NASA CR-II2I40 VIBRATION AND DAMPING OF LAMINATED, COMPOSITE-MATERIAL PLATES INCLUDING THICKNESS-SHEAR EFFECTS Final Report (Part II) NASA Research Grant NGR-37-003-055 by C.W. Bert & C.C. Siu School of Aerospace, Mechanical and Nuclear Engineering The University of Oklahoma Norman, Oklahoma 73069 Prepared for National Aeronautics -.and Space Administration Washington, D.C. \ March 1972 https://ntrs.nasa.gov/search.jsp?R=19720022240 2018-02-12T19:37:57+00:00Z
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NASA CR-II2I40
VIBRATION AND DAMPING OF
LAMINATED, COMPOSITE-MATERIAL PLATES
INCLUDING THICKNESS-SHEAR EFFECTS
Final Report (Part II)NASA Research Grant NGR-37-003-055
by
C.W. Bert & C.C. SiuSchool of Aerospace, Mechanical and Nuclear Engineering
The University of OklahomaNorman, Oklahoma 73069
Prepared for
National Aeronautics -.and Space AdministrationWashington, D.C.
9. Performing Organization Name and AddressSchool of Aerospace, Mechanical, and Nuclear EngineeringThe University of OklahomaNorman, Oklahoma 73069
12. Sponsoring Agency Name and Address
National Aeronautics and Space AdministrationWashington, D.C. 20546
3. Recipient's Catalog No.
5. Report Date
6. Performing Organization Code
8. Performing Organization Report No.
10. Work Unit No.
501-22-03-0311. Contract or Grant No.
NGR-37-003-05513. Type of Report and Period Covered
Contractor Report14. Sponsoring Agency Code
15. Supplementary Notes
16. Abstract
An analytical investigation of sinusoidally forced vibration of laminated, anisotropicplates including bending-stretching coupling, thickness-shear flexibility, all three typesof inertia effects, and material damping was conducted. The analysis begins with theanisotropic stiffness and damping constitutive relations for a single layer and proceedsthrough the analogous relations for a laminate. Then the various types of energyand work terms are derived and the problem is formulated as an eigenvalue problemby application of the extended Rayleigh-Ritz Method. The effects of thickness-sheardeformation are considered by the use of a shear correction factor analogous to thatused by Mindlin for homogeneous plates. The general analysis is applicable to plateswith any combination of natural boundary conditions at their edges.
Natural frequencies of boron/ epoxy plates calculated on the basis of two differentresonance criteria (peak-amplitude and modified Kennedy-Pancu techniques) and theassociated nodal patterns are in good agreement with previously published experimentaldata.
For sale by the National Technical Information Service, Springfield, Virginia 22151
VIBRATION AND DAMPING OF
LAMINATED, COMPOSITE-MATERIAL PLATES INCLUDING THICKNESS-
SHEAR EFFECTS
CONTENTS
Page
ABSTRACT lv
SYMBOLS vi
I. INTRODUCTION 1
1.1 Introductory Remarks 1
1.2 A Brief Survey of Selected Vibratlonal Analyses ofLaminated Plates 4
II. FORMULATION OF THE THEORY 9
2.1 Hypotheses 9
2.2 Kinematics 10
2.3 Stiffness Constitutive Relations 11
2.4 Strain Energy 15
2.5 Damping Coefficients and Dissipative Energy 17
2.6 Kinetic Energy 18
2.7 Work Done by External Forces 19
2.8 Application of the Extended Rayleigh-Ritz Method 20
2.9 Reduction to Matrix Form 23
III. DERIVATION OF THICKNESS-SHEAR FACTORS FOR LAMINATES 25
3.1 Static Approach 26
3.2 Dynamic Approach 30
IV. MODAL FUNCTIONS USED FOR VARIOUS PLATE BOUNDARY CONDITIONS 32
4.1 Simply Supported on All Edges 32
4.2 Fully Clamped on All Edges 33
