A study on non-normal torsional vibration in a crank-shaft system with many dampings by using a digital computer

A Study on Non-Normal Torsional Vibration in a Crankshaft

System with Many Dampings by Using a Digital Computer

Tokyo Machinery Works

Kosaku Kanda*

The study introduced hereunder describes numerical methods for the solution of simultaneous differential equations of free and forced vibrations of a torsional multimass system, having many dam pings such as damper, engine and propeller.

In this study, the author has made clear the characteristics of non-normal mode of vibration

such as variations of actual amplitude and stress at each part, and mode of vibration, by

numeric-ally calculating the forced vibration of a system provided with a viscous damper in the speed range from the 5th order three-node critical speed to the flank, by using an electronic digital

computer.

For instance, in the three-node vibration of the system, the specific amplitude and specific stress (amplitude and stress per unit amplitude of the damper case respectively) at each part at a

critical speed in case a damper is employed shows a larger increase than respective values of the

mode of free vibration without damping, i. e., the normal mode of vibration; and in addition the

difference between them tends to rapidly grow greater with drop of speed from the critical speed.

Therefore, the estimated values of actual stresses obtained by the normal mode of vibration

(mode-fixed) which had been generally adopted, are fairly low as compared wit/i the actual values. Also it was made clear that, as the result of the above-mentioned c/lange of mode of vibration, the speed at which the actual stress at the node becomes maximum, drops below the critical speed,

resulting in a considerable difference in tendency between the curve of actual stress and that of

actual amplitude of the damper case.

1. Introduction

Generally, in the torsional vibration of a propeller shaft system mounted on a high-speed marine diesel

engine, the major critical speeds of one-node vibration

are below the engine idling speed; and therefore they

will cause comparatively less problem. In most cases,

two- and three-node vibrations tend to become a

pro-blem. In these cases, the propeller is hardly subjected

to vibration; and the natural frequency and mode of

vibration are very close to those of the one- or two-node

vibration of the engine unit. Therefore, the engine

damping will be the main damping. But since its

damp-ing ratio is relatively small, say about 0.03, its error can be held small even when it is neglected and each

mass is regarded to make at the critical speed a so-called normal vibration of the same or the reverse phase with a fixed specific amplitude (with that at the free end of

the crankshaft taken as unit amplitude) peculiar to its

mode of vibration.

However, in case a forced vibration takes place in

a system provided with a powerful viscous damper, the

system vibrates in non-normal, mode of vibration in

which the phase of vibration of each mass differs from each other owing to its damping effect. Also, the specific amplitude of each mass and the relative specific

ampli-tude between them will be greater than the respective values of the normal mode of vibration depending on

the mode of vibration, resulting in an increased specific stress (stress per unit amplitude at the damper case) at each part. Accordingly the decrease of the actual stress

is not proportional to the decrease of the actual

am-plitude of the damper case.

In addition, the speed at which the actual stress becomes maximum is positioned off from the critical speed owing to the variation of the mode of vibration

Second Engine Designing Section. Technical Department

with change of speed from the critical speed to the flank. The actual stress at each part at this speed is consider-ably greater than the estimated value obtained from the normal mode of vibration. Therefore, the assumption

of the normal mode of vibration for estimating the

actual stresses at the critical speed and any speed

around it, may probably bring about unexpected trouble. To clarify the actual state of the non-normal mode

of vibration taking place in a system provided with a damper, the following two types of programs were

prepared for an electronic digital computer. Free vibration-energy method

In this method, the choice of the optimum damping

coefficient of a damper is carried out automatically

within the program, and equations of free vibration of

a non-normal vibration system having many viscous

dampings including the damper damping, are calculated

by numerical method, in order to obtain the natural

frequency, damping index and mode of free vibration. Then, approximations of actual amplitudes and stresses at the critical speed of each order were obtained by the

energy method, on the assumption that the mode of

forced vibration at the critical speed will be similar to that of free vibration.

Calculation of forced vibration

From the result of the above-mentioned calculation is selected the' forced excitation torque of the order with which the maximum actual stress occurs; and equations of forced vibration of the non-normal vibration system

are calculated by the numerical method to obtain the actual amplitude and stress at each part and the mode

of vibration from the critical speed to the flank. A description will be given on the result of' a study

made on the evaluation of actual stresses obtained by

numerical calculations of both free and forced vibrations are carried out, by using the above-mentioned two pro-grams, with varied dampings taken into account, on the

5th order three-node vibration of a speed,

high-output engine provided with a viscous damper.

The digital computer used in this calculation is

IBM-7044.

2. Differential equation of vibration and its solution

Major dampings of the marine engine propeller shaft system are:

(1) Engine damping

Friction damping at sliding parts as at cylinders and at bearings

Hysteresis damping of crankshaft (2) Propeller damping

(3) Damper damping

As compared with the dampings (2) and (3) which are both viscous dampings, the damping (1) is verymuch

complicated in characteristics; and therefore, it is

ex-tremely hard to determine an experimental equation for providing an equivalent viscous damping coefficient. So,

it was obtained by performing a reverse operation from the measurement of vibration amplitude of the engine. By using the equivalent viscous damping coefficient thus obtained, substitution of the equivalent vibration system as shown in Fig. i is possible. The differential equation of vibration pertaining to the mth mass is

¡mer,

+kr,(er,er,+1)+Cr,(,+1)

+C1møm,Hra*e, (m=i,2

n)

(i)

where

e,. : actual complex amplitude at mth mass (rad)

moment of inertia of mth mass (kg-cm-s2)

km : shaft stiffness between mth and m+ ith mass (kg-cm/rad)

(NOTE I,, and k,, after the reduction gear are the reduced values on the crankshaft side with the reduction ratio T,, to the crankshaft taken into account. Denoting the

respective values of the original shafting systeni as

Ij,,), and (k,,J,. the following relations can be written:

i,2=Ci,,:J, Tm2

km=EknvJs Tm2

Cr, : viscous damping coefficient for relative angu-lar velocity between mth and nz+lth mass

(kg-cm-s)

C,,' : viscous damping coefficient for absolute angu-lar velocity at mth mass (kg-cm-s)

Hr,*: combined complex torque of the ith order

harmonic components of tangential pressures of NB cylinders of phase difference Ti acting

on the crank of mass No. in (kg-cm)

Hr,*(/4)D2R(Mr,!+j11m2)Mm+jN7m (2) el H H H 3 1 13 I, k3 e. H,_2 H,1 3 1 n-2 n-I e,2 en_I- e

