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LINEAR AND AOTOPARAMETRIC ^ ODAL ANALYSIS
OF AEROELASTIC STRUCTURAL SYSTEMS
by
TOMffY DALE VTOOOALL, B . S . i n M.E.
A THESIS
IN
MECHANICAL ^GINEERING
Submitted to the Graduate Faculty of Texas Tech University in
Partial Fulfillment of the Requirements for
the Degree of
MASTER OF SCIENCE
IN
MECHANICAL ^GINEERING
Approved
Accepted
May, 1984
he-"7— '
tJa.Jt^
ACKNOWLEDGMENTS
I would like to express my sincere appreciation to Dr.
R.A. Ibrahim for his patient guidance of this thesis and to
the other members of my committee. Dr. H.J. Carper and Dr.
J.W. Oler, for their helpful criticism. I am also grateful
for the assistantship provided by Dr. J.H. Lawrence through
the Department of Mechanical Engineering and for the support
provided by the ASME Auxilary. I would also like to thank
Mr. Hun Heo for his technical contributions to this thesis.
My thanks is also given to Mrs. Rebecca Yochcun for her help
in preparing this manuscript.
11
CONTENTS
ACKNOWLEDGMENTS ii
ABSTRACT iv
LIST OF FIGURES vi
I. BACKGROUND AND SURVEY OF RECENT LITERATURE 1
Parametric Vibration 1
Nonlinear Coupling (Autoparametric Resonance) 9
Scope of Present Research 19
II. FORMULATION OF THE PROBLEM 23
III. LINEAR MODAL ANALYSIS 29
Eigenvalues 29
Mode Shapes 36
IV. AUTOPARAMETRIC MODAL INTERACTION 41
Asymptotic Approximation Solution 41
Resonance Conditions 44
Analytical Solutions 46
Numerical Solutions 53
V. CONCLUSIONS AND RECOMMENDATIONS 63
LIST OF REFERENCES 65
APPENDICES 69
A. KINETIC ENERGY DERIVATION 69
B. COEFFICIENTS 7 2
C. CSMP PROGRAM LISTINGS 76
111
ABSTRACT
This investigation deals with the linear modal emalysis and auto-
parcunetric interaction of aeroelastic systems such as an airplane
fuselage emd wing with fuel storage. The mathematical modeling is
derived by applying Lagrange's equations taking into consideration the
Christoffel symbol of the first kind to account for the nonlinear
coupling of the system coordinates, velocities, euid accelerations.
The linear modal amalysis will be obtained by considering the
linear, conservative portion of the equations of motion. The normal
mode frequencies eind the associated mode shapes are obtained in terms
of the system parameters. The main objective of the linear analysis
is to explore the critical regions of autoparametric (or internal)
resonance conditions, 2ku)j| = 0 (where k^ are integers and o) are the
normal mode frequencies). The results show that for certain system
parameters the condition of internal resonance is satisfied.
The dynamic behavior of the structure in the neighborhood of
internal resonance conditions is obtained by considering the nonlinear
coupling of the normal modes. The asymptotic approximation technique
due to Struble is employed. Three groups of internal and normal reso
nance conditions are obtained from the secular terms of the first-
order perturbational equations.. The transient cuid steady-state
responses cure obtained numerically by using the IBM Continuous System
Modeling Program (CSMP) with double precision Milne integration. The
transient response shows a build up in the interacted modes to a level
IV
which exceeds the steady-state response. In addition, the excited
mode is suppressed by virtue of the nonlinear feedback of other modes.
Under certain conditions, the steady-state response is derived cuialy-
tically.
It is concluded that the nonlinear modal analysis reveals certain
types of response characteristics which cannot be interpreted within
the framework of the linear theory of small oscillations.
LIST OF FIGURES
Figure 1.1
Figure 1.2
Figure 1.3
Figure 1.4
Figure 1,5
Figure 1,6
Figure 1.7
Figure 1.8
Figure 1 .9
Figure 1.10
Figure 3.1
Figure 3.2
Figure 3.3
Figure 3.4
Figure 3.5
Figure 3.6
Simple pendulum subject to parametric excitation,
Stability diagram for a system with parametric excitation.
Mode shape of a cantilever under combination resonance S = o) + au.
Stability boundaries of a structure under combination resonance.
An elastic pendulum as an example of an autoparametric system.
Two degree-of-freedom autoparametric system.
Chaotic response of a system with periodic excitation.
Fluid filled structure on an elastic support.
The autoparametric vibration absorber.
Schematic diagram of aeroelastic system with its coordinates.
Dependence of ov, on ^']-\/'^2 ^ ° ^ various values of a)22/< 33-
Dependence of 0)3 on ^-^^/^^ ^ ° ^ various values
of W22/'**33-
Dependence of 0)3 on w^^/uij^ for various values
of t»)22/' 33-
Variation of natural frequency combinations with (0. ./o)-- for S2/'**33 = ^'^ ^^ compared to m^.
Variation of natural frequency combinations with
'**11/ 33 ^ ° ^ ' 22/'* 33 ">-O as compared to in^*
Linear mode shape for system excited at ^ = cu ; ^ = 2 5 , * = 5 , 0 = .402, a = .112.
2
4
11
13
15
18
20
21
31
32
33
34
35
38
VI
Figure 3.7
Figure 3.8
Figure 4.1
Figure 4.2
Figure 4.3
Figure 4.4
Figure 4.5
Figure 4.6
Figure 4.7
Figure 4.8
Figure 4.9
Linear mode shape for system excited at il = 0^2' 5 = 25, * = 5, e = .402, a = .112.
Linear mode shape for system excited at J = 0)3; C = 25, $ = 5, 3 = .402, a = .112.
Steady-state response for r 3 + r23 = r^^ = nv.
Steady-state response for r23 ~ ^13 ~ ^ 33 ~ ^^'
Steady-state response for 2r-|3 = r33 = nv.
CSMP time history for fi = r33 = Ji 3 + ^23 (ni/n2 * r^/r2).
CSMP simulation of quasi-steady-state response for r 3 + r23 = r33 = fl (n^/n2 + r^/r2).
CSMP time history for i2 = r33 = 123 " ^13 (m/n2 + r^3/r23).
CSMP simulation of two mode interaction, r33 = 2r^3 = nv.
Response of normal coordinate equations motion as determined by CSMP with ^ = r33 = Ji 3 + 23' q = 2 = •0'"' = 'O ' = •°'' •
Response of general coordinate equations motion as determined by CSMP with Q = r33 = r 3 + r23, 5 = i;2 = .01, C3 = .05, e = .01 .
39
40
49
52
54
55
57
58
59
61
62
VI1
CHAPTER I
BACKGROUND AND SURVEY OF RECENT LITERATURE
Parametric Vibration
In the field of structural dynamics, many problems exist in which
an oscillating system possesses time-varying parameters. A simple pen
dulum subject to vertical, harmonic motion of its support is an
example of such a system (see Figure 1.1). The governing equation of
motion for this system is (sin0 = 0):
0 + [0) 2 _ (yQj ,2/L) cosoJt] 0 = 0 (1.1)
where dots indicate differentiation with respect to time, L is the
length of the pendulum, 0 is the angular displacement of the pendulum
from the vertical, U) = (g/L)V2 is the natural frequency of the pen
dulum, Yn ^^ ^® amplitude of the support motion, and o) is the fre
quency at which the support oscillates.
For the system described by equation (1.1), the time-variation of
the stiffness term is an explicit function of time, and the system is
said to be subject to "parametric excitation" because of the explicit
time dependence of the system parameters. Unlike forced excitation,
where the excitation appears on its own on the right-hand side of the
system equation of motion, the excitation in system (1.1) appears as a
coefficient of the system coordinate in the equation of motion. It
should be noted that the explicit function of time need not be har
monic for parametric excitation to occur. The instabilities asso
ciated with the parametrically excited, simple pendulum and other
Figure l.i. simple pendulum subject to parametri c excitation.
dulum and other systems under parametric excitation are well docu
mented [1-5]. The main feature of the dynamic behavior of such sytems
is that their equilibrium position becomes unstable if the excitation
frequency, u), is twice the natural frequency of the system, oi . The
conditon O) = 200^ is referred to as the principal parametric resonance,
which is different from normal resonance, u) = ui^, in forced vibra
tions. For parameteric instabilities to occur, a small initial con
dition must be given to the system. Figure 1.2 shows the stability-
instability boundaries for harmonic, parametric resonances.
For multi degree-of-freedom systems, like the cantilever beam
shown in Figure 1.3, combination parametric resonance can occur for
the conditions o) = l | uj jf u). I (i + j), where ui^ and u). are the n
natural frequencies of the ith and jth modes, respectively, and n is
an integer. Figure 1.4 gives the stability boundaries of a structure
under combination parametric resonance.
A linear analysis defines the regions of instability at which the
amplitudes of parametric systems increase exponentially. Such an
analysis is not adequate in determining the steady-state amplitude-
frequency response (if it exists) for parametric systems. The
amplitude-frequency response can only be determined by including the
inherent nonlinearities of the system.
An extensive survey on parametric and autoparametric vibrations
is presented by Ibrahim and Barr [4,6] . This survey includes the
mechanics of linear problems [4] as well as nonlinear problems [6]. A
continuation of this work by Ibrahim [7,8] contains reviews of the
CN
Figure 1.3. Mode shape of a cantilever under combination resonance 2 « ut " ^ 2*
Unstable Reaions
rr / r
l \
1/3 1/2 •£L CJi + CJ2
Figure 1.4. St2ibility boxindaries of a structure under combination resonsmce.
current problems of free surface of liquids in closed containers,
rods, beams, pipes, plates, shells, pendulum systems, shafts, mecha
nisms, machine components, missiles, satellites, and hydroelastic and
aeroelastic systems. In the fifth part of the review, Ibrahim and
Roberts [9] considered stochastic problems in parametric vibrations.
The trend of recent investigations is to develop more refined analyti
cal models along with numerical solutions. A summary of recent work
is presented below.