4.3 Free on All Edges 34
V. NUMERICAL RESULTS AND COMPARISON WITH RESULTS OF OTHERINVESTIGATORS 35
5.1 Thickness-Shear Factors for Laminates 35
5.2 Plate Simply Supported on All Edges 36
5.3 Plate Free on All Edges 38
VI. CONCLUSIONS 42
APPENDIXES
A. NOTATION AND TRANSFORMATION FOR ELASTIC COEFFICIENTS 44
B. MODELS AND MEASURES OF MATERIAL DAMPING 50
Bl. Mathematical Models for Material Damping 50
B2. Measures of Material Damping 62
B3. Inter-Relationships Among Various Measures of Dampingfor Homogeneous Materials 67
C. DERIVATION OF ENERGY DIFFERENCE 76
D. COMPLETE ENERGY EXPRESSIONS 79
Dl. The Energy Difference 79
D2. Equations for Minimizing the Energy Difference 80
E. DERIVATION OF THICKNESS-SHEAR FACTOR FOR THE THREE-LAYER,SYMMETRICALLY LAMINATED CASE USING THE DYNAMIC APPROACH 85
El. Dynamic Elasticity Analysis of an Individual LayerUndergoing Pure Thickness-Shear Motion 85
ii
E2. Dynamic Elasticity Analysis for Three-Layer,Symmetrically Laminated Case gg
E3. Dynamic Analysis of a Symmetrically LaminatedTimoshenko Beam Undergoing Pure Thickness-Shear Motion 88
E4. Determinations of the Thickness-Shear Factor 91
F. COMPLEX STIFFNESS COEFFICIENTS FOR A LAMINATE HAVING ALTERNATINGPLIES OF TWO DIFFERENT COMPOSITE MATERIALS 92
Fl. Introduction 92
F2. Analysis 93
G. IDENTIFICATION OF INTEGRAL FORMS 100
Gl. Trigonometric Integrals 100
G2. Combination Trigonometric-Beam Type Integrals 107
H. EVALUATION OF EXPERIMENTAL METHODS USED TO DETERMINE MATERIALDAMPING IN COMPOSITE MATERIALS 109
I. DISCUSSION ON SYMMETRY OF THE ARRAY OF CONSTITUTIVE COEFFICIENTS 123
J. COMPUTER PROGRAM DOCUMENTATION 126
REFERENCES , 128
TABLES 147
FIGURES 157
COMPUTER PROGRAM LISTING 180
iii
ABSTRACT
This report is concerned with analytical investigation of si-
nusoidally forced vibration of laminated, anisotropic plates including
bending-stretching coupling, thickness-shear flexibility, all three
types of inertia effects, and material damping.
In the analysis the effects of thickness-shear deformation are
considered by the use of a shear correction factor K, analogous to that
used by Mindlin for homogeneous plates. Two entirely different approaches
for calculating the thickness-shear factor for a laminate are presented.
Numerical examples indicate that the value of K depends on the layer
properties and the stacking sequence of the laminate.
The general analysis is applicable to plates with any combination
of natural boundary conditions at their edges. The analysis begins with
the anisotropic stiffness and damping constitutive relations for a single
layer and proceeds through the analogous relations for a laminate. Then
the various types of energy and work terms are derived and the problem is
formulated as an eigenvalue problem by application of the extended
Rayleigh-Ritz method.
The first five resonant frequencies of boron/epoxy plates with all
edges free are calculated on the basis of two different resonance criteria:
the peak-amplitude and modified Kennedy-Pancu techniques. The results show
that the resonant frequencies obtained by the two techniques differ by only
a very small amount, and are in good agreement with the results obtained
iv
both experimentally and analytically by Clary. Furthermore, the nodal
patterns obtained agree satisfactorily with those of Clary. Finally,
the damping values in this investigation are in good agreement with the
experimental ones obtained by Clary.
