Harris C.M., Piersol A.G. Harris Shock and vibration handbook

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

= [Φ

sys

]{q

sys

} (28.122)

General Method 2: Attachment of “Appendage” Substructures to a Main

Body. This method of component mode synthesis differs from General Method 1

in that all components are not considered appendages. A simple way to view this

approach is to first follow General Method 1 for all appendage substructures up to

Eq. (28.121). The boundary collector degrees-of-freedom, in this case, correspond to

those associated with a main body, which is described in terms of main body mass

and stiffness matrices [M

] and [K

], respectively. The assembled system of

appendages and main body are described as





(28.123)

where the boundary-loaded main body mass and stiffness matrices are

[M′

] = [M′

] + [M

], [K′

] = [K′

] + [K

] (28.124)

The truncated set of modes associated with the boundary-loaded main body define

the intermediate transformation





(28.125)

Application of the above transformation to Eq. (28.124) results in the following

modal equations for the system













(28.126)

If the appendages are all of the Craig-Bampton type, the above equation set reduces

to the following Benfield-Hruda

form



 



 



(28.127)

The mass coupling terms (P

, etc.) are modal participation factor matrices, which

indicate the relative level of excitation delivered to the appendages by main body

modal accelerations. This feature of the Benfield-Hruda form is the primary reason

for the enduring popularity of the method. Uncoupled system modes are finally com-

puted from the eigenvalue solution of Eq. (28.127). Component mode synthesis pro-

cedures are also applied in multilevel cascades when such a strategy is warranted.

¨q

K″

K′

K″

K′

K″

K′

K″

¨q

M″

M′

M″

M′

M″

M′

M″

K′

¨q

M′

FINITE ELEMENT MODELS 28.49

8434_Harris_28_b.qxd 09/20/2001 11:48 AM Page 28.49

DYNAMIC RESPONSE RESULTING FROM

VARIOUS ENVIRONMENTS

The response of linear structural dynamic systems to dynamic environments may be

computed by either modal or direct methods. Modal methods tend to be computa-

tionally efficient when the required number of system modes addressing the

dynamic environment frequency band are significantly smaller than the order of the

system finite element model. When this is not the case, direct methods may be more

efficient. In addition, when transient environments are brief or impulsive, direct

integration may be more efficient than modal strategies. The following discussion

provides an overview of strategies for the computation of dynamic response to var-

ious environments.

Transient Environments. General relationships detailing the modal method of

transient dynamic analysis are presented in the section entitled Application of Nor-

mal Modes in Transient Dynamic Analysis. Enhancement of the modal solution

accuracy with residual vectors and the mode acceleration method was discussed in

the sections entitled Residual Mode Vectors and Mode Acceleration and Load Trans-

formation Matrices, respectively. Direct integration methods employing implicit

explicit

numerical strategies may be advantageous when environments are of wide

bandwidth and short-lived.

Brief or Impulsive Environments. Brief or impulsive dynamic environments are

often described in terms of shock response spectra (see Chap. 23). Peak dynamic

responses and structural loads are estimated by employing approximate modal

superposition methods utilizing shock response spectra as modal weighting func-

tions.

A systematic approach to this process, which incorporates positive and nega-

tive spectra and quasi-static residual vectors, is presented in Ref. 11. Approximate

shock response spectra–based modal superposition methods are employed in earth-

quake engineering, equipment (e.g., naval shipboard subsystems) shock survivability

prediction, and related applications. This approach is especially appropriate when

standard dynamic environments are specified as shock response spectra.

Simple Harmonic Excitation. Computation of the structural dynamic response

due to simple harmonic excitation is either an end in itself or a key intermediate step

in the computation of the response to random or transient environments. In the case

of transient environments, the time-history response may be calculated through

application of Fourier transform techniques (see Chap. 23). The applied force and

displacement response, respectively, are conveniently expressed in terms of complex

exponential functions by

{F} = F

(ω)e

iωt

, {u} = {U(ω)}e

iωt

, {



u} = iω{U(ω)}e

iωt

, {ü} =−ω

{U(ω)}e

iωt

(28.128)

where ω is the forcing frequency in radians per second. Upon substitution of the

above relationships into the linear structural dynamic equations [see Eq. (28.59)],

the following algebraic matrix equation is defined.

