JPRS ID: 8962 TRANSLATION ABSORPTION OF VIBRATION ON SHIPS BY A.S. NIKIFOROV

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JPRS L/8962
3 March 1980 .
,
- Translation
- Absorptinn of Vibration on Ships
By -
A. S. Nikiforov
FBIS FOREIGN BROADCAST INFORMATION SERViCE
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JPRS L/8962
3 March 1980
ABSORPTION OF VI$RATION ON SHIPS
Leningrad VIBROPOGLOSHICHENIYE NA SUDAKH in Russian 1579 signed
to press 21 Dec 78 pp 1-184
Book by A. S. Nikiforov, Izdatel'stvo "Sudostroyeniye", 2,700
copies
- CONTENTS
PAGE
AIVNOTATION o....oo..oo..o........o.....o....o.o...a............o....
1
- PREFACE o....>.......o.......o.......o ..............o...............
2
ORIGINAL TABLE OF CONTENTS ,..o....o......o...>.o....a.a..o...oooo.0
2
INTRODUCTTON ..............o......o..........................o......
5
CHAPrER l. PAYSICAL BASES FOR ABSORFTION OF VIBRATION
7
Absorption of Vibratory Energy in Oscillating Systems With
Concentrated Parameters a........a ...............o...o..>...o.
7
- Absorption of Vibratory Er.ergy in Deformable Media o........o...
18 -
Dissipative Characteristics of Ship Machinery and Hull-Frame
_
Structures ooo.oo.o>o>o.oooo.o..oo............o.....oo........
23
CHAFI'ER 2. THE TNFLUENCE OF VIBRATION ABSORFI'ION ON VIBROACOUSTICAL
CHARACTERISTSCS OF SHIP STRUCTITRES ..o............o...
25
The Energetics Method of Describing Vibroacoustical
-
Characteristlics of Ship Structures .....o.o...o...............
25
Vibroexcitability of Structures ..........................o.o....
33
Propagation of Vibrations Through Structures
35
Sonic Radiation of Structures ..........o,........o.....oo......
37 ~
' a' LI - USSR - G FOUO]
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CONTFNTS (Continued)
pa,gc
Sound Insulation of Structures
39
CHAF`l'M 3. VIDROABSORPI LM, COATINGS FOR qIIP rTRUCTURrS..........
)11
Methods of Determining Losses of Vibrational Energy in
Oscillating Laminated Media ......oo ...............o....~...
41
Rigid Vibroabserptive Coatings..o.........,.o.o..,,.........�
50
- Stiffined Vibroabscrptive Coatings
56
Pliable Vibroabsorptive Coatings
62
Combination Vibroa.bsorptive Coatings o............oo......a...
70
CHAFI'ER 4. VIBROABSORFrIVE CONSTRUCTION MATERIALS SUITABLE FOR
USE ON SHIPS.oo.........oooo..oo......o.>oo...o.....
76
Laminated Vibroabsorptive Ma.terials o....o.......o...o.oo..,.
76
Vibroabsorptive Alloys o.o......o.o....oo...............~....
79
Non-Meta,llic Vibr�oabsorptive N1aterials , o . . . . . o , . o . .o o . . , . .
81
CHAPI'ER 5. 0'IiHER MEANS OF VIBRATION ABSORFlI'ION o... o. o. o .o .o. o o, 0
82
Local Vibration Absorberso...o.o....o.oo..o.....o.ooo..o...,0
82
Friable Vi.broabsorptive Materials o..o,.a...o........ooo.....
90
Liquid Intermediate Layers Used for Vibration Absorption .o..
93
- CHAPI'ER 6. DANPING OF VIBRATIONS OF E'LIIMENTS OF SHIP MACHINERY
AND HtTLL-FRAME STRUCTURES .............o.....o..oo..
100
Optimum Len gth of Vibroabsorptive Coatings ....a..,>.........
100
- Loss Factor in Plates Pa.rtially Faced With Vibroabsorptive
_ Coating .....o......o.........o.o....oa....aa......ooo>ooo0
106
Vibration Absorption in Ribbed Structures ..,o..,o....,,..,..
110
Effectiveness of Da,mping Rigidity Ribs Which Reinforce
Ship Structures ...o........o..ooooo.....o.....o.o.o.......
114
The Influence of a Liquid, Contiguous to a Da,mped Structure,
on Effectiveness of the Vibroabsorptive Coating oa..o..o.o,
116
- b -
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CONTIIVTS (Continued)
pa,ge
- Damping Vibration of Beams, Pipes and Other Rod Structures..
