axle displacement in the /_th crank will be received as we sum up the displacements : 1) from the force Z,, applied on ,-th crank and the over-support moments (m‘ )z, , (m'+1 )z, , derived from it; 2) from

the over-support moments (m‘+ j, (m‘+1+ j

derived from the forces Zj j=1,2..n; j *,), applied separately on the cranks of the shaft; 3) from the over-support moments (m‘ )Tj , (m,+1 )Tj , derived

from the forces Tj j=1,2..n; j *,), applied separately on the cranks of the shaft; 4) from the over-support moments (m‘) j, (+1) j, derived

from the twisting moments Mj (j=1,2...n; j *,),

applied separately on the cranks of the shaft. In this formula n stands for the number of cranks of the crank shaft. Considering the upper statements, the summary axle displacement of the i-th crank will be (j *,):

8, =[(8, )Z, + (8) ] Zv sin (a+vy1,, +^v) +

+E(8,- + Tv sin(va+vy1,j +^v)+ (11)

(8, + =(m‘) —

V ',qi V ‘ >qj El

(2 - b1)+ k R (b2 + a)

J

+ kn

Rb2 2 Jw2

R

(7)

qj El

J

2 Jw1x

2 Jw2

At symmetrical crank on table 2 we receive:

(8, )m =(m‘) — k

V ,,qj V •’qj El n 2Jw

_(m '+1) — f Oi+k^L

' ' >qj El I J n 2Jw

(8)

From the radial force Z, = 1 of rth crank for the axle displacements we receive:

(8Tz =

2E

Ra

8EJ

b1d lJw1,

(2b1 + a )-i ^ (b1 + a)-a'

2 + kn2 ^ 1 ' 2

Jw2

(9)

At symmetrical crank:

a +4ab , Rb

-------+ kn —

2 J n Jw

\

(10)

In the formulas from (7) to (10) for calculation of the overlapping of the connecting rod and the main crankpin is introduced the coefficient kn [3]. The same formulas are deduced at single forces Z, T and moment M. The full displacement from a given force (moment) will be obtained by multiplying the force (moment) by the displacement, derived by formulas from (7) to 10). For instance the axle displacement in the -th crank under the effect of the over-support moments m\ and m\+1, derived from loading of the j-th support (j *,) with force Tv sin (a + vy 1, j + )

will be:

E (8,- )M RTv Esin (va+vV1,j + ^v)

' j=2

By introducing an equivalent axial force A for the same displacement will be received:

8,= e, A (12)

where: is the axle pliability of the ith crank.

Conclusions

1. The elaborated methodology allows with precision in comparison with the experimental data to calculate the excitement of the axle vibrations in the crank shaft lines by the loads of the crank shaft, as on formula (12) with calculation of (11) we can determine the equivalent axial force for each individual crank and for each individual order of the harmonic components.

2. Neither of the referred methods for calculation of the axle vibrations in crank shaft lines (with exception to a certain extend in [4]) the influence on the load of the neighboring cranks of the crank shaft is not rendered an account.

Literature

8 = (8, +v,J Tv sin (va + vV1,j + ^v )

1. Kinosashvili R.S. and others. Determination of the forces, acting in crank shafts, Dynamics and stability of crank shafts. - Serensen, 1948.

2. Anderson G. and others. Stresses in crank shafts of

powerful diesels, Ship’s low rpm diesels, Ship building, 1967.

3. Sevastakiev V. Research on axle pliability of crank

shafts and analysis of the forces, exciting vibrations, Dissertation essay, St. Petersburg, 1970.

4. Guelemoti L. and others. Experimental research of

axle vibrations in crank shafts, Ship’s low rpm turbocharged diesels, Ship building, 1967.

5. Sevastakiev V., Ivanov N.S. Deformation

equivalent reduction of the forces, exciting axle vibrations in crank shafts, Seminar Transport and ship diesel engines, May. - Varna, 1978.

6. Sevastakiev V., Ivanov N.S., Milkov V.D.

Analysis of the excitement of axle vibrations in multi-support crank shaft with rendering an account of the pliability of the supports, Jubilee scientific session, October, 1978, TU-Varna.

7. Sevastakiev V., Ivanov N.S., Ivanov Z.D.

Determination of the “breathing” of a crank

shaft by a tensiometering, Scientific session of TU-Varna, November, 1981.

8. Sevastakiev V., Ivanov N.S. Determination of the

coefficients of influence at excitements of the axle vibrations with the help of the method for identification of mechanical systems by a calibration effect, scientific session of TU-Varna, November, 1981.

9. Sevastakiev V., Ivanov N.S., Vasilev G.p.,

Lefterov L.G. Diesel twelve cylinder engine D12500, Check Republic, Motor Simpo, 1988, as well as Sevastakiev V., Ivanov N.S. Unified order of auto tractor diesel engines with power range from 88 kW to 235 kw., Scientific session of TU-Varna, 1983.

Рецензент: Ф.И. Абрамчук, профессор, д.т.н., ХНАДУ.

Статья поступила в редакцию 20 апреля 2005 г.