[K + iωB −ω

M]{U(ω)} = {Γ

(ω) (28.129)

When F

(ω) = 1, the response quantities are called frequency response functions

(see Chap. 21). If the normal mode substitution is employed, the above equation set

is diagonalized (assuming modal viscous damping) as follows:

28.50 CHAPTER TWENTY-EIGHT, PART II

8434_Harris_28_b.qxd 09/20/2001 11:48 AM Page 28.50

{U(ω)} = [Φ]{q(ω)} {



U(ω)} = iω[Φ]{q(ω)} {Ü(ω)} =−ω

[Φ]{q(ω)}

(28.130)

(ω

+ 2iζ

ω−ω

(ω) = {Φ

}

[Γ

]{F(ω)} 1 ≤ n ≤ n

max

When the modal method is used, it is recommended that a quasi-static residual vec-

tor be employed to mitigate modal truncation errors.This is not required if the direct

method, namely, the solution of Eq. (28.129), is employed.

The modal approach to simple harmonic or frequency response analysis is com-

putationally more efficient than the direct method if the number of modes required

in a frequency band of interest (0 ≤ω≤ω

max

) is much less than the number of finite

element model degrees-of-freedom. When this is not the case, the direct method

becomes more efficient since the direct solution for {U(ω)} involves decomposition

of a sparse coefficient matrix at each forcing frequency.

When the direct solution procedure is employed, it is most convenient to describe

modal damping as complex structural damping (see Chap. 2). In this situation the

linear, frequency domain, structural dynamic equations are

[(1 + iη)K + iωB

−ω

M]{U(ω)} = {Γ

(ω) (28.131)

where the well-known approximate equivalence of structural damping loss factor, η,

and (viscous) modal damping ratio, ζ, is η≈2ζ.The advantage associated with struc-

tural damping is that the modes need not be explicitly determined in order to

account for modal damping effects. The matrix [B

] is included in the above equa-

tion to account for any known discrete viscous damping features.

An important aspect of effective frequency response analysis, regardless of

whether the modal or direct method is used, is the selection of a frequency grid for

the clear definition of harmonic response peaks. It is generally recommended that

solutions be calculated at frequency points capturing at least four points within a

modal half-power bandwidth, that is,

∆ω = 

/2 =ηω

(28.132)

This guideline suggests a logarithmic frequency grid (∆ω increases with increasing

frequency) is desirable.

Random Excitation. In the most common situations, random environments are

assumed to be associated with ergodic (see Chap. 1) processes.

The computation of

structural dynamic response to random excitation, in such a situation, utilizes

numerical results from the response to a simple harmonic excitation. If a random

environment is imposed at several discrete structural degrees-of-freedom or as sev-

eral geometric load patterns, the frequency responses associated with the individual

loads are denoted as

(ω) = U

(ω)/F

o,j

(ω) (28.133)

where these functions are computed either by the modal or direct method. There-

fore, the frequency-domain response associated with several excitations is

(ω) =



(ω) ⋅ F

o,j

(ω) (28.134)

or in matrix form

FINITE ELEMENT MODELS 28.51

8434_Harris_28_b.qxd 09/20/2001 11:48 AM Page 28.51

U(ω) = [H(ω)]{F

(ω)} (28.135)

Describing the correlated random excitations in terms of the input cross-spectral

density matrix, [G

(ω)], the response autospectral density is

(ω) = [H(ω)] ⋅ [G

(ω)] ⋅ [H(ω)]

* (28.136)

where the asterisk [ ]

* denotes the complex conjugate transpose of a matrix. Finally,

the mean square of response is calculated as the integral

= u



(



)



= 

(ω)dω (28.137)

In order to assure the accurate computation of a mean-square response, this inte-

gral must be evaluated with a frequency grid with refinement consistent with Eq.