120
Vibration Absorption in a System of Connected Spans.........
133
- Optimum Combination of Means of Vibration Absorption and
Vibration Insulation ................,o,.,................
137 -
CHAFi'ER 7- PRACTICAL USE OF MEANS OF VIBRATION ABSORPI'ION ON
_ SHIPS ...a........oo.o.o.o....o.o..........o.
141
Methods of Evaluating the Effectiveness of Vibration
-
Abs.orption in Ship Structures o.....o...o..o.......o.,....
141
The Effectiveness of Various Pat-terns of Applicati.on of
Vibroabsorptive Coatings on Ship Structures .a.........o..
149
Principles of Rational Use of Vibroabsorptive Coatings
on Ships ..........o....o .............o.,o:..oo........o..
153 -
Recommendations for Use of Means of Vibration Absorption
on Ships .o....o..a....oa ....................o.a.a.a..o..0
157
Exampies of the Use of Means of Vibration Absorption on
Ships ....a.......o........ .oo..o.....o.........oo.o......
160
CONCLUSION .oo...o...........o......oo ...................o......
165
B1BL1Vl7RtlrRl o..c.ooo.o�..��........o..oo......... o o... o ....o.. .
166
~
-c- U
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PUBLICATION DATA
English title , ABSORPTION OF VIBRATION ON SHIPS
Russian title � VIBROPOGLOSHCHEPTIYE NA SUDAKH
Author (s) . A. S. Niki_forov
Ed-itor (s)
,
Publishing House . Izdatel'stvo "Sudostroyeniye"
Place of Publication . Leningrad
Date of Publication . 1979
- Signed to press , 21 Dec 78
Copies . 2,700
COPYRIGHT , Izdatel'stvo "Sudostroyeniye", 1979
- d -
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A"lNOTATIllr1
/T-ext7 This book examines various aspects of the problem of reducing sonic
vibration which is set up in ship structures during equipment operation,
c.ausing increased air noise in compartments of the ship. It describes -
the caorkinp principle and design of vibration absorhing installations.
It makes recommendations on the rational use of ineans of vibration
absorption on ships and gives methods for evaluating their acoustical -
effectiveness. The book describes technology for manufz.cture of some
vibroabsorptive materials.
This book is intended for technician-engineers and scientific workers
who are involved with tlie questions of reducing sonic wibration and air
noise on ships. It r.iay be of interest to specialists working with the
problems of reducing the level of sonic vibration and noise j.n automobiles,
trains and aircraft. The book will be useful to students and post-
firaduates specializing in the aforementioned field of acoustics.
_ Reviewed by Professor I.I. Klyukin,
Ddctor of Technical Sciences
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PRT:FACF.
nrowth in the power and resultant vibration oF ship machinery carrie; -
~ 1aitli it increased sonic vihrations in the ship's hull-frame str+ir_tures =
= and a subsequent increase in the noise levels inside compartments of -
the ship. Development of ineasures oriented towar.d reducinp, vil>ration ~
takes two primary directions insulation of vibration and absorption
of vibration. The former has heen covered in detail in literature
[5, 241; the latter, despite the high effec.*_iveness of ineans of vibration
absorption, has not to the present been systematically e::amined.
This book getYeralizes the experience of anp?ying means of absorption
vibration on ships and other forms of transoart, as well as in industry.
In so doing it uses ciomestic and foreign Fuhlications and the authcr's
own experience in this field. However, not a11 aspects af the vibration
ahsorption problem are exa-.:�ined here in equal detail. For example,
questions of the chemical structure of vibroabsorptive materials and its
relationship to physical. and mechanical properties have been ommitted. For
informatian on these questions the work [53], in particular, is recotrmended _
to the reader. Plethods for eYPerimental study of the dissipative proper-
ties of ineans of vibration absorption, which have been adequately described =
by S.V. Rudrin in work [34], are not preser.ced. -
- 1'}ie reader may easily f.ind information on general theoretical questions,
1wo (no>2 is aupercritical damping)
- frequency w*ul takes on imaginary value, consequently movement of the eys- -
tem with such losses becomes apeziodic:
. 8,te ~0 !
x = A Le a1 o -f-
J �
(1.11)
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The loss factor value, whiah when exceeded the movement of the syetem
- become aperiodic, ie called criCical (nKp-2).
tlnder internal friction, characteristic of deformable solids, R. is
= inversely proportional to frequency
R=Ra=Ro 0; , (1:12j
where Ro is value R2 iahen w*02-W 0.
