, MITSUBISHI TECHNICAL' BULLETIN'. MTB 010033
.
mitsum§HI.TECtiNIAIL 401:JETIk No 33
actor of Helical Sprin
Introduction
It has already been described that helical compres-sion springs can fail also with the increase of additional stress due to surging, besides the defects in material. Many investigatorso)-(2) have further studied character-istics of the spring vibration and methods of damping amplitude(4) and the calculation of amplitude at surg-ing(5), while the damping factor characteristics of the spring have not been entirely examined systematically. The authors have thus studied the problem with a view to examining the influence of various factors upon the logarithmic damping factor of springs hence
ex-periments were carried out by changing the initial
height or end conditions of the several springs manu-factured by them, such as exhaust valve spring of internal combustion engine.Theoretical Treatment
The logarithmic damping factor of the helical com-pression spring can be calculated by either of the fol-lowing two methods. One is a method in which the damping factor is obtained from the damped mode of
the free vibration of springs, caused by the impact
force given to them (termed the free vibration method in this report), while the other is a method in which the damping factor is obtained from the amplitude in reso-nance when periodic motion is applied to one end of the spring (to termed the forced vibration method). Each of these methods will be separately explained hereunder.2.1 Logarithmic damping factor from
the free vibration method
Let us imagine that the stress vibration with time is measured with a wire strain gauge adhering to a cer-tain point on the wire of the clamped-clamped spring.
Let r denote the stress at a measuring point, t time, co the circular frequency of the fundamental of clamped-clamped spring, and b the resistance coefficient, then the equation of free vibration is
Damping Factor of Helical Spring
Dr. Eng., Internal Combustion Engine Laboratory, Nagasaki Technical Institute, Technical Headquarters " Internal Combustion Engine Laboratory, Nagasaki Technical Institute, Technical Headquarters
Tadashi Kushiyama*
Hisafumi Ayabe**
Abstruct
This report deals with the damping factor of helical compression springs with a constant pitch, and presents studies for the calculation of the additional stress on the spring at the time of surg-ing, which is often a main cause of failure of the spring. The result of our experiments reveals that damping effects come from three major frictions, internal and external frictions of the coil itself, the contact friction of both ends of the spring, and the friction when coils are driven home. If the logarithmic damping factor derived from the linear theory is taken into account for the clarification of experimental results, and if we consider the ratio C between the amplitude of the mth harmonics Cii, and the geometrical minimum coil clearance ar, the minimum damping factor is obtained as follows:
Ernin 0.015 when C 0.003
and emin 1.6 ()0.8 when C 0.003
dzr
dt2 +2b dt +w2r
of which the general solution is given by
r= ebt(C1 cos /w2_ b2 t+ c2 sin w2 _b2 t) ( 2 )
Now, suppose that r=r0 and dz-Idt=0 when t=0, then
r=r0e-bt cos b2 t ( 3 )
In Eq. (3) stresses when 1,62-1)2 t=0, r, 27r , is expressed by
r= roebt (4)
Thus, the resistance coefficient b can be determined from Eq. (4) if the stress vibration with time is known. And the logarithmic damping factor E is given by the following expression:
e=blf ( 5 )
where f is the frequency of spring. 2.2 Logarithmic damping factor from
the forced vibration method
The exact solution of the vibration of the helical spring was obtained by Inoue, who pointed out that the vibration could be approximately considered longitudinal vibration of the column. Now, let u be the displacement of the coil at a distance of x from the end of the spring at rest, a the propagation velocity of the elastic wave in the axial direction and b the resistance coefficient proportional to the displacement velocity duldt, then the equation of vibration is
a2u
+LO=a2--
au a2u( 6 )
at at ax2
In the case where an end of the spring is clamped and the other end is driven by a cam, Eq. (6) was already solved by A. Hussmann(2), according to which the am-plitude of additional stress during the resonance of the m th order is expressed by the following:
4r=
;b2 .N/2G; cos ( 7) ( 7 )where C., is a harmonics of the cam lift of the m th
order, v = A m e circular frequency, 2-1, 2, constants determined by each mode of vibration, G modulus of torsion of the spring material, i the specific weight of the spring wire material, 1 length of coil and g the ac-celeration of gravity.