Fig. i

Equivalent multzmass system with many viscous dam pings

Mm'{(Qst+Qt')cos(ömi+

NBi Ti)+

Qct sin(ömt+

NBi

₂

Nm={QcicoS(ömt+

NBi

It)-(Qst+Qt') sin(òmi+

NBi

Q32 : sine term of ith order harmonic component of tangential pressure due to gas pressure

per cylinder (kg/cm2)

cosine term of ith order harmonic component

of tangential pressure due to gas pressure

per cylinder (kg/cm2)

Q2' : ith order harmonic component of tangential

pressure due to inertia force per cylinder (sine term only) (kg/cm2)

ô,,2 phase difference to the ith order harmonic

component of tangential pressure acting on

No. i crank of the ith order harmonic

compo-nent of tangential pressure acting on the

crank of the mass number ivi, (rad)

multiplier for vector summation of the ith order harmonic components of tangential

pressures of NB cylinders of phase difference Ti (equally spaced) acting on one crank

sin(NB Tif2) sin (r/2) D: diameter of cylinder (cm) R: radius of crank (cm)

Q: forced circular frequency of ith order (rad/s) The general solution of Eq. (1) is

e,,=ß,,*e21+m*ei21 (5) where

2=a+jw

(6)

O*=.p+jçÇ J

a: damping index

w: natural circular frequency (rad/s)

ßr,*: actual complex amplitude at free vibration (rad)

Or,*: actual complex amplitude at forced vibration

(rad)

The first term on the right side of Eq. (5) is for the

free vibration while the second term is for the forced

vibration.

2.1 Free vibration

2.1.1 Characteristic equations for natural cir-cular frequency and damping index

The substitution of the general solution of free

vibration &r,ßm*eu& into Eq. (1) of free vibration with O on its right side gives

_(Cr,12+kr,1)ßr,j*+ {Ir,22+(Cr,1+Cr,+Cr,')2

+kmi+kr,lßr,*(Cm+1tr,)ßr,i*=0

(7)

The substitution of the ist and 2nd formulas of Eq. (6) into the above equation and its separation into real and imaginary parts give

j_(aC,,j+1.m_i»m1+wCmiÇf-',m j+ (Ir,(cx2W') +(_aC,,,1+kmi)+(aCnt+km)+HaCr,')} 'Pm - {2Ir,aw+wCr,1+wCr,+0)Cm'}Çbm_H'm

+ km)P,n+l+wCm/t'm+iJ +5CwCm-içDm-i

_(_aCr,1+kr,i)Çlir,i+(-2Ir,aW+WCmi+WCm +wCr,')ç2r,+ {Ir,(a2w2) aCm.i+ kr,,aCr,+ kr, _aCm'}çt,,,_wCr,'Pr,.i(_aCm+kr,)Ç'r,+iJ=0 (8)

For simplenes'sake, let us introduce the following

notations

a,.,=aCr,+kr,, br,wCr,

dr,

aC,,',

er,=wCr,' fr,=Ir,(C22 - w2) + am-i + am + dr, g,,, 21,,aw+ br,-1+ br,+ er, Q (3) (4) (9)

Cl - Cz

C3 C.1 C, C' C'4 1 2 e2 e3 k, k2 k ,

-+-Cn-t

-+-C,n-2 Cn-i C'n

Then Eq. (8) becomes

am_i_i+ b_1ç_1

+ bç+i+j( b jW1a,1ç1+ gr,iwn

+fb+1aç+1)=O

Theref ore,

amçom+i - brmçSm+i= a_1w_1+ b_1_1+f

gç

(10)

b+1 + aç,m+j = - b__1 - a_1ç5_1 +

+f,Th (lfl

Solution of the simultaneous equations (10) and (11) on çû,+1 and m+1 gives

au+b,,,vm

Ç°n*i

_a2+b2

bu,,+ainvm

't'm+l a,,2+bm2 where = - a_içcm_j + bm_in_i+frmçoim gnç (14)

v= -

a7,çl1+ g=w=+fç

(15)

Here, from Eqs. (12) and (13). (Ç2. Ç52), (Ç03. çl),

and +) may be expressed by a and w successively by substitution of çi=1 and =0.

Finally, when m=n, we have

cpn+1=0, 95n+i=0

therefore, from Eqs. (10) and (11), we have

a+1 bq.+1 = an_jço,_i +

g=u,=0

(16) bç+1 + a,çl+1 = - b_1ç_1 afl_lçl7_+gflçfl

+fç=v=0

(17)

Therefore, Eqs. (16) and (17) give the characteristic equations for determining the damping index a and the natural circular frequency w.

Therefore, let us write

F(a,w) =u = an_jçon_1 + bn_içn_i+fnWngnçl,i (18)

G(a,w) =v = - b,_1w_ - a_1ç5,,_j +gçn 19) 2.1.2 Successive approximation for roots

The following describes the successive

approxi-mation for the roots of the characteristic equations (18) and (19) in regard to a and w.

Let a and w be a set of roots corresponding to each other and satisfying Eqs. (18) and (19), a1 and w be the ith approximations fully near the values of a and w, and Sa1 and òw be their errors, and the following equations can be given.

a=a1+òa1, w=w+ôw Thus

F(a,w)=F(a+5a,w+ôw)=O

G(a,w)=G(a+5a1,w+5w)=O

The development of these equations in the power

series of Sa1 and 6w1 by Taylor's theorem with terms over the 2nd degree omitted gives

F(a, w)=Föa +F1ôw1 +F =0 (21) G(a, w)=G1ôa1+G1ôw1+G11=0 (22) where F=F(a1,w1). G11=G(a1,w)

=('')

\ da i= 6w

G=()

,

G.=(--)

da 5w =j

If Eqs. (21) and (22) are solved on ôa and ôw1, we obtain

oa=

F'1G. +G11F.

_{F'1G,1 GF'}

(23)

-ow1= (24)

However, the numerical calculations of the partial differential coefficients are made by using the following approximate equations.

aa1_1

F;1F. i-i

G1G1.1

G,1=

aa1

- Gil-1

G=

_{wiw1_1}

Here we may assume

Öa = a - a1 = a+1 -ôwj =ww=w1+1w Then, we have -

F1G,.+G1F.

(29)

F1G. G11F.