Solution of equation (1.1) when R-R leads to the result
rJ~o2-b~lt . _
_ x= Ae~ ~
where w*U2, 62 are the frequsncy of free attenuating oscillationa and
constant of attenuation of the system, equal to:
. 1+Y~-no .
W~l~uo 2
_ (no < 1) (1.14)
bz=(0o Yl
2 ,
- w~=sz=Wa (no > 1) ,
2
At low values of no(no2�1)
2 _ .
cu~ wo (1- $ 1 ~ ga ~ 2
/
In this case an increase of losses in the system also lowers the frequency
of the free attenuating oscillations. With rise in losa factor (no �1)
8 ~0 . (1.16)
o~ ~ ~ 2 Y,~ �
io
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- Thus at Z inversely proportional to frequency, there is no critical
damping, insofar as wo at any rto exceeds d2. -
Substituting expression (1.13) into equation (1.1) ahows that the term
describing losses in the system becomes equal,-consequently, the loases
in this case are proportional to deformation of the element of elasticity
.
X.
In some works, for example [13, 44], as losses in the system increase, -
- where R=1/m , a rise in frequency of free attenuating oscillation is
obtained. This result is derived in error, hecause x=Jw*02x"Jwpx is '
substituted into equation (1.1) instead of the abvious x-(j *-)x.
*
Herewith the solution to equation (1.1) when R=R3- Ro b'03
WO
,
83)
z=A e~~' (1�1~ .
=
� 4
where o~~ = Wo
8t = a)o 4 y �
' ~10
z
Specif ically, Muci ~ ~''~o (1- 8
2
whcrr ~0z4.
` 'b ~here
~
Thus at any 10 the constant of attenuation b3 remains less than W p,
- therefore there will be no critical damping in this case either.
rig. 2 depicts the dependence of the frequencies of attenuating oscil-
lations and constants of attenuation on no for the cases examined above.
11
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~
45
Wo~Wp a~
2
~
~
3/w, 6)
/ I
0 f 2 J tJ 0 f Zq
k'ig. 2. Frequencies of free oscillatione (a) and constants of
attenuation (b) in a sqstem r,Tith one degree of freedom
depending on loss factor.
. . . . _ . _ :
! -R=tonst; 2- R! Ro o: J- R-Ro 03 .
0
With increase in the loss factor the frequency of free attenuating
oscillations in the system drops fastest of a.ll where R=1/w, but
where R=w the drop is slowed docm. When nu0.5(nE)maX. Specifically, for the "Antivibrit-2"
material [he working temperature realm ranges AT-0=35�C. A AT value
on the order of 40�C is generally characteristic for materials of
this type known at present. This is expained by the fact that polymeric
materials have high vibration absorbing properties in a comparatively
narroca temperature realm (the vitrification realra), in which the material changes from a glass-like to a rubber-like state (see $2.)
20 ~ nat
6 - - - '
y ~T
2
i .
E
~ 10 0 �90 +20 +JD +40 1; �C -
T'ig. 14. Dependence of the modulus of losseR nE of the vibroabsorptive
material "tlntivibrit-2" on temperature T.
r To expand th'e working temperature range o� a rigid vibroabsorptive
coating, it is recomnended that two or more types of materials be used,
the maximum effectiveness of which is at different temperatLres [100].
These materials are applied to the structure being damped in layers.
There is a rigid vibroabsorptive coating which incorporates an electric -
- heating element, allowing regulation of the temperature of the material
and the v;,:I:inF temperature range of the coating [451. So-callec! '
plasticizers are used to shift the working temperature range in the
53
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54
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direction. An increASe in the amount of plaAtictzer .in thc campoAiti.on
of the vibroabsorptive material lowers ttie higli-effactiveness range of
the material and vice versa.
The physio-mechanical properties of some materials, devel.oped gnd uaed
for rigid vibroabsorptive materials, are shown in Table 2, which was
compiled from data in works [8, 9, 25, 34, 35, 56, 76, 83]. It ia
evident that our best domestic materials are not inferior to foreign
riaterials in their vibration aUsorption propertiea; epecial tmterials
are signiftcantly more effective than materials used for finish work
on stiip structures (for example, compare FCV linoleum and "Agat"
sheet material). nomestic vibration absorption materials are not
inferior to foreign materials.
