Dynamic Structural Modification

8/12/2019 Dynamic Structural Modification

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http://svd.sagepub.comThe Shock and Vibration Digest

DOI: 10.1177/0583102400032001022000; 32; 11The Shock and Vibration Digest

Yitshak M. RamDynamic Structural Modification

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Dynamic Structural Modification

Yitshak M. Ram

Yitshak M. Ram, Mechanical Engineering Department, Louisiana StateUniversity, Baton Rouge, LA 70803, USA.

ABSTRACTThispaper describes two methods for deter-mining the damped natural frequencies of a viscouslydamped vibrating system, which is changed by structuralmodification. One method involves transfer functions, theother eigenvalues and mode shapes. The transfer functionmethod is particularly suitable for problems associated withcompound systems, where modal test is conducted on thesubsystems. The modal data method is applicable for awide variety of applications in which the structural modifica-tion is embedded within the system.

IntroductionThere is a wealth of literature dealing with alternative

approaches to the problem of predicting the natural fre-quencies of a modified system using data available fromvibration measurements conducted on the unmodified sys-tem and the analytical model of the modification (seeBaldwin and Hutton [1] for a comprehensive survey).There are two main methods, however, for determining thenatural frequenciesof the modified system. One methodinvolves transfer functions (e.g., [2-5]). The other methodinvolves natural frequencies and mode shapes (e.g., [6-8]).In this paper, we develop equations that determine the

dampednatural

frequenciesof a modified

viscouslydamped system and demonstrate their use by means ofnumerical examples. An important feature of the presenta-tion is that the viscously damped case is analyzed through-out, and the case in which the mass, stiffness, and dampingmatrices cannot be diagonalizedsimultaneously is consid-ered. The results are demonstrated by examples, which canbe easily followed and reproduced.

Poles, Zeros, and Transfer Functions

Consider an n-degree-of-freedom (DOF) viscouslydamped system, which is excited by the force fi = east

applied to the ith DOF:

where e, is the ith unit vector, and dots denote derivativeswith respect to time. Then, the steady-state motion of thesystem has the form

where h,(s) is a vector independent of time. Substituting(2) in (1) gives

Hence, by Cramers rule (e.g., see Noble [9]),

where K, C, and M are the matrices obtained by deletingthe ith rows and the jth columns of K, C, and M, respec-tively, and Hil(s)is the jth element of the vector h,(s). Itthus follows from (2) that

The function His) is called the receptance transfer func-tion between the force applied to the ith DOF and the re-sulting displacement of the jth DOF, or in short thereceptance between i and j. Note that if M, C, and K aresymmetric, then the matrices obtained by deleting the jthrows and the ith columns of the mass, damping, and stiff-ness matrices are the same as those obtained by deleting

- the ith rows and the jth columns of these matrices. It thusfollows from (4) that for symmetric systems, the reciproc-ity property

holds.Of particular interest is the case of the point receptance

H;;,where the appliedforce and the resulting displacementare collocated at the ith DOF. In this case, the matrices M,C, and K are the mass, damping, and stiffness matrices,respectively, associated with the same vibratory system,subject to the constraint xi(t) = 0.

We now consider the relations betweenforce, velocity,and acceleration. Differentiating (5) with respect to t, and

noting that Hits) is independent of the time, gives

Equation (7) providesthe relation between the velocity x land the applied force f The function slg,is called the mo-bility transfer function between the force appliedto the ithDOF and the velocity response at the jth DOF. In a similarmanner, differentiating (7) with respect to time yields therelation between force and acceleration,

The function s2Hf,is the inertance between i and j. 2000 SAGE Publications. All rights reserved. Not for commercial use or unauthorized distribution.

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Alternatively, the receptance may be expressed in termsof the poles and zeros of the system. The poles of the sys-tem are the roots of the characteristic polynomial

They are the eigenvalues of the generalized eigenvalueproblem of dimension 2n,

where

and

Note that by (11) and (12),

and hence (9) and (10) share common eigenvalues,pro-vided that M is invertible. It is therefore said that equations(10) through (12) represent the first-order realization of thequadratic problem (9).