(30)

If two sets of approximations (ai, w1) and (a1_1, w1_1) are given by Eqs. (29) and (30), the next approximatiòns

(a1, w1) can be obtained. Repeat this operation till the errors can be held within the desired permissible relative values, and successive approximation to the

roots (a, w) can be done. Namely,

(30)

(31)

where CJ and

are the permissible relative errors. From the result of the above-mentioned calculation,

the following numerical values respecting the free

vibration of the equivalent vibration system can be

obtained:

natural frequency (c/mm) f=30 wir logarithmic decrement

5=2r(a/w)

real part çOni' and imaginary part çi,' of complex

specific amplitude (complex amplitude per unit amplitude at the reference mass of mass No. L) at mth mass

Wa! 1JWL2+(h2

= ,m/J WL2 -i-bL

specific amplitude (amplitude per unit amplitude at the reference mass) at rnth mass

ß =ço+çl2

phase difference of vibration of mth mass relative to the reference mass (rad)

p ,_ f

-'vsi Y,a YL

relative specific amplitude (relative amplitude per unit amplitude at the reference mass) between inth

and m+lth mass

[ß?]/(Wi Wni+i')2+(ni' _ç51f)2

specific stress (stress per degree amplitude at the

reference mass) between rnth and m+lth mass

(kg/cm2/deg.)

'ni -

kniCI3r'Jm ( r_{Z 180}

Z: sectional modulus of minimum diameter

bet-ween mth and rn+lth mass (cm3)

(Note) The reference mass (mass No. L) is generally adopted

oa

a+1a

a1 ow1 0)1+10)1 Wi Wi 12) (13)

the mass at the free end of the crankshaft. i.e. the

damper case. And in this case L=2.

2.2 Forced vibration

2.2.1 Calculation of forced vibration (method

for directly solving equations of forced vib-ration)

The records of actual measurements of torsional

vibrations of crankshaft

systems

made on many

engines show that the afore-mentioned free vibration damps in a short time, and the forced vibration only

becomes conspicuous. Therefore, it suffices if our atten-tion is paid only to the forced vibraatten-tion. Eq. (1) of the forced vibration can be solved by the similar method as

that of the free vibration. Namely, put the particular

solution of Eq. (1) as:

e=(q+jç)ei2t

(32)

then substitute it into Eq. (1) and separate into real

and imaginary parts, and the following can be obtained.

_JQ2_ k1(1,,,) + k(qÇ,,,+1)

+ C,m-i(ÇL-i - Cym(ÇÇ n+i)Q

-M,}

+ k7m(nv n+1) - C,-1(ç-1 - ç)Q +

+1)Q+C,'QN,,J =0

(33)

For simplenes'sake, let us introduce the following

notations:

!=

(34)

=Q(C,-.1+ Cr,v+C7m')

Then Eq. (33) becomes

-

+ L-1h--1+!?Thq'SL gnLcL k,çn,,+1

+ b,,,Ø,+1M,,+i( b1cp,,1 k1çi1+g,,ço,,,.

Therefore,

-

,n7n+1 -k,m_Çra_i + b m_

(35) 7in,+i+knL,Th+l = -

-

km_ia_i +ÖmÇin

(36)

Solution of the above simultaneous equations on

+i and gives kmil,m+ &m13,m (37) Çim+i Çin+i k,n2+t? (38) where

u,,,. = - k,,,.1ç,,, + n,.-i,,,-i +Jn,.,mniçL M,,,

= -

- k,,,,,,_1 +g,,,çom+fn,.Øn,. N,,, (39)

From Eqs. (37) and (38), (ç,, Ç2), (. q,),

can be expressed successively by (q,,, çi,). Finally, when m=n, we have

'Pn+i°, Ç,,+i0

Therefore, from Eqs. (35) and (36), we have

knq,n+i nn+i = k,,-1q,,,_, + nn - M,,

...(41)

nn+1 + knn+i = -

- k_1n-i+gnn

+fnNn

(Wi,ci) =0 (42)

Since the characteristic equations (41) and (42) can

be proved to be of linear equation of q,, and i,., the

following equation generally can be yielded:

(q,i, b,.)P,,q,+Qj1+R,,0

(43)

C(q,, çb,.)P,,'ç3,.+Q,,'çi+Rn'0 (44)

where P,,., Q,,., R,,, P,,', Q,,.' and R,,.' are constants which do not involve q,,. and ,.

Therefore, each coefficient can be obtained as follows:

4 =ff'(l,0)P(0,0) P(0,1)P(0,0) =P(0,0) = C(1,0) - (0,0) Q,,' =((0,l)((0,0) R,,.' =((0,0)

Whence, by solving the simultaneous equations (43) and (44), we get

- Q,,'R,,

0-i

Substitution of the values of Eqs. (46) and (47) into Eqs. (37) and (38) gives (q,2, ), (q,3, ), (q,,,, ç,,), suc-cessively.

From a result of the above-mentioned calculation,

the following numerical values of the forced vibration of the equivalent vibration system can be obtained:

actual amplitude at mth mass. (deg.)

(180/fl)

relative actual amplitude between ntth and in + lth mass (deg.)

(l80/r)

phase difference of vibration of rnth mass relative to the reference mass (mass No. L) (deg.)

= (çiç)(l80/'r)

actual stress between mth and in+lth mass

-7 i,,

[h,,:

relative actual amplitude (rad)

Z: sectional modulus (cm3)

specific amplitude (amplitude per unit amplitude at the reference mass of mass No. L) at mth mass

=

relative specific amplitude (relative amplitude per unit amplitude at the reference mass) between inth

and m+lth mass

[g'J,,,.= [8J,,,/OL

specific stress (stress per degree amplitude at the

reference mass) between mth and m+lth mass

(kg/cm2/deg.)

ria' =r,,,IeL

(Note) The reference mass (mass No. L) is generally adopted the mass at the free end of the crankshaft, i. e. the damper case. And in this case L=2.

On the basis of this calculation can be obtained the

exact actual amplitude and stress at each part and the mode of vibration in the speed range from the critical

speed to the flank.

2.2.2 Free vibration-energy method (a method for

obtaining actual amplitude and stress by the

energy method on the basis of the mode of

free vibration)

The conventional method used for obtaining the actual amplitude and stress at the critical speed in the

system provided with a viscous damper, was based on the mode of free vibration of a system having no damp-ing, i. e. normal mode of vibration. However, the mode of vibration at the critical speed in the system having a damper is considerably different from the normal mode

of vibration; and therefore the evaluation of actual

stresses obtained on the basis of the normal mode of

vibration was problematical.