SYMBOLS
A cross-sectional area
AQ area involved in shear-stress integration (fig. 8)
A.,B constants depending upon the boundary conditions (Appendix E)
Aj hereditary constants, eq. (B-21)
A.. stretching stiffness of the plate
a length of plate
a(x.) ,a(x2) wave amplitudes at positions x. and x«
a.,a _,a free-vibration amplitudes corresponding to the i-th,1 (i+l)th, and (i+n)th cycles
a initial amplitude
B. . bend ing-stretching coupling stiffness of the plate
B. . complex stiffness coefficient
b width of plate
b material damping coefficient defined in eq. (B-10)
b critical material damping coefficient
C, damping coefficient defined in eq. (B-22)
C' damping coefficient defined in eq. (B-8)
C. constants depending upon the initial conditions
C. . Cauchy elastic coefficient
C.. complex version of C..
C ' effective stiffness coefficients defined followingJ ' eq. (F-7)
C ,C characteristic parameters tabulated in ref. 44m n
c viscous damping coefficient
VI
c Kelvin-Voigt complex damping coefficient
c critical value of c
c shear wave-propagation velocitys
D total dissipative energy
D energy dissipated per unit of plate area
D.. bending and twisting stiffnesses of the plate
D.. complex version of D..
E Young's modulus
E.. . ,E_2 Young's moduli in the x,y directions
Ei(u) the exponential integral defined in eq. (B-18)
e base of the natural logarithms, e ~ 2.7183
F, damping force
F spring forceS
F exciting force amplitude
AF horizontal shear force per unit width
F . thickness-shear stiffnesses of the plate
F.. complex stiffness coefficients associated with F
|fI generalized-force column matrix
g loss tangent
g1 parameter of the Biot model
g, dimensionless geometric parameter defined in ref. 35
g. ,etc. subscripted loss tangents where the subscripts (i.e.refer to the associated stiffness; for example, g^iisignifies the loss tangent associated with thelongitudinal stretching stiffness
H factor - (h/m)
HPMF half-power magnification factor
H factor = H/u,n
vii
H ,H thickness fractions of layers a and b, respectively
h total thickness of the plate; parameter in eq. (H-2)
I integral form defined in Appendix G
i unit imaginary number =• /-I
K thickness shear factor
K factor defined in eq. (H-10)
K complex-flexibility factor defined in eq. (F-15)
K. . composite shear coefficient
k spring rate
k complex spring constant
k' stiffness coefficient associated with the total in-plane force
k11 C1 e m (C1 and m are constants)
ic Kelvin-Voigt complex stiffness defined in eq. (B-3)
L effective length of spring
L amplitude of Lagrangian energy difference
L complex-stiffness factor defined in eq. (F-16)c
-f.n natural logarithm
log logarithm with base 10
mass matrix
M stress couples, moment per unit width
MF,MF' magnification factors defined in eqs. (B-38) and (B-16)
m damping exponent defined in eq . (B-22)
m mass
tn,n cos 9, sin 0
m ,m. ,nu mass per unit of plate area, first moment of mass perunit area, second moment of mass per unit area
viii
N.. membrane stress resultants, force per unit width
n degree of heredity
Q quality factor ? reciprocal of the dimensionless bandwidth
Q. thickness-shear stress resultant
Q.. reduced stiffness coefficients transformed to platecoordinates (x,y); see eq. (A-16)
*Q. . reduced (plane-stress) stiffness coefficients with re-
spect to material symmetry axes (X,Y); defined in eq. (A-6)