We denote the roots of

by zj, zz, . . . , Z2n - 2. They are the zeros of the receptance(4). With the above definitions, we may write (4) in thefollowing form:

where on is a constant. Upon substitution of s = 0 in (4) and(15), and assuming that the system has no rigid-bodymode, we obtain

which leads to the expression of the receptance in terms ofpoles and the zeros

-

Thepoles

ofthe system characterize its dynamic

re-

sponse. They directly relate to the damped natural frequen-cies and to the rate at which the modes of motion decay. Inthe following sections, we will address t he problem of

Figure 1. The Subsystems and Their Connectors

determining the poles of a modified vibrating system basedon data (1) available from modal tests conducted on theunmodified system and (2) the characterization of themodification.

Evaluating the Damped NaturalFrequencies Using Transfer Functions

We now show how the damped natural frequenciesof acompound vibrating system may be determined knowingthe point receptance transfer functions of its components atthe interface points and the mechanical element joiningthe two components. More particularly, consider the twosubsystems A and B shown in Figure 1. Suppose that sys-tem A and system B have p and q

= n - p DOF, respec-tively. The DOF are numbered in such a way that xi,x2, ..., xp describe the dynamicsof the elements of system

A, and xp + I Xp +2 ..., Xn represent the motion in system B,as shown in Figure 1. Since, apart from this convention theDOF can be numbered arbitrarily, we may assume withoutloss of generality that system A is connected to system Bvia a connector attaching the pth DOF on system A to the(p + 1 )-theDOF on system B (see Figure 1). Three types ofconnectors will be considered, namely, a spring, a dashpot,and a mass. In this analysis, we will assume that the two sub-systems vibrate in-line and that the connectors are colinearwith the displacements of the pth and the (p + 1 )-ithDOF.We will comment later in the paper on how the results maybe used for systems vibrating in plane or in space.

The damped natural frequencies of the global systemwill be expressed in terms of the point receptance transferfunctions of the two subsystems at the interface points,prior to the modification. For convenience, we will denotethe point receptance at the interface of system A by HA.Similarly, HB denotes the point receptance at the interfaceon system B before the modification. Equation (17) showsthat the receptance is a rational function in s. We thereforedefine the numerator and denominator polynomials NA(s)and D,(s) of HA(s) as

and similarly, 2000 SAGE Publications. All rights reserved. Not for commercial use or unauthorized distribution.

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Figure 2. Two Subsystems Connected by a Spring

Connection via a Spring

Suppose that system A is connected to system B via aspring and that the global system vibrates under the influ-ence of an external force fp. Then, the free body diagramshown in Figure 2 implies that

and

Upon eliminating xpfrom (20) and (21), we obtain

The receptance of the compound system from p to p + 1 isthus

The poles of the compound system are the poles of recep-tance (23). To obtain a simple expression for determiningthe damped natural frequencies, we invoke (18),(19) andget

The damped natural frequenciesof the compound systemare thus the roots of the pole equation

Example 1

Consider the system shown in Figure 3. The free motionof system A is governed by

and the motion of system B is determined by

The characteristic equation for system A is

Figure 3. The Subsystems and Their Connector

Because system A is identical to system B, we have

The numerator polynomial of HA= HBis determined by thecharacteristic polynomial of the subsystem obtained by im-posing the constraint x2= 0 in system A, or equivalently x3=0 in system B. This constrained system, shown in Figure 4,gives

The pole equation can, therefore, be determined by (25),

which yields the following poles for the compoundsystem:

Note that the global damped system resonates under theinfluence of the harmonic force