Additional stress 4z in Eq. (7) becomes maximum at x=0 or x=1 inside the coil. Let the maximum addi-tional stress be Jr,,max , then the damping factor b is given by
1,2 r 1
b C., 27r 2G
jrm.
from Eq. (7). Approximating the vibration of the spring with longitudinal vibration of the column, on the other hand, the natural frequency of the spring fo is given by
the formula
Ad Gg
f °= 27rD2NN 27
where D is the mean coil diameter, d the wire diameter and N the number of active coils. In general, the fre-quency with damping f is lower than the frefre-quency without damping f 0. As the resistance coefficient of a usual helical compression spring b is, however, very small in comparison with the circular frequ2ncy co, we can assume to (b<w). The logarithmic damping factor in the forced vibration method is, substituting Eqs. (8) and (9) into Eq. (5), given by
d
e=2G (10)
.u-/v ar,,,max
When a spring is statically compressed by C, the mean torsional stress dr.°,
Gd
47.0C.
(11)r D'N
is used, and the logarithmic damping factor is expressed by the following simple formula:
= 277 AZIrm'n Lirmmax Or
J77Thmax/47,,c, = 2/E
That is, the maximum additional stress during the resonance of the m th order is equal to avE times the stress 47.-.0 produced when the spring is statically com-pressed by the amplitude of Cam's harmonics. This coincides with the result shown by Inoue (3) from the exact solution on the resonance of the spring.
2.3 Construction of logarithmic damping factor The logarithmic damping factor e was obtained, in either of Eqs. (5) and (12), based upon the assumption that the damping effect is uniformly distributed along the coil and is proportional to the displacement velocity of the coil, while it is supposed, that the logarithmic damping factor is chiefly constituted of the following three effects:
: Such a damping effect as to be unable to avoid
so far as the wire of the spring vibrates, that is, hysteresis of the spring wire material, air resistance, etc.
A damping effect caused by inactive coils at both ends of the spring.
$3 : A damping effect due to collision, contact, etc. between the coils themselves, this effect appear-ing if the coil clearance is slight or if amplitude in vibration of the coil is high.
The actual logarithmic damping factor of the spring can be thus given by the following expression:
E2 e3). (14)
Consequently, it seems that the logarithmic damping
2
( 8 )
factor is different depending on the form of the spring, end conditions of the spring and modes of the vibration. Although the actual damping effect is, in this way, not so simple as assumed for deriving Eqs. (5) and (12), the authors assume, in analysing the experimental re-sults, that all the damping effects could be approximately expressed by the logarithmic damping factor in Eqs.
(5) and (12).
2.4 Relation between damping effect and coil clearance
As the wire of the spring vibrates within the coil clearance, suppose that there exists some relation be-tween the damping effects and the coil clearance.
Now, put the displacement of wire u at the time when clamped-clamped spring resonances in fundamen-tal modP., in the following:
u=u0 sin cos cut (15)
1
where uo is the amplitude at the center of the spring, x the distance from one end of the spring, 1 the length of the spring, co circular frequency of fundamental and t time. The strain at the point x is given from Eq. (15) by
au
=u
ax
cos mswtThe local spring force px at the time is, denoting stiffness of the spring with k,
Pr= klex= kuor cos cos cot (17)
The maximum force pmax, which produces at the point x=0 or x=1, is expressed by the following formula:
Pmax= kit() (18)
On the other hand, the local maximum additional force pmax at both ends of the spring never exceeds the force at the solid condition pmax* of the spring. If such an external force as to produce locally higher additional force than PrilaX* consequently acts, the wires of the spring will come into contact with each other. Let a7 be the total coil clearance of the spring, then the ampli-tude at the center of the spring uomax* at the time when pmax=pmax* is given, from Eq. (18) and the relation
pmax= k aT, by
?Lomax* = arir (19)
where uomax* is the maximum amplitude within which pure sinusoidal motion can take place without mutual contact of the wires. If such an exciting force as to make uo>uomax* consequently acts, the damping effect increases due to the mutual contact of coils. It is sup-posed that the effect depends upon the relation of uo with uomax*. Therefore, the ratio between Cam's har-monics Cm, of the m th order inducing the amplitude and the total coil clearance ct7 determining ?Lomax*, that is,
C=C.,/ttr (20)
must influence the magnitude of the damping effect due to mutual contact of the coils.