The response of a two degree-of-freedom system with multifrequency,
parametric excitations was considered by Nayfeh [10]. The method of
multiple scales was utilized to investigate the stability regions of
the system under combination summed and principal resonances, com
bination difference and principal resonances, and combination summed
and difference parametric resonance conditions. For the cases in
which only one of the parametric resonance conditions was fulfilled,
the predicted results were confirmed by a previous work by Nayfeh and
Mook [11],
Yamamoto, et al [12] investigated the cases of summed-and-
differential type harmonic oscillation in a simply supported beam.
Theoretical results indicated that only the summed-type oscillation
should occur, but the experimental work showed that both summed-and-
differential harmonic oscillations occurred.
Subharmonic oscillations in a beam supported at both ends was
investigated theoretically and experimentally by Yamamoto, et al [13],
Subharmonic oscillations of orders 1/2 and 1/3 were predicted ana-
lytically. The existence of these oscillations was confirmed experi
mentally,
A study of unsymmetrical shafts was presented by Yamamoto, et al
[14] . They concluded the following: the characteristics of summed-
and-differential harmonic oscillation were similar to those of subhar
monic oscillation; oscillation phenomena which differ from those of a
round shaft occurred; and varying system parameters gave both hard and
soft spring-type solutions. The experimental results were qualitati
vely consistent with the theoretical results,
Yamamoto, et al [15] investigated subharmonic and summed-and-
differential harmonic oscillations in an unsymmetrical rotor. The
presence of nonlinearities in the unsymmetrical case caused the
results to be much different than those found in the symmetrical case.
The qualitative characteristics of the oscillations of the unsym
metrical rotor were both theoretically and experimentally the same as
the previously mentioned unsymmetrical shaft system.
Zajaczkowski [16] used the Galerkin method to obtain the solution
to the general Mathieu equation. The analytical solutions were con
firmed by numerical integration. The results were in agreement with a
previous work by Zajaczkowski and Lipinski [17],
Takahashi [18] investigated the instability of multi degree-of-
freedom, dynamic systems under parametric excitation. A vector solu
tion was expanded in a Fourier series, and the harmonic balance
technique was used to develop the characteristic equations. Numerical
results were presented for the damped and undamped Mathieu equation.
and comparison was made to results obtained by Szemplinska-Stupnicka
[19], Hsu [20], and Bolotin [5]. The results were in agreement with
those obtained from Floquet Theory, which involves the transformation
into a set of linearly independent solutions of a function which can
be represented in the form of a converging Fourier trigometric series
[5],
Struble's method was used by Stanisic [21] to study the stability
of the generalized Hill's equation with three independent parameters.
The solution, which is valid for a system equation with periodic coef
ficients, gave an explicit form of the stability regions to any order
of approximation. The first- and second-order approximations were
reasonably comparable,
Sato, et al [22] investigated the parametric response of a hori
zontal beam carrying a concentrated mass. An approximate solution was
obtained by using the Galerkin method and the harmonic balance tech
nique. The results showed that the parametric resonance occurred
along with the forced resonance because of the initial static deflec
tion of the beam,
Szemplinska-Stupnicka [19] generalized the harmonic balance
method to multi degree-of-freedom, parametric, dynamic systems. The
method was applied to a two degree-of-freedom system. The bound«ries
of the principal and combination parametric resonances were calculated
and verified by analog computer simulation.
The method of Bogoliubov-Mitropolski was used by Hsu [23] to eva
luate the response amplitude of parametric systems. Hsu considered a
system which involved time-varying damping and stiffness terms and was
subject to forced, periodic excitation. The first- and second-order
approximation of the stability regions were in agreement with the
results obtained by numerical integration.
Nonlinear Coupling (Autoparametric Resonance)
For the systems considered thus far, the parametric instabilities
have been associated with equations whose variable coefficients are
explicit functions of time, but in many instances the time variation
of system parameters is induced through implicit time dependencies,
i,e,, the variation depends on the motion of another normal mode. For
a system with two degrees-of-freedom, the equations of motion can take
the form
X + f^^ ^X-\ = ^ 1 1 ' 1 ' 1 ' 2' ^2' ^2^ (. . . . . . . , \ \ »Z)
^2 •*• ' 2 2 ^^2^^1' ^1' 1' ^2' ^2' ^2^
where, again, dots indicate differentiation with respect to time, X-j
and Xo are two normal coordinates of the system, u) and oi^ are the
corresponding linear natural frequencies, F and F2 are nonlinear
functions of the normalized coordinates, and e is a small parameter of
the system. The function, F , may contain terms of the form X.,X2/
etc, and Fo ^^V involve terms like X2X^ , etc. Both X., and X2 are
implicit functions of time and act as parametric excitations to the
first and second modes, respectively, A system whose motion can be
described by equations like equations (1 .2) is said to possess auto
parametric coupling [24], since the time variation of the system para-
10
meters is induced by the nonlinear terms involving the system coor
dinates and their derivatives with respect to time, Nonlinearities
can enter into a system model through inertial terms and elastic
restoring forces, Inertial nonlinearities can be introduced into a
system through the kinetic energy in the Lagrangian formulation of the
equations of motion. Elastic nonlinearities originate in nonlinear
strain-displacement relations.
The presence of the nonlinear terms can lead to a certain type of
instability which is referred to as internal, or autoparametric, reso
nance of the form:
n 1
kj w = 0 (1,3)
i = 1
where the k- are integers and the OJ. are the normal mode natural fre
quencies of a system of n degrees-of-freedom. These instabilities may
result in various forms of dynamic phenomena, such as amplitude jumps
and energy exchange between modes, which cannot be predicted by the
classical theory of small linear oscillations. In nonautonomous
systems, autoparametric resonance occurs when the conditions of exter
nal and internal resonance occur simultaneously. The elastic pendulum
shown in Figure 1.5 is a classic example involving autoparametric
coupling. Other structures exhibiting phenomena associated with auto
parametric resonance include a fluid filled tank on an elastic sup
port, the autoparametric vibration absorber, and an aircraft wing in
flutter [6,25,26] .
11
Figure 1.5, An elastic pendulum as an example of an autoparametric system.
12
As previously mentioned, the elastic pendulum is an excellent
example of a system which can exhibit the phenomena associated with
nonlinear (thus autoparametric) coupling, Breitenberger and Mueller
[27] used the slow fluctuation technique and obtained conditions for
the energy exchange between the r and 0 modes, which is dependent on
the system parameters. The suspension mode (r) behaved as a driven
harmonic oscillator, and the pendulum mode (0) exhibited charac
teristics similiar to the simple pendulum described by equation (1,1),
Breitenberger and Mueller noted the resemblance between the elastic
pendulum and a ship in heave, pitch, and roll. The dominant feature
of the results was the autoparametric energy transfer,
Hatwal, et al [28,29] investigated the two degree-of-freedom
system with autoparametric coupling shown in Figure 1.6. For the
forced system with the harmonic excitation, the harmonic balance
method correctly predicted the existence or absence of stable harmonic
and near harmonic solutions, as verified by numerical integration.
For certain frequency ranges, the steady-state response ceased to be
harmonic. For decreasing excitation frequency, the transition from
regions of stable, harmonic solutions to regions of unstable solutions
occurred without any amplitude jumps. In these regions more than one
amplitude-modulated, steady-state solution was possible, depending on
the initial conditions given to the system. For increasing frequency,
this transition was associated with sharp amplitude increases, and
no periodic solutions existed [28] . For the gravity controlled pen
dulum, when the excitation parameter was increased beyond a critical
13
/ // / ^ y / / / /
h P« cos
r
CJt l"^l 1
Figure 1 .6 . Two degree-of-freedom autoparsuaetric system.
14
value, the responses were found to be nonperiodic. To first-order
approximations, the solutions were found to be inadequate to describe
the stablity of the system, it was noted that the pendulum could be
used to act as a vibration absorber for the primary system when the
system was excited near the resonance condition of oc = 2a) where, o
was the natural frequency of the main mass system and OL was the
natural frequency of the pendulum system [29].
As a continuation of the analyses of the system shown in Figure
1 .6, Hatwal, et al [30] investigated the chaotic responses of the
autoparametric system. Chaotic motion is a form of motion (system
response) which is nonperiodic in nature as shown in Figure 1,7,
They showed that for certain combinations of forcing amplitude and
frequency the responses became random. The existence of this chaotic
motion was verified experimentally. The results of the numerical
integrations were used to obtain mean square values and frequency con
tent of the response. The statistical parameters were shown to be
independent of the initial conditions and the numerical integration
step size, but dependent on the values of the system parameters. The
response of the primary mass showed a strong periodic component at the
forcing frequency, and the pendulum response had a wider spectrum.
The theory of chaotic responses of deterministic, nonlinear
systems is currently being developed. The classical techniques, such
as averaging, fail to detect these motions [31], Various qualitative
analyses have shown the existence and characteristics of chaotic
motions in deterministic, nonlinear systems, but at the present, there
15
u •s u I 4J
e
09
CO
O
00
c a (0 • 4) C u 0
•H o .u •H (« •U 4J O -H IQ U u <0
0)
9
•H
16
is no theory to predict the range of system parameters for which these
motions can occur. The goal of current investigations is to develop a
chaotic motion parameter (analogous to the Reynolds number in fluid
flow) below which periodic motions would occur and above which
chaotic, nonperiodic motions would be insured. It is thought that
this "chaotic Reynolds number" is a function of driving frequency,
forcing amplitude, and damping for most mechanical systems [32],
The nonlinear response of bowed structures to combination reso
nances was investigated by Nayfeh [33], The results showed that the
difference-type combination resonances could never be excited for the
system considered. The physical conditions for the occurrence of the
summed-type internal resonance were given. The quenching of com
bination resonances was verified by numerical integration for cases
including and excluding internal resonance,
Singh [34] used a two-scale perturbation analysis to investigate
quenching in a system of van der Pol oscillators with nonlinear
coupling. The suppression of the excited mode predicted by the analy
tical technique was confirmed by digital computer simulation,
Yamamoto, et al [35-37] investigated the internal resonance in a
two degree-of-freedom, nonlinear system, Nonlinearities up to fourth
order were considered. When the natural frequencies were in the ratio
1:2, they concluded that the first and second harmonics were strong in
the vicinity of the first internal resonance condition; the response
curves, which consisted of three branches, \</ere slowly varying
periodic functions which were greatly influenced by damping. The
17
coefficients of the third-order terms were found to have a significant
effect on the form of the response curves. In the vicinity of the
higher resonance condition, when the natural frequencies were in the
ratio 1:2, they found that the type of response the system follows was
determined by the initial conditions. The amplitude response for a
single mode was similiar to that of ordinary forced oscillation; and
the response curves were, again, greatly influenced by damping. These
theoretical results were verified by analog computer simulation
[36,37], When the natural frequencies were in the ratio 2:3, they
showed that the vibratory state taken on by the system was dependent
upon the initial conditions. The response curves,in this case, con
sisted of two branches and were greatly influenced by damping. The
fourth-order term had a significant effect on the character of the
response. These results were verified by analog computer analyses, as
in the previous cases [35],
The effects of autoparametric resonance in a structure containing
a liquid (see Figure 1,8) has been investigated theoretically and
experimentally by Ibrahim and Barr [25,38]. For two-mode interaction
only, the coupling between liquid sloshing and vertical vibration was
shown to be weak. The first-order perturbation solution was not ade
quate to predict the response of the system. The second-order solu
tion gave the response of the main mass as that of a single degree-of-
freedom system. The experimental results were in agreement with the
second-order perturbation solution [38] . The experimental results for
three-mode interaction confirmed the internal resonance phenomena
18
K^/4
• 0 COCCUt
Figure 1.8. Fluid filled structure on an elastic support.
