Table Of ContentSTUDIES IN APPLIED MECHANICS
4. Variational, Incremental and Energy Methods in Solid Mechanics and Shell Theory
(Mason)
5. Mechanics of Structured Media, Parts A and Β (Selvadurai, Editor)
6. Mechanics of Material Behavior (Dvorak and Shield, Editors)
7. Mechanics of Granular Materials: New Models and Constitutive Relations (Jenkins and
Satake, Editors)
8. Probabilistic Approach to Mechanisms (Sandler)
9. Methods of Functional Analysis for Application in Solid Mechanics (Mason)
10. Boundary Integral Equation Methods in Eigenvalue Problems of Elastodynamics and Thin
Plates (Kitahara)
11. Mechanics of Material Interfaces (Selvadurai and Voyiadjis, Editors)
12. Local Effects in the Analysis of Structures (Ladevèze , Editor)
13. Ordinary Differential Equations (Kurzweil)
14. Random Vibration - Status and Recent Developments. The Stephen Harry Crandall
Festschrift (Elishakoff and Lyon, Editors)
15. Computational Methods for Predicting Material Processing Defects (Predeleanu, Editor)
16. Developments inE ngineering Mechanics (Selvadurai, Editor)
17. The Mechanics ofV ibrations ofC ylindrical Shells (Markus)
18. Theory of Plasticity and Limit Design of Plates (Sobotka)
19. Buckling of Structures-Theory and Experiment. The Josef Singer Anniversary Volume
(Elishakoff, Babcock,A rbocz and Libai, Editors)
20. Micromechanics of Granular Materials (Satake andJ enkins, Editors)
21. Plasticity .T heory and Engineering Applications (Kaliszky)
22. Stability int he Dynamics of Metal Cutting (Chiriacescu)
23. StressA nalysis by Boundary Element Methods (Balas,S ladek and Slâdek)
24. Advances int heT heory of Plates and Shells (Voyiadjis and Karamanlidis, Editors)
25. Convex Models of Uncertainty inA pplied Mechanics (Ben-Haim and Elishakoff)
26. Strength of Structural Elements (Zyczkowski, Editor)
27. Mechanics (Skalmierski)
28. Foundations of Mechanics (Zorski, Editor)
29. Mechanics ofC omposite Materials- A Unified Micromechanical Approach (Aboudi)
30. Vibrations andW aves (Kaliski)
31. Advance s in Micromechanics of Granular Materials (Shen,S atake, Mehrabadi,C hang and
Campbell, Editors)
32. Ne w Advances inC omputational Structural Mechanics (Ladevèze andZ ienkiewicz, Editors)
33. Numerica l Methods for Problems in Infinite Domains (Givoli)
34. Damag e inC omposite Materials (Voyiadjis, Editor)
35. Mechanics of Materials and Structures (Voyiadjis, Banka ndJ acobs, Editors)
36. Advanced Theories of Hypoid Gears (Wang and Ghosh)
37A. Constitutive Equations for Engineering Materials
Volume 1: Elasticity and Modeling (Chen and Saleeb)
37B. Constitutive Equations for Engineering Materials
Volume 2:P lasticity and Modeling (Chen)
38. Problems ofT echnological Plasticity (Druyanov and Nepershin)
39. Probabilistic and Convex Modelling ofA coustically Excited
Structures (Elishakoff, Lina nd Zhu)
General Advisory Editort ot his Series:
Professor Isaac Elishakoff, Centerfor Applied Stochastics Research,D epartment of Mechanical
Engineering, Florida Atlantic University, Boca Raton,F L, U.S.A.
STUDIES IN APPLIED MECHANICS 39
Probabilistic and Convex
Modelling of Acoustically
Excited Structures
I. Elishakoff
Y.K. Lin
L.P. Zhu
Center for Applied Stochastics Research
Florida Atlantic University
Boca Raton, FL, USA
ELSEVIER
Amsterdam - Lausanne - New York- Oxford - Shannon -Tokyo 1994
ELSEVIER SCIENCE B.V.
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ν
Preface
This monograph summarizes the analytical techniques for predicting the response
of linear structures to noise excitations generated by large propulsion power plants.
Emphasis is placed on beams and plates of both single-span and multi-span
configurations, common in engineering structural systems. These techniques are
developed under the premises that the acoustic pressure in such a noise field changes
randomly in space and time, and that the supporting conditions around a structure cannot
be precisely determined. The first premise dictates that the random vibration theory be
used as a basis of the analysis, and the second premise requires that the effects be
evaluated of various possible boundary conditions of the structure. For a linear system,
the required inputs in a random vibration analysis are the spectral densities and the cross-
spectral densities of the acoustic field, which must be inferred from on-site measurements.
Since the available data of such measurements are often limited to permit a traditional
statistical analysis, a new method, called convex uncertainty modeling is applied to obtain
estimates of the upper and lower bounds of the mean-square response. From a structural
safety point of view, the least favorable upper bound should be used in design.