3. Test Apparatus and Results
3.1 Spring for testing
Supposing that logarithmic damping factor of the spring is, as shown in Eq. (14), chiefly constituted of the three damping effects E2 and E3 the helical compres-sion spring shown in Fig. 1 and Table 1 was manufac-tured so that each effect could be separately tested.
The spring C), as shown in Fig. 1, with open ends
Fig. 1 Springs for testing
Table 1 Main dimensions of spring for testing
Weight
Fig. 2 Test apparatus of the free vibration method not ground, which are welded to the spring seat, is stable so that the wires do not come into contact with each other even if the amplitude of the spring is considerable. The logarithmic damping factor obtained from this test result is consequently thought to indicate such a damping effect which cannot be excluded at the time of vibration of the spring as hysteresis of the spring wire material, air resistance, etc. That is, this effect corres-ponds to ei in Eq. (14).
The springs 0 and 0, which are of the same form
as the exhaust valve spring in internal combustion
engines comprise closed end-ground at the both ends, respectively.In a test on these springs ® and (:), it
was obtained, if the amplitude of the springs is small, the logarithmic damping factor f (el, e2), which is a sum of the damping effect due to the spring @ and that due to inactive coils at both ends. If the amplitude is increased, on the other hand, the logarithmic damping factor f e2, e3) must be obtained, because the damp-ing effect due to mutual contact of the coils is added.3.2 Free vibration method
An outline of the apparatus is shown in Fig. 2. At first, a spring for testing is fitted with four bolts at any length, and, as shown in the figure, the weight is sus-pended with a fine wire near the central section of the
Adjusting bolt (>:4)
To recorders
Wire strain gauge
''Cut instantaneously
MTB 010033 APRIL 1966
0 Auto-voltage regulator C) Electromagnetic oscillogragh
0 Strain meter 0 Switch box
Fig. 3 Measuring rigs for the free vibration method
lalassetimme...
(b) Spring C) Initial setting height=16 0 mm
Fig. 4 Recordings of the stress vibration on the free vibration method
spring. If this weight is suddenly removed, the spring begins to naturally vibrate, and the stress amplitude may become damp with time. If the stress vibration is measured, the logarithmic damping factor can be deter-mined. In the test, the amplified output of a PC-5 type wire strain gauge adhering to the outside of the active coil near the ends, was measured and recorded with an electromagnetic oscillograph. The measuring rigs used is as shown in Fig. 3.
Fig. 4 shows an example of the recording. It was very rare, according to the test results, that pure damp-ing of free vibration appears as in Fig. 4 (a), and vibra-tions of higher order were included, in almost all the cases, as shown in (b). It seems that these depend upon the weight, the position at which the weight is suspended, the condition at the ends of the spring, etc. Consequently, pure oscillograms were selected for analysis as far as possible, while the vibration of a higher order was neg-lected.
3.3 Forced vibration method
A test apparatus is shown in Fig. 5. As the
vibra-tion of the exhaust valve gear system in an actual
engine most likely complicates the vibration of the spring, rocker arms were eliminated and sufficient rigi-dity was further given to the frames of the spring. Con-sequently, cam directly acts on the movable end of the spring, as shown in Fig. 5.A spring for testing is fitted with adjusting bolts in any length, and its lower end is driven with a cam. Stress of the spring was measured with a wire strain gauge in the same way as in the free vibration method. From Eq. (16), the additional stress due to surging of the spring
Number of springs 0 0 0
Initial length of spring 10 mm 250 251 240
Mean diameter of coil D mm 80 80 80
Diameter of wire d mm 10 10 9
Number of active coils 11.5 11.5 11.5
Number of inactive coils 0 3 3
AilirtlYr=119,111at.