19
(amplitude jumps and modal energy exchange) predicted analytically by
Struble's asymptotic approach [2]; however, the experimentally deter
mined amplitudes were not as great as those determined analytically
[25] .
Haxton and Barr [26] analyzed the resonant response of an auto
parametric vibration absorber which is shown in Figure 1.9. Their
results indicated that when the main mass was externally excited at a
frequency close to its natural frequency, the coupled beam suppressed
the main mass motion if ta = 2u), where, o), is the natural frequency of
the main mass sytem and (x is the natural frequency of the cantilever
portion of the system. The asymptotic approximation method of Struble
[2] was used to develop the theoretical response of the system, and a
reasonable comparison was observed between theoretical predictions and
experimental results. The main discrepancy was in that the experimen
tal amplitudes were, in general, greater than those predicted analyti
cally.
Scope of Present Research
The present investigation deals with linear and nonlinear modal
analyses of aeroelastic systems such as an airplane wing with fuel
storage. Figure 1.10 shows a typical model under vertical, harmonic
excitation. The mathematical modeling of the system is derived by
applying Larange's equations. The linear modal analysis will he
obtained by considering the linear, conservative portion of the
equations of motion. The normal mode frequencies and the associated
20
F(t)
.J
TTnTTTTTTTT
Pigur. 1.9. Th« autoparamatric vibration absorb«r.
21
es a*
(S 6
\ r
CVJ es
cvj
M O u
u CM
s
o
s
CN
cs ;^
^
i^
u
I' CO
a*
;r Csl
s
CO
-*VW— —f
L J OJ CO
u.
/ /
/ e /
s 0) 4-) <n >^ m
o •H 4-1 CO (0
i H V
o 0) ns
<4.| O
E (0
(0 •r-( • "O 03
•H (0 •U C (0 - H S T3 <U
0
o CO o
u 3 0^
22
mode shapes will be analyzed. The main objective of the linear modal
analysis is to explore the critical regions of internal resonance for
which relation (1,3) is satisfied. Should the system properties
fulfill relation (1,3), the dynamicist must conduct a nonlinear ana-
lyis to examine the associated dynamic behavior which may influence the
safe operation of the system.
The nonlinear coupling of the normal coordinates will be exa
mined by considering quadratic nonlinearities. The asymptotic
approximation technique due to Struble [2] will be employed. Three
groups of internal and external resonance conditions will be obtained
from the secular terms of the first-order perturbational equations for
typical autoparametric resonance conditions. The response of the
system will be given in both the frequency and time domains with ana
lytical and numerical results presented for both two- and three-mode
interactions.
CHAPTER II
FORMULATION OF THE PROBLEM
The response of the structure in Figure 1,10 can be determined by
considering only one-half of the model with the generalized coor
dinates q^, q2, and q as shown. The system consists of the main mass
m3, linear spring K^, and dashpot C^. The main mass carries two
coupled beams with stiffnesses K-i and K2 (lengths 1 and I2) and tip
masses m-] and m2. The kinetic energy of the structure can be derived
assuming a static deflection curve for the beams where the axial
displacement at the end of a beam is given by 3qj /51i. The kinetic
energy for the system is given by (details of the derivation are shown
in APPENDIX A):
T = 1/ [mi + m2(1 + (312/21^)2)] q 2 _^_ ^/2m2^2^
+ 1/2[m + m2 + m3] q^'^ + 3/2 m2 ^2 ^^^2
I1
+ [m-, + 1113] c[ q3 + 9m2 I2 f<?i 2^ + 5qiqiq3]
20 1 2
+ 3m2 t q i C [ t q 2 / ^ ••• ^1°^2^3 ••" ^1*^2^3 " ^1^q2^ ( 2 . 1 )
21-
+ 6m2 fq2^2^3 " '^1^2^2^ " • " —
5 1 ' 2 1 , 2
4 (m.| + m2) + 9m2l22
25 16 l i ^ .
'^^W'
+ 9m2 tq i ^2° f2^ " q i * '2^q2^
511^2
23
24
where, terms up to fourth-order are r e ta ined . Neglecting g r a v i t a
t i o n a l e f f e c t s , the p o t e n t i a l energy i s given by
V = V2[KiqT2 + K2q2^ + K3^3^1 (2.2)
Applying Lagrange's equations in terms of the generalized
coordina tes
3 3 3 8V
[jk,i]q^qj^ + 8q^ = Q.
j = 1 j=1 k=1
(2 .3)
where, [ j k , i ] i s the Chr i s to f fe l symbol of the f i r s t kind and i s given
by the expression
[ j k , i ] = 1 K j " ^ i k • ^ j k
2 L3qk ^^j ^ i -I
( 2 . 4 )
The metric tensor m^^ and the Christoffel symbol are generally func
tions of the q^r and for motion about the equilibrium configuration
they can be expanded in a Taylor series about that state. Thus from
inertial sources, quadratic, cubic, and higher-order nonlinearities
can arise. The resulting equations of motion are
"•ll " 12 " 13
n>12 " 22 0
m-| 3 0 m 33.
K 0 0
0 K2 0
0 O K 3J (2,5)
4*.
™2^ ''2
"i-
25
where,
™11
™22
™33
™12
= m 1 + m2 [1 + 2 , 2 5 ( 1 2 / 1 ^ ) 2 ]
= m.
m 13
m + m2 + m^
= 1 , 5 ( l 2 / l ^ ) m 2
™1 + ™2
(2 ,6)
and
^
(2 ,7)
^
4' = . 4 5 ( l 2 ) ( q ^ 2 ^ 2 q i q T ) / l i 2 ^ 1 ,2(q2q2 + q 2 2 ) / l 2
+ 3 (q iq2 + q2qi + q i q 2 / i o ) A i
+ 2,25(12) q iq3 /1^2 ^ 1 .5q2q3/ l i
= , 3q^q^ / l . , + 1 .2 (q2q^ / l2 + q2q3A2 " " 1 ^ / ^ l ^
+ 1 ,5q^q3/ l^
= 2,25(12) (4^2 + q^q^) /1^2 + i .2(q22 + q2q2 )A2
+ 1,5(q-,q2 + 2q., q2 + q 2 q i ) / l i
For t he l i n e a r modal a n a l y s i s , the homogeneous cases of equa t ions
(2 ,5 ) a r e c o n s i d e r e d . In order to e l i m i n a t e the e f f e c t s of l i n e a r
c o u p l i n g , equa t i ons (2 ,5) a re t ransformed i n t o normal c o o r d i n a t e s ,
{p}T _ {p - , P^, P^} where T denotes t r a n s p o s e , such t h a t
{q} = [R] {P} (2 ,8)
where [R] i s the modal mat r ix which c o n s i s t s of the normalized e i g e n
v e c t o r s
[R]
1 1 1
" l " 2 " 3
. P i P2 P3-
(2 ,9)
26
This transformation results in the equations of motion in terms of the
normalized coordinates
Xo' 3
M 11 0
0 M22 0
0 0 M 33j
^P. K ^ 0 0
0 K22 0
0 K 33J
F(t)
m-
(eo«Q(U32)
(2,10)
where.
F ( t )
P i
T
XQ
e
s
=
=
=
=
FQ COS n t
P i / ^ 0
(i^t
F0/K3
X Q A I
and
M 11
"11
a
b
a
(2,11)
1 + a(a + 2n^b + n^2) + 2(1 -f- a)pi + (1 + a
1 + n^2 0 (n/e)3 + p^2^
1 .+ (1 .5 l2Ai)^
1.5 I2/I1
m2/n*i
m3/in-j
$)p^2
(2,12)
27
and
3
0
n
i
r
Ki(i=
and
* i
=
=
=
s
=
»1 ,2)
=
I 2 / I 1
Wj/W^
t 2 / t i
K3A1
[ 4 m ^ ( l ^ / t ^ ) 3 ] / ( j ^ ^ )
= E Wi t i3 /4 i^3
^ l ^ ^ i l l P l " 1^121^2
(2,13)
P2f^112^1 - 122^2 - 132^3^-^
P3f^i13Pl + ^123^2 - 33^3^ " ^2,14)
^ill^l^ + Mi22P2^ ^ ^133^3^ +
Mil 2 1 2 •*• ^113^^3 " ^123^2^3
The L^j]^ and M .j terms of equation (2,14) are given in APPENDIX B, In
equations (2,13), E is Young's modulus and w^ and t^ are the widths and
thicknesses of the two beams, respectively. Equations (2,10) may now
be written as
P + i 13 Pl = 1 cos NT - eaV^/M^^ - 2e^^P^
P2 "*" 23^P2 ' ^2 ^°^ "" ~ °''*'2/M22 " 2^^2^2 (2,15)
P3 + r332p3 = f3 cos NT - ea'i'3/M33 - 2eC3P3
where,
(2,16)
•^13
u^i
N
f i
I S
=
s
=
0)./a)3
K i i / ^ i i
^2/0)3
CPi / (Mi^ 0132 e)
28
and damping has been inserted into normal mode equations of motion to
account for energy dissipation. It should be noted that the nonli
nearities of equations (2,15), i,e,, the H'i's of equations (2.14), are
physically attributed to the non-vanishing axial displacement of the
tip masses.