(28.132). If too coarse a frequency grid is used, the mean-square response may be

severely underestimated.

REFERENCES

1. Clough, R. W.: Proc. 2d ASCE Conf. on Elec. Comp., p. 345 (Pittsburgh) (1960).

2. MacNeal, R. H.: “Finite Elements: Their Design and Performance,” Marcel Dekker, Inc.,

New York, 1994.

3. Strang, G.: “Linear Algebra and Its Applications,” Harcourt Brace Jovanovich Publishers,

San Diego, Calif., 1988.

4. Isaacson, E., and H. B. Keller: “Analysis of Numerical Methods,” John Wiley & Sons, Inc.,

New York, 1966.

5. Przemieniecki, J. S.: “Theory of Matrix Structural Analysis,” Dover Publications, Inc., New

York, 1968.

6. Lanczos, C.: “The Variational Principles of Mechanics,” 4th ed., Dover Publications, Inc.,

New York, 1986.

7. Coppolino, R. N.: NASA CR-2662, 1975.

8. Bathe, K. J., and E. L. Wilson: Proc. ASCE, 6(98):1471 (1972).

9. Bathe, K. J., and E. L. Wilson: “Numerical Methods in Finite Element Analysis,” Prentice-

Hall, Inc., Englewood Cliffs, N.J., 1976.

10. Lyon, R. H., and R. G. DeJong: “Theory and Application of Statistical Energy Analysis,” 2d

ed., Butterworth-Heinemann, Boston, Mass., 1995.

11. Coppolino, R. N.: SAE Paper No. 841581, 1984.

12. MacNeal, R. H.: Computers in Structures, 1:581 (1971).

13. Williams, D.: Great Britain Royal Aircraft Establishment Reports, SME 3309 and 3316,

1945.

14. Coppolino, R. N.: “Combined Experimental/Analytical Modeling of Dynamic Structural

Systems,” ASME AMD-167, 79 (1985).

15. Guyan, R. J.: AIAA Journal, 3(2):380 (1965).

16. Coppolino, R. N.: Proceedings of the 16th IMAC, 1:70 (1998).

17. Lanczos, C.: J. Res. Natl. Bureau of Standards, 45:255 (1950).

28.52 CHAPTER TWENTY-EIGHT, PART II

8434_Harris_28_b.qxd 09/20/2001 11:48 AM Page 28.52

18. Craig, R. R., and M. D. D. Bampton: AIAA Journal, 6(7):1313 (1968).

19. Hurty, W. C.: AIAA Journal, 3(4):678 (1965).

20. Rubin, S.: AIAA Journal, 13(8):995 (1975).

21. Herting, D. N., and M. J. Morgan: AIAA/ASME/ASCE/AHS 20th SDM (1979).

22. Benfield, W. A., and R. F. Hruda: AIAA Journal, 9(7):1255 (1971).

23. Hadjian, A. H.: Nuclear Engineering and Design, 66(2):179 (1981).

24. Bendat, J. S., and A. G. Piersol: “Random Data Analysis and Measurement Procedures,” 3d

ed., John Wiley & Sons, Inc., New York, 2000.

FINITE ELEMENT MODELS 28.53

8434_Harris_28_b.qxd 09/20/2001 11:48 AM Page 28.53

CHAPTER 29, PART I

VIBRATION OF

STRUCTURES INDUCED

BY FLUID FLOW

R. D. Blevins

INTRODUCTION

Fluid around a structure can significantly alter the structure’s vibrational characteris-

tics.The presence of a quiescent fluid decreases the natural frequencies and increases

the damping of the structure. A dense fluid couples the vibration of elastic structures

which are adjacent to each other. Fluid flow can induce vibration. A turbulent fluid

flow exerts random pressures on a structure, and these random pressures induce a

random response. The structure can resonate with periodic components of the wake.

If a structure is sufficiently flexible, the structural deformation under the fluid load-

ing will in turn change the fluid force. The response can be unstable with very large

structural vibrations—once the fluid velocity exceeds a critical threshold value.