4
rr)
ca
(4)
Cathode ray oscillograph
Spring test apparatus
Pick-up of cam's velocity Fig. 6 Test apparatus for the forced vibration method
becomes maximum at both ends of the active coil. As the maximum additional stress is also necessary for designing a spring, it is desirable that the measuring position for stress is as near as possible to the active coil near the end. In a usual helical compression spring for exhaust valve, however, the ends ofactive coil come into contact with inactive coils, if the compressed height or the amplitude due to surging is increased, so that parts of the active coils do not fulfil its properfunction. So, PC-5 type wire strain gauges were adhered to a point about a turn from active coil at the end and to the outside of coil at the center of the spring.
In the forced vibration method, where the logarithmic damping factor is to be obtained, it is important to catch correctly the resonating revolution. In the test, an am-plified output with a strain gauge was recorded with an electromagnetic oscillograph, while stress vibration was observed by putting another output of the gauge into the cathode ray oscillograph. Moreover, endeavor was made so as to observe and record especially carefully the stress vibration near the resonating revolution. A general view of the test and measuring apparatus is shown in Fig. 6.
The cam used for testing is a symmetrical arc cam,
of which the working angle is 1200 and the maximum
lift 32.5 mm. Amplitude of harmonics obtained by the harmonics analysis of the measured cam lift curve, is
as shown in Fig. 7. As the cam
shaft revolution isvariable within the range of 100-380 rpm, it is possible to carry out test on vibration at resonance of the 8th
Test spring
Supporter
Body
Fig. 5 Test apparatus of the forced vibrationmethod
O Auto-voltage regulator 0 Strain meter
C) Electro-magnetic oscillograph
Setting nut
-Roller
Wire strain gauge Adjusting bolt --Cam To recorders 0.2 0 . 1 -E 0 1 1 1 8 9 10 11 12 13 14 Order of resonance m
Fig. 7 Harmonics of the cam's lift order or more, using the springs shown in Table 1.
In Fig. 8 are seen some typical modes selected from recordings. The cam velocity curves in the figure were simultaneously recorded so as to know the working con-dition of the cam.
Let us explain each curve in Fig. 8 as follows: In (a), the resonating stress of the spring C) of the 12th order is shown, in which the additional stress is very slight at the center of the coil, although it increases at the movable end. That is, the mode of resonance shown in Fig. 9 (a).
In (b), a mode of vibration appeared between the resonance of the 12th order and 13th is shown,in which
the harmonics of the 25th order resonates with the
second mode of the spring. This corresponds to the case, in which 2=2 in Eq. (7), while the spring resonates in such a mode as shown in Fig. 9 (b).In (c), the stress of spring C) at the cam shaft revo-lution near the middle of the resonance of the 12th order and the 13th is shown.
It is usual that the additional
stress at the other cam shaft revolutions than at the resonating point of fundamental is decreased, as shownin this figure. It seems, however, that at a certain cam shaft revolution the additional stress is, as described in (b), increased as the result of the higher harmonics re-sonance.
In (d), that at the resonating point of the spring ® of the 12th order is shown. In comparison with (a), it can be seen that there exists considerable difference in amplitude according to the end condition of the spring even if the springs have the same formand size.
In (e), that at the resonating point of the spring ® of the 11th order is shown. In comparison with (d), it can be seen that additional stresses aredifferent accord-ing to order of resonance.
In (f), that at the resonating point of the spring ® of 11th order, the same as in (e), is shown, but the mini-mum mean coil clearance a. = 0.5 mm in (f) while a,Th =
3mm in (e). From these recordings it can be seen that the less the minimum mean coil clearance is, the more decreased the additional stress at surging is.
In (g) and (h) are shown the recordings at the cam shaft revolutions that are not the resonating point of the spring C). It is interesting to note that there exists such a case, as in (g), that the additional stress during working angle of the cam is decreased in comparison with that during the un-working angle of the cam, and on the contrary, also such a case as in (h).