CHAPTER III
LINEAR MODAL ANALYSIS
Eigenvalues
For the linear modal analysis, equations (2.5) are written in the
form:
[m]{q} + [K]{q} = {f} - W (3.1)
In order to determine the eigenvalues and eigenvectors for the
system, the right-hand side of equation (3.1) is set to zero. This
gives
[m]{q} + [K]{q} = 0 (3.2)
Assuming a solution of the form q = Q sin u>t gives
UK] - X[m] {Q} = 0 (3.3)
where, X^ = 0) 2 .g j g squares of the eigenvalues of the system (i =
1,2,3), which cause the matrix which is multiplied by {Q} to be singu
lar. The eigenvalues are determined by the solution of the charac
teristic equation, which is given by
det[D(X)] = 0 (3.4)
where [D(X)] = [K] - X[m] . For the system under consideration, the
characteristic equation is
("»13 m22 - ™12^™33 " " 11"*22'"33 ^ +
tK-,in22™33 " 2 '"l1™33 " ""13 ) + K3 "*1l"'22 " "»12 >1 ^ - ^3,5)
^ 1 2"'33 " 1 3"*22 " 2 3"'l1 ^ •*" ^1^2^3 = °
where, the mij's and K^'s are given by equations (2,6) and (2.13),
respectively.
29
Equations (3.5) may be written in the form
30
y( 10/(1)33) 6 •»• 0)
L<^33J "^2
T 2
+ 1 - mi22
"'11™22
0)
L'^3j
•err - mi3'
'"ll'"33
"^3
(3.6)
"^22"
."^33. • _'^33.
2 " ^ 2 '
_ ' ^ 3 _
= 0
where.
0). '11 = Ki/miT
2 _ "^2 = ^2/ m 22
0)332 = K3/ m 33 (3.7)
P = -1 + m 12 + m^32
"»11 ""22 ™11 ""33
The frequency parameters, (n^^, are the natural frequencies if each
component of the system acts as a single degree-of-freedom. Figures
3,1 - 3,3 show the variation of the natural frequencies of the system
as functions of ^'\']/^23 *"^ ^^22/^3,
Figures 3,4 and 3,5 show the variation of several combinations of
the natural frequencies with ^•\/^2 ^"^ ***22/' 33' "^^^ critical points
are located where *-he curves (co, + a)2)/ui33, 20) /0)33, and 20)2/0)33
intersect the curve for (^y^^^S ^^ Figure 3,4, These intersections
indicate the possibility of combination resonances in the structure
when the forcing frequency is near 0)3, In Figure 3,5, the critical
points are located at the intersections of 2aV|/o»33 and (0)3 - ui^)/(ji^^
31
m
CN
CD
S
09 3 O
•H
u o
cn
c 0
O CO C 3 <U \
•O CM «= -T* <0 3
a 0) (w
o o
en <0 U 3
CO
o vO
• o m m
^ *
o
CN
.
o
32
3
n
- CN
a 9
OS 9 0
•H U
o
m en
c 0
i %4 0
0) 0 c 4)
T3 c 0) a <u Q
. pn
- P 3 \
OJ
r %i 0
(N
3
•H
33
m
3
- m
9) 9
CO 3 0
•H
ki
o
m
a c 0 m
3 lU 0
0) o e 0) T3 e 4) a 0)
a
• pn m
3 '«v
CN CN
3
<u 0
. m
V U 3
34
CO C T3 0 9>
40 « C
jQ
U
a 6 0 o
O CO O (0
o • C r-Oi 3 II
cr 4) pn
ft CN
2/ 3 ••0 U (« 0 C <4-l
%4 pn O pn
C \
o .-40 3 •
•H JS U 40 (0 -H > IS
pn
3
•H
pr 3
35
pn n
r-l (H
3
ID a
rH
o a
rH
in a
o
CD C TJ 0 «
-H V4 ^ (Q (T3 a c s
•-1 0 £ 0 e 5 en o nj >-o o • c .-0) 3 II cr 0) pn
i:i ^v r-< CM
Si^ ( J u <T3 0 C (4-1
u-i pn 0 ^
c ^. 0 f-
•J i-(C •-1 X V4 .»J (0 -H > 5
• I T
• PO
<r V4
£"
fT'
36
with 0)2/0)33. This shows the possibility of combination resonance when
the forcing frequency is near 0)2. When combination resonances like
those shown in Figures 3.4 and 3.5 exist in a system, a nonlinear ana
lysis is necessary to determine the response characterisitics of the
system.
Mode Shapes
A normalized eigenvector gives the shape of the motion of the
system at one of the eigenvalues after dividing each amplitude by the
first element. The matrix, [D(X)], of equation (3,4) may be written
ro^a^^) °ab^\)
[D(X)] = (3,8)
where, 0,^ is a 1 x 1 matrix, Dw is a b x 1 vector, D j is a 1 x b
vector, DKK is a b x b matrix, and $. is a vector containing the b
remaining elements of the normalized eigenvector. For distinct eigen
values, Dwvj is nonsingular, therefore
This gives
(*b>i = - f 5 b b < \ " - ' t £ b a < \ "
The normalized eigenvector for the i t h mode i s tiien given by
$. =
(3,9)
(3,10)
(3,11)
For this analysis, the International Math and Science Library
37
(IMSL) subroutine EIGZS was used to determine the eigenvectors as
defined by equations (3,10), For the condition where ^ = 25, $ = 5, 3
= .402, and a = ,112, the normalized eigenvectors are
1 ,ooo'
{$ } = ^-0,599
0,000
1 ,000^
{*2> = <-3,903 (3,12)
0,046
1 ,000
{$3} = < 3.117
0.444
These mode shapes are shown in Figures 3,6 - 3,8
38
Tj
<l 40 —1
0 X
v E
a; 40 CO >> M
U 0
(t-i
V CL 03
x: CO y *
0) T 0 s u fC 0)
c •r-(
^
• «£
• pn
0)
u
« CM T -
^ ,
II
a ^
CN
o ^ ,
II
•a %
in
II
•€•
% in CN
II
UJ»
39
rvj 3
II
->. "" «>
• • ^
T T
i W ~ l 0 X
a; £ <i; ^ CO >, OT
U 0
14
0) a TJ
£ 7)
u 0 £
u T3 3) C
rsj
•
II
3
« CM O ^
.
II
IX
^ in
II
^
^ in PM
!l
J '^
u
40
^^^
4J 10
T3 07 40 •H 0 X 0)
e (0 40 CO > CO
u 0 (l-l
<u a (T3
JC CO
0) TJ 0 E
U (B 0) C
•H K1
i
CN T -^ •
• II
3
^ (N
o rr .
II
T l
^ in
II
B>
% in CN
II
UJ«
00 .
pn
u 3
CHAPTER IV
AUTOPARAMETRIC MODAL INTERACTION
Asymptotic Approximation Solution
The nonlinear dynamic response of the system shown in Figure 1.10
can be determined by employing the asymptotic method of Struble [2].
This method has distinct advantages in that "it exhibits not only
resonant and non-resonant responses but also other characteristics of
a nonlinear system, including the interplay of the system parameters
and the entrainment of harmonic, subharmonic, and other responses.
In addition, since the method of Struble incorporates two classical
techniques, i,e,, the variation of parameter method and a perturbation
method, it has the advantages of both methods while avoiding the main
difficulties of each" [21], The method is outlined below and then
applied to equations (2,15),
Assume solutions of the normal modes expressed in the form
P = A(T) cos[r^3T + *^(T)] + ea + e2a2 + ...
P2 = B(T) cos[r23T + $2 * ^ + 6b + e2b2 + ... (4,1)
P3 = D(T) cos[r33T + ^^{T)] + 6d + £2^2 + ...
where, the amplitudes A, B, D, and phases *v *2' ^^ *3 ^^® slowly
varying functions of the dimensionless time parameter, T. Performing
the time derivatives of the first of equations (4.1) as indicated by
equations (2.15) gives
41
42
P-, = -A(r^3 "*• *i)sin(r^3T + $ ) + Acos(r^3T + <I» ) + ea
P = [A - A(r^3 + *T)2]cos(ri3T + ^^) + ea - (4.2)
[A*-, + 2A(r^3 + *^)]sin(r^3T + * )
Similarly
P2 = -B(r23 + *2^si'^^^23^ "*" *2^ "*" Bcos(r23T + $2^ " 1
P2 = [B - B(r23 + *2^^1cos(r23'^ + *2^ " 1 "
[B*2 + 2B(r23 + *2)]sin(r23T + $2^ ^^'^^
P3 = -D(r33 + $3)sin(r33T + $.) + Dcos(r33T + $3) + ed
P3 = [D - D(r33 + *3)2]cos(r33T^ +^3) + d' "
[D*3 + 2D(r33 + $3)]sin(r33T + $3)
where, only the first-order perturbations are retained, and r33 is
equal to unity by equation (2.16).
Equations (4.1-4.3) are substituted into equations (2.15), and
like harmonics are collected based on the order of e. The equations of
order e^ are referred to as the "variational equations". Neglecting
the slowly varying terms (A, B, b, ^^, eA, £2^, A$, etc), the
variational equations are
-2ArT3< i = A(S^2^ _ ^132)
-2Ar^3 = 0
-2Br23*2 = B(S22v2 - r232) (4,4)
-2Br23 = 0
-2Dr33*3 = D(S32v2 - r332)
-2Dr33 = 0
where, the forcing frequency, ^, has been replaced by nv and v is
43
near one of the normal mode frequencies. The S^'s are positive real
numbers such that] S^^v^ - ri32 | < £.
The terms of order £ are called the "first-order perturbational
equations," Neglecting the slowly varying terms, as before, gives
• • ^ 1 3 ^
0
0
0 0
^23^ 0
0 ^^33^.
f2 \ cos(nVT) +
^1 ^2 ^3
^1 ^2 M3
i2>
B
3 N2 N3J ( D2
'L4 L5 Lg
M4 M5 Mg
_N4 N5 Ng^
L7
A2 C O S (2ri3T + 2$^)
B 2 C O S (2r23'^ + 2*2)
D 2 C O S (2r33T + 2*3)
'2^1^13 0 °
0 2^2^23 °
0 0 2^31^33
Mo Mc M7 Mg
N7 ^8 ^<
L1O ^11 ^12'
M10 ^11 ^12
L^IO ^11 ^12.