Vibration induced by fluid flow can be classified by the nature of the fluid-

structure interaction as shown in Fig. 29.1. Effects which are largely independent of

viscosity include added mass and inertial coupling. Unsteady pressure on the sur-

face of a structure, due to either variations in the free stream flow or turbulent fluc-

tuations, induces a forced vibration response. Strong fluid-structure interaction

phenomena result when the fluid force on a structure induces a significant response

which in turn alters the fluid force. These phenomena are discussed in this section.

ADDED MASS AND INERTIAL COUPLING

If a body accelerates, decelerates, or vibrates in a fluid, then fluid is entrained by the

body. This entrainment of fluid, called the added mass or virtual mass effect, occurs

both in viscous and in inviscid, i.e., ideal, fluids. It is of practical importance when the

fluid density is comparable to the density of the structure because then the added

mass becomes a significant fraction of the total mass in dynamic motion.

Consider the rigid body shown in Fig. 29.2 which lies in a reservoir of incompress-

ible inviscid irrotational fluid.The surface S defines the surface of the body.The body

moves with velocity U(t). From ideal flow theory, it can be shown that there exists a

29.1

8434_Harris_29_b.qxd 09/20/2001 11:44 AM Page 29.1

velocity potential Φ(x, y, z, t) which is a

function of the special coordinates and

time, such that the velocity vector is the

gradient of a potential function:

V =∇Φ (29.1)

V(x, y, z, t) is the fluid velocity vector.

The potential function Φ satisfies

Laplace’s equation:

1,2

∇

Φ=0 (29.2)

The boundary condition is that on the surface of the body; the normal component of

velocity must equal the velocity of the body:

= V

◊ n on the surface S

where n is the unit outward normal vector. The pressure in the fluid is given by the

Bernoulli equation

p =−ρ − ρV

where ρ is the fluid density and V is the magnitude of V. The force exerted by the

fluid on the body is the integral of the fluid pressure over the surface.

F =



pn dS

If the fluid is of infinite extent, then the solution of these equations is consider-

ably simplified. The fluid force is

F =−ρ 

Φn dS (29.3)

∂



∂t



∂Φ



∂t

∂Φ



∂n

29.2 CHAPTER TWENTY-NINE, PART I

FIGURE 29.1 A classification of flow-induced vibration.

FIGURE 29.2 Fluid-filled region. Fluid den-

sity ρ.

8434_Harris_29_b.qxd 09/20/2001 11:44 AM Page 29.2

and flow potential can be expressed as φ=U(t)φ(x′, y′, z′), where x′, y′, and z′ are

coordinates that are fixed to the body and U is the flow velocity relative to the body.

Substituting this potential in Eq. (29.3) yields the following force:

F =−m (29.4)

where the added mass m is

m =ρ

φ dS (29.5)

The added mass force Eq. (29.3) is zero for U and Φ independent of time, i.e., for

steady translation. This is the D’Alembert paradox for an ideal inviscid fluid flow;

the fluid force is not zero for steady translation in a viscous fluid.

As an example of added mass calculation, the potential for flow over a cylinder

of radius a is

φ=U cos θ

where r = radial coordinate

θ=angular coordinate

U = flow velocity

The added mass per unit length is found from Eq. (29.5). The result is

m =ρπa

where a is the cylinder radius. This added fluid mass is equal to the mass of fluid dis-

placed by the cylinder.

In general, there will be an added mass tensor to represent the added mass for

acceleration in each of the three coordinate directions:

=ρ

and an added mass tensor for rotation about the three coordinate axes. φ

is the

potential associated with flow in the i direction. Note that the added mass tensor is

symmetric, i.e., m

= m

, but if the body is not symmetric, there is coupling between

motions in the various coordinate directions.

For example, if a body is not symmet-

ric about the X axis, acceleration in the X direction generally induces added mass

force in the Y direction and a moment as well.