4. Discussion
From the test results, the following consideration can be given on the logarithmic damping factor of the helical compression spring.
0
Cam shaft revolution per min is =240.8 rpm (a) Cam velocity
Spring
At the center of spring
OP. r.iq .aPtiYe.ççil1 JI 11111111111111111 1111
/
(b)x100 sec
n=229.2229.2 rpm Spring C)At the center of spring
MMMMAA
20[
0
111111111111111111
Near the end -^ I--1/100 sec
n=232.5 20[ 0 Spring (I20[ 0 Spring At the center ear the end
1111111111111111111 '-1/100 sec 1 1111111111111111111 rpm n=222.8 rpm At movable end 11/111 1,11iIi!i,11/111 ''/100 sec
Fig. 9 Vibration modes of spring
4.1 Free vibration method
The logarithmic damping factor obtained from the stress recording as shown in Fig. 4 and the measured frequency is plotted in Fig. 10. For x-coordinates in the figure, the minimum mean coil clearance am shown in the following formula was taken:
Minimum total coil clearance ar
am= Number of active coils N mm
In this figure, the frequencies in the springs C) and 0 are increased to some degree with the decrease of mean coil clearance am (that is, with the increase of initial compressed height of the spring), while that in the spring C) without inactive coils at both ends is ap-proximately constant regardless of am. This is perhaps due to the decrease in the number of active coils, because both ends of the active coils come into contact with the inactive coils as the result of compression.
And in such a case, it seems that the damping effect is considerable, for the coils at both ends of the active coils come into contact with the inactive coils even if the amplitude of the spring is very slight. Comparing the frequency of free vibration with the logarithmic damping factor in Fig. 10, the latter seems to be also increased at such initial length that the former is in-creased. The damping effect is increased within the range am<1 mm especially in the spring C), with the result that with the increase of compressed height some
111111111111111
Fundamental vibration 2nd order of vibration
to 20
0
"O
0 20
Fig. 8 Recordings of the spring
At movable en
II!!! IIjr,,j,Iur,IIII III ,I i:11111111111111
F-1/100 sec n-272.7 rpm 20 0 Atno end 11111111,1111111111 11111111111111111111 kiA.00 sec 1: 1,1111i ``/100 sec (h' n 253.5 rpm MTB 010033 APRIL 1966 111111111111111111111111 I/100 sec
Fig. 10 Logarithmic damping factor and natural frequency from the free vibration method
irregularity occurs in the coil clearance due to
un-uniformity of the spring material, so that contact parts of the coil with each other is increased even at any vibration. As the test was carried out while care was taken of the nonexistence of mutual contact of the coils at all on the spring 0 in Fig. 10, it seems that its loga-rithmic damping factor is chiefly due to air resistance and hysteresis of the spring wire material shown in the first term ei, the Eq. (14), that is=f(e1)= 0.0075-0.017
And from this figure, it can be seen that the loga-rithmic damping factor of the springs C) and 0 is, if am is less than 4 mm, approximately the same as that of the spring C).
As mentioned above, the logarithmic damping factor within such a range that frequency of free vibration f does not almost change with the mean coil clearance a,,, probably represents, in the free vibration method of which amplitude is small, the damping effect f() shown in the first term, Eq. (14), not withstanding the end con-dition of the spring.
4.2 Forced vibration method
Additional stress vibration of the spring in resonance is as shown in Fig. 8. In general, additional stress near the resonating revolution vibrates within one turn of the cam shaft. In almost all cases, it becomes maximum when the cam has finished its working, that is,
immedi-\
Ill
1 X:r..x.,, 4. X N... 4 6 8Minimum mean coil clearance am mm
60 50 "7. 40 c r (el n=256.1 rpm 0.05 11 0.04 g0.03 42' 0.02 2 0.01 o 0
ately after the spring is compressed by the cam and again expanded perfectly, and is considerably decreased until the cam begins to work the next time. And the amplitude of additional stress at the cam shaft revolu-tions deviated from the resonating point is small, while the mode of vibration changes with the cam shaft revo-lution, as shown in Fig. 8.