A sin (r-,3T + $ )
B sin (^23^ + ^2^
D sin (r33T + $3)
AB cos [(r^3 - r23)T + $ - $2^
AD cos [(r^3 - r33) T + $ - $3]
ED cos [(r23 - ^33) T + *2 " *3^
'AB cos [(r^3 + r23) T + ^ + ^2^
AD cos [(r^3 + r33) T + $ + $3]
BD cos [(r23 + ^33^"^ + *2 "*" " 3
(4,5)
44
with the Li's, M^'s, and N^'s given in Appendix 2,
Resonance Conditions
The first-order perturbational equations contain secular terms.
Secular terms are those terms which give rise to a resonance condition
such that the denominator of these terms approaches zero as the reso
nance condition is satisfied. These terms result in two types of
resonance conditions: external and internal. Inspection of equation
(4,5) shows that three groups of resonance conditions are possible.
Group I
When the first mode is externally excited at a frequency close to
its natural frequency (i.e,, nv = r-13), the following internal reso
nance conditions are possible
^13 " I 23 i ^33 I
^ 13 = ''/2 r23
^13 = ''/2 33 (4,6)
^13 " ^ 23
^ ^ 1 3 ^ ^ ^33
Group II
For external excitation of the second mode such that nv = r23r
the following internal resonance conditions are found to exist
45
^23 = I 13 ± r33
^23 = V2 ri3
^23 - V2 r33
^23 = 2 r^3
r^rj = 2 r 23 33
Group III
If the third mode is externally excited at nv = r33, the internal
resonance conditions are
^33 = I 13 ± ^23!
^33 = V2 rT3
^33 = V2 r23 (4,8)
^33 = 2 r^3
^33 = ^ 23
The above internal resonance conditions are classified in two
major groups: combination internal resonance, r = | r + r^ | , and
principal internal resonance, r = nr^ (where n = 2 or 0,5), it has
been shown that the resonances of each group possess their own speci
fic properties [39]; therefore, the presentation of analysis will be
restricted to the numerical and analytical solutions for the summed
and difference cases of combination resonance and one case of prin
cipal resonance where the external forcing frequency is in the neigh
borhood of r33 (i,e,, nV = r33 = 1 ) .
46
Analytical Solutions
Case 1: Summed Internal Resonance r33 = ^13 + ^23 ^^^ External
Resonance nv = r33
The secular terms of equations (4,5) corresponding to the summed
internal resonance condition are transferred to the variational
equations (equations (4,4)) which become
-2Ar^3*^ = A(S^2^ _ ri32) + £BDL9 cos 0-,
-2Ari3 = £{2CTr^3A + BDLg sin 0 }
-2Br23*2 " B(S22v2 - r23^) + EADMQ COS 0 (4,9)
-2Br23 = ^124^^23^ " ^8 ^^" ®1 1
-2Dr33*3 = D(S32v2 - r332) + £{ABNIO ^ ° ^ ® 1 " ^3 ^ ° ^ "^3!
-2Dr33 = ^(2^3r33D - ABN Q sin0^ + f3 s i n $3}
where, ©i = *i + *2 " * 3 ' I n t roduc ing the fol lowing t r ans fo rma t ions
•
0 )
T
Y
^ i
S i
f 3
33 1 ^10 1
2T
^J^3 1 ^10 1
^ 3 3 ^ - v2
^33SA3 1 10 1
2 q ri3
^ ^ 3 ! '^lol
^13 / ^33
y2
^3
( 4 , 1 0 )
47
g i v e s e q u a t i o n s ( 4 , 9 ) a s
-*^ = -S^Y + y2y3 Lg c o s 0^
y i ^ i | N ^ O |
-Yl = Tl^yi/S^ + y2y3 Lg s i n 0^
Si N^o
_ $ 2 = -S2Y + y i y 3 MQ COS0^
y2S2 I N10
- y 2 = ^2^2 /52 + y i y 3 ^ sin0^ (4,11)
S2 I ^10
-$3 = -Y + cos $3 + y^y2 N Q COS0^
yi y3 I N10
-y3 = n3y3 + sin$3 - y^y2 N^Q sin0^
N10I
The steady-state solution of equations (4,11) can be obtained by
setting the left-hand sides to zero. This results in six nonlinear
algebraic equations which are inconsistent since the actual number of
unknowns is five (y- » y2f y3 f ©1 and $3); however, these equations can
be consistent for two conditions
n = n2 = 0
or (4,12)
ni/n2 = S^/S2
The second condition of equation (4.12) is more physically realizable
than the first one. For this case, the amplitudes are given as
48
yi = ^2 / S 2 11.9
Si I Mg
2 = [y^J^^'^' ( 4 , 1 3 )
^ 3 ^ = S2N10 2 (Y2g^4 ^ ^^2)
Sl^s
Another solution is found for yi = yo = 0
y3 ^ 1
rsP^+n^^ (4,14)
which is the normal, single degree-of-freedom response.
The amplitude response, as predicted by equations (4,13 - 4,14),
is shown in Figure 4,1, There are two vertical tangencies on either
side of n = 1, One occurs (frequency increasing) at the intersection
of the two solutions for y3 (n = 0,985), At this point, the values of
y- and y2 jump from zero to the values defined by the first two of
equations (4,13), and y3 follows the curve defined by the last of
equations (4,13) which shows the typical absorbing effect of systems
with autoparametric coupling. When the forcing frequency is increased
to the point where n = 1,195, the values of Y-\ and y2 collapse to
zero, and y3 collapses to the single degree-of-freedom response
defined by equation (4,14),
For decreasing frequencies, the amplitude jumps again occur at the
intersection of the two solutions for y3 (n = 1,104), and the values of
y. , Y2' ^^^ y3 follow the response curves defined by equations (4,13),
49
0.98 1,02
Figure 4 . 1 . S t e a d y - s t a t e response for r^^ '^ ^23 ' ^33 t, = 19 .8 , « = 8, P = .549, a = .629, Q^ =
.008, (,2 = -Q^' ^3 = • ° ^ ' ^ = • ° ' *
= nv.
50
At the point where n = 0.979, the values of Y^ and y2 collapse to zero,
and y3 follows the single degree-of-freedom responses curve. The
values of n where the amplitudes jump or collapse depend primarily on
the system parameters such as damping and internal detuning.
For the case where n = ri2 = 0, the response curves are similar to
those of Figure 4,1, The only major difference is that the value of y3
is zero at n = 1, which implies that damping has a destabilizing effect,
Case 2: Difference-Type Internal Resonance ^33 ~ ^23 ~ ^13 ^^^ External
Resonance nv = r33
Following the same procedure as before, the secular terms of
equations (4,5) corresponding to the difference internal resonance con
dition are transferred to the variational equations. The resulting
equations are
-^1 = -s^Y + y2y3 L^2 ^°^ ®2
y i S i I N7
- y i = ^ y i / S i + y2y3 ^12 ^ ^ " ®2
Si I N7
$2 = - S 2 ^ + y i y 3 "11 ^ ° ^ ®2
Y2'=2 I ^7 I ( 4 , 1 5 )
- y 2 = ^2^2 /82 - y i y 3 "11 s ^ " ®2
• ^ 3
S2 I N7
- Y + cos $3 + y i y 2 N7 cos 0.
y3 ^3 I ^7
- y 3 = n3y3 + s i n $3 + yi y2 N7 s i n 0,
N7
51
where, ©2 = *i ••" *3 ~ *2* " ^ solution for y-j = y2 = 0 is the same as
that given by equation (4,14),
Equations (4,15) are similar to equations (4,13), The steady-
state response curves for the difference case with ^1/^2 = Si/S2 are
shown in Figure 4,2, The jump and collapse phenomena, seen in Figure
4,1 for the summed case, are also exhibited in the response for the
present case. For Hi = ^2 = 0, the value of y3 is zero at n = 1 ,
which is the same as for the summed condition.
Case 3: Principal Internal Resonance r33 = 2r-3 and External
Resonance nv = r33
For this case of internal resonance, the interaction takes place
between the first and third modes. Transferring the secular terms from
equation (4,5) to equations (4,4) gives
-$^ = -S^ Y + y3 Ls cos 03
Si I N 4 I
- y i = Ti^yi/Si + YyY2 L3 s i n Q^
Si I N4 I ( 4 , 1 6 )
_ $ - = -Y + c o s $ 3 + y i 2 N4 c o s 0 3
y3 ^3 I ^4
- y 3 = n3y3 + s i n * 3 - yi 2 N4 s i n 0 3
^4
where, 0 = 2 $ - *-a. I^ ^® left-hand side of equations (4.16) are
set to zero, the resulting algebraic equations are found to be com
patible, and the steady-state solutions are
52
0 , 9 6 0 . 9 8 1.0 n
1.02 1,04
F i g u r e 4 . 2 . S t e a d y - s t a t e r e s p o n s e for r23 " ^i 3 = 1 33 = - ^ . 4 = 25 , <P = 5 , ^ = . 402 , a = . 1 1 2 , Ci = . 0 0 4 , ^2 = •Ol t ^2 = •05 , £ = .01 .
53
yi 2 _ N.
8 J 2x /^2 Y^/16 + m^) {y^ + Ti3 ) i 1
(4.17)
^3 N.
L8 ^ Y 2 / 1 6 + m^
For yi = 0, the solution for y3 is the same as given by equation
(4.14).
The theoretical, steady-state response for y and y- is shown in
Figure 4.3. The jump and collapse phenomena seen in the summed and
difference internal resonance cases are also exhibited by the response
for principal internal resonance. The system response is similar to
that of the autoparametric vibration absorber [26].
Numerical Solutions
Case 1: Summed Internal Resonance Too = r-3 + r23 and External
Resonance nV = r33
When one of the conditions of equations (4.12) is not satisfied,
a numerical integration of equations (4.11), including the left-hand
terms, is required to determine whether the system achieves a steady-
state response in the time domain. The six equations can be simulated
utilizing the IBM Continuous System Modeling Program (CSMP) [40] (to
avoid division by zero, small non-zero initial conditions are assumed).
