Structural dynamics and vibrations of damped, aircraft-type structures

127 Pages·2006·8.45 MB·English

NASA Contractor Report 4424
Structural Dynamics and Vibrations
of Damped, Aircraft-Type Structures
Maurice I. Young
ViG y m ,I nc.
Hampton, Virginia
Prepared for
Langley Research Center
under Contract NASl-18585
National Aeronautics and
Space Administration
Off ice of Management
Scientific and Technical
Information Program
1992

TABLE OF CONTENTS
-Page
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.0 SUMMARY.. 1
. . . . . . . . . . . . . . . . . . . . . . . . .
2.0 NOMENCLATURE 2
,
. . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Notation.. 2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Symbols 4
. . . . . . . .
3.0 INTRODUCTION 6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.0 ANALYSIS 8
. . . . . . .
4.1 The Single Degree of Freedom Oscillator With Damping . 8
4.2 Free Vibration of Systems With Viscous or Equivalent Viscous
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Damping 13
4.3 Free Vibration With Two Different Modes Having the Same Natural
. . . . . . . . . . . . . . . . . . . . . . . . .
Frequency , 16
. .
4.4 Employing the Undamped Modes to Determine the Damped Modes 19
. . . . . . . .
4.5 Forced Vibration of Systems With Viscous Damping 21
. . .
4.5.1 Approximating the Response of Lightly Damped Systems 26
4.6 Free Vibration of Linear Systems With Dynamic Hysteresis-Viscoelastic
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Damping 27
. . . . . . . . .
4.7 Pervasive Dynamic Hyster&sViscoel&ic Damping 31
4.8 Forced Vibration of Systems With Dynamic Hysteresis . . . . . . . . 32
4.9 Derivative Operator Formulation For Systems With Dynamic
Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . 34
, ,
4.10 The Global Equations With Proportional Damping . . . . . . . - . 35
a
TABLE OF CONTENTS
(continued)
Page
5.0 APPROXIMATING THE DAMPING FRACTION FOR CONTINUOUS
SYSTEMS . . . . . . . . . . . . . . . . . - . . . . . . . . . . . 37
5.1 The Damped Rod In Axial Vibration . . . . . . . - . . . . - . . 37
5.2 Damped Structural Members Other Than Rods . . . . . . . . . . . 41
5.3 The Pervasively Damped HomogeneousIsotropic Solid . . . . . . . 43
e
6.0 SELECTED NUMERICAL RESULTS . . . . . . . . . . . . . . . . 46
a
6.1 Damping Interaction of Different Modes With Matching or Nearly
Matching Natural Frequencies . . . . . . . . . . . . . . . 46
e
6.2 Damping Raction Approximation With a Dynamic Hysteretic
Element . . . . . . . . . . . . . . . . . . . . . . . . . - . 47
,
6.3 Examples of Damped Beam Vibration . . . . . . . . . . . . . 48
6.4 Bending Vibration of a Discretely Modelled Damped Beam . . . . . 51
,
6.5 Coupled Bending and Torsion Vibrations of a Discretely
Modelled Damped Beam . . . . . . . . . . . . . . . . . . . . . 53
6.6 Vibration of a Plate With Spot Damping and Different Modes
With Matchingor Nearly Matching Natural Frequencies . . . . . . - 54
7.0 A SMART DYNAMIC VIBRATION ABSORBER: A COLLATERAL
DAMPING APPLICATIONS TECHNOLOGY . . - . . . . . . . . . . . 57
8.0 DYNAMIC STABILITY BOUNDARIES FOR BINARY SYSTEMS . . . . 59
9.0 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . 62
,
TABLE OF CONTENTS
(concluded)
Page
10.0 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
FIGURES.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
APPENDIX A: CLOSED FORM SOLUTION TO THE DAMPING
INTERACTION QUARTIC EQUATION . . . . . . . 77
APPENDIX B: AN APPROXIMATIVE SOLUTION TO THE DAMPING
INTERACTION QUARTIC EQUATION . . . . . . . . 80
APPENDIX C: DIGITAL COMPUTER SOLUTION TABULATION
FOR DAMPED BENDING VIBRATION . . . . . . . . 83
APPENDIX D: DIGITAL COMPUTER SOLUTION TABULATION
FOR DAMPED, COUPLED BENDINGTORSION
VIBRATION.. . . . . . . . . . . . . . . . . . . . 93