Additional stress immediately after the cam working is as shown in Figs. 11 and 12. And the amplitude of additional stress zlr, kg/min' at surging is the measured stress on the outside of the wire converted into that on the inside. From Fig. 11, measured on the spring C) and Fig. 12, on the spring @, it can be seen that additional stresses are different with each other depending upon the end condition of the spring or coil clearance, even on springs of the same form and size as shown in Table 1. And it can be seen also from both figures, that the additional stress at resonating point of a helical com-pression coil is clearly increased.
If deviated from the resonating point, on the other hand, the additional stress becomes rapidly lower, while its value probably does not so markedly depend upon the cam shaft revolution.
When additional stress at the adjacent resonance
order is high, the stress at intermediate frequency,
however, has a tendency to increase to some degree. And( ) Order of resonance
6
(b) 10
200 250 300 350
Cam shaft revolutions per min n rpm
Fig. 11 Additional stress of spring C)
( ) Order of resonance 30 No. of spring 0 Sign
/a
aZ
T xa
o xo°0 o 0 °x o o 0 The others A 6 ?, g x 8 am=2mm (8) (12) (11) (10) (9) (13) .. 0.5 mm (11) (10) (9) (8) (13)(12) 04) 10 20 30 40 J DM% 0 kg/mm2Fig. 13 Maximum value of additional stress the peak sometimes appears due to the resonance of
higher harmonics at an intermediate revolution, as
shown above.And the minimum coil clearances in Fig. 12 (a) and (b) are a,=2mm and 0.5mm, respectively. Such a small value in a, limits the amplitude, so that the additional stress is decreased and finally no surging becomes pos-sible to be observed.
Let irrmax be the maximum value of additional stress dr, within the test range and Armaxo be the change of stress when the spring compresses from the initial con-dition to the solid concon-dition, then the dependance of the former upon the latter is as shown in Fig. 13. From this figure it can be seen that the additional stress drrmax never exceeds Irmax0,in other words, that the less the
coil clearance at the initial condition of the spring is, the less the stress added by surging is and its value is always less than .irmax o
Next, put additional maximum stress Are,,,max and the cam's harmonic C. into Eq. (10) and we obtain the logarithmic damping factors.
The results are
sum-marized in Table 2 and are as shown in Fig. 14.X-axis in Fig. 14, which is a constant term at reso-nance of the /nth order in Eq. (10), denote exciting force on the spring, because G and d/NI32 are determined by the material and the dimension of the spring respec-tively. In this figure, the logarithmic damping factor of the spring C) is less than others and is almost con-stant in the range of lower exciting force. The loga-rithmic damping factors of the springs C) and C), which depend upon exciting force and coil clearance, do not show any difference due to an, when the exciting force is small. The dependance of this logarithmic damping factor upon C= C,Th/ar in Eq. (20) shows clear distinction
between of each spring, as shown in Fig. 15, regard-less of the initial height, the dimension of the spring and the exciting force.