Sample programs are listed in APPENDIX C. The time history response
reveals that the system is quasi-steady-state as shown in Figure 4.4.
The modal interaction is seen in the form of energy exchanges between
54
4 _
2 -
0 . 9 6
F i g u r e 4 .3
0 . 9 3
y3( y i = 0 )
1.0 n
1,02 1,04
S t e a d y - s t a t e r e s p o n s e for 2r i 3 = ^33 = nv , % -• 1 5 , * = 5 , ti = .601 , a = . 574 , ; i = .01 , C,^ = . 0 5 , £ = . 0 1 .
55
pn CM
u
+ m
vT n pn pn
u
u 0
%t
> 1
u 0 40 CD
•H X
«
. ^-*
CM U
^«. t —
u s •«• n cu 2 CO u
CM c V , ^ c ^-'
0) V4 3
• H Gu
56
the first two modes and the third mode. Another important feature is
that the peak cimplitudes during the transient period are greater than
twice the quasi-state amplitudes. This is critical in the analysis of
maximum stresses.
The response of the system as a function of internal detuning, n, is
shown in Figure 4.5. The peak amplitudes for yi and y2 are not found
in this case since there are so near the regions of parametric instabi
lity. The jump and collpase phenomena both occur at the points where
the behavior of y3 changes to the autoparametric response from the nor
mal linear resonance response and vice versa.
Case 2: Difference Type Internal Resonance ^33 ~ ^ 23 ~ ^13 ^^^ External
Resonance nv = r33
Equations (4.15), for this case of internal resonances, are simu
lated using CSMP. Figure 4.6 shows the time history for the response
of the three amplitudes yi, ^2' and y3. The response characteristics
display the same features of the previously examined summed internal
resonance condition.
Case 3: Principal Internal Resonance r^^ = 2r-3 and External
Resonance nv = r33
For this case of two mode interaction, the response is also deter-
ined by using CSMP. Figure 4.7 gives the frequency response of the
system under principal internal resonance. The jump and collapse phe
nomena occur at the transition point for y3 as in the summed case.
m
57
2.0
1.0
0.98 0.99 1.0
n
1.01 1.02
F i g u r e 4 . 5 . CSMP s i m u l a t i o n of T u a s i - s t e a d y - s t a t e r e s p o n s e f o r r i 3 + r23 = r33 = d (n^ /n2 + r ^ / r 2 ) . ^ = 1 9 . 8 , «i? = 8, c5 = . 5 4 9 , a = . 112 , (,^ = ^2 = • 0 ' ' ' C3 = . 0 5 , £ = .01 ,
58
pn
J" I
en CN
U
II
m f) u II
u 0
U -n C CN
cn "*< —I m j = »-
u « E -H-
•H •^ CN
C 04 \ r -CO -
V u 3
•H
Ln . r-i
o •
rH
LO
• o
CN
O O •
o
"g* •
o M
o m
59
4.0 -
Figure 4.7 CSMP Simulation of two mode interaction, r33 = 2r = nv. C = 15, ® = 5, 0 = .601, a = .574, ;
= !oi, 3 = -O^. = -01•
60
Case 4: Numerical Integration of Equations of Motion
Another means of evaluating the response of the system is by
direct integration of the equations of motion. Upon inspection of the
nonlinear terms (the H' 's given by equations (2,14)) of equations
(2,15), it is seen that they involve acceleration terms, P^, which can
be replaced approximately by - ^ 32 p^, if these approximate P.'s are
substituted into the nonlinearly coupled terms of equations (2,15), the
equations of motion are easily integrated with the aid of CSMP,
Figure 4.8 shows a time history of the normal mode coordinates
for the summed internal resonance condition r33 = r 3 + r23 = ^
(program listing given in APPENDIX C), Since the forcing term appears
in all three equations of motion, the normal modes oscillate at the
forcing frequency, r33. The first two modes are 180® out of phase and
appear to be exchanging energy with the third mode during the first 60
seconds of motion. Damping has little effect on the maximum amplitudes
as long as 3 is greater than Ci and i;^' ^^ ^® forcing terms for the
first two modes. Pi and P2, are set to zero, the system takes on the
normal single degree-of-freedom response, as shown by the dashed lines
in Figure 4.8.
The generalized coordinate response can be determined from the
transformations of equations (2.9) and (2.11), Figure 4,9 gives the
amplitude-time response of the general coordinates for the summed
internal resonance condition r33 = ^^3 + r23 = «. The effects of the
nonlinear coupling can be seen in the significant amplitudes of qi and
for external forcing in the q3 coordinate direction, "2
61
o c 0
•H •U 0 £ CO c 0
II
(C -n 3 m . c r »M . -0) o
I! • (V
4J -y: II
c ^ u • o - o
p-t
o ID
•H ^ u ^ in 0 o C o . . o r CD It
-H O (TJ pn e N o» VJ £ 0 c -r ^
a; o «*J c . 0 .^
E II V U CO 0! CN C -u o* 0 (D a T II CO a; to , -a: rr ^
. cr • '*
V u D
04 CN on 04
62
o in r-l
O o t H
o in
c 0
•H
mot
CO
c 0
•H ;J (C5 1"^
$
V V ITJ
c •H TJ u 0 0 o
rH fl u <0 c 4) tr
UJ 0
(U CO c 0 c:* CO (U a:
-. m CN
+ PO T —
'u
M
m rr .
O 11 •
c: II
— (*» _) —4 k 3 IP
O ^ 4 .
s en II c
pn > . vj»
-» «
c . Till
U V CN
J vJ> ••1
r II X » -- u»
(T
V U 3
-H cr
b
pn
CN
cr
m
m cr
Pu
CHAPTER V
CONCLUSIONS AND RECOMMENDATIONS
The dynamic behavior of aeroelastic structural systems with three
degrees-of-freedom of the type shown in Figure 1,10 is determined.
The investigation includes linear and nonlinear modal interaction. In
connection with the normal mode analysis, the normal mode frequencies
and mode shapes are obtained for various system parameters such as
masses and stiffnesses. The results reveal certain critical regions
for which the normal mode frequencies satisfy the internal resonance
conditions Skj w. = 0 , Under the autoparametric conditions (when the
external normal resonance and the internal resonance are satisfied),
the system dynamic response is obtained by including the nonlineari
ties inherent to the system and by employing the asymptotic approxima
tion technique due to Struble, The analysis results in three groups
of internal resonance conditions, each corresponding to normal reso
nance for one of the three normal modes.
In the present study, the steady-state and quasi-steady-state
responses are obtained in the frequency and time domains. The steady-
state response reveals amplitude jump and collapse phenomena as well
as suppression of the motion of the externally excited mode. These
results also give the range of excitation frequency for which auto
parametric instabilities can occur. In the time domain, numerical
integration shows an energy exchange between the externally excited
mode and the other modes with suppression of the externally excited
63
64
mode. The results of these numerical integrations also give the peak
amplitudes during the transient period. These peak amplitudes are
greater than twice the values of the quasi-steady-state amplitudes,
A linear modal analysis is usually adequate to predict the
response of dynamic systems. However, this study shows that if the
linear eigenvalues of the system satisfy one or more of the internal
resonance conditions, a nonlinear analysis must be conducted to exa
mine the dynamic behavior of the system.
In order to obtain more refined results, a continuation of this
investigation should include cubic and higher-order nonlinearities in
the equations of motion for which further resonance conditions may be
derived. Futhermore, there could be a possibility of multiple inter
nal resonance conditions, A more realistic analysis is to replace the
harmonic forcing function, Fgcosi t, with a broad band, random forcing
function. Finally, experimental models should be constructed for the
verification and improvement of the mathematical modeling and to
investigate the possibilities of chaotic responses of the system.
LIST OF REFERENCES
1 Struble, R.A,, "Oscillations of a pendulum under parametric excitation," Quarterly of Applied Math 21, 121-131, 1963.
2 Struble, R.A,, Nonlinear Differential Equations, New York, McGraw-Hill, 1962,
3 Skalak, R., Yarymovych, M.I., "Subharmonic oscillations of a pendulum," J. Applied Mechanics 27, 159-164, 1960.
4 Ibrahim, R.A., Barr, A.D.S,, "Parametric vibration. Part I: Mechanics of linear problems," The Shock and Vibration Digest 2£ , 1, 15-29, 1978,
5 Bolotin, V,V,, The Dynamic Stability of Elastic Systems, San Francisco, CA, Holden-Day, Inc,,1964,
6 Ibrahim, R,A,, Barr, A.D,S., "Parametric vibration. Part II: Mechanics of nonlinear problems," The Shock and Vibration Digest 10, 2, 9-24, 1978,
7 Ibrahim, R,A,, "Parametric vibration. Part III: Current problems (1)," The Shock and Vibration Digest 10, 3, 41-57, 1978,
8 Ibrahim, R,A,, "Parametric vibration. Part IV: Current problems (2)," The Shock and Vibration Digest 10, 4, 19-47, 1978.
9 Ibrahim, R.A., Roberts, J.W,, "Parametric vibration. Part V: Stochastic problems," The Shock and Vibration Digest 10, 5, 17-38, 1978,
10 Nayfeh, A,H,, "Response of two-degree-of-freedom systems to multifrequency parametric excitations," J. of Sound and Vibration 88, 1, 1-10, 1983,
11 Nayfeh, A.H,, Mook, D,T,, "Parametric excitation of linear systems having many degrees of freedom," J, of the Acoustical Society of America 62, 375-381, 1977,
12 Yamamoto, T,, Yasuda, K., Tei, N,, "Super summed and differential harmonic oscillations in a slender beam," Bulletin of the JSME 25, 204, 959-968, 1982,
65
66
13 Yamamoto, T., Yasuda, K., Aoki, K., "Subharmonic oscillations of a slender beam," Bulletin of the JSME 24, 192, 1011-1020, 1981.
14 Yamamoto, T., Ishida, Y., Ikeda, T., "Summed-and-differential harmonic oscillations of an unsymmetrical shaft," Bulletin of the JSME 24, 187, 183-191, 1981.