In this study, however, the mode of free vibration

with damper damping taken into account (the presence

of engine damping makes little difference; so that gener-ally it suffices if the damper damping only is taken into consideration) has been obtained by the method stated in

2.1. And assuming that the mode of forced vibration at the critical speed is similar to that of free vibration, the actual amplitude 020 of the damper case and the

actual stress at each part are obtained by the energy method.

Considering the general case in which the ith order

harmonic components of tangential pressures of NB

cylinders of phase difference (equally spaced) act on

one crank and the total number of cranks is N, the

energy E0 per cycle of vibration due to the excitation

torques of all cylinders can be given as follows: 'niv

Er=rcoHtt !

022

7i1=7M1 t

The energies absorbed in one cycle of vibration by varied dampings are as follows:

Viscous damper: CEdJi2tClO)[9rlJl20t2 (49) Engine damping: 'iv

)EeJtrCe'w ''

_P7,1121 0222_¡ 2ii91i J Propeller damping: [EpJtirCCpJo wß72 0212 (51) If the sum of these damping energies is equalized to

the value of Eq. (48), the actual amplitudeat damper

case 021 (deg.) can be obtained by the following equation:

N -4. (180/rr) 021= 'fl= ni t (52) CIO(ßrlJi2+ [C01J2 oi 121 + [C21J0O) t-'nt f- -lnl=lflI J where

H1: ith order resultant excitation torque due to gas and inertia force per cylinder (kgcm)

Ht=(r/4)D2RI(Q,o+ Qt')t+ Q1t2 (53)

Q,t: sine term of ith order harmonic component of

tangential pressure due to gas pressure per

cylinder (kg/cm2)

Qco: cosine term of ith order harmonic component

of tangential pressure due to gas pressure per cylinder (kg/cm2)

Qr': ith order harmonic component of tangential

pressure due to inertia pressure per cylinder

(sine term only) (kg/cm2)

ith order vector sum of specific

ampli-tudes at the engine cylinders in one

bank r 711, t111Iitj t r miv

):;

f77t' t,n = ni r t I L220=171, 1_ "7T HU3î1/sin(5,,rt+çm!)}L?11?II

specific amplitude at mth mass in free vibra-tion with damper damping taken into account

çi,':

phase difference of vibration of mth mass

relative to the damper case in free vibration

with damper damping taken into account (rad)

phase difference to ith order harmonic

com-ponent of tangential pressure acting on No. i crank of the ith order harmonic component of

tangential pressure acting on the crank of

mass number n (rad)

m1: mass number of No. i crank mN: mass number -of No. N crank

48)

(50)

multiplier for vector summation of the -ith order harmonic components of tangential pressures of

NB cylinders of phase difference Ir (equally

spaced) acting on one crank (See Eq. (4))

[C07]2: equivalent engine damping coefficient per

cylin-der row (kg-cm-s) 1f'71

L OJO

[M0]1 _ßnro'2

n:=nhl

w0: natural circular frequency in case of no

damp-ing (" normal circular frequency ") (rad/s)

Is': effective moment of inertia of the multimass

system without damping referred to the equiva-lent one-mass vibration system (" normal mass

(kg_cm_s2)

n

1r,ti3r,to' (56) In =

ß,o: specific amplitude of free vibration without

damping

dynamic magnifier to the equilibrium amplitude

0 of actual amplitude at the free end with

engine damping only taken into account; to be obtained from the result of measurement of the one- or two-node vibration of the engine unit [Cp'Jt: propeller damping coefficient (kg-cm-s)

3.5[T0]1

[COJO r_LWPJI

propeller torque (kg-cm)

angular velocity of propeller (rad/s)

specific amplitude of propeller in free vibration with damper damping taken into account damping coefficient of viscous damper (k-gcm-s)

(See Eq. (67))

w: natural circular frequency with damper

damp-ing taken into account .(rad/s)

[ßr'Ji: relative specific amplitude of inertia flywheel to the damper case in free vibration with dam-per damping taken into account

3. Regulation of viscous damper

3.1. Optimum damping coefficient of viscous damper It has been proved by calculation by a digital com-puter that the method of two-mass system based on the theory of normal coordinate is in practice fully usable for the choice of the optimum damping coefficient of a

viscous damper as shown in Fig. 2. Therefore we

decided to determine the optimum damping coefficient by this method.

We can approximately reduce the J-node vibratiön of the multimuss system having a viscous damper to the equivalent two-mass system as shown in Fig. 3.

Thus, the differential equations of motion with

regard to the forced v-ibration of this system can be

given by the following equations. 11e1+C1(e1e2)=o

(58)

I07O2C1(EJ1&2)+C'92+k27e2= {H0} e2t j where

I: moment of inertia of inertia flywheel of damper

Is': effective moment of inertia ("normal mass") of J-node vibration (see Eq. (56))

k07: effective stiffness ("normal stiffness") of

J-node vibration

k' =w02 I'

C1: damping coefficient of damper

CE': resultant engine damping coefficient

CE =C0' 59) IIL I J r: (55) (57) 5 J2 54)

6

-n

Silicone oil

Fig. 2 Viscous damper

actual complex amplitude of inertia flywheel of damper

actual complex amplitude of damper case resultant value of ith order excitation torques of all cylinders

{ì}

=e3H3{

m=m1 J t

mN 4 .1

ßn0o'

By solving Eq. (58) we get the dynamic magnifier [Majo as: 0 12 x+4p2

- rx(x(1+4pp))2+4[p

[_x{(1+Ro')p+p')j2 ] where equilibrium amplitude x=12=(Q/wo)2, r: frequency ratio

p = C1/211w0 damping ratio of damper engine damping ratio R0' = '3/Je' : inertia ratio

w0: natural circular frequency of J-node

without damping

In Eq. (61), the engine damping ratio p' is generally comparatively small, say about 0.03; and therefore even if we obtain the optimum damping ratio Popt, with p'O, its difference from the optimum value in case of p*O is very small.

If we have p'=O, Eq. (61) will be reduced to [

L0 J - x(x_l)2±4p2(i_x(Ro'+1)}2

M 2_r 02 12 x+4p°

4p2E+F 4p2G+H

and when the relation of E/G=F/H=K (constant) is

established, [Maj22=K will be obtained; namely, if R0' is determined, [Mej2 is of a fixed value irrespective of

the value of p. That is, when we draw a curve of r-[MJ2 with p as parameter, they pass through a fixed

point.