The scattering of a logarithmic damping factor is supposed to be caused by such factors as surface condi-tion of the spring end, condicondi-tion of support, etc., which are not easy to be analysed. In the spring @ in the figure, which has no inactive coil and of which both ends are welded on the spring bearings, its coils do not come into contact with each other when the amplitude of vibration is slight. The logarithmic damping factor in such modes of vibration is thought to indicate approxi-mately the damping effect due to air resistance and hysteresis of the spring wire material, its value being
= 0.017
from Fig. 15. And the logarithmic damping factor of a spring with inactive coils at both ends is small in the
200 250 300 350
Cam shaft revolutions per min rz rpm
Fig. 12 Additional stress of spring C)
(8) (II) (10) (12) (9) (14)(13) 200 250 300 350
0.20
0
02 0.4 06 08 10
Factor of exiting force 2G(d/N7y)1 Cm kern rn Fig. 14 Relation between the logarithmic damping
factor and the exciting force
same way as the spring 0. At both ends of such a
spring, it is supposed that each coil does not come into contact with each other even if the amplitude is con-siderably increased, so that the modes of vibration are almost the same as a spring without the inactive coil. This logarithmic damping factor is almost of the same value as that in the free vibration method.It can be seen from Fig. 15 that with the increase of C, $ is also increased for C 0.003, while $ = 0.015-0.045 for C50.003. The average values are as shown in full line; thus,
$=0.025 for C50.003 and
E=1.1(C), for C 0.003
It seems that for C_ .0.003 coils come into contact with each other to increase damping effect. The ampli-tude in the center of the spring in resonance can be shown from Eqs. (10) and (18) by the following formula:
2r 210= Gd
Jr.max=-T-On the other hand, the maximum amplitude with which each coil can vibrate within the total coil clearance of spring ar, is uomax*=ctrir from Eq. (19). By taking a ratio of both, the following expression is obtained:
it° ( 2r )c
UO max* k /1 ar /
e /
Taking the mean value of logarithmic damping factor
$ =0.025 for C=0.003, Uo/Uomax*= 0.75. Thus, letu (C=0.003) be the amplitude at the center of the spring for C=0.003, then it is supposed that for u0 >u (C=0.003)= 0.75arlz damping effect due to mutual contact of the coils appears, and in such a case sound due to mutual contact of the coils is heard.
= Cm/UT
Fig. 15 Logarithmic damping factor of the compression spring
Summarizing the above results, the logarithmic damping effect for C50.003 chiefly shows the damping effects el and ez in Eq. (14) and
e=f($1, $2) = 0.015-0.045 (Cs0.003)
Within this range the logarithmic damping factor, which shows considerable change, seems to be unable to be determined, for such complicated factors as end condi-tion of the spring, etc.
For C 0.003 the damping effect due to mutual
contact of coils is further added, so that the average value of logarithmic damping factor is shown by the following formula:
e= fcei, e2 $3)=1.1 (C)°." (C: k0.003)
4.3 Comparison and discussion
Although theoretical investigations on vibration characteristics of a helical compression coil have been carried out in considerable details' -00, the characteris-tics have not quite been observed experimentally.
A. Hussmann(5), who carried out extensive tests on
a small-sized spring, obtained the result shown in Fig. 16. The minimum logarithmic damping factor obtained from the frequency of the spring for testing and the curve C in this figure is as follows:
MTB 010033 APRIL 1966
Table 2 Logarithmic damping factors of springs for testing
7
No. of spring Spring 0 Spring C) Spring ®
Minimum coil clearance mm 4 3 2 1 0. 5 5 4.06 2. 25 Natural frequency f c/s 45 47 48 48.9 49.1 48 45.3 45.3 Order of reso-nance m Har-. monies cm MM 4- max ''',-kg/rom2 E zir,,,, max kg/mm2 d e r,, max kg,/mm2 E 4 max kg/mm2 e dr, max kg/ m& C 4 r, max kg/mm2 c el .m, max kg/mm2 c elr, max kg/mm2 e. 8 0.458 14 0.063 13. 4 0.068 11. 12 0.084 6.7 0. 143 4.6 0. 208 22.9 0.041 2. 2 0.04 14.75 0.061 9 0.0699 3.3 0.042 3.7 0.039 4.0 0.037 3.5 0.043 3.3 0.050 7.6 0.020 5.7 0.025 3.3 0.044 10 0.1516 6.3 0.049 8.5 0.037 7.0 0.043 6.0 0.055 4.2 0.079 21.1 0.015
-
-
10.9 0.028 11 0.1785 6.5 0.055 7.0 0.013 7.3 0.052 5.8 0.067 4.0 0.098 21.5 0.018-
-
7.1 0.051 12 0.0063 4.0 0.045-
-
4. 4 0.044 3.8 0.052 2.8 0.076 12.5 0.016-
3.7 0.049Sign Urn No.of
spring
05
0
-0
0.5 4 2 (4- x
3 4 2 ',.3) a A A IN' A It.' o o A 1.