We remark that in the above example, HA(s) and His)have been found using the mass, stiffness, and dampingmatrices of the subsystems. In structural dynamics applica-tions, they can be measured by modal tests conducted onthe subsystems (e.g., see Ewins [10]). The coefficients ofHA(s) and H,(s) can then be found using the rational func-tion approximation method described by Hildebrand [11].There is thus no need to know the complete mathematical



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Figure 4. The Constrained Subsystem

Figure 5. Two Subsystems Connected by a Dashpot

model of the unmodified system. The only required dataare the two receptances HA(s),HB(s) and the connectingspring k. -

Connection via a Dashpot

Suppose now that system A is connected to system B viaa dashpot and that the global system vibrates under theinfluence of an external force f p.Then, the free body dia-gram shown in Figure 5 implies that

and

Upon eliminatingx P fromthe above equations, we find

Themobility

fromp

top

+ 1 of thecompound system

isthus

Using (18) and (19), the pole equationof a compoundsystem, connected by a dashpot, is found:

Example 2

Consider the systems A and B of Example 1 again, butsuppose now that the connector is a dashpot of constant c =5. Then, with the same NA,DA,NB,and DBfound in Exam-ple 1, we obtain using (36)

The roots of the above equationdetermine the poles of thecompound system,

Connection via a Mass

Suppose now that system A is rigidly connected to sys-tem B via a mass and that the

compound systemvibrates

under the influence of an external force fp as shown in Fig-ure 6a.

We may consider the rigid attachment of the mass m toits right and left neighboring masses as attachments throughsprings of infinite stiffness, k ~ 00, as shown in Figure 6b.The receptance transfer function of two subsystems A andC connected by a rigid link is determined by substituting

ByNewtons second law,

and hence the receptance of the mass is

Therefore, by (37), the receptance of system A togetherwith the attached mass is

This system is then attached to system B by therightspring of infinite rigidity. Therefore, invoking (37) again,

we find

which after simplification gives

Substituting (18) and (19) in (42) yields the pole equationfor two subsystems, rigidly connected by a mass m,



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Figure 6. Subsystems Connected by a Mass

Example 3

A mass m = 5 is rigidly attached between the pth and the(p + l)-th DOF of the subsystems of Example 1. Applica-tion of (43), with the same numerator and denominatorpolynomials as in Example 1, gives the pole equation forthis case,

The poles of the global system are thus

Modelingof vibratory systems in plane or space motionleads to the mathematical description (1). Hence, theirmodifications may be handled in a similar manner to thatdescribed above. It is necessary however to ensure that themeasured transfer functions at the points of interface arecolinear with the coupling element throughout the entiremotion. If the relative orientation of the subsystems is notconstant, then the equations of motion are time dependentand a different, more complex analysis is required.

Analysis Using Modal Data

The quadraticeigenvalue problem

associated with the free motion of system (1) has 2n eigen-values

X,,~,2,...,

,znand 2n

eigenvectorsV, V2..., v2/which nontrivially solve (45). In general, the three matrices

M, C, and K cannot be diagonalizedsimultaneously bymeans of congruence transformation; that is, there is gen-erally no nonsingularmatrix Q that makes the three products

diagonal. The first-order realization (10) to (12), however,involves only two symmetric matrices, A and B. Althoughboth are

generally indefinite, theycan be

diagonalizedsi-

multaneously whenever the eigenvalues~,,,J..2 ..., J..2naredistinct. Moreover, in this case, the eigenvectorsv~, v2, ...,V 2ncan be scaled such that

and

where

and I is the identity matrix of dimension 2n.Suppose that the system (1) is modified such that the

characteristic equationassociated with a modified system is

Then, the first-order realization of (51) is given by

where A and B are given by (11) and (12),

and

It thus follows from (52) and the biorthogonal properties(47) and (48) that the pole equationfor the modified sys-tem is

This equation determines the poles of the modified systemin terms of the modal data A and U, and the characteriza-tion of the modification AM, AC, and AK. In MATLAB,for example, the solution to the pole equation (55) can befound by the software function

It should be noted that although U is generally a matrix

with complex elements, UT in the above equations isthe

transpose of U and not the conjugate transpose UH.