Therefore if p is chosen so that [M,j2 will

become maximum at this point, the damper will be

working in the optimum condition.

Accordingly the value of To which will suffice EJG= F/H will be obtained as:

Topo (63)

Consequently the resonant circular frequency [QRI in the forced vibration in the case in which the optimum

damping coefficient of the damper is chosen will be

given by

vibration

Damper inertia flywheel 02

C ase

Fig. 3 Equivalent two-mass system

[fJRJOpt=j0/_2+R0' WoW0pt (64)

where w: natural circular frequency at the optimum

damping.

Substitution of Eq. (63) into Eq. (62) leads the dyna-mic magnifier [Maj2,0pt at the optimum damping to the following expression

[Ma12.opt=1+ (65)

Next, the optimum damping ratio p0p will be

obtained by differenciating Eq. (62) by z into O and

substituting Eq. (63) as: Port

J2(1 +R0')(2+R0')

Therefore the optimum damping coefficient [C1j093 will be given by

/ 2

[Cjjop3=2pop liWO-4J _{(l+R0')(2+R01)} I wo (67)

3.2. Approximate equations for giving the

damp-ing index a and the natural circular frequency

w of multimass system free vibration when

damper damping is considered

In this section, we intend to show equations available

for giving the ist and 2nd approximations of a and w

which are the values necessary for the calculation of a

and w of the free vibration of multimass system with

damper by the successive approximation stated in 2.1.2. Let us denote

a093: damping index when C1=[C,j00 logarithmic decrement when C1=[C3j0p3

w, natural circular frequency when C1= [C11093 then

ôap0002ir( ¿sont) _[Md]o,

w0p3 _opt

From this relation, we have

W093 _[Q5jopt

a090-,

2[Maj2, opt 2[)Wej2,opt

Substitution of Eqs. (64) and (65) into the above

equation gives

aop3 _2(2+R0')3t2 68)

According to the results of calculations made on a

number of shaft systems, the relative errors s' and

e' of aopt and w0p3 given by Eqs. (68) and (63) to the exact values are

0.15< e,,,' <0.15 0<oo,,<0.001

Consequently it suffices if we make successive

cal-culations as F(a,w)*0 and G(a,w)-0 by using as the

starting values the ist and 2nd approximations given by the following equations:

Fixed (66) } (69) Inertia flywheel Case

where

[uJ1: relative velocity between the case and the

inertia flywheel

h : clearance between the case and the inertia flywheel

u7]o/h: shear rate

Rewriting as [u7Ji=r[J1 and 4s-+ds and integrating over the entire circumference of the flywheel, we have the total viscous damping torque as:

ç 2tr ç b

! [Taj1₌

₀₀

{71op r1[OJ1/h1} db'r3dçz

r r b +

₀₀

(t r2E6J1/h2} dbr2d ç2lrçr2 +2\ _\ {7)optr2[O,]i/h3)rdço.dr O r1 12b 2b r24r141 7)OPt h1 ° + _h,r23+ _h3 jE rEo =[C0J00[7J1

Therefore, the optimum viscosity it will be given by [C1) (kg_cm-2_s) -{ r13+ r23+ r21r14 i h3 0.9807[C1J50 x 10

rp{--ri3+

r23+ r24r14 (cs) (72) h3 J where

p: specific weight of silicone oil

Since the viscosity given by Eq. (72) is an effective viscosity, the nominal viscosity (temperature 25°C; shear rate 0) shall be estimated with the working temperature

(i) Type Two stroke cycle, single acting.

supercharged diesel engine

Cylinder arrangement V-600

No. of cylinders 20

Bore 150mm

Stroke 200 mm

Rated horse power/speed: 1500 PS/I 450 rpm Max. horse power/speed 2000 PS/i 600 rpm

Brake mean effective

pressure: 6.6 kg/cm2 at the rated speed

8.0 kg/cm2 at the max. speed Crank arrangement and 5 th order phase-vector diagram

Crank arrangement _{5th order phase - vector diagram}

R

1.3,54.2

opt =

a1=O.85a01. a2=l.l5a01

Wi=Wopt, w2=1.00l Wept

3.3 Optimum viscosity of viscous damper

In this section, the optimum viscosity of the

silicone oil which will give the optimum damping coef-ficient C1J0 in operating condition will be obtained.

Assuming the shape of the viscous damper as illustrated

in Fig. 2, the viscous damping torque ETa]3 at the infinitesimally small area 4s positioned at the radius r

from the center of rotation can be given by the following equation.

EV7]1

_{is r}

h

R

9,7,6,8,10

Viewed from the front of engine

Fig. 4 Specification of the tested engine

L

7,6,8,10

and shear rate taken into account.

The working temperature of the damper differs with

its location, i. e., outside or inside the crankcase. The

result of experiments on various engines of this company indicates the temperature range 80-90°C in the case in

which the damper is located inside the crankcase and

55'-65°C when it is located outside.

The mean value of shear rate [S]1 at the

circum-ference of the inertia flywheel will be given by 1 Ç a/C211)053 _{[QR] OPt r2}

.

[S3j1

_ir/[Q5J0p0

o h2 son([__ p] opt t

+[cb']3)dt( 2 ) EQRJ opt r2[O7]I

4. Study of non-normal vibration with viscous damper

At the maximum speed, 1 600 rpm, of the

light-weight, high-speed, high-output two-stroke-cycle diesel engine of the specifications shown in Fig. 4, the

equiva-lent vibration system of which is shown in Table 1,

there occurred a severe torsional vibration of high fre-quency (8 000 cpm) of 5th order three-node which reached the actual amplitude ¿92=71.3 deg. at the free end of the

crankshaft and the maximum actual stress T33=2 000

kg/cm2 at the 2nd node.

As the result of studies and experiments made on various dampers for the purpose of suppressing thi's

vibration, it was proved that the viscous damper has an

exact, excellent damping effect and high durability

against severe vibrations. Therefore, we adopted the viscous damper shown in Fig. 5.

In the study of the forced vibration of non-normal

vibration system in the case in which various dampings

are considered, we investigated the variations of the actual amplitude and stress at each part and the mode'

of vibration within the range from the 5th order three-node critical speed to the flank and clarified their charac-teristics by the method of calculation described in 2.2. 1.