Fig. 15 presents frequency characteristics of loss factors of steel
rods with layers of a rigid vibroabsorptive coating from the materials
l1-5 (L1SSR) and N1RC-0G4 (IISA) applied to them. Comparison shows that
the effectiveness of tllese materials is high (n>0.1) and comparable.
17
0,5 ~
Q3
q2 _ -
n.~ ~,v +qo
Kru f,khz
Fig. 15. Frequency characteristics of loss factors in steel rods with
vibroabsorptive coatings applied to them.
Key : A-S ma tc_rial [ 25 ](akm'2
: MI!Mi=0,39; r=20-c)
- MRC-QG4 material[56] "wm-0,33; r.:s-cf-
Foam plastic of the PCV-1 type is usually used for intermediate layers.
Its physio-mechanical Properties are as follows: G2-4x107 DIN/cm2, pti0.1,
g/cm~, nti0.02. T'rom t}ie point of view technology for applying coatings,
ttie most advanced is the use of froth-forminr polyurethane plastic foams
for intermediate .l.ayers. I�7ork [36] ahows the possibility of using f.or
- this purpose PU-101 materials, which (after it hardens) has Eti109 DzN/cm2,
p=0.12 g/cm3.
The technology for application of a rigid coating depends on its properties.
Sheet r+aterials are applied with glue (Type PN-Eh or FhPY.-519). The
surface onto which the material is to be glued must be meticulously
cleaned and primed. Special equipment for clamping the sheet plastic
ensures a high-quality bond. rlastic materials are applied by dusting,
spraying or stapling them on in layers 2-4mm thick until the required
- 55
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thickness is achieved. *tastic materials on the basis of polyvinylchloride
and polyvinylacetate are sometimes made in the �orm of a water emulAion.
After the water component dries out the layer takes on vihroabsvrpCive
properties. Warming such materials is usefiil to accelerate the drying
process. 24aterials based on epoxy polymers require a special heat treat-
ment after application, w�ithout which they do not take on vibroabsorptive
praperties. The surface of mastic materials is machined after application
to give them a decorative appearance.
$11. Stif.fened Vibroabsorptive Coatings
A stiffen vibroabsorptive coating consitutes a layer of viscoelastic
material on which a thin stiffening layer of rigid material is applied
(77]. Ttie structure of a stiffened cQating and the character of its
deformation under flexural oscillations of a damped plate are shoi,m
in Fig. 16.
a) 1 Il
3 ' 4
Z
f
b)
3 MW44
2
1
T'ig. 16. Structure oi a stiffened vibroahsorptive coating (I) an ttie
_ c}iaracter of its deformation (II): a- stiffened coatino;
- b - multilayered stiffened coating
4:ey: 1. dampecI plate; 2, viscoelastic layer; 3. stiffening layer;
A. deformation of the vibroabsorPtive material
- %hserpkion of vibratory energy in a stiffened coating is attributable
to shear deformation in the viscoelastic layer. P�.ubbers or rubber-like
Plastics, i.e. pliable niaterial, are usuall.y used f.or this layer.
nbv.tously thesp naterials must 9~lso possess high internal loss proper-
ties. The loss factor of a plate,ifaced with a stiffened vibroabsorptive
coating is most simply determined hy the wave resistance method (see $9).
~l = ~2 a2R2 --12a21a2A2 I2a3182Yo (aa~a - �ka2P2)
1 ac202'~- a3~3 12a21 a~~2 12~31a3B2Yli 'f' 92 ~lZ) ~
(3,36)
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h
where as =  ae = h ; asi - ~ ; aai = 1 +
E
h21= 2 (hi'~ h~ ; hai = 2 (hi ha) fts;
a - E3 , � yo = 1 (3.3~
- :
y~

El (1 -f- 82)2 -f- h~l2
s
= G ; , �k= ~ ~ .
8: k
E3h3k~hz
hl, hZ, hj are thickness of the damped glate, the viscoelastic layer
and the stiffening layer respectively; E1, E2, Es are the corresponding
Young's modulus values; kH kHq=-1 is the wave numher of the flexural
oscillations of the plate with its coating. The first tertn of the
expression (3.36) defines losses of energy attributable to flexure of
the viscoelastic layer; the secon that attributable to streching; the
third that attributable to shear. Insofar 2s the overall losses in
the coating are caused by shear deforr.iation, the first two terms of
expression (3.36) may be disregarded (02uk2a202. Taking the aforesaid into account, formula (3.36) takes
:;n the form
~~Yg2 (3.38)..