q normal pressure acting on plate
R density ratio (;= p^/p(2))
RMF resonant magnification factor
§ complex flexibility defined in eq. (F-22)
fS] complex stiffness matrix
T kinetic energy; period of damped oscillation (Appendix Bonly)
T. kinetic energy per unit of plate area
[T ] transformation matrix defined in eq. (A-12)
t time
t|,t~ two different specific values of time
U strain energy
U. strain energy per unit of plate areaA
U, energy dissipated per cycle
U, damping energy per cycle and per unit of volumedv
U undetermined longitudinal-displacement parametersrun
U shear strain energys
U1 shear strain energy associated with equivalent uniform,longitudinal thickness-shear strain
U strain energy per unit of plate volume
ix
u,v,w displacements in the x,y,z directions
u ,v ,w displacements of middle surface in x,y,z directionso o o
u displacement amplitude of vibration
u1 trasient solution in eq. (B-70)
u static displacement; see Appendix 8o t
V volume
V undetermined transverse-displacement parameter
v spatial attenuation constant defined in eq. (B-33)8
v temporal decay constant defined in eq . (B-30)
W total work done
W undetermined normal-displacement parametermn
X,Y,Z coordinates of the material-symmetry axes
x,y,z rectangular coordinates in the longitudinal, transverse,and thickness directions
x. ,x? two different specific values of position
Z ,Z characteristic parameters tabulated in ref. 44m' n
z,,z thickness-direction coordinates of outer and inner faces*~l of the k-th layer
•j > column matrix
[ j square matrix
normalized arguments in the x,y directions
heredity constants in eq. (B-21)
C^/C^ in Appendix E only
loss angle defined in eq . (B-27)
spatial attenuation rate defined in eq. (B-34)s
v decay rate defined in eq. (B-31)
A comples factor defined in eq. (F-22)
o logarithmic decrement defined in eq. (B-28)
6 logarithmic attenuation defined in eq. (B-32)s
e Biot parameter in eq. (B-17)
e . . strain components with respect to x,y axes
eTT strain components with respect to X,Y axesL J
e strain corresponding to longitudinal thickness-shearXZ action (Section III and Appendix E)
where c Is the shear wave-propagation velocity defined by:s
=C55/p (E-4)
However, for pure longitudinal thickness-shear motion, the dis-
placements are independent of axial position x. Thus, all derivatives
of displacements with respect to x vanish and equations (E-3) reduce
to the following single expression:
cs u'zz = u>tt
Equation (E-5) is the familiar, one-dimensional wave equation,
which is solved easily by the separation-of-variables method, with the
following solution for simple harmonic motion:
u(z,t) = (A cos Q z+B sin 0 z)(C cos At + C« sin u>t) (E-6)S 8 L £
where Q s uu/c , A and B are constants depending upon the boundaryS S
conditions, and C. and C2 are constants which depend upon the initial
conditions.
E2. Dynamic Elasticity Analysis for Three-Layer,
Symmetrically Laminated Case
Here we consider the special case of a three-layer, symmetrically
laminated member, in which the two identical outer layers are designated
by superscript 1 and the middle layer is denoted by superscript 2, as
86
shown in figure E-l. The proper boundary conditions for the anti-
symmetric modes in pure thickness-shear motion are
(z2,t) = u(2) (z2,t) ; u. 0 ;
u,z(2)(z2,t) ; u
(2) (0,t) =0
(E-7)
where z1 and z_ are dimensions shown in figure E-l.