Since the added mass acts in phase with acceleration [Eq. (29.3)], the net effect of

added mass is to increase the effective mass of the body and to decrease the natural

frequencies. In general, added mass is only important to mechanical structures in

dense fluids such as water. In gases, such as air, the added mass is ordinarily negligi-

ble except for very lightweight structures. Figure 29.3 gives added mass for various

sections and bodies in large unrestricted reservoirs. Additional tables of added mass

are given in Refs. 3 and 4.

If two structures are in close proximity, then the added mass will be a function of

the spacing between the structures and inertial coupling will be introduced between

the bodies. For example, consider a cylindrical rod centered in a fluid-filled annulus

bounded by a cylindrical cavity shown in Fig. 29.4. The radius of the rod is a and the

∂φ



∂n

+ a



∂φ



∂n

∂U



∂t

VIBRATION OF STRUCTURES INDUCED BY FLUID FLOW 29.3

8434_Harris_29_b.qxd 09/20/2001 11:44 AM Page 29.3

radius of the outer cylinder is b. The fluid forces exerted on the rod and outer cylin-

der because of their relative acceleration are

=−m¨x

+ (M

+ m)¨x

= (m + M

)¨x

− (m + M

+ M

)¨x

(29.6)

where x

, x

= displacement of inner rod and outer cylinder

, F

= force on inner rod and outer cylinder

m =ρπa

+ a

)/(b

− a

), added mass of inner rod

=ρπa

=ρπb

29.4 CHAPTER TWENTY-NINE, PART I

FIGURE 29.3 Added mass for lateral acceleration.

The acceleration is left to right. b is

the span for two-dimensional sections.

8434_Harris_29_b.qxd 09/20/2001 11:44 AM Page 29.4

These forces include not only added

mass but also inertial coupling between

the motion of the two structures. [These

equations also apply for a sphere con-

tained within a spherical cavity but here

m = (M

/2)(b

− 2a

)/(b

− a

), M

⁄3ρπa

and M

⁄3ρπb

.] Coupling is introduced

between the cylinder and the rod

through the fluid annulus. The coupling

increases with the density of the fluid

and decreases with increasing gap. If the cylinder and the rod are elastic, motion of

either structure tends to set both structures into motion.

For example, consider an array of heat exchanger tubes contained within a shell.

Water fills the shell and surrounds the tubes. If the tubes are widely spaced (more

than about two diameters between centers), then the tubes are largely uncoupled

and the effect of added mass is simply to reduce the tube natural frequencies by the

addition of fluid equal to the displaced volume of the tubes. However, if the tubes

are closely spaced, then motion of one tube sets adjacent tubes and the shell into

motion. Fluid-coupled modes of vibration will result in the tubes and the shell mov-

ing in fixed modal patterns as shown in Fig. 29.5. In Refs. 6 and 7, analysis is given for

inertial coupling of a cylinder contained eccentrically within a cylindrical cavity,

rows of cylinders, and arrays of cylinders.

VIBRATION OF STRUCTURES INDUCED BY FLUID FLOW 29.5

FIGURE 29.4 A rod in a fluid-filled annulus.

FIGURE 29.5 Coupled modes of vibration of a bank of tubes in a

dense fluid.

Added mass and inertial coupling occur in elastic and rigid bodies, but the added

complexity of elasticity and the three-dimensional motions make a closed-form

solution impossible for most elastic bodies. In the case of quasi-two-dimensional

structures (such as long span tubes or rods), the axial variation in the motion occurs

relatively slowly over the span, and two-dimensional results for sections are applica-

ble. Concentric cylindrical shells coupled by a fluid annulus are important in the

design of nuclear reactor containment vessels. Approximate solutions are required

for both the vessels and the fluid. Reviews of the analysis of fluid coupled concentric

vessels are given in Refs. 8 and 9.

Finite element numerical solutions, developed for an irrotational fluid, have

been incorporated in the NASTRAN and other computer programs to permit solu-

tion for added mass and inertial coupling.These programs solve the fluid and struc-

tural problems and then couple the results through interaction forces

(see Chap.

28, Part II).

8434_Harris_29_b.qxd 09/20/2001 11:44 AM Page 29.5