The monograph is divided into eight chapters and six appendices. Since the
natural frequencies and the associated normal modes play a central role in the random
vibration analysis of a continuous dynamical system, rather detailed discussions are
devoted to their determination. The first two chapters deal, respectively, with free
vibrations of single-span and multi-span beams. Although the materials in the first
chapter are elementary and are available in textbooks, they provide a useful reference for
the subsequent discussion of multi-span structures. Free vibrations of rectangular plates,
amendable to exact solutions, are considered in Chapter 3, including single-span plates
and multi-span plates. The interior supports of a multi-span plate are uniformly spaced,
and they can be either simple supports or elastic stiffeners. Free vibrations of rectangular
plates, not amenable to exact solutions, are treated in Chapter 4. Use is made of
Bolotin's method of dynamic edge effect, as well as a generalized version which is
applicable even when the original method is not. This generlized version is applied to
orthotopic rectangular plates with arbitrary boundary conditions, and to multi-span plates
with intermediate, uniformly spaced stiffeners.
Random vibration theory is reviewed in Chapter 5, for both discrete and
continuous structures. Applications of the random vibration theory to single-span and
multi-span beams under acoustic pressure excitation are treated in Chapter 6. The
acoustic loading model is based on physical reasoning as well as available measurements.
The probabilistic properties of the structural response are evaluated using either the
classical random vibration theory or an alternative scheme, which is more efficient for
certain structural configurations. New expressions are obtained for the normal modes of
multi-span structures, which require much less computational time when applied to either
deterministic or random vibration analysis of such structures. Chapter 7 is devoted to the
development of an improved finite element formulation for random vibration analysis.
Systematic procedures for obtaining the spectral matrices of both the excitation and the
response are presented in both the element-based and global frameworks. The element-
based spectral densities of the excitation, obtained from two approximation schemes, are
compared with the results of direct numerical evaluation. A benchmark example is given,
vi
which can also be solved exactly, and the results are useful for verifying commercially
available computer programs. Finally, a hybrid probabilistic and convex-uncertainty
modeling approach is presented in Chapter 8, in which the upper and lower bounds of the
cross-spectral densities of the acoustic excitation are obtained on the basis of measured
data, and the random vibration of a structure is treated, for the first time, as an "anti-
optimization" problem of finding the least favorable value of the mean-square response.
It is a pleasure to acknowledge our appreciation to the Kennedy Space Center for
the financial support, making the study reported in this monograph possible, and for
permission to publish the results. Our special thanks are due to Dr. Gary Lin, Mr. Raoul
Caimi, Dr. Valentine Sepcenko and Mr. Ravi N. Margasahayam for the encouragement
and numerous technical discussions during the course of the study. We are indebted to
our colleague, Dr. G. Q. Cai, who commented on the manuscript.
/. Elishakoff, Υ. Κ Lin and L. P. Zhu
Boca Raton
May, 1994
1
Chapter 1
Free Vibration of Single-Span Beams
A brief review of free vibration of uniform single-span beams is presented. Although the material is
elementary, and available in numerous textbooks, it provides a useful reference for the subsequent discussion
of the periodic beams. The presentation is made concise by use of the Krylov functions.
1.1 Basic Equations and Method of Solution
The differential equation governing small vibrations of a uniform Beraoulli-Euler beam
is given by
F T3 4 Α +Γ 2 ο = (i.i)
dx dt
where Ε = Young's modulus, J = moment of inertia, ρ = material density, A = cross-
section area, w(x,t) = transverse displacement, χ = spatial (longitudinal) coordinate, t =
time. We are interested in determining the spectrum of natural frequencies and the
associated vibration modes.
The solution of Eq. (1.1) is sought, by way of separation of variables, as follows
L2
>v(jc,i) = W(x)1{t) ()
where W(x) is a function of the axial coordinate, and J{t) is a function of time. The
function W(x) describes a mode shape of vibration.
Substitution of Eq. (1.2) into Eq. (1.1) yields
1 3
EIW«\x)T(t) + pAW(x)T{t) = 0 ( >
where the superscript (4) denotes the 4th derivative with respect to x, and each dot
denotes one differentiation with respect to time t. Eq. (1.3) can be re-written in the form
EIW«\x) _ 7(0 (1.4)
=
pAW(x) TXt)
Since the left-hand side depends only o2n JC, and th2e right-hand side only on i, these ratios
are equal to a constant, denoted by ω , where ω is to be determined.
Thus, instead of Equation (1.1) we obtain the following two equations
2
2 L 5
7(ί) + ω71(0 - Ο ( )
and
2 L 6
EIW^Xx) - ρΑωΨ(χ) = Ο ( )
Solution of Eq. (1.5) reads
L 7
7(0 «ZV*' •TV™' ( )
It is seen that the constant ω in equations (1.5) and (1.6) is the natural frequency of
vibration, and the constants of integration D and D, which are complex conjugates, may
x 2
be determined from the initial conditions set at t = 0. The natural frequency ω is
obtained from Eq. (1.6) in conjunction with boundary conditions at the two ends of the
beam, χ = 0 and χ = L.