1=- ~n2 -f- YB2 ~ ~ -F- 82 ~ t ~2~ ~ ~
where
1243I(43 , (3.39)
42'' a3P3+ 12a21aA
Formula (3.36) corresponds to the more precise expression [99J,
derived hy the comp]ex flexural rigidity nethod, where a303K 1 and
ac21a2P2wm, then this equation where n0=0 takes the
farm C"-k2~=0. Its solution ~(x)=~Oexp(-kx) has an attenuating
, ctiaracter and attests to the absence of energy transfer along the
rod.
Figure 36 shows the dependences of values of the imaginary part
oE the load on the rod jImzp and intertial resistance of the rod
- jwm, taken of the opposite sign, on uf-f/fp. It can be seen from
tne figure that at a sufficientlq low value np the modulus of
eZastic resistance of the antivibrator Imzp exceeds the modulus
of inerrial resistance of the rod wm in some frequency band Af,
situated near the resonant fraquency of the antivibrator f0.
Beyend the boundaries of Af the oscillatory process in the rod has
the cnaracter of a traveling wave with attenuating amplitude along
_ coordinate x.
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l:quation (5.8) can be rewritten as
t" -I- ks (1-1T1) 0, (5.10)
~ . P
where k 2 k 2 1 + Imz P Rez
Tj
�
tum ~ cum -a- I m zP '
t~Im
_ k� p' is the wave number of longitudinal oscillations
of the rod at zp=0.
From equation (5.10) it follows that attenuation of the amplitude
- of the traveling wave in the rod beyond the boundaries of ef will
be discribed the exponent esp(-kxn/2)� This attenuation will decrease
as it moves away from the boundary frequencies of Af. When Imzp1'2). (5.27)
ml 3(Omaa + Ys + ,3max }
maa Yo omax 72 ~
[dhere wmax=Yz both expression for rlmax agree. Figure 41 shows the
dependence of nmax on the ratio y2/wmax. With increase in yZ/wmax
values of rimax rise. It is not advantageous to design a structure
with y2/wnax>3, since Chis does not give a substantial gain in the
quantity nmax.
qma: m+/cemj ~
1'UI
D,1 F-
7 fU
Yz/wmax
Fig. 41. Dependence of maximwn loss factor of a atructure with
liquid intermediate .layer on the ratio Yz/wmax. 
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Let us analyze the dependence of the loss factor in a structure r1
on the ratio of thicknesses of the plates which compose it a3l=h3/hl.
From forriulas (5.21) and (5.22) it follows that rhis dependence
occurs at frequencies near wmax and at lower frequencies. At high
frequencies r1 does not depend on a31.
::ae indicated dependence at low frequencies, as follows from formula
(5.21), is described by parameter a, equal to (031�Y31�1)
a=ao=aa1 (1-a31~2. (5.28)
a�
e,t
0,5 aJ/ f8,
F'ig. 42. Dependence of ap on a31.
rigure ~+2 shows the c3ependence of au on a31. The greatest values
of ap, and consequently also the loss factor of the structure, will
be reached at a31max=0.45.
[JiCh increase in a31, simultaneous with rise at low frequencies of
the loss factor of the structure, wmax will decrease. This decrease
wi11 occur with an increase of a31 to value 0.45. At greater values
of a31, rise in wmax begins (Fig. 43,a).
Q) a)
~
65~
'fMQI
6) b )
tig~l < �2
z
~ J
~Mat ~
Q
f
Fig. 43. Frequencq ctiaracteristics of the loss factor of a structure
with liquid intermediate layer at various valueg of a31 (a)
and u2 (b).
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Losses of vibratory enerFy in the structure under examination dra
- attributable to movement of the viscous liquid in the apace between
plates anlong their plane. This movement occurs as a result of
changes in the spacing under flexural oscillations of the plates
_ (see Fig. 40). The amplitude of movement of the liquid, and con-
= sequently also losses of vibratory energy, will be greater as the
amolitudES of flexural oscillations of the basic plate 1 and the
- attached plate 3 differ more significantly. From this point af
= view the shape of the curve in Fig. 43,a can be explained. With
_ identical plates (a31=1) the wave mechanical resistance of the
attachpd plate z3=ja3-jbg is equal to zero (a3=b3), and flexural
oscillations are easily excited within it. The amplitudes of
- oscillations of both plates are in this case ideiitical and n$0.