Substituting equation (E-6) into equations (E-7) yields the follow-
ing set of expressions:
cos + B sin cos
+ B0 sin fl z. ; - 0 A, sin 0 z_ 4- 0 B, cos2 s 2 s 1 s 2 s 1
V ' A «:-in (V~' •* 4- O v~' R ro«i O x ~ ' z • -O V * 'A < ? i n »-)x"' ?4 o ^ a l l l H Z~ * \ t O ^ C U o ^ i ^— > — \ l n o i n i cs 2 s 2 s 2 s 2 s 1 s 1
B. cos n z. = 0 ; A_ = 0S 1 S 1 i
(E-8)
or
cos s
sin
sin 0(1)z,S fc
.s i
8 lna< 2>, 2 -
)s(2)cos n^2 )z2
0 _
/ \Al• B l
J
» = 1
0
0
0V /
(E-9)
The homogeneous system of linear algebraic equations (E-9) has a
nontrivial solution if, and only if, the determinant of its coefficient
87
matrix is equal to zero. The resulting determinants1 equation has as
its solutions the roots of the following transcendental equation:
0(1) tan nf z, - - n<2> cot CO z, - n «.) (E-10)S S £ S S £ S L
(k) (k)From the definitions of Q and c , we have
S
where
P — P^ ' /P^ ' . P = rA ' /r,' ' ft} 1O\- ^55 '^55 > K P 'P V.i-l^;
Then equation (E-10) can be expressed as follows:
tan ( C n z O / R ) = cot
where
E3. Dynamic Analysis of a Symmetrically Laminated Timoshenko
Beam Undergoing Pure Thickness-Shear Motion
*Here we consider a symmetrically laminated Timoshenko beam . The
*A beam exhibiting both thickness-shear flexibility and rotatory
inertia is generally referred to as a Timoshenko beam (refs. 41,42).
88
axial and thickness (or depth) directions are designated as the x and z
axes, respectively. Such a beam could be analyzed as a special case of
the laminated plate theory presented in Section II by merely deleting
all derivatives with respect to y. However, the beam case is so much
simpler and pure thickness-shear motion is such a simple type of motion;
therefore, it was decided to make an exact analysis for the present case.
The following kinematic relations hold throughout the entire
thickness of the laminate:
K =ilf 5 e =w +\li (E-15)XX *X,X XZ 0,X *X
The following stress-strain relations are applicable to a typical
layer "k":
l H z ; a =xx 11 xx 11 xx xz 55 xz
The bending moment and shear force, expressed on the basis of a unit
width as in plate theory (Section 2.3) are:
n
oz dz ; QJ xx
n kM = o(k)z dz ; Q =Y | ok) dzL. Jxx
k=l Vl k=l Zk-l
Substituting equations (E-15) and (E-16) into equation (E-17) and
introducing the shear factor K,c as a correction factor to be determined
later, one obtains:
Mxx = Dll*x,x ;
39
where
zkk=i Vi
The equations of motion for a symmetrically laminated Timoshenko
beam are identical in form to those governing a homogeneous Timoshenko
beam (refs. 41,42), namely:
Q = m w ;M -Q= m.* (E-20)x,x o o,tt xx, x x 2Yx,tt
where m and ou are defined in equations (28) .
Inserting equations (E-18) into equation (E-20) , one obtains the
following set of two coupled equations of motion in terms of the generalized
displacements w and \li :
K55A55 (w0,xx+*x,x> =moW
0,tt
*x.tt
For pure thickness-shear motion, w and \|i are independent of axialo x
position x, so that equations (E-21) and (E-22) uncouple and become:
Wo,tt =°
m 2 * x , t t + K 5 5 A 5 5 * x
Since equation (E-23) does not contain K,.,. and since w is not
90
in equation (E-24), we have no further need for equation (E-23).
For steady-state harmonic motion, the solution of equation (E-24)
can be expressed as follows:
s1"* (E-25)
where ^is a constant.
Substituting equation (E-25) into equation (E-24), we are led to
the following relationship:
a.2 = K A/m (E-26)
This equation is applicable to any symmetrical laminate. For the special
case of a three-layer one, using the notation depicted in figure 10 and
the definitions of A,., and m- from equations (E-19) and (28), one obtains:
u>2 = 3K55 z~2 (C )/p(1))[(C2/P) + 1-C2]
(E-27)
' [(G2/R) + 1 - Cj I"1
where g,R, and £2 are as defined previously.
E4. Determinations of the Thickness-Shear Factor
The longitudinal thickness-shear factor, K , is determined implicitly
2by equating u/ associated with the lowest non-trivial solution of equation
(E-13) to that given by equation (E-27).