We introduce a normalized frequency parameter
4 ω ρ Α
α = (1.8)
EI
so that Eq. (1.6) is simplified to
4
4
É^L - aW = 0 (1.9)
dx
We seek the solution W(x) in the form, for α * 0,
n L1
W(x) =Ce ( °)
where C is a constant, and r is the characteristic exponent to be determined. Substitution
of Eq. (1.10) into Eq. (1.9) results in the characteristic equation
4 4
r - et = 0 (111)
with roots
r = a, r = -a, r = ia, r = -ia (1.12)
l 2 3 4
Hence solution of Eq. (1.9) may be written in the following form
W(x) = C^e™ + C^-™ + C^™ + C^™ (1-13)
or
W(x) = ^sin^cut) + £ cosh(ajc) + 5sin(ajc) + Bcos(ax) (1.14)
2 3 4
The constants of integration Cj's and Bj's are related as follows
3
(1.15)
m 0
O1b0viously Eq. (1.9) is also satisfied by other linear combinations of functions e , e*,
e *, and β""", and the following set of independent combinations is often the most
convenient:
K^ax) = λ. [ cosh(ax) + cos (ax)]
K{ox) = Σ. [ sinh(ax) + sin(ax)]
2
(1.16)
K(ax) = 2. [ cosh(ax) - cos (ax)]
3
K(ox) = A [ sinh(ax) - sin (ax)]
4
These functions are called the Krylov functions and they have often been used in various
contexts, in the study of beam vibration (see e.g., Fillipov 1970, Lin and Donaldson
1969).
The Krylov functions Kfx) possess the following properties
K!(x) = K(x) , K(x) - tf(x)
H 0 4
KJ® = 1 , K(0) = K(0) = K(0) = 0
2 3 4
^'(0) = 1 , K^O) = K '(0) = KX0) = 0 (1.17)
3 4
*r"(0) = ι, ^"(0) = κ "(ο) = κ'χο) = ο
3 2 4
Κ'"(0) = 1 , ΑΓ/'ΧΟ) = ΐς"'(0) = Κ'"(0) = ο
4 3
where each prime denotes one differentiation with respect to the argument x. The
properties given in Eq. (1.17) allow for simple and straightforward construction of the
frequency determinants for beams with different boundary conditions.
Now, let the solution of Eq. (1.9) be written in the form
W(x) = A^ax) + A^ax) + AK(ax) + AK(ax) (118)
33 44
where the Aj's are constants of integration. It follows upon substituting χ = 0 in Eq.
(1.18) and making use of Eqs. (1.17)
4
A = W(0) (1.19)
1
which provides an interpretation for the constant A as the displacement at the left end of
t
the beam. Differentiating once Eq. (1.18) and noting again Eq. (1.17), we arrive at
A = AW(0) (1.20)
2 α
which shows that A is proportional to the slope of the beam at the left end.
2
Similar interpretations for the other constants A and A are possible by noting the
3 4
following expressions for the amplitudes of the bending moment and the shear force for
a uniform beam undergoing harmonic oscillation
d2W d3W
2 3
M(x) =EI W , V(x) =EI & (1.21)
dx dx
Specifically, successively differentiating expression (1.18) for W(x) and invoking Eq.
(1.17), we obtain
^ = - ± ^ ( 0 ), A = _L V(0) (1.22)
4
1.2 Boundary Conditions
Possible boundary conditions at the end of a beam are specified as follows. For a
clamped end we have:
L 2 3
W = 0, W = 0 ( )
which imply vanishing displacement and slope. For a simply supported end
L 2 4
W = 0, M = 0 ( )
which signify vanishing displacement and bending moment. For a free end
L 2 5
Μ = 0, V = 0 ( )
which imply vanishing bending moment and shearing force.
For an elastically supported left end, χ = 0,
ΕΙΨ" = -£,W
(1.26)
EIW = βΨ
2
where fa and β represent the transverse and rotational stiffness constants, respectively.
2
5
For an elastically supported right end, χ = L, where L is the length of the beam, we have
the following boundary conditions
EIW" = faW
(1.27)
EIW = -p>W
4
13 Determination of Natural Frequencies and Mode Shapes
Consider first a beam simply supported at both ends. In order that the boundary
conditions at χ = 0 are satisfied, we have, from Eqs. (1.19) and (1.24), A = A = 0.
l 3
Thus, for α * 0:
W(x) = AK(ax) AK(ax) (1.28)
22 4+4
Imposition of the boundary conditions at χ = L yields two homogeneous equations for the
two unknowns A and A
2 4
AJW.) + Α,Κ/λ) = 0
(1.29)
where
λ = aL (1.30)
and where the properties (1.17) of functions K have been used in obtaining the second
}
equation in (1.29). For nontrivial solutions, the determinant of the coefficients of A and
2
A in (1.29) must vanish, i.e.
4
Κ(λ) κ(λ)\
2 Α
(1.31)
ο
κ(λ) Κ(λ)
Α 2
or
Κ\Κ) - Κ\λ) = ο (1.32)
2 4
Substituting in the above equation the expression for K and K, given in Eq. (1.16), we
2 A
arrive at
sinh(X) sin(X) = 0 (1.33)
The possibility of