With decrease in thickness of plate 3031b3). In this case the amplitude of
. oscillations of plate 3 decrease in comparison with the amplitude
~ of the oscillations of plate 1 and ri increases. [dith further
_ decrease of a31, the inertial part of wave resistance in plate 3
will begin to predominate a3=wm3(a3yb3). Therefore wave resistance
- drops with decrease of h3, and consequently also m3.
n
. q~
qof
- 1,0 2,0 4,0 qo
_ F, Krq
Fig. 44. Loss factor of a structur.e of steel plates 3 and 2 mm thick
_ with an intermediate layer of synthetic liquid rubber SKN-26,
= calculated by formula (5.20).
Accordingly the difference in amplitudes of oscil.lations of the
plates making up the structure decreases and, consequently, so does rl.
I'ron formulas (5.24) and (5.25) it can be seen that an increase in
the viscosity of the liquid y2 in the space between the plates raises
the frequency fmax, at which the loss factor of the srructure has
maximum value (Fig. 43,b). By selection of values of the viscosity of the
liquid u2 the required value fmax can be achieved. Liquids with a
dynamic viscosity of 102-104 DTN�c/m2 are suitable for use in the
sub3ect structure.
Their basic characteristics are shown in Table 6.
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Table 6
Physio-rtechanical Characteristics of Liquid at Temperature About 20�C
~ Liquid p, g/cm3 u, DIN/cm2
I�7ater ' 1.0 10-2
_ (:lycerine 1.26 8.5
Castor oil 0.96 10.3
Silicone oil 1.0 80.0
~ Synthetic rubber SKN-26 1.6 5.3�103
It should be kept in mind that the results obtained are suitable only
f.or moderate values of viscosity, at which shear forces are not
transmitted from one plate to the other. At great?r values of viscosity
calculations of the loss factor of a structure with a liquid inter-
= mediate layer by the formulas given in this paragraph will give values
that are too low. From Fig. 4~: it ia evident that ef.f.ective damping
can be achieved by the use of liqL;d intermediate layers.
_48A
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Chapter 6. DAMPING OF VIBRATIONS OF rLEMF:NTS OF SHIP MACHINERY
AND HULL-FRAN'IE STRUCTURF.S
$ 20. Uptimum Length of Vibroabsorptive Coatings
Under actual ship conditions, where one of the basic problems is
the limitation on displacement, it is important ro find the optimt:m
variant for placement of vibroabsorptive coatings, which will yield
maximum acoustical effect for a given weight. Such an optimization
is possible because as the length of plates in hull-frame structures,
that are to be faced with coatings, varies so does the thickness of
coatings and, consequently, the loss factors.
We sha12 determir.e the optimum length for various typas of vibro-
absorptive coatings on characteristic ship hull-frame structures,
- namely, on a uniform plate and a plate reinforced with periodic
rigidity ribs. I�'e shall assume that either unidimensional (flat)
f lexural waves or bidimensional (cylindrical) waves propagate in
the plates. As this takes place there is a sector of the plate
with length LBn, faced with a vibroabeorptive coating, which lies
in the path of propagation of the flat wave. And around the source,
which excites the cyli.idrical wave, there is also a coating applied
to a sector of plate with radius I.$n. The first case corresponds
_ to application of a coating a distance away from the source of
= vibration (machinery); the second to application in direCt proximity
- to the source. Patterns for applying coatings, correspondinR to
- the indicated variants, are shown in Fig. 45.
W It is shown in $ 28 that attenuation of amplitude of a flexural wave
, in a plate does not depend on its spatial characteristics and is
determined by the length of the coating. According to formula (7.5),
the effectiveness of a vibroabsorptive coating on a uniform plate 9,
in db is
3 2,15 kNnn'qLA no
(6.1)
where n is the loss factor of a plate faced with the coating. For
a ribbed plate the analogous expression has the form
~ z 4,3 ~~~rn nnTl Lem (6�2)
aoIp
- where lp is the distance
ap is the coefficient of
through the rigidity with
between rigidity rihs reinforcing the plate;
transmission of energy of flexural waves
diffuse incidence.
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a)
4 9 N Gen J 1 d 4 Gan
~ - 4
= 'C 4 2 Len d 1 ,i 4 Len
la
Fig. 45. Patterns for application of a vibroabsorptive coating on ~
a hull-frame structure: a. �or flat wave; b. for cylindrical =
wave.
Key: 1. uniform plate; 2. ribbed plate; 3. coating; 4. source of
vibration.