91
APPENDIX F
COMPLEX STIFFNESS COEFFICIENTS FOR A LAMINATE HAVING
ALTERNATING PLIES OF TWO DIFFERENT COMPOSITE MATERIALS*
Fl. Introduction
The theory presented in Section II can be used to calculate the
stiffness and damping coefficients for an arbitrary laminate, i.e. one
consisting of any number of plies of any thickness, material, and orienta-
tion. However, in many cases of practical importance, laminates are de-
signed to have many plies of two alternating materials. In such cases,
the approximate approach used in refs. 29-31 is sufficiently accurate.
Although this approach was originated for application to wave propagation in
an infinite medium, it is applicable also to plates. In the latter con-
figuration, it has been verified experimentally in several instances. By
comparison with static experimental results, Rose and Tshirschnitz (refer-
ence 92 ) found that it gave good predictions of the in-plane elastic
modulus and in-plane shear modulus. Also Achenbach and Zerbe (ref. 32)
found that it gave a frequency versus wavelength relationship for longi-
tudinal vibration of a laminated beam which was in excellent agreement
with experimental results.
In all of the work mentioned above, all of the individual layers were
isotropic. However, in many cases of increasing importance, at least one
of the sets of layers may be made of composite materials. Some examples
*After completion of this derivation, the work of Chou et al.. (Referencecame to the attention of the authors. In their work, Chou et al. derived.purely elastic equations which are analogous to the complex ones derived here,
92
are as follows:
1. Alternating layers of an isotropic material and an orthotropic
material. Examples: armor plate consisting of alternating layers of a
hard ceramic material (isotropic) and a lossy glass fiber-epoxy matrix
composite (orthotropic); a laminate consisting of alternate layers of high-
modulus orthotropic composite material (such as boron-epoxy or graphite-
epoxy) and low-modulus, high-damping polymer (to increase the damping cap-
acity of the laminate).
2. Alternating layers of two different orthotropic composite materials,
Example: Boron-epoxy (for high stiffness) and glass-epoxy (for low cost).
The lamination scheme may be either parallel ply (unidirectional) or cross
ply.
3. Alternating layers of the same composite material and thickness,
but oriented alternately at 4« and -9, where 0 < 9 < 90°. This is the so-
called angle-ply lamination arrangement, which is very popular in a variety
of aerospace structures.
An original derivation is presented here which is applicable to
determination of both the stiffness and damping of the above three classes
of laminates. It may be considered to be a generalization of the work of
refs. 29-31; necessarily the resulting equations reduce to theirs in the
case when both materials are isotropic.
F2. Analysis
The bases for the present analysis, as well as that of refs. 29-
31, are the following hypotheses:
1. Strains in the plane of the laminations are equal.
93
2. Stresses in the direction normal to lamination planes are equal.
Hypothesis 1 leads to the Voigt upper-bound estimate (reference 93)
of the equivalent macroscopic properties of a two-phase material having
arbitrary geometrical configuration of individual constituents. Except
for the presence of an additional Poisson's ratio effects term, this is
known also as the "rule of mixtures". Hypothesis 2 leads to the correspond-
Reuss lower-bound estimate (reference 94), which is almost identical with
the so-called "inverse rule of mixtures".
Here we consider a medium consisting of repeating alternating
layers denoted by "a" and "b", as shown in figure F-l. The plane of
the laminations is designated as the xy plane and z is the normal to this
plane. Then the two hypotheses mentioned above can be stated mathematically
as follows in contracted notation:
e(3) = e(b> = 6 0=1,2,6) (F-l)
o.(a) = Oi(b) = a. U=3,4,5) (F-2)
where the e. are strain components, the a. are stress components, and
superscripts (a) and (b) refer to layers "a" and "b". Subscripts 1 and 2
refer to normal strain (or normal stress) in the plane of the layers, 3
refers to thickness-normal effects, 4 and 5 refer to thickness shear, and
6 refers to in-plane shear. This notation is consistent with that most
widely used in the field of composite-material mechanics (ref. 4) but
differs from the mixed notation used in the body of this report.