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Example 4

We now solve Example 1 again, but this time using thepole equation (55). Here, we have

and

The matrix U that simultaneously diagonalized A and Bsuch that (47) and (48) hold is given by (49), where

and

Application of the pole equation (55) leads to the samepoles of the modified system as obtained in Example 1.

One may wonder why only two receptance transferfunctions, HAand H~, are needed to determine the poles ofthe modified

systemin

Example 1,whereas a

completemodal matrix is required to solve the same problem inExample 4. Closer inspection reveals that with AA givenby (58), only the second and third rows of U play a role inthe matrix product U~AAU. These rows precisely deter-mine HA and HB and vice versa.

Concluding Remarks

The damped natural frequenciesof a modified vibratorysystem can be predicted using receptances measured byvibration tests conducted on the unmodified system. Wehave presented equationsthat determine the damped natu-ral frequenciesof the modified system in terms of the pointreceptances at the interface points with the basic couplingdevices: a spring, dashpot, and mass. The more generalcase in which the coupler itself is a vibrating system,which includes a combination of these basic elements, canbe handled in a similar manner. It may require, however,analysis on a case-by-casebasis, which may be tedious.

Alternatively, the modification may be expressed in termsof the incremental mass, damping, and stiffness matrices

AM, AC, and AK. The relevant rows of the modal matrixmay be calculated from the measured point receptances.Then, a simpleequation, which can be solved by a stan-dard eigenvalue routine, may be invoked to determine the

damped natural frequenciesof the modified system. Theadditional effort needed to build the part of the modalmatrix from the measured receptances provides the benefitof solving the modification problem by a unified approach,which requires no special analysis. In a sense, it demon-strates the general feature of the matrix approach, wheremany, apparentlydifferent, problems are transformed intoone canonic form and solved via a common routine.

References

[1] Baldwin, J. F., and Hutton, S. G., 1985, "Natural Modes of ModifiedStructures," AIAA Journal, Vol. 23, 1737-1743.

[2]Gladwell, G.M.L., and

Bishop,R.E.D., 1960, "Interior

Receptancesof Beams," Journal of Mechanical Engineering Science, Vol. 2, 1-13.[3] Carfagni, M., 1991, "Sub-Structuring as a Modal Analysis Design

Tool," International Journal of Analyt ical and Experimental Modal Analysis, Vol. 6, 251-270.

[4] Simpson, A., 1974, "Scanning Krons Determinant," Quarterly Jour-nal of Mechanics and Applied Mathematics, Vol. 27, 27-43.

[5] Tsuei, Y. G., Yee, E.K.L., and Lin, A.C.Y., 1991, "PhysicalInterpre-tation and Application of Modal Force Technique," InternationalJournal of Analytical and Experimental Modal Analysis, Vol. 6, 237-250.

[6] Elliott, K. B., and Mitchell, L. D., 1985, "Realistic Structural Modi-fication : Theoretical Development," in 3rd International Modal Analy-sis Conference, 471-467.

[7] Ram, Y. M., Braun, S. G., and Blech, J., 1988, "Structural Modifica-tion in Truncated Systems by the Rayleigh-Ritz Method," Journal ofSound and

Vibration,Vol.

125,203-209.



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[8] Zhang, Q., Wang, W., Allemang, R. J., and Brown, D. L., 1988,"Prediction of Mass Modification for Desired Natural Frequencies,"in6th International Modal Analysis Conference, 1026-1032.

[9] Noble, B., 1969, Applied Linear Algebra, Prentice Hall, EnglewoodCliffs, NJ.

[10] Ewins, D. J., 1986, Modal Testing: Theory and Practice, John Wiley& Sons, New York.

[11] Hildebrand, F. B., 1974, Introduction to Numerical Analysis, 2nd ed.,McGraw-Hill, New York.


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Dynamic Structural Modification