It can be seen from r-xpressiens (6.1) and (6.2) that maximum ef.fective-
ness will correspond to the grea[est value af parameters r1LB and
Y nLBr. Insofar as rl and I,Bn at a given mass of the coatingNHn are
functions of the thickness if the coating h, our problem amounts to
findinfi the solution to equations:
for a uniform plate
8 [tj (Jt) Lnn (h)1 = 0; (6.3)
ah
for a ribbed plate
a Ij~~l ~h) Lsn (h)] 0, (6.4) -
ah
_ in which psrameters nl.Bn and V__n_LBn have maximum values.
j,*e shall begin examination of the problem with a rigid vibroabsorptive
_ coar_ing, for which ratios between coating length LBn and coating thick-
ness h2 are as follows: _
--for a flat wave
Mnn = Lnnff nnP21l2; (6.5)
- for a cylindrical wave
m e� = nLo Pzhz, (6.6)
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_ where P2 is density of the coating material; HBn is the width af
the faced plate. The loss factor of the coating is determined by
formula (3.31). By substituting formulas (3.31), (6.1), (6.2),
(6.5) aud (6.6) into expressions (6.3) and (6e4) it is not difficult
to derive an equation for determining a2MT, at which the effective-
ness of the coating reaches maximum:
where
ValuQS of coefficient n are as follows:
In a uniform pla te for waves:
flat .................3
cylindrical.......... 6
_ In a ribbed plate for waves:
flat .................3/2
- cylindrical.......... 3
Dependences of 02 an a2 at the indicated values of n are ahown in
Fig. 46. FiRure 47 show~s dependences of effectiveness 3 on a2.
Analysis of Figures 46 and 47 show the foll.owing.
In the first case (n=3) at 82>2/3 the dependence of 9 on a2 (h2)
has no maximums, asymptotically approaching final and zero value
when a2-0 and a2--. At the R2?/d
t
I ~
~
0 C9zonr at 0 :Y?ent 012
3q6 3 3,461 4
~ 3~~ ~z Z~~
I c
i~
0 di a2ont maz az 0 a2o,,r a2
azonr min
Fig. 47. Dependences of the effectiveness of vibroabsorF+rive coatinge
on az=h2/hl.
I:ey: 1. Flat wave in a uniform plate; 2. cylindrical wave ir a uniform
plate; 3. f lat wave in a ribbed plate; 4. cylindrical wave in
a ribbed plate.
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When 02E2S2.
Damping of longitudinal oacillations of a tube by a atiffened coating
can be determined by the wave mechanical resistance method (aee $ 9)
in the following form: (6.50)
~ 0 +as) 0 +ai +vaJ'
where _ S,E9 ~ E,h, ,
Y S,EI Elhl . g's = G'S' G' ;
4E3S3k~h2 ~ 4E3k2
nh2h3
kH is the wave number of longitudinal oscillations of the tube.
1
In this case, when g2=g2nrrr=(l4'y)- '2 the loss factor has maximum
equal to nmSX=T12Y(2'F-y-F'2g-12pTIT) -l.
The loss factor of a longitudinally-oscillating tube with a pliable
coating is calculated by formula (3.44) by inserting ml=rii(ndl)-1
and k2=icc2. And as in the preceding case, the greatest value of
loss factor npl here will be at the frequency of first thickness
resonance of the coating, determined by formula (3.48) where
k2-kC2. Factor npl is calcualted by formula. (3.49) when m1-M1(ndi)-1
is inserted.
tdith torsional oscillations of a tube with a rigid coating its loss
factor ia determined by the wave mechanical resistance method
~t, *~G,lp' _ n2G2d2h2 (d2+hs) +taGshs. (6.51)
~ + G, lvi ~ Glivi Gldih, (di h21 1 Gl,`t
1
G21ns
The loss factor of a tube with a stiffened coating which undergoes
torsional oscillations will be calculated by the deformation energq =
method (see $ 9). .
According to formula (3.5) the loss factor can be written as
. . VIsW nor s (6.52)
~ WnoT 1+ Wnots + Wnor 31
where n2 is the loss factor of the viscoel.astic material; TiTnOT 1(1s1,2,3)
is the potential energy built up in the tube, the viscoelastic layer
- and the stiffening layer respectively. It is obvious in this case
that damping is attributable to shear deformation of the viscoelastic
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layer. The values of the potential energy included in formula (6.52) _
should be calculated with allowances made for the deflection angle
of sections of the tube and the outer layers, which takes ulace
under torsional oscillation of the tube (Fig. 57). For the angle
of deflecCion of sections of the tube one can write
91(x) = 81 sin kKx, (6.53)
where kK is the torsional wave number in the tube with vibroabsorptive
material applied; Ot is amplitude of the deflection angle.