As a consequence of the complex, orthotropic version of Hooke's law
94
and static equilibrium, equations (F-l) and (F-2) imply the following re-
lations :
O. = H(a>o.(aWb)0.(b)J J J
1,2,6) (F-3)
(1-3.4.5) (F-4)
where H and H are the thickness fractions of layers "a" and "b",
respectively. Thus,
(F-5)
The complex, orthotropic version of Hooke's law, alluded to above,
holds for each type of layer as follows:
,<»"a"'
°3
"4
°C5
o\ X
>• =
p(k) =(k) s(k) n n = (k)~Cll °12 C13 ° ° C16
g(k) g(k) =(k) -(k)L12 22 °23 U U C26
-(k) =(k) =(k) n = (k)C13 L23 L33 ° ° C36
o o o c.(k) c.(k) o44 45
o o o cfk) c^k) o^»5 55
s(k) =(k) =(k) =(k)L16 26 ^36 °66
^
el
62
e3
e(k)64
(k)e c5
66
(k=a,b)
(F-6)
-(k)where the C.. are the complex stiffness coefficients.
95
f \al°2
°3
(T
3 .
=
\ >
c ' r ' r ' r ' ' r '11 °12 C13 C13 C16
C1 r ' r ' r ' ' r*°12 C22 °23 C23 C26
C(a) C(a) C(a) 0 r(a)°13 C23 C33 ° C36
c(b) c(b> o c(b) c(b)13 ^23 U L33 °36
•<
r \el
e2
(a) se3
(b)e3
, 66 ,
where
(a)
C!, = H(a) C'
; i,j = 11, 12, 16, 22, 26
• - H(b)r (b>-
j ~ " °ij ;
= 13, 23
13, 23
(F-7)
Combining equations (F-4) and the last two of equations (F-7), we
0 •z «t • m -• in «Z N O f*> + ^ wu n ~ <* O « o u O n* O < D -J O O
UJ3D D* Z< tart
< *••
M Z< O< U
<#Az o•*• 4O N "* UJ 1in o «• || M
0 N IIII UJ —u o «
N'JJO+mNMOJo+4-NCM
— UJ1 O* II* NII UJ
* O
n
*
(
*-».rt;
«*
UJ4««• Z
PO •X M• II
**II 0n coN- OUJ O
•rt
M
ro
N
*
X
*rt
l
l
N
^
N
*<M
**
N
*rt»1wUJ
*mo4>fON_
UJIItoN
UJ
UJDZ
*+, **
~* Hf^ Z# o* u
**^•rt««
N
*o•rt
««.
UJ4fin•oIIN(M
UJ
(VJ
**•rt
1
*
N
•UJ
*in•o^Z M
• rjfu NII --> UJ
II0 N0» t\)
NO -O UJ
<v
**•rt%^
N
*•rt%^
UJ
*in•iN(MN.«UJIINrjN
U
•
»4
S^
^
*N(VIIM
•rt
UJII"0N.4
UJ
(M
*#
1-}
M«
(VJ
**->^^N%i
***,->^^
UJ»m•
z oZ 1• N
(\l f\|II N~1 •*
UJ0 IIo rsj•rt (M
NO -
«*l»
«^1T
**N1^•>^
N^^
*NfWN•>UJ4>
")N•rt
UJ uJ< 3II Z"1 tartNl K-• ZUJ O
CM N* UJ# tf-, (flr*i *N M- 0UJ Z^ ID
+ inro ^N X-« nUJ Nw *rt
II UJ
rO </5N II— ^UJ U.(ft (n
^u.w*2«^O•rt
^4
•
»0*l*
UJ.tart
a
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EA
R
FA
CT
OR
K =
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7.4
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•Hin•C\itart•
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Zo
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a0 0u. o
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181
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