15 Yamamoto, T., Y., Ikeda, T., Yamada, M., "Subharmonic and Summed-and-Differential Harmonic Oscillations of an Unsymmetrical Rotor," Bulletin of the JSME 24, 187, 192-199, 1981,
16 Zajaczkowski, J., "An approximate method of analysis of parametric vibration," J. of Sound and Vibration 79, 4, 581-588, 1981 .
17 Zajaczkowski, J., Lipinski, J,, "Instability of the motion of a beam of periodically varying length," J. of Sound and Vibration 63, 1, 9-18, 1979,
18 Takahashi, K,, "An approach to investigate the instability of the raultiple-degree-of freedom parametric dynamic systems," J. of Sound and Vibration 78, 4, 519-529, 1981,
19 Szemplinska-Stupnicka, W,, "The generalized harmonic balanace method for determining the combination resonance in the parametric dynamic systems," J. of Sound and Vibration 58, 3, 347-361, 1978,
20 Hsu, C , "On the parametric excitation of dynamic systems having multiple degrees of freedom," J, of Applied Mechanics 30, 363-372, 1963,
21 Stanisic, M.M., "On the stability of generalized Hill's equation with three independent parameters," International J, of Non-linear Mechanics 15, 6, 485-496, 1980.
22 Sato, K,, Siato, H,, et al, "The parametric response of a horizontal beam carrying a concentrated mass under gravity," J, of Applied Mechanics, Transactions of the ASME 45, 3, 643-648, 1978.
23 Hsu, C,S,, "On nonlinear parametric excitation problems," Advances in Applied Mechanics 17, 245-301, 1977,
24 Minrosky, N,, Nonlinear Oscillations, Van Nostrand, 1962,
67
25
26
27
Ibrahim, R.A., Barr, A.D.S,, "Autoparametric resonance in a structure containing a liquid. Part ll: Three mode inter-action," J, of Sound and Vibration 42, 2, 181-200, 1975,
Haxton, R.S,, Barr, A,D,S., "The autoparametric vibration absorber," J, of Engineering for Industry, Transactions of the ASME 94, 1 , 119-125, 1972, ~ ~
Breitenberger, E,, Mueller, R,D,, "The elastic pendulum: A nonlinear paradigm," J, of Mathematical Physics 22, 6, 1196-1210, 1981, "~"
28 Hatwal, H., Mallik, A.K., Ghosh, A., "Forced nonlinear oscillations of an autoparametric system - Part 1: Chaotic Responses," J. of Applied Mechanics 50, 3, 657-662, 1983,
29 Hatwal, H,, Mallik, A.K,, Ghosh, A,, "Nonlinear vibrations of a harmonically excited autoparametric system,' J, of Sound and Vibration 81, 2, 153-164, 1982,
30 Hatwal, H., Mallik, A.K., Ghosh, A., "Forced nonlinear oscillations of an autoparametric system - Part 2: Chaotic Responses," J. of Applied Mechanics 50, 3, 663-668, 1983.
31 Moon, F.C., Holmes, P.J., "A magnetoelastic strange attractor," J. of Sound and Vibration 65, 2, 275-296, 1979.
32 Moon, F.C., "Experiments on chaotic motions of a forced nonlinear oscillator: Strange attractors," ASME J. of Applied Mechanics 47, 638-644, 1980,
3 3 Nayfeh, A,H,, "Combination resonances in the nonlinear response of bowed structures to a harmonic excitation," J, of Sound and Vibration 9£, 4, 457-470, 1983,
34 Singh, Y,P,, "Quenching in a system of van der Pol oscillators with non-linear coupling," J, of Sound and Vibration 77, 4, 445-453, 1981 ,
35 Yamamoto, T,, Yasuda, K,, Nagasaka, I,, "On the internal resonance in a two-degree-of-freedom system (when the natural frequencies are in the ratio 2:3)," Bulletin of the JSME 22, 171, 1274-1283, 1979,
36 Yamamoto, T,, Yasuda, K,, Nagasaka, I., "On the internal resonance in a nonlinear two-degree-of-freedom system (forced vibrations near the higher point when the natural frequencies are in the ratio 1:2)," Bulletin of the JSME 20, 147, 1093-1100, 1977.
68
37 Yamamoto, T,, Yasuda, K,, Nagasaka, I,, "On the internal resonance in a nonlinear two-degree-of-freedom system (forced vibrations near the lower resonance point when the natural frequencies are in the ratio 1:2)," Bulletin of the JSME 20, 140, 168-175, 1977,
38 Ibrahim, R.A,, Barr, A.D.S,, "Autoparametric resonance in a structure containing a liquid. Part I: Two mode interaction," J, of Sound and Vibration 42, 2, 159-179, 1975,
39 Kovalchuk, P,S,, "Nonlinear resonances in mechanical self-oscillatory systems," Soviet Applied Mechanics 12, 10, 1040-1046, 1977,
40 Speckha r t , F ,H , , A Guide to Using CSMP - The Continuous Modeling Program, P r e n t i c e - H a l l , Englewood C l i f f s , N , J . , 1976,
APPENDIX A
KINETIC ENERGY DERIVATION
1) Position vectors of mi and m2 (See Figure 1,10 for reference)
Pi = ^ 1 + ^3) i + h^' i.
P2 = (q3 + qi + AV + q2 sin 0 + h2' cos 0) j_
- (Ah + q2 cos 0 - h2' sin 0) £
where
hi ' = 22i!.
51l
h2 ' = 3q22
5I2
Av = I2 (1 - cos 0)
Ah = I2 s i n 0 - 3q^2
51l
s i n 0 i 0 = 3qi
2I1
cos 0 = 1 - ^ + . . . = 1 - 9qi 2
2 81i2
2) V e l o c i t y Vectors
Vi = Pi = ^^1 + ^3) 1 + 6q^qi i
r ^2 = ?2 = ± - J 1 3 + ^1 + 3q^q2 + 9l2qi2 ±_J U2 - 1 - 3q^q2 +
d t L 2I1 SI , 2
69
70
^ q i ^ 2 + 2q iq2q2) > i
I O I 1 I 2
^3 - 4 l + _ ! . ( q i q 2 + q 2 q i ) + 9 I2 q^c^^ + 6q2q 2
2 I1 41^2 5^^
" 27 ( q i q i q 2 ^ + q i ^ q 2 ^ 2 ^ ( i
201i2i2 J
3) K i n e t i c Energy
T = 1; mi [V^2^^^^2 j^ j ^ 1 ^ m 2 [ V 2 2 j . + ^^^jj + ^/2m2^2^
S u b s t i t u t i n g the exp re s s ions for Vi and V2 i n t o T and r e t a i n i n g
up t o f o u r t h o rde r terms g i v e s :
T = 1/2[mi + m2(1 + 2 ,25 l22 / i ^ 2 j j ^ ^ 2 ^ ^/2m2Cf2^
+ 1/2(mi + ni2 + m3)q32 + 2 "*2 "'•2 * 1 ^2 •*" ^"'l "*• " '3^ '^ l^3
2 l l
+ ^ m2 ^2 ^ ^ 1 ^ 2 ^ + 5 q i q i q 3 )
20 l i 2
+ 3m2 ( 1 / 5 q i q i q 2 + ^ i ^ 2 ^ 3 " ^1^2^3 " ^1^q2^
21 1
71
+ 12 m2 ^<l2'^2^3 " ^1^2^2^
10 l 2
+ 9 4 (m^ + itij) + 9 ^2 I2 U 2
lll^ L25
^^'i^' 16 li^J
+ 9 nij (q iq2^2^ + qi^2^q2^
5 1 , 1 1- 2
APPENDIX B
COEFFICIENTS
L. ., and M. ., for equa t ions 2,14 ^ i j k i:]k ^
' i j k = I - 9 ^ + 1 1 1 n^n^^ + 3n. + ,3n^
+ 2 .25p^3 + LSn^p^ j
+ n i [ . 3 + I : : ! n . (1 + p^) + 1.5Pkj 3
+ p ( 2 . 2 5 3 + 1 . 5 ( n j + n ^ ) + l i i n ^ n ^ ^ j
M i l l = {'^^^ + l i 2 n3 2 + 3nA - 1 .2ni
+ P i f 2 . 2 5 3 + 3n^ + J j d ^ l ^ j
"^ijk = (-^^ "" 1 : 1 ^j"k - 3(n. + nj^)j
- 2 . 4 n i - P i ( 4 . 5 3 . 3(n . . n , ) . 1 ^ n -n^)
coefficients for Equations (4,5)
L l = cxr^32 [L^ i i - M^ill
2Mii
•23" '"^122 ^ = <xro,2 [ L , „ - M^22^
2Mii
L3 = Q^ SS" '^133
2Mii
= a r , , 2 [ L , . ^ - M^33^