Further, we made a study comparing the result of

the aforementioned calculations both to the solution by the energy method based on the non-normal mode of free

Table i Equivalent vibration system

(73)

Note (1) The damping coefficient of the propeller differs with its speeds. It is to be obtained from Eq. (57).

(2) Reduction gear ratio r,,=O. 625

m Name of mass

I,o ¡toe/lO7 Coo Coo' Zoo

(kg-cm -s2) (kg-cm/rad) (kg-cm-s) (kg-cm-s) (cm°) i Damper flywheel 21.28 0 14517.7 O 2 Damper case 12.45 4.695 0 O 178 3 No. 1 Cyl. 8.11 3.626 0 496 178 4 No. 2 " 4.77 3.636 O 496 178 5 No. 3 " 4.77 3.636 (I 496 178 6 No. 4 " 4.77 3.636 0 496 178 7 No. O o _{8.11 3.049 0 496 178} 8 No. 6 o 8. 11 3.636 0 496 178 9 No. 7 o' 4.77 3.636 0 496 178 10 No. 8 o, 4.77 3.636 0 496 178 11 No. 9 _{" 4.77 3.636} 0 496 178 12 No. 10 ,, 8.11 5.181 0 496 178 13 Gears 6.60 2.174 0 0 172 14 Reverse gears 65.00 1.754 0 0 172 15 Reduction gears 19.00 0. 0315 O 0 196 16 Propeller 66.80 0 _{Note I}

LO) O) "J CN LO O' CN 0.05 Unit : cfi 0.03 5 03 75

Moment of inertia of flywheel: 2L28 kg-cm-s2

Mass ratio of flywheel : 0.3179 (for three node vibration)

Damping coefficient : 14517.7 kg-cm-s

Viscosity of silicone oil

when in operation 4.8X104 cs.

when not in operation 100X 10 cs.

Fig. 5 Viscous damper

vibration with damper damping taken into consideration as described in 2.2.2. and to the normal mode ofvibration

by Holzer's method in conventional general use. The

following deals with the result of the comparative

study.

4.1 Conditions of calculation

1) Relation between engine speed and power

Power at each speed was calculated by the propeller

law. Namely,

B =2000 (nc/i 600)

where B,: power of engine (ps)

nc: engine speed (rpm)

Order of excitation torque: 5th order

The 5th order harmonic component of tangential

pressure used is the result of harmonic analysis of the measured indicator diagram by a digital computer and is shown in Fig. 6.

Various dampings

Calculations were made on the four cases with the following each damping taken into accout.

1° No damping

2° Engine and propeller dampings 30 Damper damping only

40 _{All dampings of damper, engine and propeller}

(Note 1): The equivalent engine damping coecient per cylinder row io the value obtained by reverse calculation from the measured 5th order three-node vibration: C,'='494 kg-cm-s.

(Note 2): Since the propeller damping effect in case of three-node vibration of this system is very small, the case of 2° is hereafter referred to in the following description as "in the case in which engine damping is considered".

4.2. Consideration on the result of calculation

(Note): Symbols used in the following

description:-Values for forced vibration: "-" is attached on each symbol.

Example: ¡L'.FOL" nL, etc. (See 2.2)

Values for free vibration: Each symbol of values of normal vibration of a system having no damping was formerly provided with "o" as the subscript (e.g., so,ßno,etc.); however, this will be omitted and symbols without

will be used for the values based on the mode of free

vibration irrespective of the presence of dampings. Example: O,,. ¡9,,, rm.', re,, etc. (See 2.1.)

4.2.1. 5th order three-onde critical speed: ñ5 (1) In the case when no damper is employed

The 5th order three-node critical speed fi5 in the forced vibration in the case in which engine damping is

consid-ered is 1620rpm and this is rather 10 rpm higher than the critical speed n5=l 610 rpm calculated from the

frequency of normal vibration (8051 cpm) without damp-ing. (See Table 2) This is due to the following facts.

8

10

1 2 3 4 5 6 7 8 9 10 1112 Indicated mean effective pressure P0, kg/cm2

Fig. 6 Harmonic components of tangential pressure due to

gas pressure only. Two stroke cycle, high speed diesel engine

10 Engine power is accordant to propeller law.

2° In the vicinity of ñ5 the square sum ,O)I2 of the

specific amplitude ' in the engine decreases and the resultant engine damping coefficientC1=C,gC2 tends to rectilinearly decrease.

(2) In the case when a damper is employed

The 5 th order three-node critical speed ñ5 in the case

in which all dampings are taken into consideration is

1 560 rpm (1 540 rpm when torque is fixed). This is 36rpm (2.4%) higher rather than ñ5= 1 524 rpm (1 518 rpm when torque fixed) which is the value in the case when damper damping only is taken into consideration. (See Table 2) This is for the reason described in (1)-2°.

However, the natural frequency is 7511 cpm in the

case when all dampings are employed and is slightly below 7 517 cpm which is the value in the case when

damper damping only is employed.

4.2.2 Mode of vibration and actual stress at

the 5th order three-node critical speed In the case when no damper is employed

The mode of forced vibration where the engine dam-ping is considered makes almost no difference from the normal mode of vibration; and therefore the evaluation

of actual stresses obtained on the basis of the normal

mode of vibration is of sufficient accuracy for practical use. (See Fig. 7)

In the case when a damper is employed

There is a relatively small difference in specific

amplitude ', relative specific amplitude $O'JITh and

specific stress f,,1' between the mode of the forced vibra-tion in the case in which the viscous damper is mounted

at the free end, viz., when the use of all dampings is

taken into account, and the mode of the forced vibration in the case in which damper damping only is considered, but there is a cosiderable difference between the mode

of forced vibration and the normal mode of vibration.

(see Fig. 7)

Namely, the rate of increase of f' to

' at the

principal parts is 78% at the damper attachment (f2tfr2'

AA1

11111111 VAIl

IIIIIIIV4I1V

uiuiiiri

Hhli1Pll

lIl,1ipp,r

6

IIPfAl4L

o

i.iaE!

E u o. S o o u 4 3

Table 2 Actual amplitudes and stresses at 5th order

three node critical speed

=1546/868), 19% at the ist node (f41/r4'=l 757/1 481) and

12% at the 2nd node (f/c' = 1 726/1 543). This terdency is remarkable, particularly in the vicinity of the damper attachment.