V
1.0 SUMMARY
Engineering preliminary design methods for approximating and predicting the effects of
viscous or equivalent viscous-type damping treatments on the free and forced vibration of
lightly damped aircraft-type structures are developed. Similar developments are presented
for dynamic hysteresis-viscoelwtic-type damping treatments. It is shown by both engineering
analysis and numerical illustrations that the intermodal coupling of the undamped modes
arising from the introduction of damping may be neglected in applying these preliminary
design methods, except when dissimilar modes of these lightly damped, complex aircraft-type
structures have identical or nearly identical natural frequencies. In such cases it is shown
that a relatively simple, additional interaction calculation between pairs of modes exhibiting
this “modal resonance” phenomenon suffices in the prediction of interacting modal damping
fractions. The accuracy of the methods is shown to be very good to excellent, depending on
the normal naturd frequency separation of the system modes, thereby permitting a relatively
simple preliminary design approach. This approach is shown to be a natural precursor
to elaborate finite element, digital computer design computations in evaluating the type,
quantity and location of damping treatments. It is expected that in many instances these
simplified computations will supplant the more elaborate ones.
2.0 NOMENCLATURE
2.1 Notation
A cross-sectional area of rod, in2; an arbitrary constant
a coefficient in interaction quartic equation; membrane dimension, inch
B coefficient in resolvent cubic equation; an arbitrary constant
b coefficient in interaction quartic equation; membrane dimension, inch
C coefficient in resolvent cubic equation; damping coefficient, lb-sec/in
coefficient in interaction quartic equation; damping coefficient ,
C
lb-sec/in
D plate flexural rigidity, bin2; coefficient in resolvent cubic equation
d coefficient in interaction quartic equation; damping coefficient
lb-sec/in
E Young’s modulus of elasticity, lb/in2
F function symbol; force
f force, lb
G shear modulus of elasticity, lb/in2
I area moment of inertia, in4; mass moment of inertia, Ib-sec2-in
i ordinal number; a subscript
J torsional section constant, in4
f l
j ordinal number; complex operator,
K stiffness, spring rate, lb/in
k stiffness, spring rate, lb/in
2
M mass, 1b-sec2/in
m mass, lb-sec2/in; an ordinal number; a subscript
N an ordinal number; aspect ratio
n an ordinal number; a subscript
naught, a subscript
0
P generalized coordinate
Q a response quantity; a generalized force
Q a generalized coordinate
R a response quantity; a parameter
r an ordinal number; a subscript; a ratio
S an ordinal number; a subscript
T membrane tension, lb/in
t time, sec
a displacement, inch
U
21 a displacement, inch
W a displacement, inch
II: a Cartesian coordinate, inch
Y a Cartesian coordinate, inch
z a Cartesian coordinate, inch
a! an ordinal number; a subscript; a constant
P an ordinal number; a subscript; a constant
Y a parameter
3
a
dilatation
s
a damping coefficient,l b-sec/in; perturbation symbol
c
fraction of critical damping
77 loss factor
e
torsional displacement, radian
11, char act eristi c number
P mass per unit length, lb-sec2/in2; mass per unit area, lb-sec2/in3
v frequency ratio
axial coordinate position
mass density, lb-sec2/in4; frequency ratio
damping per unit length, lb-sec/in2
f2 forcing frequency, rad/sec
frequency, rad/sec
W
2.2 Symbols
dot, differentiation with respect to time
-
bar, amplitude of
*
asterisk, complex conjugate
V2 del squared, the Laplacian operator
-# arrow, vector quantity
tilde, a modified quantity
J ( 1 integral, the integral of ( )
I I the magnitude of, the determinant of
4