I'or the angle of deflection of sections of a stiffened layer, conse-
quently, we have
ea (x) = 0a sin kKx. (6.54)
Displacement of the outer surface of the viscoelastic layer relative
to its inner surface, taking (6.53) and (6.54) into account, is ;
R� Oa (x) = R(ei-ee) sin kKx, (6.55)
ivhere P. is the outer radius of the section through the tube. -
e
Fip. 57. Pattern of deforr.iations of a stiffened coating on a tube subjected to torsional oscillations.
Let us calculate the potential energy WnoT i(i=1,2,3) for a piece
of tuhe with lengtn 1aaK/4-n/2k (aK is length of the torsional iqave in the tube). We shall bring the left edge of this piece into _
coincidence with the beginning of coordinate x=0. The potential
energy of an element of the tube with length dx, placed at a
disr,ance x from the beginning of the coordinates, is equal to
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= dWnoT a = O1l ~2az,' ' (6-56)
where Ipl is the po.lar moment of inertia of a section through the
tube; G1 is the modulus of shear of the tube material; d0 is the
angle of torsion of this element, equal to
as (dz)~
d0 = 01 (z -F dx) - O! (z) = 91 412 � (6.57)
By inserting (6.57) into (6.56) we find that
- Gl,pln
d~nor 1- 8~y � (6.58)
From (6.58) it is not difficult to derive value
~
~
W nor t= [IW noT I= G,Ip102jnkK
4 � (Fi.J9~
0
To determine T+TnoT2 we shall isolate from the viscoelastic layer
an element with volume
dV = dxhsRdO ' (6.60)
and detetmine the potential energy dWMr2 within it.
: dV
dWiloT 2= 2 G$e'dV = G: (R~:) 9 (6.61)
2h2
where G2 and h2 are modulus of sher and thickness of the viscoelastic
layer; e is shear deformation in the isolated element when the tube
is deflected at angle 0. Inserting (6.60) into (6.61) and integrating
" the result by 6 and x, we find
t sn
, . n2G101Rs
Wnorz=~ ~ '~Wnors= � (6.62)
9kKh!
ou
_ Energy WTVr3 can be found much the same as WnoTl
- W _ "G3ip3e3kK (6.63)
nor9- 4 ~ .
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where c:3 is modulus of sheur of the material in the st.tffening layer;
Ip3 is the polar moment of inertia of a section through the stiffen-
ing layer.
It will be noted that the equivalent rigidity of the stiffening
layer rElative to the tangential forces is equal to
ia(R' . (6.64).
- C' nGslp3kpg
= The analogous value for the viscoelastic layer is
14 G)I{9kK (6.65)
CZ = n3GaR �
Using (6.64) and (6.65) through 61, one can express values 82 and
03 in the following form, assuming that the rigidities C2 and C3
act in parallel
ol .
Ua - 1-+- go' ,
09 _ e1ga , (6.66)
i -I- g: ~
where C3 S,G, Ga
ga Cs SaG3kKh2 GakKh2ha
S2 and S3 are the areas of sections through the viscoelastic and
stiff.ening layers respectively.
Inserting (6.59), (6.62) and (6.63) and iising (6.66) we find from
(6.52) the sought for val.ue r,:
lv (g,g7)
~ 0 -f- b'x) (1 +gs -f- g: Y) ~ where G! G d h (d2 li2G li
,
3p3 _ 3 3 3` 3+ 3~ ~ 316
Y~
Giini Gidtii, (di+~'i) Gtf'l
Analysis of formula (6.67) shows that at loiq frequencies the factor
n`I'ncreases in proportion to w2, and then having reached maximum rlmax
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at 92an.i., it begins to decrease in inverse proportian to w2.
The value Timax can be found by differentiating (6.67) by $2
and equating the result to zero
l
glonr - +Y � (6�6$)
1
Wherein
tlmaa = 7hY ~ (6.69)
2 + Y -f; 282onr
The loss factor under torsional oscillations of a tube, faced on
its outer surface with a pliable coating, is determined in work [30]
_ 2 sli rlv - ill si n 2v
' G.7O)
Ti - 112 2 sh �;_v il_ sin 2v ly2v%=(cos2 v-}- ch ~hv) (
wheie 1112=2pI111IInPAR3, v=kC2h2.
Analysis of this foneula shows that at low frequencies (kc2h2