Consequently, in case a damper is mounted, there may be a problem with the evaluation of actual stress

by using the normal mode of vibration. (See Table 2) For instance, the actual stress o2 at the damper at-tachment obtained by the normal mode of vibration using the actual amplitude of forced vibration #=0.3196 deg. is 868 x 0.3196= 278 kg/cm2. This value remains within the permissible vibration stress 290 kg/cm2 for continuous operation in a crankshaft made of carbon steel, specified by the Japanese Marine Corporation. The actual stress r2 of the forced vibration, however, is 494 kg/cm2, which

R everse

Damper Da mPer gears Reduclion t!ywheel case Cylinders V type)

Gears gears

ác

P r sp eller16

3 4 5 6 7 8 9 1011 1213

®®®®®o o

ö

-Mode of sib. with all

dampings csnsidered

1553rpm

Ø/

Normal mode of sib.

(without damping)

1610rpm Mode of vib with damping

effect st dampen only considered i 524 rpm Mode of sib. with engine & ---'

propeller dam Dings considered 1620 npm

Fig. 7 Modes of forced vibration of 5th order three node at

critical speed considering various dampings is the value requiring the use of alloy steel.

4.2.3.

Variations of actual amplitude, mode of

vibration and actual stress with the change

of speed from the 5th order three-node critic-al speed to the flank

Actual amplitude of damper case:

The actual amplitude #2 gets smaller with decrease

of engine speed from the 5th order three-node critical

speed ñ5, becoming the minimum at 1 220 rpm

irrespec-tive of the presence of a damper. Subsequently, with

drop of engine speed, it gradually increases untill at the 5th order two-node critical speed. On the other

hand, the decrement of 02 with increase of engine speed

from the critical speed ñ5 is gentler than that with

decrease of the speed from n5. Thus the 5th order three-node resonance curve is asymmetrical. This tendency is conspicuous particularly when a damper is used. The

engine speed of 1220rpm is the "transitional speed" where the mode of three-node vibration changes to

that of two-node vibration. (See Fig. 8 and 9)

Specific amplitude: ßrm', relative specific amplitude: I:gr'Jns, and specific stress: rm9

1° [ßr'Jns and f suC at the damper attachment decrease nearly rectilinearly with drop of engine speed from ñ5.

20

g,r, [yr'Jns, and fj at other parts rapidly

in-crease with drop of engine speed from ñ5, growing to the maximum at 1 220 rpm. Thence, it rapidly decreases

until at the 5th order two-node critical speed, drawing a similar curve as the resonance curve. (See Fig. 8 and 9)

The above-mentioned variations are of the same

tendency irrespective of the presence of a damper. Phase difference of each mass to the demper case:

ns,

In the forced vibration of a system without dam-ping, each mass always vibrates with the same or the reverse phase within the range from the critical speed to the flank. However, phase differences ç' when the

engine damping or all dampings are considered, increase

with drop of engine speed from n5.

(In Fig. 10, the

radius vectors which represent 62 and ÇirrO

are of left

turn). At 1 220rpm, they reach:

çZ;53=27Odeg., ç14=90deg. on both sides of the 2nd node and

çi=9O deg., çZ= 270 deg. on both sides of 3rd node.

When the engine speed passes through this speed, the mode of vibration shifts continuously from

three-node to two-three-node condition. They keep increasing with decrease of engine speed. (See Fig. 10)

Actual stess: Treu

The difference between the specific stress rm'

obtained by the mode of forced vibration and rrm'

by the normal mode of vibration grows greater as the

9 Dampings _Forced vibration Free vibration-energy method Engine Damper 1< X 1610 1610 Critical _{O x 1620} speeds X _Q 1524 1CO3 (rpm) o o 1560 1502 Actual X X = = amplitudes at Q x 1.3014 [1.2811) X _Q 0.5009 0.4781 damper case (deg) _o

₀

_0.3196 0. 2775 (0.2831) X X 868 At O X 874 damper attach-ment X Q . 1491 1472 O Q 1546 1483 X 1< 1401 Specific stresses ist Q X 1455 X _Q 1791 1805 (kg/cm/deg.) node O Q 1757 1830 X X 1543 2nd Q X 1480 node X Q 1877 1865 O Q 1726 1897 X X At O X 1143 [1112] damper attach-ment X _Q 762 704 411 O

0

₍₄₁₇₎ X X Actual stresses Ist O X 1893 [1897) X Q 915 863 (kg/cm) node 508 O 0 562 (511) X X 2nd 0 X 1926 [1977] node X _Q 959 892 527 O Q _[528]

Dampings: considered: Q, not considered: X

At damper attachment m= 2-3

ist node m= 4-5

2nd node m 13-14

J: Values obtained on the basis of the normal mode of vibration

): Values obtained on the basis of the mode of

vibration with damper damping only taken into account

1.4 1.3 1.2 1.1 0.9 0.7 0.6 0.5 0.4 0.3 0.2 0.1

lo

2 cao 5m 5000

Fig. 8 Actual amplitude and actual stresses of forced vibration without damper (with engine damping considered)

5000 E 10000 E 5000 5th c ri ti 156 Damper flywheel m:1 Damper Gears case 3 4 5 6 7 e 910 U 1213/14 2 ,-Cylinders--.--. I_-.i ,Reverse gears is le Propeller 450rpm 5th ordertwo-ne critical speed 587rpm _650rpm 0000

-Fig. lo Variations of actual amplitude 0m and phase angle Çm'

of forced vibration with change of speed from 5th

order three-node to 5 1h order two-node critical speed, in case with damper (with all dampings considered)

this engine, f is 2.2 times higher than (498/224=2.2);

and therefore the evaluation of the actual stress by

using the normal mode of vibration may possibly result in the erroneously selected material.

When no damper is emloyed, the resonance curve is

sharp; and therefore the actual stress

becomes maximum at the speedñ5=1620 rpm. However, in case a damper is employed, as the resonance curve forms a

gentle curve, the speed at which the actual stress

reaches its maximum, being affected by the variation of the mode of vibration, is shifted to a point slightly

off from ñ5. The speed where the actual stress becomes maximum tends to be shifted up above n5 at the damper attachment because of dfm'/dflc>O and down below it

at other parts because of df,j/dn0 <0. This tendency appears most remarkably particularly at the 2nd node.