72
73
2M11
L5 « 0^23^ ^^122 + ^ 2 2 ^
2Ml1
2M 11
r^-^rool L7 « g t ^ 1 2 l ' ' l 3 + Ll 12^23 - Ml 12^13^23
2Mii
L8 2 - Mi i - j r - i - j r , , ]
Lg -
g t L ^ B i ^ n ^ + Ll 13^33^ - Ml 13^131:33
2Mii
g [Li32^23^ + Li23^33^ " ^i 23^23^33^
2Mii
rool L i e = « t L l j i ' n ^ + Ll 12^23^ + Ml 12^13^23
2Mi i
L11 = a t L i 3 i r ^ 3 ^ + Li13^33 r-,^2 + Ml 13^13^33]
2Mii
L12 a tLi32r232 + Li 23^33^ + Mi 23^^23^33^
Mi
M'
M3
2Mii
^ 2 3 ^ ^^211 - ^211
2M22
gr 2 [L.,,0 - M222J 23 ^""222
2M22
= (xr 23^ ^^233 ^23 3^
2M22
74
M4
Ms
M,
M-
°^23^ ^^211 + ^^^^
2M 22
M, 8
°°^23^ fL222 + ^222]
2M22
^ 2 3 ^ [L233 + M233]
2M22
- 2 _ ^ ^ 2 1 ^ 1 3 ^ + L2i2 '^23^ - M2i2J^i3^23^
2M22
- 2 L - t L 2 3 i r ^ 3 2 + L 2 i 3 ^ 3 3 ^ " M 2 i 3 r i 3 r 3 3 ]
2M22
Mg = _ a _ [ L . 3 , r „ 2 ^ 232 ' '23 + L223^33 - M223^23'^33^
2M 22
M 10
M 11
M 12
Ni
-^L-_ fL22 i r^32 + L212^23^ + M212^13^23^
2M22
-5L— fL23 i r^32 + L 2 i 3 r 3 3 2 + M2i3 r i3 r23 ]
2M22
- ^ i — ^^232^23^ + L223^33^ + M223^23^33^
2M22
^ 3 3 ^ f ^ l l - M311]
N-
N3
2M33
°^33^ ^ ^ 2 2
2M 33
°^33^ f^333
^322^
^333^
2M33
75
N4 =
N5
N,
°^^33^ f ^ 3 l 1 + ^ 3 ^ ^ ]
2M33
^ ^ 3 3 ' tL322 H- M322]
2M33
°^33^ f^333 + M333]
2M
N 7 =
N8 =
33
_ o c _ [ L 3 2 i r i 3 2 + L 3 i 2 r 2 3 2 - M 3 i 2 r i 3 r 2 3 ]
2M33
_ 2 L - fL33 i r^32 + L 3 i 3 r 3 3 2 . M3i3r^3r33]
2M33
Ng = _ a _ _ [ L . . , r , ^ 2 ^ 332^23 + L323r33 " M323^23^33^
2M 33
N 10
N 11
N 12
- 2 — fL32 i r 32 + L312^232 + M3i2 r i3 r23 ]
2M33
_ a _ _ [ L 3 3 ^ r , 3 2 + L 3 i 3 r 3 3 2 + M3i3r^3r33]
2M33
- 2 — ^^332 ' '23^ + L323^33^ + M323^23^33^
2M33
APPENDIX C
CSMP PROGRAM LISTINGS
CSMP Program L i s t i n g f o r s i m u l a t i o n of e q u a t i o n s ( 4 . 1 1 )
/ / JOB CWP$TTT,3259tt30, l5) ,»W000ALL« //CSMP EXEC CSMP PARAMETER Rl« .4^96553 ,R2« .5503A488 PARAMETER GAM « ( - 3 f - 2 . 8 , - 2 . 7 , - 2 . 6 , - 2 , - l , 0 , 1 , 2 , 2 . 6 , 2 . 7, 2 . 8 . 3) PARAMETER L 5 l « - . 0 1 6 2 7 6 1 8 1 2 , L 5 2 « - . 0 0 7 4 9 8 3 3 1 1 , L 5 3 = - , 0 0 6 0 0 2 7 2 7 1 PARAMETER NNl = . 1 0 0 0 0 0 , NN2 « . 1 0 0 0 0 0 , NN3 =*200000 INCON AlO « O.OIOO, A20 » O.OIOO, A30 » O.OIOO INCON PIO - 0 . 0 0 0 0 , P20 > 0 . 0 0 0 0 , P30 » 0.0000 THTA s PI • P2 - P3 A l 3 INTGRL(A10,A10) A2 " INrGRLCA20,A20) A3 = INTGRL(A30,A3DJ P I « INTGRL(P10,P1D) ?2 * INTGRL(P20,P2D) P3 » INTGRL«P30,P30J
AID = - f N N l * A l / R l • A2*A3«L51*SIN(THTA) / (ABS(L53) *RU} A20 = - (NN2*A2 /R2 • A1*A3*L52*SINiTHTA) / (ABStL53) • R 2 n A3D = - (NN3*A3 • S IN(P3) - A l *A2* l .53*SIN( THTA) / I ABS (L 53) IJ PIO = - ( -R1*GAM • A2*A3*L51*COS(THIA) / (A l *R l *ABS(L53 ) ) ) P20 = - l -R2»GAM • A1*A3*L52*C0S(THTA)/(A2*R2»ABS(L53))J P3D = - ( -GAM *• C0S(P3) /A3 • Al*A2*L53*CUS(THrA ) / ( A3*ABS ( L53) ) ) *
TIMER OELT "0.0010, FINTIN « 50.00, OUTOEL ».2000 PRTPLOT AllAl) PRTPLOT A2(A2) PRTPLOT A3(A3) ENU STOP ENOJOB //
76
CSMP Program L i s t i n g for s imulat ion of equations ( 2 . 1 5 ) .
77
JOB CWPSTTT,3259,,30,15J,«WCQ0ALL» EXEC CSMP
. 4 4 9 6 5 5 3 , R2 » .5503448f 1 9 . 7 9 9 9 9 2 9 , EP « . 0 1 1 . 9 3 1 2 0 5 9 , M2 » 4 . 1 9 1 9 2 0 6 , • 6 2 8 8 3 1 6 , B » .5489877 - . 0 2 7 6 1 8 5 , N2 « - . 0 8 6 3 5 6 2 , N3 - . 0 3 2 5 8 4 7 , 0 2 =» . 3 4 0 0 7 2 7 , 03 *
I! II PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER INCON PIO INCON PDIO
Rl « EX = Ml « AL » Nl « QI « ETl L I U « M i l l = L121 = M121 = L131 = M131 = L211 = M211 = L221 = M221 = L231 = M231 = L311 = M311 = L321 = M32i = L331 = M331 = = . 0 1 0 ,
= 0 . 0 0 ,
0 . 0 5 , ET2 . 3 2 2 7 8 9 , • 1 6 1 3 9 5 , . 1 5 9 1 7 9 , . 0 7 9 5 8 9 , 4 . 4 1 9 9 4 9 , 2 .209975 . 7 4 1 5 2 9 , . 6 6 1 9 3 9 , . 5 5 3 7 0 2 , . 5 4 8 6 2 2 , 5 .445121 3 .499651 . 7 4 8 9 3 6 ,
= 0.05, L112 =« M112 = L122 ' M122 = L132 «
, M132 = L212 a M212 » L222 » M222 « , L232 = 9 M232 « L312 =
0 .50 L113 M113 L123 M123
ET3 « ( . 7 4 1 5 2 9 , . 0 7 9 5 8 9 , . 5 5 3 7 0 2 , • 0 0 5 0 3 0 ,
5 . 4 4 5 1 2 1 , Ll 1 .945470 , Ml
1 .097244 , L21 . 5 4 8 6 2 2 , H213 . 8 9 0 8 2 1 , L223 . 4 4 5 4 1 1 , M223
6 . 2 6 6 5 1 4 , L2 3 . 1 3 3 2 5 7 , M2
1 .946354 , L31
GAMl = - ( P1*R1**2 ) • ( - ( P2*R2**2 ) • ( - ( P3 ) • (
M l l l * P D l * * 2 + ¥
- 1 . 4 6 1 0 3 8 , M312 =* - 1 . 5 5 3 2 9 7 , • 392172 , L322 = U 4 9 5 9 2 8 , L32 - 1 ^ 5 5 3 2 9 7 , M322 « - 1 ^ 6 3 7 3 2 8 , 9 . 6 8 3 0 8 1 , L332 « 1 3 . 2 2 5 9 9 1 , L . 8 4 9 3 3 4 , M332 = . 5 5 1 0 2 7 , M333
P20 a . 0 1 0 , P30 * . 0 1 0 P020 ' 0 . 0 0 , PD30 = 0 .00
L111*P1 + L121«P2 *• LI L112*P1 • L122*P2 • L l L113*P1 • L123»P2 + Ll
M122*PD2**2 > M133*P03*
W3 » 1.6207
M3 s 5.2363615
1.4433017 .0910431 , . 2 5 , . 7 5 ) =» .748936 = 2.2099 75 « .392172 = 1.9454 70
33 = 9.683081 33 ' 8.833748 3 ' 1.946354
= 3.4996 51 = 1.495928 ^ 3.133257
33 = 13.225991 33 = 12.674964 3 = 1.698666 M313 « .849334 3 ^ 1.102054 M323 =« .551027 333 ' 16.639141
= 8.319571
( (
CAM2 = - ( - ( - ( •f • f
•••
M112 *• M121 )*P01*PD2 M123 > M132 ) *P02*P03 P1*R1**2 ) • ( L211*P1 • L221*P2 P2*R2**2 ) • ( L212*P1 • L222*P2 P3 ) • ( L213*P1 • L223*P2
M 2 1 l * P 0 1 * * 2 • M222«PD2**2 • ( M212 * M221 ) *PDl*PD2 • ( ( M223 • M232 J*PD1*P03
3 i *P3 » . . . 32*P3 ) . . . 33*P3 ) . . . • 2 . . .
• ( M113 *• M131 )*P01*P03
> L2 ^ L2 • L2
M233*PD3* M213 * M2
3 l * P 3 ) . . . 32*P3 ) . . . 33«P3 } . . . •2 . . . 31 J*PD1*PD3
78
GAM3 « - ( P1*R1*«2 ) • ( L 3 l l * P l f L321*P2 • L331«P3 I . . . - ( P2«R2^*2 ) • ( L312*P1 • L322*P2 • L332*P3 ) . . . - ( P3 ) • ( L313*P1 • L323*P2 *• L333*P3 ) . . . • M311*PDl**2 • M322*P02**2 • M333*PD3**2 . . . • ( M312 «- M321 )*PDl*PD2 • C M313 • M331 )*PD1*P03 . *• [ M323 «• M332 )*P02*PD3
P I « INTGRL(P10,PD1) P2 « INTGRL(P20,P02) P3 = INTGRHP30,P03) PDl = INTGRL(PD10,PDD1) P02 = INTGRL(P020,P002) P03 * INTGRL(P030,P0D3) F l a EX*QI / (M1*W3**2 ) F2 « EX*Q2/(M2*W3**2) F3 « EX*Q3/(M3*W3**2) PODl = - P 1 * R 1 * * 2 • F1*C0S(TIME) - EP*AL*CAM1/M1 - 2«ET1*P01 PDD2 ' - P 2 * R 2 * * 2 f F2*C0S(TIME) - EP*AL*GAM2/M2 - 2*ET2*P07 PD03 = - P 3 • F3«C0S(TIME) - EP*AL*GAM3/M3 - 2*ET3*P03 TIMER OELT = . 0 0 0 1 , FINTIM = 5 0 . 0 , QUTDEL = .250 PRTPLOT P l ( P l ) PRTPLOT P2(P2) PRTPLOT P3(P3) END STOP ENDJOB