Sign am No. ofspring
0 5 CO 0 0.5 1
2
0/
x 3 4 A 2 (.3) 0 A 0 0-
The others X 0 I' a x o-o i II o 00 oa oc? og,0 th)8. o 0.001 0.005 0.01 0.05 0.1 as -a so f3 0.05Fig. 16 Minimum damping factors by A. Hussmann") emin=0.0055-0.03 for C.=0-0.04 mm And a tendency is observed, in which emin (bmin) is gradually increased with the increase of C..
As sometimes emin=0.0075 for a small value of Cam's harmonics from the above-mentioned test result, the results of both coincide. Taking into consideration the fact that resistance coefficient b does not change only with C., it is assumed that the change of b in Fig. 16 is due to the increase of resistance coefficient, as the result of mutual contact near the inactive coils because of the increase of additional amplitude with the increase of C.. Consequently, it is impossible to apply the result in Fig. 16 to the designing of other springs as it is.
Inoue and others obtained the following logarithmic damping factor by the forced vibration method and the free vibration method, respectively.
e=0.03 by forced vibration method and
e =0.02-0.07 by free vibration method, which approximately coincide with the test results by the authors.
5. Conclusion
The test results can be summarized as follows: 5.1 Phenomena of vibration of spring
The maximum stress at surging of the spring never exceeds the stress of the spring in the solid condition.
Additional stress at surging, which is sometimes un-neglectable even at the resonating point of the 10th order or more, is determined by the magnitude of Cam's harmonics and the logarithmic damping factor at that time.
The additional stress at other revolutions than the resonance revolution is very slight regardless of the cam shaft revolution.
5.2 Logarithmic damping factor
(1) Logarithmic damping factor of a helical
compres-7 cn 6 0.03 0.01 0.02 Cam's harmonics Cm mm 0.04
sion spring can be put in order with C=C.lar , where C. is Cam's harmonics and ar is the total coil clearance at that time when height of the spring at work becomes minimum.
When is very low the logarithmic damping factor
shows the damping effects e, due to air resistance of the spring and hysteresis of the spring wire material. In the free vibration method e= f(e,)= 0.0075-0.017, while in the forced vibration method f (e)=0.015-0.017,
thus both values approximately coinciding.
In a spring with inactive coils at both ends, the
damping effects $2, due to the influence of the inactive coils, is further added. Then the logarithmic damping factor is e = f(,e2) = 0.015-0.045, which cannot be un-conditionally defined because it is determined by such complicated factors as the form of inactive coils, sup-porting condition, the condition between spring and spring seat, etc.With the increase of spring amplitude, the loga-rithmic damping factor is increased because of the mutual contact of the coils. In a helical compression spring, this influence appear for the amplitude at the center of coil u00.75 (arl7c), that is, =0.003 or more.
For the range of 0.003, consequently, the damping effect E3 due to mutual contact of the coils is added, and the following experimental formula can be obtained from the average value of logarithmic damping factor:
e=f(e,., e2/ 3)=1.1 ()0.65
Summarizing the above-mentioned, the logarithmic damping factor of a helical compression spring with closed ends, as for the average value of the experimental results, is,
e=0.025 for 5 0.003 and
1.1 (c)0 65 for
while it is as for the lowest value,
emin=0.015 for 5 0.003 and emin= 1.6 (C)0.8 for
When designing, it is safe to adopt the lowest loga-rithmic damping factor. The exhaust valve spring for our UE engine, which was designed utilizing this result, has proved to be excellent.
References
Shimizu and other two: Transaction of the Japanese Society
of Mechanical Engineers, 27-179 (July 1961), 1119.
Inoue and Yoshinaga: Transaction of the Japan Society of
Mechanical Engineers, 27-179 (July 1961), 1130.
Shimizu and other two: Transaction of the Japan Society of
Mechanical Engineers, 27-179 (July 1961), 1138.
Tokue: Transaction of the Japan Society of Mechanical
Engi-neers, 19-87 (1953) 24: 19-112 (1955) 958.
A. Hussmann: Jahrbuch der Deutschen Luftfahrtforschung 1938
(119)
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