At this node, the actual stress f13 at the critical speed ñ5=l56Orpon is 552kg/cm2, while the speed at which the actual stress becomes maximum is 1 500 rpm with the maximum stress of f 1= 623 kg/cm2. (See Fig. 8, 9

and 10, Table 2)

4.2.4. Free vibration-energy method

(1) The mode of free vibration with damper damping

only taken into account makes almost no difference from

VUE 900-5th order critical speed 1620rpm lnree-node I 700 600 500 380 200 Actual amplitude stress amplitude stress(stress at damper case per degree i --o-- Actual -- -4---- Specific 5th arder tws-ntde critical speed an dampen Case) j )600rpnd L Transitional speed , i,_.\ 5 .1220;:) 700

11

4

j -r 00

-k..Y'

'í.--T),\

i 5, 9..

'9O5

I

6øOi0

I

Reductioe gears 1700rpm irder three-node al speed )rpm ØØ 1440 rpm

jiL

11111111

II

-L

-- -

ir1

11111

0250rpm 1350rpm 1150rpm

--,i.ui

T ansit onal speed

1220rpm

IiiiI4hI1I

2

an darnpn case

per degree amplitude at damper case) --- clual anrplitide

--o--- Actual stress ---i.--- Specific stress)stress

700

5th order tel-node cnlical - (587rpm) 0, speed Transitional speed (1220rpm) critical

/r

5th orden three-node speed) r 560npm) 500

1111

:N

'b

II

300

200ata

100 400 600 1000 1200 1400 I 600 Enne speed n, rpm

Fig. 9 Actual ampliude and actual stresses of forced vibration with damper (with all dampings considered)

speed near the "transitional speed" 1 220rpmoff from

ñ5, owing to the variation of fm(m3) as described

earlier in (2)-2°. Therefore the use of the normal mode

of vibration for the evaluation of actual stress at each

part at the flank speed will cause a big error. Fig. 11 shows a comparison of the actual stress f at the 2nd node in the forced vibration with all damp ings considered with the actual stress r = ' O obtained by

assuming the normal mode of vibration and using the actual amplitude 02

at the damper case obtained

from the calculation of the forced vibration.

It states

that if compared at the cruising speed, 1 400 rpm, of 1 600 i 400 600 800 1000 5200 Engine speed n, rpm al 0.6 05 10.4 E 0.3 0.2 OI

500 I-400 o I 00 1200 1300 1400 1500 1600 1700 Engine speed flc rpm

stress at the 2nd node obtained from the calculation of forced vibration.

Dia : stress at the 2nd node obtained by assuming normal mode of

vibration, using actual amplitude i of damper case obtained

from the calculation of forced vibration. rii=rii'O2=1 543 (

Fig. 11 Comparison of the actual stress at the 2nd node

obtained from the calculation of forced vibration with that obtained by assuming normal mode of vibration, in case with damper (with all dampings considered)

that with all dampings involving the engine damping

taken into consideration. (See specific stresses shown in Table 2)

(2) The above-mentioned two modes of free

vibra-tion are of full practice for the evaluavibra-tion of actual amplitude and stress at the critical speed as shown in

Table 2. However, it is not suitable to use these modes

of vibration for the evaluation of actual stress at the

flank speed.

Result of measurements

The measurements conducted at sea on this propeller shaft system have proved that as shown in Fig. 12 the 5th order three-node critical speed and the curve of the

actual amplitude (resonance curve) nearly agree with

the result of calculation of the forced vibration and that

the method of calculation of the forced vibration as

related in this study are fully useful in practice. Conclusion

We have performed numerical calculation, by using

a digital computer, of the forced vibration, at speeds ranging from the 5th order three-node critical speed to the flank, of the non-normal vibration system with many such dampings as viscous damper, engine and propeller dampings, and further added a study to the

evaluation of actual stresses based on the mode of free

vibration without dampingthe normal mode of vib-ratiOnin general use. The result of the study can be

summarized as

follows:-(1) The mode of forced vibration at the critical speed is extremely close to the normal mode of vibration in

the case in which no damper is used but differs

consid-erably when a damper is mounted; and accordingly it

0.1

0.4

0.3

0.2

Torsiograph used New high speed type Geiger torsiograph

Calculated value

5th order three-node

Reprinting or reproduction without written permission prohibited.

We would appreciate receiving technical literoture published by you.

141

"

I!

Speed mas. at stress ,' / 1500rpm

W5th

order three-node cr tical speed 1 560rpm 1100 1200 1 300 1 400 1 500 1 600 Engine speed n, rpm

Fig. 12 Result of measurement

is of no accuracy for the evaluation of actual stresses

to be made by assuming the normal mode of vibration.

This tendency is noticeable at the damper attachment

in particular. According to an example of this calcula-tion, the actual stress obtained from the calculation of

the forced vibration is 1.78 times higher than that obtained by assuming the normal mode of vibration,

when compared at same amplitude of the damper case.

In general, there exists a " transitional speed " at

which the mode of vibration varies from (J-1)-node to J-node condition, between the critical speeds of the ith

order (J-1)-node and the ith order J-node. And the

amplitude and stress at each part per unit amplitude of

the damper case draw similar curves as the resonance

curve, which arrive at their maximums at the transitional speed between both critical speeds. (However, at

dam-per attachment the stress dam-per unit amplitude of the

damper case increases rectilinearly with the increase of

speed.) The aforementioned variation is of a similar tendency regardless of the presence of a damper. In either case, as the speed changes to the flank, the dif-ferences in amplitude and stress per unit amplitude of the damper case from those obtained by assuming the normal mode of vibration grow greater; thus the

estimated value of the actual stress by the normal mode of vibration becomes of great difference, unsuitable for practical use, possibly inducing an unexpected

miscalcu-lation.

From the characteristics stated earlier, it has been known that the more the transitinal speed is situated

near the critical speed, the more the speed at which the actual stress becomes maximum will be located closer

to the transitional speed off from the critical speed.

This tendency is remarkable particularly when a

dam-per is employed. And consequently the actual stress

curve and the resonance curve indicate fairly different tendencies between them.

The actual amplitude and stress at the critical

speed obtained by the energy method on the basis of

the mode of free vibration with damper damping taken

into account, proved fully practicable for the system having a damper. However, the assumption of the

normal mode of vibration as generally done is not

desirable.

Finally, the author wishes to express his appreciation to Dr. Shigeru Watanabe, Professor of Tokyo University, for many constructive suggestions.

I. w 300 'n 200 100 0.3 0.2 E 0.1 700 600 E